UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Heat exchanger fouling by precipitation of calcium phosphates Singh, Atmajeet 1992

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-ubc_1992_fall_singh_atmajeet.pdf [ 5.73MB ]
Metadata
JSON: 831-1.0058511.json
JSON-LD: 831-1.0058511-ld.json
RDF/XML (Pretty): 831-1.0058511-rdf.xml
RDF/JSON: 831-1.0058511-rdf.json
Turtle: 831-1.0058511-turtle.txt
N-Triples: 831-1.0058511-rdf-ntriples.txt
Original Record: 831-1.0058511-source.json
Full Text
831-1.0058511-fulltext.txt
Citation
831-1.0058511.ris

Full Text

HEAT EXCHANGER FOULING BY PRECIPITATION OF CALCIUM PHOSPHATES By ATMAJEET SINGH B.Tech., Indian Institute of Technology, New Delhi, INDIA A THESIS IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES ( Department of Chemical Engineering ) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October 1992 ® Atmajeet Singh, 1992 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. I 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 writ ten permission. Department of C^^H\CM- ^t<C,Q • The University of British Columbia Vancouver, Canada DE-6 (2/88) ABSTRACT The fouling characteristics of artificially hardened waters containing calcium and orthophosphate ions were investigated. The water was circulated through a steam heated double pipe heat exchanger under turbulent flow conditions. Inlet water temperatures were about 20 to 25°C and the phosphate content about 100 ppm. Experiments were carried out to ascertain the effects of pH, velocity, surface temperature and concentrations of the calcium and the orthophosphate on the fouling process. The fouling behaviour was recorded in terms of the fouling resistance-time curve which was calculated from the time dependence of the overall heat transfer coefficients. Under the constant heat flux condition imposed, the fouling resistance was generally linear in time; however, asymptotic fouling was observed in some cases. The fouling rate was found to be a very strong function of the pH over the range 6.5 to 8.0. The rate was also found to be exponentially dependent on the calculated surface temperature. The initial fouling rate increased linearly with the velocity and with the concentrations of scale forming species. Deposit thickness increased with heated length of the stainless steel test section. In most cases, the deposit was a dense crystalline scale, which had a Ca/P molar ratio of 1.35±0.15. The results are discussed in terms of the ionic equilibrium involved and the solubility and crystallisation kinetics from the literature. It is apparent that the rate of phosphate precipitation is influenced both by surface reaction and mass transfer in the range of the variables studied. Findings are compared with the few prior phosphate fouling studies from the literature. Scale density is reported, and its thermal conductivity estimated using the fouling resistance measurements. TABLE QF CONTENTS Abstract i i List of Tables vii List of Figures viii Acknowledgements xii 1. INTRODUCTION 1 2. THEORY 2.1 Definitions of Solubility and Supersaturation 8 2.2 General Mechanism of Phosphate Fouling in a Heat Exchanger 9 2.3 Heat Transfer in a Double Pipe Heat Exchanger 10 2.4 Types of Fouling Curves 18 2.5 Chemistry of Calcium Phosphates: Equilibrium Considerations 20 2.6 Distribution of Different Ionic Forms of the Phosphate Species 28 2.7 Solid Forms of Calcium Phosphates 35 2.8 Kinetics of Crystallisation of Calcium Phosphates 37 2.9 Related Studies in the Area of Phosphate Fouling 44 3. EXPERIMENTAL APPARATUS 3.1 Water Flow Loop 47 3.2 pH Control System 49 3.3 Steam Supply System 50 3.4 Storage and Supply Tanks 50 3.5 Heat Exchanger 51 3.6 Temperature Measurements 53 4. EXPERIMENTAL PROCEDURES 4.1 General Strategy 55 4.2 Calcium Content of the Solutions 56 4.3 Phosphate Content of the Solutions 57 4.4 Step-by-step Procedure of a Typical Scaling Run 58 4.5 Cleaning of the Heat Exchanger 62 5. RESULTS AND DISCUSSIONS 5.1 General Overview 64 5.2 Variation of the Clean Overall Heat Transfer Coefficient with the Heat Transfer rate 65 5.3 Fouling Resistance versus Time 66 5.4 Reproducibility of the Results 68 5.5 Explanation for the Type of the Fouling Behaviour 83 5.6 Effect of pH on the Initial Fouling Rate 84 5.7 Effect of Velocity on the Initial Fouling Rate 91 5.8 Effect of Surface Temperature on the Initial Fouling Rate 95 