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Model investigation of initial fouling rates of protein solutions in heat transfer equipment Rose, Ian C. 1999

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M O D E L INVESTIGATION OF INITIAL FOULING RATES OF PROTEIN SOLUTIONS IN H E A T TRANSFER EQUIPMENT  by  Ian C. Rose B.E. (1993) University of Auckland, N.Z.  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF T H E REQUIREMENTS FOR T H E D E G R E E OF DOCTOR OF PHILOSOPHY  in T H E F A C U L T Y OF G R A D U A T E STUDIES Department of Chemical and Bio-Resource Engineering  We accept this thesis as conforming to the required standard  T H E UNIVERSITY OF BRITISH COLUMBIA April, 1999. © I a n Rose, 1999.  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. 1 further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.  Department of  <?vj 7j e- R^a^r^  CK^ICAI  (  The University of British Columbia Vancouver, Canada Date  DE-6 (2/88)  /  Apr-'i \  tftl  -C^^e.*^^  ii  Abstract As protein solutions are heated, denaturation and aggregation processes give rise to deposition on the heated surface. The present study undertakes the problem of understanding how process variables such as fluid velocity and temperature affect the initial fouling rate. Previous studies of chemical reaction fouling for other systems have demonstrated contrasting behaviour of the initial fouling rate with respect to fluid velocity: some investigations report an increase, others a decrease, and still others both an increase followed by a decrease of initial fouling rate with increasing fluid velocity. In this work a theoretical model for initial chemical reaction fouling in turbulent flow, where attachment is treated as a physico-chemical rate process in series with mass transfer (Epstein, 1994), was examined. According to the model, mass transfer is directly proportional to the friction velocity, and attachment is inversely proportional to the square of this velocity. Therefore, at a given wall temperature, it follows that if the initial fouling rate is mass transfer controlled (low fluid velocity), the deposition flux increases as the fluid velocity increases. If, however, the initial fouling rate is attachment controlled (high fluid velocity), the deposition flux will decrease as the fluid velocity increases. Therefore as the velocity is lowered the initial fouling rate goes through a maximum at a given wall temperature. In addition, this maximum initial fouling rate can be expected to increase and to move towards higher critical velocities as the wall temperature increases. Two separate experimental studies were performed, one using a 1 wt. % whey protein solution at pH 6, and the other a 1 wt. % lysozyme solution at pH 8. These experiments were performed over film Reynolds numbers of 2950 - 22730, clean inside wall temperatures of  Ill  59 - 102°C and bulk temperatures of 30 - 57°C in a 9.017 mm i.d. electrically heated, stainless steel tube. The above features of the model were qualitatively demonstrated with both protein solutions, i.e. a maximum in experimental initial fouling rate at a given wall temperature over a range of fluid velocities, and an increase in the maximum rate and in the corresponding critical velocity as the wall temperature increased. Lysozyme fouling results showed that as the mass flux increased from 200 kg/m s to 2  1101 kg/m s, the fouling activation energy, AEf, increased from 29 kJ/mol to 118 kJ/mol. This 2  observation was consistent with the model, since the optimum model prediction, with an average absolute percent deviation of 23.3 %, was obtained with a kinetic reaction order of 0.75 and chemical activation energy, AE, of 161 kJ/mol. Thus at low velocity, when mass transfer dominated, AEf was low, but as the velocity was increased and chemical attachment became increasingly more important, AEf increased, but never to the value for the pure chemical reaction, AE, since mass transfer could never be entirely neglected. Whey protein modeling results, after rejection of renegade data points, showed an optimum solution, with an average absolute percent deviation of 24.5 % from the fit of the model, with a kinetic reaction order of 0.99 and a chemical activation energy, AE, of 201 kJ/mol. These values were compatible with the kinetic parameters for whey protein denaturation in the literature. Using estimates of the deposit physical properties from the whey protein fouling experiments, the dimensionless mass transfer constant (k') for both protein fouling experiments were found to lie between the isothermal value of Metzner and Friend (1958), and the non-  iv  isothermal value of Vasak and Epstein (1996), where all physical properties had been evaluated at the wall temperature. In contrast, the present work evaluated the physical properties in the mass transfer term at the film temperature, and therefore an intermediate estimate of k' was to be expected. In general, both protein fouling studies show results that conform in the mass transfer control region with the Epstein (1994) model, but in the attachment control region, the inverse dependence of the initial fouling rate on the friction velocity is even greater than the second power dependence predicted by that model.  V  Table of Contents Abstract  ii  Table of Contents  v  List of Tables  ix  List of Figures  xii  Acknowledgments  xix  1.  Introduction  1  1.1  Fouling of Heat Exchangers  1  1.2  Fouling Categories 1.2.1 Factors Affecting Fouling Objectives  4 7 8  1.3 2.  Literature Review  11  2.1  11 13 14 18 19 24 26 26 29 32 37  2.2  2.3  2.4  3.  Chemical Reaction Fouling 2.1.1 Effect of Wall and Bulk Temperatures 2.1.2 Effect of Fluid Velocity Theoretical Fouling Model 2.2.1 Mathematical Model Development (Epstein, 1994) 2.2.2 Application of Model Milk Based Fluid Fouling 2.3.1 Fouling Components of Milk 2.3.2 Factors Affecting Fouling 2.3.3 Biochemistry of Whey Protein Solutions 2.3.4 P-lactoglobulin and a-lactalbumin Denaturation Kinetics 2.3.5 Previous Fouling Studies into the Controlling Mechanisms of Deposition 2.3.6 Whey Protein Deposit Properties Lysozyme Fouling 2.4.1 Properties of Lysozyme 2.4.2 Effect of Electrostatic Forces on Fouling 2.4.3 Thermal Inactivation Kinetics of Lysozyme  39 47 50 50 51 52  Experimental Apparatus and Methods  55  3.1  55 55 61  Tube Fouling Unit (TFU) 3.1.1 T F U Apparatus (Wilson, 1994) 3.1.2 Wall Temperature Measurement  vi  3.2  3.3  4.  81  4.1 4.2  81 84 85 86 89 104 112 125 137 137 138 139 146 149 152 155  4.4  Data Handling Methods Whey Protein Fouling 4.2.1 Physical Properties 4.2.2 Microbial Contamination 4.2.3 Velocity Effect 4.2.4 Effect of Particulate Material 4.2.5 Deposit Morphology 4.2.6 Physical Properties of Whey Protein Fouling Deposits Lysozyme Fouling 4.3.1 Physical Properties 4.3.2 Particulate Material 4.3.3 Steady State Conditions and the Measurement of Fouling 4.3.4 Effect of Fluid Velocity 4.3.5 Effect of pH on the Fouling Behaviour 4.3.6 Effect of Enzymatic Activity Kinetic Compensation Effect (KCE)  Mathematical Modeling and Discussion  157  5.1 5.2  157 159 159 162 172 172 175 182 184 188 192  5.3  5.4 6.  63 65 69 70 74 77 77  Experimental Results and Discussion  4.3  5.  3.1.3 Data Collection 3.1.4 T F U Operating Procedures Whey Protein Solutions 3.2.1 Microbial Contamination Testing Procedure 3.2.2 Deposit Property Analysis Lysozyme 3.3.1 Enzymatic Activity Assay  Development of F O R T R A N 77 Program Modeling Whey Protein Solution Fouling 5.2.1 Input Data 5.2.2 Mathematical Model Predictions Modeling Lysozyme Fouling 5.3.1 Input Data 5.3.2 Mathematical Model Predictions 5.3.3 The Optimum Model Solution 5.3.4 Elimination of Low Reynolds Number Data 5.3.5 Discussion of the Model Response Interpretation of Optimum Model Predictions  Conclusions  199  6.1 6.2  199 202  Whey Protein Solution Fouling Lysozyme Solution Fouling  vii  7.  Recommendations for Further Study  206  Nomenclature  208  References  214  Appendix 1  Calibrations  221  Appendix 2  Acid Catalyzed 2-Furaldehyde Fouling  223  Literature Review A2.1.1 2-Furaldehyde as a Decomposition Product A2.1.2 2-Furaldehyde Decomposition Kinetics Experimental Apparatus and Methods A2.2.1 Stirred Cell Reactor (SCR) A2.2.2 U V Spectrophotometer A2.2.3 Fluid Physical Properties A2.2.4 Tube Fouling Unit (TFU) Kinetic Experiments Fouling Experiments Application of Experimental Results to Model  223 223 227 228 228 230 232 234 236 239 243  Whey Protein Solutions  246  Composition of WPC-80 Powder Fouling Resistance for T F U 200 Experiments used for Deposit Property Analysis Thermocouple Location with respect to Cut Tube Sections Velocity and Temperature Distribution Calculations for Whey Protein Fouling Experiments  246  Lysozyme Solutions  259  Analysis of Food Grade Lysozyme Chloride (Powder) Enzymatic Assay of Lysozyme Spectrophotometer Data for Substrate and Blank Runs  259 259 266  Mathematical Modeling  267  Algorithm for Levenberg-Marquardt Non-Linear Curve Fitting Method F O R T R A N Program used for Non-Linear Multi-Parameter Regression of Experimental Data According to Epstein's Mathematical Model (1994) Whey Protein Solution Input Data Lysozyme Solution Input Data Statistical Analyses of Optimum Model Fit of Whey Protein Data to Experimental Results  267  A2.1  A2.2  A2.3 A2.4 A2.5 Appendix 3 A3.1 A3.2 A3.3 A3.4  Appendix 4 A4.1 A4.2 A4.3 Appendix 5 A5.1 A5.2 A5.3 A5.4 A5.5  246 248 248  269 277 278 280  viii  A5.6  Statistical Analyses of Optimum Model Fit of Lysozyme Data to Experimental Results  281  Appendix 6  Kinetic Compensation Effect  283  Appendix 7  Experimental Uncertainty  290  Uncertainty in the Calculated Heat Transfer Coefficient Uncertainty in the Measured Initial Fouling Rate Experimental Error caused by forcing data to fit the Arrhenius Expression Uncertainty in Time Averaged Mass Flux  290 292 294 297  A7.1 A7.2 A7.3 A7.4  List of Tables 2.3.1  Milk composition (Taylor, 1992)  26  2.3.2  Composition of whey proteins in milk (Taylor, 1992)  27  2.3.3  Kinetic parameters for heat denaturation of P-lactoglobulin and cc-lactalbumin (Lyster, 1970) [kj: s' , k : l g V ]  37  2.3.4  Kinetic parameters for heat denaturation of P-lactoglobulin and a-lactalbumin (Dannenberg and Kessler, 1988)  38  2.3.5  Reaction kinetic data for p-lactoglobulin in skim milk  39  2.3.6  Comparison of deposit solid weight fractions (Davies et al., 1997)  49  2.4.1  Physical properties of lysozyme and a-lactalbumin (Haynes and Norde, 1994b)  50  2.4.2  Denaturation temperature of lysozyme (Pfeil and Privalov, 1976)  51  2.4.3  Effect of pH on the activation energy (Makki, 1996)  54  2.4.4  Reaction order with respect to time of thermal inactivation of lysozyme (Makki ,1996)  54  3.1.1  T F U safety measures  59  3.2.1  Microbial growth at 37°C using petrifilm trays  71  3.2.2  Microbial growth at 21 °C using petrifilm trays  72  3.2.3  Microbial growth at 37°C using agar plates  73  3.2.4  Microbial growth at 21 °C using agar plates  73  3.2.5  Tubes used for deposit analysis  75  3.2.6  T F U thermocouple locations  76  4.2.1  Microbial content of whey protein solutions  87  4.2.2  Microbial content of water supply  88  4.2.3  Potential sources of microbial contamination  88  1  1  2  4.2.4  Arrhenius parameters for whey protein fouling experiments  98  4.2.5  Filtering results of 1 wt. % whey protein solution after mixing  4.2.6  Typical whey protein fouling experimental conditions  109  4.2.7  Comparison of experimental and particulate deposition rates  110  4.2.8  Non-linear least squares parameters for T F U 220 series of experiments  111  4.2.9  Deposit sample results from the whey protein fouling experiments  123  107  4.2.10 Deposit coverage and thickness results for T F U 200 series  128  4.2.11 Summary of deposit coverage and thickness results  129  4.3.1  Particle size distribution in a 1 wt. % lysozyme solution at pH 8  139  4.3.2  Comparison of experimental steady state temperatures with water  141  4.3.3  Arrhenius parameters for lysozyme fouling experiments  143  4.3.4  Operating parameters to study the effect of pH  150  4.3.5  Results of enzymatic assay of lysozyme for T F U 310 (G = 297 kg/m s)  153  5.2.1  Spreadsheet used to evaluate the constants C and B for whey protein modeling  161  5.2.2  Modeling results using all whey protein experimental data and input file Model2.dat (Film temperature used for both mass transfer and chemical attachment)  164  5.2.3  Modeling results using all whey protein experimental data and input file Model3.dat (Film temperature used for mass transfer and wall temperature used for chemical attachment)  165  5.3.1  Spreadsheet used to evaluate the constants C and B for lysozyme modeling  173  5.3.2  Modeling results using all lysozyme experimental data and input file Model5.dat (Film temperature used for both mass transfer and chemical attachment)  175  5.3.3  Statistical analyses of experimental and predicted lysozyme fouling results  178  2  5.3.4  Modeling results using all experimental data and input file Model6.dat (Film temperature used for mass transfer and wall temperature used for chemical attachment)  182  5.4.1  Optimum model solutions  194  5.4.2  Summary of k' and k" values for whey protein and lysozyme fouling data  196  A2.2.1 UV Absorbance of furan derivatives (Fritz and Schenk, 1966)  231  A2.3.1 Summary of Stirred Cell Reactor (SCR) experiments  237  A2.4.1 Overall 2-furaldehyde fouling experiment summary  240  A2.4.2 Individual 2-furaldehyde fouling experiment summary  242  A3.4.1 Comparison of deposit, sublayer and buffer layer thicknesses at axial positions x = 48 - 713 mm  255  A3.4.2 Comparison of reaction volumes of TFU 211 and TFU 209  257  A6.1  Styrene polymerization Data of Crittenden et al. (1987a)  286  A6.2  Effect of re-parameterization upon the Arrhenius parameters for whey protein and lysozyme fouling experiments  288  A7.1  Experimental uncertainty in the heat applied to the test section ( Q)  291  A7.2  Experimental uncertainty (95 % confidence interval) for TFU 207  292  (G = 531.7 kg/m s, Q = 3840 W, a . = 50.3 W) 2  Q  A7.3  Experimental uncertainty (95 % confidence interval) for TFU 304  292  (G = 523.0 kg/m s, Q= 2798 W, a . = 32.7 W) 2  Q  A7.4  Percentage experimental uncertainty (95 % confidence interval) in initial fouling 293 rate for each thermocouple and experiment average for all fouling experiments  A7.5  Error (AAD) from forcing experimental data to fit the Arrhenius expression  295  A7.6  Mass flux uncertainty for all fouling experiments  297  List of Figures 1.1.1  Temperature profiles and heat transfer resistances  2  1.1.2  Fouling curves  3  2.1.1  The heat denaturation of p-lactoglobulin (Roefs and Dekruif, 1994)  12  2.1.2  Effect of mass velocity on initial fouling rate at ten wall temperatures. Circles are experimental fouling results of Crittenden et al. (1987a) for the solution polymerization of styrene in kerosene. Lines represent the best fit regression to the experimental data.  15  2.3.1  Variation in amount of deposit on a plate heat exchanger with age of milk (open circles: separated milk, solid circles: homogenized milk), (Burton, 1968)  29  2.3.2  Variation in deposit formation on a plate heat exchanger with season for the bulk 30 milk of two herds (Burton, 1968)  2.3.3  Tertiary structure of P -lactoglobulin (Papiz et al., 1986)  33  2.3.4  Protein denaturation and aggregation (Fryer et al., 1989)  34  2.3.5  Possible pathways of deposit formation (Fryer et al., 1989)  34  2.3.6  The structure of p-lactoglobulin (Visser and Jeurnink, 1997)  36  2.3.7  Plot of fouling resistance against wet deposit coverage (Davies et al., 1997)  49  2.4.1  Effect of pH on lysozyme adsorption to a positively charged surface (Haynes and Norde, 1994)  52  3.1.1  Schematic of Tube Fouling Unit (TFU) apparatus  56  3.3.1  Absorbance of Micrococcus lysodeikticus suspension and blank runs  3.3.2  Enzymatic assay for lysozyme (Dilution: 600 x)  80  3.3.3  Determination of the maximum linear rate for sample 1 (t=0)  80  4.1.1  Axial temperature profiles for T F U 303 (G = 879 kg/m s)  82  4.1.2  Driving temperature difference at x = 660 mm for T F U 303 (G = 879 kg/m s)  82  4.1.3  Initial fouling rate at x = 660 mm for T F U 303 (G = 879 kg/m s)  84  79  2  2  2  xiii  4.2.1a Ratio of whey protein solution to water density at pH 6.2  86  4.2.lb Kinematic viscosity of 1 wt. % whey protein solution at pH 6.2 4.2.2  Inside wall temperature profiles for T F U 209 ( G = 882 kg/m s)  86 91  2  4.2.3a Clean inside wall temperatures @ G = 221 kg/m s  91  4.2.3b Clean inside wall temperatures @ G = 532 kg/m s  92  4.2.3c Clean inside wall temperatures @ G = 1275 kg/m s  92  2  4.2.4  Determination of initial fouling rate at x = 550 mm for T F U 209 (G = 882 kg/m s) 2  4.2.5  94  Initial and accelerated fouling rates at x = 660 mm for T F U 212 (G = 400 kg/m s)  96  4.2.6  Linear least squares regression for T F U 207 (G = 532 kg/m s)  96  4.2.7  Non-linear least squares regression for T F U 207 (G = 532 kg/m s)  4.2.8  Effect of Reynolds number on fouling activation energy  2  2  Open circles : Gotham (1990), 1 % protein, protein inlet 73°C and outlet 83°C Solid squares : WPC-80 experimental results (T ) = 68 - 102°C, T = 31-57°C ' w>i  c  b  4.2.9  97  Comparison of initial and accelerated fouling rates for low mass flux whey protein fouling experiments  4.2.10aEffect of mass flux on initial fouling rate for 1 % whey protein solution at  99 100  102  (T ,i) = 9 2 ° C , T = 31 - 5 7 ° C w  c  b  4.2.1 ObEffect of mass flux on initial fouling rate for 1 % whey protein solution at  102  (T i) = 88°C, T = 31 - 5 7 ° C w>  c  b  4.2.10cEffect of mass flux on initial fouling rate for 1 % whey protein solution at  102  (T )c = 8 4 ° C , T = 31 - 5 7 ° C W;i  b  4.2.11 Fouling results for T F U 204 and T F U 212 ( G * 400 kg/m s) 2  4.2.12 Single mode particle size distribution for 1 wt. % whey protein solution (replicates)  103 105  4.2.13 Particle size distribution for 1 wt. % whey protein solutions 4.2.14 Effect of centrifugation on the initial fouling rate for T F U 220 series (TFU 221 centrifuged to remove particulates, T F U 222/223 contain « 18 % of WPC as particulates) 4.2.15 Photograph of fouled tube sections from T F U 208 (G = 284 kg/m s) 2  4.2.16a  S E M of TFU209, x « 713 mm, Magnification: lOOx  4.2.16b  S E M of TFU209, x « 713 mm, Magnification: 400x  4.2.16c  S E M of TFU209, x « 713 mm, Magnification: lOOOx  4.2.16d  S E M of TFU209, x « 713 mm, Magnification: 4000x  4.2.17a  S E M of TFU211, x « 713 mm, Magnification: lOOx  4.2.17b  S E M of TFU211, x « 713 mm, Magnification: lOOx  4.2.17c  S E M of TFU211, x « 713 mm, Magnification: 400x  4.2.17d  S E M of TFU211, x « 713 mm, Magnification: 1 OOOx  4.2.17e  S E M of TFU211, x « 713 mm, Magnification: 4000x  4.2.18a  S E M of WPC-80, Magnification: 80x  4.2.18b  S E M of WPC-80, Magnification: 200x  4.2.18c  S E M of WPC-80, Magnification: 1 OOOx  4.2.19 Deposit study along length of tube for T F U 207 (G = 532 kg/m s) 2  4.2.20 Deposit coverage results for T F U 200 series of experiments 4.2.21 Deposit thickness results for T F U 200 series of experiments 4.2.22 Deposit coverage and thickness results for each experiment 4.2.23 Re-analysis of T F U 210 4.2.24 Dependence of physical properties on temperature (All data)  4.2.25 Dependence of physical properties on temperature (Subject to fouling Biot number criterion)  133  4.2.26 Dependence of physical properties on temperature (Subject to fouling Biot number criterion and no TFU210)  134  4.2.27 Thermal conductivity of dairy products (Rao and Rizvi, 1995)  135  4.2.28 Estimation of an average deposit density for T F U 200 experiments  135  4.3.1a Density of 1 wt. % lysozyme solution at pH 8  137  4.3.1b Dynamic viscosity of 1 wt. % lysozyme solution at pH 8  138  4.3.2  Inside wall temperature profiles for T F U 304(G = 523 kg/m s)  140  4.3.3  Determination of initial fouling rate at x = 713 mm for T F U 304 (G = 523 kg/m s)  142  4.3.4  Non-linear least squares regression for T F U 304 (G = 523 kg/m s)  142  4.3.5  AEf and ln(A) dependence on mass flux for T F U 300 series of experiments atpH8  144  2  2  2  4.3.6a Effect of mass flux on initial fouling rate for 1 % lysozyme solution at  146  (T ,j)c = 65°C, T b = 30 - 52°C w  4.3.6b Effect of mass flux on initial fouling rate for 1 % lysozyme solution at (T ,i) = 70°C, T = 30 - 52°C  147  4.3.6c Effect of mass flux on initial fouling rate for 1 % lysozyme solution at (T ) = 7 5 C , T = 3 0 - 5 2 C  147  4.3.6d Effect of mass flux on initial fouling rate for 1 % lysozyme solution at (T ,i) = 80°C, T = 30 - 52°C  147  4.3.7  Arrhenius type plot for T F U 306 and T F U 310 (Repeat Experiments)  149  4.3.8  Effect of pH on Fouling Rate at ( T ) = 80°C  151  4.3.9  Effect of pH on Fouling Rate at thermocouple 10 (x = 713 mm)  152  w  c  b  o  Wii  w  c  c  o  b  b  wJ  c  4.3.10 Average lysozyme enzymatic activity for T F U 310  154  4.4.1  156  Mutual dependence of Arrhenius parameters  xvi  5.2.1  Experimental data used for modeling whey protein fouling (59 data points)  160  5.2.2  Friction factor correlations  162  5.2.3  Non-existence of an optimum solution for all 59 experimental data points using Model3.dat  166  5.2.4  Best model solution for n = 2, N = 59  166  5.2.5  Optimum reaction order for whey protein solution fouling (47 data points)  167  5.2.6  Best solution for n = 0.99, N = 47  169  5.2.7  Effect of wall temperature on model solution  170  5.2.8  Comparison of model predictions to WPC experimental data obtained from the Arrhenius type equations  171  5.3.1  Optimum reaction order when film temperature is used for the physical properties for mass transfer and chemical attachment terms in the mathematical model  176  5.3.2  Best solution for reaction order, n = 0.58 (N = 86)  177  5.3.3  Optimum reaction order for lysozyme fouling when film temperature is used for mass transfer term and surface temperature is used for chemical attachment term  181  5.3.4  Best solution for reaction order = 0.57 (N= 86)  181  5.3.5  Effect of wall temperature on model solution for lysozyme and comparison to experimental fouling data  184  5.3.6  Optimum reaction order when the film temperature is used for the physical properties for the mass transfer term and the wall temperature is used for the physical properties for the chemical attachment term in the mathematical model. T F U 309 neglected  185  5.3.7  Best solution for n = 0.75, N - 77  186  5.3.8  Comparison of model predictions to experimental data obtained from the Arrhenius type equations  187  5.3.9  The controlling limits of Equation (5.3.1)  189  xvi 5.3.10a  Fit of Equation (5.3.1) to experimental data at T = 80° C  190  5.3.10b  Fit of Equation (5.3.1) to experimental data at T = 80° C  190  5.3.10c  Fit of Equation (5.3.1) to experimental data at T = 80° C  191  5.4.1  w  w  w  Comparison between experimental and model predicted initial fouling rates for two mass fluxes for whey protein fouling. (a) T F U 211 (G = 221.4 kg/m s) (b) T F U 205 (G = 1055.6 kg/m s) 2  5.4.2  193  2  Comparison between experimental and model predicted initial fouling rates for a low mass flux lysozyme fouling experiment (TFU 306, G = 296.9 kg/m s)  193  A1.1  Tube Fouling Unit (TFU) current calibration  221  A1.2  Low flow rate (LFR) rotameter calibration  221  A l .3  Mid flow rate (MFR) rotameter calibration  222  A l .4  High flow rate (HFR) rotameter calibration  222  2  A2.1.1 Effect of 2-furaldehyde on browning rate of sugar solutions (Tan et al., 1950)  224  A2.1.2 Schematic of 2-furaldehyde reactions (Dunlop and Peters, 1953)  224  A2.1.3 Marcusson's (1925) proposed acid catalyzed decomposition mechanism of 2-furaldehyde (cited in Dunlop and Peters, 1953)  226  A2.1.4 Schematic representing the loss of resonance stability in acidic conditions  226  A2.2.1 Schematic diagram of the Stirred Cell Reactor (SCR) apparatus  228  A2.2.2 Calibration of U V Spectrophotometer  231  A2.2.3 Kinematic viscosity of a 1 wt. % 2-furaldehyde solution A2.3.1 aDecomposition reaction of 2-furaldehyde in SCR 006 A2.3.1 bKinetic description of 2-furaldehyde decomposition in S C R 006  232 237 23 8  A2.3.2 Photograph of 2-furaldehyde samples from experiment SCR006 @ 169°C  23 8  A2.3.3 Temperature dependence of acid catalyzed decomposition of 2-furaldehyde  239  A2.4.1 Arrhenius plot for T F U 100 series (3-point plot)  241  xviii  A2.4.2 Arrhenius Plot for T F U 100 series (24-point plot) T F U 101,102 and 103: open points are local data, 3 closed points are averages  241  A2.5.1 Solution of Equation (2.2.25) to find A E for T F U 100 Experiments  244  A3.4.1 Axial temperature profiles for T F U 209, V vera e = 0.8860 m/s  253  A3.4.2 Radial temperature and velocity profile for T/C 10 T F U 209 Vaverage = 0.8860 m/s  253  A3.4.3 Axial temperature profiles for T F U 211, V r a g e 0.2225 m/s  254  A3.4.4 Radial temperature and velocity profile for T/C 10 T F U 211 Vaverage = 0.2225 m/s  254  A3.4.5 Comparison of viscous and buffer layer thicknesses for T F U 209 & 211  255  A3.4.6 Thickness of whey protein solutions with T > 65°C  256  A3.4.7 Schematic of film thickness for deposition in the test section  256  A6.1  A schematic of the mutual dependence of the Arrhenius parameters  284  A6.2  Arrhenius plot for perfectly correlated data from whey protein fouling  a  g  =  ave  experiments  285  Experimental Arrhenius plot for whey protein fouling experiments  285  A6.4a Fouling Arrhenius plot for whey protein solutions after re-parameterization  287  A6.4b Fouling Arrhenius plot for lysozyme solutions after re-parameterization  287  A6.5  The lack of mutual dependence of ln(A°) and AE after re-parameterization  289  A7.1  Arrhenius type expression for T F U 209 with error bars  296  A7.2  Arrhenius type expression for T F U 306 with error bars  296  A6.3  xix  Acknowledgments I would like to thank all members of the Department of Chemical Engineering for their friendship and encouragement throughout the duration of this study. In particular, I wish to express sincere gratitude to my research supervisors, Dr. Norman Epstein and Dr. Paul Watkinson, for the support, patience and guidance that they have shown over the past  five and a  half years. Valuable information concerning whey protein solution fouling and operation of the Tube Fouling Unit, provided by Dr. Ian Wilson, is truly appreciated. Support from the Chemical Engineering office, stores, and technical advice from the workshop are truly appreciated. In addition, I would like to thank Dr. Louise Creagh for her help with the lysozyme enzymatic activity assay, the Department of Food Science for providing the resources to investigate the microbial content of whey protein solutions, and the assistance of the Department of Metals and Materials for the use of the scanning electron microscope. Financial support of the University of British Columbia (University Graduate Fellowships and Webster Fellowship) and the Natural Sciences and Engineering Research Council of Canada are gratefully acknowledged.  1: Introduction  1.  1  Introduction Heat exchanger fouling can be defined as the accumulation of undesirable (Epstein, 1983)  or thermally insulating material on a heat transfer surface. This material has the effect of decreasing the heat transfer efficiency of the equipment, resulting in reduced production time, increased maintenance and a loss of potential income. Despite the presence of fouling in industry it is difficult to get accurate, reliable fouling data, and hence lab and pilot scale units offer the ideal environment to study such a phenomenon. The goal of fouling research is therefore to understand the mechanisms of fouling and develop models of the fouling process to move towards eliminating fouling from industry. 1.1  Fouling of Heat Exchangers Fouling is usually monitored by measuring the mass of deposit per unit heat transfer  surface area, the deposit thickness or the thermal fouling resistance with time. The relationship between the deposit coverage, mf, deposit thickness, Xf, the fouling resistance, Rf, and the deposit physical properties (pf, A,f) is given by dx dm d R = - = —Tf  f  L  f  T  (1-1.1)  For a given fluid, the fouling process is usually a function of heat exchanger geometry, surface material, temperature, fluid velocity and fouling material properties. The various resistances to heat transfer encountered as heat flows from a hot fluid to a cold fluid through a planar surface are shown in Figure 1.1.1. Equation (1.1.2) illustrates calculation of the fouling resistance when the inside heat transfer area differs from the outside heat transfer area.  1: Introduction  2  ^ =R 1  R  1 +  W +  -  -  AR -  = R R +R 1 +  N  2  4 n .4  w +  _i-R -J-R 1  a +  2  (1.1.2)  2  A, Rf=  iJ7"ij:  =R,,+  ~ ' R  Wall  Wall • Deposit  \ lb  •VI  Rf! R  w  Rf;  Fouled  Clean  Figure 1.1.1: Temperature profiles and heat transfer resistances The fouling resistance (Rf) is usually obtained by subtracting the thermal resistance at time zero, when the surface is assumed to be clean, from the corresponding thermal resistance at time t. At constant heat flux, for fouling on one side of a surface with one thermocouple measuring the wall temperature, T , and another the bulk temperature, T , then: w  R  D  T - T  w,c  w  f =  :  (1.1.3)  q An induction period may occur in which there is no measurable fouling. In turbulent conditions, negative values of Rf may be calculated at small times because the initial increase in surface roughness increases the convective heat transfer coefficient, which will temporarily mask the increase in resistance due to the deposit.  3  1: Introduction  The decrease in heat transfer performance is therefore measured by the fouling resistance. In design, selection of an appropriate fouling resistance is usually based upon the fluid velocity, temperatures and fluid properties. Estimates for shell and tube exchangers are typically based on the Standards of the Tubular Exchanger Manufacturers Association (TEMA) and are usually fixed values independent of time. For plate heat exchangers, fouling factors are considerably lower due to the increased fluid velocity, producing scouring effects and the absence of stagnant zones (Bond, 1981). However, in industrial practice, constant values are still used. This single value estimate of Rf implies that asymptotic fouling is assumed, while many fluids do not actually achieve this asymptotic state. Frustration is often expressed by designers or operators over the use of these steady-state estimates of fouling resistance in the design of heat exchangers, because this over-design factor can often exacerbate the deposit formation process. Figure 1.1.2 shows that after an induction period there are generally four types of fouling curves:  Time Figure 1.1.2; Fouling curves  1: Introduction  4  1. Linear :  A linear increase in fouling resistance with time (constant rate).  2. Falling Rate:  A decrease in net deposition rate with time. The fouling curve does not attain a maximum.  3. Asymptotic:  A decrease in the net deposition rate with time. The fouling curve approaches an asymptote.  4.  Saw Tooth:  A linear increase in fouling resistance with time, interrupted by periodic shedding of deposit due to deposit weakening.  1.2  Fouling Categories The five primary categories of fouling as outlined by Epstein (1983) include:  •  Crystallization Although treated by Hewitt et al. (1994) as two separate categories, precipitation and freezing fouling can both be regarded as crystallization fouling. a) Precipitation Fouling This refers to crystallization from solution of dissolved substances onto the heat transfer surface, and is sometimes called scaling. Normal solubility salts precipitate on cooled surfaces, while the more troublesome inverse solubility salts precipitate on heated surfaces. b) Solidification Fouling This is freezing of a pure liquid (for example, icing) or the higher melting point constituents of a multi-component solution onto a sub-cooled surface.  1: Introduction •  5  Particulate Fouling This is the accumulation of finely divided solids suspended in the process fluid onto the heat transfer surface. In a minority of cases deposition is controlled by gravity and the process is referred to as sedimentation fouling.  •  Chemical Reaction Fouling Deposits which occur at the heat transfer surface are the result of chemical reactions in the flowing fluid in which the surface material is not a reactant.  •  Corrosion Fouling This is due to the accumulation of indigenous corrosion products on the heat transfer surface.  •  Biological Fouling This process is due to the attachment of macro-organisms and / or micro-organisms to a heat transfer surface along with the adherent slimes often generated by the latter.  It is important to note that although these categories classify the system, they do not denote the rate-governing process. The progression of fouling has also been broken down into five sequential events giving rise to the 5 X 5 matrix (Epstein, 1983):  •  Initiation This event is associated with an induction period (td) before any measurable fouling has  occurred. For all modes of fouling, many investigators have reported that tj decreases as the surface roughness increases. The roughness projections provide additional sites for nucleation, adsorption and chemical surface activity. Surface roughness also decreases the thickness of the viscous sublayer and hence increases eddy transport to the wall. It has also been shown that an  6  1: Introduction  increase in surface temperature decreases the induction period (Troup and Richardson, 1978), while no clear effect of fluid velocity has been determined.  • . Transport The fouling material (or its precursor) must be transported from the bulk fluid to the wall, where its concentration decreases. This event is therefore a mass transfer phenomenon, where the concentration difference is the driving force for transport.  •  Attachment Attachment of the fouling species to the wall follows the transport of the foulant or the  key component to the wall region, where the deposit is actually formed.  •  Removal This may or may not begin right after deposition has started. That it does is an assumption  implicit in the following removal model originally proposed by Kern and Seaton (1959) and further developed by Taborek et al. (1972): • bx m m = w  r  f  f  (1.1.4)  The removal flux, m , is assumed to be directly proportional to the mass of deposit per unit r  surface area (mf) and the shear stress (T ), and inversely proportional to the deposit strength (T ). w  •  Aging Aging of the deposit starts as soon as it has been deposited on the heat transfer surface.  The aging process may include changes in crystal or chemical structure by dehydration or polymerization.  1: Introduction 1.2.1  7  Factors Affecting Fouling The main variables that affect the rate at which deposit builds up on the heat transfer  surface include wall temperature, fluid velocity and foulant (or precursor) concentration. The fluid velocity affects the fluid shear. Increasing the velocity will increase the shearing action, which may result in decreasing the attachment rate and in foulant removal. However, if the system involves mass transfer or diffusion, increasing the velocity will increase the diffusion of the foulant (or precursor) down the concentration gradient toward the surface. This situation is complicated by the fact that as the velocity increases the convective heat transfer coefficient increases, lowering the surface temperature and thereby offsetting the resistance effect of the deposit layer. Under conditions where deposition is controlled by mass transfer, the effect of  wall  temperature is felt only through the precursor diffusivity, and therefore its contribution is small. However, when the deposition process is controlled by chemical reactions, and therefore an attachment mechanism, the reaction rate is strongly dependent on surface temperature as displayed by the use of the Arrhenius expression. Under these conditions the fouling rate increases exponentially with wall temperature. Concentrations of foulant (or precursor) at the surface will depend on concentration gradients throughout the heat exchanger, but generally the higher the concentration of foulant material, the greater potential for fouling. This study examines these key parameters, using aqueous systems where native proteins undergo denaturation and aggregation reactions, to investigate further the applicability of a specific mathematical model to chemical reaction fouling.  1: Introduction 1.3  8  Objectives This study aims to pursue the understanding of fouling mechanisms by examining and  testing a model for predicting initial fouling rates over a range of fluid temperatures and velocities. The model to be tested was developed by Epstein (1994) for chemical reaction fouling, where mass transfer is treated as a rate process in series with chemical attachment. For a given fluid, chemical system and a fixed wall temperature, the chemical attachment coefficient is assumed to be inversely proportional to the square of the friction velocity, while the mass transfer coefficient is directly proportional to this velocity. Therefore, it follows that, if the initial fouling rate is mass transfer controlled, e.g. at low fluid velocities, then as the velocity increases, the deposition flux increases. If however, the initial fouling rate is chemical attachment controlled, e.g. at high fluid velocities, then as the fluid velocity increases, the deposition flux will decrease. It is this concept, that there is a maximum initial fouling rate at a given wall temperature for a range of fluid velocities which demonstrates a shift from mass transfer to chemical attachment control, that is to be examined in this work. The model also predicts that this maximum rate and the fluid velocity at which it occurs both increase as the wall temperature is increased. Three experimental systems were employed to validate this model. The first system involved the acid catalyzed decomposition of 2-fiiraldehyde to a higher molecular weight insoluble polymer that deposits on the heat transfer surface. After a successful determination of the appropriate kinetic parameters, the fouling experiments achieved only limited success, due to the presence of corrosion. It was this presence of two competing mechanisms that complicated  1: Introduction  9  the fouling results, and lead to unsafe experimental conditions. The fouling experiments were therefore terminated and are presented in Appendix 2 for completeness. Two further successful experimental studies have been performed using dilute solutions of whey protein concentrate, and of purified chicken egg white, namely lysozyme. Both systems involved protein denaturation and aggregation reactions. Therefore the specific objectives of this work are:  1. To investigate the effect of fluid velocity, over a range of wall temperatures, on the initial fouling rate. A wide range of fluid velocities is required to observe both an increase and decrease in initial fouling rate with increasing fluid velocity.  2. To quantitatively assess the consistency of the experimental data for at least two experimental systems with the predictions of the theoretical model of Epstein (1994).  3. To execute experiments under conditions such that the dominant deposition method is through chemical reaction fouling as assumed in the model, rather than precipitation, particulate, biological or corrosion fouling.  4. To determine the validity of the model assumption of constant foulant properties (pf, Xf), and whether significant amounts of deposit aging occur over the duration of experiments with protein fouling.  1: Introduction  10  5. To compare the kinetic activation energy and reaction order obtained as the best fit from the mathematical model to literature and/or experimentally determined kinetic values.  Section 2 provides a review of the pertinent literature material available on the modeling of chemical reaction fouling and on the behavior of the protein solutions to be studied. Section 3 covers the experimental apparatus and methods performed to achieve the results that are presented in Section 4. Included in Section 4 is a discussion of the relevant experimental results primarily from the Tube Fouling Unit (TFU) required to satisfy the above listed objectives. Section 5 discusses quantitatively the goodness of fit of the experimental results to the mathematical model and indicates the success and / or failure of the mathematical model when a system like the T F U is used with protein solutions. Section 6 summarizes the experimental and model findings and indicates the value of the study.  2: Literature Review  11  2.  Literature Review  2.1  Chemical Reaction Fouling Chemical reaction fouling as defined by Watkinson (1992), is a deposition process in  which a chemical reaction or series of reactions either form the deposit directly on a surface or is involved in forming precursors which subsequently give rise to a deposit. Unlike corrosion fouling, this reaction does not take place with the wall itself. Considerable experimental work has been performed in industries where this type of fouling occurs. These include the petrochemical industry where hydrocarbons undergo chemical degradation, the food and dairy industry, where food products will spoil when exposed to heat, and also the pharmaceutical industry. However, in the petrochemical industry, plant shut-downs due to fouling may occur on a yearly basis, while in the food and dairy industry this occurs daily. Mechanisms of chemical reaction fouling tend to vary, but fouling in hydrocarbon streams at moderate temperatures in the presence of traces of oxygen proceeds via autoxidative polymerization, propagated by free radical chains (Watkinson, 1992). Taylor (1969), cited in Braun and Hausler (1976), postulated: 0 C10H24  + traces S, N  2  -> 0  C10H22S0.3N0.05O1.0  A soluble oxidation product  C 1H59S0.5N0.5O7.5  Insoluble polymer, M W = 400 - 600 g/mol  2  ~£  3  For food products, the primary sources of food spoiling are proteins, in particular whey proteins. The mechanism proposed by Roefs and DeKruif (1994) also follows the typical free radical polymerization mechanism of initiation, propagation and termination. However, rather than the formation of soluble oxidation products, the protein transforms from the stable native  2: Literature Review  12  protein (B) to a denatured protein (B*) by a series of molecular level chain unfolding steps, to an insoluble, aggregated form (Bi*, Bj*, Bj+j) of multiple denatured proteins joined together through the readily available disulfide bridges and sulphydryl groups. This is shown schematically in Figure 2.1.1.  Figure 2.1.1 : The heat denaturation of B-lactoglobulin (Roefs & De Kruif, 1994)  2: Literature Review 2.1.1  13  Effect of Wall and Bulk Temperatures For systems which involve chemical reactions, wall and bulk temperatures are clearly  important variables. For a given fluid composition, the rate of fouling increases with wall temperature and typically follows the Arrhenius form of Rf = A e x p ( - A E / R T jover a wide 0  f  W  range of wall temperatures. Large changes in the fouling activation energy, AEf, can often be attributed to a change in the dominant reactions, where values of AEf have typically been reported in the range of 20 - 180 kJ/mol. For a recirculated gas oil system (Watkinson, 1968) initial fouling rates were correlated as follows:  (2.1.1) demonstrating a reciprocal dependence upon velocity. As stated by Watkinson (1992), an activation energy of the order of 20 kJ/mol represents a physical process, or the combined effect of several chemical reactions. Values greater than 40 kJ/mol are assumed to be in the chemical reaction regime. Bulk temperature has a strong effect on the controlling fouling mechanisms. As discussed by Watkinson and Wilson (1997), when the concentration of fouling precursors in the bulk fluid is small, deposition is governed by the transport of reactants to the hot wall, reaction at the wall and then subsequent attachment. Although other factors such as bulk oxygen concentrations can have an effect, the situation of low precursor concentration would correspond to low bulk temperatures incapable of producing fouling precursors. However, when the concentration of fouling precursors in the bulk fluid is large (corresponding to high bulk temperatures with  2: Literature Review  14  significant reaction rates), deposition will be governed by precursor generation in the bulk fluid, mass transfer of precursors to the hot wall, chemical reaction at the wall and finally attachment. The transition between bulk fluid reaction control and thermal boundary layer control has also been discussed by Fryer et al. (1988) and Belmar-Beiny et al. (1993) in their consideration of milk fluid fouling. Protein denaturation and aggregation occurs rapidly above 70 - 75°C, such that wall and bulk reaction control can occur in different parts of the same heat exchanger if the bulk fluid temperature exceeds this critical temperature. However, in the experiments of Asomaning (1997) with 10 % heavy oil in fuel oil, an increased bulk temperature resulted in a decreased fouling rate when all other conditions were held constant. This was explained in terms of a decrease in concentration of suspended asphaltene solids which occurred as the bulk temperature and hence solubility increased. Fouling was likely not controlled by a chemical reaction in this case. 2.1.2  Effect of Fluid Velocity If the fouling deposition process is controlled solely by a chemical reaction, which is not  influenced by mass transfer, the fouling rate should be independent of fluid velocity. However, reports are contradictory. In fact, as shown by Crittenden et al. (1987a) with their 1 % v/v styrene in kerosene polymerization system, the velocity effects can in fact be very complex. At low wall temperatures the initial fouling rate is virtually independent of velocity (though both small  increases followed by small decreases of Rf  0  with increasing mass velocity are discernible),  whereas at high wall temperatures, the rate increases with increasing velocity, i.e. mass transfer appears especially important at the higher temperatures. These results are presented in Figure  2.1.2.  2: Literature Review  100  300  500  700 -2-1  Figure 2.1.2: Effect of mass velocity on initial fouling rate at ten wall temperatures. Circles are experimental fouling results of Crittenden et al. (1987a) for the solution polymerization of styrene in kerosene. Lines represent the best fit regression to the experimental data. It should be noted that elucidating the effect of increasing velocity without considering the compensating effect of a decrease in wall temperature and its effect on the Reynolds number can lead to erroneous conclusions. Modeling can obviously be very complex. The Watkinson / Epstein model (1970) for gas oil and sand-water slurry fouling involves transfer, adhesion and release steps. The initial fouling rate, which obviously involves no release mechanism, is given by  2: Literature Review  16  where <> | is the mass flux of the fouling precursor and S is the sticking probability of the foulant material to the wall: <> t = k (C -C ) m  K=  b  w  (2.1-2)  2 / /3  11.8Sc a exp 2  S=—  /  V  RT y w  V f 2  Combining these equations, the initial fouling rate is given by dR dt  (  f  C  b "  c  w )  f-AE  (2.1.3)  t=o  From above it is clear that this last equation is in approximate agreement with Equation (2.1.1), which correlates the experimental results of Watkinson and Epstein (1970). Note, however, that this model predicts that the initial fouling rate decreases with increasing fluid velocity. This will not always be the case; various studies have shown an increase and/or a decrease in the initial fouling rate with increasing velocity. Crittenden et al. (1987a) performed model experiments to study chemical reaction fouling using a dilute solution of styrene in kerosene. These experimental results are presented in Figure 2.1.2 (Re = 1000 - 4500). From their model experiments and the observation of a mixed velocity effect on the fouling rate, Crittenden et al. (1987a) concluded that at relatively high wall temperatures and low fluid velocities mass transfer effects control the overall rate of fouling. Their results at their highest flow rates and their three lowest wall temperatures produced activation energies only 30 % lower than that for pure styrene polymerization, suggesting that under these conditions the overall deposition rate is mainly controlled by reaction kinetics.  2: Literature Review  17  To model these observations, Crittenden et al. (1987b) considered mass transfer plus chemical reaction as a potential deposition process and allowed for transport of foulant from the wall back into the bulk fluid, i.e.  dR dt  f  1=0  I  pX {  I  K  (2.1.4)  fWi  {  kT k~ +  where k is the reaction rate constant and Cf, is the interfacial deposit concentration. Obviously r  this last term is not quantifiable, and is therefore very difficult to estimate. Although the model appeared to fit the experimental results qualitatively in the extremes of kinetic and mass transfer control, Crittenden et al. (1987b) achieved only limited success in the prediction of the velocity effect. Nijsing (1964) assumed that fouling from an organic nuclear reactor coolant was caused by the instantaneous reaction of a precursor (A) to a product (B) which crystallized rapidly when compared with its diffusion rate to the surface. Assuming that the physical properties were not temperature dependent, that the reaction rate was instantaneous above a critical temperature (T ) c  and that the diffusivities of A and B were equal, the average rate of deposition was given by: Average rate of deposition oc [ C D R e b  08 7 5  Sc  0 3 3  ]/d  (2.1.5)  Here it is clear that the deposition was mass transfer controlled and increased with increasing fluid velocity. Paterson and Fryer (1988) developed a model for fouling by milk based fluids (Re = 2000 - 5000) in the surface temperature range of 85 - 110 °C. They treated the boundary layer as a differential chemical reactor and used a sticking probability approach, which like Watkinson and  2: Literature Review  18  Epstein (1970) showed a decrease in fouling rate with an increase in velocity. The underlying assumption in this model was that the reaction occurred throughout the region of the fluid hot enough to support significant reaction rates (Reb = 2000 - 5000): dBi dt  a  5  = —exp  "-AE /  (2.1.6)  f  -  /  R  T  w .  To date no single mathematical model for chemical reaction fouling has been capable of explaining the variation in experimental observations caused by two of the most fundamental variables, namely fluid velocity and surface temperature. 2.2  Theoretical Fouling Model The following model proposed by Epstein (1994) was developed for initial chemical  reaction fouling of a heat transfer surface where surface attachment was treated as a rate process in series with mass transfer. This model focuses on the initial fouling rate, R f , where time zero 0  is taken as the time when measurable fouling can first be detected. The experimental data used to model this process were those of the 1% v/v styrene in kerosene experiments of Crittenden et al. (1987a) shown as Figure 2.1.2. These experiments had previously been modeled (Crittenden et al., 1987b) as a mass transfer process in series with chemical reaction, with mass transfer control at low flow rates and high temperatures, and surface reaction control at high flow rates and low temperatures. However, the model, given by Equation (2.1.4), achieved only limited success, not only because the predicted Rf values were 0  considerably higher than the experimental values, but more importantly because the model failed to show a maximum in any of the predicted curves of Rf versus fluid velocity. 0  2: Literature Review  19  The Epstein model has also been applied to particulate fouling (Vasak et al., 1995) and has been proposed for precipitation fouling (Epstein, 1993). 2.2.1  Mathematical Model Development (Epstein, 1994) Consider chemical attachment modeled as a rate process in series with mass transfer. The  mass transfer rate of the precursor to the heat transfer surface is given by $ =k (C -C ) m  b  (2.2.1)  w  The rate at which precursor reactant is converted to foulant product which attaches to the heat transfer surface is given by <|) = k C :  (2.2.2)  a  where n is the reaction order. Equations (2.2.1) and (2.2.2) can then be combined as follows: c  b  = ( c  b  _  r  - c  b  ) + c  w  - * • k K  w  *  -'  (2 2 3)  k C -' n  m  K  a ^ w  Rearranging Equation (2.2.3), C  1  b  1  •+ k  a  c  „ - i  r  w  (2.2.4) V  J  or  • = !  C  " j  -+ -  k  m  (2-2.5) n-1  k C a  w  Mass Transfer For turbulent liquid solution flow parallel to a heat transfer surface, and assuming fully developed velocity and concentration layers, then in good approximation (Treybal, 1980), k =V./k, m  (2.2.6)  2: Literature Review  20  where k,=k'Sc  (2.2.7)  2/3  For isothermal fluid flow, k' = 11.8 (Metzner and Friend, 1958). Chemical Attachment Near the wall of a pipe, a very thin viscous sublayer exists with a linear relationship between v and y as given by the universal velocity distribution equation (Knudsen and Katz., 1958), v  V.  Near the heat transfer surface, the fluid residence time 9 is inversely proportional to velocity v within the viscous sublayer, and therefore  eoc-V yv.  2  P  The chemical attachment coefficient, k , at the surface may be written as being proportional to a  the product of an Arrhenius term and the fluid residence time near the surface. The Arrhenius term describes the strong dependence of the reaction rate constant, k , on surface temperature, r  while the fluid residence time allows for the fact that the longer the fouling material spends at the heat transfer surface, the greater the probability that it will deposit.  k^a  AE/RT  e  »e  Consequently, for an arbitrarily small but fixed value of y, k  o c - i W ^ pV. 2  Therefore  (2.2.8)  21  2: Literature Review  e  -AE/RT  K=~  w  (2.2.9) ^  T~ k"pV  a  2  ;  Examination of Equation (2.2.9) shows that for a givenfluid,chemical system and fixed wall  temperature,  k  - =M 7  ( 2  - 2  1 0 )  where u  k  AE/RT„.  k =  (2.2.11)  2  Substituting Equation (2.2.6) into Equation (2.2.1), Equation (2.2.10) into Equation (2.2.2), and both Equations (2.2.6) and (2.2.10) into Equation (2.2.4) yields: C -C  1  (2.2.12)  C»=k V <|>  (2.2.13)  b  =k <|)V.-  1  w  2  2  ^  = k V;'+k V. C^ 2  1  (2.2.14)  n  2  <P Equation (2.2.14) shows § increasing and then decreasing as V* increases. Substituting Equation (2.2.13) into Equation (2.2.14) gives:  ^  = k.V; + k V. ((|>k V. p 1  2  2  2  2  (2.2.15)  which can be rearranged to produce  111 C = k,V;'(|) + k "V."(|)  n  b  2  (2.2.16)  The fouling resistance, R , is given by the following equation, assuming average foulant physical f  properties:  22  2: Literature Review  X  (2.2.17)  pX  {  {  {  From this it follows that the fouling rate, Rf (= d R / d t ) , is dependent on the rate of precursor f  converted to foulant and deposited on the heat transfer surface. Therefore the relationship of initial fouling rate, R f , to § is G  Rf,  m<|)  (2.2.18)  kfPf  where the stoichiometric factor m represents the mass of fouling deposit per mass of precursor transported to and reacted at the wall. Rearranging Equation (2.2.18) and substituting into Equation (2.2.16) for (j) yields  ^ L ]  m /  R  k.v;  1  fo  (R  Vm /  fc  k)« V"  (2.2.19)  2  Making use of Equations (2.2.7) and (2.2.11), X p/  r  C =  Rfo k ' S c V . /3  b  V  m j  +  f  V  \Y  n  m J  l  R f o  (2.2.20)  For a given fluid, chemical system and average foulant properties,  ) k' = constant = k m J ^fPf m  (2.2.21)  (k")^ = constant = k  Therefore, Equation (2.2.20) becomes /  C  b  =kRf Sc^Vr 0  1  +k^R  f 0  ^  AE/  pe ' "  N /n  (2.2.22)  23  2: Literature Review  The initial fouling rate (Rf ), the bulk concentration (Cb), the Schmidt number (Sc) and 0  the friction velocity (V*) can be obtained from experimental data, while the activation energy (AE) and the reaction order (n) must be obtained from literature or appropriate reaction kinetic experiments. Once these and the fluid properties (p, r\) are available, a non-linear multiparameter least squares regression can be employed to determine the best fit of the parameters k and k for all conditions of variable wall temperature (T ) w  and fluid velocity.  Therefore, from this discussion, it is obvious that k and k, as well as k'and k" , are universal constants for the given system, while ki and k.2 will vary with wall temperature. Once Equation (2.2.22) has been solved for k and k, it can be used again to determine model values of Rf  0  for  all velocities and temperatures, enabling a comparison of experimental and model initial fouling rates. The true test of the model is whether a single value of k and k (and A E and n) can • correlate all the data over a wide range of V* (including both a rise and fall in Rf with V*) and 0  for more than one value of T . w  From Equation (2.2.22) it is clear that an ideal system for assessing the model would be one for which kinetic data are available from the literature, or at least is conducive to experimentation where the rate of reaction, order of the reaction and the activation energy can be determined. If kinetic data cannot be determined, then the activation energy, A E , and reaction order, n, will be unknown and can be added to the parameters of Equation (2.2.22) to be solved using a least squares analysis. The original method of evaluating the model constants used by Epstein (1994) involved selection of the single maximum initial fouling rate for a given wall temperature, which allowed  24  2: Literature Review  evaluation of the universal constants from the single measurement at the chosen condition. The present method is different and more reliable than that used by Epstein (1994), as it takes advantage of the fact that performing a regression analysis using all the measured data will lead to more accurate values of the universal constants. 2.2.2  Application of Model As discussed above, the critical equation which determines the success or failure of the  model is Equation (2.2.22). This equation corresponds to Equation (1) of Vasak and Epstein (1996), where they discuss the regression analysis method of chemical reaction fouling via this model. Vasak and Epstein (1996) consider the method previously used by Epstein (1994) to analyze the styrene polymerization experimental system of Crittenden et al. (1987a) and concluded that the goodness of fit is best measured by minimizing the variance across all data points after performing the regression analysis. The sample variance is calculated using the following equation:  (2.2.23)  The activation energy (AE) and reaction order (n) can be obtained from kinetic experiments or, as with Crittenden et al. (1987b), the activation energy from the high velocity and low wall temperature fouling experiments can be used. To compare different sets of experimental data, the variance, root mean square deviation ( R M S ) and average absolute deviation (AAD) offer various methods of comparison. The average absolute deviation is given by  AAD = 100x£ | R i=l  f 0  )  Rfo calc  expt  )  'N expt  Ji  (2.2.24)  2: Literature Review  25  For goodness of fit purposes, Vasak and Epstein (1996) favored minimizing the variance because, unlike the root mean square and average absolute deviation, it gives less weight to deviations between predicted and experimental fouling rates for low values of experimental fouling rate, where the experimental error was the greatest. If the chemical reaction is first order, i.e. n = 1, then Equation (2.2.22) reduces to an even simpler form, C  kSc  b  -rR  =  kpe  2 / 3  -T7— + — * V  AE/RT  »V.  2  (2-2-25)  ^  ^ f o  and the following explicit equation for Rf can be determined: 0  C  Rf = 0  kSc V.  -  (2.2.26)  k p e ^ V  2/3  +  2  T]  However, for all other reaction orders an implicit solution to this model equation is required, as shown in Equation (2.2.22).  2: Literature Review 2.3  26  Milk Based Fluid Fouling Deposits which form on heat transfer surfaces when milk products are heated are of  considerable interest to the dairy industry (Burton, 1961, 1965, 1968) not only for understanding the cleaning and removal stages of the process, but because deposit build-up may be the limiting factor for the length of a production run. For most heat exchangers used in the dairy industry, cleaning on a daily basis is common practice (Visser and Jeurnink, 1997).  2.3.1  Fouling Components of Milk Table 2.3.1 shows a typical milk composition, with the 4.2 % of proteins and minerals  dominating as the fouling material.  Table 2.3.1: Milk composition (Taylor, 1992) Component  % In Milk  Water  86  Lactose  5  Fat  4.7  Proteins  3.5  Minerals  0.7  Proteins are very large molecules, with high molecular weights, comprised of a large number of amino acids joined together in a specific sequence. Milk proteins can be broadly classified into two groups: casein proteins (2.8 % of milk) and whey proteins (0.7 % of milk). Casein proteins make up approximately 80% of the protein in milk, and are present in the form of colloidal particles called micelles, in the size range of 40 to 300 um. Milk of good quality typically has a pH around 6.7; if this pH is lowered toward the isoelectric point of caseins (pH  27  2: Literature Review  4.6), the micelles will coagulate and precipitate from solution. Casein proteins, however, are not temperature sensitive. Whey proteins make up approximately 20% of the proteins in milk. The molecules have a globular shape and are present in solution (Taylor, 1992). When milk is heated to 60°C these proteins will begin to denature, upon which their shape and structure changes dramatically. The temperature at which these proteins begin to denature will depend upon the operating conditions, but in general this occurs around 60 - 70°C. Whey protein composition is given in Table 2.3.2.  Table 2.3.2: Composition of whey proteins in milk (Taylor, 1992) Protein  Isoelectric  Molecular  Number of  % of Whey  Point  Weight  Amino Acids  Protein  162  50  123  22  18300  B-Iactoglobulin A  5.3-5.5 36600  P-lactoglobulin B a-lactalbumin  4.2-4.5  94000  Immunoglobulins (G, A, M)  14200  4.6 - 6.0  400000  12  900000 Bovine Serum Albumin  5.1  69000  5  4000  Proteose  3.3-3.7 Peptone  582  10  40000  Minerals in milk are present as inorganic salts in solution (sodium, potassium, calcium, magnesium, chloride, phosphate and citrate). These salts dominate deposition above 100°C,  2: Literature Review  28  although they will be present in a proteinaceous deposit at lower temperatures. Early literature indicated that mineral salts, mostly calcium phosphate, were the first species to be deposited during fouling; however, more recently there is agreement that the first layer to be deposited is a monolayer of adsorbed protein (De Jong et al., 1992; Visser and Jeurnink, 1997). Visser and Jeurnink (1997) and Belmar-Beiny and Fryer (1993) have indicated that, in the early stages of an experiment, whey proteins adsorb onto a metal surface to form a monolayer at room temperature. Since mineral salts rely upon inverse solubility and hence elevated temperature for precipitation, it follows that proteins will always be the first material to deposit on the heat transfer surface. According to Burton (1961) there are at least two reactions in milk that can result in fouling. The first is the denaturation of proteins, which is responsible for the Type A , proteinaceous deposit. The second is due to the inverse solubility of salts like calcium phosphate, which with rising temperatures lead to the formation of type B deposits. The lower temperature deposit (type A), which comprises the largest amount, is a soft, voluminous, curd-like material, white, or cream-like in color, which may overlay a harder base probably caused by over-heating at the surface (Gordon et al., 1969). These deposits are largely made up of protein (50 - 60%) but also contain 30 - 35% mineral matter. The fat content, however, is low, in the range of 4 - 8%. At higher temperatures the deposit changes to a more brittle and gritty deposit, which is gray in color. This deposit (type B) has a lot higher ash content (about 70%) and a corresponding lower protein content (15 - 20%) than the lower temperature deposits. The fat content is similar in both deposit types. Model studies (Paterson and Fryer, 1988) have shown induction times of the order of thirty minutes in plate heat exchangers before noticeable deposit formation takes place. This may be a consequence of the formation of the protein monolayer and hence, the reactions necessary  2: Literature Review  29  for protein-protein aggregation and subsequent deposition. 2.3.2  Factors Affecting Fouling Reproducible fouling studies are difficult to achieve using milk (due to both seasonal and  feedstock variations), making it difficult to perform a systematic study. In addition, variation in results may be due to differing milk quality, pH, pretreatment, dissolved gases, variable heat transfer surfaces or changing operating conditions. I. Milk Quality Examples of deposit variations with variations in milk quality are shown in Figures 2.3.1 and 2.3.2:  220 r  '  0  10  20  30  40  50  Time after mUking. h Figure 2.3.1: Variation in amount of deposit on a plate heat exchanger with age of milk (open circles: separated milk, solid circles: homogenized milk) (Burton, 1968) Fouling can be strongly enhanced if dissolved gas (air) is released from within milk at temperatures greater than 40°C (Jeurnink, 1995). Burton (1968) suggested that this is only significant if the bubbles- separate at the heat transfer surface, thus acting as nucleation sites for deposit growth. This problem can be eliminated by operating the test equipment under pressure  30  2: Literature Review  to ensure that the dissolved air remains in solution.  30 Herd A o o. 20  © 10 E < 1 1 1 t 1 1 1I J I I L J 1 1 ( May J««e J«(jr A<ig.SepcOct.Nov. D e c Jan. Feb. Mir. Apr. May Jane l<Ay Aug.Sepc T96S •< 1966-  Herd B  £ 10 o o  © 10 ©  4  0  J  1 « « ' < j I 1 I t t 1 j i July A « g ^ e p t . O c c . N o v J 5 e c J»a. Feb. Mir. Apr. May i « a e July Aog.Sepc. 196S >"H1966 =—r— f  i  Figure 2.3.2: Variation in deposit formation on a plate heat exchanger with season for the bulk milk of two herds (Burton, 1968) II. pH Gordon et al. (1969) performed a systematic study of numerous variables on the accumulation rate of milk deposits. They showed that decreasing the pH towards the isoelectric point increased the amount of deposit. This trend was strongest at low mass flow rates, but was apparent in all cases (2500 - 7272 kg/h). The range of solution pH values tested was 6.03 - 6.91. Skudder et al. (1986) showed that decreasing the pH from 6.67 to 6.0 or 5.5 significantly increased the differential pressure drop on a U H T (ultra high temperature) plate heat exchanger. This was a manifestation of an increased deposition rate of denatured protein when operating at temperatures up to 140°C. It was also noted that the deposition of minerals decreased with a decrease in pH, implying the increased stability of salts with decreased pH.  31  2: Literature Review III. Pre-Treatment  Burton (1968) discussed the feasibility of pre-holding milk at temperatures above 65°C. According to Bell and Sanders (1944) this pretreatment produces a reduction in pressure drop in a steam heated exchanger. In addition, they noted that the higher the temperature of pretreatment, the greater the reduction in deposit quantities (minimum deposition at 95°C). Work using plate heat exchangers has enabled visual inspection of this phenomenon and revealed that the amount of type A deposit was reduced after pre-holding, due to earlier denaturation of the soluble proteins, such that the deposits that do result are of mainly the type B category. Visser and Jeurnink (1997) suggested that both pre-holding and reconstituting milk based fluids leads to a substantial reduction in fouling of the heat transfer surface. This is because the pre-treatment of these solutions causes earlier denaturation of some of the whey proteins, resulting in a less active protein capable of participating in the fouling mechanism.  IV. Velocity Gordon et al. (1969) studied the effect of the mass flow rate for a series of experiments at a constant milk outlet temperature of 82°C, using a fixed volume of milk. A logarithmic decay relationship between the mass of deposit and the mass flow rate of feed resulted: log(g deposit) = 7.6937 - log(2.0019(kg/h))  (2.3.1)  This study covered mass flow rates from 771 kg/h (Re > 6000) to 8165 kg/h through a pipe diameter of 3.81 cm. Fryer (1987) suggested that after the induction period, food fouling is no longer controlled by a surface reaction. Rather it is controlled by a bulk reaction in the hot regions of the fluid, with this reaction then producing solid material that adheres to the surface. Therefore to  2: Literature Review  32  minimize fouling one should minimize the volume of fluid that is hot enough for the reaction to occur and the residence time of the protein in that volume. From this rationale, with all other conditions held constant, increasing the bulk fluid velocity to decrease the volume of fluid that is hot enough to be denatured, would decrease the amount of deposition and hence the fouling rate. V. Bulk and Surface Temperatures Research has been performed to determine whether deposit formation is dependent upon surface and bulk reactions, or by mass transfer of the fouling species to the heat transfer surface. Recent work by Fryer and Slater (1984) and Belmar-Beiny et al. (1993) indicates that deposit formation is controlled by the volume of fluid near the heat transfer surface hot enough to sustain significant reaction rates, i.e. in the viscous sublayer. Hence, when high bulk temperatures are employed, both bulk and surface reaction rates appear significant in the amount of deposit formation. This would suggest a complex relationship between bulk temperature, surface temperature and fluid velocity. 2.3.3  Biochemistry of Whey Protein Solutions To avoid many of the previously mentioned problems associated with milk fouling, such  as different types of deposit depending on temperature, reconstituted whey protein solutions serve as an ideal model fluid for milk based studies. Of the whey proteins, P-lactoglobulin represents the most abundant part of the total serum protein (approximately 50%). Figure 2.3.3 shows a schematic of the structure of a p-lactoglobulin molecule.  2: Literature Review  33  Figure 2.3.3: Tertiary structure of P-lactoglobulin (Papiz et al., 1986) At a neutral pH, below 40°C p-lactoglobulin is a dimer of two identical sub-units, each of which has a molecular weight of 18300 g/mol and contains two disulfide (S-S) bridges and one free (-SH) group, all normally unreactive. On heating between 60 and 70°C the dimer dissociates and large conformational changes occur, which are accompanied by unmasking of the (-SH) group. This change allows protein molecules to interact with each other through the (-SH) group and the (S-S) bridge (Lalande et al., 1985). Above 70°C these reactions become irreversible and polymers of high molecular weight may result. According to Bott (1990) protein degradation is a two-stage process. On heating, the complex three dimensional structure of a protein normally held together by intramolecular disulfide bridges, ion pair interactions and Van der Waals forces, is distorted. First, the protein unfolds; this process is called denaturation. The denaturation stage is monomolecular, and may  34  2: Literature Review  be reversible, in that when this protein cools, its structure returns to its original state. As the secondary and tertiary structure of the protein unfolds, more of its inner structure is able to interact with other molecules. Aggregation of proteins into insoluble clumps via intermolecular bridging reactions thus occurs readily and is irreversible. A simplified schematic from Fryer et al. (1989) is shown in Figure 2.3.4. Native protein  ^  Denatured protein  Hydrophobic core  Aggregated protein  ^ Disulphide bridge SH Sulphydryt group  Figure 2.3.4: Protein denaturation and aggregation (Fryer et al., 1989) Figure 2.3.5 shows a summary of the possible mechanism paths for denaturation and aggregation of whey proteins. Dcaaturatioa  Aggregation  BUUC  WALL LAYER  1  t  Surface  "It  adsorption  " It  Figure 2.3.5: Possible pathways of deposit formation (Fryer et al., 1989)  2: Literature Review  35  Skudder et al. (1981) have shown that the addition of free sulphydryl (-SH) groups (by the addition of L-cysteine & HC1) increased the total amount of deposit, and the removal of free (-SH)  groups (by the addition of iodates, which oxidize the (-SH)  groups) showed the  correspondingly decreasing trend. Gotham (1990) showed that it was protein aggregation, rather than denaturation, that is the rate-determining step leading to the deposition of proteinaceous material. Although Plactoglobulin becomes more thermally unstable above pH 6.25, i.e. denaturation is greater, fouling in pasteurization and ultra high temperature equipment decreases rapidly above pH 6.6. If P-lactoglobulin denaturation were the limiting step, the decreasing thermo-stability at higher pH should lead to increased deposit formation; however, the opposite is true. Although the rate of Plactoglobulin denaturation increases above pH 6.25, the amount of molecular unfolding which takes place during denaturation is reduced, resulting in a lower concentration of free (-SH) groups. However, P-lactoglobulin aggregation increases significantly below pH 6.0, reflecting increased molecular unfolding during denaturation, resulting in increased disulfide exchange reactions. The pH dependence of deposition is very similar to the effect of pH on P-lactoglobulin aggregation. This suggests that although denaturation is necessary for fouling to occur, the stage that leads to fouling is the formation of insoluble aggregates. The pH of whey protein solutions has a very strong effect on the heat stability of its key protein, namely P-lactoglobulin. Its behavior is illustrated in Figure 2.3.6. At a pH of less than 2, and at room temperature, p-lactoglobulin is monomeric owing to strong electrostatic repulsive forces between each molecule and the stainless steel surface. At a pH of 4.65, P-lactoglobulin is close to its isoelectric point (pH = 5.13) and hence has little  2: Literature Review  36  surface charge. Therefore it is not surprising that it is octameric at room temperature. Upon heating, the protein starts to aggregate and precipitate immediately. Fouling in these conditions is severe due to the absence of an electrostatic barrier. At pH values of 5.5 - 6.5, P-lactoglobulin dissolved in water is a dimer. Upon heating, the dimer dissociates into a monomer at 50°C, and at temperatures greater than 60°C, the (-SH) group is exposed to the solution by unfolding and becoming reactive. At pH values greater than 7 very little coagulation of P-lactoglobulin occurs.  fi,  —  fi-fi.. trrev. I dcoxL I riow, very icoclcrtted np«f in cold  ( I  3.58 «<n  J (  I reversible I transfboaatiod  |OCtunecpuX|^ft  36 734  (  3  371  4  <€S b i  5 H  1 6  i  7.5 1  7  8  P  ~l i J 9  Figure 2.3.6: The structure of B-lactoglobulin (Visser and Jeurnink, 1997) From differential scanning calorimetry measurements (Visser and Jeurnink, 1997), the denaturation temperature TD (the temperature at which 50 % of the native protein has unfolded) for p-lactoglobulin is strongly affected by pH. At pH = 4.0, T = 82.5°C, while at pH = 8.0, T D  D  =  72.5°C.  Heat denaturation of the P-lactoglobulin molecule is of most interest at pH = 6 - 7, since this is the range of industrial application. The denaturation and aggregation of a protein is a two-  37  2: Literature Review  stage consecutive process, comprising the unfolding of the molecule followed by aggregation. The first step is reversible and follows first order kinetics, while the second step is irreversible and follows second order kinetics. Overall, a reaction order of 1.5 is reported to adequately describe the process (Visser and Jeurnink, 1997). 2.3.4  P-Lactoglobulin and oc-Lactalbumin Denaturation Kinetics Lyster (1970) reported an in depth study of the denaturation kinetics of cc-lactalbumin and  P-lactoglobulin in heated skim milk, over a wide range of fluid temperatures. He used small sample volumes (0.05 ml) of skim milk in sealed capillary tubes immersed in water (68 - 100°C) or oil (90 - 155°C) baths. Analysis for concentrations of the two major proteins was achieved using established immunodiffusion methods. The kinetic results (Table 2.3.3) for both proteins show very strong temperature dependencies, with a change in activation energy at 90°C. In addition, Lyster (1970) found that, for both proteins, the rate of denaturation at either 78 or 100°C was independent of pH, where the experiments were performed within the pH range of 6.2 - 6.9. Table 2.3.3: Kinetic parameters for heat denaturation of (3-lactoglobulin and a-lactalbumin (Lyster, 1970) fki: s' , fo: 1 g'VM 1  k = Aexp(-AE(kJ/ mol)/RT(K))  Temperature (°C)  Reaction Order  a-la  = 1.412-10 exp(-68.92/RT)  90- 150°C  1  P-lg  k = 9.546-10 exp(-54.79/RT)  90- 135°C  2  68 - 9 0 ° C  2  7  5  2  k = 8.814 -IO exp(-277.85/RT) 37  2  Dannenberg and Kessler (1988) used fresh, skimmed raw milk of pH 6.67 - 6.72 in a  38  2: Literature Review  tubular heat exchanger (1.3 mm id, wall thickness 0.14 mm) and a thermostatically controlled holding section to hold the fluid at 70 - 150°C for periods of 2 - 5400 s. The samples of milk were analyzed and the kinetics determined, as summarized in Table 2.3.4.  Table 2.3.4: Kinetic parameters for heat denaturation of fi-lactoglobulin and a-lactalbumin (Dannenberg and Kessler, 1988)  P-lgA  p-lgB  a-la  n  Temp (°C)  AE(kJ/moI)  ln(A)  r  1.5  70-90  265.21  84.16  0.996  95-150  54.07  14.41  0.994  70-90  279.96  89.43  0.995  95-150  47.75  12.66  0.999  70-80  268.56  84.92  0.997  85-150  69.01  16.95  0.999  1.5  1.0  2  According to Lalande et al. (1985), the kinetics of the irreversible denaturation of Plactoglobulin follow second order rate equations and show a sharp change in activation energy at about 90°C.  Below this temperature the activation energy is approximately 300 kJ/mol, but  above this it is about 40 kJ/mol. Lalande et al. (1985) inferred from these results that the irreversible denaturation of P-lg in milk (i.e. in the presence of other proteins containing disulfide bonds) is a complex reaction consisting of at least two consecutive reactions with different temperature dependencies. From the previous discussion on free (-SH) groups and their apparent effect on the rate of denaturation, Lalande et al. (1985) suggested that the disulfide reaction with the (-SH) group was the rate-determining step.  2: Literature Review  39  2.3.5  Previous Fouling Studies into the Controlling Mechanisms of Deposition  1.  De Jong et al. (1992) discussed the experimental relationship between the denaturation  of {3-lactoglobulin and the deposition of milk constituents on a plate heat exchanger using skim milk in the temperature range of 70 - 122°C. The denaturation and aggregation of P-lactoglobulin was described as a consecutive reaction of unfolding and aggregation steps (Native, Unfolded, Aggregated). The rates of formation and disappearance of these species were given by: dC, dt dCj  ~dT dC  -  K  k C -k C p  —- kC  dt  (2.3.2)  U ^ N  K  q  (2.3.3)  q  (2.3.4)  A ^ U  The reaction rate constants ku and kA are related to the absolute temperature by the Arrhenius equation k = Aexp(-AF^ )  (2.3.5)  T  The kinetic data for the reaction used by De Jong et al. (1992) were those of De Wit and Klarenbeek (1988) (Table 2.3.5) and Dannenberg and Kessler (1988) (Table 2.3.4). Table 2.3.5: Reaction kinetic data for P-lactoglobulin in skim milk Reference  Reaction  Temperature  Order  AE (kJ/mol)  ln A  De Wit and Klarenbeek, 1988  Unfolding  70-90 (°C)  1  261  86.41  De Wit and Klarenbeek, 1988  Aggregation  70-90 (°C)  2  312  99.32  From this information, it is realized that applying an n  th  order attachment process  (Epstein, 1994), based on Cb (measured as CN), is an oversimplification of the fouling  40  2: Literature Review mechanism if unfolding and aggregation are the key steps.  The fouling process was treated as a heterogeneous adsorption reaction of milk constituents at the surface with mass transfer and reaction in series. The mass transfer equation is given by:  J  F  "x 5^  u  C u  n  '  i m  (2.3.6)  )  where xp is the fraction of p-lactoglobulin in the deposit, 8 the thickness of the concentration boundary layer, and C i t the local concentration of U near the surface in equilibrium with the Uj  n  concentration Cu* in the deposit layer at the interface. The reaction rate equation is given by:  R "=-kQ F  (2.3.7)  mt  At steady state the transport rate (JF) and reaction rate ( - R  F  ) are equal, so the unknown  interfacial concentration can be eliminated, resulting in  J  F  =  D  Vk;  x8 p  (2.3.8)  De Jong et al. (1992) invoked the Hatta criterion, which determines whether fouling is controlled by chemical reaction or mass transfer. The square of the Hatta number is the ratio of the maximum chemical reaction rate to the maximum diffusion transport rate:  Ha  2  =  kcr,  D  (2.3.9)  x8 p  When Ha < 0.3, the kinetic reaction rate completely limits the adsorption rate and when Ha > 2, the rate of mass transfer limits the adsorption process. The diffusion coefficient of the key  41  2: Literature Review constituent was estimated from the Wilke-Chang (1955) equation, D = 1.3xl(T —^-r T|V °'  (2.3.10)  17  6  F  where D has units of m /s, T has units of K, and n has units of kg/m.s. Vp is the molar volume of 2  the adsorbed species and is given by  V  F  = N  A  V  (2.3.11)  ^ 7 i d ;  o  The thickness of the concentration boundary layer was estimated from the Sherwood number:  Sh k  k„d" :  D - °  6=— Sh  (2.3.12) '  V  In the range 2000 < Re < 100000 (Experiments were conducted at Re « 12000) and Sc > 0.7, the following relationship was used (De Jong et al., 1992): Sh = 0.027Re Sc 08  (2.3.13)  033  Substituting for k according to its Arrhenius temperature dependence, Equation (2.3.8) becomes  InJp = InA -  AE  + nlnj RT v F  R  _  X  P  6  D  J  F  A  (2.3.14)  j  Therefore, the constants A , AE and n were determined by a multi-parameter regression analysis from the experimental results. The concentration of unfolded P-lactoglobulin at any point across the heat exchanger was estimated from the kinetic data in Table 2.3.5. However, since the "particle" diameter of the adsorbing milk components were unknown, a series of particle sizes were selected, and the most appropriate size was that which had the greatest coefficient of correlation, r . From this, the Hatta number was evaluated and the parameters determined from  2: Literature Review  42  the best fit multi-parameter regression. De Jong et al. (1992) concluded that a mean diameter of adsorbed milk components of > 50 nm gave no agreement with the experimental results. The highest correlation coefficients were achieved with particle diameters smaller than 10 nm and very small Hatta numbers. This meant that their model was most accurate when they assumed a reaction controlled fouling process, such that Cu,im  =  Cu The corresponding results for this solution were, n = 1.2, A = 1 x 10"  4  (m /kg s), A E = 18.8 kJ/mol and r = 0.926 (r - 0.857). Note that the reaction order is 16  02  2  consistent with the previous discussion and note also the very low activation energy for the process. Given the high bulk temperatures (up to 110°C) and chemical reaction control, a relatively low activation energy is not too surprising. 2.  Fryer and Slater (1984) conducted experiments on a differential section of a stainless  steel tube (4.6 mm id, 9 cm long) which was encased in a brass block. It was found that the fouling rate of reconstituted skim milk in the post-induction period between wall temperatures of 90°C and 110°C, bulk temperatures of 60°C, and Reynolds numbers between 650 and 5760, could be correlated by: dBi  k ""-AE 1 + Bi = — expi R T +BiT dt Re d  1  w  Bi is the fouling Biot number (RfU ), T c  w  and T  b  k Bi  (2.3.15)  f  b  are the wall and fluid bulk temperatures,  respectively, and kd and kf are experimental constants. The activation energy was found to be 89 ± 5 kJ/mol and kd and k were equal to 4.85 x 10 f  13  s' and 1.31 x 10" s" . The removal term of 1  3  1  this equation is of the Kern-Seaton type without the shear stress factor, while the deposition term contains an auto-retarding step to account for the decrease in reaction rate due to the reduction in  2: Literature Review  43  deposit-fluid interface temperature as the deposit builds up. Paterson and Fryer (1988), using reaction engineering theory, proposed an inverse proportionality to fluid velocity (Re) with an Arrhenius dependence on interfacial wall temperature for deposition. They maintained that models based on sequential fouling, i.e. foulant precursor mass transfer to the wall followed by surface reaction, cannot explain the decline in initial fouling rate with increased fluid velocity. However, they suggested that fouling is controlled by a single, determinable, reaction, throughout the region of the fluid which was hot enough to support significant reaction rates. With this assumption, they also assumed that the conversion of the limiting species in the single reaction was low, which was what they found experimentally. Paterson and Fryer (1988) assumed that the only region hot enough for protein denaturation would be in the viscous sub-layer. Thus they considered that the mass transfer of foulant material would be from the viscous sub-layer to the wall, and therefore they focused on the reaction within the viscous sub-layer to develop the above equation. In conclusion, Fryer and Slater (1984) say there is no need to assume that the rate controlling step in reaction fouling is necessarily the wall reaction, but rather in this case a reaction occurring in the viscous sub-layer, the size, temperature and velocity of which controls the fouling rate. 3.  Belmar-Beiny et al. (1993) performed experiments with whey protein solutions in a  tubular heat exchanger, and interpreted their results by considering the amount of deposit to be proportional to the volume of fluid that was hot enough to produce denatured and aggregated protein. Belmar-Beiny et al. (1993) discuss previous work with skim milk fouling by Fryer and  44  2: Literature Review  Slater (1984) and their model equation (Equation 2.3.15). They too cannot reconcile this equation with a wall reaction alone, and continue to explain it through the reaction engineering approach, showing that Equation (2.3.15) could be produced by assuming a homogeneous reaction in the viscous sub-layer. Two effects of flow were considered to account for the variation in initial fouling rate with increasing Reynolds number: (I)  The change in the volume of the reactor; as the velocity increases, the thickness of the thermal boundary layer, and hence the effective volume of fluid hot enough to react, decreases. Given that the thermal boundary layer for protein solutions will likely fall within the viscous sublayer (y < 5), the changing thickness of the thermal boundary layer +  can be estimated from (Appendix 3.4)  5 where the momentum boundary layer thickness (8) is given by the thickness of the viscous sublayer,  (II)  A decrease in the sticking probability results due to increased shear forces with increased velocity.  If deposit formation results from a sequential combination of mass transfer and chemical reactions, as suggested above, the slowest of these will be the rate controlling step. Therefore Belmar-Beiny et al. (1993) aimed to identify this step. They considered two possible cases: (1)  If fouling is mass transfer controlled, the slowest process will be the transfer of reacted protein to the wall. The rate of deposit formation will not be a strong function of  45  2: Literature Review temperature, but it will increase with fluid velocity. (2)  If fouling is reaction controlled, deposit formation will be a function of the temperature where the controlling reaction takes place. Reactions in a number of different locations could control the process: (a)  If fouling is controlled by surface reaction, deposition will occur wherever the wall temperature is hot enough for denaturation and aggregation, regardless of the bulk temperature. The process will be a function of the wall temperature, but not of the bulk temperature.  (b)  If the controlling reaction for fouling takes place away from the wall, then there are two possibilities: (i)  If the wall temperature is hot enough for protein denaturation and aggregation, but the bulk temperature is such that the native protein is thermally stable, fouling will result from deposition of protein which has denatured and aggregated in the thermal boundary layer adjacent to the wall.  (ii)  If both the thermal boundary layer and the turbulent core are hot enough for protein denaturation and aggregation, protein denatured and aggregated in both regions will contribute to deposit formation.  If a surface reaction alone is responsible for fouling, the rate of fouling should increase with wall temperature and decrease with fluid velocity. If bulk processes contribute, then the rate of fouling should increase when the bulk fluid temperature increases, and therefore be a function of both bulk temperature and fluid velocity. Belmar-Beiny et al. (1993) showed that, along the length of the tube, the amount of  2: Literature Review  46  deposit (g/m ) increased, correlating with temperature. This suggested that the process is not 2  mass transfer controlled but rather is reaction controlled. The question now became: is it bulk reaction, surface reaction, or some combination of the two ? Activation energies for each experiment were obtained from  exp  "-AE " f  deposit at inlet  RT/  deposit at outlet  " -AE ~ exp j^-p outlet  n,et  _  (2.3.16)  f  It was shown that the fouling activation energy initially decreased from 100 to 50 kJ/mol as the Reynolds number increased from 1800 to 5200. This was then followed by a step increase in activation energy to 200 kJ/mol for Reynolds numbers ranging from 6250 to 9000. If a wall reaction alone were involved, the fouling activation energy would have been constant or would have gradually increased with Reynolds number. However, there was no systematic increase and Belmar-Beiny et al. (1993) concluded that bulk reactions must have been involved. They then invoked a similar analogy to that of Paterson and Fryer (1988), by considering the volume of fluid hot enough to react. This means that at the tube inlet, deposit is produced from the fraction of the tube cross-section occupied by the thermal boundary layer of the fluid, while at the outlet this becomes the entire cross-section of the tube. This results in deposit at inlet deposit at outlet  ratio of cross-sectional areas occupied by respective thermal boundary layers  1-  60  r|  (2.3.17)  = 1-  60  Re,E  2: Literature Review  47  Equation (2.3.17) is contrary to Equation (10) of Belmar-Beiny et al. (1993), which along with their definition for y (= y(x / r ) ) ° ) was in error. However, Equation (2.3.17) was tested over +  5  w  their range of Reynolds numbers and generated identical results to their Figure 9. Good agreement was achieved between experimental results and this model equation. These results suggest that for high bulk temperature experiments, bulk and surface reactions are both responsible for the overall fouling process. 2.3.6  Whey Protein Deposit Properties Delplace and Leuliet (1995) modeled the thermal conductivity of the fouling deposit with  time from their whey protein fouling resistance measurements. They determined a decreasing thermal conductivity with time for all experiments. This decrease was explained by assuming that the deposit at the wall surface was cooking, and therefore aging, while the deposit at the fluid interface did not change. The result would be layers of different thermal conductivity within the deposit, causing a changing average thermal conductivity with time, given by X (t) = 3.73e(  2510  ^ +0.27  (2.3.18)  where t is in seconds and Xf is in W/m.K. Other workers (Rao and Rizvi, 1995) have reported thermal conductivities of similar dairy products in the range of 0.4 - 0.6 W/m.K. Davies et al. (1997) discussed the physical properties of whey protein deposits. The product pfA,f of the matrix of denatured protein, mineral scale and solution could be expected to lie anywhere between that of water (618 W.kg.m^K"  1  at 30°C) and mineral scales (5500  W.kg.m^K" ). There is little information in the literature on the thermophysical properties of 1  milk related deposits, or of proteins, and the effect of aging on these properties. Under the operating conditions used, Davies et al. (1997) explain that the limiting step in whey protein  2: Literature Review  48  fouling changes as the fluid bulk temperature exceeds 75°C. Below this temperature, fouling is controlled by protein denaturation, aggregation and deposition at the heat exchange surface, i.e. surface reaction control. Above this temperature, deposition is controlled by protein denaturation in the bulk solution and formation of aggregates that subsequently attach to the wall, i.e. bulk reaction control. This change in mechanism is known to produce different types of deposit structures. Davies et al. (1997) generated fouled tubes using 3.5 wt. % whey protein solutions in a counter-current tube heat exchanger, and then used a heat flux sensor to measure the thermal resistance prior to sectioning for gravimetric analysis. Under conditions of surface reaction control (bulk temperatures < 75°C), the deposit was relatively smooth and consisted of small aggregates of denatured proteins. Deposits generated under bulk reaction control (bulk temperatures > 75°C) contained significantly larger aggregates in a less densely packed arrangement that is consistent with bulk generation followed by adhesion to the surface. Their results show that bulk reaction control has a lower solid fraction deposit. The results of their thermal fouling measurements are presented in Figure 2.3.7. As discussed by Davies et al. (1997), there is significant scatter, particularly at the higher deposit coverages. However, there is a visible trend, and a linear regression performed on data in the surface reaction control section, i.e. solid weight fraction of 0.5 - 0.7 in the deposit, gave a value of reciprocal slope (pfAf) = 470 W.kg.m^K" , which is less than that of water (600 1  W.kg.m^K" ). A n independent estimate of pfXf for the deposit can be made, given that the solid 1  and solution densities are both similar to that of water. A volume (and hence mass) fraction average can be obtained using the thermal conductivity of protein (0.27 W/m.K) and water (0.60 W/m.K) and a mean solid fraction of 0.6. Hence a value of pfA,f = 402 W.kgm" .K" is estimated. 4  1  2: Literature Review  49  The data in the lower right hand side, below the regression line in Figure 2.3.7, show the results for oil temperatures of 108°C (the fouling device was operated in countercurrent mode with heating oil on the shell side and whey protein solution on the tube side), and hence bulk reaction control, for comparison. 0.0012  o.oo 10 ^  0.0008  o o  J  0.0006  <J  oc J? 0.0004  & 0.0002  0.0000 0  tOO  200  300  400  500  600  Wet Deposit Coverage (g/itr]  Figure 2.3.7: Plot of fouling resistance against wet deposit coverage (Davies et al., 1997) The effect of the controlling fouling mechanism on deposit solid weight fraction is shown Table 2.3.6: Comparison of deposit solid weight fractions (Davies et al., 1997) in Table 2.3.6, along with the corresponding experimental conditions. Fouling Mechanism  Operating Conditions  Solid Weight Fraction  Surface reaction control  Ton = 88°C, Re = 5000  0.50-0.67  Ton = 98°C, Re = 10000  0.56-0.70  T n = 1 0 8 ° C , Re = 5000  0.28-0.34  Ton = 98°C, Re = 5000  0.32-0.38  Bulk reaction control  0  (axial distance > 0.8 m)  2: Literature Review  2.4  50  Lysozyme Fouling One of the key factors affecting the results from whey protein fouling experiments is the  complicating effect of several proteins (P-lactoglobulin,  a-lactalbumin and BSA) to the  denaturation, aggregation and deposition steps of the fouling process. To eliminate this unknown factor in protein fouling, a pure, heat sensitive, soluble and relatively cheap protein was required. Lysozyme, although not a constituent of whey protein, appeared to fill this gap.  2.4.1  Properties of Lysozyme Table 2.4.1 indicates some characteristics of lysozyme and compares these with the  corresponding properties of a-lactalbumin.  Table 2.4.1: Physical properties of lysozyme and a-lactalbumin (Haynes and Norde, 1994b)  a-lactalbumin  lysozyme  14200  14600  3.7x3.2x2.5  3.0x3.0x4.6  1.06 x 10'  1.04 x 10"  Molar Mass (g/mol) Dimensions (nm) Diffusion Coefficient (m /s) 2  4.3  Isoelectric Point (pi)  10  10  11.1  The very alkaline isoelectric point of lysozyme indicates that under neutral conditions lysozyme possesses a positive charge. This is the opposite to that of the dominant whey protein molecules, where P-lactoglobulin and a-lactalbumin were both negatively charged under the same conditions. Table 2.4.2 shows the denaturation temperature of lysozyme over a range of pH values. This information, taken from Pfeil and Privalov (1976), was determined from the denaturational Gibbs energy change (AG (T,pH)= G°' (T,pH)- G (T,pH)), i.e. when A G = 0. D  D  0,N  D  2: Literature Review  51  Table 2.4.2: Denaturation temperature of lysozyme (Pfeil and Privalov, 1976) pH  Denaturation Temperature (°C)  2  53  3  69  4  76  5  78  6  78  7  78  Clearly, once the pH exceeds 4 there is very little change in the denaturation temperature. Hence one would expect for pH 8 the denaturation temperature would also be 78°C. In fact Pfeil and Privalov (1976) state that" In the case of lysozyme at pH > 4.5 the denaturation temperature TD is independent ofpH.... ". 2 A.2  Effect of Electrostatic Forces on Fouling According to Nassauer and Kessler (1985), the charge on a stainless steel surface in an  ionic solution is positive due to the presence of chromium ions: Cr + 3H 0 -> Cr(OH) + 3H + 3e +  2  3  For whey protein solutions, there is a strong electrostatic attraction between the stainless steel surface and the protein molecules. However, for lysozyme, the opposite trend is true. At an initially low (acidic) pH, a lysozyme molecule possesses a strong positive charge, resulting in little adsorption of the molecule to the stainless steel surface. However, as the pH increases (towards the isoelectric point, pi = 11.1), the strength of the charge on the molecule reduces, and hence adsorption increases. This is shown schematically in Figure 2.4.1.  2: Literature Review  52  At the isoelectric point the globular protein charges are balanced by the surrounding surface to produce the highest packing density available for a native protein. At a pH slightly higher than the isoelectric point for lysozyme, one notes that the adsorption plateau increases slightly. This is to be expected, since the protein molecule will distort and denature to balance the charges between the surface and the protein. Therefore from this discussion and given that the practical pH range for our fouling study is pH = 5 - 9, one would expect protein adsorption to increase as the pH increases.  Solution Gbnoertration (kg'rtf)  pH  Figure 2.4.1: Effect of pH on lysozyme adsorption to a positively charged surface (Haynes and Norde, 1994b) In addition to this, the lysozyme molecule is more soluble at acidic pH values, i.e. further from its isoelectric point. Thus, at these pH values, even if the protein denatures (a decrease in protein activity) it may not necessarily attach to the wall surface. Therefore this effect enhances the fact that adsorption will be greater at a higher pH.  2.4.3  Thermal Inactivation Kinetics of Lysozyme Makki (1996) set out to determine the irreversible thermal inactivation kinetics of  lysozyme over a wide pH range (4.2 - 9.0) and temperatures from 73 to 100°C. Chicken egg white lysozyme with an enzymatic activity of 41,400 units/mg protein was used. Stock solutions  2: Literature Review  53  of the enzyme were prepared by dissolving 4.3 mg of lysozyme into 50 ml of the appropriate buffer for each required pH. This corresponded to approximately 3400 units/ml or 86 ppm. For pH 9 the initial lysozyme concentration was doubled. Makki (1996) discussed the methods available for identification and quantification of lysozyme. He used a turbidimetric assay which is based on spectrophotometric measurements of the clearing of a turbid suspension of Micrococcus lysodeikticus in the presence of lysozyme. This method measures the enzymatic activity of the native protein and it is assumed that this is a measure of the concentration of the native protein. This method has the advantage that it is simple, rapid and highly sensitive. The relationship between the logarithm of lysozyme activity (a measure of the concentration of native protein) and incubation time appeared linear, and hence consistent with first order kinetics. Samples were analyzed either immediately or held in a refrigerator for 1 - 2 days. Table 2.4.4 illustrates the results. Makki (1996) also showed that lysozyme is more heat stable in acidic solutions, i.e. farthest from its isoelectric point. Table 2.4.3 shows the effect of pH on the activation energy of the inactivation process in buffer solutions. A relationship between the kinetic reaction rate constant, k, and two independent variables, T and pH, in the pH range 5.2 - 7.2 and the temperature range 73 - 100°C, was also observed: ln(k) ,T pH  = 32.90 - 1.62 x 10 /T + 1.19(pH) 4  (r = 0.975) 2  2: Literature Review  54  Table 2.4.3: Effect of pH on the activation energy (Makki, 1996) PH  AE (kJ/mol)  4.20  110.5  5.20  151.1  6.24  134.0  7.20  122.3  8.10  50.2  9.00  78.7  Table 2.4.4: Reaction order with respect to time of thermal inactivation of lysozyme (Makki, 1996) pH 4.20  5.20  6.24  7.20  8.10  9.00  Temperature (°C)  75 82 91 100 75 82 91 96 73 81 84 93 95 75 82 91 95 75 82 91 95 75 80 85 91  Best n  0.8-0.9 1.0-1.1 0.5 1.4-1.6 2.0 1.9-2.0 2.0 1.4-1.8 0.5-0.8 2.0 1.6-1.7 0.8 1.5 0.5-0.7 0.7-0.8 0.9-1.0 0.8-0.9 1.0 1.3 1.1 0.9-1.0 1.2 1.1 1.1 1.1  r for n  r for n =1  Reaction Completion (%)  0.975 0.950 0.904 0.983 0.780 0.920 0.889 0.957 0.731 0.919 0.846 0.993 0.966 0.985 0.986 0.981 0.981 0.982 0.888 0.940 0.984 0.942 0.961 0.982 0.853  0.974 0.950 0.895 0.975 0.741 0.911 0.853 0.954 0.730 0.899 0.841 0.990 0.952 0.981 0.981 0.981 0.977 0.982 0.864 0.938 0.984 0.929 0.958 0.979 0.848  75 52 39 77 50 37 47 50 27 40 48 86 80 63 86 90 95 94 96 90 94 95 93 96 97  2  2  3: Experimental Apparatus and Methods 3.  55  Experimental Apparatus and Methods The following section describes the experimental methods employed to study fouling  using the selected model fluids. These model fluids included a 1 wt. % aqueous solution of whey protein at pH 6, and a 1 wt. % aqueous solution of lysozyme at pH 8. These protein solutions were selected for their ease of use, and, in the case of the former, for its application to understanding the mechanisms of whey protein fouling in the dairy industry. In addition, these solutions could be studied over a wide range of fluid velocities at relatively low surface temperatures. Complications caused by several protein constituents and hence competing chemical reactions, and by the presence of particulate material, in whey protein solutions led to the adoption of lysozyme, a pure, more soluble protein, for comparative and further verification purposes. The composition and specification of the supply powders for the protein solutions are provided in Appendices 3.1 and 4.1. It should be noted that the ash content of the whey protein (Appendix 3.1) is (3.5 + 0.3) %. 3.1  Tube Fouling Unit (TFU)  3.1.1  TFU Apparatus (Wilson, 1994) The apparatus shown in Figure 3.1.1 is a modified version of Watkinson's (1968) design,  adapted to permit removable test sections for easier deposit examination. The test section consists of a 1.83 m (6 ft) long 304L stainless steel tube (OD = 9.525 mm, ID = 9.017 mm, thickness = 0.254 mm) heated by alternating current along 0.772 m of its central section. The power to the heated section was kept essentially constant during a run so that it could be operated at a constant heat flux, such that as fouling proceeded, the deposit-fluid interface temperature would remain constant while the wall temperature increased. For each experiment, the heat flux  3: Experimental Apparatus and Methods  56  a p p l i e d to the test section w a s adjusted to a c h i e v e a range o f outside w a l l temperatures  in  approximate agreement w i t h experiments o f d i f f e r i n g f l u i d v e l o c i t y .  Primary Cooler  TI  {X}-| D r a i n  Auxiliary Coolers  MM  Test Section  Rotameters  TI TI  .Or.  J L A L B a n d Heater  E3  I  D P Orifice  Drain as  Drain  PSW: Pressure Switch  PRV: Pressure Relief Valve  TI: Thermocouple  LAL: Low-Level Alarm  Figure 3.1.1: Schematic of Tube Fouling Unit (TFU) apparatus A l l materials i n contact w i t h the f l u i d are m a d e f r o m stainless steel. T h e m o d e l fluids were stored at the required b u l k temperature i n a 65 L h o l d i n g tank, then passed t h r o u g h a p u m p ,  3: Experimental Apparatus and Methods  57  an orifice plate, the heated test section, a number of cold water coolers, one of two rotameters, and finally returned to the tank through a small diameter orifice, to aid mixing. Hoke globe valves were used to split the flow between the test section and the bypass line, which ensured good mixing, especially when low flow rates were used in the test section. The return lines both terminated well below the liquid level in the tank, thus ensuring minimum gas entrainment. A mesh strainer located upstream of the pump could be switched on line if particulate matter became a problem. However, this was bypassed for all experiments. The tank initially included filling and draining ports, a low level alarm floating device and a 3 kW low heat flux heater to heat the holding tank to the required bulk temperature prior to an experiment. Due to early corrosion problems (resulting from 0.1 N HC1 used as a catalyst in the 2-furaldehyde experiments, see Appendix 2), the immersion heater was replaced with a 20foot by 2-inch, 2-kW band heating tape rated to a surface temperature of 260°C (500°F). This tape was wrapped around the lower half of the holding tank. The tank contents could be maintained at temperatures of 100°C ± 2°C by an Omega C N 911 controller. At bulk temperatures of 30°C, this controller was not used. The system was capable of being pressurized to 790 kPa (100 psig). Either nitrogen or building air was used. The gas flow was monitored by the rotameter at the control panel, and gas flowed up from the bottom of the tank through the liquid into the available head space until the desired pressure was achieved, after which the gas was shut off. Pressure was primarily used to eliminate the possibility of boiling as the wall temperature rose. The tank was also fitted with a pressure relief valve, set to 859 kPa (110 psig), a pressure vent line, fed out the side of the laboratory, and along with the piping the tank was insulated with aluminum backed fiberglass insulation. The piping section between the pump and  3: Experimental Apparatus and Methods  58  the orifice plate was also fitted with heating tapes connected to a variable transformer in the event that, at higher temperatures, insulation losses became significant. For the protein solutions, the system was pressurized to 308 kPa (30 psig) using building air, and the aluminum backed fiberglass insulation along the piping was removed to help maintain bulk temperatures in the holding tank as low as 30°C. Liquid flow rates were monitored using one of three calibrated (using water) rotameters. These included High Flow Rotameter (HFR):  Re i t  = 15000 - 40000  Mid Flow Rotameter (MFR):  Re i t  = 2000 - 20000  Low Flow Rotameter (LFR):  Re i =1300-4300  in  in  in  e  e  et  The temperatures of the bulk liquid entering and exiting the heated section were monitored by thermocouples mounted in T-pieces as close to the entrance and exit of the heated section as practicable. The pressure drop across the heated section was monitored by a differential pressure transducer, again connected as close as possible to the entrance and exit of the heated section. The coolers removed the heat supplied to the test section. The primary cooler consisted of a 2.44 m long x 25.4 mm diameter copper tube around the outside of the 12.7 mm diameter stainless steel tube carrying the process fluid. Cold water was passed counter-current to the process fluid and was controlled using a rotameter and globe valve located at the control panel. Two auxiliary coolers downstream of the primary cooler were operated independently of the primary cooler and were used intermittently. An additional co-current water cooler was, however, inserted on the bypass line to remove any additional heat added to the fluid from the pump. This  3: Experimental Apparatus and Methods  59  cooler was critical in maintaining bulk temperatures of the protein solutions in the holding tank at 30°C. Table 3.1.1 highlights the safety features installed into the T F U apparatus to enable safe operation of the equipment.  Table 3.1.1: TFU safety measures Hazard  Cause  Trip  Action  Overpressure  Blockage/Explosion  PSW/PRV  System Shut Down  Fluid Leaks / Runs Dry  Leakage/Rupture  LAL  System Shut Down  Overheated Liquid  Cooling Water Failure  TAH on cooler outlet  System Shut Down  Tube Burn Out  Flow Stopped  TAH on tube surface  Heating Off  Thermocouple Damage  Tube Too Hot  TAH on tube surface  Heating Off  Power Surges  Power Failure/Restored  Reset Relays  Power Stays Off  Despite these safety features it was still possible for tube burnout to occur at the electrical connections, causing fluid to leak across these connections (hence large fluctuations in electrical resistance) and down the length of the test section. Because the flow hadn't stopped, the tube temperature was within the acceptable range and the T A H (temperature alarm high) alarm did not respond. In this case a manual over-ride was required to terminate the experiment. From this realization it was established that heat fluxes corresponding to greater than 90 % of the maximum variac output were not advisable. The test section is electrical resistance heated by alternating current (up to 20 V , 300 A). The tube lengths constructed of drawn 304L stainless steel were supplied by Greenvilles Tube Corp., Clarksville, Arkansas (Wilson, 1994). The tube had a nominal thickness of 10 thousandths  3: Experimental Apparatus and Methods  60  of an inch, but actual values and weights would vary from tube to tube. The tube was connected to the fouling rig by Swagelok dielectric fittings (with an electrical resistance of 10 M Q @ 10 V D C and a leakage current of 1 uA) which ensured electrical isolation from the rest of the apparatus, and were capable of a holding capacity of 13884 kPa (2000 psig). Previously, Swagelock Teflon fittings had been used to provide the electrical isolation; however, these fittings were only rated to a holding capacity of 274 kPa (25 psig) at 100°C. It could be argued that they should be rated to a bursting pressure, since it is the thermocouples and the electrical connections that hold the tube in place. Hence, two Swagelock dielectric fittings were installed. These fittings are typically installed on impulse lines ahead of monitoring stations in natural gas pipelines with the fitting interrupting the cathodic flow while still permitting fluid flow. A thermoplastic insulator provides the high dielectric strength over a range of conditions and a viton o-ring provides the primary fluid seal. Current is supplied to the test section from a mains 208 V A C supply via a power variac and a 208-19 V step-down transformer. The step-down transformer was connected to the test section by a pair of number three welding cables bolted to 10 mm thick copper busbars. The first busbar is located 546 mm (60 tube diameters) from the upstream fitting to ensure a fully developed velocity profile in the heated section under turbulent conditions. The second busbar marked the end of the heated section and was positioned 772 mm downstream. For turbulent flow the required entry length is given by 10 < ^/j < 60, while for laminar flow, ^/j = 0.05Re. Clearly under laminar conditions, Re < 2300, it will sometimes be difficult to achieve an adequate entry length (Incropera and DeWitt, 1990).  3: Experimental Apparatus and Methods  .  61  The voltage across the test section, V, was measured using an A C panel meter and displayed on the control panel. The current, I, was measured using a current transformer on one set of welding cables and was calibrated using an Amprobe A C D - 9 current meter (Appendix 1): /(Amps) = -0.4322+ 149.71 (ammeter reading)  r = 0.9998 2  3.1.1  The power to the heated section was calculated assuming a power factor of unity (Watkinson, 1968) as equal to VI, and then converted to heat flux using the nominal tube thickness of 10 thousandths of an inch: Q/TtdjL = q (W/m ) = F//(TI(0.0090 17x0.772)) = 45.73 VI 2  3.1.2  where L is the length of the heated section in metres. As previously discussed by Wilson (1994), there were variations in the tube thickness from tube to tube; however, the nominal thickness of 10 thousandths of an inch was considered the best overall estimate. 3.1.2  Wall Temperature Measurement To enable a test section to be removed and sectioned after each experiment it was  necessary to use removable wall thermocouples. This differs from Watkinson's (1968) setup, where silver soldered thermocouples were permanently fixed to the test section and were used to measure the tube outer wall temperatures. The method employed by Wilson (1994) involved thin film temperature sensors compressed against the side of the tube inside a clamping block which was screwed against the outside of the tube. The thin film sensors were essentially ribbon thermocouples mounted in a polyimide (Kapton™) carrier which provided electrical resistance up to 270°C, upon which the polyimide would begin to degrade. The temperature  sensors were K type 20112 foil  thermocouples manufactured and purchased from RDF Corporation. Several designs were  3: Experimental Apparatus and Methods  62  initially considered (Wilson, 1994) but here only the optimal configuration will be discussed. The supporting frame comprised two solid blocks of Lava which were screwed together to compress the thermocouple against the outside wall of the tube. This configuration was extremely sensitive to the test section assembly. To ensure accurate temperature readings, the block had to be positioned perfectly perpendicular to the tube with no air gaps. The slightest misalignment resulted in wall temperature variations from one thermocouple to another. The lead from the thermocouple had to be mounted and maintained normal to the direction of the tube, since the alternating current's magnetic field could cause significant noise and hence contribute to fluctuations in the thermocouple signal. Ten thermocouples were located along the length of the heated section at 48, 110, 163, 221, 330, 440, 550, 605, 660 and 713 mm from the start of the 772 mm heated length. For experimental systems that exhibit characteristics which are strong functions of the wall temperature, the axial temperature profile provides an excellent description of the system behaviour. Between each thermocouple a lava collar was placed around the tube and held in place by a wrap of glass fiber. The heated section was tightly wrapped by two layers of aluminum backed fiberglass blanket which were secured by removable straps. The entry and exit lengths were also insulated using a wrap of glass fiber and then covered with a cylinder of fiberglass insulation, again fastened by removable straps. The entire test section was enclosed by a steel cover which prevented accidental contact with live surfaces and prevented the release of chemicals into the path of the operator in the event of an emergency shutdown.  