5.9 Effect of Concentration on the Initial Fouling Rate 96 5.10 Characteristics of Deposit 98 5.11 Comparison with on Previous Studies 104 5.12 Calculation of the Fouling Rate if not Limited by Surface Reaction 107 5.13 Correlations for Prediction of the Initial Fouling Rates 110 6. CONCLUSIONS AND RECOMMENDATIONS 8.1 Conclusions 115 8.2 Recommendations for the Future Work 117 7. NOMENCLATURE 120 8. REFERENCES 130 APPENDIX I : CALIBRATION CURVES 1.1 Calibration Curve of the Rotameter 135 1.2 Calibration Curve for the Determination of Total Orthophosphate 135 APPENDIX n : SAMPLE CALCULATIONS n. 1 Calculation of the Rate of the Heat Transfer 138 n.2 Determination of the Dirty Overall Heat Transfer Coefficient 139 n.3 Determination of the Fouling Resistance 140 n.4 Determination of the Theoretical Overall Heat Transfer Coefficient 140 n.5 Determination of the Wall Temperature and the Scale Surface Temperature 143 II.6 Determination of the Reynolds and Schmidt Numbers 145 n.7 Determination of the Density and Thermal Conductivity of the Scale 148 n.8 Determination of the Total System Volume 150 II.9 Water Analysis Calculations 152 n.lO Determination of the Theoretical and Experimental Pressure Drops and the 153 Theoretical Shear Stress n . l l Numerical Example Using Run 2 160 APPENDIX m : COMPUTER PROGRAMS 168 APPENDIX IV : FOULING CURVES 182 LIST OF TABLES 3-1 Dimensions and Thermal Conductivities of the Tubes 51 5-1 Summary of the Slopes of the Heat Transfer Rate Versus the Clean 66 Overall Heat Transfer Coefficient Curves 5-2 Summary of the Results 80 5-3 Summary of Concentrations and Activity Coefficients 82 5-4 Shapes of the Fouling Curves 85 5-5 Summary of the ICP Analysis of the Deposit 103 5-6 Ca/P Molar Ratios for Various Calcium Phosphates 103 5-7 Determination of the Fouling Rates if Limited by Diffusion 109 5-8 Comparison of Predicted Versus Experimental Initial Fouling Rates 114 APPENDIX n n-1 Comparison of Experimental and Theoretical Clean Overall 144 Heat Transfer Coefficients n-2 Determination of the Density of the Scale 151 n-3 Determination of the Thermal Conductivity of the Scale 151 n-4 Water Analysis Data 154 n-5 Determination of the Total System Volume 165 n-6 Experimental Pressure Drop as a Function of Velocity 167 LIST OF FIGURES 2-1 Types of Fouling Curves 19 2-2 Distribution of Various Phosphate Species as a Function of pH 31 2-3 Logarithmic Concentiation Diagram for the Phosphate Species 32 2-4 Solubilities of Various Phosphates as a Function of pH 40 2- 5 Total Calcium Versus Time in Growth of HAP Crystals 41 3- 1 Process Flow Diagram of the Apparatus 48 3-2 Schematic Diagram of the Double Pipe Heat Exchanger 52 3-3 Front Photograph of the Apparatus 54 5-1 Variation of Water Chemistry over Time 69 5-2 Effect of tiie Heat Transfer Rate on the Clean Overall Heat Transfer 70 Coefficient at Low Velocity 5-3 Effect of the Heat Transfer Rate on the Clean Overall Heat Transfer 71 Coefficient at Medium Velocity 5-4 Effect of the Heat Transfer Rate on the Clean Overall Heat Transfer 72 Coefficient at High Velocity 5-5 Effect of the Heat Transfer Rate on the Clean Overall Heat Transfer 73 Coefficient at Low Velocity ; Repeat Run 5-6 Effect of tiie Heat Transfer Rate on the Clean Overall Heat Transfer 74 Coefficient at Medium Velocity ; Repeat Run 5-7 Effect of the Heat Transfer Rate on the Clean Overall Heat Transfer 75 Coefficient at High Velocity ; Repeat Run 5-8 Run 3 Corrected and Uncorrected Fouling Resistance Versus Time 76 5-9 Run 11 Fouling Resistance Versus Time 77 5-10 Run 21 Fouling Resistance Versus Time 78 5-11 Run 23 Fouling Resistance Versus Time 79 5-12 Effect of pH on the Initial Fouling Rate at High Velocity 86 5-13 Effect of pH on the Initial Fouling Rate at Low Velocity 87 5-14 Log of Initial Fouling Rate Versus pH at High Velocity 88 5-15 Log of Initial Fouling Rate Versus pH at Low Velocity 89 5-16 Effect of Velocity on the Fouling Rate 93 5-17 Qualitative Effect of Vdocity on the Initial Fouling Rate 94 5-18 Effect of Surface Tempaature on the Initial Fouling Rate 97 5-19 Effect of Total Calcium Concentration on the Initial Fouling Rate 99 5-20 Effect of Total Phosphate Concentration on the Initial Fouling Rate 100 5-21 Effect of Calculated PC^" Concentration on the Initial 101 Fouling Rate at pH 6.5 to 8.0 5-22 Plot of Initial Fouling Rate Versus Experimental Parameters 112 5-23 