3: Experimental Apparatus and Methods  63  A high temperature alarm thermocouple of similar construction to the previously mentioned thermocouples was positioned close to the hotter end of the test section and served only as an alarm at the control panel rather than being data logged on the computer. 3.1.3  Data Collection Experimental data were collected using LabTech notebook, a datalogging software used  on a PC along with a DAS-8 analog/digital interface board. All data were saved to the PC and later transferred to Excel where a spreadsheet enabled easy data analysis. Fluid bulk and wall temperatures were measured using a multiplexer. On average the variation in output due to these measuring methods was approximately ± 1°C. The tank bulk temperature was recorded as 1 m V / ° C analog output from the tank heater controller; this was then amplified using an operating amplifier and the output fed to the DAS-8. This temperature was considerably more stable and estimated errors were in the range of ± 0.5°C. The orifice plate and tube section differential pressures were obtained as analog outputs from the DP-350 differential pressure transducers. Again these signals were filtered through an operational amplifier and then connected to the DAS-8. The current through the heated section was measured at the control panel and automatic data logging was disconnected due to significant fluctuations in the signal. Since the system was operated at constant heat flux, both the voltage and current were recorded from the control panel. The voltage was also displayed at the control panel. All significant temperature and pressure drop data were logged and displayed on the PC screen such that the progress of the experiment was easily monitored. To avoid noise in thermocouple signals the data sampling filtered out the 60 Hz noise by sampling every six  3: Experimental Apparatus and Methods  64  seconds and calculating a moving point average over the previous minute. This average was then logged as the value over the mid-point of the averaging time period. This procedure reduced thermocouple fluctuations to approximately ± 2°C . Sampling every one minute was considered sufficient where experiments would typically last 4 - 1 2 hours, j The heat flux applied to the test section was assumed to be uniform since the electrical properties of the stainless steel tube are not a strong function of temperature. The electrical  resistance, R = — , where a = 71.7(1+0.00094 T(°C)) u,Qcm. Therefore given that the electrical A  resistivity is a very weak function of temperature, and assuming that the cross-sectional area remains constant (not the case for 2-furaldehyde fouling, where corrosion and chemical reaction fouling progressed simultaneously, see Appendix 2) the resistance will remain essentially constant (providing a uniform heat flux to the test section) and was calculated using Equation (3.1.2). The heat transferred to the fluid was calculated using  Q = ^C (T p  b i 0 u t  -T  b i n  )  3.1.3  and the results from Equations (3.1.2) and (3.1.3) were compared to estimate the heat loss. The specific heat capacity C  p  for a 1 wt. % solution was assumed to equal that of water and was  evaluated at the average bulk temperature, Tb = (T ,i + Tb, t)/2. These two estimates from the b  n  ou  heat balance were on average within 4.7 % of each other (Equation (3.1.2) gave consistently  greater values of Q than Equation (3.1.3)) and considered to be in reasonable agreement. The  difference in Q by these two equations was on average 141 W, which can be accounted for by the range of uncertainty of the thermocouple readings (± 1°C). At high heat fluxes this difference would be 2 - 3 %, while at low heat fluxes, generally the case at the lower fluid velocities, there  3: Experimental Apparatus and Methods  65  could be up to a 10 % variation in heat flux estimates. The value of the heat flux and thus heat transferred to the liquid was calculated from Equation (3.1.2), since Equation (3.1.3) involved assumptions about the physical properties of the fluid. However, given that these equations were generally in good agreement, the heat losses were assumed insignificant. Except at low fluid flow rates (using the L F R rotameter) the mass flow rate of process fluid was typically very stable once steady state had been achieved. This set-point was then monitored and recorded manually. Using Equations (3.1.2) and (3.1.3) as a check on the heat balance (and hence the rotameter calibration), this method proved satisfactory. 3.1.4  T F U Operating Procedures The key difference in the handling of the T F U compared to other fouling rigs involves the  removable test section, namely its preparation, cleanliness and treatment before and after an experiment.  The following protocol shows the stringent methods employed to ensure  reproducible conditions from one experiment to another. 1. For all protein experiments the test section was first soaked in a bath of hot water and detergent (Tergazyme) overnight to remove any grease or residual dirt that may be present. 2. The test section was then rinsed with cold water to remove the grease and detergent. 3. The inside and outside of the test section was cleaned using acetone with several bottle cleaning brushes. The test section was then allowed to dry. 4. The test section and fittings were weighed to obtain a clean weight. 5. The thermocouples attached to the supporting frame were loosened from their connections. 6. The union to the heated section at the top of the test section was removed and the tube installed. The union was then re-assembled.  3: Experimental Apparatus and Methods  66  7. After ensuring that all thermocouples were loose, the tube was tightened at the electrical connections. 8. The top of the tube was secured to the equipment. 9. All thermocouples were attached to the tube but left semi-loose. 10. The T F U was pressure tested to above the required operating pressure for the experiment and tested with Snoop. 11. With the thermocouples still loose, the ceramic insulators were attached between thermocouples. Thermocouples were then aligned and tightened for the experiment. 12. Thin white fiberglass which acted as the second layer of insulation was wrapped over the ceramic insulators. 13. Two lengths of insulation were installed over the entrance and exit lengths of the tube. 14. The third layer of insulation, two layers of aluminum backed fiberglass, were used at each thermocouple location. Two long cylinders of insulation were used at the entrance and exit lengths. 15. Metal guards were installed around the tube to protect the operator from electric current or a chemical leak. 16. For tube disassembly, steps 5 - 1 5 were performed in reverse order. Special attention was paid to avoid dislodging any material deposited on the inside of the tube. 17. The tube was then carefully removed and allowed to drip dry in the laboratory. To commence a run, the pump was started, the flow rate and operating pressure adjusted to the desired set point, and the variac set to the required power setting. The datalogger was started, and once thermal pseudo-steady state was achieved (approximately 15 minutes), the critical control parameters were monitored and followed for the duration of the experiment, i.e.  3: Experimental Apparatus and Methods  67  cooling water flow rate and temperature, model fluid flow rate, surface temperatures, voltage and current, and bulk fluid temperature. Cyclic variations in heat flux throughout an experiment required the occasional adjustment of the variac to ensure operation at a constant heat flux. For the protein experiments several liquid samples were taken to follow the pH throughout the experiment. These samples were taken from the test section inlet, thus providing an indication of any changes that may be occurring in the bulk fluid. These experiments were performed unbuffered at designated pH values, and hence if there were significant amounts of protein denaturation in the bulk fluid, the pH would change. Of course, this assumes that the natural buffering capacity of the protein solution and mineral content (in the case of whey protein solutions) will be rapidly exhausted. Bulk temperatures were maintained as low as practicable (30°C bulk inlet temperature) to try and avoid significant bulk reactions. For whey protein solutions the bulk temperature typically rose from 30°C at the test section inlet to 55°C at the outlet. For lysozyme solutions the temperature rise was typically from 30°C to 50°C. As previously reported by Fryer (1987), bulk denaturation only really becomes significant above bulk temperatures of 60°C. In comparison to the 2-furaldehyde experiments, the protein fouling experiments were very fast. The duration of the experiment was limited to a maximum surface temperature of 125°C at 308 kPa (30 psig), i.e. the anticipated onset of boiling of the aqueous fluid. Unfortunately some experiments (at high heat flux) were prematurely terminated due to tube burn-out. To terminate an experiment, the power to the test section was slowly reduced to zero over five minutes to allow the test section to cool to the bulk temperature, thereby preventing the  3: Experimental Apparatus and Methods  68  deposit from suffering thermal shock and becoming dislodged. Then the pump was stopped and the system pressure slowly reduced to atmospheric pressure. The holding tank contents were emptied through the drainage port and the test section removed to dry in the laboratory. T F U cleaning protocol Given that protein solutions were the model fluids, a standard protocol similar to that used in the dairy industry was employed: 1. As soon as practicable after an experiment, the protein solution was drained from the T F U , the insulation removed and the test section disconnected. 2. 4 0 - 6 0 liters of distilled water were recirculated through the T F U for approximately 1/2 hour to remove the remaining protein solution. The contents were then drained. 3.  60 liters of detergent solution (2 g/1 Tergazyme in water) were pumped and recirculated for approximately 4 hours until the bulk temperature reached 50°C. At this time the contents were drained immediately.  4. 60 liters of distilled water were recirculated through the T F U for approximately 1/2 hour to remove the detergent solution. 5. 60 liters 0.01 N HC1 were added and recirculated through the T F U for approximately 4 hours until the bulk temperature reached 50°C. At this time the contents were drained immediately. 6. 60 liters of distilled water were recirculated through the T F U for approximately 1/2 hour as a final rinse solution. 7. The T F U was air dried overnight.  3: Experimental Apparatus and Methods 3.2  ;  69  Whey Protein Solutions Whey protein concentrate powder is difficult to mix without the formation of clumps of  powder at the top or bottom of the mixing vessel. To make 60 liters of 1 wt. % solution, 748 g of WPC-80 was used. To help prevent mixing problems a concentrated 10 % solution was prepared in the mixing vessel and then later transferred to the T F U , made up to 60 liters and recirculated for 1/2 hour prior to an experiment. The following protocol identifies the procedure used for the T F U 200 and T F U 210 series of experiments. For the T F U 220 series of experiments, a protocol similar to that for lysozyme was followed, which is explained in section 3.3. 1. 748 ± 1 g WPC-80 powder was weighed into two equal batches. 2. Two 3.5 liter beakers were filled with WPC and approximately 2 liters of deionised water. 3. This mixture was stirred at 800 - 1000 rpm for 30 minutes. The mixing impeller was maintained in the lower 1/3 of the beaker to promote mixing and prevent splashing, air entrapment and vortices. 4. Beakers were filled with deionised water to the 3 liter mark and mixed at 800 - 1000 rpm for 30 minutes. 5. When all the powder appeared to have dissolved (usually after 40 minutes), the pH was measured. 6. The solution was acidified with 0.1 N HNO3 to pH = 6.2-6.3. Approximate volume = 250 ml. 7. Total mixing time per beaker was kept constant at 1 hour for all experiments to ensure no variation.  3: Experimental Apparatus and Methods 3.2.1  70  Microbial Contamination Testing Procedure Dilute solutions of whey protein concentrate, like other food products when stored at  temperatures near 37°C for long periods of time, become susceptible to microbial growth and therefore contamination. Therefore a proposed bulk temperature of 30°C posed some concern. With the growth of microbes, the solution runs the risk of becoming acidic (a potential detection method); however, with the accuracy of the equipment used for pH measurement, this is not likely to be detectable. The pH drop could also be masked by pH changes due to denaturation and aggregation of the whey protein itself. If substantial levels of microbes build up within the solution they could affect the physical properties of the solution. It was recommended that to avoid contamination, bulk temperatures of 5 - 7°C be used, employing an ice jacket on the outside of the holding tank, or to use anti-microbial agents such as K2Cr207, NaN3, or Cl" ions. Neither of these options was desirable because 1. At low bulk temperatures the equipment would not be able to provide enough power to achieve wall temperatures of 90 - 95°C. 2. Adding an anti-microbial agent could affect the fouling mechanism and hence the kinetics of the denaturation / aggregation steps. This would make the chemistry of the process more complicated. It was therefore decided to test the whey protein solutions at selected time intervals over an experiment to determine whether microbial levels and growth rates were significant and likely to affect the experimental results. Initially, agar plates and petrifilm trays were incubated at 21°C and 37°C in aerobic and anaerobic conditions to determine which type of microbes grew the most within the TFU. Agar  3: Experimental Apparatus and Methods  71  plates involved considerably more preparation. 23.5 g of plate count agar was dissolved in 1 L of distilled water. DIFCO Bacto-peptone, which is used in bacteriological culture media, and is very hygroscopic, was made up to 0.1 wt % with distilled water. The plate count agar was poured onto the trays and, along with the Bacto-peptone and the vials and syringe tips, was sterilized. Since the fouling experiments are in the psychrophilic (low temperature) and mesophilic (medium temperature) temperature ranges, these categories include the type of microbes that one would expect to be present. Tables 3.2.1 - 3.2.4 review the preliminary results from one experiment to determine the best method of testing the whey protein solutions for microbial growth. The samples were placed in 21°C and 37°C incubators for 12 hours, and then examined.  Table 3.2.1: Microbial growth at 37°C using petrifilm trays Sample #  Petrifilm Anaerobic 37°C (0)"  (-D  1  Petrifilm Aerobic 37°C (0)  CFU  (-1) CFU  1  TNTC*  105**  TNTC  TNTC  2  TNTC  110  TNTC  TNTC  3  TNTC  110  TNTC  TNTC  4  TNTC  95  TNTC  TNTC  5  TNTC  130  TNTC  TNTC  6  TNTC  120  TNTC  TNTC  7  TNTC  130  TNTC  TNTC  8  TNTC  95  TNTC  TNTC  H  * TNTC : Too numerous to count  ^  I  Zero dilution  1  One order of magnitude dilution  ** CFU/g of supply powder = (CFU x Dilution Factor x 80.212)  3: Experimental Apparatus and Methods  72  The first observation to be made is that aerobic growth is greater than anaerobic growth. This is understandable, since the T F U is under 308 kPa (30 psig) of air, encouraging aerobic growth. The temperature of incubation seemed to have little effect on the results; however, 37°C is within the operating temperature range (i.e. bulk temperature = 30 - 55°C), and the results are achieved more quickly at this incubation temperature. Samples 1 - 8  were taken throughout the duration of a fouling experiment, and Tables  3.2.1, 3.2.2 and 3.2.4 all suggest that the level of contamination in the solution was present from the onset of the experiment, with no significant growth rate. Table 3.2.2: Microbial growth at 21°C using petrifilm trays Sample #  Petrifilm Anaerobic 21°C (0)  (-1)  Petrifilm Aerobic 21°C (0)  (-1)  1  TNTC  72  TNTC  TNTC  2  TNTC  55  TNTC  TNTC  3  TNTC  43  TNTC  TNTC  4  TNTC  39  TNTC  TNTC  5  TNTC  36  TNTC  TNTC  6  TNTC  48  TNTC  TNTC  7  TNTC  39  TNTC  TNTC  8  TNTC  44  TNTC  TNTC  Petrifilms showed the same trends as the agar plates, and since they are easier to use, they were employed in subsequent tests. Also, it appears that tests at 37 °C give the most microbial  3: Experimental Apparatus and Methods  73  growth, and hence the analysis will be restricted to aerobic tests at this temperature. The microbial test results from the fouling experiments will be presented in Section 4.2.2. Table 3.2.3: Microbial growth at 37°C using agar plates (desire 30 - 300 counts) Sample #  Agar Plates Anaerobic 37°C (0)  (-1)  Agar Plates Aerobic 37°C (0)  (-1)  1  TNTC  TNTC  TNTC  TNTC  2  TNTC  TNTC  TNTC  TNTC  3  TNTC  TNTC  TNTC  TNTC  4  TNTC  TNTC  TNTC  TNTC  5  TNTC  TNTC  TNTC  TNTC  6  TNTC  TNTC  TNTC  TNTC  7 .  TNTC  TNTC  TNTC  TNTC  8  TNTC  TNTC  TNTC  TNTC  Table 3.2.4: Microbial growth at 21°C using agar plates (desire 30 - 300 counts) Sample #  Agar Plates Anaerobic 21°C (0)  (-1)  Agar Plates Aerobic 21°C (0)  (-1)  1  250  80  TNTC  TNTC  2  260  TNTC  TNTC  3  200  40  TNTC  TNTC  4  250  TNTC  TNTC  TNTC  5  TNTC  50  TNTC  TNTC  6  150  TNTC  TNTC  7  150  TNTC  TNTC  8  90  TNTC  TNTC  60  3: Experimental Apparatus and Methods  3.2.2  2A  Deposit Property Analysis  The objectives of this part of the work were as follows: 1. To estimate values of the deposit thermal conductivity and density, and to determine whether these properties are a function of wall temperature. If so, this would suggest that the normal method of determining an average value for a given experiment would not be applicable, and hence the fouling resistance versus deposit coverage or fouling resistance versus deposit thickness plots would not follow the commonly assumed linear relationship. In addition, the results would indicate whether or not a single value of A,f and pf used in solution of the mathematical model is a valid approximation. 2. To determine if there is a correlation between fouling resistance and deposit temperature, in which case there would likely be an aging effect of the deposit over the duration of an experiment. 3. To determine whether the deposit properties can provide an explanation for the observed accelerated fouling rates in some of the low fluid velocity experiments. For these reasons, seven test sections were studied taking advantage of the range of fluid mass fluxes, as shown in Table 3.2.5. As well, several of the samples were examined using the Scanning Electron Microscope (SEM) to compare the deposit morphology under accelerated and non-accelerated fouling conditions. The following protocol was used to evaluate the physical properties of the deposit. As previously discussed, results from up to 10 individual thermocouples can be accumulated from one experiment, and therefore up to 10 mass deposition and thickness results can be utilized.  3: Experimental Apparatus and Methods  75  This procedure, although approximate, will elucidate values of deposit thermal conductivity, X(, and density, pf, for the fouling deposit generated from a WPC-80 powder. Table 3.2.5: Tubes used for deposit analysis TFU  Mass Flux (kg/m s)  Accelerated Fouling ?  211  221.4  T/C 9 & 10  208  284.1  T/C 9 & 10  204  401.4  T/C 9 & 10  207  531.7  NO  209  882.3  NO  205  1055.6  NO  210  1274.9  NO  2  Protocol for determining whey protein fouling deposit physical properties 1. Determine dry weight of fouled tube (± 0.0001 g). 2.  Section and cut tube into approximately 50.8 mm (2-inch) sections.  3. Measure the exact length of each 50.8 mm section. 4. Performing duplicates and averaging, estimate with calipers the thickness of the deposit at the tube section inlet and outlet. Establish a zero thickness (each tube diameter will vary slightly) using the clean, non-fouled section at the tube inlet. 5. Put each section of interest (the heated test section) into the desiccator and dry until constant weight. This process will typically take 48 hours. At this point remove and place in a sealed container any sample required for S E M analysis. 6. Heat 1 liter of 1 N K O H solution to 90°C using a hot plate and magnetic stirrer.  3: Experimental Apparatus and Methods  76  7. Place each section into the K O H solution for approximately 4 minutes to remove the deposited, denatured protein. Visually check each section for signs of remaining deposit. 8.  Soak each section in hot water for 2 minutes and rinse with distilled water.  9. Dry for up to 48 hours in a desiccator until the clean sections reach constant weight. 10. Record total deposit mass. Table 3.2.6 shows the thermocouple locations along the heated length of the tube. Appendix 3.4 shows where these thermocouples lie relative to the cut test sections. Table 3.2.6: TFU thermocouple locations Axial location (mm) downstream of 546-mm entrance length Thermocouple 1  48  Thermocouple 2  110  Thermocouple 3  163  Thermocouple 4  221  Thermocouple 5  330  Thermocouple 6  440  Thermocouple 7  550  Thermocouple 8  605  Thermocouple 9  660  Thermocouple 10  713  Heated length  772  Exit length  512  3: Experimental Apparatus and Methods  77  After determining the deposit coverage (mf) at a given fouling resistance, the product  XfPf  can be determined from  R  f  =  _ 2l_ n  =  ^fPf  i^L  (3.2.1)  ^f  Also, by estimating deposit thickness (xf) and plotting this against the fouling resistance, the value of Xf can be determined, enabling an estimate of both Xf and pf of the fouling deposit.  3.3  Lysozyme In a similar manner to the T F U 220 series of experiments with whey protein, the  lysozyme solutions (TFU 300) were prepared in a 60 liter mixing vessel and transferred to the TFU. This preparation method and mixing time were exactly the same as for the whey protein experiments, except that 0.1 N NaOH was used to adjust the pH of the solution to 8 (or in some experiments pH 5 and 6.5). The preparation and mixing time was also constant at 1 hour to ensure reproducible conditions. 3.3.1  Enzymatic Activity Assay The enzymatic assay for lysozyme (Appendix 4) obtained from the Sigma Chemical  Company homepage (www.sigald.sial.com) can be used to determine the enzymatic activity and the effect of prolonged heat treatment and shear on the denaturation of the native protein. Activity is proportional to the concentration of native protein in solution, and therefore provides an indication as to whether significant denaturation of the protein has occurred throughout the duration of a fouling experiment. The following steps outline the procedure followed. 1. 8.9826 g potassium phosphate (monobasic, anhydrous) were weighed and dissolved in 1 liter of deionised water and the pH adjusted to 6.24 using 1 M K O H . This resulted in a 66 mM  3: Experimental Apparatus and Methods  78  potassium phosphate buffer, at room temperature. This was called reagent A and the solution was sterilized at 140°C for approximately 30 minutes. 2.. 0.0167g of Micrococcus lysodeikticus (Sigma) was dissolved in 100 ml of reagent A to produce approximately a 0.015 % (w/v) cell suspension. Solubility of this substrate has proven to have a large effect on the results of the analysis, with the slightest amount of remaining undissolved particles having a large effect on the absorbance. The absorbance at 450 nm for this substrate was between 0.6 and 0.7, as recommended by Sigma. This solution was called reagent B. Previously, another supply of substrate was used, and following exactly the same procedure, the absorbance of this substrate only reached 0.4 - 0.5. The activity of the substrate was found to be very sensitive to storage and treatment. 3. 60 liters of a 1 wt. % solution of lysozyme (Inovatech) was prepared as per the whey protein fouling experiments and then run through the equipment for approximately 7 hours (TFU 310). Samples were withdrawn periodically and immediately placed in a refrigerator (« 5 °C) where they were stored overnight until testing the following day. Before analysis, these samples were diluted 600 times in reagent A to achieve the required concentration of 200 400 Units/ml of lysozyme. This was achieved by initially performing a 1:6 dilution followed by a 1:100 dilution in reagent A. In addition, it was determined necessary to sterilize all equipment that was used to analyze for lysozyme activity. If there was any trace of protein contamination in the Micrococcus lysodeikticus suspension, then the absorbance of the suspension would decrease rapidly and render itself useless by the time the tests were carried out. The success of the sterilization is shown in Figure 3.3.1 (raw data in Appendix 4.4).  3: Experimental Apparatus and Methods  79  0.65 0.64 l fi ftfi D  ©  0.63  -  A A A  A  A  A A A A , i-A-A-A-A-A-*A-A-*, .A^A-A^A^AnAA-AA^, ,-A-A-A-A-A-A—A-A-i ,  Jiooooooooo< I O O O O O O O O O < 5 | X X X X X X X X ) :xxxxxxxxx)  a  D D  fi lj] a  A A A A A A A A ^ i-*-AAnA-A-A-A^A-A-i  I O O O O O O O O O C I O O O O O O O O O I ) O O O O . 0 0 0 < I X X X X X X X X X ) I X X X X X X X X X ) i X X X X X X X X X ) r  0  X  0.62 < 0.61 0.60 50  100  150  200  250  300  Time (s) •  Substratel  • Substrate 2 A Blank 1 X B I a n k 2 O Blank 3  Figure 3.3.1: Absorbance of Micrococcus lysodeikticus suspension and blank runs Immediately after preparation of the suspension, a sample of the substrate was placed in the spectrophotometer and its absorbance at 450 nm recorded for 5 minutes. Two identical samples of the substrate were tested and a constant absorbance over a 5 minute period at 0.635 and 0.634 indicated that the substrate was stable (i.e. no significant contamination or settling particles) and also that the absorbance was within the range recommended by Sigma. Duplicates were performed as standard practice, and in cases of considerable variation between results, triplicates were performed. In any sample, 2.5 ml of substrate was added to a cuvette, with 0.1 ml of lysozyme solution added for a test and 0.1 ml of reagent A for a blank. The samples were mixed immediately by inversion and the decrease in absorbance recorded for 5 minutes. Figure 3.3.2 shows the results achieved for sample 1 from T F U 310. According to Sigma (Appendix 4.1), the delta A450 /minute is determined from the nm  maximum linear rate for both the test and blank over the 5 minute period. It was suggested that this maximum rate was typically at the beginning of the test. Figure 3.3.3 confirms this.  3: Experimental Apparatus and Methods  N  0.61 ^ 0.57 E e 55 0.53 u  e re X! o Vi  X)  B B  80  U J B  , 0 B  °U  U  , 0  0.49  0g  D  ° °g u  D  0.45 0.41 0.37 0  50  100  150  200  250  300  Time (s) • i gib |  Figure 3.3.2: Enzymatic activity assay for lysozyme (Dilution: 600 x) The maximum linear rate is determined by performing a linear regression over the maximum possible number of linearly aligned data points (for at least 60 s) to maximize the value of the coefficient of correlation. The results from experimental fouling data of T F U 310 are shown in Section 4.3.6. 0.60 0.58 £  0  5  6  £ 0.54 » 0.52  y _ .0^000'>49x-t~0:59f 731  u  e I  i  0.50  o Vi  < 0.48 0.46  ' --  0.44 0  10  20  30  40  50  60  70  80  90  100 110 120 130 140 150  Time (s) Figure 3.3.3: Determination of the maximum linear rate for sample 1 (t = 0)  4: Experimental Results and Discussion 4.  _81  Experimental Results All fouling experiments were performed at a constant bulk concentration of 1 wt. %  protein dissolved in aqueous solution. For the purpose of modeling, and implicitly assumed in the application of Epstein's (1994) mathematical model, it is shown in Section 4.3.6 that the concentration of protein in the bulk solution remained essentially constant over the duration of an experiment. It was thus shown, by estimating the number of tube passes per run and by assuming that any decline in protein concentration occurred in the test section, that the bulk concentration decrease along the length of the test section was negligible. 4.1  Data Handling Methods The thermal fouling results were measured from up to 10 thermocouples located at  various axial positions along the length of the tube. Both whey protein and lysozyme systems had fouling rates that were strongly dependent upon surface temperature, so local fouling results were used. However, an integrated heat transfer coefficient and temperature was calculated for the 2fiiraldehyde experiments, where fouling was not so temperature dependent (Appendix 2.2.4). The following procedure outlines how local and integrated fouling results were obtained. Prior to any calculations, a plot of outside wall temperature (T  w>0  ) versus thermocouple  axial location at selected times was constructed to determine the individual performance of each thermocouple throughout an experiment.  Slight misalignment of a thermocouple during  assembly, or simply age, might render that thermocouple unreliable for a given experiment, and therefore to be discarded from calculations. Figure 4.1.1 shows an example plot. Figure 4.1.1 clearly shows the strong wall temperature effect of lysozyme fouling along the length of the test section.  4: Experimental Results and Discussion  0  100  200  82  300  400  500  600  700  800  Axial Location (mm) Figure 4.1.1: Axial temperature profiles for T F U 303 (G = 879 kg/m s) 2  Procedure to determine the initial fouling rate 1.  initially, all thermocouple temperatures were plotted and an approximate time to arrive at  steady state was determined. This allows one to evaluate the steady state conditions. 2.  For a given thermocouple, the region before commencement of fouling, i.e. the delay time  td, when the driving force (T i - Tt,) was constant, was determined. This is illustrated in Figure Wj  4.1.2. 70 65  V  60 r \ ©  H *  55 50 45  Unsteady  9r  40 35 30 25 1000  2000  3000  4000  5000  6000  7000  8000  Time (s) Figure 4.1.2: Driving temperature difference at x = 660 mm for TFU 303 (G = 879 kg/m s) 2  4: Experimental Results and Discussion 3.  83  The local heat transfer coefficient for this steady state region was taken as the clean heat  transfer coefficient, UC ) (  U (x) = q/[(T , ) (x)-T (x) c  w  i  c  b  (4.1.1)  For whey protein fouling, a steady state period was not always present, making difficult the estimation of the clean outside wall temperature and heat transfer coefficient. The bulk temperature at position x was determined by assuming that the bulk temperature increases linearly under conditions of a uniform heat flux across the length of the tube, and was therefore calculated as  T  b ( ) = b.in+[T ,ou,-T ,in]|; X  T  b  b  (4.1.2)  where L is the length of the heated section and x is the location of the thermocouple of interest. The inside wall temperature was calculated using the analytical solution of the steady state heat conduction equation for a long, hollow cylinder with uniform heat generation and an adiabatic outer wall:  (4.1.3)  Depending upon the exact operating conditions, the variation between the inside and outside wall temperature could be between 0.5°C and 2°C. 4.  For the time period in which fouling clearly occurred, the fouling resistance, Rf, was  determined from  (4.1.4) where  4: Experimental Results and Discussion  U(x,t) =  84  T  w>i  (4.1.5)  (x)-T (x) b  and a linear regression performed to determine the initial fouling rate, R f , given by 0  • d( Rfo = — dt U(x,t).  (4.1.6)  This is shown in Figure 4.1.3, and assumes that the convective heat transfer coefficient does not change with respect to time, e.g. as a result of surface roughness effects. The first point was chosen such that there was no systematic deviation from the regression at time values below this initial point. The small amount of fouling that occurs before the linear rate has been ignored in this calculation. 0.35 o Heat up data points 0.30  y = 3.06E-05x + 9.06E-02  a. 0.25 *—*  0.20  5 0.15 0.10 0  1000  2000  3000  4000  5000  6000  7000  8000  Time (s)  Figure 4.1.3: Initial fouling rate at x = 660 mm for TFU 303 (G = 879 kg/m s) 2  4.2  Whey Protein Fouling Twelve fouling experiments with a 1 wt. % solution of whey protein concentrate (WPC-  80) in distilled (and deionised) water were performed over a range of fluid mass fluxes and wall temperatures at a constant pH (6.2 - 6.3). Nine experiments were performed to study the velocity  4: Experimental Results and Discussion  85  effect on the initial fouling rate between inside wall temperatures of 60 and 102° C, while the remaining three experiments were performed at similar operating conditions to study the effect of particulate material, and to determine experimental reproducibility. 4.2.1  Physical Properties WPC-80 is a powder consisting of 80 wt. % protein. To obtain a 1 wt. % solution of whey  protein, 748 g WPC-80 were dissolved in water to obtain 60 liters of solution. The physical properties (T|, v, and p) of this 1 wt. % solution were investigated over the typical operating temperature range. A Cannon-Fenske viscometer was used to estimate the kinematic viscosity, and specific gravity bottles were used to determine the density. Figure 4.2.1 shows the results for these two properties. The ratio of the density of whey protein solution to water was found to be almost constant at an average value of 1.0035. Therefore, the solution density is calculated as the density of water multiplied by this factor. The kinematic viscosity of the whey protein solution is a strong function of temperature and, as shown by comparison to experimental and literature values of water, behaves in a slightly different manner with increasing temperature. The kinematic viscosity is given by v(centistokes) =  0.0001278(T( c)) -0.023814(T(°C)) + 1.510165 o  2  (4.2.1)  4: Experimental Results and Discussion  86  50  60  70  80  90  Temperature (°C)  Figure 4.2.1a: Ratio of whey protein solution to water density at pH 6.2 1.0 0.9 — Whey protein solution  A O  -  \ •4-1  Water (test)  JU v>  «  e  J  \ ^  R  2  = 0.997211  06 CJ  Water (Perry and Green, 1984)  C  2  ^  y = 0.000128x -0.023814x+ 1.510165  0.5  P A  A  0.4  o~~~~ A  — A  0.3 20  30  40  50  60  70  80  90  100  Temperature (°C)  Figure 4.2.1b: Kinematic viscosity of 1 wt. % whey protein solution at pH 6.2 As expected, the dilute whey protein solution was slightly more dense and viscous than water. This is consistent with a pure solution containing a small fraction of dissolved particles. 4.2.2  Microbial Contamination 12.5 g WPC-80 powder was dissolved in 1 L of deionised water to give a 1 wt. % whey  protein solution. The solution was tested for microbial contamination at various dilutions (where  87  4: Experimental Results and Discussion  a dilution of 3 is 1 ml diluted to 1000 ml), as was the deionised water without any solute. Table 4.2.1 shows the results.  Table 4 . 2 . 1 : Microbial content of whey protein solutions Sample  Dilution  Colony Forming Units (CFU)  Whey Protein  0  TNTC*  1  180  3  2  5  0  7  0  0  TNTC  1  300  Deionised Water  * Too numerous to count  These results are somewhat surprising, in that they suggest the deionised water is causing the microbial contamination. Comparing the 180 CFU/petrifilm obtained for an order of magnitude dilution of the initial whey protein solution, which corresponds to 144,390 CFU/g, to the 1500 CFU/g in the specifications of the supplied powder, implies that the microbial content has increased by 100 times. It is unlikely that this is due to the powder. Therefore the next step involved testing the available water supplies to try to isolate the cause of contamination at various dilutions. These results are recorded in Table 4.2.2. The results show that the holding bottle has some, albeit very little, microbial content; however, the water has been proven to be clean; therefore the contamination must be arising in the preparation of the protein solutions. It is unlikely to be the powder, since this is considered fit  4: Experimental Results and Discussion  88  for human consumption. Table 4.2.3 presents an analysis for an experiment of all the possible sources of contamination.  Table 4.2.2: Microbial content of water supply Sample  (0)  (1)  (3)  Deionised Water: Lang Room  0  -  -  Distilled Water: Lang Room  0  -  -  Deionised Water: Holding Bottle  51  -  -  WPC not in T F U  TNTC  TNTC  360  TNTC  TNTC  235  TNTC  TNTC  290  Table 4.2.3: Potential sources of microbial contamination Sample  (0)  (2)  (4)  Deionised Water: Bottle  24  -  -  Deionised Water: Lang Room  0  -  -  WPC Concentrated Solution: Pre-TFU  TNTC  175  -  Final Rinse Water  TNTC  TNTC  33  Mixing WPC in T F U  TNTC  TNTC  27  First Sample in Exp.  TNTC  TNTC  31  Last Sample in Exp.  TNTC  TNTC  41  From all of these experiments it appears that microbial contamination is present from the beginning, with possible sources of contamination being the powder, the mixing equipment, and the T F U . However, the promising result from this test is that during the approximately six hours  4: Experimental Results and Discussion  89  of an experiment, there is no microbial growth. This suggests that bio-films are unlikely to be laid down on the tube, and based on a simple calculation of their size and population, these microbial species will not affect the physical properties of the solution. 4.2.3  Velocity Effect Due to the longitudinal temperature gradient at the surface of the tube (caused by the  solution being heated as it passes through the tube), the downstream thermocouples showed the highest temperatures; they therefore exhibited the highest rates of fouling and thus limited the duration of an experiment. Because clean inside wall temperatures were as high as 102°C, it was necessary to maintain some over-pressure on the test section to prevent the onset of boiling as the wall temperature rose due to fouling. When the test section pressure was maintained at 308 kPa (30 psig), measured wall temperatures were allowed to rise to 125°C before terminating the experiment. Because of this temperature restriction, the low temperature regions of the tube gave very low, and sometimes barely measurable, initial fouling rates. For this reason, and to simplify data analysis, the criterion used by Paterson and Fryer (1988), where the induction period is defined as ending when the fouling deposit Biot number, Bi = RfU = (U(/U)-l = 0.05, was c  invoked. Thus any data where the local heat transfer coefficient did not decrease by more than 5 % of the clean heat transfer coefficient by the end of the experiment were neglected. Reynolds numbers based on local fluid properties were varied from 3400 to 25700 to provide adequate data for a complete study on the velocity effect of initial fouling rates. Batches of whey protein were prepared in concentrated solutions (10 wt. %) from WPC80 powder, to ensure a well mixed solution without clumping. After mixing, the concentrated solution was acidified to a pH range of 6.2 - 6.3 with 0.1 N HC1, and then added to the holding  4: Experimental Results and Discussion  90  tank and diluted to the desired 1 % concentration. Prior to the addition of heat to the test section, the solution was circulated for 30 minutes to ensure a thoroughly well mixed chemical system. The power to the test section was then applied to achieve the operating conditions required. Heating up the test section to steady state took approximately 5-15 minutes. As previously discussed, fouling was primarily measured thermally, and the local heat transfer coefficient at a given thermocouple location was determined from Equation 4.1.5, the fouling resistance from Equation 4.1.4, and the initial fouling rate from Equation 4.1.6. Figure 4.2.2 shows a plot of the inside wall temperature obtained from the four highest temperature surface thermocouples for T F U 209, which indicates a strong temperature effect on the initial fouling rate. All thermocouples in Figure 4.2.2 show a linear increase in wall temperature with time after the brief heat-up period. This makes the analysis for initial fouling rate straight-forward. In the region where the temperature is first increasing at a constant rate, the corresponding plot of reciprocal heat transfer coefficient versus time is linearly regressed and the slope is equal to the initial fouling rate. As shown in Figure 4.2.2, estimation of the clean inside wall temperature, i.e. the wall temperature after the mixing period (1800 s) and the initial heating up period but before fouling becomes significant, was difficult with the high temperature thermocouples. It was less difficult with the lower temperature thermocouples because the fouling data points were more easily distinguishable from those in the heating up period. Note that the clean inside wall temperature (T ,i) = deposit-fluid interface temperature, provided that the film coefficient from the deposit w  c  to the fluid equals that for the clean metal surface to the fluid.  4: Experimental Results and Discussion  91  130 • T/C 7  120  • T/C  8  T/C  9  A  X T/C 10  110 ^ 100 a?  H  90 80 70 0  1000  2000  3000  4000  5000  6000  Time (s)  7000  8000  9000 10000 11000  Figure 4.2.2: Inside wall temperature profiles for T F U 209 ( G = 882 kg/m s) 2  A n attempt was made to check these clean wall temperature values by repeating the experimental conditions with water as the test fluid. Unexpectedly, a discrepancy was noted between identical runs. At high mass fluxes (G > 532 kg/m s), there was at most a 2 - 3°C wall temperature difference, with the wall temperature of the protein solution being higher. At the lower mass fluxes however, temperature differences were up to 10 - 15°C. See Figure 4.2.3. no 100 WPC  90 G  80  b  60  7 0  Water  50 40 30 0  100  200  300  400  500  600  700  800  Axial Location (mm) Figure 4.2.3a: Clean inside wall temperatures (a), G = 221 kg/m s 2  4: Experimental Results and Discussion  92  110 WPC  100 90 80  Water  70 b  60 50 40 30 100  200  300  400  500  600  700  800  Axial Location (mm) Figure 4.2.3b: Clean inside wall temperatures (a), G = 532 kg/m s n o  100 WPC  90 U  80  ~  70  b  6  Water  0  50 40 30 100  200  300  400  500  600  700  800  Axial Location (mm) Figure 4.2.3c: Clean inside wall temperatures (a), G = 1275 kg/m s 2  As previously suggested (Section 4.2.1), it was discovered that the viscosity of the protein solution was 10 - 15 % greater than that of water. This would cause the convective heat transfer coefficient in the turbulent region to be sufficiently lower, such that the wall temperature of the protein solution was 2 - 3°C higher. From the Sieder-Tate correlation for turbulent flow conditions, 0.14  Nu = ^  = 0.027Re°- Pr°- f^ ^ 8  33  (4.2.5)  4: Experimental Results and Discussion  93  ^Uc^nLrnLr  ( - -6) 4  2  From an energy balance, 1  —  U  c water  (T  wwater  —T j — U ^^T b  c  w w p c  —T j  (4.2.7)  b  Substituting Equation (4.2.7) into (4-2.6) and rearranging, 0.33 / *H b.wpc  T w,wpc  b.water )  \ 0.14  w.wpc V^w, water V  (T , w  w a t e r  -T ) + T b  b  (4.2.8)  Equation (4.2.8) was applied to the experimental data, and was shown to account for a 2 - 3°C difference between T  w > w p c  and  T  w > w a t e  r-  This difference accounts for the discrepancy at mass fluxes  > 500 kg/m s, but when a similar Sieder-Tate expression is used for laminar to transition flow, 2  the wall temperature is much less dependent upon bulk fluid viscosity, and hence does not explain the larger differences at low mass fluxes. Other physical properties such as fluid density, p (Section 4.2.1), thermal conductivity, X, and specific heat capacity, C , were investigated by p  consultation with the literature (Fernandez-Martin, 1972a, b & c), which indicated no significant deviation of these properties for various milk types with 1% total solids from that of water. At one stage, it was thought that in the 30 minute time period of mixing in the holding tank before heating the test section, physical adsorption of the denatured protein to the tube wall was occurring. However, this supposition was tested by weighing and then visually inspecting the inside surface of the test section for an experiment that was terminated after the mixing step, i.e. before heating, and proven not to be the case. The only remaining explanation is that, at low mass fluxes, fouling starts before steady state is achieved. Clean heat transfer coefficients were estimated using heat transfer correlations (Gnielinski (1976) correlation for turbulent flow and  94  4: Experimental Results and Discussion  Schlunder (Gnielinski, 1976) correlation for laminar flow) over the range of experimental conditions. The average absolute percent deviation between the correlation and experimental coefficients was 10 - 15 % depending upon the correlation used, which could account for up to a 6°C wall temperature difference. Despite scatter between experimental and correlation estimates of the clean heat transfer coefficient, the results showed no clear relationship that would explain the varying wall temperature difference with increasing mass flux. Therefore, the experimental estimates of the clean inside wall temperature were used in this study. The initial fouling rates are determined from the slope of the reciprocal heat transfer coefficient vs. time plot, as shown in Figure 4.2.4. Note in this case the absence of any clear delay period prior to the onset of fouling. 0.22 o Heat up data points 0.20  g  0.18 y = 6.68E-06x+ 1.44E-01  E,  £ 0.16  0.14 0.12 0  1000  2000  3000  4000  5000  6000  7000  8000  9000  10000 11000  Time (s)  Figure 4.2.4: Determination of initial fouling rate at x = 550 mm for TFU 209 (G = 882 kg/m s) 2  Figure 4.2.5 shows an increasing fouling rate after the initial constant rate, an effect which occurred at low Reynolds numbers (TFU 211, 208, 204 and 212) and high wall temperatures (x = 713 mm, and sometimes 660 and 605 mm). There are potentially two  4: Experimental Results and Discussion  95  explanations for this phenomenon. The first is the formation of a mobile gel-like layer (highly viscous) between the deposit and fluid interface (Delplace, 1997) which, although plausible, is difficult to prove. A second, more likely explanation, is that the accelerated fouling rates are caused by the onset of an aggregation mechanism that becomes significant once enough aggregates form in the volume of fluid near the wall, which is hot enough to sustain significant reaction rates. This aggregation mechanism need not necessarily occur in the bulk fluid. Increased fouling rates have also been observed by Wilson (1994) using an indene in kerosene chemical system in the same equipment. In that case, the increased fouling rates were shown to be correlated to the formation of polyperoxide agglomerates in the fluid bulk. Also, from this explanation, it would follow that this phenomenon would be observed to a greater extent at the higher temperature thermocouples and for low mass fluxes, when the fluid has a greater residence time in the heated test section. Results from the deposit morphology study (Section 4.2.5), and consideration of the velocity and temperature distributions (Section A3.4), may provide insight into the cause of this phenomenon. However, it is only the initial (lower) fouling rate, as shown in Figure 4.2.5, that is of interest for testing the mathematical model. The possibility exists that the calculated initial fouling rate is really a measure of the approach to pseudo-steady state that occurs within the induction period, and that the accelerated fouling rate should therefore be interpreted as the initial fouling rate. Although unlikely, this possibility is discussed later.  96  4: Experimental Results and Discussion  0  2000  4000  6000  8000  10000  12000  14000  16000  18000  Time (s)  Figure 4.2.5: Initial and accelerated fouling rates at x = 660 mm for T F U 212 (G = 400 kg/m s) 2  Therefore from any one experiment it was possible for 10 thermocouples to describe the system behaviour over a range of wall temperatures. Figure 4.2.6 shows the linear least squares regression for T F U 207 fitted to the Arrhenius type equation Rf = Aexp  -AE  f  0  R( w,)c T  1.0E-04 0.00274  0.00276  0.00278  0.0028  0.00282  0.00284  0.00286  0.00288  5 en s  1.0E-05  o fa  y = 2.2497E+10e- -  1 2579E+04x  1.0E-06 l/(T ) (K ) 1  Wii  c  Figure 4.2.6: Linear least squares regression for T F U 207 (G = 532 kg/m s) 2  However, using a linear least squares regression linearises the experimental data as l n ( R ) versus ln(l/(T j) ) and therefore gives approximately equal weight to high and low fo  Wj  c  4: Experimental Results and Discussion  97  fouling rate data. One would have more confidence in the higher initial fouling rates which occur at the higher wall temperatures. From Figure 4.2.6 the apparent fouling activation energy is 104.6 kJ/mol. Figure 4.2.7 shows the same data, where a non-linear least squares regression, which minimizes the sum of the squares of the residuals of R  rather than of l n ( R ) , has been used.  fo  fo  The higher fouling rate data is thus given greater weight than the lower fouling rate data, which  could show values of R  f 0  one order of magnitude smaller. The corresponding activation energy  was 120.6 kJ/mol. Since the non-linear regression placed greater emphasis on the higher fouling rate data, it was decided that this regression method be used for all Arrhenius type equations in this work. The Arrhenius parameters by both methods for all whey protein fouling experiments are shown in Table 4.2.4. l.OE-04 0.00274  0.00276  0.00278  0.00280  0.00282  0.00284  0.00286  0.00288  • c  1.0E-06 l/(T ) WII  (KT ) 1  C  Figure 4.2.7: Non-linear least squares regression for TFU 207 (G = 532 kg/m s) 2  4: Experimental Results and Discussion  98  Table 4 . 2 . 4 : Arrhenius parameters for whey protein fouling experiments Linear ln(A)  Linear AE (kJ/mol)  Non-linear ln(A)  Non-linear AEf (kJ/mol)  0.96  3.5  45  4.7  48.3  284.1  1.57  33.0  135  30.9  128  212  400.4  2.27  61.7  221  49.9  186  204  401.4  2.14  18.8  93  17.9  90.5  207  531.7  3.84  23.9  105  29.2  121  206  699.4  5.08  72.2  251  59.7  213  209  882.3  5.45  64.4  229  57.7  208  205  1055.6  5.65  35.0  141  37.4  148  210  1274.9  5.87  14.8  83.5  14.5  82.4  221*  699.4  4.60  90.3  306  78.8  271  222  703.1  4.67  97.8  329  82.5  282  223  704.1  4.78  71.0  248  64.4  228  TFU  G (kg/m s)  211  221.4  208  2  Q (kW)  f  * Centrifuged at 50,000 rpm to remove particulate material  Experiments T F U 204 - 212 were performed with the same supply of whey protein (Lot # 120296), while experiments T F U 221 - 223 were performed using a supply of slightly varying composition (Lot # A702) to determine the effect of particulate material in the reconstituted whey protein "solution". The specifications of these two batches are presented in Appendix 3.1. T F U 206, T F U 222 and T F U 223 were performed under identical experimental conditions using the two different batches of whey protein powder. Despite the large variation in fouling activation energy values over the range of experimental conditions, these values are reasonably consistent with the purely kinetic activation  4: Experimental Results and Discussion  99  energy for the denaturation of P-lactoglobulin (the primary fouling component of whey protein), which in the temperature range of 70 - 90°C is reported as 265 - 280 kJ/mol and in the temperature range of 95 - 150°C as 54 - 48 kJ/mol (Dannenberg and Kessler, 1988; Lyster, 1970). Fouling activation energies have been previously reported as ranging from 60 to 250 kJ/mol when bulk reactions are involved (Belmar-Beiny et al., 1993), and 89 kJ/mol for skim milk with a relatively low bulk temperature of 60°C (Paterson and Fryer, 1988). For a series of whey protein fouling experiments at constant bulk inlet and bulk outlet temperatures, Gotham (1990) evaluated fouling activation energies over a range of Reynolds numbers using Equation (2.3.16). These results along with the fouling activation energies from the whey protein experiments performed in this work are shown in Figure 4.2.8. 250 • WPC-80  3  o Gotham  200 150  100  <  50 0 0  2000  4000  6000  8000  10000  12000  14000  Reynolds Number  Figure 4.2.8: Effect of Reynolds number on fouling activation energy Open circles : Gotham (1990). 1 % protein, protein inlet 73°C and outlet 83°C Solid Squares : WPC-80 experimental results (Tw iV = 68 - 102°C, T = 31 - 57°C h  Despite the unsystematic variation in fouling activation energy with Reynolds number (evaluated at bulk inlet conditions), there is some agreement between both sets of results. Gotham (1990) suggested that since the activation energy was not constant over a range of Reynolds numbers, wall processes do not always govern fouling, and that both bulk and wall processes must be  4: Experimental Results and Discussion  100  important in whey protein fouling. In this work, these processes are considered to be mass transfer from the bulk to, and chemical attachment in, the vicinity of the wall. As indicated earlier, there was some uncertainty in the low mass flux fouling experiments (G = 221.4, 284.1, 400.4 and 401.4 kg/m s) about how to interpret the data that demonstrated 2  accelerated fouling rates. The fouling Arrhenius type plots for each of the low mass flux experiments are shown in Figure 4.2.9 for all thermocouples that satisfied the fouling Biot number criterion. Also plotted are the few accelerated fouling rates that developed after the initial low fouling rate. I.E-03 0.0026  l.E-03 0.0027  0.0028  0.0029  0.0030  0.00267  0.00270  0.00273  0.00276  0.00279  5 ^ l.E-04  l.E-04  E, V CQ  y = 1.08E+02e" i  K u  l.E-05  .E l.E-05  l.E-06  l.E-06 l/fr.jMKr ) 1  (a) TFU 211 (G = 221.4 kg/m s)  (b) T F U 208 (G = 284.1 kg/m s)  2  l.E-04 0.00274  0.00276  0.00278  0.00280  l.E-05 H  2  0.00282  l.E-04 0.00274  0.00276  0.00278  0.00280  0.00282  [ l.E-05  l.E-06  l.E-06  • Initial Rate • Accelerated Rate  y = 6.18E+07e"'  l/(T.j).(K"')  (c) TFU 212 (G = 400.4 kg/m s) 2  (d) T F U 204 (G = 401.4 kg/m s) 2  Figure 4.2.9: Comparison of initial and accelerated fouling rates for the low mass flux whey protein fouling experiments Figure 4.2.9 suggests that the currently used low, initial fouling rates correlate well in the Arrhenius manner over the whole temperature range, while the accelerated fouling rates appear to  4: Experimental Results and Discussion  .  101  belong to a different family of results, indicating the existence of a different deposition mechanism. Therefore, for this study the lower, initial fouling rates appear to be the appropriate rates to use for investigating the velocity effect on the initial fouling rate of whey protein solutions. From Table 4.2.4 it is possible to use the Arrhenius expression to determine R a t a fo  given value of (T i) for each experiment and hence investigate the true effect of velocity on the Wj  c  initial fouling rate. Thus Figure 4.2.10 shows three sets of results at surface temperatures where the initial fouling rate is strongly dependent upon the velocity. In all three cases one can see the presence of a maximum deposition rate at a critical mass flux as predicted by the model. Also, as the clean inside wall temperature decreases the value of the maximum R  f0  decreases, and the  location of the maximum shifts towards a decreasing mass flux, both trends being consistent with the model. The three plots indicate how strong a dependence the fouling rate has on wall temperature, with appreciable differences observed for only 4°C wall temperature changes. Once the wall temperature drops to approximately 80°C, there is very little distinction between initial fouling rates at different mass fluxes, with experimental scatter probably playing a more significant role. Included in Figure 4.2.10 is an estimate of the error in the initial fouling rate caused by forcing the experimental data to fit an Arrhenius type expression. The error bars in the ordinate are the average absolute deviation (AAD) between experimental results and the non-linear least squares regression for each experiment which is used to construct Figure 4.2.10. The error bars plotted on the abscissa are the 95 % confidence intervals for the mass flux over the duration of the experiment. A summary of the calculations and their rationale are given in Appendix 7.  102  4: Experimental Results and Discussion 3.5E-05 r 3.0E-05 -  I  2.SE-05 -  %  2.0E-0S  «ec c  1.5E-05 -  "=  b'S  1.0E-05 5.0E-06 O.OE+00 200  400  600  1000  800  1200  1400  Mass Flux (kg/ms) J  Figure 4.2.10a: Effect of mass flux on initial fouling rate for 1 % whey protein solution at (T i) = 92°C, T = 31 - 57°C Wl  f  h  3.5E-05 r  I V  3.0E-05 2.5E-0S 2.0E-05  CB  «ec  c 1.5E-05 "3  (2  1.0E-05  '=  5.0E-06 0.0E+00 0  200  400  600  800  1000  1200  1400  Mass Flux (kg/ms) 2  Figure 4.2.10b: Effect of mass flux on initial fouling rate for 1 % whev protein solution at ( T = 88°C, T = 31 - 57°C w  h  3.5E-05 3.0E-05 ^ «  2.5E-05 2.0E-05  ec  =  3 O  b  «  1.5E-05 [1  1.0E-05  B  HE-1  5.0E-06 O.OE+00  200  400  600  800  1000  1200  1400  Mass Flux (kg/ms) 2  Figure 4.2.10c: Effect of mass flux on initial fouling rate for 1 % whev protein solution at ( T ) = 84°C, T = 31 - 57°C wi  f  h  4: Experimental Results and Discussion  103  Figure 4.2.10 shows an incongruous drop in initial fouling rate at 400 kg/m s; for this reason it was decided to perform a repeat experiment at this mass flux to check the reproducibility and make sure that the fouling rate is in fact this low. From Table 4.2.4 it is seen that T F U 204 (the original run) and T F U 212 (the repeat run) were both performed at about 400 kg/m s. It is thus surprising to note significant variation in fouling activation energies and pre2  exponential factors for these two experiments, given that at the same wall temperatures, as in Figure 4.2.10, the fouling rates are mainly comparable. The fouling results for these two experiments are shown in Figure 4.2.11. Clearly there is some experimental variation in the measured initial fouling rates; however, this occurs mainly at the highest wall temperatures, and the experimental spread is still not excessive (26 - 32 % deviation for (T j) = 81.9 - 89.6°C). Wj  c  This leads into a discussion about the kinetic compensation effect (KCE), and how large changes in experimental activation energies may be compensated by a corresponding change in the preexponential factor, when experimental results change only slightly. This will be discussed in greater detail in Section 4.4 and Appendix 6. 1.4E-05 • TFU 204 • TFU 212  1.2E-05  5 j2  1.0E-05  I  8.0E-06  ...•••a  ai OJ)  =s  6.0E-06  o  « 4.0E-06 '3 2.0E-06 0.0E+00 81  82  83  84  85  86  87  88  89  90  91  92  (T ) (°C) Wii  c  Figure 4.2.11: Fouling results for TFU 204 and T F U 212 (G « 400 kg/m s) 2  4: Experimental Results and Discussion 4.2.4  104  Effect of Particulate Material A recent study (Elliot et al., 1997) on the effect of shear on the behaviour of a whey  protein "solution", raised the question as to whether the whey protein was present in solution or as a suspension. These authors showed that whey protein "solutions" exhibited a bi-modal particle size distribution in the range of 0.1 - 10 pm, with the size distribution increasing to 10 100 pm as the mixing time increased. To determine whether whey protein concentrates behave as a solution or as a colloidal suspension it was necessary to quantify the amount and size distribution of particulate material in these so-called "solutions". There are numerous techniques used to evaluate particle size distributions, ranging from sieving to microscopy to sedimentation. The technique employed here utilized a Malvern Zetasizer 3, which can measure particle sizes in the range of 5 - 5000 nm, using the method of photon correlation spectroscopy (PCS), commonly called dynamic light scattering. Particles analyzed in this size range are typically colloids, and one of their characteristics is that they are in constant random thermal, or Brownian, motion. This motion causes the intensity of light scattered from the particles to vary with time. Large particles move more slowly than small particles, so that the rate of fluctuation of the light scattered from them is lower. PCS uses the rate of change of these light fluctuations to determine the size distribution of the particles scattering light. The standard mode of operation for the Zetasizer is to search for a unimodal particle size distribution, which, as shown in Figure 4.2.12, can mask the effects of multimodal size distributions. This feature is controlled by the band width setting. When the band width is 1 it searches for a single mode particle size distribution and has a high resolution; however, when the  4: Experimental Results and Discussion  105  band width is set to 1000 it can pick up numerous peaks (assuming there is not too much noise), but has little resolution. It was recommended in the equipment manual that a band width of 100 would satisfy both requirements. Results are shown in Figure 4.2.13, where the same tests are performed with band widths of 50 and 100. Despite some variation in intensity, they both show bimodal size distributions with peaks at approximately 0.31 pm and 0.65 pm. 12  r  Particle Size (nm)  Figure 4.2.13: Particle size distribution for 1 wt. % whey protein solutions  4: Experimental Results and Discussion  106  The second requirement for the Zetasizer was to dilute the "solution" to a slightly opaque color to prevent too many particles being present in the analysis, which again could interfere with the reproducibility of the results. A dilution of 2 ml of solution to 100 ml by adding distilled water produced satisfactory results. Another issue to be considered was that the particle size distribution was a function of the shear history of the whey protein concentrate. Analyses were performed after 60 and after 105 minutes of mixing the whey protein solutions. It was found that there was no significant change in the particle size distributions between these two times of mixing, suggesting that mixing time employed during the fouling experiments had no significant effect on particle size distribution. In addition to particle size distribution, the Malvern Zetasizer was capable of measuring their zeta potential. The zeta potential, however, was determined for a diluted solution, so caution should be taken in interpretation of this value. It was determined that the WPC colloids had a negative zeta potential of -37.6 ± 1 . 8 mV. Similarly, the diffusion coefficient of the particles was estimated to have a mean value of 0.137 x 10" cm /s at a constant temperature of 7  2  25°C. The diffusion coefficient can also be estimated from the Stokes-Einstein equation, given by  D =-  ^  (4.2.9)  Given that the mean diffusion coefficient is known from the above measurements, the mean particle size can be estimated, and as indicated from the Zetasizer, is in fact 0.354 pm. Of course, implicit in this equation is the assumption that the scattering particles are spheres (which is  4: Experimental Results and Discussion  107  reasonable since whey proteins are generally globular), and also that a dilute suspension is used such that particle - particle interactions can be neglected. After establishing the approximate particle size range, it was necessary to attempt to quantify the amount of particulate material. 37.4 g of WPC-80 powder was dissolved into 3 liters of solution and 0.1 N HC1 was added to achieve pH 6.3 and mixed at 600 R P M for one hour. The sample volume was then filtered successively through filter papers having four different pore sizes. As is evident from Table 4.2.5, the filtering ability was reduced as the filter pore size was reduced, to such an extent that at 0.1 p.m only 30 ml of sample volume could be filtered over a time period of two hours. Hence, the accuracy and reliability of this method is somewhat questionable.  Table 4.2.5: Filtering results of 1 wt. % whey protein solution after mixing Sample  Clean  Constant  Particulate  Volume  Particulate  weight (g)  weight (g)  weight (g)  filtered (ml)  content (g/1)  Whatman 4 (20 - 25um)  0.2176  0.2184  0.0008  900  0.00089  Whatman 1 (Hum)  0.6287  0.6424  0.0137  900  0.01522  (1.2 um)  0.3638  0.4365  0.0727  100  0.72700  (0.1 um)  0.5122  0.5586  0.0464  30  1.54667  If one considers the mass of particulate material on the basis of one liter, 2.29 g of the whey protein powder (18.4 %) exists as particulate material, which suggests that particulate deposition rates could become significant. Looking qualitatively at Table 4.2.5, the filtration results are in general agreement with those from the Zetasizer, in that the majority of the  4: Experimental Results and Discussion  108  particulate material lies in the 0.1 - 1.2 p.m size range. Hence particle movement is likely controlled by Brownian motion. Therefore, with seemingly significant levels of particulate material, the maximum particle deposition rates were estimated to determine whether this is comparable to the experimental initial fouling rates. The nature of the "motion" of these particles in suspension can be evaluated by making use of the following equation for the non-dimensional particle relaxation time (Epstein, 1988): 2 J2  p  18v  As an approximation, assume that p = p, since the exact density of the particulate material is p  unknown. Since t  + p  < 0.1 (Table 4.2.6), the transport of particles is clearly in the diffusion region for  all operating conditions. If mass transfer control is assumed, i.e. operation at the left of the maximum of the experimental initial fouling rate curves, the Stokes-Einstein equation can be used to estimate the diffusivity of WPC:  D  1 =  3  8  1  0  " < 37trid I  3  T  K  >  (4.2.1.)  p  D is then used to determine the Schmidt number, Sc = v/D. The mass transfer coefficient can be estimated for isothermal flow from the Metzner-Friend version of the Reichardt Analogy, which reduces at very high Schmidt numbers to (Epstein, 1988)  K=  * 11.8 S c  (4.2.12)  V  y  / 3  and then the initial fouling rate can be determined from  109  4: Experimental Results and Discussion  (4-2.13)  R f o = V ^  where Afpf is assumed to equal 221.7 kgW/m K from Section 4.2.5, and Cp is the concentration 4  of particles in the bulk fluid (2.29 kg/m ). The results are shown in Table 4.2.7, where they are compared with experimental values.  Table 4.2.6: Typical whey protein fouling experimental conditions Bulk Temperature (T )  50°C  Particle Concentration (Cp)  2.29 kg/m  D  3  WPC Kinematic Viscosity (v )  0.639 .10" m /s  WPC Density (p )  991.5 kg/m  6  D  b  2  3  WPC Dynamic Viscosity (r| )  634.1 .10" kg/ms  Mass Flux (G)  401.4 kg/m s  699.4 kg/m s  Reynolds Number (Re)  5708  9946  Friction factor (f = 0.25(0.791n(Re) -1.64V )  0.00930  0.00790  Friction Velocity (V*)  0.02761 m/s  0.04433 m/s  t  p  (d = 0.2 pm)  0.000004  0.000011  t  p  (d = 1.0 pm)  0.000104  0.000267  6  D  2  2  +  p  +  p  2  Thus, the computed maximum possible particle deposition rates (i.e. assuming perfect stickability) appear to be quite significant, and hence there is a clear need to determine whether the mechanism responsible for deposition here is actually chemical reaction, as it has been modeled, or in fact some mixed process of particle deposition combined with chemical reaction.  4: Experimental Results and Discussion  110  Table 4.2.7: Comparison of experimental and particulate deposition rates G  D  Sc  R (m K/kJ) 2  Rf„(m K/kJ)  Experiment*  Particulate  f0  (kg/m s)  (m/s)  (m /s)  2  2  2  Contribution of Particulate Deposition (%)  401  699  0.2  3.73 .lO  171457  0.749 .10"  (1.5-2.5). 10"  7.74 .10"  6  31-52  1.0  7.46 .10'  857287  0.256 .10"  It  2.64 .10"  11-18  0.2  3.73 .10'  12  171457  1.210 .10"  (1.2-2.8).10  1.25 .10"  45-100  1.0  7.46.10"  857287  0.413 AO-  It  4.27.10"  15-36  -12  13  13  6  6  6  5  6  5  6  5  6  Estimates taken from wall temperature results of 88 - 95°C  As indicated in the discussion of Table 4.2.4, T F U 221, T F U 222 and T F U 223 were performed to investigate the effect of particulate material on the deposition rate, and the latter two also to check on the experimental reproducibility. All three experiments were performed at essentially the same operating conditions ( G = 700 kg/m s, q =214 kW/m ) with nearly identical preparation procedures. The "solution" for T F U 221, however, was prepared in the usual manner and then centrifuged at 50,000 rpm in a continuous Sharpies centrifuge for approximately six hours to remove any particulate matter. The sludge effluent that resulted weighed 90 g, which included some water content. As usual with these fouling experiments, there were slight variations in temperatures from one experiment to another, so the best comparison of the results is in the Arrhenius type plot for all of the experimental data. T F U 222 and T F U 223 were run under almost identical operating conditions to give an indication of experimental reproducibility, and as a comparison to T F U 221, for which the feed was centrifuged to remove the particulate material. A summary of the Arrhenius parameters from the non-linear least squares regression is presented in Table 4.2.8. Any comparison of this data with  4: Experimental Results and Discussion  HI  the T F U 200 series of experiments should be done with caution, since a different composition of whey protein powder was used (Appendix 3.1). The Arrhenius plots with the experimental data are presented in Figure 4.2.14.  Table 4.2.8: Non-linear least squares parameters for TFU 220 series of experiments ln(A)  AE (kJ/mol)  210.2  78.8  271  703.1  213.5  82.5  282  704.1  218.5  64.4  228  TFU  G (kg/m s)  221  699.4  222 223  2  q (kW/m ) 2  f  The temperature ranges of the experimental plots in Figure 4.2.14 are in close agreement; however, the small differences appear to be due to the variations in heat flux. T F U 221 had the lowest heat flux, and therefore the correspondingly lowest series of wall temperatures. T F U 223, on the other hand, had the highest heat flux, and hence the highest wall temperatures. However, note that the variations in heat flux and wall temperatures are minimal (< 2°C for the latter). From Figure 4.2.14 it appears that the fouling effect of the particulate material in T F U 222 and 223 is negligible, where one notes that there is considerably more variation from one experiment to another through scatter. The experimental data in Figure 4.2.14 and the fouling Arrhenius parameters in Table 4.2.8 suggest that the data for the three runs could be described by just one Arrhenius type expression, without introducing any additional uncertainties. Thus, the role of the particulate material in the fouling mechanism for these experiments appears to be negligible and one is justified to assume that the dominant deposition mechanism is chemical reaction fouling.  112  4: Experimental Results and Discussion  I  1E-04 0.00272  E  Pi OX)  0.00273  0.00274  0.00275  0.00276  0.00277  0.00278  0.00279  0.0028  0.00281  A" — - . ^- . A  •xP  1E-05  a "3 o X  TFU221  TFU221 Regression  D  TFU222  TFU222 Regression  A  TFU223  —  TFU223 Regression  1E-06 1/(T ,,) (K" ) 1  W  C  Figure 4.2.14: Effect of centrifugation on the initial fouling rate for T F U 220 series (TFU 221 centrifuged to remove particulates) (TFU 222/223 contain » 18 % of WPC as particulates) 4.2.5  Deposit Morphology Study of the deposit morphology is useful to help answer several questions: •  What is the nature of the deposit ? Is it uniform and homogeneous, does deposit aging occur and is its appearance the typical milkstone deposit as described by Burton (1968)?  •  Do the deposit characteristics change from one experiment to another, and if so does this variation explain the existence of the increasing fouling rate at low fluid velocities ?  •  What is the nature of the deposit along the length of the test section ? Is the amount of deposit at a given fluid velocity only wall temperature dependent, or does the bulk temperature also have a significant effect ?  •  Is the deposit composition comparable to the protein in the supply powder, or are other fouling mechanisms also having an effect ?  113  4: Experimental Results and Discussion  These questions concerning the deposit material and its morphology are discussed below. Figure 4.2.15 is a photograph of an opened test section, cut into approximately 50.4 mm (2-inch) lengths, showing the nature of the off-white to creamy deposit along some of the sections of the fouled tube. The fluid flow direction in the photograph is indicated by increasing section numbers. Clearly, in the unheated entry (sections 1 - 1 1 ) and exit (sections 27 - 36) lengths there is no deposition, while the hottest sections (23 - 26) show visible amounts of deposit. This shows that the amount of protein deposition is a strong function of wall temperature, and the fact that there is a sudden stop in deposition at the end of the heated section (last part of section 26) suggests that the contribution of a bulk deposition mechanism is very small. In addition, sections 1 4 - 1 7 show no visible signs of deposition because the wall temperature here is too low to sustain significant chemical reactions.  m  m  IL-1  31 Eft ? . M  U  w f c  Figure 4.2.15: Photograph of fouled tube sections from TFU 208 (G = 284 kg/m s) 2  4: Experimental Results and Discussion  114  Note the striations of protein deposit in the direction of fluid flow along the length of the tube in sections 23 and 24. Clearly, the deposit layer is not homogeneous or uniform along the entire length of the heated section. This is to be expected, since the threshold of protein denaturation (60 - 70°C) occurs at some point along the heated section. Therefore this nonhomogeneity should be considered when the effective deposit thickness is used to evaluate the deposit physical properties (kf and pf). Figures 4.2.16 - 4.2.18 show some of the scanning electron micrographs from samples of two different experiments. To help gain an understanding of the deposit morphology, samples from experiments TFU 209 (which did not display accelerated fouling, G = 882 kg/m s) and TFU 2  211 (which did display accelerated fouling, G = 221 kg/m s) were prepared for scanning electron microscope (SEM) analysis. To avoid damaging the deposit structure, samples were prepared and mounted in cross-section and thus analyzed in-situ. Each sample taken of the fouled tube was a 50.4 mm stainless steel length of 9.525 mm OD with approximate wall thickness of 0.254 - 0.305 mm. Approximately 12.7 mm of each length was sectioned using a tube cutter (this method was selected because it had the least impact on the tube and deposit) and the ends were finely polished using 200 - 600 nm "wetordry" paper. This ensured that a flush finish was achieved for gold sputtering and thus promoted greater coverage and hence an easier analysis. Gold sputtering is used so that the electron beam does not charge the sample during the analysis. Organic samples, however, are particularly prone to charging because of the difficulty of getting complete coverage. Note the onset of charging in the top of the micrograph of Figure 4.2.16a.  4: Experimental Results and Discussion  115  Given that a short cylinder was gold sputtered, not all of the inside surface was covered, and hence most of the analysis was restricted to the upper quarter, where good coverage was achieved. Charging, however, was only a problem for TFU 211, where deposit coverage of the test section was greater and somewhat less uniform. This can be seen in Figure 4.2.17. Analysis of most of the samples was performed at a 15 - 20° tilt away from the vertical axis to enable examination along the inside of the tube. Several magnifications were photographed and are discussed below. Figure 4.2.16a shows a very regular, uniform deposit for TFU 209. Cracking is clearly visible at the lower magnifications (lOOx and 400x) and probably occurred as the sample was dried. All tubes were drip dried in the laboratory after an experiment; however, for SEM analysis, drying was achieved to constant weight in a silica filled desiccator. The deposit appears compact, and across the length of the sample is uniform. At higher magnifications (lOOOx and 4000x), there is clearly some rippling at the deposit surface, but the relative differences between peaks and troughs appears to be very small. Figure 4.2.17, however, tells a different story. In contrast to Figure 4.2.16a, Figure 4.2.17a, which was taken perpendicular to the tube wall (also shown in the micrograph), shows a deposit of variable thickness and character along the circumference of the tube. Figure 4.2.17b was taken at exactly the same angle as Figure 4.2.16a; however, given the variation in deposit coverage and hence projections from the tube wall, this photograph is difficult to interpret. At a magnification of 400x (Figure 4.2.17c) the picture becomes a little clearer and shows what appears to be a more compact deposit along the inside of the tube wall (consistent with TFU 209) followed by a more "fluffy" or "cauliflower-like" deposit on top. This is shown more clearly at a  4: Experimental Results and Discussion  116  m a g n i f i c a t i o n o f lOOOx, F i g u r e 4.2.167. A t a m a g n i f i c a t i o n o f 4 0 0 0 x ( F i g u r e 4.2.17e), where the f o c u s was s o l e l y o n the " f l u f f y "  deposit, one c a n clearly see the " c a u l i f l o w e r "  nature o f the  proteinaceous deposit. T h i s deposit nature has often b e e n o b s e r v e d b y other researchers ( B u r t o n , 1968;  Davies  et  al.,  1997)  when  dealing with fluid  bulk  temperatures  that a p p r o a c h  the  denaturation temperature o f the w h e y protein constituents. T h e a b o v e observations c o u l d e x p l a i n w h y i n T F U 211  one observes an initially  low  f o u l i n g rate, f o l l o w e d b y a later increased rate due to the f o r m a t i o n o f this m o r e v o l u m i n o u s deposit. T h i s p r o b a b l y results from a d e p o s i t i o n m e c h a n i s m , where after a g i v e n t i m e p e r i o d , the concentration o f aggregates i n the b o u n d a r y layer f l u i d b e c o m e s significant a n d they deposit at the heat transfer surface. A s d i s c u s s e d later, the b u l k temperature, a n d its effect i n the c a l c u l a t i o n o f the f i l m temperature, is s h o w n to be m o r e important than first anticipated, e s p e c i a l l y at l o w mass fluxes where the d e p o s i t i o n m e c h a n i s m is s e e m i n g l y not restricted to the tube surface.  F i g u r e 4.2.18 s h o w s the nature o f the s u p p l y w h e y protein p o w d e r u s e d for the t w e l v e f o u l i n g experiments. T h e a s s u m p t i o n o f a spherical particle appears reasonable, a l t h o u g h there is considerable size variation. T h i s c o u l d be due to partial aggregation f r o m the p o w d e r process, i n a d d i t i o n to the o r i g i n a l size variation o f the s u p p l y p o w d e r .  drying  Figure 4.2.16b: S E M of TFU209. x « 713 mm Magnification: 400x  Figure 4.2.16d: S E M of TFU209, x « 713 mm. Magnification: 4000x  Figure 4.2.17a: S E M of TFU211, x « 713 mm. Magnification: lOOx  Figure 4.2.17b: S E M of TFU211. x « 713 mm. Magnification: lOOx  4: Experimental Results and Discussion  Figure 4.2.17d: S E M of TFU211. x « 713 mm. Magnification: lOOOx  120  Figure 4.2.18a: S E M of WPC-80. Magnification: 80x  4: Experimental Results and Discussion  Figure 4.2.18c: S E M of WPC-80. Magnification: IQOOx  122  4: Experimental Results and Discussion  123  As indicated in Table 2.3.2, whey protein powder is comprised of several constituents; however, the predominant proteins are p-lactoglobulin (50 %) and a-lactalbumin (22 %). These are the proteins that are assumed to dominate deposition. However, do these proteins really predominate in the deposit that is laid down on the tube wall? In fact, the whey protein powder is 80 % protein, with the remaining 20 % comprising ash, fat, moisture and lactose. It is conceivable that the deposition mechanism is controlled by the inverse solubility of the mineral content in the powder. Therefore several samples were sent to Canadian Inovatech Laboratories for ash, protein and gel electrophoresis analysis. Due to the limited amount of material available from the fouling experiments, composite samples were prepared for analysis, and the results are shown in Table 4.2.9.  Table 4.2.9: Deposit sample results from the whey protein fouling experiments Sample  Source  T F U 200  TFU204, 207,212.  series  Section 26  T F U 220  TFU221,222,223.  series  Sample  Protein  Ash  Gel  Mass  Content  Content  Electrophoresis  0.1727 g  89.8 %  0.31 %  No  0.4680 g  89.5 %  2.32 %  Yes  Sections 24, 25, 26.  Table 4.2.9 shows that the deposit protein content is higher than that of the powder protein content of 80.1 % (and 80.3%), and that the protein content results from both samples are extremely consistent, indicating that the dominant deposition mechanism is associated with the protein content of the feed (Appendix 3.1). There is some variation in the ash content of the two  4: Experimental Results and Discussion  124  deposit samples; however, both deposits have lower ash contents than the whey protein powders (3.2 and 3.8 %), again indicating that mineral deposition is relatively insignificant. The remaining 8 - 10 % deposit mass would likely be comprised of moisture (a rigorous drying procedure was not performed in the lab), fat (up to approximately 5 %) and lactose (also up to approximately 5 %). Gel electrophoresis was used to try to identify the major protein constituents of the protein deposits. Due to a very limited solubility, it was difficult to quantify the protein percentages, but according to Canadian Inovatech, after silver staining they could identify {3lactoglobulin and a-lactalbumin as the dominant proteins in the deposit. It was also indicated that there were trace amounts of other unidentified proteins, likely bovine serum albumin, immunoglobulins and protease / peptone. From the observations in this section, it appears that the fouling deposit is representative of the whey protein powder, and is in agreement with the observations made by Burton (1968) that the deposit is a voluminous, curd-like material, white or cream in appearance. The above mentioned analyses were performed using the standard methods listed below: •  A S H : A modified version of the A O A C Official method 942.05. 0.8 g of sample was weighed into a porcelain crucible and added to 0.1 N sulfuric acid. This mixture was placed into a temperature controlled furnace at 600°C for five hours. The crucible was then transferred to a desiccator, cooled and weighed, reporting % ash to the second decimal place.  4: Experimental Results and Discussion  •  125  PROTEIN : A O A C Official method 990.03. Nitrogen freed by combustion at high temperature in pure oxygen is measured by thermal . conductivity detection and converted to equivalent protein by an appropriate numerical factor. Three factors must be considered for this method of analysis: 1. The furnace must be maintained at 850°C for pyrolysis of the sample in pure oxygen. 2. The isolation system must trap liberated nitrogen for measurement by thermal conductivity detector. 3. Detector required to measure % nitrogen (wt/wt). The protein % = % N x 6.25.  •  PROTEIN CONSTITUENTS: Polvacrvlamide gel electrophoresis.  4.2.6  Physical Properties of Whey Protein Fouling Deposits Following the measurement procedures described in Section 3.2.2, Figure 4.2.19 shows a  typical plot of the deposit coverage and deposit thickness along the length of a fouled tube.  Heated Section  Lg-gn^QnPPP-cLg-B 0  200  400  600  0.00 800  1000  1200  1400  1600  1800  2000  Position along length of tube (mm) Figure 4.2.19: Deposit study along length of tube for T F U 207 (G = 532 kg/m s) 2  4: Experimental Results and Discussion  126  The heated section started at 546 mm from the tube inlet, and deposition is clearly visible shortly after 600 mm. Similarly the heated section ends at 1318 mm and the amount of deposit reduces rapidly. Note that the deposit coverage method is more accurate than the deposit thickness method, where reading the caliper thickness is considerably more subjective, and therefore very much an approximate technique. However, the same trends are clearly observed using both methods. A summary of the deposit coverage and thickness results along the heated section are shown in Table 4.2.10. Figures 4.2.20 and 4.2.21 show the deposit coverage and thickness results for all tubes tested. The straight lines on these graphs assume constant values of Xf and pf for each experiment; the results appear to satisfy this assumption. However, the results do display a dependency on mass flux, i.e. as the mass flux increases, the slopes of the straight lines and hence both pfXf and Xf increase. Figure 4.2.22 shows the deposit coverage and deposit thickness results for each experiment, resulting in unique values for Xf and pf. All regressions except those for T F U 210 were forced through the origin in order to satisfy Equation (3.2.1). All linear regressions fit the experimental data well with the exception of that for T F U 210. Towards the end of T F U 210 the lower electrical connection started to arc at the tube, causing the formation of a small pin-hole. This rapidly increased the temperature at the entrance to the heated section and presumably caused greater amounts of deposition along this part of the tube. In addition, the tube itself was slightly deformed making it difficult to estimate the deposit thickness. This experiment was also performed at a high mass flux (1275 kg/m s) and only a slight temperature rise was observed. 2  Therefore one would expect very little measurable deposit. Thus, given the arcing, the difficult  4: Experimental Results and Discussion  127  nature of the deposit thickness measurements and very little deposition, the small negative slope is not surprising. The fact that the deposit coverage has an intercept can also be explained by the arcing, which resulted in increased deposit formation. Whether this intercept is solely due to arcing or some other physical process is not clear. It is possible that at this higher velocity, where experiments are run for longer times, some level of physical adsorption could occur along the entire length of the test section. However, physical adsorption was not in general found to be significant. Figure 4.2.23 shows the data for T F U 210 re-analyzed, considering just the 4 highest temperatures. Essentially the plot of Xf versus Rf has a zero slope, indicating unmeasurable A,f, while the deposit coverage indicates a slope more in line with previous experiments. 60  r  F o u l i n g Resistance ( m K / k \ V ) 2  Figure 4.2.20: Deposit coverage results for TFU 200 series of experiments 0.07  r  Fouling Resistance ( m K / k W ) 2  Figure 4.2.21: Deposit thickness results for TFU 200 series of experiments  128  4: Experimental Results and Discussion Table 4.2.10: Deposit coverage and thickness results for TFU 200 series Length H . T Area +/- 0.05  T F U 207  TFU211  TFU204  TFU209  TFU205  TFU210  TFU208  Fouled  Clean  +/• 0.0001 +;. 0.0001  Deposit  Rf  Coverage Thickness  +/- 0.0002  T  »A>v«  Twjin.|  T.jnuui  G  +/-0.5  Coverage/Rr  Thickness/Rr  (shaded area fails Lo meet Bi criterion)  (mm)  (m ) 0.001391  (g) 3.7165  <g> 0.0020  (gW)  (thou)  49.10  (g) 3.7185  (m'K/kW)  1.44  0.05  0.0013  76.5  CQ  CQ  CQ  50.70  0.001436  3.8552  3.8500  0.0052  0.0163  3.62  0.20  0.0051  80.5  76.3  84.8  50.80  0.001439  3.8918  3.8810  0.0108  0.0409  7.50  0.31  0.0079  89.7  83.5  50.40  0.001428  3.8723  3.8539  0.0184  0.0600  12.89  0.68  0.0171  88.8  50.00  0.001416  3.8564  3.8334  0.0230  0.0708  16.24  0.75  0.0191  50.40  0.001428  3.8622  3.8315  0.0307  0.0950  21.50  1.06  49.90  0.001414  3.8240  3.7879  0.0361  0.1019  25.54  50.40 49.60  0.001428  3.8990 3.8672 3.8844  3.8588 3.8236  0.0402 0.0436  0.1049 0.1487  49.70  0.001405 0.001408  3.8352  0.0492  49.15  0.001392  3.6826  3.6815  0.0011  50.20  0.001422  3.7505  3.7491 .  50.90  0.001442  3.7987 .  49.60  0.001405  50.60  (mm)  (kg/m's)  (kgW/rnV)  (W/mK)  532  205.42  0 181  532  222.12  0.312  96.0  532  183.50  0.194  82.0  95.7  532  214.79  0.286  92.6  84.7  100.6  532  229.36  0.269  0.0270  98.5  87.8  109.3  532  226.35  0.284  1.58  0.0400  99.3  88.1  110.5  532  250.62  0.393  28.16 31.03  1.81 2.00  0.0460 0.0508  986 106.8  87.3 91.0  110.0 122.6  532 532  268.42 208.68  0.439 0.342  0.1442  34.95  2.38  0.0603  105 1  89.8  120.5  532  242.34  0.418  0.0437  0.79  0.08  0.0019  59.7  59.4  600  221  18 08  0044  0.0014  0.0442  0.98  0.13  0.0032  68.5  67.8  69.1  221  22.27  0.072  3.7970  0.0017  1.18  0.00  0.0000  75.9  74.9  76.8  221  17.81  0.000  3.7298  3.7222  0.0076  0.0662 0.0866  5.41  0.50  0.0127  76.3  75.2  77.4  221  62.46  0.147  0.001433  3.8303  3.8171  0.0132  0.1175  9.21  0.63  0.0159  84.6  82.6  86.7  221  78.37  0.135  50.40  0.001428  3.8376  3 8234  0.0142  0.1320  9.95  0.73  0.0184  91.5  89.1  94.0  221  75.35  0.140  51.00  0.001445  3.8954  3.8775  0.0179  0.1659  12.39  1.00  0.0254  95.9  92.4  99.4  221  74.68  49.65  0.001406  3.7983  3.7746  0.0237  0.2146  16.85  1.15  0.0292  98.9  94.5  103.3  221  78.52  0.153 0.136  49.90  0.001414  3.8510'  3.8142  0.0368  0.3730  26.03  2.00  0.0508  109.0  100.9  117.1  221  69.80  0.136  49.60  0.001405  3.8271  3.7720  0.0551  0.5263  39.22  2.50  0.0635  112.9  101.5  124.3  221  74.51  49.20  0.001394  3.7205  3.7183  0.0022  0.0016  1.58  0.13  0.0032  65.8  65.3  66.3  401  986.56  0.121 1 <=h4  274.38 '  0713  303 08  0 547  1  0.0070  75.7  77.4  51.20  0.001450  3.8757  3.8718  0.0039  0.0098  2.69  0.28  0.0070  70.8  70.2  71.4  401  49.20  0.001394  3.7003  3.6954  0.0049  0.0116  3.52  0.25  0.0064  76.9  75.9  77.8  401  49.95  0.001415  3.7657  3.7532  0.0125  0.0196  8.83  0.50  0.0127  79.2  77.8 '  80.6  401  49.50  0.001402  3.7185  3.6953  0.0232  0.0537  16.55  0.63  0.0159  84.4  81.9  86.9  401  50.45  0.001429  3.7981  3.7661  0.0320  0.0739  22.39  0.90  0.0229  89.6  85.7  93.5  401  51.30  0.001453  3.8731  3.8361  0.0370  0.0741  25.46  1.03  0.0260  89.2  85.0  93.3  401  343.60  0.351  49.10  0.001391  3.7123  3.6716  0.0407  0.0960  29.26  1.38  0.0349  90.9  85.5  96.3  401  304.81  0.364  49.50  0.001402  3.7709  3.7253  0.0456  •  .450.72;,.;, ^uxO-648'-^'; 308.10 6.296 302.99 0.309  0.1531  32.52  1.45  0.0368  97.4  89.1  105.6  401  212.41  0.241  50.50  0.001431  3.8417  3.7818  0.0599  0.1708  41.87  1.75  0.0445  97.9  88.6  107.2  401  245.15  0.260  49.85  0.001412  3.8059  3.8036  0.0023  0.0025  1.63  0.00  0.0000  71.7  70.1  73.4  882  651 -19  0 000  50.00  0.001416  3.8260  3.8225  0.0035  0.0070  2.47  0.00  0.0000  77.1  75.6  78.6  882  35101  . 0.000  50.60  0.001433  3.8684  3.8641  0.0043  0.0076  3.00  0.00  0.0000  81.3  80.2 '  82.3  882  394.72  50.15  0.001421  3.8343  3.8273  0.0070  0.0073  4.93  0.00  0.0000  81.8  80.5  83.0  882  674.98  0.000  50.90  0.001442  3.8749  3.8644  0.0105  0.0151  7.28  0.10  0.0025  85.1  82.4 "  87.8  882  482~26  0.168  49.90  0.001414  3.7848  3.7680  0.0168  0.0388  11.88  0.15  0.0038  90.8  86.4  95.2  882  306.31  0.098  50.20  0.001422  3.7854  3.7577  0.0277  0.0554  19.48  0.35  0.0089  93.4  86.7  100.2  882  351.60  0.160  49.30  0001397  3:7196  3.6851  0.0345  0.0746  24.70  0.59  0.0149  97.3  88.2  106.4  882  331.15  0.200  49.35  0.001398  3.7469  3.7042  0.0427  0.1048  30.54  0.81  0.0206  104.5  91.6  117.4  882  291.45  0.197  50.25  0.001423  3.8665  3.8116  0.0549  0.1298  38.57  1.44  0.0365  107.4  91.4  123.4  882  297.13  0.281  48.90  0.001385  3.6569  3.6550  0.0019  0.0048  1.37  0.08  0.0019  73.5  73.4  73.6  ' 1056  285.75  0 397  50.00  0.001416  3.7364  3.7344  0.0020  0.0112  1.41  0.05  0.0013  75.6  74.3  76.9  1056  126.08  ' 0. U3-  50.25  0.001423  3.7568  3.7542  0.0026  0.0086  1.83  0.15  0.0038  78.8  77.6  80.0  1056  212.39  0.443 0.473  :  ,  ' 0.000  :  50.10  0.001419  3.7390'  3.7356.  0.0034  0.0094  2.40  0.18  0.0044  78.6  77.8  79.5  1056  254.86  0.001426  3.7812  3.7771  0.0041  0.0096  2.87  0.18  0.0044  81.6  80.7  82.5  1056  299.43  0.463  50.35  0.001426  3.7696  3.7637  0.0059  0.0152  4.14  0.20  0.0051  85.1  83.2  87.1  1056  272 14  0.334  51.05  0.001446.  3.8084  3.7992  0.0092  0.0205  6.36  0.28  0.0070  85.5  83.2  87.8  1056  310.33  0.341  49.45  0.001401  3.6946  3.6840  0.0106  0.0255  7.57  0.40  0.0102  85.5  82.6  88.3  1056  296.40  0.398  50.10  0.001419  3.7559  3.7434  0.0125  0.0311  8.81  0.45  0.0114  91.2  87.3  95.1  1056  283.20  0.368  50.60  0.001433  3.7880  3.7722  0.0158  0.0428  11.02  0.48  0.0121  92.7  87.7  97.8  1056  257.54  0.282  49.80  0.001411  3.8853  3.8840  0.0013  0.0000  0.92  0.40  0.0102  63.2  63.1  63.4  1275  49.55  0.001404  3.8566  3.8551  0.0015  0.0000  1.07  0.30  0.0076  65.7  65.9  65.6  1275  50.10  0.001419  3.8838  3.8821  0.0017  0.0000  1.20  0.33  0.0083  69.8  69.5  70.1  1275  49.40  0.001399  3.8279  3.8256  0.0023  0.0016  1.64  0.38  0.0095  69.1  69.0  69.1  1275  50.45  0.001429  3.9104  3.9078  0.0026  0.0017  1.82  0.35  0.0089  72.2  72.1  72.3  1275  1070.17  5 229  50.25  0.001423  3.8762'  3.8726  0.0036  0.0016  2.53  0.25  0.0064  75.2  75.1  75.4  1275  1580.65  3.969  49.15  0.001392  3.7544  3.7501  0.0043  0.0044  3.09  0.23  0.0057  76.7  76.5 •  76.9  1275  701.91  SI.10  0.001448  3.8817  3.8766  0.0051  0.0034  3.52  0.25  0.0064  77.7  76.8  78.6  1275  1036.24  1*868  49.55  0.001404  3.7674  3.7619  0.0055  0.0065  3.92  0.20  0.0051  82.2  81.8 '  82.7  1275  602.83  0.782  50.15  0.001421  3.8041  3.7977  0.0064  0.0081  4.51  0.25  0.0064  82.6  81.5 .  83.6  1275  556.18  0.784  50.10  0.001419  3.6468  3.6454  0.0014  0.0087  0.99  0.35  0.0089  58.5  58.7  58.3  284  113 39  50.85  0.001440  3.6895  3.6881  0.0014  0.0037  0.97  0.13  0 0032  66.3  65.7 '  66.9  284  262 68  50.50  0.001431  3.6269  3.6242  0.0027  0.0116  1.89  0.25  0.0064  72.4  72.0  72.7  284  162 71  49.65  0.001406  3.4823  3.4781  0.0042  0.0131  2.99  0.13  0.0032  75.6  75.3  76.0  284  227.95-/".  49 70  0.001408  3.4839  3.4705  0.0134  0.0276  9.52  0.00  0.0000  83.5  82.5  84.6  284  344.85  51.20  0.001450  3.5682  3.5484  0.0198  0.0515  13.65  0.58  0.0146  90.0  88.2  91.8  284  265.08  51.70  0.001465  3.6544  3.6286  0.0258  0.0705  17.62  0.83  0.0210  95.3  92.5  98.0  284  249.88  0.297  50,30  0.001425  36159  3.5810  0.0349  0.1385  24.49  1.30  0.0330  98.1  93.4  102.8  284  176.85  0.238  49.35  0.001398  3.6010  3.5521  0.0489  0.2519  34.98  2.00  0.0508  106.8  979  115.8  284  138.86  0.202  50.90  0.001442  3.7465  3.6692  0.0773  0.3901  53.61  2.25  0.0572  111.2  97.6  124.8  284  137.43  0.147  327.50  0.572  Average  1027.23.  '  :  50.35  jHniiflilJ!!  " .  ' ' ''5 953 - ,  "'-1,299 \ •„  1 022 - V ' 0 858  .s  0 547 ' '  0 242  N  •' o.ooo ,- : 0.284  4: Experimental Results and Discussion  129  Table 4.2.11: Summary of deposit coverage and thickness results TFU  G  Re(T )  XfPt  Pf (kgW/m K) (W/mK) (kg/m ) 72.75 0.128 570 145.32 0.173 841  b  (kg/m s) 221.4 284.1  2274-2918 2936-3950  401.4 531.7 882.3 1055.6 1274.9  4127-5705 5495-8174 9124-12976 10905-15060 13146-17144  4  2  211 208 204 207 209 205 210*  3  257.10 232.98 307.06 270.89 431.63  0.281 0.364 0.227 0.331  914 641 1355 818  -  Estimation of deposit physical properties for T F U 211  40  0.10 0.09  35  y = 72.746x  30  0.08 If  4  25  •—r\  o - - ^ y ^ O . l I76x  > 20 o U 15  0.07  \  0.06  |  0.05 o 0.04 H  W  0.03  10  |  0.02 &  5  0.01 6"PT  0 0.0  0.00 0.1  0.2  0.3  0.4  0.5  0.6  Fouling Resistance (m K/k\V) 2  Estimation of deposit physical properties for T F U 208 60  0.14 • J  ft 2  o U •fa  o  50  0.10  •  0  a.  1  & 10  0.12  0.08  30 2  = 145.32x  A  4 0  •  l  •  p  ^>  o  —o  —  0.06 y = 0.172  0.04 o 0.02 0.00  0.00  0.05  0.10  0.15  0.20  0.25  Fouling Resistance (m K/k\V) 2  0.30  0.35  0.40  D.  4: Experimental Results and Discussion  Estimation of deposit physical properties for T F U 204  45 ~  130  c Q  0.07  V = 257.1  E 35 en ^ 30 es 2 25 o O  0.08  40  •  • •  20  0.05  0.03  o  10  0.04  o  o  •  15  0.06  •  y * "D.2812X  0.02 o  1  cu  0.01  5 0  a  0.00  0.00  0.02  0.04  0.06  0.08  0.10  0.12  0.14  0.16  0.18  Fouling Resistance (m K/k\V) 2  Estimation of deposit physcical properties for T F U 207  40 35  •  30  y = 232.98x  0.10 0.09 0.08 ? — 0.07 ~ Cfl  en 25  O  > 20 o U 15  —o  O  -  y = 0 3636x  --  o  10  1——<  5  "  0.00  0.02  0.06 1 0.05 I 0.04 H 0.03 i 0.02 J" 0.01 0.00  --  3  0.04  0.06  0.08  0.10  0.12  0.14  0.16  Fouling Resistance (m K/k\V) 2  Estimation of deposit physical properties for T F U 209  40 - 35 E  «4—  S° 25 20  0.00  •  g  • — —"  ? E, 0.04 ¥ 0) B  0.03 « O y = 0.2 Z66x  10  0  o  •  15  5  -- 0.05  y = 307. 06x  M 30 O O  0.06  a. 0.01 r%  -•  o  0.02 %  0  0.00 0.02  0.04  0.06  0.08  0.10  Fouling Resistance (m K/k\V) 2  0.12  0.14  4: Experimental Results and Discussion  131  Estimation of deposit physical properties for TFU 205 0.025  12 1  i  10  ~3b an  8  >o  6  « u  V  0.015 o  4  ft  „  1•  0.010 H  y = 0.3312x  2  Q  0.020 £  J = 270.89x  0.005 o.  o 0.00  0.01  0.02  0.03  0.04  0.000 0.05  Fouling Resistance (m K/k\V) 2  5.0  5  4.0  Estimation of deposit physical properties for TFU 210  0.020  •  y = 431.63x+ 1.2431  0.015  bX)  o  c  2 3.0  3  --  2.0  o ft <u Q  o  1.0  ~o  •.  y = -0.4307x + 0.00 i6  o  0.005 g ft  0.0 0.000  0.002  0.004  0.006  0.010 «  0.008  0.000 0.010  Fouling Resistance (m K/k\V) 2  Figure 4.2.22: Deposit coverage and thickness results for each experiment Estimation of deposit physical properties for TFU 210 5.0 4.0 41  an  0.020  <-  •  0.015  y = 249.77x -i2.36  c  g 3.0  3 o ft <u Q  o  2.0 1.0  ? o y = -o.o: 86x + 0.006  •  0.0 0.000  0.002  0.004  0.006  0.008  0.010 « H 0.005  o a.  0.000 0.010  Fouling Resistance (m K/kW) 2  Figure 4.2.23: Re-analysis of T F U 210 All of the deposit density and thermal conductivity results are shown in Table 4.2.11. Assuming a solids density similar to that of water (995.7 kg/m at 30°C) (Davies et al., 1997), 3  one may calculate the volume fraction of dry solids from this apparent density value (units: kg of  4: Experimental Results and Discussion  132  dry deposit per m of wet deposit) as pf /995.7. The resulting volume fractions of dry solids are 3  0.57, 0.84, 0.92, 0.64, >1, and 0.82. Clearly these results show considerable variation in the deposit voidage from one experiment to another. Figures 4.2.24, 4.2.25 and 4.2.26 throw more light on the data. Deposit coverage and deposit thickness results for each thermocouple were divided by the corresponding final fouling resistance for the duration of the experiment (Appendix 3.2) and plotted against the average deposit temperature. This deposit temperature was determined by averaging the inside wall temperature at time zero and at the end of the experiment. The wall temperature at time zero was assumed to be the deposit-fluid interface temperature throughout the experiment since the process was at a constant heat flux. Figure 4.2.24 shows the results for all experimental data points. The data points at the top of Figure 4.2.24 are due to small deposit coverages, accompanied by even smaller fouling resistances. Since the fouling resistance is in the denominator, as this value tends to zero (as in T F U 210) the deposit coverage / fouling resistance ratio tends to infinity. Using the fouling Biot number criterion (Bi > 0.05), much of the scatter is eliminated, as shown in Figure 4.2.25. However, Figure 4.2.25 still shows several data points with large values of  Afpf  and  X, {  which are primarily from T F U 210 (as shown in Table 4.2.10)  and are due to a small fouling resistance. Eliminating the T F U 210 results from the plot, for reasons explained earlier, yields Figure 4.2.26, which shows no clear dependence of the deposit thermal conductivity or density on deposit temperature. This indicates that there is not likely to be a significant aging effect.  4: Experimental Results and Discussion  133  1600  u  e co  6  1400  5  1200 BX)  c  4  1000 800  2 £  3  600  ©-  tu  >  O  400  O  200  c o  2 1  e3 O c  o c.  0 Q  0  Q  6D C  60  50  70  80  90  100  110  120  Average deposit temperature (°C) Figure 4.2.24: Dependence of physical properties on temperature (All data) 1200  2.0 tu  u  1.8 g  I  CO  1000  1-6 I  V)  tu  'vi  tu W)  IF £ e  800 1.4  600  bio  M em « a, U  >  o ex  '—•  400  272.7 kgW/m"K j 0.8 J  200  u  0.340 W/mK  1  0.6 2  0 50  60  70  80  90  100  110  120 0.4  Average deposit temperature (°C) Figure 4.2.25: Dependence of physical properties on temperature (Subject to fouling Biot number criterion)  Q. <u  0.2 a o.o  Despite considerable scatter in Figure 4.2.26 (Xf = 0.0 - 0.5 W/m.K), the average deposit thermal conductivity for these experiments was 0.26 W/m.K, corresponding very closely to the asymptotic value of 0.27 W/m.K of Delplace and Leuliet (1995). The corresponding averages for XfQf and pf were 221.7 kg.W/m K and 853 kg/m , respectively (corresponding to a solids volume 4  3  4: Experimental Results and Discussion  134  fraction of 0.86, i.e. a voidage of or, when wet, a water content of just 14 %). Rao and Rizvi (1995) reported thermal conductivities (Figure 4.2.27) for dairy products ranging from 0.25 to 0.65 W/m.K, with the exact value depending upon the water content. At a water content of 14 %, the thermal conductivity of any dairy product could be expected to fall within the range of 0.2 0.3 W/m.K since, as illustrated in Figure 4.2.27, as the water content of the deposit decreases, the thermal conductivity also decreases. 1200 u B  «  1000  w>  800  2.0 1.8 g 1-6 I cu  IF  to E Sl  CU)  1.4 M L  600  > o u  o a. cu  N  I * cu O  f= U  2  0.8 ^  400  221.7 kgW/m"K |  0.6 £ 0.4  200  a  0.260 W/mK 50  60  70  0.2 O  Q •  fi 0  0.0 80  90  100  110  120  Average deposit temperature (°C)  Figure 4.2.26: Dependence of physical properties on temperature (Subject to fouling Biot number criterion and no TFU210) In addition to the above, all of the measured deposit thickness results were plotted against the deposit coverage results to determine a more direct estimate of the average deposit density for this set of experiments. These results are shown in Figure 4.2.28. From the reciprocal of the slope of the best fit line, the average deposit density was determined to be 805 kg/m . This is in 3  reasonable agreement with the 853 kg/m determined as 221.7/0.260 from Figure 4.2.26, given the scatter in this figure. Concern about using this averaging method involves the effect of the data points which deviate significantly from the average value. When all data points are given  4: Experimental Results and Discussion  135  equal weight, these highly scattered points tended to artificially inflate the average value. The method shown in Figure 4.2.28 avoids the strong influence of these data points, thus obviating the previous requirement (in Figures 4.2.24 - 4.2.27) to filter data with unusually high values of PfAf and X{.  1.24  I  1.0  |  0.8  •1 0.4 •]  q 4 U'M  0.2  ~i  10  1  1  1  20  30  40  J  1  SO  60  ^  70  1  1  80  90  j  —  100  VATER C<H<T£KTy Z  Figure 4.2.27: Thermal conductivity of dairy products (Rao and Rizvi, 1995)  10  20  30  40  50  Deposit Coverage (g/m ) 2  Figure 4.2.28: Estimation of an average deposit density for T F U 200 experiments  60  4: Experimental Results and Discussion  136  Figure 4.2.28 was found to produce the same slope regardless of whether data subject to the fouling Biot number criterion were included or discarded. The latter had little effect because all of this data lay near the origin of the line where it was forced to pass through the origin. The question which remains unanswered is why there appears to be a velocity effect on the thermal conductivity and density of the fouling deposit, shown previously in Table 4.2.11. There are two possible explanations. The first explanation is that perhaps there is a slight deposit aging effect after all. If the deposit was to age with time and/or temperature, then the protein would be converted to a more carbonaceous form with a corresponding increase in thermal conductivity and density. This increase has been observed (Table  4.2.11) but with a  corresponding increase in mass flux rather than time or wall temperature. From the experimental data there was generally a decrease in deposit mass as the fluid mass flux increased, with more voluminous deposit formation at lower mass fluxes. Appendix 3.4 shows that at low mass fluxes, up to 37 % of the total cross-sectional area at the hot end of the test section is capable of sustaining deposition reactions, in a highly viscous sub-layer and buffer layer, resulting in a more voluminous deposit. Aging of this thicker deposit, would be considerably slower than for a thin, smooth deposit formed at higher mass fluxes. Therefore at low mass fluxes, a lower deposit thermal conductivity and density could be expected. A second possibility could be the effect of void spaces within the deposit. As the fluid mass flux increases, the replenishment of liquid in the void spaces will also increase, causing a corresponding increase in heat transfer coefficient, and decrease in the observed fouling resistance. Therefore, for a given deposit coverage and thickness, an increased deposit thermal conductivity and density would result.  4: Experimental Results and Discussion  4.3  137  Lysozyme Fouling Eleven fouling experiments using a 1 wt. % solution of lysozyme in distilled water were  performed over a range of fluid velocities, wall temperatures and pH values. Nine of the experiments were at pH 8, with fluid mass fluxes between 200 and 1101 kg/m s, while the 2  remaining two were at a mass flux of 700 kg/m s and pH values of 5 and 6.5. A 1 wt. % solution 2  was selected to be consistent with the whey protein experiments in Section 4.2.  4.3.1  Physical Properties Although the physical properties of a 1 wt. % solution of lysozyme in distilled water at  pH 8 are likely to be close to those of pure water, the viscosity (r\ and v) and fluid density (p) were investigated over the temperature range applicable to the fouling experiments. A CannonFenske viscometer was used to estimate the kinematic viscosity, and specific gravity bottles were used to determine the density. Figure 4.3.1 shows the results and a comparison to the properties of water.  1005  Solution Temperature (°C)  Figure 4.3.1a: Density of 1 wt. % lysozyme solution at pH 8  4: Experimental Results and Discussion  138  The density of the 1 wt. % lysozyme solution is well represented by p(kg/m ) = - 0 . 0 0 3 2 ( T ( ° C ) ) -0.1084T( C) +1004.4 3  2  (4.3.1)  o  1.0E-03  Solution Temperature (°C)  Figure 4.3.1b: Dynamic viscosity of 1 wt. % lysozyme solution at pH 8 and the dynamic viscosity of the solution is given by r)(kg/m.s) = - 3 . 4 6 x l 0 - ( T ( ° C ) ) + 7.66x1 0 ( T ( C ) ) - 6 . 0 2 x 1 0 - T ( ° C ) +2.09x10~ 9  3  _ 7  O  2  5  3  (4.3.2)  Figure 4.3.1 shows that the lysozyme solution is slightly more dense and viscous than distilled water, which again, like whey protein solutions, is consistent with a solution containing a small fraction of dissolved particles.  4.3.2  Particulate Material As with whey protein solutions, concern was raised about the presence of particulate  material in suspension and its effect on the foulant deposition mechanism. Unlike whey protein solutions, lysozyme was a clear, colorless solution leading to the belief that there were very few undissolved particles. However, this belief was tested.  4: Experimental Results and Discussion  139  9.95 grams of lysozyme powder were dissolved in distilled water to give 1 liter of a 1 wt. % lysozyme solution. The pH was adjusted to 8 and the entire volume was then filtered over a range of filter sizes with the results shown in Table 4.3.1. Table 4.3.1: Particle size distribution in a 1 wt. % lysozyme solution at pH 8 Filter Size  Clean filter weight (g)  Dirty filter weight (g)  Solids (g/1)  Whatman 4 (20 - 25 pm)  0.2185  0.2194  0.0009  Whatman 1 (Hum)  0.2021  0.2064  0.0043  1.2 pm  0.0729  0.0753  0.0024  0.1 pm  0.1024  0.1031  0.0007  TOTAL  0.5959  0.6042  0.0083  Therefore, given that 9.95 grams of lysozyme were dissolved in 1 liter of water, only 0.083 % of this lysozyme was suspended as particulate material. Hence, this result shows there were no significant particulate levels in the lysozyme solutions, which can therefore be described as pure solutions. 4.3.3  Steady State Conditions and the Measurement of Fouling As previously discussed, fouling was primarily measured thermally, and the local heat  transfer coefficient at a given thermocouple location was determined from U(x,t) = q / ( T ( x ) - T ( x ) ) wi  (4.3.3)  b  where q was evaluated from the voltage and current applied to the test section divided by the heat transfer area, A, Tb from the bulk inlet and outlet fluid temperatures and T i from the Wj  measured thermocouple temperature. The fouling resistance R was determined from f  140  4: Experimental Results and Discussion  R,=  V , 'U(x,t)  1/ /U (x)  (4.3.4)  e  and the initial fouling rate from  (4.3.5)  Rfo =• dt U(x,t)y V  Figure 4.3.2 shows a plot of the four highest temperature surface thermocouples for T F U 304, which indicates a strong wall temperature effect after a short induction time. 120 110 100 90 80 70 0  2000  4000  6000  8000  10000  12000  Time (s) [ • T / C 7 DT/C 8 A T / C 9 X T / C 10 |  Figure 4.3.2: Inside wall temperature profiles for TFU 304 (G " 523 kg/m s) Unlike some of the whey protein fouling experiments, the lysozyme solutions rose to a steady state temperature for a period of time before the onset of fouling. This fact enabled a much easier estimation of the clean inside wall temperature for each thermocouple along the length of the tube. Before each experiment, approximate steady state conditions at a given mass flux were determined using distilled water, so that the heat up period during the experiment was short, but still accurate.  4: Experimental Results and Discussion  141.  For T F U 303 and T F U 304 a comparison was performed between steady state conditions using water and steady state conditions using the test solution. It was determined that on average the difference between surface temperatures at identical positions was approximately 0.5°C. This was a marked contrast with the larger differences determined for the whey protein experiments. The results are shown in Table 4.3.2. Table 4.3.2: Comparison of experimental steady state temperatures with water (TFU 303, Q = 4.42 kW, T F U 304, Q = 2.80 kW) T/C#  Axial  TFU 303  Difference  TFU 304  Difference  Location  (T i)c(°C)  (°C)  (T ) (°C)  (°Q  (mm)  Lysozyme  Water  1  48  63.89  62.69  2  110  66.42  3  163  4  W)  Wii  c  Lysozyme  Water  +1.20  63.81  63.55  +0.26  65.32  +1.10  68.53  67.79  +0.74  67.80  67.62  +0.18  72.80  72.11  +0.69  221  68.30  67.88  +0.42  71.58  70.96  +0.62  5  330  68.70  68.73  -0.03  74.48  73.79  +0.69  6  440  74.89  74.34  +0.55  76.76  76.33  +0.43  7  550  75.26  74.96  +0.30  77.31  76.98  +0.33  8  605  74.95  74.71  +0.24  77.26  77.11  +0.15  9  660  76.74  76.59  +0.15  78.08  77.22  +0.86  10  713  79.33  78.73  +0.60  79.47  78.80  +0.67  The initial fouling rate for each thermocouple was determined from the point where fouling had clearly started until either the completion of the experiment, or non-conformity to a  4: Experimental Results and Discussion  142  linear fouling rate. Figure 4.3.3 shows a plot of reciprocal heat transfer coefficient for thermocouple 10 from T F U 304 and the resulting initial fouling rate.  2000  4000  8000  6000  10000  12000  Time (s) Figure 4.3.3: Determination of initial fouling rate at x = 713 mm for TFU 304 (G = 523 kg/m s) 2  1.0E-04 0.00282  0.00285  0.00288  0.00291  0.00294  0.00297  cu M  en1.0E-05  y = 7.1161E+07e-  I0013E+04x  e  "a ©  1.0E-06 l/(T ) (K" ) 1  Wji  c  Figure 4.3.4: Non-linear least squares regression for TFU 304 (G = 523 kg/m s) 2  As with the whey protein experiments, any thermocouple not satisfying the fouling Biot number criterion (Bi = R U f  c  =(U /U)-1>0.05) was rejected. In addition, any obviously C  4: Experimental Results and Discussion  143  erroneous data points (e.g. due to malfunctioning thermocouples) were neglected. From any one experiment (at a fixed mass flux) it was possible for 10 thermocouples to describe the system behaviour over a range of wall temperatures. This behaviour is shown in Figure 4.3.4, where a non-linear  least  -AE  (Rfo = Aexp  squares  regression  is  used  to  fit  the  Arrhenius  type  equation  ) to the fouling results from T F U 304.  f  R( w,i) T  c  From this regression one is able to determine the fouling activation energy and preexponential constant, and therefore to describe the system over the range of applicable wall temperatures. These parameters and operating conditions are shown in Table 4.3.3.  Table 4.3.3: Arrhenius parameters for lysozyme fouling experiments TFU  G  T (°C) b  (kg/m s) 2  Re (in)  (kW)  f  Q  pH  Deposit Mass (g)  (kJ/mol)  AEf  In (A)  (Tw,i)c (°Q  309  199.5  31-42  2951  0.67  8.0  0.79  29.35  -1.56  59-84  306  296.9  31-50  4557  1.71  8.0  2.21  45.76  5.03  64-82  310  297.2  31-50  4529  1.71  8.0  2.48  49.01  6.08  63-81  307  412.3  31-50  6254  2.35  8.0  1.23  84.85  18.59  63-80  304  523.0  31-49  8011  2.80  8.0  0.83  83.25  18.08  64-79  301  700.5  30-51  10821  4.06  8.0  0.79  86.13  19.16  65-80  303  878.7  31-49  13960  4.42  8.0  0.59  93.77  21.79  66-79  308  991.3  31-49  16074  4.96  8.0  0.73  115.24  28.22  68-79  305  1101.2  31-49  17685  5.45  8.0  0.21  118.17  29.17  68-77  302  696.5  31-52  11095  4.10  5.0  0.03  326.61  97.08  81-86  311  698.0  31-52  11136  4.11  6.5  0.10  144.91  36.61  68-81  4: Experimental Results and Discussion  144  Table 4.3.3 shows that measurable fouling was recorded at temperatures below the denaturation temperature (78°C) for these conditions. However, the denaturation temperature is a measure of unfolding of 50 % of the native protein and is therefore not a threshold value for deposition. This fact is also reflected in the thermal inactivation kinetics reported by Makki (1996) over a range of temperatures from 73 to 100°C. The T F U 300 series of experiments at pH = 8.0 show a much narrower range of fouling activation energies (29 - 118 kJ/mol) compared to the whey protein experiments (44 - 251 kJ/mol). However, it is interesting to note the similar dependence of both AEf and ln(A) on mass flux over an essentially constant wall temperature range (Table 4.3.3). These results are presented in Figure 4.3.5. 