Plot of Experimental Versus Predicted Initial Fouling Rates 113 APPENDIX I I-l Calibration Curve for the Rotameter 136 1-2 Calibration Curve for the Determination of the Total Phosphate 137 APPENDIX n n- l Temperature Gradients and Heat Transfer Resistances 146 n-2 Effect of the Velocity on the Theoretical and Experimental Pressure Drop 158 n-3 Effect of the Velocity on the Calculated Shear Stress 159 APPENDIX m m- l Computer Program to Calculate the Experimental Fouling Resistances 170 as a Function of Time in-2 Computer Program to Calculate the Concentration of Various Species 173 and the Activity Coefficients in-3 Computer Program to Calculate the Concentration of Various Phosphate 176 Species (assuming all activity coefficients to be unity) III- 4 Computer Program to Calculate the Clean Overall Heat Transfer Coefficient 177 in-5 Computer Program to Calculate the Shear Stress and the Pressure Drop 179 in-6 Computer Program to Calculate the Shear Stress and the Pressure Drop as a 181 Function of the Velocity APPENDIX rv rV-1 Run 2 Fouling Resistance Versus Time 182 rV-2 Run 3 Fouling Resistance Versus Time 183 rV-3 Run 4 Fouling Resistance Versus Time 184 rV-4 Run 5 Fouling Resistance Versus Time 185 rV-5 Run 6 Fouling Resistance Versus Time 186 IV- 6 Run 7 Fouling Resistance Versus Time 187 IV-7 Run 8 Fouling Resistance Versus Time 188 rV-8 Run 9 Fouling Resistance Versus Time 189 IV-9 Run 10 Fouling Resistance Versus Time 190 rV-lO Run 12 Fouling Resistance Versus Time 191 r v - l l Run 13 Fouling Resistance Versus Time 192 IV-12 Run 14 Fouling Resistance Versus Time 193 rV-13 Run 15 Fouling Resistance Versus Time 194 rV-14 Run 16 Fouling Resistance Versus Time 195 IV-15 Run 17 Fouling Resistance Versus Time 196 IV-16 Run 18 Fouling Resistance Versus Time 197 rV-17 Run 19 Fouling Resistance Versus Time 198 rV-18 Run 20 Fouling Resistance Versus Time 199 rV-19 Run 22 Fouling Resistance Versus Time 200 rV-20 Run 24 Fouling Resistance Versus Time 201 rV-21 Run 25 Fouling Resistance Versus Time 202 rV-22 Run 26 Fouling Resistance Versus Time 203 IV-23 Run 27 Fouling Resistance Versus Time 204 IV-24 Run 28 Fouling Resistance Versus Time 205 IV-25 Run 29 Fouling Resistance Versus Time 206 IV-26 Run 30 Fouling Resistance Versus Time 207 IV-27 Run 31 Fouling Resistance Versus Time 208 IV-28 Run 32 Fouling Resistance Versus Time 209 rV-29 Run 33 Fouling Resistance Versus Time 210 rV-30 Output of Computer Program m-l 211 rV-31 Output of Computer Program in-2 212 ACKNOWLEDGEMENTS I would like to express my indebtedness to Dr. A.P. Watkinson for his beneficial guidance and support throughout the duration of this research work. I am also thankful to the workshop staff and the electrical workshop for their help in installation and maintenance jobs. Special thanks are also due to Mr. Horace Lam of the stores for his help in ordering and procuring chemicals and parts. I am grateful to my wife Paramjeet for her unlimited cooperation and care to help me in completing this project. Financial support provided by NSERC is also gratefully acknowledged. §1. INTRODUCTION In general, heat exchanger fouling can be defined as the phenomenon of any undesirable deposition or solidification on the heat transfer surface, caused by the presence in the fluids of substances that either tend to accumulate at or react with the surface. Thermal fouling may be promoted by a temperature gradient near the heat transfer surface. The term scaling is usually used to describe the formation of a dense, crystalline deposit, which adheres well to the heat transfer surface. Fouling is a more general term and can include deposition of non-crystalline as well as crystalline solids. Fouling has been classified into six different types by Epstein (1), based on the mechanism by which it occurs. These are as follows: 1. Precipitation Fouling : the precipitation, on the heat transfer surface, of normal or inverse solubility solutes due to supersaturation. 2. Particulate Fouling : the deposition of suspended solids that are already present in a fluid bulk onto a heat transfer surface caused by the concentration gradient and/or gravity. 3. Chemical Reaction Fouling : the formation of solid products due to chemical reactions which occur at the heat transfer surface. The metallic surface may act as a catalyst but not as a reactant. 