120  200  400  600  800  1000  Mass Flux (kg/m s)  Figure 4.3.5: AEf and ln(A) dependence on mass flux for TFU 300 series of experiments at pH 8  1200  4: Experimental Results and Discussion  145  The increase of AEf with mass flux (or fluid velocity) is consistent with the mathematical model. Taking Equation (2.2.26) where n = 1, and assuming constant fluid physical properties and fully rough flow (f = constant), the model in its simplest form becomes  (4.3.6)  For small fluid velocities, i.e. when mass transfer controls,  (4.3.7) indicating that the fouling activation energy must approach zero (fluid physical property variations with temperature would yield a non-zero but relatively small value of AEf). For high fluid velocities, i.e. when chemical attachment controls,  (4.3.8)  which, in turn, shows that the fouling activation energy approaches the purely kinetic activation energy. Therefore, by consideration of the mathematical model in the limits of the fluid velocity, it is expected that the observed fouling activation energy will increase with fluid velocity and approach the purely kinetic activation energy. For this series of experiments it was determined that if an Arrhenius type expression were  attached to the mass transfer term in Equation (4.3.6), e.g. by putting a = a'e  / ( '<, thus R  Tf  allowing for temperature dependence of diffusion, a diffusional activation energy of 23.5 kJ/mol would be obtained. This value is significantly smaller than the corresponding purely kinetic  4: Experimental Results and Discussion  146  activation energy of 140.4 kJ/mol. These two activation energies bracket, and are therefore consistent with, the experimental fouling activation energies reported in Table 4.3.3.  4.3.4  Effect of Fluid Velocity The fouling Arrhenius parameters in Table 4.3.3 allow a qualitative view of how a 1 wt.%  solution of lysozyme fouls over a range of mass fluxes at fixed wall temperatures. Typically the wall temperature varied from 65 to 80°C, and Figure 4.3.6 displays how the fouling behaviour varied over this range of clean inside wall temperatures. Error bars corresponding to the average absolute percent deviation between experimental data and the best fit regression are plotted on the ordinate. On the absicssa, the 95% confidence interval of the fluid mass flux is plotted, again displaying small uncertainty. A calculation summary and its rationale are given in Appendix 7. The model of Epstein (1994) predicts that at a given wall temperature a maximum initial fouling rate exists over a range of mass fluxes. This feature is clearly demonstrated in Figure 4.3.6. Also, note that as the wall temperature increases the value of the maximum initial fouling rate increases, and the location of this maximum shifts towards an increasing mass flux. Both of these trends are also qualitatively consistent with the model.  400  600  800  1000  1200  Mass Flux (kg/m's)  Figure 4.3.6a: Effect of mass flux on initial fouling rate for 1 % lysozyme solution at ( T ) = 65°C, T = 30 - 52°C wi  f  h  4: Experimental Results and Discussion  200  147  400  600  1000  800  1200  Mass Flux (kg/m s) 2  Figure 4.3.6b: Effect of massfluxon initial fouling rate for 1 % lysozyme solution at (T i) = 70°C, T = 30 - 52°C Wi  f  h  3.0E-05  « 5.0E-06 0.0E+00 200  400  600  800  1000  1200  Mass Flux (kg/m s) 2  Figure 4.3.6c: Effect of massfluxon initial fouling rate for 1 % lysozyme solution at (T ^ = 75°C, T = 30 - 52°C w  c  h  4.5E-05 i 4.0E-05 -  I HE  3.5E-05 3.0E-05 -  OS 2.5E-05 en _c 2.0E-05 "3 o 1.5E-05 -  X  th  is s  1.0E-05 5.0E-06 0.0E+00 400  600  800  1000  1200  Mass Flux (kg/m s) 2  Figure 4.3.6d: Effect of massfluxon initial fouling rate for 1 % lysozyme solution at (T X = 80°C, T = 30 - 52°C w  h  4: Experimental Results and Discussion  148  A repeat experiment was performed at a mass flux of 297 kg/m s for two reasons. Firstly, 2  at a clean inside wall temperature of 65°C in Figure 4.3.6, the fouling rate at 297 kg/m s appears 2  somewhat higher than the rest of the trend would predict, and therefore repeating this experiment would indicate if the initial experiment was in error. Secondly, it was necessary to determine the repeatability of these experiments, and therefore establish, at least qualitatively, the extent of experimental scatter that can be expected. Figure 4.3.7 displays the fouling Arrhenius curves for T F U 306 and T F U 310 performed at 297 kg/m s. 2  Figure 4.3.7 shows that the results from T F U 306 and T F U 310 are very reproducible (percentage difference decreased from 10 % at 63 °C to 4 % at 81°C, when the best fit Arrhenius type equations were compared), and this is in part reflected by the similar activation energies obtained for these two experiments, shown previously in Table 4.3.3. However, this figure does introduce an interesting feature that was observed at low mass fluxes. This is the nonconformance of the experimental data at the left of the plot to the exponential Arrhenius type function. This type of response consistently occurred (TFU 306, 309, 310) at the highest wall temperatures of the low mass flux experiments. Whether this is due to a laminar to turbulent Reynolds number transition (Re-let = 3043, 2044, 3046 respectively) as the bulk fluid heats up, is undetermined; however, this irregularity does have an effect on the results in Figure 4.3.6. From the Arrhenius type equations, the expected low temperature points will show a considerably higher fouling rate than was experimentally determined. Therefore in Figure 4.3.6a, the fouling rates at 297 kg/m s should actually be closer to 1 x 10" m K/kJ, which would be 2  5  2  more consistent with the expected model trend. This is clearly shown in Figure 4.3.6a.  4: Experimental Results and Discussion  149  As with the whey protein fouling results, a composite deposit sample from the T F U 300 series of experiments at pH 8 was analyzed by Canadian Inovatech Laboratories for protein content and identification. Results showed that the deposit was 93.5 % protein, and gel electrophoresis identified Lysozyme as the only protein present. These results confirm that lysozyme is the dominant deposit. 1E-04 0.00280 0.00282 0.00284 0.00286 0.00288 0.00290 0.00292 0.00294 0.00296 0.00298  E  DC  1E-05  a "3 o  O  TFU310  •  TFU306  310 Regression  306 Regression  1E-06 i/(T )c W)i  (ic ) 1  Figure 4.3.7: Arrhenius type plot for TFU 306 and TFU 310 (Repeat Experiments) 4.3.5  Effect of pH on the Fouling Behaviour As indicated in Table 4.3.3, three fouling experiments were performed at the same mass  flux and temperature conditions to study the effect of pH on the initial fouling rate. Table 4.3.4 highlights the operating conditions and the resulting fouling Arrhenius parameters. It is clear that as the pH decreased from 8 to 5, the activation energy of the fouling process increased. This was confirmed by observation throughout the experiments, where the duration of the experiment increased and wall temperature elevation (hence the extent of fouling) decreased, with decreasing pH. Despite mass and heat fluxes being nearly identical, there were  4: Experimental Results and Discussion  ISO  some variations in the initial inside wall temperature. These can mainly be attributed to variations in tube thickness from one experiment to another, and to the contact of the thermocouple with the tube. However, the bulk temperature elevation was consistent for all three experiments, indicating that thermocouple comparisons at given axial locations from one experiment to another would be valid since the bulk conditions would be identical. Table TFU  4.3.4: Operating parameters to study the effect of p H  G  pH  (kg/m s)  Heat Flux  T  (kW/m ) 2  2  Tw,i  AEf  (°Q  CO  (kJ/mol)  b  ln(A)  301  700.5  8.0  186  30-51  65-80  86.13  19.16  311  698.0  6.5  188  31-52  68-81  144.9  36.61  302  696.5  5.0  187  31-52  68-86  326.6  97.08  Figure 4.3.8 shows the effect of pH on the wall temperature increase from three different experiments. These thermocouples were chosen to match (as closely as possible) an inside wall temperature of 80°C. This result shows that as the pH decreases, the rate and extent of fouling decreases to almost undetectable levels. In addition, the induction time increases with decreasing pH. However, to enable comparison at (T i) = 80°C, three different axial locations were Wi  c  selected (TFU 302 thermocouple 6, T F U 311 thermocouple 9, T F U 301 thermocouple 10). Hence one might argue that variations in bulk temperature in the axial direction would interfere with the comparison based purely on pH considerations, especially if the fouling rate was considered to be proportional to the volume of fluid hot enough to undergo chemical reaction. This additional consideration would suggest that, irrespective of pH variations, T F U 301 would  4: Experimental Results and Discussion  151  have a greater fouling rate than T F U 311 which in turn would have a greater fouling rate than T F U 302, which in fact was observed. Hence interpretation of the results from Figure 4.3.8 requires some caution. 125 |  0  1  4000  8000  12000  16000  20000  24000  28000  Time (s) | - T F U 3 0 2 T / C 6 p H 5 OTFU311 T / C 9 p H 6 . 5 A T F U 3 0 1 T / C 1 0 p H i T ]  Figure 4.3.8: Effect of p H on Fouling Rate at (T .) = 80°C Wi  f  Figure 4.3.9 was therefore plotted by selecting thermocouple 10 from all three experiments and therefore almost identical bulk temperatures. Although the bulk conditions are the same, there is some variation in the initial inside wall temperature, where thermocouple 10 from T F U 302 shows approximately 86°C while T F U 301 and T F U 311 read approximately 81°C. Despite T F U 302 having the highest wall temperature, it still has the lowest temperature rise and therefore the lowest fouling rate. Thus, from Figure 4.3.9, it is clear that it is the decrease in pH rather than any bulk temperature effect which is primarily responsible for decreasing the rate and extent of fouling. In addition, Figures 4.3.8 and 4.3.9 show that as the pH increases the induction time decreases. For thermocouples satisfying the fouling Biot number criterion, both the lysozyme and whey protein results indicated that, at a constant fluid mass flux, as the wall  4: Experimental Results and Discussion  152  (and bulk) temperature increased, the induction time decreased. However, neither set of experiments demonstrated a clear velocity effect upon the induction time. 125 ,  0  ,  4000  8000  12000  16000  20000  24000  28000  Time (s) | - T F U 3 0 2 T/C 10 p H 5 OTFU311 T/C 10 p H 6.5 A T F U 3 0 1 T/C 10 p H 8 |  Figure 4.3.9: Effect of pH on Fouling Rate at thermocouple 10 (x = 713 mm). As discussed in Section 2.4, this effect of pH is consistent with the lysozyme molecule becoming less stable as it approaches the isoelectric point (pi = 11.1). At more acidic conditions these molecules retain an increasingly more positive charge and are therefore progressively less attracted to the positively charged stainless steel surface.  4.3.6  Effect of Enzymatic Activity As discussed in Section 3.3, the enzymatic activity of a lysozyme solution is a measure of  the concentration of native proteins still present in the bulk solution. Table 4.3.5 shows the results of the assay over the duration of T F U 310, where the slope of the graph (e.g. Figure 3.3.3) presented in Section 3.3.1 is used to calculate the enzymatic activity. The average activity for the lysozyme samples taken from T F U 310 appears to decrease throughout the course of a 7 hour experiment, one of the longest experiments performed at pH 8. These results are shown clearly in Figure 4.3.10.  4: Experimental Results and Discussion  153  Table 4.3.5: Results of enzymatic assay of lysozyme for TFU 310 (G = 297 kg/m s) 2  Sample  Time (s)  Slope  r  1  0  0.000949  0.9992  341640  lb  0  0.000972  0.9988  349920  2  4890  0.000920  0.9990  331200  2b  4890  0.000988  0.9990  355680  2c  4890  0.000943  0.9992  339480  3  10080  0.000926  0.9992  333360  3b  10080  0.000944  0.9990  339840  4  14580  0.000889  0.9986  320040  4b  14580  0.000933  0.9990  335880  4c  14580  0.000960  0.9993  345600  5  19230  0.000926  0.9988  333360  5b  19230  0.000934  0.9990  336240  6  21810  0.000872  0.9986  313920  6b  21810  0.000917  0.9989  330120  6c  21810  0.000893  0.9987  321480  7  23850  0.000898  0.9990  323280  7b  23850  0.000914  0.9988  329040  2  Activity (Units/ml)  Average Activity (Units/ml)  345780  342120  336600  333840  334800  321840  326160  Given that the specification of the lysozyme powder (Appendix 4.1) was an activity of at least 22800 Units/mg (required specification for food grade lysozyme), corresponding to 226860  4: Experimental Results and Discussion  154  Units/ml, the results found here show an activity well above this value. However, the method performed in this work is a slight variation on the method used by Inovatech (Appendix 4.2) to arrive at their result of 22800 Units/mg. 360000 350000 + S 340000 s  330000 y = -0.869x + 346172 R = 0.8597  % 320000  2  310000 300000 0  5000  10000  15000  20000  25000  Time (s) Figure 4.3.10: Average lysozyme enzymatic activity for T F U 310 Figure 4.3.10 highlights several interesting points. The error bars show that there is some scatter in the activity when the same sample is analyzed as a repeat. Note in Figure 3.3.2 the slightest variation in slope between samples 1 and lb produced significant variation in activity. It is believed that this difference is magnified by the fact that a 600 x dilution had to be achieved to perform the assay. Using the best fit least squares regression, the decrease in enzymatic activity of lysozyme for T F U 310 was approximately 6 %. Hence this result suggests that there was a decrease in concentration of native protein from 9.95 g/1 (assuming all of the initial powder is in the native form) to 9.35 g/1.  4: Experimental Results and Discussion  155  A 6 % decrease in native protein concentration over the course of a 7 hour experiment is not believed to significantly affect the nature of the results achieved, and hence the assumption of a constant bulk concentration applied to Epstein's (1994) mathematical model is valid. 4.4  Kinetic Compensation Effect (KCE) Upon completion of both the whey protein and lysozyme fouling experiments, it was  noted that there was a mutual dependence of the Arrhenius parameters (pre-exponential factor, A , and fouling activation energy, AEf) when a non-linear least squares regression was used to fit an Arrhenius type equation to the experimental data:  Rf = Aexp 0  ^-AE ^ f  (4.4.1)  V RT /  That is, when the activation energy was low, the pre-exponential factor was also low. The results of the non-linear least squares regression of the experimental data are presented in Tables 4.2.4 and 4.3.3 for the whey protein and lysozyme experiments, respectively. These results along with the styrene in kerosene polymerization data of Crittenden et al. (1987a) are shown in Figure 4.4.1. As shown in Appendix 6, the mutual dependence of the Arrhenius parameters can be eliminated if A and AEf are re-evaluated in the manner suggested by Koga (1994), such that the Arrhenius type equation is modified to  -AE  Rfo=Ae  R(Twj)  -AEf  -AEf  i  i  -AE  r  < =Ae  R R ( T  ^°  = A°e  i  f  i  ( w. T  (4.4.2)  The overall result has no effect on the fouling activation energies, but eliminates the mutual dependence of these parameters by calculation of a new pre-exponential factor (A ). The 0  4: Experimental Results and Discussion  156  mutual dependence of the Arrhenius parameters for these series of experiments appears to result when an Arrhenius type equation is used to analyze data far from the origin.  60 50  V  • WPC A  Lysozyme  •  Styrene  40  Lysozyme  Linear ( W P C ) Linear (Lysozyme)  3, s  30 20 10  WPC y = 0.3357x- 12.1597 R = 0.9992  y = 0.3437x - 10.8919 R = 0.9980 2  2  L i n e a r (Styrene)  Styrene y = 0.3322x-11.1720 R = 0.9967. 2  0 -10  20  40  60  80  100  120  140  160  180  AE (kJ/mol) Figure 4.4.1: Mutual dependence of Arrhenius parameters f  200  220  5: Mathematical Modeling and Discussion  157  5.  Mathematical Modeling and Discussion  5.1  Development of FORTRAN 77 Program The model equation, Equation (2.2.22), required to either validate or disprove Epstein's  mathematical model (1994) is repeated below:  (2.2.22)  The experimental results from either the whey protein or lysozyme solution fouling experiments were used to evaluate best fit values of the constants k,k,AEand n in the above implicit equation (assuming n ^ 1). The precursor diffusivity required to calculate the Schmidt number in Equation (2.2.22) is unknown. In the Stokes-Einstein relationship for a dilute suspension of spheres (Section 4.2), the solution diffusivity is related to the viscosity and temperature by  and hence the precursor diffusivity is proportional to the solution temperature divided by the viscosity:  (5.1.1)  The same approach was employed by Wilke and Chang (1955) for dilute solutions, where the solute diffusivity was correlated to the properties of the solvent, solute and solution. Therefore Equation (2.2.22) is modified using Equation (5.1.1) and becomes  k, Rf  0  vr +k|Rf 1  0  (5.1.2)  5: Mathematical Modeling and Discussion  158  where  (5.1.3)  D 2/3  For each experimental data point (i.e. Rfo, T , T and V.), Equation (5.1.2) can be simplified to: b  w  kR  f 0  ~|/n  AE/  C^k.RfcC  +  Be  / r t  (5.1.4)  »  where 2/3  C=  ^PT;,  v.  and  (5.1.5)  B  These expressions for C and B are unique to each data point and are used as the input data for the mathematical program to solve Equation (5.1.4) for the unknowns, k,, k, A E , andn. This input data is shown in Table 5.2.1 for whey protein fouling, and Table 5.3.1 for lysozyme fouling. Solution of Equation (5.1.4) for N sets of data and j variables ( k i , k , A E , n ) can be achieved using a non-linear least squares regression, by minimizing c in 2  N  a = 2  2\  R  f o  (expt) - Rfo (model)  i=l  (N-j)  (5.1.6)  The Levenberg-Marquardt method (Press et al., 1986), as previously employed by Vasak and Epstein (1996), uses a combination of Newton's method and a steepest-descent method to perform non-linear curve-fitting to the experimental data. This procedure was employed here  5: Mathematical Modeling and Discussion  159  using F O R T R A N 77. The algorithm for the Levenberg-Marquardt method and the corresponding computer program are shown in Appendices 5.1 and 5.2 respectively. Once the best solution of k i, k, and AE has been evaluated for each reaction order, n (i.e. from the minimum sum of the squares of the residuals), the optimal solution for all data can be determined by the overall minimum of the sum of the squares of the residuals for A L L reaction orders. For this solution of the vector a (k i, k, AE and n), the model is then capable of predicting initial fouling rates at given wall and bulk temperatures, over a range of fluid velocities, and therefore can be quantitatively compared to the experimental data. 5.2  Modeling Whey Protein Solution Fouling Potentially ninety data points from nine whey protein fouling experiments at pH 6.2 - 6.3  using ten surface thermocouples were available for modeling. A considerable number of data points either failed to meet the fouling Biot number criterion (Bi > 0.05) or produced obviously erroneous results (due to malfunctioning thermocouples), and therefore 59 data points were initially modeled. 5.2.1  Input Data To perform a variety of analyses, several input files were prepared. These were achieved  by converting raw experimental data from text delimited files to Microsoft Excel spreadsheets for data manipulation. A sample spreadsheet used to evaluate the constants C and B for Equations (5.1.4) and (5.1.5) is shown in Table 5.2.1. These files were subsequently saved as comma delimited text files which F O R T R A N was capable of reading. The following data files were used.  5: Mathematical Modeling and Discussion  160  MODEL2.DAT: A l l 59 experimental data points. The film temperature was used for both mass transfer and chemical attachment terms. MODEL3.DAT: A l l 59 experimental data points. The film temperature was used for mass transfer, and wall temperature was used for chemical attachment. MODEL4.DAT: 47 experimental data points. Analysis using temperatures as in M O D E L 3 . D A T revealed two experiments (TFU 204 and T F U 212) with considerable data scattered from the observed trend. These data prevented the model from converging to an optimum solution and were deleted from the input file to permit further analysis. A summary of experimental data used in the analysis is shown in Figure 5.2.1. As indicated above, T F U 204 and T F U 212 demonstrate unusually low fouling rates compared to experimental data for mass fluxes both above and below values for these runs. The Reynolds number based on local bulk properties ranged from 2310 to 16993, and clean inside wall temperatures varied from 68 to 102°C. The bulk inlet temperature was approximately 30°C. 3.0E-05 jg  TFU 211 T F U 208  2.5E-05  rn  £  TFU207 B  O  2.0E-05  TFU206  • •  T F U 209 n  BX)  1.5E-05  •  a " £©3 1.0E-05  o o B  a  •  5.0E-06  • TFU204/212 5 n •  •  D  • • •  TFU205 • • T F U 210  O.OE+00 0  200  400  600  800  1000  1200  1400  Mass Flux (kg/ms) 2  Figure 5.2.1: Experimental data used for modeling whey protein fouling (59 data points)  5: Mathematical Modeling and Discussion  161  Table 5.2.1: Spreadsheet used to evaluate the constants C and B for whey protein modeling TJFU T . / Q 204 81.9 85.7 85.0 85.5  89.1 89.6 205 77.7 77.8 80,8 83.2 83.3 827 87.3 87.7 206 849 844 87.2 886 884 91.0 90.5 207 763 83.6 821 847 87.8 881 87.3 91.0 89.8 203 882 925 935 986 97.6 209 825 86.4 867 882 9L6 9L4 210 768 81.9 81.5 211 67.9 75.0 75.2 826 89.2 924 94.6 100.9 101.5 212 83.4 863 860 85.8 883 901  355.1 3588 3582 3587 3623 3627 3508 3509 353.9 355,3 3564 355.8 360.4 3608 3580 357.5 3603 361.7 361.5 364.1 363.6 349.4 3567 355.2 357.8 3609 3612 360.4 364.1 3629 361.3 365.6 365.6 371.7 370.7 355.6 359.5 359.8 361.3 364.7 364.5 349.9 355.0 351.6 341.0 3481 3483 355.7 3623 365.5 367.7 374.0 374.6 356.6 359.5 359.1 3589 361.5 363.3  TXQ 39.7 427 45.6 47.1 486 5Q0 35.5 37.0 39.9 429 45.9 47.3 488 502 385 421 45.8 49.5 51.4 53.2 55.0 345 362 382 418 45.4 49.1 5Q9 528 545 421 448 461 47.5 488 4Q7 43.9 47.2 488 504 520 445 45.8 46.9 33.3 344 35.6 37.9  40.1 424 43.5 44.7 45.8 40.5 43.6 466 482 49.7 51.2  T«K) T{K) XCQ 3128 333.9 6Q8 315.8 337.3 642 3187 3384 65.3 320.2 339.4 663 321.7 3420 688 323.1 3429 69.8 3086 329.7 566 3101 3305 57.4 3D.0 333.5 603 3160 3362 a o 319.0 337.7 64.6 320.4 3381 65.0 321.9 341.2 680 323.3 3421 689 311.6 334.8 61.7 3152 3364 S2 3189 339.6 665 3226 3422 69.0 324.5 343.0 69.9 3263 345.2 721 3281 345.9 727 307.6 3285 554 309.3 333.0 59.9 311.3 333.3 60.1 314.9 3364 63.2 3185 339.7 666 3222 341.7 686 324.0 3422 69.1 325.9 345.0 719 327.6 345.3 721 315.2 3383 65.1 317.9 341.8 686 319.2 3429 69.8 320.6 3462 73.0 321.9 3463 732 313.8 334.7 616 317.0 3383 65.1 3203 340.1 669 321.9 341.6 685 323.5 344.1 7L0 325.1 344.8 7L7 317.6 333.8 606 3189 337.0 63.8 320.0 3373 642 305.4 323.7 506 307.5 327.8 54.7 3087 3285 55.4 311.0 333.4 60.2 3D.2 337.8 646 315.5 340.5 67.4 316.6 3422 69.0 317.8 345.9 728 3189 345.8 73.6 313.7 335.1 620 3167 3381 64.9 319.8 339.5 663 321.3 340.1 67.0 3228 3422 69.0 324.3 343.8 707  ROigfW) 973.75 971.32 971.74 971.41 969.02 96870 97646 97640 974.51 97297 97290 97329 970.28 970.01 971.86 97219 970.34 969.41 959.54 967.78 96812 977.32 97271 973.68 971.99 959.94 969.74 970.28 967.78 96860 969.67 965.75 96505 96246 963.18 973.42 970.87 97)67 969.67 967.37 967.51 977.02 973.81 974.06 98230 97812 97800 973.36 969.00 96582 965.29 960.81 960.37 97278 970.91 971.12 971.24 969.55 96834  n(kgfai) prfkgV) 4.073BO4 985.18 3.977504 984.36 3.991504 983.74 3.98Q604 983.18 3.920BO4 981.74 3.914B04 981.20 4.226E04 98837 4222604 987.97 4.111604 985.44 4.038EO4 985.00 4.035BO4 984.15 4.052604 983.93 3.947604 98221 3.940BO4 981.70 3.995E04 985.72 4.007BO4 984.89 3.949604 983.09 3.927B04 981.64 3.930504 981.15 3.901BO4 979.86 3.905BO4 979.47 4386504 98897 4.027604 98568 4069604 98555 40X604 984.89 3.939504 983.04 3.934604 981.90 3.947604 981.61 3.901604 979.98 3.912604 979.83 3.933604 983.85 3.892604 981.87 3.8S9604 981.21 3.912604 979.29 3.932504 979.20 4058604 985.78 3.S63504 983.85 3.958504 98284 3.933504 981.96 3.897504 980.51 3.898604 983.09 4264604 98628 4.076504 984.56 4.088504 984.37 4751604 991.27 4346504 989.32 4.337504 98897 4055504 98549 3919504 984.12 3.892604 98258 3.889604 981.64 3.943604 979.44 3.953504 97893 4030504 985.55 3.964604 983.93 3.971504 983.17 3.975504 98280 3.930504 981.63 3.908504 980.68  v*(nVs)diy<ll(nrK<J) f C T&HM VC<J,)PbOsM Qkg'iif's) V-talKs) 5.281504 6787 995.89 401.41 04031 0.00876 O.Q2668 288509 3.355507 5.012604 7147 994.71 401.41 Q4035 O.00863 0.02651 3.97509 3.132507 4.928504 7273 993.50 401.41 0.4040 0.00859 0.02647 3.97609 3.061607 4.857504 7380 99285 401.41 0.4043 0.00855 0.02643 4.26509 3.002507 4.687504 7641 99218 401.41 0.4046 000846 0.02631 5.48509 2864507 4.629604 7738 991.55 401.41 Q4048 0.00843 0.02628 5.66509 2816507 5.662504 16660 997.42 1055.63 1.0584 0.00683 005184 1.66609 1.595607 5.586504 16887 99589 1055.63 1.0589 0.00680 0.05177 209509 1.571507 5.323604 17714 995.81 1055.63 1.0501 0.00672 006145 232609 1.473607 5.101504 18478 994.63 1055.63 1.0613 0.00665 0.06118 3.07609 1.392607 4.983604 18925 993.37 1055.63 1.0627 0.00661 0.06108 3.85609 1.348507 4.953604 19046 99276 1055.63 1.0633 0.00660 0.06107 4.35609 1.337607 4.741504 19877 99209 1055.63 1.0640 Q00652 005077 5.77609 1.261507 4.683604 20126 991.46 1055.63 1.0647 000650 0.05071 7.84509 1.240607 5.210604 11975 936.34 69937 0.7)19 0.00745 0.04285 5.41509 2048507 5.086504 12274 994.95 699.37 07029 0.00741 004277 5.21509 1.982607 4.846604 12878 993.41 699.37 07040 000731 0.04256 1.03608 1.858E07 4.677604 13347 991.78 699.37 0.7052 0.00724 0.04243 1.54508 1.771507 4.624504 13504 99090 65937 0.7058 0.00722 0.04240 1.59608 1.743507 4.496501 13882 990.04 699.37 07064 Q00716 0.04228 207508 1.678507 4.461604 13999 989.17 699.37 07070 0.00715 0.04227 222608 1.659607 5.778604 8224 997.76 531.65 05328 0.00829 0.03430 5.55509 2959507 5352504 8847 997.18 531.65 Q5332 000811 0.03395 862609 2654507 5.340604 8888 59545 531.65 05335 000810 003396 1.06608 2678507 5.086504 9330 995.07 531.65 05343 Q00799 003377 1.18608 2511507 4.839504 9802 993.59 531.65 0.5351 0.00788 0.03359 1.43508 235CB07 4705504 10085 991.96 531.65 05360 0.00782 003351 1.60608 2262507 4673504 10159 991.13 531.65 05364 0.00780 Q03351 1.64508 2240507 4.507604 10526 99024 531.65 05369 000773 003337 236508 2134607 4.493604 10556 989.41 531.65 0.5373 0.00772 0.03338 232508 2124507 4942604 5125 99495 284.07 02855 000953 0.01971 696509 4.128607 4.702504 5382 993.84 284.07 0.2858 000939 0.01958 1.03608 3.866507 4630504 5455 993.29 284.07 0.2860 0.C0935 0.01955 1.58508 3.787507 4.445504 5685 99267 28407 0.2862 0.00923 0.01944 209608 3.589B07 4437604 5698 99209 284.07 0.2863 0.00922 0.01945 229608 3.578507 5.218504 15097 995.50 88227 0.8863 0.00701 0.05246 231609 1.677507 4.942604 15929 994.22 88227 0.8874 000691 0.05215 4.34609 1.550507 4815604 16357 99281 88227 Q8887 0.00685 0.05204 668609 1.505507 4712604 16712 99209 88227 0.8893 000682 0.05194 9.49609 1.452507 4558604 17261 991.36 88227 08900 000676 005175 1.35608 1.359507 4.518504 17420 99062 88227 08906 0.00575 0.05173 1.73508 1.381507 5.297504 21533 993.97 1274.85 1.2826 0.00640 Q07253 9.44610 1.240607 5.039604 22608 993.41 1274.85 1.2833 0.00632 0.07211 1.30509 1.160507 5.013604 22734 99294 1274.85 1.2839 000631 0.07210 1.53509 1.152507 6279504 3158 99816 221.43 0.2218 0.01115 0.01657 4.63509 6925607 5.847504 3386 997.80 221.43 0.2219 0.01089 0.01638 656509 6.325607 5.778604 3427 997.39 221.43 02220 0.01085 0.01635 7.30509 6227507 5331604 3707 99656 221.43 02222 0.01057 0.01615 9.39509 5.618607 4.979504 3953 995.74 221.43 0.2224 0.01034 0.01599 1.06508 5.144607 4784504 4122 994.83 221.43 0.2226 0.01021 0.01590 135508 4.883607 4.677504 4215 994.38 221.43 02227 0.010D 0.01585 1.39508 4.740607 4.458504 4414 993.88 221.43 0.2228 0.00998 0.01574 1.88506 4.453607 4.413604 4458 993.41 221.43 0.2229 0.00995 0.01572 223508 4.393607 5.183604 6895 995.55 400.42 0.4022 0.00872 O.0B656 211509 3281507 4.954504 7212 994.33 400.42 0.4027 0.00861 0.02642 3.70509 3.091607 4.856604 7361 993.03 400.42 0.4032 0.00856 0.02637 4.65509 3.008507 4.810604 7435 99236 40042 0.4035 0.00853 0.02635 5.49509 2969607 4.676504 7643 991.67 400.42 0.4038 0.00845 0.02626 7.39E09 2860607 4.576504 7808 990.98 400.42 04041 000841 0.02620 873509 2778607 !  B 1701.8 1715.9 17054 1705.5 17120 1709.8 88358 8823.3 8951.1 9019.9 8993.6 8957.4 90768 90727 4457.5 4439.1 44512 4443.5 44353 44342 4429.4 26826 2784.6 27593 2771.7 2777.8 2767.8 2759.7 27622 27588 9582 9528 949.4 929.8 933.4 6601.1 6661.8 6643.2 6650.4 66483 6641.8 120528 12425.5 123864 567.4 603.6 603.0 626.2 6320 6280 623.6 603.5 600.4 17026 17089 1701.0 1697.1 1701.7 1700.7  5: Mathematical Modeling and Discussion 5.2.2  162  Mathematical Model Predictions Modeling the experimental data required consideration of two factors related to  temperature dependence of physical properties: Effect of non-isothermal flow on friction factor Friction factors are typically correlated for isothermal flow; therefore applying the Gnielinski (1976) equation, f = 0.25(1.821og Re-1.64)  (5.1.7)  -2  10  to non-isothermal data caused concern about its applicability. For dG/t) < 2100, McAdams (1954) recommended evaluation of the friction factor for hydrocarbon streams in non-isothermal flow at T ' , where T' = Tb + (T -Tb)/4. For dG/n > 2100, evaluation of the friction factor was w  recommended at T " , where T" = Tb + (T -Tb)/2. w  In essence, using such film temperatures accounts for the non-isothermality of the fluid flow. Since all experimental data in the present work were at dG/n > 2100 the film temperature was calculated as the arithmetic mean of T and Tb. w  0.013 0.012  Blasius:  f=0.0791Re -°(  Re > 4000  25)  GnieUnski: f = 0.25(1.821og Re -1.64)"  2  c  o  0.011  1()  Re > 2300  1 0.010 |  0.009  u  £ 0.008 0.007 0.006 5000  10000  15000 Re  Figure 5.2.2: Friction factor correlations  20000  25000  5: Mathematical Modeling and Discussion  163  The Gnielinski friction factor correlation shown in Figure 5.2.2 was employed in this work since this correlation, unlike the more conventional Blasius equation, also shown in Figure 5.2.2, is reported to be valid to the low end of the transition Reynolds numbers. This will be used to evaluate the fluid friction velocity in Equation (5.1.5). Effect of fluid physical properties on the model results Another important consideration in Equation (5.1.2) was whether the bulk temperature, wall temperature or some mathematical average of these two temperatures should be used to evaluate the fluid physical properties. Since mass transfer of the fouling precursor to the vicinity of the heat transfer surface occurs in the momentum boundary layer, between the bulk solution and the surface, the logical temperature to use was the film temperature. In this case the arithmetic average film temperature, Tf = {(Tb+ T )/2} was employed to evaluate the fluid density, viscosity and Reynolds number. w  The chemical attachment component was more complicated. The exponential term is assumed to describe a surface phenomenon; however, there was uncertainty as to whether the physical properties should be evaluated at the film or wall temperature.  (1) Film temperature for both mass transfer and chemical attachment terms Using all experimental data subject to the fouling Biot number criterion (Bi > 0.05), Table 5.2.2 shows that as the reaction order increases the sum of the squares of the residuals decreases and hence the "goodness offit" increases when the film temperature is employed for both mass transfer and chemical attachment terms. As shown in Figure 5.2.3, this "goodness of fit"  increases in an asymptotic fashion, indicating that no realistic optimum solution exists.  5: Mathematical Modeling and Discussion  164  Reaction orders above three are unlikely, but higher orders were used in the search for an optimum. Table 5.2.2: Modeling results using all whey protein experimental data and input file Model2.dat (Film temperature used for both mass transfer and chemical attachment) (UNITS: k =  kg  (.•0/ vVKs  m K s J  Order n  k, (x lO )  k  0.5  2.0182  1.0  / n  A E (kJ/mol)  5  )  80  0.2366 x 10"  311.8  Sum of squares of residuals • for Rfo (m K/J) (x 10 ) 1.3540  1.7012  0.1903 x l O  -39  310.7  1.1643  1.5  1.5242  0.8089 x 10"  320.0  1.0582  2.0  1.3936  0.1247 x 10"  20  331.2  0.9937  2.5  1.2882  0.7450 x l O "  17  341.3  0.9576  3.0  1.1996  0.2915 x 10"  14  349.9  0.9316  4.0  1.0566  0.7128 x 10"  11  363.2  0.9001  5.0  0.9449  0.1003 x 10'  372.6  0.8823  10.0  0.6178  0.4768 x lO"  394.0  0.8521  -15  2  27  4  8  2  15  (2) Film temperature for mass transfer and wall temperature for chemical attachment Table 5.2.3 highlights the modeling results using M O D E L 3 . D A T as the input file. Comparison of Table 5.2.2 with Table 5.2.3 shows that a better model solution is always obtained using the wall temperature to evaluate the physical properties for the chemical attachment term, and the film temperature for the mass transfer term. The differences in sums of  165  S: Mathematical Modeling and Discussion  squares are small at a given reaction order, and the fit is about the same to two significant figures. Table 5.2.3: Modeling results using all whev protein experimental data and input file Model3.dat (Film temperature used for mass transfer and wall temperature used for chemical attachment)  -  k  (UNITS: k =  ( n +  ^  w  u  e  -  , k, =  ( ks* ^ °  ,  )  Order n  k, (x 10 )  k  0.5  2.0151  0.6840 x 10"  295.9  Sum of squares of residuals • for Rfo (m K/J) (x 10 ) 1.3444  1.0  1.6949  0.2858 x 10"  38  295.1  1.1541  1.5  1.5163  0.2183 x 10"  25  304.7  1.0485  2.0  1.3848  0.1483 x 10'  19  315.8  0.9874  2.5  1.2788  0.5486 x 10'  16  325.8  0.9492  3.0  1.1898  0.1566 x 10"  13  334.3  0.9236  4.0  1.0463  0.2581 x l O  - 1 0  347.3  0.8926  5.0  0.9345  0.2859 x l O "  08  356.4  0.8752  10.0  0.6086  0.8252 x 10"  377.1  0.8456  15  AE (kJ/mol)  2  76  04  2  15  More importantly, neither method illustrates an optimal reaction order for a model analysis containing all experimental data that satisfy the fouling Biot number criterion. Reaction orders previously reported in the literature are in the range of 1.0 - 2.5 (Dannenberg and Kessler, 1988; DeWitt and Klarenbeek, 1988; De Jong et al., 1992). Figure 5.2.3 shows that the sum of the squares of the residuals reduces in asymptotic fashion as the reaction order increases. Kinetic  S: Mathematical Modeling and Discussion  166  reaction orders do not generally exceed three; however, a wide range was tested here (up to n 100) in search of an optimum.  1.4E-15 S  1.3E-15  ja  1.2E-15  -+->  § S3 1.1E-15 %S <u  1.0E-15 9.0E-16  S  8.0E-16 10 Reaction order (n) Figure 5.2.3: Non-existence of an optimum solution for all 59 experimental data points using Model3.dat Figure 5.2.4 shows the best solution to all experimental data for a reaction order, n = 2.  I  2.5E-08 o 204  -  206 208 X 210 0 212 •  2.0E-08  •  X  •  205 207 209 211 Y=X  M 1.5E-08 S  .2  1.0E-08  °8P°°  'S  ~  5.0E-09  T3 O  0.0E+00 0.0E+00  5.0E-09  1.0E-08  1.5E-08  2.0E-08  Experimental Initial Fouling Rate (m K/J) 2  Figure 5.2.4: Best model solution for n = 2, N = 59  2.5E-08  5: Mathematical Modeling and Discussion  167  Clearly there is considerable scatter around the 45° line. The data well above the line (shown as open circles) originate from experiments T F U 204 and T F U 212, which contribute to much of this scatter. This scatter prevents determination of an optimum solution over a range of credible reaction orders. For this reason, a third input file (MODEL4.DAT) was created excluding the data from T F U 204 and T F U 212 to determine the validity and application of the aforementioned model. For this input file, the film temperature was used to evaluate the physical properties for the mass transfer term, and the wall temperature was used to evaluate the physical properties for the chemical attachment term. The results are shown in Figure 5.2.5. 3.30E-16 s  3.25E-16  >CU  °§ £  3  &  w  3.20E-16  3.15E-16  cn cu PS  ° S s  3.10E-16  CO  3.05E-16 0.0  0.5  1.0  1.5  2.0  2.5  Reaction order (n) Figure 5.2.5: Optimum reaction order for whey protein solution fouling (47 data points) The optimal solution for this set of data occurs at a reaction order n = 0.99, activation  energy A E = 200.9 kJ/mol, k, = 1.3616 x 10  and k =0.5552 x10  15  U ' K s V  kg  •24  m K 3  2.01  1 0 1  s  1 0 1  These reaction order and activation energy results fall well within the range suggested from kinetic experiments. Lyster (1970), and more recently Dannenberg and Kessler (1988),  5: Mathematical Modeling and Discussion  168  determined kinetic reaction orders and activation energies for a-lactalbumin and P-lactoglobulin. They showed that for P-lactoglobulin the reaction order was 1.5 - 2.0, and the activation energy decreased from approximately 280 kJ/mol at 68 - 90°C, to approximately 48 kJ/mol at 90 150°C. For a-lactalbumin, the reaction order was 1.0, and the activation energy decreased from 270 kJ/mol at 70 - 80°C, to approximately 69 kJ/mol at 85 - 150°C. It is usually assumed that plactoglobulin ( T D « 81°C at neutral pH) is the dominant species in whey protein denaturation and aggregation reactions, due to its higher concentration in whey protein. However, an optimum reaction order of approximately 1.0 from these modeling results suggests that the thermal denaturation and aggregation of a-lactalbumin ( T D « 64°C at neutral pH) may be more significant than first anticipated. A similar analysis was performed using the film temperature to evaluate the physical properties for both the mass transfer and chemical attachment terms for this reduced set of 47 data points. However, the overall minimum sum of squares of the residuals at a reaction order of 0.96 was 3.23693 x 10" (m K/J) , which is 4.7 % greater than the minimum shown in Figure 16  2  2  5.2.5. A plot comparing the experimental and model predictions for initial fouling rate is shown in Figure 5.2.6. It is clear that there is considerably less scatter around the 45° line than in Figure 5.2.4. This latter set of results provides partial justification for eliminating T F U 204 and T F U 212 from the input data file to achieve the optimum solution. Not only has a minimum sum of squares been successfully achieved, but the results are comparable to those reported in the literature for both the activation energy and reaction order for the purely kinetic denaturation and aggregation steps of P-lactoglobulin and a-lactalbumin.  169  5: Mathematical Modeling and Discussion  I S  2.5E-08  * X  2.0E-08  •  205 207 209 211  -  206 208 X 210 -Y=X 4  Pi ex 1.5E-08 D  ^  1.0E-O8  is -  tu T3  5.0E-O9  O  O.OE+00 0.0E+00  5.0E-09  1.0E-08  1.5E-08  2.0E-08  2.5E-08  Experimental Initial Fouling Rate (mK/J) Figure 5.2.6: Best solution for n = 0.99, N = 47 Figure 5.2.7 shows the strong effect of wall temperature on the initial fouling rate over a range of fluid velocities, according to the model predictions. Predictions are shown at T  w  = 80,  88 and 95°C. Also, note the strong effect of bulk temperature: shown at each wall temperature are predictions for bulk temperatures of 31 and 57°C, which were the minimum and maximum values used in the experiments. In this analysis the physical properties for the mass transfer term are evaluated at the film temperature, while those for the chemical attachment term use the clean inside wall temperature. From Figure 5.2.7, results at a constant wall temperature show that the bulk temperature has a considerably greater effect in the mass transfer controlled region than in the attachment region. Hence, although bulk temperatures were kept low to eliminate bulk reactions, the bulk temperature influences the film temperature, and hence the balance between mass transfer and chemical attachment.  5: Mathematical Modeling and Discussion  170  The cut-off Reynolds number (Re = 2300), shown approximately in Figures 5.2.7 and 5.2.8, was calculated using film properties for an intermediate wall (88°C) and bulk (30°C) temperature combination. Using Equation (5.1.6), the variance for the optimal solution was 0.7025 x 10"  11  (m K/kJ) . As shown in Appendix 5.5, the average absolute percent deviation (AAD) and root mean square percent deviation (RMS) were greater than those reported by Vasak and Epstein (1996). The A A D was 24.5 % for this work, which compares with 16.7 % in the latter study using all of the styrene in kerosene polymerization data of Crittenden et al. (1987a).  3.0E-08 5  T„ = 57°C  '  £ 2.5E-08  T = 95°C W  T = 31°C B  "S 2.0E-08 Pi  SD  3oO 1.5E-08 ta  : i I.OE-08 -a J 5.0E-09 CJ  0.0E+00  r  <2300  200  400  600  800  1000  1200  1400  1600  Mass Flux (kg/m s)  Figure 5.2.7: Effect of wall temperature on model solution Although slightly inferior to that of Vasak and Epstein (1996), the model fit of the whey protein solution fouling data displays the same trends and observations that are crucial for model verification (Epstein, 1994), i.e. a maximum in predicted initial fouling rate with mass flux, and a shift of the location of the maximum to higher mass fluxes as the wall temperature is increased.  S: Mathematical Modeling and Discussion  171  Figure 5.2.8 is a comparison plot of the experimental whey protein data from the Arrhenius plots of Figure 4.2.10 to the corresponding optimum model prediction from the 47 data points o f M O D E L 4 . D A T . The greatest deviations between experiment and predictions occur at the highest wall temperature, and at a low mass flux of 284 kg/m s. However, despite these deviations, the model and the data points follow a similar trajectory. 3.0E-08  ?  * E  2.5E-08  1 PS  2.0E-08  • Tw = 92°C A Tw = 88°C • Tw = 84°C  T„ = 31  DX)  T = 92°C  c •"S 1.5E-08 o ta ;| "3  1.0E-08  •g  5.0E-09  w  Re <2300  O.OE+00 0  200  400  600  800  1000  1200  Mass Flux (kg/m s) Figure 5.2.8: Comparison of model predictions to WPC experimental data obtained from the Arrhenius type equations  1400  1600  5: Mathematical Modeling and Discussion 5.3  172  Modeling Lysozyme Fouling Potentially,  ninety data points corresponding to the ten local wall  temperature  measurements from nine lysozyme fouling experiments at pH 8 were available for modeling. In comparison to the whey protein solution experiments, fouling for this solution was considerably more rapid, and hence Rf values were larger. It is therefore not surprising that 86 of the possible 90 data points satisfied the fouling Biot number criterion (Bi > 0.05) and were therefore used for modeling purposes. The F O R T R A N program developed for modeling whey protein fouling was used here with few modifications. As with whey protein fouling, it was uncertain whether the film temperature should be used to evaluate the physical properties for both the mass transfer and the chemical attachment terms, or whether the film temperature should be used for mass transfer and the wall temperature for chemical attachment. Therefore this problem was again addressed as two separate cases to determine the best solution for all reaction orders by minimizing the variance in Equation (5.1.6). 5.3.1  Input Data To determine the best combination of bulk and wall temperatures to evaluate the fluid  physical properties, two input files were developed. Table 5.3.1 shows values from a sample spreadsheet from which the data were collected and transferred for use by the program. This spreadsheet involved the necessary calculations to evaluate the constants C and B for each data point as described by Equations (5.1.4) and (5.1.5). T  W;i  is the clean inside wall temperature, T  b  the corresponding bulk temperature at a given thermocouple location, and G the time average mass flux over the duration of the experiment.  5: Mathematical Modeling and Discussion  173  Table 5.3.1: Spreadsheet used to evaluate the constants C and B for lysozyme modeling TFU  Tw,ifC)  Tw,i(K)  T.fC)  301  64.90 70.44 73.74 73.94 74.10 76.51 77.23 77.85 77.88 79.84 66.42 67.80 68.30 68 70 74.89 75.26 74.95 76.74 79.33 63 81 68.53 72.80 71.58 74.48 76.76 77.31 77.26 78.08 79.47 67.86 70.56 70.40 71.70 74.07 74.73 74.81 76.05 77.24 63.60 68.09 72.43 74.66 76.63 81.12 81.66 81.02 80.68 79.93 62.59 67.43 71.34 71 69 74.51 7677 77.16 77.41 79.21 80.11 68.46 71.73 70.57 72.28 74.47 75.74 76.67 76.62 78.92 59.06 64.25 65.72 70.57 76.73 79.50 80.33 82.47 84.10 62.67 66.74 71.50 73.02 77.35 80 15 79.69 80.17 80.11 80.95  338.05 343.59 346.89 347.09 347.25 349.66 350.38 351.00 351.03 352.99 339.57 340.95 341.45 341.85 348.04 348.41 348.10 349.89 352.48 336.96 341.68 345.95 344.73 347.63 349.91 350.46 350.41 351.23 352.62 341.01 343.71 343.55 344.85 347.22 347.88 347.96 349.20 350.39 336.75 341.24 345.58 347.81 349.78 354.27 354.81 354.17 353.83 353.08 335.74 340.58 344.49 344.84 347.66 349.92 350.31 350.56 352.36 353.26 341.61 344.88 343.72 345.43 347.62 348.89 349.82 349.77 352.07 332.21 337.40 338.87 343.72 349.88 352.65 353.48 355.62 357.25 335.82 339.89 344.65 346.17 350.50 353.30 352.84 353.32 353.26 354.10  31.57 33.25 34.69 36.26 39.21 42.19 45.17 46.66 48.15 49.58 33.26 34.51 35.88 38.45 41.04 43.64 44.94 46.23 47.48. 31.69 33.20 34.49 35.90 38.54 41.22 43.89 45.22 46.56 47.85 33.14 34.38 35.75 38.32 40.90 43.49 44.79 46.08 47.33 32.12 33.67 34.98 36.43 39.14 41.87 44.61 45.98 47.35 48.67 31.83 33.39 34.73 36.19 38.93 41.71 44.48 45.86 47.25 48.59 33.72 34.98 36.36 38.94 41.55 44.16 45.46 46.77 48.03 32.63 33.41 34.27 35.88 37.50 39.12 39.93 40.75 41.53 32.23 33.77 35.09 36.53 39.25 41.98 44.72 46.09 47.45 48.77  T.(K) T K K ) TrfC)  304.72 306.40 307.84 309.41 312.36 315.34 318.32 319.81 321.30 322.73 306.41 307.66 309.03 311.60 314.19 316.79 318.09 319.38 320.63 304.84 306.35 307.64 309.05 311.69 314.37 317.04 318.37 319.71 321.00 306.29 307.53 308.90 311.47 314.05 316.64 317.94 319.23 320.48 305.27 306.82 308.13 309.58 312.29 315.02 317.76 319.13 320.50 321.82 304.98 306.54 307.88 309.34 312.08 314.86 317.63 319.01 320.40 321.74 306.87 308.13 309.51 312.09 314.70 317.31 318.61 319.92 321.18 305.78 306.56  321.39 325.00 327.37 328.25 329.81 332.50 334.35 335.41 336.17 337.86 322.99 324.31 325.24 326.73 331.12 332.60 333.10 334.64 336.56 320.90 324.02 326.80 326.89 329.66 332.14 333.75 334.39 335.47 336.81 323.65 325.62 326.23 328.16 330.64 332.26 332.95 334.22 335.44 321.01 324.03 326.86 328.70 331.04 334.65 336.29 336.65 337.17 337.45 320.36 323.56 326.19 327.09 329.87 332.39 333.97 334.79 336.38 337.50 324.24 326.51 326.62 328.76 331.16 333.10 334.22 334.85 336.63 319.00 321.98  307.42 309.03 310.65 312.27 313.08 313.90 314.68 305.38 306.92 308.24 309.68 312.40 315.13 317.87 319.24 320.60 321.92  323.15 326.38 330.27 332.46 333.28 334.76 335.97 32060 323.41 326.45 327.93 331.45 334.22 335.36 336.28 336.93 338.01  48.24 51.85 54.22 55.10 56.66 59.35 61.20 62.26 63.02 64.71 49.84 51.16 52.09 53.58 57.97 59.45 59.95 61.49 63.41 47.75 50.87 53.65 53.74 56.51 58.99 60.60 61.24 62.32 63.66 50.50 52.47 53.08 55.01 57.49 59.11 59.80 61.07 62.29 47.86 50.88 53.71 55.55 57.89 61.50 63.14 63.50 64.02 64.30 47.21 50.41 53.04 53.94 56.72 59.24 60.82 61.64 63.23 64.35 51.09 53.36 53.47 55.61 58.01 59.95 61.07 61.70 63.48 45.85 48.83 50.00 53.23 57.12 59.31 60.13 61.61 62.82 47.45 50.26 53.30 54.78 58.30 61.07 62.21 63.13 63.78 64.86  pAg/n.')  991.7 990.2 989.1 988.7 988.0 986.7 985.8 985.2 984.9 984.0 991.0 990.5 990.1 989.4 987.4 986.6 986.4 985.6 984.7 991.9 990.6 989.4 989.3 988.1 986.9 986.1 985.8 985.2 984.5 990.8 989.9 989.6 988.8 987.6 986.8 986.5 985.8 985.2 991.9 990.6 989.3 988.5 987.4 985.6 984.8 984.6 984.3 984.2 992.2 990.8 989.7 989.2 988.0 986.7 986.0 985.6 984.8 984.2 990.5 989.5 989.5 988.5 987.3 986.4 985.8 985.5 984.6 992.7 991.5 991.0 989.6 987.8 986.7 986.3 985.6 985.0 992.1 990.9 989.5 988.9 987.2 985.8 985.3 984.8 984.5 983.9  C (keta'i)  n,(k£/ms) RefT,) 5.801E-04 5.457E-04 5.264E-04 5.198E-04 5.089E-04  4.920E-04 4.8I7E-04 4.762E<4 4.724E-04  4.645E-04 5.640E-04 5.518E-04 5.436E-04 5.314E-04 5.004E-04 4.9I4E-04 4.886E-04 4.802E-04 4.705E-04 5.853E-04 5.544E-04 5.308E-04 5.301E-04 5.098E-04 4.94IE-04 4.849E-04 4.815E-04 4.759E-04 4.693E-04 5.S78E-04 5.404E-04 5.354E-04 5.204E-04 5.034M4 4.934E-04 4.894E-04 4.824E-04 4.760E-O4 5.84IE-04 5.543E-04 5.303B-O4 5.165 E-04 5.009&-04 4.801E-O4 4.718E-04  4.701 E-04  4.676E-04 4.663&04 5.911E-04 5.586E-04 5.357E-04 5.285E-04 5.084E-04 4.926E-04 4.837E-04  4.794E-04 4.714E-04 4.661 E-04 5.524E-04 5.33IE-04 5.322E-04 5.161E-04 5.001E-04 4.885E-04 4.824E-04 4.79IE-04 4.702E-04 6.067E-04 5.740E-O4 5.625E-04 5.342E-04 S.058E-O4 4.922E-04 4.875E-04 4.795E-04 4.734E-04 5.885E-04 5.601 E-04 5.336E-04 5.222E-04 4.983E-04 4.824E-04 4.764 E-04 4.719E-04 4.687E-04 4.638E-04  10821 11492 11907 12060 12322 12742 13018 13168 13277 13501 13960 14267 14483 14820 15720 16011 16108 16386 16713 8011 8450 8818 8835 9181 9472 9654 9725 9839 9975 17685 18247 18420 18950 19585 19987 20154 20444 20714 4557 4798 5011 5143 5303 5528 5626 5649 5680 5699 6254 6613 6891 6985 7258 7490 7630 7700 7829 7917 16074 16645 16680 17200 17747 18168 18398 18530 18872 2951 3116 3179 3345 3528 3623 3658 3717 3764 4529 4755 4987 5095 5335 5509 5580 5635 5674 5735  997.8 997.3 996.8 996.3 995.2 994.1 993.0 992.4 991.8 991.2 997.3 996.8 996.4 995.5 994.6 993.6 993.1 992.5 992.0 997.8 997.3 996.9 996.4 995.5 994.5 993.5 993.0 992.4 991.9 997.3 996.9 996.4 995.5 994.6 993.6 993.1 992.6 992.1 997.6 997.1 996.7 996.2 995.3 994.3 993.2 992.7 992.1 991.5 997.7 997.2 9968 996.3 995.3 994.3 993.2 992.7 992.1 991.6 997.1 996.7 996.2 995.3 994.4 993.4 992.9 992.3 991.8 997.5 997.2 996.9 996.4 995.8 995.3 995.0 994.7 994.4 997.6 997.1 996.7 996.2 995.2 994.2 993.2 992.6 992.1 991.5  700.46 700.46 700.46 700.46 700.46 700.46 700.46 700.46 700.46 700.46 878.67 878.67 878.67 878.67 878.67 878.67 878.67 878.67 878.67 523.04 523.04 523.04 523.04 523.04 523.04 523.04 523.04 523.04 523.04 1101.20 1101.20 1101.20 1101.20  1101.20 1101.20 1101.20 1101.20 1101.20 296.91 296.91 296.91 296.91 296.91 296.91 296.91 296.91 296.91 296.91 412.32 412.32 412.32 412.32 412.32 412.32 412.32 412.32 412.32 412.32 991.26 991.26 991.26 991.26 991.26 991.26 991.26 991.26 991.26 199.51 199.51 199.51 199.51 199.51 199 51 199.51 199.51 199.51 297.22 297.22 297.22 297.22 297.22 297.22 297.22 297.22 297.22 297.22  V-local (m/s) 0.7020 0.7024 0.7027 0.7031 0.7038 0.7046 0.7054 0.7058 0.7063 0.7067 0.8811 0.8814 0.8819 0.8826 0.8835 0.8844 0.8848 0.8853 0.8857 0.5242 0.5245 0.5247 0.5249 0.5254 0.5259 0.5265 0.5267 0.5270 0.5273 1.1042 1.1046 1.1051 1.1061 1.1072 1.1083 1.1088 1.1094 1.1100 0.2976 0.2978 0.2979 0.2980 0.2983 0.2986 0.2989 0.2991 0.2993 0.2994 0.4133 0.4135 0.4137 0.4139 0.4143 0.4147 0.4151 0.4154 0.4156 0.4158 0.9941 0.9946 0.9950 0.9959 0.9969 0.9979 0.9984 0.9989 0.9994 0.2000 0.2001 0.2001 0.2002 0.2003 0.2005 0.2005 0.2006 0.2006 0.2979 0.2981 0.2982 0.2984 0.2987 0.2990 0.2993 0.2994 0.2996 0.2998  r 0.00768 0.00755 0.00747 0.00745 0.00740 0.00734 0.00729 0.00727 0.00726 0.00722 0.00716 0.00712 0.00709 0.00705 0.00694 0.00690 0.00689 0.00686 0.00683 0.00836 0.00823 0.00813 0.00812 0.00803 0.00796 0.00792 0.00790 0.00788 0.00785 0.00673 0.00667 0.00666 0.00661 0.00655 0.00652 0.00651 0.00648 0.00646 0.00991 0.00975 0.00961 0.00953 0.00944 0.00932 0.00927 0.00926 0.00925 0.00924 0.00899 0.00884 0.00873 0.00870 0.00860 0.00852 0.00847 0.00845 0.00841 0.00838 0.00690 0.00683 0.00683 0.00678 0.00672 0.00668 0.00666 0.00665 0.00661 0.01142 0.01121 0.01113 0.01094 0.01075 001065 0.01062 0.01056 0.01052 0.00993 0.00977 0.00963 0.00956 0.00943 0.00933 0.00930 0.00927 0.00925 0.00922  v* (mis) dR/dt (m'K/J) 0.04349 1.10E-08 0.04315 I.71E-08 0.04295 2.06E-08 0.04290 2.14E-0S 0.04282 2.42E-08 0.04267 2.55E-08 0.04260 3.05E-O8 0.04256 3.38E-08 0.04254 3.56E-08 0.04247 3.56E-08 0.05272 6.97E-09 0.05259 1.16E-08 0.05251 1.34E-08 0.05240 1.66E-08 0.05203 2.3IE-08 0.05196 2.57E-08 0.05195 2.64E-08 0.05185 3.06E-08 0.05174 3.52E-08 0.03389 8.8SE-09 0.03364 1.44E-08 0.03345 1.82E-08 0.03345 1.80E-08 2.07E-08 0.03330 0.03319 2.48E-08 0.03313 2.72E-08 0.03312 2.91E-08 0.03308 3.10E-O8 3.32E-08 0.03303 0.06404 3.85E-09 0.06381 5.11E-09 0.06376 5.12E-09 6.09E-09 0.06358 0.06337 6.80E-O9 0.06327 7.85E-09 0.06324 8.72E-09 0.06315 1.06E-08 0.06308 1.12E-08 0.02095 8.63E-09 0.02079 1.35E-08 0.02065 1.96E-08 0.02058 2.32E-08 0.02050 2.61E-08 0.02039 2.57E-08 0.02036 2.36E-08 0.02036 2.63E-08 0.02035 2.74E-08 0.02035 2.87E-08 0.02771 6.65E-09 0.02749 1.31E-08 0.02733 I.71E-08 0.02729 1.72E-08 0.02716 2.O0E-O8 0.02706 2.42E-08 2.54E-08 0.02702 0.02700 2.72E-08 3.15E-08 0.02695 0.02692 3.53E-08 0.05838 4.53E-09 0.05814 5.85E-09 0.05815 5.85E-09 0.05797 6.70E-09 0.05779 8.10E-O9 0.05767 9.82E-09 0.05761 I.12E-08 0.05759 1.26E-08 0.05748 1.40E-O8 0.01511 2.33E-09 0.01498 4.89E-09 0.01493 6.33E-09 0.01481 9.02E-O9 0.01469 1.I5E-08 0.01463 I.I3E-08 0.01461 1.19E-08 0.01458 8.79E-09 6.78E-09 0.01455 0.02099 8.04E-09 0.O2084 J.24E-08 0.02069 1.74E-08 2.00E-08 0.02063 0.02050 2.48E-08 0.02042 2.38E-08 0.02041 I.96E-08 0.02039 2.41E-08 2.59E-08 0.02038 0.02036 2.76E-08  C 2.3845E-07 2.2009E-07 2.0983E-O7 2.0625E-07 2.0035E-O7 1.9I32E-07 1.8574E-07 1.8278E-07 1.807IE-O7 I.7647E-07 1.8890E-07 1.8349E-07 1.7984E-07 I.7439E-07 I.6087E-07 I.5686E-07 I.5557E-07 1.5I89E-07 1.4769E-07 3.0990E-07 2.8882E-07 2.7279E-07 2.72I6E-07 2.5837E-07 2.4760E-07 2.4122E-07 2.3879E-07 2.3494E-07 2.3046E-07 1.5304E-07 1.4673E-07 1.448SE-07 1.3943E-07 1.3326E-07 1.2959E-07 1.28UE-07 I.2557E-07 1.2327E-07 4.9990E-07 4.6725E-07 4.4I19E-07 4.2614E-07 4.0889E-07 3.8624E-07 3.7698E-07 3.7489E-07 3.7208E-07 3.7046E-07 3.8446E-07 3.5732E-07 3.3829E-07 3.3222E-07 3.I550E-07 3.0228E-07 2.9470E-07 2.9102E-07 2.8435E-07 2.7990E-O7 I.6553E-07 1.5792E-07 1.5751E-07 I.5108E-07 1.4472E-07 1.4011E-07 1.3767E-07 1.3631E-07 1.3282E-07 7.3148E-07 6.8193E-07 6.6450E-07 6.2164E-07 5.7906E-07 5.5849E-07 5.5133E-07 5 3926E-07 5.3002E-07 5.0425E-07 4.7310E-07 4.4429E-07 4.3I77E-07 4.0574E-07 3.8832E-07 3.8I55E-07 3.7644E-07 3.729IE-07 3.6739E-07  B 3233.07 3377.93 3467 05 3501.34 3560.50 3652.45 3714.52 3748.13 3773.17 3821.60 4883.94 4964.31 5021.69 5112.00 5342.19 5420.66 5448.11 5519.68 5603.05 1946.16 2021.92 2084.93 2088.91 2149.04 2199.62 2232.26 2245.39 2265.52 2289.07 7285.01 7457.89 7514.30 7680.86 7878.58 8006.86 8061.13 8151.33 8235.49 745.09 772.15 795.69 810.40 828.54 853.42 864.89 867.93 871.81 874.36 1288.46 1340.13 1379.85 1393.87 1433.38 1466.99 1488.10 1498.66 1517.14 1530.14 6111.90 6273.04 6286.23 6436.27 6593.31 6715.59 6782.11 6822.06 6918.20 373.73 387.43 392.70 406.37 421.21 429.05 431.91 436.70 440.48 742.76 768.27 793.96 806.16 832.90 852.39 861.09 867.44 872.12 879.09  5: Mathematical Modeling and Discussion  174  Reynolds numbers based on film temperatures varied from 2900 to 21,000, and clean inside wall temperatures varied from 59 to 84°C. This provided a wide range of experimental conditions for modeling purposes. A third input file was later developed to eliminate the data from experiment T F U 309, where Reynolds numbers based on local bulk conditions were in the range of 2100 - 2700 and considered to be outside the range of application for the Gnielinski (1976) friction factor equation and the Metzner and Friend (1958) mass transfer correlation. All experiments had a 20°C bulk temperature rise, except for T F U 309, which showed only a 10°C bulk temperature rise. Also, this experiment displayed an unusual Arrhenius relationship (Section 4.2), which will be discussed in more detail in Section 5.3.4. The following data files were used: MODEL5.DAT: A l l 86 experimental data points; reduced from 90 due to the fouling Biot number criterion. The film temperature was used for both mass transfer and chemical attachment terms. MODEL6.DAT: A l l 86 experimental data points. The film temperature was used for mass transfer, and wall temperature was used for chemical attachment. MODEL7.DAT: 77 data points (TFU 309 eliminated due to low Reynolds numbers). The film temperature was used for mass transfer, and wall temperature was used for chemical attachment. The input data used for modeling lysozyme solution fouling are shown in Appendix 5.4.  