4. Corrosion Fouling : the reaction of the heat transfer surface with the process fluid resulting in corrosion products. The corrosion may enhance formation of the fouled layer by other mechanisms. 5. Biofouling : the accumulation and possible growth of biological organisms and their byproducts on a heat exchanger surface. 6. Freezing Fouling : the solidification of a pure liquid or components of a liquid as an outcome of a heat transfer surface being cooled below the freezing point of the fluid or any of its components. Hasson (2) defines precipitation fouling as " the phenomenon of a solid layer deposition on a heat transfer surface, arising primarily from the presence of dissolved inorganic salts in the flowing solution which exhibit supersaturation under the process conditions " . Precipitation fouling is quite common in industrial cooling water systems (3,4). The most common deposits which occur in cooling water systems are calcium carbonate, calcium sulphate, calcium phosphate, magnesium silicate, and silica (5). Among the less frequent ones are iron oxide, zinc phosphate, calcium fluoride and iron carbonate. Calcium carbonate is by far the most common scale found in cooling water systems. The degree of scaling depends on calcium hardness and bicarbonate alkalinity. Calcium sulphate occurs less commonly than calcium carbonate. If carbonates and sulphates are both present, only when all the carbonate is precipitated will calcium deposit as a sulphate. Magnesium silicate scale is quite rare and is formed as a two step process : magnesium hydroxide precipitates first, which reacts with suspended and dissolved silica to form a dense, hard-to-remove scale. Since silica solubility increases with temperature and pH, this type of scale is found in the cold regions of heat exchangers. Silica will generally precipitate only if its concentration is above 150 ppm; this value constitutes a threshold limit for cooling water systems (5). In industrial heat transfer operations, significant fouling can occur because of deposition of calcium orthophosphate salts (3). The problem of phosphate in waters can be due to high concentration factors used in recirculating cooling water systems due to local water shortage; use of low quality make-up water, for example, an end product of a sewage treatment facility; and use of waters with residues from organic phosphonate corrosion and fouling inhibitors. Also, river or lake water can accumulate phosphates from agricultural run-off. Several phosphate-containing organic phosphonates (-P0(0H)2) and polyphosphates (e.g. NajPaOy, sodium pyrophosphate) are used in cooling water systems to reduce corrosion (9,12). Some of these are as follows: 1) 1-hydroxyethylidene-1-1-diphosphonic acid (HEDP) 2) Amine tris methylene phosphonic acid (AMP) 3) Ethylene diamine tetramethylene phosphonic acid (EDTP) 4) Hexamethylene diamine tetramethylene phosphonic acid (HMTD) These compounds are effective scaling inhibitors for calcium carbonate and calcium sulphate scaling. In addition to that, these phosphonates also act as corrosion inhibitors in many cooling water applications. However, these compounds can be oxidised by the different oxidisers e.g. chlorine present in the cooling water. The products of their oxidation are orthophosphate ions which can give rise to the problem of phosphate scaling in water containing significant amounts of calcium hardness. Under normal conditions, the organic phosphonates are less likely to degrade into orthophosphate ions than are the inorganic polyphosphates, e.g. Na3P04, that are used for the same purpose. The fouling of heat exchangers caused by precipitation of inverse solubility salts, such as calcium phosphates, gives rise to various deleterious effects. The efficiency of heat exchange can be reduced significantly, resulting in energy losses. Cleaning costs can be high and production rates reduce by shutdowns. There may be an increase in the pumping costs due to increased surface roughness caused by fouling and due to reduced cross-sectional area available for flow. Also, fouling may enhance corrosion of the heat transfer surface. Phosphate fouling can occur with relatively low levels of soluble phosphates (5-10 ppm as PO4 ~) in industries employing condensers and evaporators (3,4). This can result in huge financial costs due to production interruptions, labour costs for cleaning, etc. For example, the