5: Mathematical Modeling and Discussion  5.3.2  175  Mathematical Model Predictions  (1) Film temperature for both mass transfer and chemical attachment terms Firstly consider the case where the film temperature is used to evaluate the physical properties for both terms in Equation (5.1.5). Table 5.3.2 and Figure 5.3.1 highlight these results. Table 5.3.2: Modeling results using all lysozyme experimental data and input file ModeI5.dat (Film temperature used for both mass transfer and chemical attachment) -  (UNITS: k =  ke  m K/ s n  Order n  k, (x 10 )  0.40  1.1810  0.50  -  A  k,=  g  / n  ( ke «  4  -  V  Vm Ks /  3  )  AE(kJ/mol)  Sum of squares of residuals  0.2390 x 10"  53  172.1  for Rfo (x 10 ) (m K/J) 1.7221  1.1501  0.4227 x 10  -43  174.3  1.6913  0.55  1.1355  0.2260 xlO  -39  175.3  1.6856  0.57  1.1298  0.4613 x 10"  175.7  1.6848  0.58  1.1270  0.1930 x 10"  175.9  1.6847  0.59  1.1242  0.7701 x lO"  176.1  1.6848  0.60  1.1214  0.2937 x 10"  176.3  1.6851  0.65  1.1077  0.1294 x 10"  33  177.3  1.6887  0.70  1.0943  0.2446 xlO"  31  178.3  1.6959  0.80  1.0684  0.1312 xlO"  180.0  1.7150  0.90  1.0433  0.1345 x 10"  181.5  1.7471  1.00  1.0191  0.2728 x 10"  22  182.9  1.7803  1.50  0.9088  0.6671 x lO  -15  187.1  1.9570  2.00  0.8156  0.5092 x 10'  189.0  2.1079  15  k  • 15  38  37  37  36  27  24  11  2  2  5: Mathematical Modeling and Discussion  0.4  0.5  176  0.6  0.7  0.9  0.8  1.0  Reaction order (n)  Figure 5.3.1: Optimum reaction order when film temperature is used for the physical properties for mass transfer and chemical attachment terms in the mathematical model From Table 5.3.2 and Figure 5.3.1, the optimum solution occurs when the reaction order IS  9  9  is 0.58 and the sum of the squares of the residuals = 1.6847 x 10" (m K/J) . The corresponding kg  optimum activation energy is 175.9 kJ/mol, k i = 1.1270 x 10 15 ^ m  10  •37  8  4  K s V  \  and k = 0.1930 x  2.72  kg , m'K s 7 2 , / 2  17 2  . Notice, from Table 5.3.2, how insensitive k i is to the reaction order, but how  very sensitive k is to the reaction order. This reflects the strong dependence of the chemical attachment term to the overall kinetic reaction order. From Figure 5.3.1 there is little variation in the sum of squares of the residuals for reaction orders of 0.54 to 0.62. A comparison plot between experimental and calculated initial fouling rate data for the best model solution is shown in Figure 5.3.2. Individual model results and their "goodness of fit" used to plot Figure 5.3.2 are shown in Table 5.3.3 where the variance, average absolute percent deviation (AAD) and root mean square percent deviation (RMS) can be compared. A s discussed by Vasak and Epstein (1996), the variance is considered the better criterion for "goodness of fit"  5: Mathematical Modeling and Discussion  177  «  because it gives less weight to deviations between predicted and measured values of R  values of R , where experimental uncertainty is the greatest, than it does for high R 4.0E-08 f o  O.OE+00 5.0E-09  1.0E-08  1.5E-08  2.0E-08  2.5E-08  3.0E-08  fo  fo  for low  .  3.5E-08 4.0E-08  Experimental Initial Fouling Rate (m K/J) z  Figure 5.3.2: Best solution for reaction order, n = 0.58 (N = 86) It is worth noting that there is considerable scatter between the model and experimental initial fouling rate data for T F U 309 (shaded data in Table 5.3.3). There are several observations from this experiment which deserve comments. Firstly, considering all 9 thermocouples, there is an average deviation of 98 % between experimental and model results. At low fouling rates, the absolute differences between results are enhanced when expressed as percentages. Secondly, looking at the fouling Arrhenius type plot for this experiment (characteristics discussed in Section 4.3 and shown in Figure 4.3.7), it is clear that the relationship was not of the usual Arrhenius behaviour. The Reynolds number for this experiment (based on fluid properties evaluated at the bulk temperature) increased from a laminar flow at the test section inlet (Re = 2100) to a transitional or slightly turbulent flow at the test section outlet (Re = 2700), so that  5: Mathematical Modeling and Discussion  178  irregularities in the fouling response may have resulted due to this change in flow patterns as the fluid progressed along the length of the test section. Table 5.3.3: Statistical analyses of experimental and predicted lysozyme fouling results Parameters: k. = 1.12702 x 10  k = 1.93032 x 10  15  (*0  expt  (m K/J) 2  model  (in K/J)  AE = 175.9 kJ/mol  38  -I Rfo  Rfo  exptv  (m K/J) 2  Reaction order = 0.58  ' expt  model  2 s  expt ecpt  1 10E-0 8 1 71E-08 2 06E-08 2 14E-08 2 42E-08 2 55E-08 3 05E-08 3 38E-08 3 56E-08 3 56E-08 6 97E-09 1 1 6 E-0 8 1 34E-08 1 66E-08 2 3 1E-08 2 57E-08 2 64E-08 3 06E-08 3 52E-08 8 88E-09 1 44-E-08 1 82E-08 1 80E-08 2 07E-08 2 48E-08 2 72E-08 2 91E-08 3 10E-08 3 32E-08 3 85E-09 5 1 1 E-09 5 12E-09 6 09E-09 6 80E-09 7 85E-09 8 72E-09 1 06E-08 1 12E-08 8 63E-09 1 35E-08 1 96E-08 2 32E-08 2 6 1 E-08 2 57E-08  5 1 1 1 1 2 2 2 2 3 4 5 6 6 1 1 1 2 2 6 1 2 1 2 2 2 2 3 3 4 6 6 7 1 1 1 1 1 1 1 1 1 2 2  29E-09 22E-08 88E-08 92E-08 96E-08 56E-08 75E-08 9 1 E-08 92E-08 47E-08 75E-09 95E-09 4 1 E-09 76E-09 68E-08 77E-08 69E-08 1 4E-08 92E-08 7 1 E -09 30E-08 07E-08 8 3 E -0 8 3 9 E -0 8 82E-08 9 5 E -0 8 96E-08 1 0E-08 32E-08 1 9E-09 5 1 E-09 36E-09 76E-09 1 0E-08 22E-08 23E-08 47E-08 74E-08 1 1 E-08 56E-08 87E-08 99E-08 1 1 E-08 2 7 E -0 8  3.26E-1 7 2 3 8E-17 3 .40E-1 8 4 89E-18 2 11 E-l 7 5 62E-2 1 9 .1 O E - 1 8 2 1 7E-1 7 4 10E-17 8 70E-19 4 93E-1 8 3 1 9E-1 7 4 89E-1 7 9 69E-17 3 95E-17 6 47E-1 7 9 03E-17 8 40E-1 7 3 60E-1 7 4 71 E-l 8 1 99E-18 6 34E-18 1 08E-19 1 05E-1 7 1 1 9E-1 7 5 49E-1 8 2 04E-19 1 69E-22 7 84E-22 1 14E-19 1 96E-18 1 55E-1 8 2 79E-18 1 79E-17 1 86E-17 1 28E-17 1 65E-1 7 3 85E-1 7 6 1 3E-1 8 4 35E-1 8 7 85E-19 1 07E-17 2 53E-1 7 9 10E-1 8  0.5 1 8 7 0.2852 0.0895 0.1034 .1897 .0029 .0989 .1378 .1800 .0262 .3187 .4 8 7 3 .5217 .5930 0.272 1 0.3 1 2 9 .3599 .2995 .1705 .2443 0.09 8 1 0.13 84 0.0 1 8 2 0.1565 0.1389 0.0862 0.01 55 0.0004 0.0008 0.0879 0.2743 0.242 9 0.2742 0 . 6 2 13 0.5492 0.4097 0.3 83 1 0.5 5 3 8 .2868 .1544 .045 2 . 1 4 11 .1926 .1174  .2 6 9 0 .0813 .0 0 8 0 .0107 .0 3 6 0 .0000 .0098 .0 1 9 0 . 0 3 24 .0007 . 1 0 15 0.2 3 74 0.2722 0.3516 0.0740 .0 9 7 9 .12 9 5 .0 8 9 7 .029 1 .0 5 9 7 0.0096 0.0191 0.0003 0.024 5 0.0 1 93 0.0074 0.0002 0.0000 0.0000 0.0077 0.0 7 5 2 0.0 5 9 0 0.0 7 5 2 0.3 8 6 0 0.3016 .1679 .1468 .3 0 6 6 .0 8 2 2 .0 2 3 9 .0 0 2 0 .0 1 9 9 .0 3 7 1 .0138  J  5:>:  179  Mathematical Modeling and Discussion  -  2 2 2 2 6 1 1 1 2 2 2 2 3 3 4 5 5 6 8 9 1 1 1 2 4 6 9 1 I 1 8 6 8 1 1 2 2 2 1 2 2 2  36E -08 6 3 E -0 8 74E-08 87E-08 65E-09 3 1 E -0 8 7 1 E -0 8 7 2 E -0 8 0 0 E -0 8 42E-08 S4E-08 7 2 E -0 8 15E-08 53 E-08 53E-09 85E-09 85E-09 7 0 E -0 9 10E-09 82E-09 1 2 E -0 8 26E-08 4 0 E -0 8 3 3 I. - 0 9  r  8 9 -0 9 3 3 f- -0 9 02E-09 1 5 I -0 8 1 3 E -0 8 1 9 E -OS 7 9 E -09  r-  78 09 04E-09 2 4 E -0 8 74 E -0 8 00E-08 4 8 E -0 8 3 8 E-08 9 6 E -0 8 4 1 E-08 5 9 E -0 8 76 E -0 8  2 2 2 2 7 1 1 2 2 2 2 2 2 3 5 9 7 9 1 1 1 1 2 8 1 1 1 1 1  1 1 1 1 1 1 1 2 2 2 2 2 2  3 3 E -0 8 3 4 E -0 8 3 5 E -0 8 3 5 E -0 8 4 1 E-0 9 3 8 E -0 8 9 5 E -0 8 0 1 E-0 8 3 9 E -0 8 6 5 E -0 8 7 3 E -0 8 7 8 E -0 8 9 3 E -0 8 0 1 E-0 8 4 6 E -0 9 2 3 E -0 9 5 9 E -0 9 8 1 E-0 9 3 6 E -0 8 6 3 E -0 8 8 4 E -0 8 8 3 E -0 8 4 8 E -0 8 3 4 C -09 1 6 E-0 8 2 3 I -0 8 •, 9 I -0 8 5 2 L -0 8 5 8 E -0 8 6 0 L -0 8 i -0 8 64 -0 8 66 0 0 E -0 8 4 4 E -0 8 8 3 E -0 8 9 3 E -0 8 1 3 E -0 8 2 5 E -0 8 2 9 E -0 8 3 2 E -0 8 3 4 E -0 8 3 8 E -0 8  r r  1 20E-1 9 8 7 0 E -1 8 1 5 2 E -1 7 2 6 6 E -1 7 5 76 E -19 4 60E-1 9 5 8 4 E -1 8 8 1 6 E -1 8 1 55E-I 7 5 29E-1 8 3 6 IE-1 8 3 2 5 E -1 9 4 7 7 E -1 8 2 7 0 E -1 7 8 7 0 E -1 9 1 1 4 E -1 7 3 02E-1 8 9 6 9 E -1 8 3 0 0 E -1 7 4 1 6 E -1 7 5 1 9 E -1 7 3 3 1 E -1 7 1 1 8E-1 6 3 61 -1 7 4 4 6 E -1 7 3 5 5 U -1 7 2 3 7 -1 7 ; 1 3 6E-17' \ 2 0 0 E -1 7 > > 1 67E-l"7 5 72E-1 7 9 74'E 1 7 J 3 94E-1 8 " 4 02E-1 8 7 26E-1 9 5 46E-1 9 1 2 1 E -1 7 1 67E-1 8 1 0 6 E -1 7 8 1 5E-1 9 6 2 0 E -1 8 1 44E-1 7  0 .0147 0 .1122 0 .14 2 4 0 17 9 6 0 .1142 0 0 5 18 0 14 13 0 16 6 0 0 19 7 0 0 .0 9 5 1 0 074 8 0 02 10 0 0693 0 14 7 1 0 205 8 0 5780 0 297 1 0 4647 0 675 7 0 6567 0 643 0 0 4 5 63 0 7744 .• 5 7 9 4 1 36 5 8 0 9 4 17- • 0 540 1 0 3203 0 3 96 1 , 0 343 4 860 8 . ; 1 4 55 3 0 24 70 0 16 16 0 04 90 0 0370 0 1402 0 05 43 0 1662 0 037 5 0 096 1 0 13 74  r.  fc  :  5  111  :  AAD  Variance  2.0298 x 10 (mK/J) -17  2  2  30.7051 %  0 .0002 0 .0126 0 02 03 0 03 2 3 0 0 130 0 002 7 0 .02 00 0 .0276 0 .03 8 8 0 .0090 0 00 5 6 0 0 004 0 .0 04 8 0 .02 16 0 0424 0 .3 3 4 0 0 088 3 0 .2159 0 .4 56 5 0 43 1 3 0 4 13 5 0 .2 0 8 2 0 5996 ; 6.65 3 5 1 865 5 0 8868 29 \ 7 1 0,2 6 1569 ,1 18 0 7 40 9  ' "5 0 0 0 0 0 0 0 0 0  l"l  7 9 0 6 10 02 6 1 0024 00 14 0 19 7 003 0 02 7 6 00 14 0092 0 189  RMS  47.9806 %  Despite T F U 309 contributing to much of the scatter between experimental and model fouling rates, the average absolute deviation for all of the experimental data was 30.7 %. This is considerably higher than the 16.7 % found in Vasak and Epstein's (1996) modeling of the styrene polymerization data of Crittenden et al. (1987a). (2) Film temperature for mass transfer and wall temperature for chemical attachment In this analysis, using the wall temperature to calculate the fluid physical properties in the chemical attachment term suggests that the reaction and attachment processes occur at the tube  5: Mathematical Modeling and Discussion  180  wall. However, when the film temperature was used for both terms, the attachment process was assumed to be governed by reactions within the vicinity of the tube surface, i.e. in the viscous sub-layer and buffer layer. If one refers to Equation (2.2.22) it is apparent that the optimum model solution is strongly affected by the fluid physical properties. Table 5.3.4 and Figure 5.3.3 highlight the modeling results when the film temperature is used to evaluate the physical properties for the mass transfer term, and the wall temperature is used for the chemical attachment term. Note the small variation in optimum activation energy (165-182 kJ/mol) over a range of reaction orders. For this set of modeling conditions the optimum solution is achieved when the reaction order is 0.57. The sum of the squares of  residuals = 1.6880 x 10" (m K/J) , A E = 169.1 kJ/mol, ki = 1.1280 x 10 15  2  2  15  (  i™4 \V* kg<  km Ks y 8  ,-36  0.1951 x 10"  kg — r — , , m  . Figure 5.3.3 shows that there is little variation in the sum of the  3 1.75 1.75 K  and k  5  s  squares of the residuals for reaction orders between 0.55 and 0.59. A parity plot of the rates at the optimum reaction order for all experimental data is given in Figure 5.3.4, for the same temperature assumptions as in Figure 5.3.3. The individual results for this solution are very similar to the results when the film temperature was used for both terms. However, the variance for all 86 data points was slightly higher at 0.2034 x 10" (m K/kJ) , the A A D was 31.0 % and the RMS was 48.8 %. Hence, using 10  2  2  a combination of film and wall temperatures to evaluate the "goodness of fit" resulted in a slightly worse correlation. Therefore best results were achieved when the film temperature was used for both mass transfer and chemical attachment terms, suggesting a film temperature driven process within the wall layer (y < 30). +  5: Mathematical Modeling and Discussion  181  1.80E-15 •g '« cu u  1.78E-15  JS  1.76E-15  £ 53 1.74E-15  S £ cr 2 -** *o  E s  w  1.72E-15 1.70E-15 H 1.68E-15  0.4  0.5  0.6  0.7  0.8  0.9  1.0  Reaction order (n)  Figure 5.3.3: Optimum reaction order for lysozyme fouling when film temperature is used for mass transfer term and surface temperature is used for chemical attachment term 4.0E-08 301 304 306 X 308 o 310 o •  -  •  •  A  303 305 307 309 Y=X  0.0E+00 0.0E+00  5.0E-09  1.0E-08  1.5E-08  2.0E-08  2.5E-08  3.0E-08  3.5E-08 4.0E-08  Experimental Initial Fouling Rate (m K/J) Figure 5.3.4: Best solution for reaction order = 0.57 (N= 86) 2  5: Mathematical Modeling and Discussion  182  Table 5.3.4: Modeling results using all experimental data and input file Model6.dat (Film temperature used for mass transfer and wall temperature used for chemical attachment) kBe  (UNITS: k= Order n  k, ( x l O )  0.40  1.1794  0.45  /  n  -  i  kg  l  , k ,= - f - r  )  A E (kJ/mol)  Sum of squares of residuals  0.5663 x 10"  165.4  for Rfo (x 10 ) (m K/J) 1.7212  1.1636  0.1459 x 10"  45  166.5  1.7035  0.50  1.1484  0.3118 x l O  -41  167.6  1.6929  0.54  1.1366  0.2425 x lO" *  168.5  1.6889  0.56  1.1309  0.4758 x 10"  37  168.9  1.6881  0.57  1.1280  0.1951 x 10'  36  169.1  1.6880  0.58  1.1252  0.7632 x 10  -36  169.3  1.6881  0.60  1.1196  0.1022 x 10'  169.7  1.6889  0.65  1.1058  0.3394 x 10"  32  170.7  1.6935  0.75  1.0791  0.3965 x l O  -28  172.6  1.7121  1.00  1.0163  0.2285 x 10"  21  176.3  1.7888  1.50  0.9050  0.2836 x 10  -14  180.4  1.9658  2.00  0.8111  0.1539 x 10"  10  182.2  2.1148  1 5  k  • 15  5.3.3  51  3  34  2  2  The Optimum Model Solution The best model fit to the lysozyme fouling experimental data occurs when the film  temperature is used for both mass transfer and chemical attachment. The minimized variance is 0.2030 x 10" (m K/kJ) , and the corresponding activation energy is 175.9 kJ/mol. This resulting 10  2  2  5: Mathematical Modeling and Discussion  183  kinetic activation energy is much higher than the fouling activation energies (29 - 118 kJ/mol) reported in Table 4.3.3. This result makes sense because the combined chemical reaction, mass transfer and attachments steps have the effect of reducing the overall energy barrier. Note, as discussed in Section 4.3, as the fluid velocity increases the fouling activation energy approaches, but never equals, the value of this purely kinetic activation energy. Figure 5.3.5 shows the best model solution to the experimental data over a range of fluid velocities and wall and bulk temperatures. As with the whey protein results, one notes the strong effect of wall temperature on the initial fouling rate at mass velocities greater than 200 kg/m s. 2  There are three sets of lines, at T  w  = 70, 75 and 80°C. Also note the effect of bulk temperature.  Shown at each wall temperature are the maximum and minimum bulk temperatures of 30 and 52°C used in the experiments. In comparison to Figure 5.2.7 (the optimum model solution for whey protein fouling), use of the film temperature to calculate the physical properties for both mass transfer and chemical attachment terms causes an interesting cross-over in initial fouling rate curves between minimum and maximum bulk temperatures at a constant wall temperature. This effect is due entirely to the temperature dependency of the physical properties. Furthermore, as T  w  increases, the crossover occurs at higher mass flux values. Although the differences in  initial fouling rates at high mass fluxes are not as significant as in the mass transfer controlled region (G < 600 kg/m s at T 2  w  = 80°C), the implication is that under identical operating  conditions of mass flux and wall temperature at sufficiently high mass fluxes, a lower bulk temperature can (although not always) lead to slightly higher initial fouling rates. Compared to whey protein fouling, one notes that the maximum fouling rates are considerably higher (factor of three) for lysozyme solutions at the same temperatures, and-that  5: Mathematical Modeling and Discussion  184  fouling occurs at a much lower temperature. This may be influenced by the nature of the proteins, the relative proximity to their isoelectric point and hence the solution pH. However, despite this, it is interesting to note that the maximum initial fouling rates at the highest clean wall temperature for both systems appear to occur at the same mass flux, approximately 600 kg/m s, 2  which corresponds to a film Reynolds number of about 11,200 for this system. 4.0E-08 f$  • Tw = 70°C  3.5E-08  T =  A Tw = 75°C  h  %  3.0E-08 -|  • Tw = 80°C  T = h  tS  2.5E-08 ^  S  % © te «  2.0E-08 1.5E-08  -«-»  B  ~  l.OE-08  o S  Re <2300 X  5.0E-09 0.0E+00 0  200  400  600  800  1000  1200  1400  1600  Mass Flux (kg/ms) Figure 5.3.5: Effect of wall temperature on model solution for lysozyme and comparison to experimental fouling data 2  5.3.4  Elimination of Low Reynolds Number Data Some experimental data have bulk Reynolds numbers in the range of 2100 - 2700 (TFU  309, G = 199.51 kg/m s). The Gnielinski friction factor correlation is valid down to Re = 2300 2  (compared to 4000 for the Blasius equation) and the linear dependence of the mass transfer coefficient on bulk fluid velocity is valid for Re > 4000 (Metzner and Friend, 1958). To eliminate the data that possibly display laminar flow, the F O R T R A N program (MODEL7.DAT) was run excluding T F U 309.  5: Mathematical Modeling and Discussion  185  For this reduced set of data (N = 77) it was determined that the best fit resulted when the film temperature was used to evaluate the physical properties for the mass transfer term, and the wall temperature was used to evaluate the physical properties for the chemical attachment term. This is contrary to Section 5.3.3, where all experimental data points were used. The results are shown in Figure 5.3.6. 1.300E-15  2  1.295E-15  «  1.290E-15  I 52 tr ~ 1.285E-15 W <3J  ® a  1.280E-15  1.275E-15 0.6  0.7  0.8  0.9  1.0  Reaction order (n) Figure 5.3.6: Optimum reaction order when the film temperature is used for the physical properties for the mass transfer term and the wall temperature is used for the physical properties for the chemical attachment term in the mathematical model. T F U 309 neglected The optimum reaction order for this set of data is 0.75 and the corresponding activation energy has reduced to 161.4 kJ/mol. A plot of the fouling rate data for the optimum solution is shown in Figure 5.3.7. The variance for the 77 data points is 0.1725 x 10" (m K/kJ) , and the 10  2  2  average absolute percent deviation = 23.3 %. Eliminating the low Reynolds number data where viscous flow patterns are likely significant produces a far better correlation with the mathematical model of Epstein (1994). The A A D for the optimum lysozyme model prediction (23.3 %), where a pure protein solution was used as the model fluid to simplify the number of possible chemical reactions, is smaller than the A A D for the optimum whey protein model  186  5: Mathematical Modeling and Discussion  prediction (24.5 %), which involves two reacting proteins. However, neither of these results correlate quite as well as those obtained by Vasak and Epstein (1996). Comparing Figure 5.3.4 to Figure 5.3.7, the latter plot has a significantly improved correlation between experimental and model-predicted fouling rates at the low fouling rates (< 1.5 x 10" m K/J), which typically correspond to the lower Reynolds numbers. 8  2  4.0E-08 3.5E-08 \  3.0E-08  £  b ad  2.5E-08  =3 2.0E-08 o fa .2 1.5E-08 "3 ^ 1.0E-08 "o £  5.0E-09 0.0E+00 O.OE+00  5.0E-09  1.0E-08  1.5E-08  2.0E-08  2.5E-08  3.0E-08  3.5E-08  4.0E-08  Experimental Initial Fouling Rate (m K/J) 2  Figure 5.3.7: Best solution for n = 0.75, N = 77 Figure 5.3.8 shows a comparison plot between the optimum model prediction and the 77 experimental data points using the film and wall temperature combination detailed above. Note the different effect of bulk temperature upon the initial fouling rate for all data to the right of the maximum, i.e. the chemical attachment dominated region. As previously mentioned, these results show that the cross-over present in Figure 5.3.5 arises from the use of the film temperature to evaluate the physical properties of the chemical attachment term.  5: Mathematical Modeling and Discussion  187  4.0E-08 3.5E-08  • Tw = 70°C T =  • Tw = 75°C  b  1=, 3.0E-08  • Tw = 80°C  T„ =  « 2.5E-08 c 1 2.0E-08 o te 3 1.5E-08  '5 ~ ©  l.OE-08  S  5.0E-09  R e < 2300  0.0E+00  200  400  600  800  1000  1200  1400  1600  Mass Flux (kg/m s) 2  Figure 5.3.8: Comparison of model predictions to experimental data obtained from the Arrhenius type equations The optimum kinetic parameters obtained from the mathematical model agree reasonably well with the range of values reported by Makki (1996) in Tables 2.4.3 and 2.4.4. When the mathematical model was run with a reaction order n = 1, the kinetic value reported by Makki (1996), the average absolute percent deviation increased slightly from 23.3 % to 23.7 %. Interpolating the activation energy data from Table 2.4.3 for pH 8.0, thereby fixing AE = 101 kJ/mol, then with n = 1 again, the average absolute percent deviation increased to 30.2 %. However, Makki (1996) reported an unsystematic variation in activation energy with pH and thus there is little confidence in this interpolated value. Therefore the optimum model activation energy, 161.4 kJ/mol, should be compared to the range of reported literature values, i.e. AE = 50.2-151.1 kJ/mol.  188  5: Mathematical Modeling and Discussion 5.3.5  Discussion of the Model Response To this point, the mathematical model has been assessed by how well the experimental  data fits the model predictions by means of minimizing the variance for each set of operating conditions. It has been noted that the optimum kinetic reaction orders for whey protein and lysozyme solutions lie between 0.6 and 1.0. To analyze the nature of the model response and to simplify this approach, a first order kinetic reaction will be assumed, such that the model equation takes on the form  C kSc — - = R * b b  V  2 / 3  kpe^V. +— ^  2  (2.2.25)  It will also be assumed that the system is at a constant bulk composition and wall temperature, and that the fluid physical properties are constant from one friction velocity to another. By definition, V* can be replaced by  and V by G/p. Equation (2.2.25) can therefore be  simplified for fully rough flow (f = constant) to take the form  Rfo = - — - + BG G  (5.3.1) 2  where, in the limits this equation takes on the form of either mass transfer or chemical attachment control as shown in Figure.5.3.9. Therefore at any mass flux between these limits, the result will be a graph comprising both characteristics. For this analysis, a comparison will be made to the experimental results at T = 80°C in w  Figure 4.3.6. From Equation (5.3.1) it follows that if two conditions of G and Rf are known, 0  then the constants A and B can be solved for, and the equation will be capable of describing the  189  5: Mathematical Modeling and Discussion  system over a range of mass fluxes. Note, however, that this will not be an optimized solution, but rather one way of discussing the nature of the model response to the experimental data.  Mass Flux (kg/m s)  Mass Flux (kg/m s)  2  2  Figure 5.3.9: The controlling limits of Equation (5.3.1) Three separate cases will be studied and compared to the experimental data for lysozyme at a clean, inside wall temperature of 80°C.  (I) Specifying two initial fouling rates, one on either side of the expected maximum from Fig 4.3.6d: I.  G = 297 kg/m s, R/ = 2.5 x 10' m K/kJ (two fouling results at the same mass flux)  II.  G = 1101 kg/m s, Rf = 1.5 x Iff m K/kJ (fouling rate at maximum mass flux)  2  5  2  0  0  Solving Equation (5.3.1) for these two sets of input parameters yields A = 1.0648 x 10  10  (kg /m s K) and B = 47.02 (m /kg.K). The corresponding plot of experimental data and model 2  2  3  4  result is shown in Figure 5.3.10a. The model maximum is well to the left and below the experimental data. Note the long tail to the right of the maximum in the chemical attachment region, suggesting the difficulty of the model equation to fit a sudden decrease in experimental initial fouling rates.  5: Mathematical Modeling and Discussion  190  4.0E-05 §" 3.5E-05 3.0E-05 -\ 2  2.5E-05  *  2.0E-05 H  ~  1.5E-05  fe  1.0E-05  Sc  5.0E-06 O.OE+00 0  200  400 600 800 1000 1200 Mass Flux (kg/m s) Figure 5.3.10a: Fit of Equation (5.3.1) to experimental data at T w = 80° C 2  (II) Specify two initial fouling rates to the left of the expected maximum from Figure 4.3.6d: I.  G = 297 kg/m s R = 2.5 x W m K/kJ  II.  G = 523 kg/m s R = 3.5xI0- m K/kJ  2  5  2  fo  2  5  2  fo  Following an identical solution procedure to (I), A = 1.1193 x 10 (kg /m s K) and B = 26.21 10  2  2  3  (m /kg.K). The corresponding plot is shown as Figure 5.3.10b. 4  4.0E-05 I*  3.5E-05  "  3.0E-05  g  ¥  2.5E-05  OS  M 2.0E-05 % 1.5E-05 fa  5c 93  1.0E-05 5.0E-06 0.0E+00 0  200  400 600 800 1000 Mass Flux (kg/m s)  1200  2  Figure 5.3.10b: Fit of Equation (5.3.1) to experimental data at T  w  = 80° C  With this approach, as with Figure 5.3.10a, the maximum in the model solution appears to occur at a mass flux of 500 - 600 kg/m s, somewhat lower than determined experimentally. It appears 2  5: Mathematical Modeling and Discussion  191  difficult to obtain a model solution that will fit the experimental data at mass fluxes of 879, 991 and 1101 kg/m s. 2  (Ill) Determine the best model solution to all nine experimental data points at  = 80 °C:  Minimizing the sum of the squares of the residuals between experimental and model initial fouling rates, A = 1.0974 x 10 (kg /m s K) and B = 29.11 (m /kg.K). This solution is 10  2  2  3  4  very similar to the result of the second case, and the corresponding plot is shown in Figure 5.3.10c. 4.0E-05 3.5E-05 ^  3.0E-05  S  2.5E-05  5. 2.0E-05 % 1.5E-05 fa  1.0E-05  03  S  5.0E-06  ~  0.0E+00  0  200  400 600 800 1000 Mass Flux (kg/m s)  1200  J  Figure 5.3.10c: Fit of Equation (5.3.1) to experimental data at T = 80° C w  By this rather simplified approach, the model fit presented as Figure 5.3.5 appears to be the optimum solution to all of the experimental data, and a rapid tailing off in initial fouling rate in the chemical attachment region is not generated by the model, given the dependency of the initial fouling rate on fluid velocity. The velocity dependency of the chemical attachment term in Equation (2.2.25) was varied for comparative purposes. A much improved fit to the experimental data was achieved when the velocity term was raised to the third power instead of the second. However, no theoretical justification for this power dependency was apparent.  5: Mathematical Modeling and Discussion  192  This discussion is intended as an insight into the model response and how well the model fits the experimental data. By no means is this simplified solution an optimum; not only is the reaction assumed to be first order, but the friction factor has played no part in the analysis. In addition, only some of the experimental data have been utilized. This analysis has shown that Epstein's mathematical model (1994), at a constant wall temperature and physical properties, shows a velocity effect in conformity with experiment at low fluid velocities (under mass transfer control), but a more subdued reduction (tailing off) of the fouling rate at higher fluid velocities (under chemical attachment control) than is exhibited by the whey protein and lysozyme fouling data in this study. A similar increased velocity dependence in the attachment controlled region has previously been noted by Vasak et al. (1995) in their treatment of particle deposition measurements in laminar and turbulent flows. They measured zero particle deposition rates at high Reynolds numbers where presumably the shear effects were too great for particle attachment to occur. The Epstein model, however, as illustrated in its simplified first order form, Equation (2.2.26), predicts that under conditions of attachment control, the initial fouling rate will decrease asymptotically with increasing fluid velocity, rather than decrease to zero at a finite velocity. The latter event implies that above some critical Reynolds number attachment ceases to occur. 5.4  Interpretation of Optimum Model Predictions Analysis of the mathematical model in Sections 5.2 and 5.3 was performed by comparing  experimental data and model predictions at constant, clean inside wall temperatures over a range of fluid velocities. Figure 5.4.1 shows a comparison between the optimal model predictions and experimental data for two whey protein fouling experiments over a range of wall temperatures at constant fluid velocities (one low velocity and one high velocity).  5: Mathematical Modeling and Discussion l.E-04 0.00265  0.00275  0.00285  193  0.00295 (  l.E-OS 0.00276 «.  0.00279  0.00282  0.00285  0.00288  e, n  a...E-OS Initial Rate  tt.  - Arrhenius Correlation l.E-06  •n  - Optimum Model Prediction  l.E-06  Initial Rate - Arrhenius Correlation - Optimum Model Prediction l/CT.jMIO  l/(T»j) (K"') c  (a) T F U 211 (G = 221.4 kg/m s)  (b) T F U 205 ( G = 1055.6 kg/m s)  2  2  Figure 5.4.1: Comparison between experimental and model predicted initial fouling rates for two massfluxesfor whey protein fouling Figure 5.4.1 shows that at low mass flux experiments, over a range of wall temperatures where both mass transfer and chemical attachment are important, the mathematical model prediction does not strictly follow the Arrhenius type expression. In comparison, at a high mass flux, Figure 5.4.1b shows that the model prediction is described well by an Arrhenius type expression, and hence is dominated by the exponential term associated with attachment. At low wall temperature, the sharp increase in the slope of Figure 5.4.1a, and hence of the activation energy (AEf increases from 48 to approximately 129 kJ/mol) also reflects the increased kinetic activation energy for whey protein solutions at lower solution temperatures (Table 2.3.3 and 2.3.4). This feature was absent for the lysozyme experiments, where the pure kinetic parameters did not display any such temperature dependence. This is shown in Figure 5.4.2 for comparison. l.E-04 0.00280  0.00285  0.00290  0.00295  0.00300  , l.E-05  •  l.E-06  Initial Rate Arrhenius Correlation Optimum Model Prediction 1/(T ) (K') WJ  C  Figure 5.4.2: Comparison between experimental and model predicted initial fouling rates for a low mass flux lysozyme fouling experiment (TFU 306, G = 296.9 kg/m s) 2  194  5: Mathematical Modeling and Discussion  Following the discussion of Sections 5.2 and 5.3, Table 5.4.1 shows the best modeling results achieved for whey protein and lysozyme fouling. Table 5.4.1: Optimum model solutions Whey Protein  Lysozyme  Reaction order, n  0.99  0.75  Kinetic activation energy (kJ/mol)  200.9  161.4  47  77  Number of data points  1.3616 x 10  1.0104 x 10  0.5552 x 10"  0.7116 x IO  0.7025 x 10  0.1725 x 10"  24.5  23.3  15  k  f ^  f  15  'lm Ks J 5  8  24  n+1/  -  -26  kg * m  3 X X K  s  -11  Variance (m K/kJ) 2  2  AAD (%)  10  From the solutions to the model it is possible to compare values of the constants shown in Table 5.4.1 to those reported by Epstein (1994), Vasak and Epstein (1996), and to accepted isothermal values (Metzner and Friend, 1958). The following discussion illustrates how values of k' and k" are extracted from the experimental results. From the work of Wilke and Chang (1955), the molecular diffusivity of a dilute aqueous solution can be estimated from 'm ^ 2  D AB  V s )  = 1.272xl0"  17  T(K) kg | o . ' m ^ V m. s. KmolJ 3  v  6  A  (5.4.1)  5: Mathematical Modeling and Discussion  195  The molecular volume ( V A ) of the solute particles, assuming they are spherical, can be estimated from  V  A  =N  A  V  ^7td  3  (5.4.2)  p  where N A V is the Avogadro number. Whey proteins and lysozyme are generally globular in nature: According to De Jong et al. (1992), where fouling of milk heat transfer surfaces was assumed to involve heterogeneous adsorption of the constituents, successful modeling was only achieved when adsorbing particle diameters of less that 10 nm was assumed. Visser and Jeurnink (1997) illustrated that at a neutral pH, the P-lactoglobulin molecule was a dimer with dimensions of 3.58 x 6.93 nm. If the dimer dissociates at elevated temperatures into two monomers, a particle diameter of 3.58 nm can be used for whey protein. Lysozyme, according to Haynes and Norde (1995), has molecular dimensions of 3.0 x 3.0 x 4.6 nm. Equation (5.1.1) assumed that the diffusivity was proportional to the solution temperature divided by viscosity at the relevant film conditions, and therefore the constant that resulted was lumped into the mass transfer constant k, and defined as k, (Equation 5.1.3). Once the molecular volume has been estimated for each system, k can be evaluated. From Section 2.2, k and k are related to k' and k" by the deposit physical properties, as illustrated in Equation (2.2.21). From a study of the whey protein fouling deposit properties (Section 4.2.6), the product p X {  {  was estimated as 221.7 kg /m Ks . If m (the mass of fouling 2  2  3  deposit per mass of precursor transported to and reacted at the wall) is taken as 1, a solution for k' and k" can be estimated. This is summarized in Table 5.4.2.  5: Mathematical Modeling and Discussion  196  ;  The molecular volumes calculated in Table 5.4.2 for lysozyme and P-lactoglobulin agree reasonably well with literature values of 0.01066 and 0.01399 m /mol, respectively (Hinz, 1986). 3  A fouling deposit property study for lysozyme was not performed; however, if similar values for the deposit properties were assumed, k' would be of the same order of magnitude as for the whey proteins. Table 5.4.2: Summary of k' and k" values for whey protein and lysozyme fouling data  Volume Dimension (nm) V (m /mol) 3  A  D in Equation (5.1.1) 0  D (m /s) 2  Whey Protein  Lysozyme  3.58  3.00 x 3.00 x 4.60  0.01447  0.01301  1.6151 x 10  1.7216 x 10"  -16  (1.101- 1.417) x l O '  16  1 0  (0.905 - 1.255) x l O "  4355 - 2850  6752 - 3757  40382  31269  221.7  221.7'  k' from Equation (2.2.21)  182.1  141.0  k" from Equation (2.2.21)  4.3776 x 10'  Sc k from Equation (5.1.3)  27  10  1.2836x1a  23  assumed, since information not available  In Epstein's mathematical model (1994) development, k' was evaluated as 502.3. It was explained that the magnitude of this value (compared to 11.8 determined by Metzner and Friend (1958) for isothermal systems with high Schmidt numbers) was due to the non-isothermality of Crittenden et al.'s (1987a) experimental system. In addition, Epstein (1994) indicated an  5: Mathematical Modeling and Discussion  197  overestimation of the mass transfer coefficient due to the use of wall temperatures to evaluate the fluid physical properties. However, given that the constant 502.3 was used in the denominator for the mass transfer coefficient, it was argued that this effect was somewhat compensated for. Vasak and Epstein (1996) re-analyzed Crittenden et al.'s (1987a) experimental data by minimizing the variance using a multi-parameter non-linear least squares regression. This analysis achieved similar results and the best estimate of k' was revised to 481. In this work, again employing a non-isothermal fouling apparatus, it is shown that the constant of 11.8 (Metzner and Friend, 1958) does not apply. However, k' estimates of 182 and 141 are considerably smaller than suggested above for styrene polymerization fouling. In this work, the physical properties were accounted for by consideration of the bulk, film and wall temperatures. Hence, it is expected that an estimate of k' would lie between 11.8 and 481. In comparison, k", the chemical attachment coefficient is considerably smaller than reported by Vasak and Epstein (1996). The best fit kinetic activation energies for these systems are quite large and therefore k" would compensate for this to some extent. In comparison, Vasak and Epstein (1996) estimated k" as 70.9 x 10"  5  k g ' V m '  5  5  for Crittenden et al.'s (1987b) styrene  in kerosene polymerization data. However, given that this work involves a different chemical reaction(s), it is to be expected that a different value (with different units due to the different order, n, of the reaction) of the chemical attachment coefficient would result. This Section has shown that mathematical modeling of the whey protein and lysozyme solution fouling data of Section 4, using Epstein's (1994) theoretical model for chemical reaction fouling, has been reasonably successful. Although the experimental data displays a poorer model fit than the previously mentioned styrene in kerosene polymerization data (Crittenden et al., 1987a & b), the fundamental features of a maximum initial fouling rate, a velocity dependence,  5: Mathematical Modeling and Discussion  198  and wall temperature dependence, were all displayed. In addition, the bulk temperature was shown to have a significant effect on the model-predicted initial fouling rate. This temperature effect was particularly significant in the mass transfer controlled region, where an elevated bulk temperature (52°C compared to 30°C) due to processing conditions could increase the initial fouling rate by up to 15 %. In addition to this, selection of the appropriate fluid temperature (film versus wall) in the chemical attachment controlled region was shown to have a significant effect on the nature of the model predictions. The experimental data showed a more rapid decrease in initial fouling rate following the maximum than was predicted by the model.  199  6: Conclusions 6.  Conclusions The effect of fluid velocity and of wall and bulk temperature on the initial fouling rate of  1 wt. % whey protein and 1 wt. % lysozyme solutions was investigated and used to verify a mathematical model for initial rate of chemical reaction fouling (Epstein, 1994). Whey protein fouling experiments were performed over the following range of experimental conditions: Ref = 3160 - 22730, T ,j = 68 - 102°C, T = 31 - 57°C. The lysozyme fouling experiments were w  b  performed under similar conditions: Re = 2950 - 20715, T ,i = 59 - 84°C, T = 30 - 52°C. f  w  b  For any given experiment, local fouling results were in most cases well represented by an  Arrhenius type equation (Rf = Aexp  ) at a given mass flux and range of clean,  0  /  R  (  T  . j ) .  inside wall temperatures. The fouling Biot number criterion (Bi > 0.05) previously invoked by Paterson and Fryer (1988) was the main criterion used in deciding on which experimental data to include in the data analysis. From each experiment a fouling activation energy was determined.  6.1  Whey Protein Solution Fouling Reconstituted whey protein solutions contained approximately 18 wt. % (of the 1 wt. %)  particulate material, displaying a bi-modal particle size distribution with modes at 0.31 and 0.65 pm. A series of experiments showed that this particulate material had no significant effect on the initial fouling rate. In addition, chemical analyses of the deposit material showed that the composition was approximately 90% protein, consisting predominantly of P-lactoglobulin and oclactalbumin. Therefore the dominant deposition mechanism was the mass transfer in solution and chemical attachment of denatured and aggregated protein to the test section wall, i.e. nonparticulate chemical reaction fouling.  6: Conclusions  200  Examination of the morphology of the deposit revealed that it was not always homogeneous; in particular, the test sections with little fouling demonstrated deposit striations in the. direction of fluid flow. The deposit was an off-white to cream curd-like material, as reported by Burton (1968). The deposit morphology (SEM) of the high wall temperature sections of two experiments was compared. A much more voluminous, cauliflower-like deposit resulted at the lower mass flux experiments, where a thicker wall layer was present. It was shown that at low mass fluxes, up to 33% of the cross-sectional area was occupied by fluid that was hot enough to sustain deposition reactions. Since deposit formation for all experiments occurred completely within the hot wall layer, it was reasoned that the thickness of this layer rather than a bulk mechanism was responsible for this more voluminous deposit of large sized aggregates. Although the bulk temperature was maintained below 60°C to eliminate bulk reactions, mathematical modeling showed that this temperature (in its effect on the film temperature and on the fluid physical properties) was more significant than first anticipated, consistent with the observations of Paterson and Fryer (1988) and Belmar-Beiny et al. (1993). A deposit property study allowed evaluation of the deposit density and thermal conductivity. For axial locations with significant (homogeneous) deposit coverage, these properties were essentially independent of wall temperature and displayed no significant aging. The deposit density and thermal conductivity were approximated as 852 kg/m and 0.26 W/m.K, 3  respectively. These results corresponded to a water content (i.e. voidage) of 14%, which was in good agreement with values for other dairy products reported by Rao and Rizvi (1995). Evaluation of these properties enabled a quantitative comparison of the mass transfer constant  6: Conclusions  201  resulting from the optimum model prediction with those of Vasak and Epstein (1996), and of Metzner and Friend (1958). From the twelve fouling experiments, overall activation energies of 48.3 to 282 kJ/mol resulted. These values were in rough agreement with the range of kinetic activation energies in the literature (Dannenberg and Kessler, 1988; Lyster, 1970). It was expected that as the mass flux increased the fouling activation energy would increase and approach the kinetic activation energy. However, this trend was not unambiguously observed, probably due to the existence of more than one protein deposition reaction (P-lactoglobulin and a-lactalbumin), of variable reaction kinetics (n and AE), and of the different temperature ranges in which these kinetics applied. Qualitatively, it was shown that at a constant wall (and bulk) temperature there was a maximum initial fouling rate over a range of mass fluxes. This maximum rate increased with increasing wall temperature, and the mass flux at which the maximum occurred also increased with increasing wall temperature. These observations were consistent with the mathematical model. Quantitatively, the best modeling results from Equation (5.1.2) were achieved when the film temperature was used to evaluate the fluid physical properties associated with the mass transfer term, and the wall temperature was used to evaluate the physical properties associated with the chemical attachment term. The optimum kinetic reaction order and activation energy were determined to be 0.99 and 201 kJ/mol, respectively, which agree with or are bracketed by the wide range of kinetic values reported in the literature. The average absolute percent deviation between the optimum solution and the experimental results was 24.5 %, which was larger than  202  6: Conclusions  the 16.7 % achieved by Vasak and Epstein (1996) with the experimental data of Crittenden et al. (1987a). The mass transfer coefficient, k,, was 1.3616 x 10  attachment coefficient, k, was 0.5552 x 10"  24  (kg  2 0 1  /m K 3  15  1 0 1  s  (kg /m Ks ) 4  1 0 1  2  5  173  and the chemical  ) . Using the results from the  deposit property study, k', the dimensionless mass transfer constant, was estimated as 182. This compares to 11.8 from Metzner and Friend (1958) for isothermal conditions, and 481 by Vasak and Epstein (1996) for the non-isothermal styrene-in-kerosene polymerization data (Crittenden et al., 1987a), where all physical properties were calculated at the wall temperature. A n estimate of k'  for this experimental study was expected to lie between 11.8 and 481, since these  experiments were not performed at isothermal conditions, but the physical properties were accounted for by consideration of both the bulk and wall temperatures. 6.2  Lysozyme Solution Fouling A series of fouling experiments at near identical operating conditions was performed to  study the effect of pH (5 - 8) on the initial fouling rate. As the pH increased toward the isoelectric point of lysozyme (pi = 11.1), the fouling rate increased. This is consistent with the lysozyme molecule becoming less stable, possessing less positive charge and therefore becoming more attracted to the positively charged stainless steel surface. The bulk concentration of native protein throughout the duration of an experiment was followed by measuring the enzymatic activity (proportional to the native protein concentration). Over the course of a seven hour experiment, a 6 % decrease in native protein concentration was observed, which reduced the lysozyme concentration from 9.95 g/1 to 9.35 g/1. This reduction would have little effect on the test of the mathematical model and therefore the assumption of a constant bulk concentration was justified.  6: Conclusions Characterization  203 of  the  lysozyme  fouling  mechanism  was  considerably  more  straightforward than for the whey protein system. The lysozyme solutions contained essentially no particulate material ( < 0.1 wt. % of the supply powder), and the deposit composition was approximately 94 % protein, of which lysozyme was the only detectable protein. Therefore deposition could be attributed with confidence to the chemical reaction fouling of denatured and aggregated protein. Using the best fit Arrhenius type equations for each experiment at a given mass velocity, the same features as for the whey protein experiments were demonstrated, again qualitatively consistent with the mathematical model. From the nine fouling experiments performed at pH 8, the fouling activation energies varied from 29.4 to 118 kJ/mol, a considerably narrower range than for the whey protein experiments. These two values are in approximate agreement with the reported kinetic activation energies for pH values of 7.2 and 8.1 (Makki,  1996). As the mass flux increased, the  experimental fouling activation energy increased from 29.4 kJ/mol at 200 kg/m s to 118 kJ/mol 2  at 1101 kg/m s. This observation is consistent with the mathematical model. 2  Both the whey protein and lysozyme fouling experiments showed an increase in initial fouling rate followed by a decrease as the mass flux was increased. In the case of the whey protein experiments, the decrease (at high mass fluxes) was more unambiguous than the increase. In both cases, the decrease in initial fouling rate was more precipitous than predicted by the model. In the case of the lysozyme experiments, the decrease in initial fouling rate was especially abrupt. Both studies demonstrate the validity of the model. Quantitatively, the best modeling results were achieved when the film temperature was used to evaluate the physical properties associated with the mass transfer term, and the wall  204  6: Conclusions  temperature was used to evaluate the physical properties associated with the chemical attachment term. This was similar to the whey protein results. The optimum kinetic reaction order and activation energy were determined to be 0.75 and 161 kJ/mol, respectively. This compares reasonably well to the kinetic activation energy range of 50.2 - 151.1 kJ/mol reported by Makki (1996) as bracketing values of A E which vary unsystematically with pH (see Table 2.4.3). The average absolute percent deviation between the optimum solution and the experimental results was 23.3 %. The optimum fit for the lysozyme fouling results was slightly better than the fit for the whey protein fouling results; however, the fit of experimental data to the model predictions was again inferior to that achieved by Vasak and Epstein (1996). The mass transfer coefficient, k,, was 1.0104 x 10 10  (kg  Im K  s  15  (kg /m Ks ) 4  2  5  1/3  and the chemical attachment coefficient, k, was 0.7116 x  ). From the whey protein deposit property study, k', the dimensionless  mass transfer constant, was estimated as 141. This approximate result agrees reasonably well with the corresponding whey protein result of 182. The lysozyme modeling results showed that the model prediction is strongly dependent upon the temperature used to evaluate the fluid physical properties. For a system that is strongly dependent on these physical properties, in the high velocity chemical attachment controlled region and at a constant wall temperature, a lower bulk temperature may theoretically lead to a slightly higher initial fouling rate, in contrast to the lower initial fouling rate at lower velocities. Fouling results from both the whey protein and lysozyme experiments demonstrated a mutual dependence of the Arrhenius parameters. This dependence was due to the Kinetic Compensation Effect, and was a result of the manner in which experimental data was treated. Data analysis in fouling studies usually involve regressions far from the origin. It is concluded  205  6: Conclusions  that an improved method to represent this data and eliminate the mutual dependence would be to -AE  R  utilize the equation, R f = A ° e 0  1  f  l( w,i)  1  T  c  ( J.\  I  T  c  , where ( T J wi  c  is the isokinetic temperature,  corresponding approximately to the average experimental inside wall temperature.  7: Recommendations for Further Study 7.  206  Recommendations for Further Study During this attempt to validate Epstein's (1994) mathematical model for initial chemical  reaction fouling a number of experimental variables were tested. Several questions were raised over the duration of this work concerning the mathematical model and the fouling equipment that could be further investigated in the future.  1. Epstein's (1994) mathematical fouling model was developed using Crittenden et al.' s (1987a) styrene-in-kerosene polymerization data (chemical reaction fouling) and further verified here using two dilute protein solutions that underwent protein denaturation and aggregation reactions (again chemical reaction fouling). It has also been verified for isothermal colloidal particulate fouling under attractive double layer conditions (Vasak et al., 1995). This model can probably be applied as well to other types of fouling, especially precipitation fouling, and a separate study on the effect of process variables (fluid velocity, wall and bulk temperature) should be undertaken in order to test this model for that type of fouling.  2. Film temperatures used to calculate the solution physical properties were assumed to be the mathematical average of the wall and bulk temperature at any given axial position along the length of the tube. It would be worthwhile performing a study to determine the best way, for predictive purposes, of averaging the temperatures at which the non-isothermal mass transfer and chemical reaction processes occur, and also to determine the consequent influence on the modeling results.  3. Further development of techniques to estimate the physical properties of the fouling deposit would enable a more accurate estimate of the dimensionless mass transfer and chemical  7: Recommendations for Further Study  207  attachment constants (in the initial mass deposition flux rather than the initial fouling rate equations) for non-isothermal experimental systems.  4.  If the Tube Fouling Unit (TFU) apparatus is to be used for further studies, some modifications are recommended: •  The T F U voltage and current signals are recorded manually. These were removed from the data logging system because they were found to be the source of considerable noise. A new datalogging setup would improve this situation. It may be worthwhile to replace the old version of LabTech notebook software, by writing a programme in Visual Basic tailored to the needs of the T F U setup.  •  The thermocouple mountings are showing signs of age, and starting to deteriorate. Correct assembly of these mountings during an experiment is critical to its success. For a fouling system that is sensitive to the wall temperature, these thermocouples can be a major source of error. Re-design of this configuration could markedly improve the quality of results.  5.  The kinetic compensation effect was found to be present in the data analysis of this nonisothermal fouling study. Although treated in a manner to eliminate the mutual dependence of the two Arrhenius parameters for this study, there may be improved methods of treating fouling data and hence the wall temperature effect. A n investigation into the cause of this mutual dependence, and how best to represent the experimental data, could prove valuable.  208  Nomenclature Nomenclature ai.5  Constants in Section 2.1.2  a  Solution vector from F O R T R A N program  AAD  Average absolute deviation (%)  A  Area  A  Arrhenius pre-exponential factor  A  0  m  2  Modified Arrhenius pre-exponential factor  B  Constant defined by Equation (5.1.5a)  b  Stoichiometric constant in Equation (1.1.4)  Bi  Fouling Biot number (RfU )  C  Constant defined by Equation (5.1.5b)  (kg /s.m.K )  Cb  Bulk concentration of reactant  kg/m  C  Specific heat capacity  kJ/molK  Cp  Bulk particle concentration  kg/m  d  Tube diameter  m  dp  Particle diameter  m  D  Diffusivity  m/s  Constant in Equation (5.1.1)  kg.m/s K  AE  Activation energy  J/mol  f  Friction factor = 2 x / p V  G  Fluid mass flux or mass velocity  kg/m s  AG  Gibbs free energy  kJ/kg  D  p  0  s"  1  c  3  2  w  209  Nomenclature  Ha  Hatta number ( H a = 2  kC" " ) — C x5 u  I -  Current  j  Number of unknown parameters  JF  Mass flux of adsorbing milk constituents  kg/m s  Chemical attachment coefficient  m^/kg""^  k  a  A  2  -i  kd, kf Deposition and removal rate constants k  r  Kinetic reaction rate constant  n=l, k: s" , n=2, k: 1/g.s  k  m  Mass transfer coefficient  m/s  k  Mass transfer constant defined in Equation (2.2.1)  kg /m Ks  ki  Lumped mass transfer constant defined in Equation (5.1.3) (kg s/m K)  1  2  2  4  8  9  —  Chemical attachment constant defined in Equation (2.2.1)  k  3  "3  kg lm Ks  when n = 1  (kg /m Ks) 3  k'  Dimensionless mass transfer constant in Equation (2 .2.7)  1/3  6  whenn = 2  1/2  "Xv\  9  9  9  k"  Dimensional chemical attachment constant  ki  Dimensionless constant in Equation (2.2.6)  k  Constant in Equation (2.2.10)  kg " s /m  Boltzmann's constant  1.3805 x 10" J/K  L  Length of test section  m  m m  Mass flow rate of fluid Mass of fouling deposit per mass of precursor transported to and reacted at the wall  2  K  b  kg " s lm '  n  1  3  3n  23  kg/s kg/kg  210  Nomenclature rrif  Deposit coverage  g/m  m  Mass  MW  Molecular weight  n  Kinetic reaction order  N  Number of experimental data points  NAV  Avogadro number  Nu  Nusselt number (UdA,)  P  Pressure  Pr  Prandtl Number (r|C A)  q  Heat  flux  2  kg/m s g/mol  6.022x 10  mol  kPa p  flux  W/m  2  • Q  Heat rate  W  R  Electrical resistance  Q  R  Universal gas constant  8.3143 J/molK  Rf  Fouling resistance  m K/W  Production rate of adsorbed milk components  kg/m s  R  F  2  Re  Reynolds number (Gd/r|)  R  Initial fouling rate  fo  m K/J 2  S  Sticking probability in Equation (2.1.2)  Sc  Schmidt number (r)/pD)  Sh  Sherwood number (k d/D)  t  Time  m  s  211  Nomenclature t  + p  Dimensionless time  T  Temperature  °C or K  Tf.  Film temperature  °C or K  U  Heat transfer coefficient  W/m K  u  +  Dimensionless local velocity (u = v/V•) +  V  Voltage  V  v  Local fluid velocity  (m/s)  V  Bulk fluid velocity  m/s  V*  Friction velocity (= V J fy)  m/s  x  Axial distance along test section Also generic abscissa  mm  Xj  mol fraction of species i in the liquid phase  mol/mol  Xf  Deposit thickness  m  xp  Fraction of P-lactoglobulin in deposit  y  Distance from tube wall Also generic ordinate  y  +  f  m  Dimensionless distance (y = ypV*/r|) +  Greek Symbols 8  Boundary layer thickness  m  A  Difference / drop  T  Surface concentration of adsorbed protein species  mg/m  o)  Mass flux of fouling precursor  kg/m s  n  Dynamic viscosity  kg/m.s  212  Nomenclature 0  Fluid residence time  s  X  Thermal conductivity  W/m.K  v .  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Friend, "Theoretical Analogies Between Heat, Mass and Momentum Transfer and Modifications for Fluids of High Prandtl or Schmidt Numbers," Can. J. Chem. Engineering, pp. 235 - 240 (Dec. 1958). Nassauer, J and H.G Kessler, "Adsorption of Proteins and Microorganisms at Stainless Steel Surfaces," Food Engineering and Process Applications, Transport Phenomena, 1 (1985). Nijsing, R., "Diffusional and Kinetic Phenomena Associated with Fouling," Report EUR 543. e, Euratom (1964). Papiz, M.Z., L. Sawyer, E.E. Eliopoulos, A.C.T. North, J.B.C. Findlay, R. Sivaprasadarao, T.A. Jones, M . E . Newcomer and P.J. Kraulis, "The Structure of p-lactoglobulin and its Similarity to Plasma Retinol-Binding Protein," NATURE, 324, pp. 383 - 385 (27 November 1986). Paterson, W.R. and P.J. Fryer, " A Reaction Engineering Approach to the Analysis of Fouling," Chem. Eng. Sci., 43, No. 7, pp. 1714 - 1717 (1988). Perry, R.H. and D. Green, "Perry's Chemical Engineers' Handbook," 6 Edition, New York, McGraw-Hill, p. 3-75 (1984). th  Pfeil, W. and P.L. Privalov, "Thermodynamic Investigations of Proteins, III. Thermodynamic Description of Lysozyme," Biophysical Chemistry, 4, pp. 41 - 50 (1976). Press, W . H . , B.P. Flannery, S.A. Teukolsky and W . T . Vetterling, Numerical Recipes, Cambridge University Press, pp. 523 - 528 (1986). Rao, M.A. and S.S.H. Rizvi, "Engineering Properties of Food," Marcel Dekker, New York, p 117(1995). Reklaitis, G.V., "Introduction to Material and Energy Balances," John Wiley and Sons, New York, (1983). Roefs, P.F.M., and K . G . DeKruif, " A Model for the Denaturation and Aggregation of Plactoglobulin," Eur. J. Biochem., 226, pp. 883 - 889 (1994). Skudder P.J., B.E. Brooker, A.D. Bonsey and N.R. Alvarez-Guerrero, "Effect of pH on the Formation of Deposit from Milk on Heated Surfaces During Ultra High Temperature Processing," J. Dairy Res., 53, pp. 75 - 87 (1986).  References  219  Skudder, P.J., E . L . Thomas, J.A. Pavey and A . G . Perkin, "Effects of Adding Potassium Iodate to Milk before U H T Treatment. I. Reduction in the Amount of Deposit on the Heated Surface," J. Dairy Res., 48, pp. 99 - 113 (1981). Smith, J.M. and H.C. Van Ness, "Introduction to Chemical Engineering Thermodynamics," Fourth Edition, McGraw-Hill, pp. 418 - 420 (1987). Tan, T., M . Wolfrom and A . Langer, "Chemical Interactions of Amino Compounds and Sugars. V. Comparative Studies with D-Xylose and 2-Furaldehyde," Am. J. ChE., 72, pp. 5090 5095 (November 1950). Taylor, M . "Composition and Structure of Milk," NZDRIDIGTP Notes, 1992. Taylor, W.F., "Technical Report # AFAPL-TR-69-70," Esso Research and Engineering Company, Air Force Aero Propulsion Laboratory, Air Force Systems Command, September (1969). Toyoda, J., P.J.R. Schreier and P.J. Fryer, " A Computational Model for Reaction Fouling from Whey Protein Solutions," Fouling and cleaning in food processing, Department of Chemical Engineering, University of Cambridge, Cambridge, U K , pp. 191 -196 (1994). Treybal, "Mass Transfer Operations," Chemical Engineering Series, 3 Montreal, pp. 72-75 (1980).  