losses at a 2000 MW power station because of phosphate fouling were estimated to be 6 million dollars for 1975 (3). Fouling of heat exchangers by all causes has been estimated by Smith and Dirks to cost somewhere between 4.2 and 10 billion dollars per year in the U.S. alone (6). Thackery (7) suggested that fouling costs in the U.K. were about 0.73-1.20 billion dollars in 1979. Phosphate fouling is of considerable concern to the dairy industry also. During milk processing, calcium phosphates are deposited on plate heat exchangers, which along with denatured proteins can aggravate the problem. Following are some approaches to deal with precipitation fouling (8): 1. Development of theoretical models for quantitative prediction of fouling rates under different sets of conditions. These could identify conditions to minimise fouling. 2. Modification of the flow pattern and/or equipment geometry to minimise fouling, e.g. increasing the velocity by recirculation, bypassing the process fluid, using fluidised bed heat exchangers instead of shell and tube type, etc. 3. On-line control of fouling using antifoulants or additives that minimise deposit formation. 4. Mechanical and chemical methods to remove deposits after they have formed. The major factors which affect precipitation fouling have been well recognised by many authors and are as follows: 1. Water chemistry : the concentration of ionic species that participate in the precipitation process and pH that determines the distribution of different dissolved species significantly influences the fouling process. For a definite water chemistry, varying pH can significantly alter the fouling rate since the distribution of anionic species changes, and the solubility limit may be exceeded. 2. Surface and bulk temperatures : since the depositing salts in most cooling water flow situations have inverse solubility, an increase in surface temperature can increase the fouling rate. In the case where deposition is purely mass transfer controlled, increasing the temperature would produce little change. 3. Velocity : the effect of velocity on the fouling rate depends on the type of fouling and must be established by experiments. In general, increasing the velocity tends to decrease the fouling rate (6). 4. Nature of the heating surface : the tube material becomes most important in the case of corrosion fouling, but could affect precipitation fouling as well. Surface finish can affect the nucleation delay time, the attachment of the deposits, and the labour costs of the cleaning operation. In general, excluding surface roughness effects, if the tube material doesn't act either as a catalyst or as a reactant, it should have little effect on the rate of fouling. In a study on phosphate fouling done by Knudsen et. al. (9), it was found that different metal surfaces (stainless steel, 90/10 copper nickel and admiralty brass) had practically no effect on the fouling rate. Carbon steel in the beginning of the fouling process showed a higher rate of fouling due to formation of a corrosion film. Since precipitation fouling is a cosdy phenomenon in many types of industry e.g. food and kindred products, wood and paper products, chemical and allied industries, petroleum refineries and electricity generation, it is worthwhile to quantitatively assess the effects of various variables on the fouling rate and to develop theoretical models of the process. This has been done to a considerable extent for deposition of calcium carbonate, calcium sulphate and some other salts. However, previous studies in phosphate deposition are rather limited. In this work an attempt has been made to interpret results from prior experimental findings, to establish new experimental data, and to interpret all these results using fundamentals of precipitation equilibria and kinetics, heat transfer and fluid flow. § 2. THEORY 2.1 Definitions of Solubility and Supersaturation Solubility of a salt in a fluid is defined as the maximum concentration of the salt in that fluid that can remain for an infinite time in a dissolved state without being precipitated. Solubility is a function of temperature, pH and the nature and extent of other components present. For most salts, the solubility increases with temperature, but for some salts, e.g., calcium carbonate and phosphates, it decreases with temperature. These salts are said to have inverse solubility characteristics. The solubility product is defined as the product of the concentration of all the ions raised to their respective stoichiometric coefficients at the saturation condition. According to the "common ion effect", precipitation of a salt can take place if the solubility product is exceeded. A solution is said to be supersaturated under such conditions. Thus, one of the prerequisites of precipitation fouling is creation of a supersaturation potential. Hasson (2) lists various cases that may lead to supersaturation: 1. A dissolved salt is concentrated beyond its solubility limits at a particular temperature. 