rd  Edition, McGraw-Hill,  Troup, D.H. and J.A. Richardson, "Scale Nucleation on a Heat Transfer Surface and its Prevention," Chem. Eng. Commun., 2, pp. 167-180 (1978). Vasak, F., B.D. Bowen, C.Y. Chen, F Kastanek and N. Epstein, "Fine Particle Deposition in Laminar and Turbulent Flows," Can. J. Chem Engineering, 73, pp. 785 - 792 (1995). Vasak, F. and N. Epstein, "Regression Analysis of a Chemical Reaction Fouling Model," Can. J. Chem Engineering, 74, pp. 173 - 175 (Feb. 1996). Visser, J. and Th. J.M. Jeurnink, "Fouling of Heat Exchangers in the Dairy Industry," Experimental Thermal and Fluid Science, pp. 407 - 424 (1997). Watkinson, A.P., "Particulate Fouling of Sensible Heat Exchangers," Ph.D. Thesis, The University of British Columbia (1968). Watkinson, A.P., "Chemical Reaction Fouling of Organic Fluids," Chem. Eng. Technol. 15, pp. 82 - 90 (1992). Watkinson, A.P. and N. Epstein, "Particulate Fouling of Sensible Heat Exchangers," HE 1.6, Heat Transfer, (1970).  References  220  Watkinson, A.P. and D.L Wilson, "Chemical Reaction Fouling: A Review," Experimental Thermal and Fluid Science, 14, pp. 361 - 374 (1997). Wilke, C R . and P. Chang, "Correlation of Diffusion Coefficients in Dilute Solutions," A.I.ChE. Journal, pp. 264 - 270 (June 1955). Williams, D. And A Dunlop, "Kinetics of Furfural Destruction in Acidic Aqueous Media," I.E.C., 40, pp. 239 - 241 (February 1948). Wilson, D.L, "Model Experiments of Autoxidation Reaction Fouling," Ph.D. Thesis, The University of British Columbia (1994).  221  Appendix 1: Calibrations Appendix 1: Calibrations  91.6  10.4  T  -I  1  1  50  70  90  1  110  1  130  1  150  1  170  1  190  Rotameter Setting (mm) Figure A1.2: Low flow rate (LFR) rotameter calibration  1  210  j 230  1  250  Appendix 1: Calibrations  222  Appendix 2: Acid catalyzed 2-furaldehvde fouling Appendix 2 A2.1  223  Acid Catalyzed 2-Furaldehyde Fouling  Literature Review  ^  2-furaldehyde (furfural) is a colorless, liquid aldehyde with a pungent, aromatic odor reminiscent of almonds. Chemical reaction fouling involves reactions which lead to precursors which become insoluble in the fluid or at the wall because of molecular weight increases, or changes in polarity. The following review indicates the reasons 2-furaldehyde was selected as a model fluid for this study.  A2.1.1  2-Furaldehyde as a Decomposition Product 2-furaldehyde is a decomposition product of sugar and is present in most foods in various  concentrations. Detection of 2-furaldehyde is an indication of the deterioration of food products due to heat and / or storage time. Therefore, in this sense, the presence of 2-furaldehyde is considered a precursor to browning reactions that occur in food spoilage, and a candidate fouling precursor. Some previous research on 2-furaldehyde decomposition is discussed below. In a series of experiments where fructose and xylose were heated, the rate of coloration was measured spectroscopically and determined to have an activation energy of 105 - 126 kJ/mol (Del Pilar Buerna et al., 1987). When successive amounts of 2-furaldehyde were substituted (Tan et al., 1950) into samples containing xylose and glycine, the solutions started to brown at a faster rate, as shown in Figure A2.1.1. This indicates that 2-furaldehyde promotes browning, and hence decomposition reactions.  Appendix 2: Acid catalyzed 2-furaldehydc fouling  224  Effect of gradual substitution of 2-furaldehydc •oc D-jxjrtosc oa the rate of coloratioa ia the sugar solutioa: cucrc A. cquimolar (0.250 if) solutioa of D-xylose and thrAtK it reflux temperature; B. 10% of D-xylose ia A «U(itutcd by 2-furaldehydc; C. 60% substituted; D . 1<M% lubstituted. Ixunetroa (Model <02 E) photoelectric •Aximetcr. Essentially similar results were obtained Vrepeatincthis in a. phosphate buffer at £ H 4.9.  60 100 150 Tunc ia minutes.  200  Figure A2.1.1: Effect of 2-furaldehyde on browning rate of sugar solutions (Tan et al.,1950) Dunlop and Peters (1953) developed the following scheme (Figure A2.1.2) indicating the type of decomposition reactions that 2-furaldehyde undergoes. Resins or polymers are produced by all four reaction routes. 2-Furaldehyde  •  *  Autoxidation  Water & Acid  t  1  Acid, No Water  ® Acidic Products Formic Acid & Resin (Some Polymers)  © Resin  No Acid, No Water  ® Non-Acidic Resin & Water  Figure A2.1.2: Schematic of 2-furaldehyde reactions (Dunlop & Peters, 1953) When stored open to air, 2-furaldehyde darkens and undergoes autoxidation (A), albeit very slowly. The addition of tertiary amines, sodium carbonate and water inhibits the formation  Appendix 2: Acid catalyzed 2-furaldehyde fouling  225  of acid and hence color. By a series of experiments that measured the uptake of oxygen it was noted that autoinhibition occurred at the later stages of the reaction with autoxidation stopping at 7 -. 8 % 2-furaldehyde conversion. Because of the complexity of the 2-furaldehyde resinification reaction (Kirk-Othmer, 1980), the intermediates of pathway (D) have not easily been identified. Conditions of polymerization with aqueous or anhydrous, inert or oxygen atmospheres, all affect the composition of the polymer. More recent work, however, has been performed on the thermal resinification of 2furaldehyde and the prediction of black, insoluble resins and their intermediates (Gandini, 1977). Galego and Gandini (1975), in the absence of air, with heat to 150 - 230°C, identified several complex intermediates, but were unable to predict an ultimate mechanism. It was observed that basic substances and trace amounts of water decrease the rate of resinification. From an industrial standpoint the rate of 2-furaldehyde decomposition was considered to be very slow and essentially thermally stable (Dunlop & Peters, 1940). The resinification reactions (B & C) which could lead to fouling deposits are accelerated by heat, but even at 230°C many hours are required to produce detectable changes in the physical properties of 2-furaldehyde (Dunlop & Peters, 1953). Acidic materials however, accelerate the rate of resinification. Marcusson (1925) proposed the mechanism shown in Figure A2.1.3. The catalytic action of acids may be attributed to the loss of resonance stability of the 2-furaldehyde molecules as described by the scheme in Figure A2.1.4. This is due to the action of hydrogen ions.  226  Appendix 2: Acid catalyzed 2-furaldehvde fouling  CH2  H0 2  -HCOOH H O / HC1  CHO CHO -H 0  [H ] +  CH2 CH2 HCOH HCOH CH  CH  2  z  CHO CHO Humins "  !HO  " Humic Acids "  Aromatic of Lignin type  Figure A 2.1.3: Marcusson's (1925) proposed acid catalyzed decomposition mechanism of 2-furaldehyde (cited in Dunlop and Peters, 1953) Given that resinification is catalyzed by acids, it is not surprising that it is inhibited by alkalis, with strong alkalis inducing a different mechanism of decomposition. The exact structure of the resin is unknown, but according to the scheme proposed by Marcusson (1925), 1 mole of formic acid should be produced per mole of 2-furaldehyde decomposed. However, the most ever observed was 2/3 mole (Williams & Dunlop, 1948).  Figure A2.1.4: Schematic representing the loss of resonance stability in acidic conditions  227  Appendix 2: Acid catalyzed 2-furaldehvde fouling  A2.1.2  2-Furaldehyde Decomposition Kinetics In an acidic (H2SO4), aqueous solution of 2-furaldehyde, the estimation of 2-furaldehyde  content was made using the Hughes-Acree method (Williams & Dunlop, 1948). Previous published data reported an activation energy for this reaction of 84 kJ/mol and the following expression: logk = ^ ^ +7.145 T(K)  A2.1.1  Therefore d[2-furaldehyde]  =  ^ _ ^ 2  F E  D E H Y D E  | J^Q] H +  A2.1.2  but the concentration of acid and water remains essentially constant and therefore follows a pseudo first order reaction: d[2 - furaldehydel , , -i — ± = k[2 - furaldehyde]  A2.1.3  In another set of experiments, 25 g of 2-furaldehyde were added to a refluxing solution of 3.9 N HC1 (Hurd and Isenhour, 1932). After 3 hours, the mixture was cooled and 15 g of black precipitate recovered by filtration. The filtrate contained 6.6 g of unreacted 2-furaldehyde. Therefore 15 g of precipitate had been formed from 18.4 g 2-furaldehyde. Their conclusion was that strongly dissociating acids like HC1 and  H2SO4  promoted the resinification reaction. The  resin-forming reaction was considered to be a candidate for this fouling study.  Appendix 2: Acid catalyzed 2-furaldehvde fouling  A2.2  228  Experimental Apparatus and Methods  A2.2.1 Stirred Cell Reactor (SCR) The acid catalyzed thermal polymerization reactions of dilute solutions (1 wt. %) of 2furaldehyde in 0.1 N HC1 were studied in the batch reactor shown in Figure A2.2.1. The 2.3 liter reactor was constructed from 316 stainless steel and consisted of a stirrer (speed controlled by a G K H S-12 motor controller) to ensure complete mixing of the contents, a K-type thermocouple, and a pressure relief valve set to 150 psig to protect against over pressurizing the reactor. Due to the nature of the chemical system a 0 - 150 psig pressure gauge was installed to enable a visual check on the pressure.  Motor  Figure A2.2.1: Schematic diagram of the Stirred Cell Reactor (SCR) apparatus  Appendix 2: Acid catalyzed 2-furaldehyde fouling  .  229  A 1/8" sample line was used with a needle valve to remove samples at required times for analysis. Samples were collected in 20 ml disposable vials, placed immediately into an ice bath and stored in a dark cupboard until analysis. Prior to any experiment this sample line was used to purge nitrogen through the reactor for five minutes to eliminate as much air from the system as possible and therefore minimize the possibility of autoxidation. During an experiment, however, this line remained closed. The reactor was mounted into a stabilized mineral oil bath (boiling point = 360°C) which contained two 500 Watt Chromalox immersion heaters capable of heating the bath to approximately 200°C, and a stirrer. The oil bath was kept in a fume hood. A n experiment was initiated by heating the oil bath to approximately 10 - 20°C above the desired reaction temperature. This process would take approximately two hours, during which time the reactor was charged with the test fluid and purged with nitrogen. The reactor was then immersed into the bath, marking time zero for the experiment, where the temperature and pressure were recorded. Samples were taken every 20- 30 minutes while the oil bath and reactor stabilized to the desired experimental conditions. Depending on the reactor temperature, the length of an experiment varied, but typically with a reactor temperature of 169°C, it took 60 minutes to reach thermal equilibrium, and then isothermal conditions would run for another 160 minutes. An experiment was terminated once particulate matter was clearly visible in the sample vial. This indicated that the acid catalyzed polymerization reaction was proceeding and therefore quantifiable amounts of 2-furaldehyde had decomposed. This is shown in Figure A2.3.2. Once the experiment was terminated, the samples were stored in an ice bath overnight in a dark cupboard for analysis the following day. The oil bath and stirrer were turned off and the  Appendix 2: Acid catalyzed 2-furaldehyde fouling  230  reactor was removed and placed in the fume hood to cool down. The contents were examined, removed and disposed of the following day. Due to the toxic nature of 2-furaldehyde, protective clothing and butyl gloves were used. Initially the reactor was rinsed with water to remove any particulate matter and dilute any remaining solution. The reactor was then removed from the fume hood, thoroughly rinsed with acetone and scrubbed to remove any residue. Once the entire reactor was clean it was returned to the fume hood and allowed to dry.  A2.2.2 U V Spectrophotometer The amount of light absorbed by a species in solution will depend on the number of ions or molecules of the species in the light path of the photon beam. It follows that more light will be absorbed as the concentration of the absorbing species increases. Using a Varian 2390 UV-Vis NIR Spectrophotometer capable of wavelengths between 185 and 3152 nm, a range of wavelengths were employed to determine the maximum absorbance of dilute 2-furaldehyde solutions in 0.1 NHC1. The maximum concentration of 2-furaldehyde that the Spectrophotometer could detect was 25 ppm, when using quartz cells. The maximum absorbance occurred at approximately 270 nm, which agreed with literature values (Mackinney and Temmer, 1948), as shown in Table A2.2.1. Therefore to avoid detecting trace amounts of furan or hydroxymethylfurfural a wavelength of 270 nm was used. However, levulinic acid (CH3COCH2CH2CO2H) is also a decomposition product of acid catalyzed thermal polymerization of 2-furaldehyde, but since the true mechanism of the reaction was not determined, the exact quantities of this product were not known. Because it has the same absorbance wavelength as the major peak of 2-furaldehyde, trace amounts could easily be lost in the base of the 2-furaldehyde peak.  Appendix 2: Acid catalyzed 2-furaldehvde fouling  231  Table A2.2.1: UV Absorbance of furan derivatives (Fritz and Schenk, 1966) ^min  E (l.mol^cm' ) 1  Levulinic Acid  270  240  25.1  Furan  290  270  1.4  2-furaldehyde  272.5  240  14800  Hydroxymethyl Furfural  282.5  245  16900  2-furaldehyde however, has a second smaller absorption peak at 219 nm, where levulinic acid doesn't absorb. Therefore the ratios of these two peaks were compared as a function of time. If this ratio increased, then significant amounts of levulinic acid would have been produced. However, this ratio decreased with sampling time, which indicated that no significant amounts of levulinic acid were produced. The following standards were prepared to develop a calibration curve: 20, 15, 10, 5 and 1 ppm. The result of the linear least squares regression gave rise to the following calibration: A = 0.156 [2-furaldehyde(ppm)] + 0.094  r = 1,000 2  3.5  0  2  4  6  8  10  12  14  16  Concentration of 2-furaldehyde (ppm)  Figure A2.2.2: Calibration of UV Spectrophotometer  18  20  22  Appendix 2: Acid catalyzed 2-furaldehyde fouling  232  As a result, experimental samples of 1 wt. % 2-furaldehyde were diluted 500 times to produce a maximum possible sample concentration of 20 ppm.  A2.2.3 Fluid Physical Properties Solution Viscosity The viscosity, r\, of the 1 wt. % solution was determined from kinematic viscosity measurements and literature density values. A size 50 (kinematic viscosity range = 0.8 - 4.0 centistokes) Cannon-Fenske Opaque (Reverse-Flow) Viscometer was immersed in a Fenske viscometer •oil bath, and the kinematic viscosity determined at several temperatures. Figure A2.2.3 shows that the viscosity results for water are comparable to the literature values. It was also shown that for dilute 2-furaldehyde solutions, the kinematic viscosity was essentially identical to that of water. Hence it was assumed that the density could also be approximated by the density of water. 1.4  0.2  r  1  10  '  '  '  >  '  '  1  1  20  30  40  50  60  70  80  90  1  100  Temperature (°C)  Figure A2.2.3: Kinematic viscosity of a 1 wt. % 2-furaldehyde solution This approximation was shown to be valid, after a simple thermodynamic consideration. By treating the solution as ideal, the molar volume of the system is purely additive, based on the  233  Appendix 2: Acid catalyzed 2-furaldehvde fouling  components (Smith & Van Ness, 1987). By this approximation, and the assumption that the excess volume of mixing is zero, it was shown that at low concentrations, the solution takes on the properties of water, indicating a maximum possible error of less than one percent. Solution vapor pressure When dealing with a two-component system, it is important to consider both components to evaluate the vapor pressures and compositions of the solution at elevated temperatures. Assuming no interaction between 2-furaldehyde and dilute hydrochloric acid, one can approach this by employing the Antoine equation. Assuming the mixture is in the liquid phase, one can evaluate the bubble point as a function of composition. From Raoult's law,  y P = x Pr i  i  psat  =>Yi Ji  =-p— ii  And  Yi + y = 1  L  x  (A2.2.1)  2  p sat  p sat  => — X . + - ^ — X ,  P  '  p sat  o^-Z  P  =1  2  p sat 1 +  - i - ( l - Z ) =l 1  The saturation pressures of 2-furaldehyde and water can be approximated as follows (Reklaitis, 1983): P^u^de  = exp[16.7802-5365.88/(T + 5.6186)] (A2.2.2)  P^ter  = exp[16.5362 - 3985.44/ ( T - 38.9974)]  Combining these equations, the total pressure P is given by P = Z,exp[16.7802 - 5365.88 / (T + 5.6186)] + (1 - Z , )exp[16.5362 - 3985.44 / ( T - 38.9974)] (A2.2.3)  Appendix 2: Acid catalyzed 2-furaldehvde fouling  234  where Zi is the mole fraction of 2-furaldehyde and T is the temperature (K). It can be shown that the vapor pressure of 2-furaldehyde is slightly less than that of water, and therefore the vapor pressure of the system will never be greater than that of water. Hence pressure considerations are simplified by use of the water vapor pressure. For dilute aqueous solutions of 2-furaldehyde, this organic compound contributes little to the physical properties of the system. A2.2.4 Tube Fouling Unit (TFU) Many of the calculations introduced in Sections 3.1 and 4.1 also apply to acid catalyzed 2-furaldehyde fouling, but in these experiments fouling was not a strong function of wall temperature and therefore overall heat transfer coefficient calculations were also performed. Calculation of the overall inside heat transfer coefficient From a heat balance over a differential increment of tube, d Q =cL4 U (T (x)-T (x)) = mC dT i  i  wi  b  T (T (x)-T (x)) bl  wi  b  p  4 . 0 mC  b  p  Therefore integrating the area under the plot of  1  versus Tb enables  determination of Uj.  (A2.2.5)  Appendix 2: Acid catalyzed 2-furaldehyde fouling  235  Specific heat capacity values (C ) of the solution were approximated by those of water, and p  therefore given the potential for slight errors in these values, mC was replaced by Q/AT p  b>  i.e. by  K7/(T ut - T i n ) , in Equation (A2.2.5). Heat losses were assumed to be insignificant. b>0  b >  .-. U j (t) =  VI  -.  :  o  * Area under curve (W/m K) 2  — r  (A2.2.6)  ( b,o«, - T b . i n J T t d i L T  It was determined that a plot of the reciprocal of the inside wall minus bulk temperature versus bulk temperature produced essentially a straight line. Therefore, a linear regression was performed to enable easier integration. Using this technique, it was a matter of performing this calculation at each time interval to evaluate the overall heat transfer coefficient as it changed throughout the experiment. Calculation of the average initial inside wall temperature The following procedure was used: 1.  Over a relatively short time period at steady state, the time-average inside wall  temperature for each thermocouple was determined, and a linear regression performed: T ( x ) = ax + b wl  2.  (A2.2.7)  The space-time average inside wall temperature was determined from  ?T .i(x)dx w  T  w,i = "Ss  (A2.2.8)  Jdx xl  3.  Since the overall average inside wall temperature could be approximated by the above  formula, the integration was performed over the length of the test section.  236  Appendix 2: Acid catalyzed 2-furaldehyde fouling  Additional safety precautions Due to toxicity of the chemical system, and to the high temperatures and pressures of operation, a shield and safety curtain were installed to encase the apparatus. A n extraction fan inside the curtain was utilized such that if there was an accidental chemical release, the operator had protection and the toxic fumes would be vented outside the building. The T F U was tested to 100°C and 790 kPa with 100% heat flux for several hours, and no leaks or problems with the equipment were detected.  A2.3  Kinetic Experiments The purpose of these kinetic experiments was to verify the literature data which indicated  that acid catalyzed decomposition of 2-furaldehyde is a first order reaction and has a chemical activation energy of 84 kJ/mol (Williams and Dunlop, 1948). Six experiments were performed to establish the operating window available for this model solution. A reproducible method to analyze for the decomposition of 2-furaldehyde was developed. A summary of the results is shown in Table A2.3.1. Table A2.3.1 shows the kinetic results from the U V Spectrophotometer to be more reliable than those from the G C . Experiments SCR003, 004 and 005 indicate that the reaction in the presence of oxygen proceeds faster than in a nitrogen environment. However, given the proposed decomposition mechanism, the autoxidation reaction had to be avoided. Figure A2.3.1 shows the decomposition results for SCR006, and the first order kinetics used to describe the reaction. With a correlation coefficient (r ) of 0.97, the linear regression fit is 2  considered reasonable. Consideration of other reaction orders produced no better correlation. In  237  Appendix 2: Acid catalyzed 2-furaldehvde fouling  the regression of the experimental data, the first three data points were ignored, because these were taken during the initial heat-up period of the SCR. Table A2.3.1: Summary of Stirred Cell Reactor (SCR) experiments Atmosphere  Exp.  [2-furaldehyde]  T (°C)  t (min)  Analysis  k (min )  1  1  (g/100 ml) SCR001  Air  1.002  124  220  GC  -  SCR002  Air  1.002  115  300  GC  0.00  SCR003  Air  0.995  144  280  GC  2.10xl0-  SCR004  Nitrogen  1.000  150  290  GC  1.90xl0"  SCR004  Nitrogen  1.000  150  290  uv  1.45xl0"  SCR005  Nitrogen  1.008  130  300  uv  7.79X10"  SCR006  Nitrogen  1.007  169  180  uv  2.78x10"  Literature  Nitrogen  1.000  160  210  Hughes*  2.16xl0"  3  3  4  3  3  * Hughes-Acree method of analysis Average temperature over the duration of the experiment 1  1.0  j  |  0.9  1  0.8 \  0.4  H  r  1  1  1  1  1  1  1  1  0  20  40  60  80  100  120  140  160  180  •  3  1  i  200  220  Time (minutes)  Figure A2.3.1a: Decomposition reaction of 2-furaldehvde in SCR006  238  Appendix 2 : Acid catalyzed 2-furaldcliydc fouling  40  60  80  100  120  140  160  ISO  200  220  Time (minutes)  Figure A2.3.1b: Kinetic description of 2-furaldehvde decomposition in SCR006 Figure A2.3.2 shows the color degradation of the solution throughout SCR006. As discussed in section A 2 . 1 , the degradation of 2-furaldehyde goes through a series of color transformations from a clear solution when freshly distilled to a yellow to brown clear solution to something that is murky brown to black, with visible particulate matter suspended in solution.  V . o  aa  ho  to  mo  *>  too  wo  3,f,»/it  i »  »—  jscm.oo*  US  Utfl  T.n<fc  USS OS  }  <r_.  !Sfi—iSS  i «/.oo..i  <U£>  m  Figure A2.3.2: Photograph of 2-furaldehvde samples from SCR006 (a), 169°C  Appendix 2: Acid catalyzed 2-furaldehvde fouling  239  Analysis of the U V experimental results (Figure A2.3.3) show that first order kinetics adequately describe the decomposition reaction, with an activation energy of 48 kJ/mol. These results apply to temperatures of 100 -170°C. Figure A2.3.3 shows a summary of these results and enables a comparison between the experimental and literature results. Note that the one data point from the literature using HC1 as the catalyst fits the experimental data achieved in this study. Also note that the results with  H2SO4  are different than that with HC1, but this may be  related to the fact that the normality of the acidic solutions could have been incorrectly interpreted (Williams and Dunlop, 1948). l.OE-01 0.00205  0.00210  0.00215  0.00220  0.00225  0.00230  l.OE-02  0.00235  0.00240  0.00245 0.00250  y=1296.7e S783.6x R = 0.9961 2  B  1.0E-03 • O  Experiment Literature (HCI) Expon. (Experiment) . Expon. (Literature (H2S04))  1.0E-04 1/T(K) Figure A2.3.3: Temperature dependence of acid catalyzed decomposition of 2-furaldehvde For the experimental conditions described above the kinetic activation energy AE = 48.1 kJ/mol (c.f. 83.6 kJ/mol) and the pre-exponential factor, A = 1297 min" (c.f. 1.396 x 10 min' ). 1  A2.4  7  1  Fouling Experiments Due to unforeseen corrosion problems, only three fouling experiments were completed.  The presence of corrosion is of interest, because it poses a problem that is commonly observed in industry, i.e. fouling may occur by a combination of mechanisms, including corrosion.  Appendix 2: Acid catalyzed 2-furaldehyde fouling  240  The three fouling experiments were performed at essentially the same mass flux (304.2 kg/m s). This enables a somewhat limited examination into the effect of wall temperature on the 2  combined corrosion and chemical reaction fouling problem. Table A2.4.1 highlights the operating conditions used.  Table A2.4.1: Overall 2-furaldehvde fouling experiment summary [2-furaldehyde] (g/100 ml)  101  TFU  G (kg/m s)  P (kPa)  py. (°C)  (m K/kJ)  1.003 ±0.003  304.2  480  109.9  1.007 x 10'  102  1.000 ±0.003  304.2  715  139.6  103  0.997 ± 0.003  304.2  659  132.1  2  T (°Q  Induction Time (s)  6  99 -107  49500  2.083 x IO  -6  100-127  18,480  1.691 x IO  -6  99 -122  5,000-10000*  b  Rfo 2  * Induction time very difficult to determine due to uncertain onset of combined fouling mechanisms  Given that the bulk temperature of the solution was maintained at 100°C (± 2 °C) and that the mass flux was essentially constant, the Reynolds numbers for all three experiments calculated at the test section inlet were the same at 9720. A fouling Arrhenius plot of this data is shown in Figure A2.4.1. The fouling activation energy = 31.7 kJ/mol, which is considerably smaller than the purely kinetic activation energy of 48.1 kJ/mol. Given the level of corrosion experienced during the fouling experiments, an experimental activation energy (from the kinetic experiments of Appendix 2.3) of 48.1 kJ/mol compared to 83.6 kJ/mol reported by Williams and Dunlop (1948) is understandable. It is possible that dissolved metal ions in the SCR could have been catalyzing the thermal polymerization steps. From these three averaged experimental results, the Arrhenius expression from Figure A2.4.1 is given by  Appendix 2: Acid catalyzed 2-furaldehvde fouling  R l.OE-05 0.00240  fo  0.00245  241  = 0.021exp[-31674 / R ( T ) j  (A2.4.1)  wi  0.00250  0.00255  0.00260  0.00265  R  „ 1.0E-06  _c -  y = 0.0209e- R = 0.9973  38M 71  "3  2  o  fa  1.0E-07 l/(T ) (K") 1  Wli  c  Figure A2.4.1: Arrhenius plot for TFU 100 series (3-Point Plot) The foregoing analysis was based on the average fouling rate and wall temperature integrated along the length of the tube. A different approach is to use all of the meaningful local experimental data, but use a least squares regression to determine the trend. This is shown in Figure A2.4.2, and the data shown in Table A2.4.2. l.OE-05 0.00235  0.00240  0.00245  0.00250  0.00255  0.00260  0.00265  5 C3  05 ex  1.0E-06  c  y = 0.002 le-2891.3x R = 0.5133  o fa  :  1.0E-07 l/(T ) (K" ) 1  w>i  c  Figure A2.4.2 Arrhenius Plot for TFU 100 series (24-Point Plot) TFU 101,102 and 103: 24 open points are local data, 3 closed points are averages  Appendix 2: Acid catalyzed 2-furaldehvde fouling  242  The resulting Arrhenius equation is somewhat different from Equation A2.4.1. R  = 0.002exp[-24038 / R(T . ) ]  f0  W  (A2.4.2)  c  Caution should be displayed in interpretation of the results shown here; however it is clear that there is a wall temperature effect in this combined chemical reaction / corrosion fouling process of the acid catalyzed decomposition of 2-furaldehyde. Table A2.4.2: Individual 2-furaldehvde fouling experiment summary  (U  CU  Location  TFU101  #  (mm)  (m K/kJ) xlO  1  48  1.690  104.9  2.220  124.6  0.770  123.6  2  110  -  -  2.440  127.3  -  -  3  163  0.995  108.7  -  -  -  -  4  221  1.080  106.9  2.080  134.1  1.110  129.8  5  330  1.040  106.8  2.110  139.4  1.430  131.3  6  440  0.958  110.8  2.380  144.5  1.650  132.4  7  550  -  -  1.980  145.8  1.530  134.6  8  605  1.060  110.5  -  -  2.440  135.8  9  660  1.300  114.0  2.250  151.3  2.340  140.7  10  713  0.897  113.3  1.900  151.3  2.430  140.7  All  -  1.007  109.9  2.083  139.6  1.691  132.1  2  TFU102  CU  T/C  (m K/kJ) xlO 2  6  CO  6  TFU103 (m K/kJ) xlO 2  CO  6  CO  243  Appendix 2: Acid catalyzed 2-furaldehvde fouling  A2.5 Application of Experimental Results to Model A s discussed i n A p p e n d i x 2.4, the results o f these experiments are i n c o n c l u s i v e because an. insufficient n u m b e r were p e r f o r m e d . H o w e v e r , i f first order kinetics are a s s u m e d ( A p p e n d i x 2.3), then it is p o s s i b l e to p e r f o r m a p r e l i m i n a r y analysis. E q u a t i o n (2.2.25) w i t h n =  1 was  utilized:  C  kSc  h  2 / 3  —^ =  kpe  A E f / R T  »V.  2  +—  R  V  *  (2.2.25) 11  ^fo  T h e o n l y u n k n o w n s are the t w o constants, k  a n d k , a n d the kinetic activation energy  ( A E ) . A l t h o u g h the latter activation energy was determined i n the kinetic experiments,  this  parameter was left as a n u n k n o w n i n the m o d e l . T h e r e f o r e , three u n k n o w n s required a m i n i m u m o f three equations a n d hence all three averaged experimental results to determine a solution. Table  A2.4.1  illustrates  the  experimental  results  utilized  here.  All  three  fouling  experiments were p e r f o r m e d at a constant mass flux. A t this m a s s f l u x , the f o u l i n g rates were averaged over a range o f temperatures, a n d the relative change i n b u l k temperature  between  experiments was s m a l l . F o r these reasons, variations i n p h y s i c a l properties w e r e c o n s i d e r e d s m a l l  and l u m p e d into the constants, k  and  k  o f E q u a t i o n (2.2.25). A n equation i m p l i c i t i n A E  resulted a n d was s o l v e d u s i n g a spreadsheet. A  c h e m i c a l activation energy o f 25.9  kJ/mol  resulted, w h i c h is clearly an inaccurate answer, since this predicts an activation energy l o w e r than recorded for both the f o u l i n g (31.7 k J / m o l ) and kinetic experiments (48.1 k J / m o l ) , w h i c h appears unreasonable.  T h e m o d e l solution for this data was s h o w n to be sensitive to all experimental results, and although there  appeared to be little scatter i n the f o u l i n g results o f F i g u r e A 2 . 4 . 1 ,  small  244  Appendix 2: Acid catalyzed 2-furaldehvde fouling  deviations from the Arrhenius line affected the model solution. Therefore this analysis was repeated by taking Rf values directly from the Arrhenius plot at the three experimental values of 0  T -. Solution for the activation energy is shown in Figure A2.5.1. w  From this brief analysis an activation energy of A E = 45.8 kJ/mol was determined with only a small variation in the data used for the model analysis. This result is comparable to the 48.1 kJ/mol from the kinetic experiments. These results show that solution of the mathematical model is extremely sensitive when only a few experimental results are available. 0.08  -0.08 A E (J/mol)  Figure A2.5.1: Solution of Equation (2.2.25) to find A E for T F U 100 Experiments Consistent with the kinetic results from Appendix 2.4 the same model fit was performed considering all 24 data points from the three fouling experiments. This solution was determined by minimizing the sum of the squares of the residuals between the calculated and experimental initial fouling rates. The optimum solution for all data points occurred at A E = 42.7 kJ/mol. A model solution was attempted by considering variation in physical properties and fluid velocity along the length of the test section for each experiment. However, no solution leading to  Appendix 2: Acid catalyzed 2-furaldehvde fouling  245  a positive activation energy was achieved. Again, this indicates the sensitivity of the model solution when such few experimental data are available. From this study the chemical activation energy for HC1 catalyzed decomposition of 2furaldehyde has been estimated as 48.1 kJ/mol from the kinetic experiments, and 42.7 - 45.8 kJ/mol from the simplified modeling of the fouling experiments.  246  Appendix 3: Whey protein solutions  Appendix 3 Whey Protein Solutions A3.1 Composition of WPC-80 Powder 120296 Good Good Good 76.4 % 80.1 % 4.6 % 3.2 % 4.5 % 7.99 < 0.1 ppm < 0.1 ppm < 5000/g <10/g neg. in 11 g neg. in 25 g  Lot Number Flavor and odor Color Texture Crude protein (N x 6.38) Wet Basis Crude protein QN x 6.38) Dry Basis Moisture Ash Crude Fat Lactose PH Arsenic Heavy Metals (Cd, Pb, Hg) Standard plate count Coliform Staphylococcus Aureus Salmonella A3.2  A702 Good Good Good 76.3 % 80.3 % 5.0 % 3.8 % 6.9 % 4.5 % 7.05 < 0.1 ppm < 0.1 ppm 900/g <10/g neg. in 25 g  Fouling Resistances for T F U 200 Experiments Used for Deposit Property Analysis TFU  U (kW/m K)  U (kW/m K)  208  2.786 2.274 1.967 1.873 1.674 1.567 1.514 1.596 1.512 1.584  2.720 2.255 1.923 1.828 1.600 1.450 1.368 1.307 1.095 0.979  R (m K/kW) End of exp. 0.0087 0.0037 0.0116 0.0131 0.0276 0.0515 0.0705 0.1385 0.2519 0.3901  207  4.065 3.721 3.502 3.819 3.883 3.920 4.215 4.487 4.278 4.599  3.953 3.508 3.063 3.107 3.046 2.856 2.949 3.051 2.615 2.765  0.0070 0.0163 0.0409 0.0600 0.0708 0.0950 0.1019 0.1049 0.1487 0.1442  211  1.634 1.285 1.095 1.108  1.525 1.216 1.021 1.011  0.0437 0.0442 0.0662 0.0866  * *  2  c  2  f  2  f  Appendix 3: Whey protein solutions  247  0.989 0.902 0.867 0.853 0.778 0.784  0.886 0.806 0.758 0.721 0.603 0.555  0.1175 0.1320 0.1659 0.2146 0.3730 0.5263  2931 2750 2437 2435 2373 2325 2528 2627 2466 2605  2917 2678 2370 2324 2105 1984 2129 2098 1790 1803  0.0016 0.0098 0.0116 0.0196 0.0537 0.0739 0.0741 0.0960 0.1531 0.1708  209  6.121 5.901 5.508 5.661 5.720 5.897 6.340 6.362 6.081 6.355  6.029 5.665 5.288 5.436 5.266 4.798 4.691 4.315 3.714 3.483  0.0025 0.0070 0.0076 0.0073 0.0151 0.0388 0.0554 0.0746 0.1048 0.1298  205  6.337 6.432 6.143 6.350 6.362 6.417 6.903 7.304 6.708 6.882  6.151 5.999 5.835 5.991 5.997 5.847 6.047 6.164 5.549 5.315  0.0048 0.0112 0.0086 0.0094 0.0096 0.0152 0.0205 0.0253 0.0311 0.0428  210  8.619 8.254 7.667 8.083 7.924 7.792 8.016 8.244 7.386 7.697  7.981 7.820 7.696 7.741 8.021 7.050 7.243  0.0016 0.0017 0.0016 0.0044 0.0034 0.0065 0.0081  * *  204  * * *  * Thermocouple displayed accelerated fouling rates  Appendix 3: Whev protein solutions  A3.3  248  Thermocouple Location with respect to Cut Test Sections  Section #  Location (mm)  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18  0.0-50.8 50.8-101.6 101.6-152.4 152.4-203.2 203.2-254.0 254.0-304.8 304.8-355.6 355.6-406.4 406.4-457.2 457.2-508.0 508.0-558.8 558.8-609.6 609.6-660.4 660.4-711.2 711.2-762.0 762.0-812.8 812.8-863.6 863.6-914.4  Thermocouple  Section #  Location (mm)  19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36  914.4-965.2 965.2-1016.0 1016.0-1066.8 1066.8-1117.6 1117.6-1168.4 1168.4-1219.2 1219.2-1270.0 1270.0-1320.8 1320.8-1371.6 1371.6-1422.4 1422.4-1473.2 1473.2-1524.0 1524.0-1574.8 1574.8-1625.6 1625.6-1676.4 1676.4-1727.2 1727.2-1778.0 1778.0-1828.8  1 @ 594 mm 2 @ 656 mm 3 @ 709 mm 4 @ 767 mm 5 @ 876 mm  Thermocouple  6 @ 986 mm 7 @ 1096 mm 8 @ 1151 mm 9 (Sj 1206 mm 10 @ 1259 mm  A3.4 Velocity and Temperature Distribution Calculations for Whey Protein Fouling Experiments The goals of this work were: 1. To calculate the thickness of the viscous sublayer and buffer layer for a high and a low flow fluid velocity (882 and 221 kg/m s). 2  2. To examine the calculated temperature  profiles throughout  the test section,  particularly in the viscous sublayer and buffer layer, in order to estimate what volume of fluid is capable of undergoing deposition reactions. From the universal velocity distribution: (a) Viscous Sublayer: u = y +  0<y <5  +  +  (b) Buffer Layer:  u = 51n y - 3.05  5<y <30  (c) Turbulent core:  u = 2.51ny +5.5  30<y  +  +  +  +  +  +  249  Appendix 3 : Whey protein solutions  u =v /V., +  x  V.  =vVf72,  f -0.25(1.821ogio(Re)-1.64)-  2  y = ypV./ri = (y/d)Re VrV2 +  The thickness of the viscous sublayer or buffer layer can be evaluated by setting y = 8y or 5 B , i.e. y = 5 or 30 respectively, in +  ^  _ V o r B _  yvorB  d  ReVf72  The temperature profiles in these layers can be evaluated using the Martinelli analogy (1947) between heat and momentum transfer under conditions of constant and uniform heat flux. A number of assumptions were involved in using this analogy: •  Fully developed thermal and velocity profiles. Fortunately, for the thermocouples of most interest (thermocouples 9 and 10 near the tube exit where x = 660 mm, and 713 mm, respectively), the flow is likely to be fully developed. Hydrodynamic Entry Length: (Incropera and DeWitt, 1990) Turbulent Re: 10 < x/d < 60 : CONDITION SATISFIED Laminar Re:  x/d = 0.05 Re @ Re = 2000, x = 901.7 mm. CONDITION SATISFIED B Y T/C 6 (986 mm)  Thermal Entry length: (Incropera and DeWitt, 1990) Turbulent Re: x/d = 10 : CONDITION SATISFIED Laminar Re:  x/d = 0.05 RePr @ Re = 2000, Pr = 3.5, x = 3156 mm.  Appendix 3: Whey protein solutions  250  CONDITION N E V E R SATISFIED Clearly the thermal boundary layer is not developed for purely laminar experiments. The Martinelli analogy (1947) is only valid for turbulent flow. According to Georgiadis (1998) and Toyoda et al. (1994) the thermal boundary layer thickness in turbulent flow is related to the momentum boundary layer thickness by:  5  where the momentum boundary layer thickness is given by the thickness of the viscous sublayer. Therefore, since 1 < Pr < 10 the thermal boundary layer will likely fall within the viscous sublayer. •  The properties of the fluid to which heat is being transferred are postulated to be independent of temperature. However, for this analysis, the physical properties were evaluated at an average temperature (film temperature) between the points of interest. Thus, for calculating the viscous sublayer thickness, the wall temperature and the temperature at the edge of the viscous sublayer were averaged; for calculating the thickness of the buffer layer, the temperatures at the edge of the viscous sublayer and the edge of the buffer layer were averaged; for calculating the contributions of the turbulent core, an average temperature was taken between the edge of the buffer layer and the center line temperature.  •  The eddy diffusivity for heat and momentum were assumed to be equal and the ratio therefore taken to be unity.  Appendix 3: Whey protein solutions  251  (a) Viscous sublayer (0<y <5) +  ccPr  T„, - T  Re ccPr + ln(l + 5aPr) + 0.5Fln V60 where a = SH/SM =1-0 F = ratio of the total thermal resistance of the turbulent core due to molecular and eddy conduction, to the thermal resistance of the turbulent core due to eddy conduction only. That is, it is a measure of the importance of molecular conduction in the turbulent core.  T.., - T T -X b w  T  -T w  Pr  c  V T. - T  Pr + ln(l + 5Pr) + 0 . 5 F l n ( ^  ^  (b) Buffer Layer (5<y <30) +  T -T  T  T.., - T ,b  VT  -T w  Pr + lri l + Pr ^--1 V Vy, )  c  - T .  Re Pr + ln(l + 5Pr) + 0.5Flnf— '/£  V60V^2  (c) Turbulent Core (30<y ) +  T.., - T  T. - T  T.., - T ,b  V T.., - T ,  Pr + ln(l + 5Pr) + 0.5Fln  Re  f/U.  Re Pr + ln(l + 5Pr) + 0.5Fln V60  The function F can be calculated according to the following two equations, but unless the Prandtl number is very low (< 0.01) this factor is approximately 1.0. For this work, F was assumed to equal 1.0.  Appendix 3: Whey protein solutions  252  + Vi +  V r 20T lrJ 1 1-V1+20X F  =  Re  2mf^  J  0  -1  -4T+20X  +  It  where X can be evaluated from  and yi is the thickness of the viscous sublayer and y2 is the distance from the pipe wall to the edge of the buffer layer. The Peclet number is defined as the product of the Reynolds and Prandtl numbers based on fluid properties in the turbulent core.  The temperature correction  f1  _ T w  VT -T.  A c  can be estimated from a figure provided by  W  Martinelli (1947), or from the temperature and velocity distributions by considering the following integral:  f\v  "^b T. - T  where V / V  M A X  and A T / A T  5.5 + 2.51n V  max  T.„ - T  J  " V' .  " T,„ ~ - T„  m a x  °  V  0  V _  R  {—^-rdr  are given by  f jr^ Re  r7T  Re 5.5 + 2.51n V2  rdr  w  AT AT „  Pr + ln(l + 5Pr) + 0.51n  'Re_r_ 60 r  Pr + ln(l + 5Pr) + 0.51n V60  ^  253  Appendix 3: Whev protein solutions  The temperature and velocity distributions for T F U 209 and T F U 211 are shown in Figures A3.4.1 - A3.4.4, where T(Viscous) = T(y = 5), T(Buffer) = T(y = 30) and T(Centre) = T(y/r = +  +  1.0). 100 .»  90 80  >»  »•  .a  •  CT70  | 60 £ so  40 30 20  100  300  200  400  500  600  700  800  Axial Location (mm) • Tw  - o - T(Viscous)  - » - T(Buffer)  -Tb  Figure A3.4.1: Axial temperature profiles for TFU 209,  V  - •* - T(Centre) avP  ra.»f  = 0.8860 m/s  100 90  1.2 •  Temp (x = 713 mm)  •  Velocity (m/s)  1.0 0.8 ~ 0.6 2 > 0.4 t0.2  50  40 0.0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  y/(r„)  Figure A3.4.2: Radial temperature and velocity profile for T/C 10 TFU 209, V = 0.8860 m/s a v f r a g f  1.0  0.0  fe  Appendix 3: Whey protein solutions  254  110 r  20  I  '  '  '  '  '  '  0  100  200  300  400  500  600  1  1  700  800  Axial Location (mm) |  --»--Tw  — - o — T(Viscous)  --»-T(Buffer)  —I—Tb  - * - T(Centre)  Figure A3.4.3; Axial temperature profiles for TFU 211, V , avf  0.0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  agP  0.8  = 0.2225 m/s  0.9  1.0  ylr.  Figure A3.4.4: Radial temperature and velocity profile for T/C 10 TFU 211, V = 0.2225 m/s a v P r a g f  From Figures A3.4.2 and A3.4.4, one notes that the viscous sublayer and buffer layer for T F U 211 occupies up to 20 % of the radial distance of the test section, compared to just 5 % for  Appendix 3: Whey protein solutions  255  T F U 209. This result suggests that the wall layer plays a significant part in the heat and mass transfer and therefore in the fouling results for this low flow experiment. The deposit and layer thicknesses are given in Table A3.4.1. Although not protruding into the buffer layer, the deposit in T F U 209 reaches much closer to that layer than in T F U 211, and therefore is likely to experience considerably more turbulence. Deposit formation in T F U 211, however, occurs in a thick viscous sub-layer followed by a very significant buffer layer. Table A3.4.1: Comparison of deposit, sublayer and buffer layer thicknesses at axial positions x = 48 - 713 mm Experiment  TFU 211  TFU 209  Viscous sublayer thickness Buffer layer thickness  0.18-0.13 mm 1.34-0.94 mm 0 - 0.065 mm  0.05 - 0.04 mm 0.38 - 0.28 mm 0 - 0.038 mm  Deposit thickness 1.4 1.2 1.0 0.8 h .§ 0.6 u 0.4 H  0.2 0.0 100  200  300  400  500  600  700  800  Axial Location (mm) - TFU 209 Viscous —•— TFU 209 Buffer —A— TFU 211 Viscous —O— TFU 211 Buffer  Figure A3.4.5: Comparison of viscous and buffer layer thicknesses for TFU 209 & 211 Deposition is reported to commence (Belmar-Beiny et a l , 1993) once temperatures exceed 65°C; therefore a calculation was performed to estimate the thickness of the volume of  256  Appendix 3: Whey protein solutions  fluid at which the temperature exceeds 65°C. This calculation is shown along the length of the test section in Figure A3.4.6. 0.30 r  -0.05  • -0.10 L  A x i a l L o c a t i o n (mm)  Figure A3.4.6: Thickness of whey protein solutions with T> 65°C  L = 0.713 m  r= 4.5085 mm  • j  Ar Centre Line  dx Figure A3.4.7: Schematic of film thickness for deposition in the test section The regressions in Figure A3.4.6 can be used to estimate the test section volume that can undergo these reactions, as displayed in the schematic in Figure A3.4.7.  Appendix 3: Whey protein solutions  257  Ar(TFU211) = 4.5085 .10" -1.10 .10" x - 3.05 .10"* x  (m)  Ar(TFU209) =4.5085 .IO" - 4.93 .IO x - 1.27 .10" x  (m)  3  4  3  The volume of the test section = wfh  =  Volume of the test section below 65°C  =  -5  4  5  0.713  3  2  j dV = J 7t(Ar) dx 0  =  2  4.5530619 .10" m L  where Ar  2  0  a - bx - cx, as given above.  The results are shown below in Table A3.4.2. Table A3.4.2: Comparison of reaction volumes of T F U 211 and T F U 209 Experiment  T F U 211  Viiquid > 65°C (HI ) Volume % capable of deposition along tube length 3  2.5223 .IO"  T F U 209 1.0743 .10"  5.54 %  2.36 %  6  6  The volume of fluid capable of deposition reactions for T F U 211 is 2.35 times the volume capable of deposition for T F U 209. Also, this comparison has been integrated along the length of the tube. More significant differences would be observed at the hotter sections of the tube, towards thermocouples 8-10. Bearing in mind the difference in deposit morphology observed with the S E M photographs for thermocouples 9 and 10 from these two experiments (Figures 4.2.16 and 4.2.17), the more voluminous nature of the T F U 211 deposit may be explained. Rather than bulk reactions being responsible for the observed deposit nature and accelerated fouling rates, the above calculations suggest that these reactions could occur completely within the wall region, which occupies up to 37 % (Figure A3.4.5) of the cross sectional area at the hot end of the test section in experiment T F U 211.  Appendix 3: Whey protein solutions  258  In comparison, a similar analysis was performed for the lysozyme fouling experiments. As with the whey protein experiments, it was assumed that deposition was significant at bulk temperatures greater than 65°C. Experimental conditions of low and high mass fluxes were considered (200 and 1100 kg/m s). Given that lower wall and bulk temperatures were used in this 2  series of experiments, the volume of fluid at temperatures greater than 65°C was estimated to be almost completely within the viscous sublayer. At 200 kg/m s, the thickness of this volume of fluid at the hot end of the test section was 0.15 mm, occupying only 6.5 % of the cross sectional area. Although deposit morphology was not studied for this series of experiments, this thin reaction zone over a range of mass velocities goes some way to explaining the absence of subsequently increased fouling rates (and hence larger sized aggregates) in the lysozyme fouling experiments, compared to the whey protein experiments.  259  Appendix 4: Lysozyme solutions  Appendix 4 A4.1  Lysozyme Solutions  Analysis of Food Grade Lysozyme Chloride (Powder)  Lot# Source Appearance Ash Moisture PH Solubility Bulk density Chloride Standard plate count Yeast / Mold Storage Activity  A4.2  Specification Chicken egg white Dialyzed, white amorphous powder Max. 2.0% Max. 6.0% 3.5-4.5 Min. 95% T (1.5% solution O D Min. 0.5 g/ml  6 4 0  nm)  Max 3.5% < 100 CFU/g < 10 CFU/g Original package at cool temperature Min. 95% (min. 22800 Shugar Units/mg)  A8062  990A  0.44% 5.65% 4.01 99.7% 0.5  0.71% 5.09% 4.35  test passed < 100/g  99.5% 0.5 test passed < 100/g  <10/g  <10/g  test passed  test passed  Enzymatic Assay of Lysozyme Two methods for the enzymatic assay of lysozyme are given below, one by Sigma Co.  and the other by Canadian Inovatech Inc.  Appendix 4: Lysozyme solutions  260  ^  SIGMA  Sigma Quality Control Test Procedure Enzymatic Assay of L Y S O Z Y M E 1 (EC 3.2.1.17) SIGMA PRODUCT NUMBERS: L l 129, L2879, L6255, L6394, L6876, L7001, L7773, L8402 PRINCIPLE: Micrococcus lysodeikticus Cells (Intact)' vsozyme^ Micrococcus lysodeikticus Cells (Lysed)  CONDITIONS: T = 25°C, pH = 6.24, A450nm, Light path = 1 cm M E T H O D : Turbidimetric Rate Determination REAGENTS: A. 66raMPotassium Phosphate Buffer, pH 6.24 at 25°C (Prepare 100 ml in deionized water using Potassium Phosphate, Monobasic, Anhydrous, Sigma Prod. No. P-5379. Adjust to pH 6.24 at 25°C with 1 M KOH.) B. 0.015% (w/v) Micrococcus lysodeikticus Cell Suspension (Substrate) (Prepare 25 ml in Reagent A using Micrococcus lysodeikticus, A T C C 4698 lyophilized cells, Sigma Prod. No. M-3770. The A450nm of this suspension should be between 0.6 and 0.7.) C. Lysozyme Enzyme Solution (Immediately before use, prepare a solution containing 200 - 400 units/ml of lysozyme in cold Reagent A.) PROCEDURE: Pipette (in milliliters) the following reagents into suitable cuvettes: Test Blank Reagent B (Substrate) 2.50 2.50 Equilibrate to 25°C. Monitor the A450nm until constant, using a suitably thermostatted spectrophotometer. Then add:  Appendix 4: Lysozyme solutions  261  Reagent C (Enz Soln) 0.10 Reagent A (Buffer)  0.10  Immediately mix by inversion and record the decrease in A450nm for approximately 5 minutes. Obtain the delta A450nm/minute using the maximum linear rate for both the Test and Blank. CALCULATIONS: (AA450nm/min Test - ZWUsonm/min Blank)(df)  Units/ml enzyme =  :  (0.001) (0.1) df= Dilution factor 0.001 = Change in absorbance at Attonm as per the Unit Definition 0.1 = Volume (in milliliter) of enzyme used units/ml enzyme Units/mg solid = mg solid/ml enzyme units/ml enzyme Units/mg protein = mg protein/ml enzyme  UNIT DEFINITION: One unit will produce a delta A450nm of 0.001 per minute at pH 6.24 at 25°C using a suspension of Micrococcus lysodeikticus as substrate, in a 2.6 ml reaction mixture. FINAL ASSAY C O N C E N T R A T I O N : In a 2.60 ml reaction mix, the final concentrations are 66 mM potassium phosphate, 0.014% (w/v) Micrococcus lysodeikticus cell suspension and 20 - 40 units lysozyme. REFERENCE: Shugar, D. (1952) Biochimica et Biophysica Acta 8, 302-309 NOTES: 1. This assay procedure is not to be used to assay Lysozyme, Bovine Recombinant Expressed in Pichia pastoris, Sigma Prod. No. L-9772, Lysozyme, Human, Recombinant Expressed in Pichia pastoris, Sigma Prod. No. L-2026, and Lysozyme Insoluble Enzyme on Agarose, Sigma Prod. No. L-1129. 2. This assay is based on the cited reference.  Appendix 4: Lysozyme solutions  3.  262  Where Sigma Product or Stock numbers are specified, equivalent reagents may be  substituted. Sigma warrants that the above procedure information is currently utilized at Sigma and that all Sigma-Aldrich, Inc. products conform to the information in this and other Sigma-Aldrich, Inc. publications. Purchaser must determine the suitability of the information and produces) for their particular use. Additional terms and conditions may apply. Please see reverse side of the invoice or packing slip.  Appendix 4: Lysozyme solutions  263  InKvafech I LT_i  v Wl I VrV^I I  I n n o v a t i v e  E g g  T e c h n o l o g y  +  TEST ID ACTVO1-01  PERCENT ACTIVITY O N L Y S O Z Y M E POWDER  IMPORTANT: Make sure all apparatus are cleaned by heat sterilization. Work area should be absolutely clean. Any residual Lysozyme contamination anywhere will upset the results dramatically. 1. REAGENTS - Deionized Water Sterilize by boiling. -Buffer A Dissolve 10.4 grams of NaH P0 in one litre of deionized water, prepared above. 2  4  - Buffer B Dissolve 9.465 grams Na HP0 in one litre of deionized water, prepared above. 2  4  - Buffer C a) Mix 815 rnilfflitres of Buffer A to 185 nullilitres of Buffer B. b) Check an aliquot of the buffer for pH, (this is to prevent contamination) pH should be at 6.2. Adjust accordingly by adding either Buffer A or B. - Micrococcus Lysodeikticus Substrate a) Prepare substrate solution by adding Micrococcus Lysodeikticus to 100 milliltres of Buffer C. This should give an absorbance reading of 1.8 ± 0.1 (450 nanometres). Adjust solution by-adding more Micrococcus or Buffer to obtain the required absorbance. b) Let the substrate sit in a warm area (approximately 30°C) for one half hour before use. c) Check for purity using the spectrophotometer procedure. There should be a very rninimal absorbance change over a three rninute period. If contarninated, remake the substrate solution. - Freeze Dried Lysozyme Standard  Canadian Inovatcch Inc. 31212 Pcardonvilie Road Abbotsford, British Columbia Canada V2T6K8  Telephone: (604] 85743695 1-800-665-3447 [EGGS] Facsimile: [604] 857-2679  Appendix 4: Lysozyme solutions  InKvatech  264  Innovative  Egg  Technology  +  2. APPARATUS - 100 rnillilitre volumetric flasks - Pipettors and tips 1-5rnillilitrepipettor and tips 100 microlitre pipettor and tips - Analytical Balance with + 0.0001 gram precision - Spectrophotometer - see instrument manual for setup - Glass or plastic cuvettes (1 centimeter square). 3. PREPARATION OF LYSOZYME POWDER NOTE: PREPARE FREEZE DRIED LYSOZYME STANDARD ALONGSIDE YOUR SAMPLE. a) b) c) d) e) f)  Add 50 milligrams of sample (record weight) to a 100 niillilitre volumetric flask. Dissolve the sample in approximately 50 millilitres of Buffer C. Stir to dissolve. Bring to volume and mix thoroughly. Transfer 3 millilitres to a second 100rnillilitrevolumetric flask. Bring up to volume with Buffer C and mix thoroughly. The initial concentration of this solution is 0.05%. The final concentration is 0.015%.  NOTE: FOR LYSOZYME STANDARD, THE SOLUTION AT STEP 3 MAY BE FROZEN IN SMALL QUANTITIES FOR FUTURE USE. STEPS (d) and (e) WOULD THEN BE CARRIED OUT. 4. ACTjrvrrY ASSAY a) Place cuvette into spectrophotometer and adjust zero. b) Add 2.9rnillilitreof prepared substrate using 1-5 niillilitre pipettor. Initial absorbance reading should be 1.8 ± 0.1. c) Add 100 microlitres of the working standard to the substrate; mix well. d) Set spectrophotometer with the following parameters; 450 nanometers, Absorbance, Time length: three minutes. Start recording. e) Thefirstminute readings should be disregarded, as this allows the test to equilibrate. f) Repeat steps (a) through (e) for your samples. Repeat each standard and sample at least three times to ensure accuracy and precision. >  Canadian Inovatcch Inc. 31212 Pcardonville Road Abbotsford, British Columbia Canada V2T6K8  Telephone: 1604] 857-0695 1-800-665-3447 (EGGS] Facsimile: (6041857-2679  Appendix 4: Lysozyme solutions  A4.3  266  Spectrophotometer Data for Substrate and Blank Runs Time (seconds) 0 5 1 0 1 5 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 11 0 11 5 120 125 130 135 140 145 150 155 160 165 170 175 180 185 190 195 200 205 2 1 0 2 1 5 220 225 230 23 5 240 245 250 255 260 265 270 275 280 285  Substrate 1 0 634 0 634 0 634 0 634 0 635 0 635 0 635 0 635 0 635 0 635 0 635 0 635 0 635 0 635 0 635 0 635 0 635 0 635 0 635 0 635 0 635 0 635 0 635 0 635 0 635 0 635 0 635 0 635 0 635 0 635 0 635 0 635 0 635 0 635 0 635 0 635 0 635 0 635 0 635 0 635 0 635 0 635 0 635 0 63 5 0 635 0 635 0 63 5 0 635 0 635 0 63 5 0 635 0 635 0 0 0 0 0 0  63 5 63 5 635 .635 .63 5 .635  S u b s t r a te 2 0 633 0 633 0 63 3 0 63 3 0 63 3 0 63 3 0 633 0 634 0 634 0 634 0 634 0 63 3 0 634 0 634 0 634 0 63 4 0 634 0 634 0 634 0 634 0 63 4 0 634 0 634 0 634 0 634 0 63 4 0 634 0 634 0 634 0 634 0 634 0 634 0 634 0 634 0 634 0 634 0 634 0 634 0 634 0 634 0 635 0 635 0 635 0 634 0 635 0 635 0 634 0 634 0 634 0 635 0 634 0 635 0 0 0 0 0 0  634 635 634 .634 .635 .63 5  Blank 1 0 628 0 629 0 630 0 630 0 630 0 630 0 630 0 630 0 630 0 630 0 630 0 630 0 630 0 630 0 630 0 630 0 630 0 630 0 630 0 630 0 630 0 630 0 630 0 630 0 630 0 630 0 630 0 630 0 630 0 630 0 630 0 630 0 630 0 630 0 630 0 630 0 630 0 63 1 0 630 0 630 0 630 0 630 0 630 0 630 0 630 0 630 0 630 0 630 0 630 0 630 0 630 0 63 1 0 630 0 630 0 630 0 630 0 .630 0 .630  Blank 2 0 624 0 624 0 625 0 625 0 625 0 625 0 625 0 625 0 625 0 625 0 625 0 625 0 625 0 625 0 625 0 625 0 625 0 625 0 625 0 625 0 625 0 625 0 625 0 625 0 625 0 625 0 625 0 625 0 625 0 625 0 625 0 625 0 625 0 625 0 625 0 625 0 625 0 625 0 625 0 625 0 625 0 625 0 625 0 625 0 625 0 625 0 625 0 625 0 62 5 0 625 0 625 0 625 0 0 0 0 0 0  625 625 625 .625 .62 5 .624  Blank 3 0 626 0 627 0 627 0 627 0 627 0 627 0 627 0 627 0 627 0 627 0 627 0 627 0 627 0 627 0 627 0 627 0 627 0 627 0 627 0 627 0 627 0 627 0 627 0 627 0 627 0 627 0 627 0 627 0 627 0 627 0 627 0 627 0 627 0 627 0 627 0 627 0 627 0 627 0 627 0 627 0 627 0 627 0 627 0 627 0 627 0 626 0 627 0 627 0 626 0 627 0 627 0 626 0 626 0 626 0 626 0 626 0 626 0 626  Appendix 5: Mathematical Modeling  Appendix 5 A5.1  267  Mathematical Modeling  Algorithm for Levenberg-Marquardt Non-Linear Curve Fitting Method  i. .The function to be solved can be described as y = f(x,a), where xandyare experimental data and a is a vector of unknown constants. ii. Guess initial values of parameter a . For this problem, at a given reaction order (n), the vector a consists of k i , k , and A E , hence a-(l)=ki, a(2)=k,and a(3) = AE. Optimum solutions were determined at fixed, discrete reaction orders (n) to best fit all experimental data (x, y). A n optimum solution is determined by a minimized sum of the squares of the residuals between model predictions and experimental initial fouling rates. iii. Read input data from experimental files for all experiments, i.e. values of C and B for each data point (see sections 5.2 and 5.3). iv. Specify an initial value of the scaling factor, X. X controls whether the steepest descent method or Newton's method is used to find the optimum solution. The steepest descent method is typically used (X > 1) far from the optimum solution to find a local minimum in the sum of the squares. Newton's method (X < 1) is used for a more rapid convergence on the optimum solution. Initially X is set equal to 1 to enable the program to move easily between the two methods. v.  Choose initial guesses of the vector a . Well chosen first estimates of a will give a rapid solution to the problem, while poor estimates may take several thousands of iterations, or worse, X, will increase beyond the maximum permissible value and never converge. This is particularly true when a solution to the problem is very dependent upon one value of a .  vi. Calculate the sum of the squares for the data from the initial estimates,  Appendix 5: Mathematical Modeling  268  S=Z[y -f(x ,a)] k  k  where yk is the experimental initial fouling rate and f(xk, a) is the predicted initial fouling rate. vii. Evaluate the matrix a (curvature matrix) and the vector /? (gradient matrix) at a , to be used for evaluation of the next a : N  df(x ,a)  k=i  da  k  af(x ,a) da  ;  Pi= i:|[y -fl[x ,a)]. k  k  k  k  i = 1,2,3 ...n j=l,2,3...n  ;  5f(x ,a) k  da  '  J i=l,2,3...n  viii. Evaluate the augmented coefficient matrix, a' a--' =a (l  + X)  s  =a  ifi^i^n +1  ; ; IJ  J  = P\ ix. Solve linear equations oc^ .8a; = c c  in+1  i f i = J5tn + 1  ifj = n + l for 8aj, i = 1,2,...n.  x. Calculate a w,i = aj + 8 a , i = l,2,...n ne  xi. Calculate S  ;  N  n e w  = l[y k=r  2  k  -f(x ,a k  n e w  )l J  xii. if Snew < S then X = 0.1 x X, aj= a w,i, i  =  ne  xiii. if Snew > S then X = 10 x X, return to (viii)  xiv.STOP when max  l,2,...n, S = S w, return to (vii) ne  Appendix 5: Mathematical Modeling  269  A5.2  FORTRAN Program used for Non-Linear Multi-Parameter Regression of Experimental Data According to Epstein's Mathematical Model (1994). **********************************#******************************************* * * * * * * *  Propose: Determines the best-fit parameters, k', k" and E for a given n which yield the minimum sum-of-squares-of-differences between the predictions of Esptein's implicit initial fouling rate equation and experimental results. Equation (2.2.22) takes the form: Cb = k'C(Rfo) + k"[B expCE/RTsni/ordeOJCRfoyXl/order) order C&B E Rfo Ts k' & k"  * * * * *  Main program: Calls the Marquardt method subroutine MARQRT to iteratively determine the best-fit parameter values and then prints out thefittedparameters along with a comparison of the original and predicted y-values.  implicit real*8(a-h,o-z) parameter (md = 77) dimension rfo(77) common cb,c(md),b(md),ts(md),r_expt(md),a(3),order external tunc, dfunc *  Specify the number of experimental data points used.  * * *  (a)Whey Protein 47 data points if TFU 204 & 212 neglected. 59 data points if all experiments are included  * * *  (b)Lysozyme 86 data points if all experiments are included 77 data points if TFU 309 is neglected data m /77/ data n,eps,maxit/3,l .d-6,50000/ Input initial guesses of solution, and experimental conditions a(l)= l.dl5 a(3)=l.d5 cb = 9.95d0 write(V) 'Input the Chemical Reaction Order (0.5 - 2.5 ?)' read(*,*)order write(*,*)'Input initial guess of a(2), i.e. k" '  * *  read(V) a(2)  Read input experimental data file (a) Whey Protein model3.dat => Tf for C and Ts for B model2.dat => Tf for both C and B model4.dat => Same as model3.dat except TFU 204/212 deleted (Here we have only 47 data points => change programme slightly) (b) Lysozyme model5.dat => Tf for both C and B model6.dat => Tf for C and Ts for B model7.dat => No TFU 309 open(9,file= 'model3.dat') open(9,file= 'model2.dat') open(9,file= 'model4.dat') open(9,file= 'model5.dat') open(9,file= 'model6.dat') open(9,file= 'model7.dat') do i - l,md read(9,*)c(i),b(i),ts(i),r_expt(i)  •  is the reaction order of the chemical reaction are individual experimental data points is the activation energy (J/mol) is the initial fouling rate (m2K/J) is the clean surface temperature (K) are functions of the mass transfer and chemical attachment constants, respectively  ******************************************************************************  * * *  * * * * * *  < 1  * ' '  Appendix 5: Mathematical Modeling  * * *  10 20 30 35  40 50 * * *  270  end do call MARQRT (func,dfunc,m,n,eps,maxif) Open outputfileto write solution results open (10,file= 'results.dat') write (10,10) format Px.'Parameters;'/) write (10,20) (i, a(i), i= 1,3) format(3xX.'l.')-,13x,dl3.6) write(10,30) order format (3x,"Reaction Order =',fl0.3) write(10,35) format^^x^xperimental'.Sx, 'Model'/) sum_sq = O.dO do k = l,m y_pred = func(k) sumsq = sumsq + (r_expt(k)-y_pred)**(2.d0) write (10,40)r_expt(k),y_pred format (2el3.5) end do write (10,50) sum_sq format (/3x,'Sum-of-squares =',dl3.6) close (10) Now solve the model equation with the given parameters for a range of surface temperatures and velocities  call MODPLOT (a,order,cb,rfo,st,bt) open (11, file = 'results2.dat') write(ll,60)st,bt 60 format(/,3x,'ModeI Plot for Ts =',f5.2,' and Tb =',£5.2,//) write(l 1,100X35 d0*i,rfo(i),i=l,77) 100format(3x,'Mass Flux =',f8.2,5x,'Rfo ='el3.6) close(ll) stop end ******************************************************************************  * This subroutine is used to regress the model and given a(l) and * a(2) parameters for a model fit at a constant surface and bulk ' * temperature and a range of fluid velocities. * ****************************************************************************** 1  subroutine MODPLOT (a,order,cb,rfo,ts,tb) implicit real*8(a-h,o-z) real*8 nuus,muus,nuuf,muuf dimension a(3),rfo(77) * * •  Define the model functions f(x) =(a(l)*c*x)+a(2) (b*x*expon)»*(l.d0/order) - cb df(x)=(l d0'order)'(x"((l.d0/order)-l.d0))»a(2)'(b»expon) + **(l.d0/order)+a(l)*c data eps,dx,xf /l .d-6,1 .d-9,1 .d0/ ,  * * *  Input required surface and bulk temperature predictions. The film temperature is assumed to be the arithmetic average of the bulk and surface temperature at any given location. ts = 80.d0 tb = 55.d0 tf=(ts+tb)/2.d0  * * *  Calculate surface physical properties (density, kinematic viscosity and dynamice viscosity). Remember to change the formulas depending on whether the data is whey protein or lysozyme solutions rhos=-(0.0032d0)*ts *(2.d0)-(0.1086d0)»ts+l 004.4d0 nuus=(-3.31d-12)ts"(3.d0)+(7.41d-10)*ts**(2.d0H5.88d-8)*ts + +2.07d-6 muus=nuus*rhos ,  ,  *  Calculatefilmphysical properties rhof=-(0.0032d0)*tf*(2.d0)-(0.1086d0)*tf+1004.4d0 nuuH-3.31d-12)*tf *(3.d0)+(7.41d-10)*tf'»(2.d0H5.88d-8)'tf + +2.07d-6 muuf=nuuf*rhof ,  ,  Appendix 5: Mathematical Modeling * *  Calculate bulk density rhob=-(0.0032d0)*tb**(2.d0H0.1086d0)*tb+1004.4d0  *  * * *  Solve for the model equation for the Initial Fouling Rate for all velocities of interest flux=0.d0 Define the exponential term since this will always be constant, expon=DEXP(a(3)/((ts+273.15)*8.314d0)) do i = 1,77 Increments of 0.05 m/s flux=flux+35.d0  * *  •  * * *  Determine surface and film Reynolds numbers, film friction factor and hence friction velocity res=flux*(9.017d-3)/muus ref=flux*(9.017d-3)/muuf fnc=0.25d0(1.82d0LOG10(re0-164d0)"(-2.d0) vstar=(flux/rhob)»((fric/2.d0)"(0.5d0)) ,  * * *  ,  Determine the constants c and b for each velocity. Remember to change b depending on the input file  c=(l •d0/vstar)»(((muuf"2.d0)/(rhof (tf+273.15d0))) "(2.d0/3.d0)) b=(vstar"(2.d0))*(mo£'muuf) * Guess initial starting solution xl=l.d-20 yl=fHxl) * Use incremental search to find the interval containing a single * root 20 x2=xl+dx y2=f(x2) if[yl'y2gt.0.d0)then iffx2.gt.xf)then ,  +  write(V)' X2 > XF  stop else xl=x2 yi=y2 goto 20 endif endif * * * 30  Use Linear interpolation to obtain a closer approximation of the root X3  x3=(xl»y2-x2*yl)/(y2-yl) y3=f(x3) ifiy3.eq.0.d0)then x=x3 goto 60 endif * Use Newton Raphson to obtain converged root x4 40 x4=x3-y3/dfl;x3) ifl[x4.ge.xl .and.x4.1e.x2)then y4=f(x4) if(y4.eq.0.d0)then x=x4 goto 60 endif if(dabs((x4-x3)/x4).le.eps)then x=x4 goto 60 else x3=x4 y3=y4 goto 40 endif else * Use bisection to reduce root interval if newton raphson fails do50ihalf=l,3 x3=(xl+x2)/2.d0 y3=fi;x3) if(y3.eq.0.d0)then x=x4  271  Appendix 5: Mathematical Modeling  272  goto 60 endif if(dabs((x3-xl)/x3).lt.eps)then x=x3 goto 60 endif if(yl*y3.gt.0.d0)then xl=x3 yi=y3 else x2=x3 y2=y3 endif 50 continue goto 30 endif 60 rfo(i)=x end do return end * ' * *  subroutine MARQRT: Purpose: Uses the Marquardt's method to iteratively determine the best fit parameters for a non-linear fitting equation.  • ' 1  '  * Arguments: ' * tunc user-supplied function that defines the fitting equation * dfunc user-supplied function that defines the partial derivs. ' * of f relative to each fitting parameter ' * m number of experimental data points (77) ' * a one-dim. array of parameters of fitting equation; ' * guessed values supplied as input, best-fit values ' * returned as output ' * eps relative convergence criterion for each a(i) ' * maxit maximum number of Marquardt iterations allowed ' ***«****«***********«***************************************«**«******** ****** 1  subroutine MARQRT (func,dfunc,mn,eps,maxit) implicit real*8(a-h,o-z) real*8 lamda, lamdap integer flag parameter (nd = 3) dimension da(nd), a_o!d(nd), alpha(nd,nd+l), diag(nd) common cb,c(77),b(77),ts(77),r_expt(77),a(3),order external func, dfunc )  lamda = 1 .d0 damax = l.dlO iter = 0 flag=l ssq = SUMSQ(func,m) do while (da_max.gt.eps) iter = iter + 1 if (iter .gt. maxit) then write(6,5) maxit, damax 5 format( 1 x,'Marquardts method failed to converge in',i6, + ' iterationsV/5x,'da_max =',dl3.6) stop end if if (flag eq. l)then call COEFF(func,dfunc,m,n,alpha) do i = 1, n diag(i) = alpha(i.i) end do end if lamdap = lamda + 1 .d0 do i = 1,n alpha(i,i) = diag(i)*lamdap end do call GAUSS(alpha, n, nd, nd+1, da, rnorm, ierror) da_max__new = O.dO do i = l,n a_old(i) = a(i) a(i) = a(i) + da(i) if(a(i).lt.0.d0)then a(i)=0.5d0*a_old(i) endif da_max_new = dmaxl(da_max_new,dabs(da(i)/a_old(i)))  273  Appendix 5: Mathematical Modeling end do ssq_new = SUMSQ(func,m) if(ssq_new .It. ssq) then lamda = O.ldO'Iamda ssq = ssq_new damax = damaxnew flag=l else do i = l,n a(i) = aold(i) end do lamda = lO.dO'lamda flag = 0 end if * Write to the screen the individual iteration results. write(*,*) iter,lamda,ssq_new 10 format(5x, dl 1.6, 5x, d9.4, 5x, flO.O) end do return end ****************************************************************************** * subroutine COEFF: ' * Purpose: " * Determines the elements of the augmented "curvature matrix" * * alpha required by Newton's method. Note that alpha(i j), ' * (i= 1 :n, j-1 :n) are approximated onlyfromthe first partials ' * offunc, not from the second partials. ' ****************************************************************************** subroutine COEFF (func,dfunc,m,n,alpha) implicit real*8(a-h,o-z) common cb,c(77),b(77),ts(77),r_expt(77),a(3),order parameter (nd = 3, md = 77) dimension alpha(nd,nd+l), diff(md), dfda(nd,md) np = n+1 do k= I, m diff(k) = r_expt(k)-func(k) do i - 1, n dfda(i,k) = dfunc(i,k) end do end do do i = 1, n do j = i, np alpha(ij) = O.dO end do end do do i = 1, n do j = i, n do ic = 1, m alpha(i j) = alpha(i j) + dfda(i,k)'dfda(j,k) end do if (j .ne. i) then alpha(j,i) = alpha(i j) end if end do do k = 1, m alpha(i,np) = alpha(i,np) + diff(k)dfda(i,k) end do end do return end ****************************************************************************** * function SUMSQ: < * Purpose: < * Evaluates the sum-of-squares of differences for any set of ' * parameter values a. ' ****************************************************************************** ,  double precision function SUMSQ(func,m) implicit real*8(a-h,o-z) common cb,c(77),b(77),ts(77),r_expt(77),a(3),order SUMSQ = O.dO do k = 1, m SUMSQ = SUMSQ + (r_expt(k)-func(k))"(2.d0) end do return end tt****************************************************************************  *  function func:  *  Appendix 5: Mathematical Modeling  274  * Purpose: ' * Calls the appropriate routine to rootfindthe implicit ' * equation * •«•*•**••«**•*«****•••**#•••**#*•********•*•*«**•***•*«*•*••••**********•**•** double precision function fiuic(k) implicit real*8(a-h,o-z) common cb,c(77),b(77),ts(77),r_expt(77),a(3),order call NEWRAP(k,Rf) func=Rf return end *••*•#«**«•••••«•••***•*******«****••«**#****•*****••*•*****•********«*******#  * Subroutine Newton Raphson: ' * Purpose: * * Root solving the implicit equation of initial fouling rate " «****«•#«•••••«•«•*•***«***«•«•**••**•**••***#*••***•**•**•*•*********+**•**** subroutine NEWRAP (k,x) implicit real*8(a-h,o-z) common cb,c(77),b(77),ts(77),r_expt(77),a(3),order f(x)=(a(l)*c(k)*x)+a(2)*(b(k)«x exp(a(3)/(8.314d0»ts(k)))) + **(l.dO/order)-cb df(xHldO/order)(x"((l.dO/order)-l.dO))*a(2)*(b(k) + *exp(a(3)/(8.314d0*ts(k))))"(l.d0/order) + a(l) c(k) data eps,dx,xf /l.d-6,l.d-9,l.d07 ,  ,  ,  * * *  Specify the starting guess of the model initial fouling rate. This will normally be put equal to the experimental initial fouling rate.  xold = r_expt(k) 10 x = xold - f(xoId)/df(xold) if(x.lt.0.d0)then •  * * *  If the predicted initial fouling rate is less than zero, we need to use a more elaborate method to find the root.  xl=l.d-20 yl=ftxl) * Use incremental search tofindthe interval containing a single * root 20 x2=xl+dx y2=f[x2) if(yl'y2gt0.d0)then if(x2.gt.xf)then write(V)'X2>XF stop else xl=x2 yl=y2 goto 20 endif endif * Use Linear interpolation to obtain a closer approximation of the * root X3 30 xSKxi'yZ-^'yiyCyZ-yi) y3=f(x3) if(y3.eq.0.d0)then x=x3 goto 60 endif * Use Newton Raphson to obtain converged root x4 40 x4=x3-y3/df?x3) if(x4.ge.xl .and.x4.1e.x2)then y4=f(x4) ifty4.eq.0.d0)then x=x4 goto 60 endif ifl;dabs((x4-x3)/x4).le.eps)then x=x4 goto 60 else x3=x4 y3=y4 goto 40 endif else * Use bisection to reduce root interval if newton raphson fails  Appendix 5: Mathematical Modeling  275  do 50 ihalf=l,3 x3=(xl+x2)/2.d0 y3=fi>3) if(y3.eq.0.d0)then x~ x4 goto 60 endif if(dabs((x3-xl)/x3).lt.eps)then x=x3 goto 60 endif if(yly3.gt.0.d0)then xl=x3 yl=y3 else x2=x3 y2=y3 endif 50 continue goto 30 endif endif if (dabs((x-xold)/x).le.eps)Uien return endif xold = x goto 10 ,  60 return end ****************************************************************************** * function dfunc: * * Purpose: < * Defines the partials of func with respect to each a(i), (i=l:n) ****************************************************************************** double precision function dfunc(i,k) implicit real*8(a-h,o-z) common cb,c(77),b(77),ts(77),r_expt(77),a(3),order  1  * *  Use finite difference (central difference) to approximate the derivative temp=a(i) h=l.d-6*a(i)  a(iH(')  +h  fl=func(k) a(i)=a(i>2.d0h f2=func(k) a(i)=temp dfunc=<n-f2)/(2.d0*h) return end ,  «******«*««*«*******«*******«**«**********************************************  * * * *  Subroutine Gauss Purpose: Uses Gauss elimination with scaled partial pivot selection to solve simultaneous linear equations of form [A]'{X(={C}.  * * * * * * * * * * *  Arguments: A N NDR NDC X RNORM IERROR  Augmented coefficient matrix containing all coefficients and R.H.S. constants of equations to be solved. Number of equations to be solved. First (row) dimension of A in calling program. Second (column) dimension of A in calling program. Solution vector. Measure of size of residual vector [C]-{A}*{X}. Error flag. =1 Successful Gauss elimination. =2 Zero diagonal entry after pivot selection.  1  * ' ' ' ' 1  ' ' ' ' 1  ' '  '  tt****************************************************************************  *  SUBROUTINE GAUSS(A,N,NDR,NDC,X,RNORM,IERROR) IMPLICIT REAL«8(A-H,0-Z) DIMENSION A(NDR,NDC),X(N),B(3,4) NM=N-1 NP=N+1 Set up working matrix B  Appendix 5: Mathematical Modeling DO 20 I=1,N  DO 10J=1,NP B(I,J)=A(I,J) 10 CONTINUE 20 CONTINUE * Carry out elimination process N-1 times DO70K=l,NM KP=K+1 »  * * *  30 *  40 * *  *  Search for largest coefficient in column K, rows K through N IPIVOT is the row index of the largest coefficient BIG=ABS(B(K,K)) rPIVOT=K DO 30 I=KP,N AB=ABS(B(I,K)) IF(AB.GT.BIG)THEN BIG=AB IPIVOT=I ENDIF CONTINUE Interchange rows K and IPIVOT if IPIVOT.NE.K IF(IPIVOT.NE.K)THEN DO 40 J=K,NP TEMP=B(IPIVOT,J) B(IPIVOT,J)=B(K,J) B(K,J)=TEMP CONTINUE ENDIF Check diagonal element for a zero entry B(K,K) = 0 causes an abnormal return with ierror = 2 IF(B(K,K).EQ.0.D0)THEN IERROR = 2 RETURN ENDIF Eliminate B(I,K) from rows K+l through N DO60I=KP,N QUOT=B(I,K)/B(K,K) B(I,K)=O.D0 DO 50 J=KP,NP Bfl,J)=Bfl,J)-QUOTB(KJ) CONTINUE CONTINUE CONTINUE IF(B(N,N).EQ.0.D0)THEN IERROR = 2 RETURN ENDIF Back substitute to find solution vector X(N)=B(N,NP)/B(N,N) DO90I=NM,l,-l SUM=0.D0 DO 80 J=I+1,N SUM=SUM+B(I,J)'X(J) CONTINUE ,  50 60 70  *  80  x(i)=(B(i,NP)-suMyB(i,i) 90 CONTINUE * * *  Calculate norm of residual vector, C-A*X Normal return with IERROR=l  RSQ=0.D0 DO 1101=1,N SUM=0.D0 DO 100 J=1,N SUM=SUM+A(I,J)«X(J) 100 CONTINUE RSQ=RSO+(DABS(A(I,NP>SUM))**(2.dO) 110CONTINUE RNORM=DSQRT(RSQ) IERROR=l RETURN END ************************************************•****•*•*•***••******•**•••**  276  277  Appendix 5: Mathematical Modeling  A5.3  Whey Protein Solution Input Data  Units: C =  ,B = s ~ , T 1  VsmK  MODEL3.DAT  MODEL2.DAT B  Rfo  3.3548E-07,1329.25,355.1,2.88E-09 3.1322E-07,1379.93,358.8,3.97E-09 3.0609E-07,1398.96,358.2,3 97E-09 3.0017E-07,1414.51,358.7,4.26E-09 2.8642E-07,1450.40,362.3,5.48E-09 2.8160E-07,1464.41,362.7,5.66E-09 1.5993E-07,6676.58,350.8,1.66E-09 1.5708E-07,6747.38,350.9,2.09E-09 1.4733E-07,6997.22,353.9,2.32E-09 1.3920E-07,7227.80,356.3,3.07E-09 1.3481E-07,7367.44,356.4,3.85E-09 1.3367E-07,7407.55,355.8,4.36E-09 1.2611 E-07,7649.81,360.4,5.77E-09 1.2399E-07,7726.23,360.8,7.84E-09 2.0483E-07,3474.85,358.0,5.41E-09 1.9823E-07,3543.18,357.5,5.21E-09 1.8583E-07.3674.82.360.3.1.03E-08 1.7707E-07,3778.37,361.7,1.54E-08 1.7430E-07,3814.67,361.5,1.59E-08 1.6782E-07,3895.39,364.1,2.07E-08 1.6592E-07,3923.42,363.6,2.22E-08 2.9687E-07,2013.75,349.4,5.56E-09 2.6941E-07,2121.65,356.7,8.62E-09 2.678 lE-07,2130.59,355.2,1.06E-08 2.5107E-07,2208.77,357.8,1.18E-08 2.3499E-07.2291.67,360.9,1.43E-08 2.2619E-07.2343.31,361.2,1.6E-08 2.2400E-07,2358.03,360.4,1.64E-08 2.1339E-07,2420.85,364.1.2.36E-08 2.1235E-07.2429.91,362.9,2.32E-08 4.1275E-07,773.63,361.3,6.96E-09 3.8662E-07,800.96,365.6,1.03E-08 3.7875E-07,809.95,366.6,1.58E-08 3.5888E-07,832.69,371.7,2.09E-08 3.5784E-07,834.63,370.7,2.29E-08 1.6772E-07,5198.39,355.6,2.31E-09 1.5603E-07.5413.8 l,359.5,4.34E-09 1.5056E-07,5529.17,359.8,6.68E-09 1.4623E-07.5621.28,361.3.9.49E-09 1.3988E-07,5760.45,364.7,1.35E-08 1.3815E-07,5804.53,364.5,1.73E-08 1.2396E-07,9793.78,349.9,9.44E-10 1.1605E-07,10159.76,355.0,1.3E-09 1.1519E-07,10207.90,3 54.6,1.53E-09 6.9250E-07,433.26,341.0,4.63E-09 6.3247E-07.453.83.348.1.6.56E-09 6.2270E-07,457.70,348.3,7.3E-09 5.6179E-07,482.72,355.7,9.39E-09 5.1443E-07.505.20,362.3,1.06E-08 4.8831E-07,519.23,365.5,1.35E-08 4.7400E-07,527.30,367.7,1.39E-08 4.4526E-07,544.17,374.0,1.88E-08 4.3925E-07.548.15,374.6,2.23E-08 3.2807E-07,1341.28,356.6,2.11E-09 3.0909E-07,1385.87,359.5,3.7E-09 3.0080E-07,1408.16,359.1.4.66E-09 2.9688E-07,1419.13,358.9,5.49E-09 2.8599E-07,1448.07,361.5.7.39E-09 2.7784E-07,1471.07,363.3,8.73E-09  3.3548E-07,1701.81,355.1,2.88E-09 3.1322E-07,1715.94,358.8,3.97E-09 3.0609E-07,1706.40,358.2,3.97E-09 3.0017E-07,1705.47,358.7,4.26E-09 2.8642E-07.1711.95,362.3,5.48E-09 2.8160E-07,1709.79,362.7,5.66E-09 1.5993E-07,8836.78,350.8,1.66E-09 1.5708E-07.8823.31,350.9,2.09E-09 1.4733E-07.8951.06,353.9,2.32E-09 1.3920E-07,9019.93,356.3,3.07E-09 1.3481E-07,8993.60,356.4,3.85E-09 1.3367E-07,8957.45,355.8,4.36E-09 1.2611E-07,9076.78,360.4,5.77E-09 1.2399E-07,9072.65,360.8,7.84E-09 2.0483E-07,4467.54,358.0,5.41E-09 1.9823E-07,4439.09,357.5,5.21E-09 1.8583E-07.4451.16,360.3,1.03E-08 1.7707E-07,4443.49,361.7,1.54E-08 1.7430E-07,4435.26 361.5,1.59E-08 1.6782E-07,4434.20,364. l,2.07E-08 1.6592E-07,4429.40,363.6,2.22E-08 2.9687E-07.2682.59.349.4.5.56E-09 2.6941E-07,2784.63,356.7,8.62E-09 2.6781 E-07,2759.30,355.2,1.06E-08 2.5107E-07.2771.67,357.8,1.18E-08 2.3499E-07,2777.83,360.9,1.43E-08 2.2619E-07,2767.84,361.2,1.60E-08 2.2400E-07,2759.72,360.4,1.64E-08 2.1339E-07,2762.18,364.1,2.36E-08 2.1235E-07,2758.78,362.9,2.32E-08 4.1275E-07.958.20.361.3.6.96E-09 3.8662E-07,952.79,365.6,1.03E-08 3.7875E-07,949.40,366.6,1.58E-08 3.5888E-07,929.83,371,7,2.09E-08 3.5784E-07,933.36,370.7,2.29E-08 1.6772E-07,6601.15,355.6,2.31E-09 1.5603E-07.6661.76.359.5.4.34E-09 1.5056E-07,6643.22,359.8,6.68E-09 1.4623E-07.6650.43.361.3.9.49E-09 1.3988E-07,6648.34,364.7,1.35E-08 1.3815E-07.6641.80,364.5,1.73E-08 1.2396E-07,12052.79,349.9,9.44E-10 1.1605E-07,12425.48,355.0,1.30E-09 1.1519E-07,12386.41,354.6,1.53E-09 6.9250E-07,567.45,341.0.4.63E-09 6.3247E-07,603.63,348.1,6.56E-09 6.2270E-07,603.01,348.3,7.30E-09 5.6179E-07,626.25,355.7,9.39E-09 5.1443E-07,631.97,362.3,1.06E-08 4.883 lE-07,627.96,365.5,1.35E-08 4.7400E-07,623.61,367.7,1.39E-08 4.4526E-07,603.53,374.0,1.88E-08 4.3925E-07,600.36,374.6,2.23E-08 3.2807E-07,1702.58,356.6,2.11E-09 3.0909E-07,1708.87,359.5,3.70E-09 3.0080E-07,1700.98,359.1,4.66E-09 2.9688E-07,1697.09,358.9,5.49E-09 2.8599E-07,1701.68,361.5.7.39E-09 2.7784E-07,1700.69,363.3,8.73E-09 )  J  MODEL4.DAT R fo  B  = K, Rfo = m K / J 2  w  B  T  w  R  f  1.5993E-07,8836.78,350.8,1.66E-09 1.5708E-07,8823.31,350.9,2.09E-09 1.4733E-07.8951.06,353.9,2.32E-09 1.3920E-07,9019.93,356.3,3.07E-09 1.348 lE-07,8993.60,356.4,3.85E-09 1.3367E-07,8957.45,355.8,4.36E-09 1.2611E-07,9076.78,360.4,5.77E-09 1.2399E-07,9072.65,360.8,7.84E-09 2.0483E-07,4467.54,358.0,5.41E-09 1.9823E-07,4439.09,357.5,5.21 E-09 1.8583E-07.4451.16,360.3,1.03E-08 1.7707E-07,4443.49,361.7,1.54E-08 1.7430E-07,4435.26,361.5,1.59E-08 1.6782E-07,4434.20,364.1.2.07E-08 1.6592E-07,4429.40,363.6,2.22E-08 2.9687E-07,2682.59,349.4,5.56E-09 2.6941 E-07,2784.63,356.7,8.62E-09 2.6781 E-07,2759.30,355.2,1.06E-08 2.5107E-07,2771.67,357.8,1.18E-08 2.3499E-07,2777.83,360.9,1.43E-08 2.2619E-07,2767.84,361.2,1.60E-08 2.2400E-07,2759.72,360.4,1.64E-08 2.1339E-07,2762.18,364.1,2.36E-08 2.1235E-07,2758.78,362.9,2.32E-08 4.1275E-07,958.20,361.3.6.96E-09 3.8662E-07,952.79,365.6,I.03E-08 3.7875E-07.949.40,366.6,1.58E-08 3.5888E-07,929.83,371.7,2.09E-08 3.5784E-07,933.36,370.7,2.29E-08 1.6772E-07,6601.15,355.6,2.31 E-09 1.5603E-07,6661.76,359.5,4.34E-09 1.5056E-07,6643.22,359.8,6.68E-09 1.4623E-07,6650.43,361.3.9.49E-09 1.3988E-07,6648.34,364.7,1.35E-08 1.3815E-07.6641.80,364.5,1.73E-08 1.2396E-07,12052.79,349.9,9.44E-10 1.1605E-07,12425.48,355.0,1.30E-09 1.1519E-07.12386.41,354.6,1.53E-09 6.9250E-07,567.45,341.0.4.63E-09 6.3247E-07,603.63,348.1.6.56E-09 6.2270E-07,603.01,348.3,7.30E-09 5.6179E-07,626.25,355.7,9.39E-09 5.1443E-07,631.97,362.3,1.06E-08 4.8831E-07,627.96,365.5,1.35E-08 4.7400E-07.623.61,367.7,1.39E-08 4.4526E-07,603.53,374.0,1.88E-08 4.3925E-07,600.36,374.6,2.23E-08  Appendix 5: Mathematical Modeling  A5.4  278  Lysozyme Solution Input Data Units: C =  (  1  \'  X 1  B = s" ,T =K, 1  w  VsmK  Rfo=m K/J 2  /  MODEL5.DAT  MODEL6.DAT  MODEL7.DAT  B Rfo 2.3845E-07,3233.07,338.1,1.1 OE-08 2.2009E-07,3377.93,343.6,1.71E-08 2.0983E-07,3467.05,346.9,2.06E-08 2.0625E-07,3501.34,347.1,2.14E-08 2.0035E-07,3560.50,347.3,2.42E-08 1.9132E-07,3652.45,349.7,2.55E-08 1.8574E-07,3714.52,350.4,3.05E-08 1.8278E-07,3748.13,351.0.3.38E-08 1.8071E-07,3773.17,351.0,3.56E-08 1.7647E-07,3821.60,353.0,3.56E-08 1.8890E-07,4883.94,339.6,6.97E-09 1.8349E-07,4964.31,341.0,1.16E-08 1.7984E-07.5021.69,341.5,1J4E-08 1.7439E-07.5112.00,341.9,1.66E-08 1.6087E-07.5342.19,348.0,2.31 E-08 1.5686E-07,5420.66,348.4,2.57E-08 1.5557E-07,5448.11,348.1,2.64E-08 1.5189E-07.5519.68,349.9,3.06E-08 1.4769E-07,5603.05,352.5,3.52E-08 3.0990E-07,1946.16,337.0,8.88E-09 2.8882E-07.2021.92,341.7,1.44E-08 2.7279E-07,2084.93,346.0,1.82E-08 2.7216E-07,2088.91,344.7,1.80E-08 2.5837E-07.2149.04,347.6,2.07E-08 2.4760E-07.2199.62.349.9.2.48E-08 2.4122E-07,2232.26,350.5,2.72E-08 2.3879E-07,2245.39,350.4,2:91E-08 2.3494E-07,2265.52,351.2,3.10E-08 2.3046E-07,2289.07,352.6,3.32E-08 1.5304E-07,7285.01,341.0,3.85E-09 1.4673E-07,7457.89343.7,5.11E-09 1.4488E-07.7514.30,343.6,5.12E-09 1.3943E-07,7680.86,344.9,6.09E-09 1.3326E-07,7878.58,347.2,6.80E-09 1.2959E-07,8006.86,347.9,7.85E-09 1.2811 E-07,8061.13,348.0,8.72E-09 1.2557E-07,8151.33,349.2,1.06E-08 1.2327E-07,823 5.49,3 50.4,1.12E-08 4.9990E-07,745.09,336.8,8.63E-09 4.6725E-07,772.15,341.2,1.35E-08 4.4119E-07,795.69,345.6,1.96E-08 4.2614E-07.810.40,347.8,2.32E-08 4.0889E-07,828.54,349.8,2.61E-08 3.8624E-07,853.42,354.3,2.57E-08 3.7698E-07,864.89,354.8,2.36E-08 3.7489E-07,867.93,354.2,2.63E-08 3.7208E-07,871.81,353.8,2.74E-08 3.7046E-07,874.36,353.1,2.87E-08 3.8446E-07,1288.46,335.7,6.65E-09 3.5732E-07,1340.13,340.6,1.31 E-08 3.3829E-07,1379.85,344.5,1.71 E-08 3.3222E-07,1393.87,344.8,1.72E-08 3.1550E-07,1433.38,347.7,2.00E-08 3.0228E-07,1466.99,349.9,2.42E-08  B T Rfo 2.3845E-07,4013.89,338.1,1.10E-08 2.2009E-07,4141.18,343.6,1.71 E-08 2.0983E-07,4213.52,346.9,2.06E-08 2.0625E-07,4210.15,347.1,2.14E-08 2.0035E-07,4199.97,347.3,2.42E-08 1.9132E-07,4253.59,349.7,2.55E-08 1.8574E-07,4264.95,350.4,3.05E-08 1.8278E-07,4279.69,351.0.3.38E-08 1.8071E-07,4276.93,351.0.3.56E-08 1.7647E-07,4338.45,353.0,3.56E-08 1.8890E-07,5979.80,339.6,6.97E-09 1.8349E-07,6020.45,341.0,1.16E-08 1.7984E-07.6027.11,341.5,1.34E-08 1.7439E-07.6021.39,341.9,1.66E-08 1.6087E-07,6239.69,348.0,2.31E-08 1.5686E-07.6241.1 l,348.4,2.57E-08 1.5557E-07,6222.36,348. l,2.64E-08 1.5189E-07,6292.59,349.9,3.06E-08 1.4769E-07,6409.33,352.5,3.52E-08 3.0990E-07,2412.38,337.0,8.88E-09 2.8882E-07,2478.49,341.7,1.44E-08 2.7279E-07,2535.44,346.0,1.82E-08 2.7216E-07.2512.37,344.7,1.80E-08 2.5837E-07,2547.46,347.6,2.07E-08 2.4760E-07,2577.62,349.9,2.48E-08 2.4122E-07.2581.26,350.5,2.72E-08 2.3879E-07,2577.69 350.4,2.91E-08 2.3494E-07.2590.31,351.2,3.1 OE-08 2.3046E-07,2615.42,352.6,3.32E-08 1.5304E-07,8932.76,341.0,3.85E-09 1.4673E-07.9064.91,343.7,5.11 E-09 1.4488E-07,9039.87,343.6,5.12E-09 1.3943E-07,9083.94,344.9,6.09E-09 1.3326E-07.9195.35,347.2,6.80E-09 1.2959E-07,9214.96,347.9,7.85E-09 1.2811 E-07,9211.16,348.0,8.72E-09 1.2557E-07,9280.37,349.2,1.06E-08 1.2327E-07,9351.46,350.4,1.12E-08 4.9990E-07,919.93,336.8,8.63E-09 4.6725E-07,942.83,341.2,1.35E-08 4.4119E-07,963.96,345.6,1.96E-08 4.2614E-07,974.25,347.8,2.32E-08 4.0889E-07,982.62,349.8,2.61 E-08 3.8624E-07,1012.21,354.3,2.57E-08 3.7698E-07,1014.50,354.8,2.36E-08 3.7489E-07,1007.97,354.2,2.63E-08 3.7208E-07,1004.17,353.8,2.74E-08 3.7046E-07,997.24,353.1,2.87E-08 3.8446E-07,1593.71,335.7,6.65E-09 3.5732E-07,1639.69,340.6,1.31 E-08 3.3829E-07,1673.49,344.5,1.71 E-08 3.3222E-07,1673.11,344.8,1.72E-08 3.1550E-07,1694.98.347.7.2.00E-08 3.0228E-07,1714.30,349.9,2.42E-08  B Rfo 2.3845E-07,4013.89,338.1,1.10E-08 2.2009E-07.4141.18,343.6,1.71 E-08 2.0983E-07,4213.52,346.9,2.06E-08 2.0625E-07.4210.15,347.1,2.14E-08 2.0035E-07.4199.97,347.3,2.42E-08 1.9132E-07,4253.59,349.7,2.55E-08 1.8574E-07,4264.95,350.4,3.05E-08 1.8278E-07,4279.69,351.0.3.38E-08 1.8071 E-07,4276.93,351.0.3.56E-08 1.7647E-07,4338.45,353.0,3.56E-08 1.8890E-07,5979.80,339.6,6.97E-09 1.8349E-07,6020.45,341.0,1.16E-08 1.7984E-07,6027.11,341.5,1.34E-08 1.7439E-07.6021.39,341.9,1.66E-08 1.6087E-07,6239.69,348.0,2.31E-08 1.5686E-07.6241.11,348.4,2.57E-08 1.5557E-07,6222.36,348.1,2.64E-08 1.5189E-07,6292.59,349.9,3.06E-08 1.4769E-07,6409.33,352.5,3.52E-08 3.0990E-07.2412.38,337.0,8.88E-09 2.8882E-07,2478.49,341.7,1.44E-08 2.7279E-07,2535.44,346.0,1.82E-08 2.7216E-07.2512.37,344.7,1.80E-08 2.5837E-07,2547.46,347.6,2.07E-08 2.4760E-07,2577.62,349.9,2.48E-08 2.4122E-07.2581.26,350.5,2.72E-08 2.3879E-07,2577.69,350.4,2.91E-08 2.3494E-07.2590.31,351.2,3.10E-08 2.3046E-07,2615.42,352.6,3.32E-08 1.5304E-07,8932.76,341.0,3.85E-09 1.4673E-07,9064.91,343.7,5.11E-09 1.4488E-07,9039.87,343.6,5.12E-09 1.3943E-07,9083.94,344.9,6.09E-09 1.3326E-07.9195.35,347.2,6.80E-09 1.2959E-07,9214.96,347.9,7.85E-09 1.2811 E-07,9211.16,348.0,8.72E-09 1.2557E-07,9280.37,349.2,1.06E-08 1.2327E-07,9351.46,350.4,1.12E-08 4.9990E-07,919.93,336.8,8.63E-09 4.6725E-07,942.83,341.2,1.35E-08 4.4 U9E-07,963.96,345.6,1.96E-08 4.2614E-07,974.25,347.8,2.32E-08 4.0889E-07,982.62,349.8,2.61E-08 3.8624E-07,1012.21,354.3,2.57E-08 3.7698E-07,1014.50,354.8,2.36E-08 3.7489E-07,1007.97,354.2,2.63E-08 3.7208E-07,1004.17,353.8,2.74E-08 3.7046E-07,997.24 353.1,2.87E-08 3.8446E-07,1593.71,335.7,6.65E-09 3.5732E-07,1639.69,340.6,1.31E-08 3.3829E-07,1673.49,344.5,1.71 E-08 3.3222E-07,1673.11,344.8,1.72E-08 3.1550E-07,1694.98,347.7,2.00E-08 3.0228E-07,1714.30,349.9,2.42E-08  )  w  >  >  Appendix 5: Mathematical Modeling  2.9470E-07,1488.10,350.3,2.54E-08 2.9102E-07,1498.66,350.6,2.72E-08 2.8435E-07,1517.14,352.4,3.15E-08 2.7990E-07,1530.14,353.3,3.53E-08 1.6553E-07.6111.90,341.6,4.53E-09 1.5792E-07,6273.04,344.9,5.85E-09 1.5751E-07,6286.23,343.7,5.85E-09 1.5108E-07,6436.27,345.4,6.70E-09 1.4472E-07,6593.31,347.6,8.10E-09 1.401 lE-07,6715.59,348.9,9.82E-09 1.3767E-07,6782.11,349.8,1.12E-08 1.3631 E-07,6822.06,349.8,1.26E-08 1.3282E-07.6918.20,352.1,1.40E-08 7.3148E-07,373.73,332.2,2.33E-09 6.8193E-07,387.43,337.4,4.89E-09 6.6450E-07,392.70,338.9,6.33E-09 6.2164E-07,406.37,343.7.9.02E-09 5.7906E-07.421.21,349.9,1.15E-08 5.5849E-07,429.05,352.7,1.13E-08 5.5133E-07.431.91,353.5,1.19E-08 5.3926E-07,436.70,355.6,8.79E-09 5.3002E-07,440.48,357.3,6.78E-09 5.0425E-07,742.76,335.8,8.04E-09 4.7310E-07,768.27,339.9,1.24E-08 4.4429E-07,793.96,344.7,1.74E-08 4.3177E-07.806.16,346.2,2.00E-08 4.0574E-07,832.90,350.5,2.48E-08 3.8832E-07,852.39,353.3,2.38E-08 3.8155E-07,861.09,352.8,1.96E-08 3.7644E-07,867.44,353.3,2.41E-08 3.7291 E-07,872.12,3 53.3,2.59E-08 3.6739E-07,879.09,354.1,2.76E-08  279  2.9470E-07,1714.50,350.3,2.54E-08 2.9102E-07,1715.56,350.6,2.72E-08 2.8435E-07,1736.50,352.4,3.15E-08 2.7990E-07,1747.33,353.3,3.53E-08 1.6553E-07,7460.83,341.6,4.53E-09 1.5792E-07,7595.88,344.9,5.85E-09 1.5751 E-07,7529.73,343.7,5.85E-09 1.5108E-07,7585.60,345.4,6.70E-09 1.4472E-07,7670.84,347.6,8.10E-09 1.4011 E-07,7719.03,348.9,9.82E-09 1.3767E-07.7761.42,349.8,1.12E-08 1.363 lE-07,7752.61,349.8,1.26E-08 1.3282E-07,7878.99,352.1,1.40E-08 7.3148E-07,456.57,332.2,2.33E-09 6.8193E-07.473.18,337.4,4.89E-09 6.6450E-07,476.62,338.9,6.33E-09 6.2164E-07,488.46,343.7,9.02E-09 5.7906E-07,504.72,349.9,1.15E-08 5.5849E-07,513.13352.7,1.13E-08 5.5133E-07.515.83,353.5,1.19E-08 5.3926E-07,524.56,355.6,8.79E-09 5.3002E-07,532.26,357.3,6.78E-09 5.0425E-07,915.46,335.8,8.04E-09 4.7310E-07,936.84,339.9,1.24E-08 4.4429E-07,960.47,344.7,1.74E-08 4.3177E-07,966.57,346.2,2.00E-08 4.0574E-07,988.89,350.5,2.48E-08 3.8832E-07,1006.07,353.3,2.38E-08 3.8155E-07,1000.08,352.8,1.96E-08 3.7644E-07,1002.74,353.3,2.41E-08 3.7291E-07,1001.28,353.3,2.59E-08 3.6739E-07,1007.28,3 54.1.2.76E-08  2.9470E- 07,1714.50,350.3,2.54E-08 2.9102E- 07,1715,56,350.6,2.72E-08 2.8435E- 07,1736,50,352.4,3. 15E-08 2.7990E- 07,1747.33,353.3,3.53E-08 1.6553E- 07,7460,83,341.6,4.53E-09 1.5792E- 07,7595.88,344.9,5.85E-09 1.5751E- 07,7529,73,343.7,5.85E-09 1.5108E- 07,7585.60,345.4,6.70E-09 1.4472E- 07,7670.84,347.6,8. 10E-09 1.401 IE-07,7719.03,348.9,9.82E-09 1.3767E- 07,7761.42,349.8,1. 12E-08 1.3631E- 07,7752.61,349.8,1,26E-08 1.3282E- 07,7878.99,352.1,1.40E-08  )  5.0425E-•07,915.46,335.8,8.04E-09 4.7310E-07,936.84,339.9,1.24E-08 4.4429E-•07,960.47,344.7,1.74E-08 4.3177E-•07,966.57,346.2,2.00E-08 4.0574E- 07,988.89,350.5,2.48E-08 3.8832E- 07,1006.07,353.3,2.38E-08 3.8155E- 07,1000.08,352.8,1.96E-08 3.7644E-•07,1002.74,353.3,2.41E-08 3.7291E- 07,1001.28,353.3,2.59E-08 3.6739E-•07,1007.28,354.1,2.76E-08  Appendix S: Mathematical Modeling  A5.5  280  Statistical Analyses of Optimum Model Fit of Whey Protein Data to Experimental Results P a r a m eters: M ass t r a n s f e r c o e f f i c i e n t C h e m i c a l attachment coefficient C h em i c a l a c t i v a t i o n e n e r g y Reaction order, n Experimental Initial Fouling Rate (m K/J) 1.66E-09 2.09E-09 2.32E-09 3.07E-09 3.85E-09 4.36E-09 5.77E-09 7.84E-09 5.41E-09 5.21E-09 1.03E-08 1.54E-08 1.59E-08 2.07E-08 2.22E-08 5.56E-09 8.62E-09 1.06E-08 1.I8E-08 1.43E-08 1.60E-08 1.64E-08 2.36E-08 2.32E-08 6.96E-09 1.03E-08 1.58E-08 2.09E-08 2.29E-08 2.31E-09 4.34E-09 6.68E-09 9.49E-09 1.35E-08 1.73E-08 9.44E-10 1.30E-09 1.53E-09 4.63E-09 6.56E-09 7.30E-09 9.39E-09 1.06E-08 1.35E-08 1.39E-08 1.88E-08 2.23E-08  .S55177D-24 . 2 0 0 9 2 0 D + 06 0.99 M o d e l Initial Fouling Rate (m K/J) 1.33E-09 1.36E-09 2.36E-09 3.63E-09 3.71E-09 3.35E-09 7.33E-09 7.84E-09 8.35E-09 7.85E-09 1.18E-08 1.43E-08 1.4 I E - 0 8 1.92E-08 1.83E-08 3.02E-09 9.10E-09 7.52E-09 1.07E-08 1.54E-08 1.61 E - 0 8 1.50E-08 2.1 1 E - 0 8 1.94E-08 1.50E-08 1.74E-08 1.80E-08 1.98E-08 1.97E-08 4.21E-09 8.02E-09 8.47E-09 1.07E-08 1.71 E - 0 8 1.67E-08 8.32E-10 2.13E-09 1.99E-09 2.31E-09 5.86E-09 6.03E-09 1.04E-08 1.32E-08 1 43E-08 1.50E-08 1.63E-08 1.65E-08  2  2  S u m-of-squares ( m K / J ) = A v e r a g e a b s o l u t e d e v i a t i o n (% ) R o o t m e a n s q u a r e d e v i a t i o n (%) = 2  . 1 3 6 1 6 4 D + 16  2  A A D  (kgVm'Ks ) 5  kg  2 0 1  /m K s (J/mol) 3  1 0 1  R M S  ( E x p r e s s e d as f r a c t i o n s ) 0.1988 0.0395 0.3493 0.1220 0.0172 0.0003 0.1824 0.0333 0.0364 0.0013 0.2317 0.0537 0.2704 0.073 1 0.0000 0.0000 0.5434 0.2953 0.5067 0.2568 0.1456 0.0212 0.0714 0.005 1 0.1132 0.0128 0.0725 0.0053 0.1757 0.0309 0.4568 0.2087 0.0557 0.003 1 0.2906 0.0844 0.0932 0.0087 0.0769 0.0059 0.0062 0.0000 0.0854 0.0073 0.1059 0.01 12 0.1638 0.0268 1.1552 1.3344 0.6893 0.4752 0.1392 0.0194 0.0526 0.0028 0.1397 0.0195 0.8225 0.6765 0.8479 0.7190 0.2680 0.071 8 0.1275 0.0163 0.2667 0.071 1 0.0347 0.0012 0.1186 0.0141 0.6385 0.4076 0.3007 0.0904 0.501 1 0.251 1 0.1067 0.01 14 0.1740 0.0303 0.1076 0.01 16 0.2453 0.0602 0.0593 0.0035 0.0791 0.0063 0.1330 0.0177 0.260 1 0.0676  .309099D-15 24.5030 34.7806  1  Appendix 5: Mathematical Modeling  A5.6  281  Statistical Analyses of Optimum Model Fit of Lysozyme Data to Experimental Results  Parameters: Mass transfer coefficient Chemical attachment coefficient Chemical activation energy Reaction order, n Experimental (m K/J) 1.10E-08 1.71E-08 2.06E-08 2.14E-08 2.42E-08 2.55E-08 3.05E-08 3.38E-08 3.56E-08 3.56E-08 6.97E-09 1.16E-08 1.34E-08 1.66E-08 2.31E-08 2.57E-08 2.64E-08 3.06E-08 3.52E-08 8.88E-09 1.44E-08 1.82E-08 1.80E-08 2.07E-08 2.48E-08 2.72E-08 2.91E-08 3.10E-08 3.32E-08 3.85E-09 5.11E-09 5.12E-09 6.09E-09 6.80E-09 7.85E-09 8.72E-09 1.06E-08 2  1.0104E+15 7.1161E-27 1.6144E+05 0.75 Model (m K/J) 5.81E-09 1.24E-08 1.84E-08 1.89E-08 1.95E-08 2.50E-08 2.70E-08 2.86E-08 2.87E-08 3.38E-08 5.20E-09 6.43E-09 6.94E-09 7.39E-09 1.69E-08 1.78E-08 1.72E-08 2.13E-08 2.82E-08 7.37E-09 1.33E-08 2.04E-08 1.84E-08 2.37E-08 2.81E-08 2.95E-08 2.96E-08 3.11E-08 3.35E-08 4.51E-09 6.78E-09 6.70E-09 8.14E-09 1.13E-08 1.25E-08 1.27E-08 1.49E-08 2  (kg /m Ks ) kg^/n^K'^V" (J/mol) 4  8  AAD RMS (Expressed as fractions) 0.4718 0.2226 0.2747 0.0754 0.1092 0.0119 0.1183 0.0140 0.1943 0.0378 0.0180 0.0003 0.1153 0.0133 0.1544 0.0238 0.1926 0.0371 0.0500 0.0025 0.2533 0.0642 0.4459 0.1988 0.4823 0.2326 0.5545 0.3075 0.2705 0.0732 0.3071 0.0943 0.3475 0.1207 0.3024 0.0914 0.1990 0.0396 0.1703 0.0290 0.0752 0.0057 0.1231 0.0152 0.0201 0.0004 0.1464 0.0214 0.1321 0.0175 0.0847 0.0072 0.0156 0.0002 0.0033 0.0000 0.0097 0.0001 0.1725 0.0298 0.3275 0.1072 0.3095 0.0958 0.3369 0.1135 0.6622 0.4385 0.5916 0.3500 0.4557 0.2077 0.4073 0.1659  5  1/3  282  Appendix 5: Mathematical Modeling 1.12E-08 8.63E-09 1.35E-08 1.96E-08 2.32E-08 2.61E-08 2.57E-08 2.36E-08 2.63E-08 2.74E-08 2.87E-08 6.65E-09 1.31E-08 1.71E-08 1.72E-08 2.00E-08 2.42E-08 2.54E-08 2.72E-08 3.15E-08 3.53E-08 4.53E-09 5.85E-09 5.85E-09 6.70E-09 8.10E-09 9.82E-09 1.12E-08 1.26E-08 1.40E-08 8.04E-09 1.24E-08 1.74E-08 2.00E-08 2.48E-08 2.38E-08 1.96E-08 2.41E-08 2.59E-08 2.76E-08  1.74E-08 1.16E-08 1.60E-08 1.96E-08 2.11E-08 2.26E-08 2.48E-08 2.54E-08 2.55E-08 2.56E-08 2.56E-08 8.16E-09 1.42E-08 1.96E-08 2.02E-08 2.43E-08 2.72E-08 2.82E-08 2.87E-08 3.06E-08 3.16E-08 5.84E-09 9.44E-09 7.98E-09 1.02E-08 1.38E-08 1.64E-08 1.84E-08 1.84E-08 2.42E-08 1.07E-08 1.48E-08 1.90E-08 2.02E-08 2.29E-08 2.45E-08 2.48E-08 2.52E-08 2.55E-08 2.60E-08  Sum-of-squares (m K/J) = Average absolute deviation (%) = Root mean square deviation (%) = 2  2  0.5571 0.3488 0.1864 0.0013 0.0896 0.1346 0.0363 0.0772 0.0313 0.0657 0.1085 0.2266 0.0802 0.1464 0.1747 0.2150 0.1255 0.1084 0.0552 0.0275 0.1037 0.2887 0.6132 0.3648 0.5231 0.7046 0.6708 0.6421 0.4640 0.7282 0.3255 0.1964 0.0917 0.0113 0.0753 0.0289 0.2662 0.0469 0.0170 0.0594  0.3104 0.1216 0.0347 0.0000 0.0080 0.0181 0.0013 0.0060 0.0010 0.0043 0.0118 0.0513 0.0064 0.0214 0.0305 0.0462 0.0157 0.0118 0.0030 0.0008 0.0108 0.0833 0.3761 0.1331 0.2737 0.4964 0.4499 0.4123 0.2153 0.5303 0.1059 0.0386 0.0084 0.0001 0.0057 0.0008 0.0708 0.0022 0.0003 0.0035  1.27660E-15 23.2766 30.5538  Appendix 6: Kinetic Compensation Effect  Appendix 6  283  Kinetic Compensation Effect  As discussed by Froment (1975) and more recently by Koga (1994), the Kinetic Compensation Effect (KCE) has been observed in numerous kinetic studies of solid state reactions and heterogeneous catalysis reactions. The exact explanation of this effect is the subject of continuing research and is beyond the scope of this study. According to Koga (1994), sometimes the mutual dependence of the Arrhenius parameters can be treated in such a way as to eliminate dependence. Although not usually reported in fouling experiments, it will be shown that this effect commonly occurs when the simple Arrhenius type equation is used to describe a somewhat complicated chemical process. The fouling Arrhenius type plots (resulting from the non-linear least squares regression) were represented by equations of the form of Equation (4.4.1). Froment (1975) proposed that to reduce the correlation between A and AEf, the following re-parameterization may be useful, such that the data are moved and analyzed near the origin:  i  Rfo=Ae ^=Ae I  R ( T  i  -AE  raJ  ™^  i^  I  r  R  (MJ  = A°e  Thus A is a constant, called the isokinetic rate, and {j™*) 0  m  e  (A6.1)  isokinetic temperature. This  c  mutual dependence of the Arrhenius parameters is shown schematically in Figure A6.1. Bearing in mind that each experiment was performed separately and is therefore an independent set of results, the activation energy and pre-exponential factors were plotted against each other as previously shown in Figure 4.4.1. Their correlation indicates that these two parameters (A and AE) are strongly related by the following equations: ln(Awp ) = 0.3357 (AE )wp (kJ/mol)-12.1964 C  f  C  (A6.2)  Appendix 6: Kinetic Compensation Effect  284  ln(Ai ) = 0.3439 (AE ), (kJ/mol) -10.8958 ys  f  (A6.3)  ys  Isokinetic Point  < AE,  Kinetic Compensation Effect 1/T„  Correlation Analysis  Correlation Analysis  Figure A6.1: A schematic of the mutual dependence of the Arrhenius parameters According to Koga (1994), ln(A) = AJE/R Tj + lnk S0  (A6.4)  iso  Comparing Equations (A6.2), (A6.3) and (A6.4), the isokinetic temperature (Ti ) for whey S0  protein fouling is 85.1°C and the isokinetic rate constant is 5.05 x 10" m K/kJ. For lysozyme fouling the corresponding values are 76.6°C and 1.85 x 10" m K/kJ. For both systems, these isokinetic temperatures and rates correspond approximately to the average temperature and fouling rate observed on the test sections. Using Equation (A6.2) to eliminate either AEf or A from Equation (4.4.1) for whey protein solutions, the initial fouling rate becomes dependent upon only one of the Arrhenius parameters. Plotting the initial fouling rate thus computed, versus reciprocal inside wall temperature (T ;) as in Figure A6.2, shows how experimental data which adheres perfectly to w>  c  the kinetic compensation effect (as illustrated in Figure A6.1) would correlate. Figure A6.3, in  Appendix 6: Kinetic Compensation Effect  285  contrast, shows Arrhenius correlations based on the actual experimental data, when Equation (A6.2) is not substituted into Equation (4.4.1).  0.00295  1E-07 l/<T ,i) (K- ) 1  w  c  Figure A6.2: Arrhenius plot for perfectly correlated data from whey protein fouling experiments 1E-03 0.00265  0.00270  0.00275  0.00280  0.00285  0.00290  0.00295  1E-04  M 1E-05 e  o  1E-06  1E-07 1/fT.A (K ) 1  Figure A6.3: Experimental Arrhenius plot for whey protein fouling experiments Before continuing the re-parameterization procedure, it is interesting to look at the work of Crittenden et al. (1987a) and their styrene in kerosene polymerization results. The Arrhenius regressions (non-linear least squares) are shown in Table A6.1 and Figure 4.4.1.  286  Appendix 6: Kinetic Compensation Effect  Table A6.1: Styrene Polymerization Data of Crittenden et al. (1987a) Mass Flux (kg/m s)  A  134  4.30 x 10"  24.7  194  4.21 x 10"  24.4  251  7.09 x 10"  25.7  313  1.00 x 10"  26.7  376  4.75 x 10"  23.9  442  5.63 x 10"  2  24.7  512  1.28 x lO  -1  27.4  579  1.39 x 10°  34.6  649  6.59 x 10°  39.3  2  AE (kJ/mol) f  2  2  2  1  2  Figure 4.4.1 and Table A6.1 shows how well correlated the activation energy and preexponential factor are for a completely different set of experimental conditions and equipment. It is also interesting to note how similar the data regressions are: ln(A) = 0.332 AEf -11.171  Crittenden et al. (1987a)  ln(A) = 0.336 AEf -12.196  This work, WPC  ln(A) = 0.344 AEf - 10.896  This work, lysozyme  This information implies that if one were to plot all of the above data, the points could be well correlated by a single line, over a range of activation energies. Continuing with the re-parameterization, if one considers Equation (A6.1) and plots the  initial fouling rate versus  the fouling activation energy (AEf) and pre-  Appendix 6: Kinetic Compensation Effect  287  exponential factor (A°) can be determined. From Figure A6.4, performing this re-analysis has had the effect of shifting the Arrhenius plots toward the origin. , -0.OQOJ5  1 -0.00010  1 HS-03-0.00005 0.00000  0.00005  0.00010  0.00015  0.00020  1E-04  s as CUD  c o ta  Figure A6.4a: Fouling Arrhenius plot for whey protein solutions after re-parameterization  -0.00015  -0.00010  -0.00005  0.00000  0.00005  0.00010  0.00015  0.00020  1E-04 E,  a a  ai a o fa  "" " O —  309  306  310  307  1E-06 •{  -- + - 304 - -O- - 301 --*>- 303  308  1E-07  •305  Figure A6.4b: Fouling Arrhenius plot for lysozyme solutions after re-parameterization Table A6.2 and Figure A6.5 show that A ° is independent of AEf for both whey protein and lysozyme solutions, with the average value corresponding approximately to the isokinetic rate. Therefore, representing the Arrhenius type equation in the form suggested by Equation (A6.1) eliminates the mutual dependence of the Arrhenius parameters, indicating that the kinetic  Appendix 6: Kinetic Compensation Effect  288  compensation effect for these experiments is likely a result of the regression analysis being performed far from the origin.  Table A6.2: Effect of re-parameterization upon the Arrhenius parameters for whev protein and lysozyme fouling experiment TFU  A  AEf (kJ/mol)  Sum of squares (m K/kJ)  211  1.0838 x 10  9.9689 x 10"  48.3  1.0007 xlO"  11  9  208  2.3594 x 10  5.0879 x 10  -6  128  1.2765 xlO'  11  5  212  4.3169 x 10  3.8420 x l O  -6  186  3.6340 xlO'  12  6  204  6.1831 x 10  3.9121 x 10  90.5  7.7654 x 10'  207  4.7410 x l O  1.2370 xlO'  5  121  2.1106 x 10'  206  8.7210 x 10  7.2582 x 10  -6  213  1.5941 x lO  209  9.9777 x 10  4.5633 x 10"  208  1.3904 xlO-"  6  205  1.6859 x 10  4.9315 xlO"  148  2.8933 x lO'  12  8  210  1.8954 x 10  82.4  3.5476 x l O  1 4  3  309  2.1018 xlO"  29.4  4.5961 x 10'  9  306  1.5491 x l O  45.8  6.8227 x 10"  11  10  310  4.4403 x 10  2.0915 x 10"  5  49.0  5.0326x 10'"  10  307  1.3422 x 10  8  2.5080 x lO  -5  85.2  1.1329 x lO  10  304  8.4451 x 10  2.6013 x 10'  83.7  1.1794 x 10'  301  2.5003 x 10  2.8374 x 10"  86.7  3.7207 x 10"  303  2.8936 x 10  9  2.8390 xlO"  5  93.8  3.5516 x 10"  u  9  308  1.8498 x 10  1.1003 xlO"  5  115  3.1590 x 10"  12  9  305  7.2136 x 10  1.0342 xlO'  5  119  2.0257 x 10'  12  9  A° 2  13  21  7  12  25  24  16  6  -6  6  6  1.8451 xlO"  6  1  8.6344 x l O  6  -6  2.2204 x 10'  2  5  2  7  8  12  12  5  5  2  14  11  -11  11  -11  11  11  2  #ofT/C's  6 9 7  10 10  Appendix 6: Kinetic Compensation Effect  289  1E-04 50  100  150  200  250  • WPC • LYS  < 1E-05  1E-06 AE (kJ/mol) f  Figure A6.5: The lack of mutual dependence of ln(A°) and AEf after re-parameterization  Appendix 7: Experimental Uncertainty  Appendix 7 A7.1  290  Experimental Uncertainty  Uncertainty in the Calculated Heat Transfer Coefficient A n estimate of the experimental uncertainty in the heat transfer coefficient (the reciprocal  of which is used to determine the initial fouling rate) can be calculated using Equation (4.1.1). This estimate evaluates the uncertainty in the clean heat transfer coefficient using the clean, inside wall temperature.  U (x) = 9/>4[(T ) (x)-T (x) e  WJ  e  (4.1.1)  b  The standard deviation of the heat transfer coefficient at any axial location can be calculated via Equation (A7.1), where the uncertainty in the heat transfer area (A) is assumed to be negligible.  (dU a. + — Q v dA 2  c  s  5U„  (  au  v  V,) -Q  +  'au^  (A7.1)  au.  (A7.2)  c  VdQj The standard deviation of the heat transfer rate applied to the test section ( Q ) and of the bulk temperature (Tb) were estimated over the duration of the experiment, while the standard deviation of the clean inside wall temperature was estimated over the 5 -10 minutes in which the clean, inside wall temperature was determined. The standard deviation for Q , shown in Table A7.1 for all experiments, was obtained from the standard deviation of the voltage (V) and current (I) applied to the test section. These results show that there was very little variation in heat flux with time for all experiments.  291  Appendix 7: Experimental Uncertainty  Table A7.1: Experimental uncertainty in the heat applied to the test section (Q) TFU 301 •303 304305 306 307 308 . 309 310  V 16.68 17.44 13.82 19.16 10.85 12.86 18.49 6.91 10.84  a(V) 0.04 0.04 0.09 0.05 0.05 0.03 0.05 0.04 0.05  0.24 0.23 0.65 0.26 0.46 0.23 0.27 0.58 0.46  TFU 204 205 206 207 208 209 210 211 212  V 12.27 19.80 18.78 16.01 10.65 19.15 19.74 8.15 12.42  o-(V) 0.08 0.11 0.05 0.08 0.03 0.04 0.05 0.04 0.02  0.65 0.56 0.27 0.50 0.28 0.21 0.25 0.49 0.16  %  %  r 1.629 1.697 1.355 1.903 1.053 1.224 1.793 0.649 1.056  0.010 0.009 0.007 . 0.013 0.005 0.005 0.006 0.004 0.012  I 243.45 253.63 202.42 284.47 157.21 182.81 268.00 96.73 157.66  afl) 1.50 1.35 1.05 1.95 0.75 0.75 0.90 0.60 1.80  I' 1.170 1.910 1.810 1.605 0.990 1.903 1.990 0.788 1.222  a(T) 0.011 0.011 0.010 0.013 0.006 0.008 0.015 0.006 0.008  I 174.73 285.51 270.54 239.85 147.78 284.47 297.49 117.54 182.51  <T(I)  %  1.65 1.65 1.50 1.95 0.90 1.20 2.25 0.90 1.20  0.94 0.58 0.55 0.81 0.61 0.42 0.75 0.76 0.66  %  0.61 0.53 0.52 0.68 0.48 0.41 0.34 0.62 1.14  Q(W) 4060.7 4423.2 2797.5 5450.4 1705.8 2351.0 4955.3 668.4 1709.1  o(Q) 34.7 33.6 32.7 51.5 16.0 15.1 30.0 8.0 27.4  0.85 0.76 1.17 0.95 0.94 0.64 0.61 1.20 1.60  Q(W) 2143.9 5653.2 5080.8 3840.0 1573.9 5447.5 5872.5 957.9 2266.8  <*(Q) 34.2 64.0 41.6 50.3 14.0 34.3 59.2 12.0 18.5  1.59 1.13 0.82 1.31 0.89 0.63 1.01 1.26 0.82  %  %  Tables A7.2 and A7.3 show the magnitude of experimental uncertainty in the heat transfer coefficient for each thermocouple in a typical experiment, for whey protein and lysozyme fouling, respectively. Both tables show variation in experimental uncertainty from one thermocouple to another (3.8 - 7.2% for T F U 207); however, the average experimental uncertainty in the heat transfer coefficient for both whey protein (5.0%) and lysozyme (4.3%) experiments appears to be relatively small. It is worth noting that the standard deviation for the clean inside wall temperature for the whey protein experiment was greater than that for the lysozyme experiment, which is likely due to the difficulty in obtaining a good estimate of the clean inside wall temperature for whey protein.  Appendix 7: Experimental Uncertainty  292  Table XI.2: Experimental uncertainty (95 % confidence interval) for TFU 207 (G = 531.7 kg/m s, Q= 3840 W, a . = 50.3 W) 2  T/C 1 '2  3 4 5 6 7 8 9 10  T ,i( C) 75.66 76.25 83.50 82.00 84.69 87.75 88.07 87.29 90.95 89.75 0  w  o((Tw^)c)(°C) 0.58 0.95 0.75 0.76 0.74 0.76 0.41 0.89 0.85 1.13  T (°C) 32.34 34.40 36.17 38.10 41.73 45.39 49.05 50.88 52.71 54.47 b  CT(T )( C) 0.32 0.31 0.31 0.31 0.32 0.35 0.39 0.41 0.43 0.46 0  b  8f/5Q 1.0556 1.0926 0.9661 1.0416 1.0644 1.0795 1.1719 1.2559 1.1958 1.2961  -5f/6T 93.57 100.26 78.38 91.11 95.14 97.86 115.33 132.45 120.08 141.07 w  Q  8f/8T U ( W / m K ) 93.57 4053 100.26 4196 78.38 3710 91.11 4000 95.14 4087 97.86 4145 115.33 4500 132.45 4823 120.08 4592 141.07 4977 2  b  c  o ( U ) +/-1.96 U 81.6 160.0 114.3 224.0 80.1 156.9 91.3 179.0 93.5 183.3 98.2 192.6 87.9 172.4 144.3 282.9 129.2 253.3 184.0 360.7 c  c  Average  % 3.9 5.3 4.2 4.5 4.5 4.6 3.8 5.9 5.5 7.2 5.0  Table A7.3: Experimental uncertainty (95 % confidence interval) for TFU 304 (G = 523.0 kg/m s, Q= 2798 W, a . = 32.7 W) 2  Q T/C 1 2  3 4 5 6 7 8 9 10  T»j(°C) 63.81 68.53 72.80 71.58 74.48 76.76 77.31 77.26 78.08 79.47  a(fTwj)c)( C) 0.68 0.43 0.61 0.70 0.43 0.45 0.44 0.45 0.62 0.55 0  T (°C) 31.69 33.20 34.49 35.90 38.54 41.22 43.89 45.22 46.56 47.85 b  o(Tb)(°C) 0.33 0.31 0.30 0.29 0.29 0.29 0.31 0.33 0.35 0.37  5f/8Q 1.4236 1.2943 1.1936 1.2816 1.2723 1.2866 1.3682 1.4272 1.4507 1.4461  -8f/8T 123.99 102.48 87.16 100.48 99.03 101.28 114.53 124.61 128.76 127.94 w  8f/8T U ( W / m K) a ( U ) +/-1.96 U 123.99 3983 104.6 205.1 3621 102.48 68.9 135.0 87.16 3339 70.9 139.1 100.48 3585 86.9 170.3 99.03 3559 66.1 129.6 101.28 3599 68.6 134.5 114.53 3828 76.2 149.3 124.61 83.7 164.1 3993 128.76 4058 103.2 202.3 127.94 4046 97.1 190.3 b  c  c  Average  A7.2  c  % 5.1 3.7 4.2 4.8 3.6 3.7 3.9 4.1 5.0 4.7 4.3  Uncertainty in the Measured Initial Fouling Rate From any one experiment up to ten thermal fouling results (depending upon the number  of thermocouples that satisfied the fouling Biot number criterion) were achieved by performing a linear regression of the reciprocal heat transfer coefficient with time. Here, a regression analysis was performed on all fouling results to determine the 95% confidence interval for each rate. Results for individual thermocouples and experimental averages are shown in Table A7.4. Table A7.4 shows that the confidence intervals for the lysozyme fouling experiments (TFU 300) are considerably smaller than for the corresponding whey protein experiments (TFU  Appendix 7: Experimental Uncertainty  293  200). This is expected, since lysozyme initial fouling rates were greater, demonstrating less spread in the reciprocal heat transfer coefficient (compare Figure 4.1.3 to Figure 4.2.4). Table A7.4: Percentage experimental uncertainty (95 % confidence interval) in initial fouling rate for each thermocouple and experiment average for all fouling experiments TFU  G  TCI  TC2  TC3  TC4  TC5  TC6  TC7  TC8  TC9  T C 10  Exp.  (kg/m s) 2  204  401.4  -  -  -  -  6.9  3.8  3.3  3.8  6.2  5.1  4.9  205  1055.6  -  -  30.1  18.7  17.2  11.1  6.2  7.1  7.5  3.6  12.7  206  699.4  -  -  -  22.2  18.6  12.6  5.8  5.7  4.8  3.6  10.5  207  531.7  -  25.7  14.0  9.4  7.6  3.5  '2.5  2.4  3.0  3.0  7.9  208  284.1  -  -  -  -  -  15.2  14.6  10.8  15.3  29.3  14.2  209  882.3  -  -  -  -  9.1  5.3  2.7  2.2  2.2  1.2  3.8  210  1274.9  -  -  -  -  -  -  -  29.2  26.2  15.0  23.5  211  221.5  -  24.4  15.9  13.0  10.8  11.3  6.7  8.6  18.6  20.6  14.4  212  400.4  -  -  -  -  10.9  8.6  4.9  4.7  8.4  14.2  8.6  301  700.5  4.5  3.5  2.9  2.3  2.1  2.7  1.6  1.5  1.7  1.7  2.5  303  878.7  -  16.2  5.2  5.2  3.0  2.6  2.3  2.3  2.0  1.7  4.5  304  523.0  6.1  4.2  1.6  1.7  1.4  1.2  1.1  1.0  1.3  1.2  2.1  305  1101.2  -  5.5  3.9  3.5  2.8  2.2  1.9  1.8  1.9  0.9  2.7  306  296.9  3.6  3.0  1.5  1.3  1.5  1.2  1.3  1.1  1.1  1.4  1.7  307  412.3  10.7  3.1  2.3  2.3  1.5  1.7  1.2  1.1  1.0  0.8  2.6  308  991.3  -  3.3  2.4  2.2  1.6  1.5  1.4  1.8  1.6  0.7  1.8  309  199.5  -  11.6  5.9  5.2  3.8  3.5  5.3  5.9  13.8  18.9  8.2  310  297.2  3.0  2.4  1.1  1.0  0.8  0.8  1.0  0.8  1.2  0.7  1.3  Table A7.4 shows a high percentage uncertainty for T F U 210. This experiment was performed at a high mass flux, and relatively small fouling rates resulted. This observation is also reflected in the fact that only three thermocouples satisfied the fouling Biot number criterion.  Appendix 7: Experimental Uncertainty  294  Typically, the lower temperature thermocouples (TC 1 - T C 4) have greater uncertainty in the fouling rate, while the high temperature thermocouples (in particular, the lysozyme fouling experiments) had only 1 - 2% uncertainty at the 95% confidence interval. Anomalies to this trend were some of the whey protein experiments (TFU 211,208 and 212) which displayed accelerated fouling rates (Section 4.2), and hence lower, initial fouling rates. With the exception of T F U 310, Table A7.4 shows that the experimental uncertainty in the initial fouling rate is small, and in the case of lysozyme fouling is smaller than the experimental uncertainty in the heat transfer coefficient. Figures A7.1 and A7.2 display the experimental uncertainty in the initial fouling rate as error bars in the ordinate.  A7.3  Experimental Error caused by forcing data to fit the Arrhenius Expression Sections 4.2 and 4.3 utilize the Arrhenius expression to correlate the experimental fouling  data, thus enabling data interpolation and hence an estimate of R& at specific wall temperatures at a given mass flux. This allows a qualitative comparison of the experimental fouling results to the features of the mathematical model. However, forcing the data to fit the Arrhenius expression causes some error in the resulting data points (Figures 4.2.10 and 4.3.6). This was investigated by estimating the average absolute deviation (AAD) between the experimental fouling data and the non-linear least squares regression for each experiment. The results are shown in Table A7.5. Comparing the experimental average uncertainty of Table A7.4 to Table A7.5, the error introduced by forcing the experimental data to fit the Arrhenius expression is generally, although not always, greater than the uncertainty in the measurement of the initial fouling rate. This point is clearly shown in Figures A7.1 and A7.2, where the vertical deviation between the best fit regression line and each data point is greater than the uncertainty in Rf of that data point. T F U 0  Appendix 7: Experimental Uncertainty  295  209 and T F U 306 visibly demonstrate this difference, while in other experiments the difference was not so obvious. Therefore, the average absolute percent deviation (AAD) between the experimental data points and the non-linear least squares regression are the values used in the ordinate error bars of Figures 4.2.10 and 4.3.6.  Table A7.5: Error (AAD) from forcing experimental data to fit the Arrhenius expression TFU  G  TC 1  TC 2  TC 3  TC 4  TC 5  TC 6  TC 7  TC  8 TC 9  TC 10  Exp.  (kg/m s) 2  204  401.4  -  -  -  -  3.8  3.5  2.0  4.5  0.0  0.7  2.4  205  1055.6  -  -  3.0  16.3  15.5  23.1  1.0  19.7  15.1  10.3  13.0  206  699.4  -  -  -  28.1  20.4  5.8  6.5  12.6  11.1  5.9  12.9  207  531.7  -  20.1  19.5  18.0  0.0  16.8  8.1  3.7  0.4  10.3  10.8  208  284.1  -  -  -  -  -  5.3  17.5  14.6  7.2  5.7  10.0  209  882.3  -  -  -  -  17.8  31.6  7.6  13.1  17.0  12.1  16.5  210  1274.9  -  -  -  -  -  -  -  0.9  9.2  9.8  6.6  211  221.5  -  5.4  5.5  14.0  5.4  12.3  2.2  8.6  4.8  9.4  7.5  212  400.4  -  -  -  -  36.0  27.8  3.9  21.1  9.3  4.2  17.1  301  700.5  6.7  1.6  8.8  6.6  4.5  11.4  1.0  5.9  10.5  5.5  6.2  303  878.7  -  56.4  7.5  2.4  18.1  5.9  1.5  6.9  5.2  4.5  12.0  304  523.0  0.4  7.4  5.2  4.0  6.4  7.1  2.2  4.9  4.6  0.3  4.2  305  1101.2  -  4.3  0.0  2.1  3.8  14.2  6.9  2.9  7.6  0.4  4.7  306  296.9  40.8  11.6  5.9  11.9  14.4  6.1  18.3  3.2  2.4  9.9  12.5  307  412.3  12.6  12.0  5.3  2.9  6.1  6.0  4.3  0.5  0.3  4.3  5.4  308  991.3  -  4.5  8.7  5.1  1.2  7.8  2.8  0.2  11.5  3.2  5.0  309  199.5  -  118.1  22.4  1.1  19.6  24.4  16.7  19.1  16.4  57.9  32.8  310  297.2  29.9  4.0  5.9  11.7  12.1  4.6  24.3  3.4  4.1  6.3  10.6  Also included in Figures A7.1 and A7.2 are estimates of the uncertainty in the reciprocal wall temperature  measurements, with both plots showing significant wall  temperature  uncertainty. For the whey protein fouling experiments this was due to the difficulty in the  Appendix 7: Experimental Uncertainty  296  measurement of the clean inside wall temperature (typically ± 1.5°C), while the uncertainty for the lysozyme fouling experiments, while still significant, was due mainly to fluctuations in the temperature measurement of the thermocouples (typically ± 1.0°C). The corresponding uncertainties in the reciprocal wall temperatures in Figures A7.1 and A7.2 appear unduly large due to the narrow temperature range encompassed by these figures and hence the large abscissa scales involved. 1E-04 0.00272  0.00274  0.00276  0.00278  0.00280  0.00282  y = 1.09E+25e  J  E  1E-05 o to  1E-06 1/(T A (K ) 1  W  Figure A7.1 Arrhenius type expression for TFU 209 with error bars 1E-04 0.00280  0.00284  0.00288  0.00292  0.00296  r-i •i. • . - . — i OS BD C  1E-05 y=1.52E+02e -  5 50E+<)3x  O  1E-06  J  l/CT^MK- ) 1  Figure A7.2 Arrhenius  expression for TFU 306 with error bars  0.00300  297  Appendix 7: Experimental Uncertainty  Although the experimental errors and uncertainties in this work are significant, it appears that in general the Arrhenius expression correlates the experimental data reasonably well. T F U 306 (Figure A7.2), T F U 309 and T F U 310 are exceptions to this generalization. In addition, the uncertainty in the measurement of the clean, inside wall temperature appears to be the most significant factor affecting these results.  A7.4  Uncertainty in Time Averaged Mass Flux The uncertainty in the mass flux for each experiment was determined from the time  averaged mass flux and its standard deviation for the duration of the experiment, from which the 95 % confidence interval was estimated. The results in Table A7.6 are used as the error bars in the abscissa of Figures 4.2.10 and 4.3.6. These results show that the experimental uncertainty in the mass flux over the duration of an experiment is very small.  Table A7.6: Mass flux uncertainty for all fouling experiments TFU 204 205 206 207 208 209 210 211 212 301 303 304 305 306 307 308 309 310  G (kg/m s)  ± A G (kg/m s) to 95 % confidence interval  401.4  8.8 (2.2%)  1055.6  34.6 (3.3%) 22.0 (3.1%)  2  699.4 531.7  2  284.1  15.7 (3.0%) 5.9(2.1%)  882.3  9.4(1.1%)  1274.9 221.5 400.5  7.4 (0.6%) 3.1 (1.4%) 3.2 (0.8%)  700.5 878.7 523.0  8.0(1.1%) 10.7(1.2%) 7.2(1.4%)  1101.2  9.5 (0.9%) 6.1 (2.1%) 5.0(1.2%)  296.9 412.3 991.3 199.5 297.2  8.4 (0.8%) 1.3 (0.7%) 6.2(2.1%)  

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