2. A solution containing a dissolved salt is either cooled below or heated above its normal solubility temperature, depending on whether the salt shows normal or inverse solubility with temperature. This can occur locally in a system. 3. Streams of different compositions are mixed together so that the ionic product of one of the potentially scale forming salts exceeds its solubility product at that temperature. 2.2 General Mechanism of Precipitation Fouling in a Heat Exchanger Considering the situation where water containing potential scale forming ions such as calcium and orthophosphate flows adjacent to a hot heat transfer surface, local supersaturation may occur because of the temperature profile that exists near the heat transfer surface. The metal surface provides an excellent site for nucleation. The following steps may be involved in the subsequent scaling process: 1. Diffusion of the calcium and phosphate ions from the bulk to the solid-liquid interface. 2. Nucleation and precipitation of calcium phosphate salts on the heat transfer surface. These processes may or may not be physically and chemically well-defined. In other words, the deposit may be crystalline or non-crystalline in nature, and of a definite or variable stoichiometric composition. 3. Removal phenomena may occur due to shear stresses applied by the fluid on the depositing layer or by thermal stresses caused by temperature variations over time. 4. Aging phenomena that can either harden or loosen the deposit may occur, for example, by the increasing temperature. Phase transformation of the crystals can also take place during the process. The deposits can also become detached due to thermal stresses caused by temperature variations over time. 2.3 Heat Transfer in a Double Pipe Heat Exchanger (DPHE) During Fouling As a vertical double pipe heat exchanger is used to monitor the fouling process in this work, it is of interest to analyze the heat transfer process for this situation where steam is on the tube side and water is flowing countercurrently in the annular space. This unit is run at a constant heat flux with time as fouling proceeds, by raising the pressure on the steam side. The following assumptions are made: 1. Steady state operation occurs in the clean condition and pseudo-steady state operation occurs in the fouling condition. 2. One-dimensional radial heat conduction takes place through the tube wall. Axial heat losses are neglected. 3. Radiation heat transfer is neglected. 4. The heat exchanger is perfectly insulated. 5. The steam is pure having no air mixed with it. The condensate drains freely from the tube side of the heat exchanger, resulting in no build-up. 6. On the water side, the entry effects on the boundary layer are neglected. 7. The fouling is purely precipitative in nature and any entrainment or particulate fouling caused by either bulk precipitation, recycle of removed particulates or foreign contaminants in water is negligible. 8. The fouled layer is of uniform thickness, bulk density, chemical composition, and thermal conductivity. The tube is uniformly covered with the foulant layer during a run. 9. The thermal conductivity of the fouled layer is not significantly influenced by changes in temperature or moisture occlusion. This is because the calculated thermal conductivity corresponds to that of a wet deposit whose temperature varies from the tube-scale interface temperature to the scale-water interface temperature. The heat flow in a clean heat exchanger can be expressed by: Q = ^oK^^u. (2-1) where: Q = the rate of heat transfer Uo= overall heat transfer coefficients based on the outside area AT|^= log mean temperature difference The log mean temperature difference is determined by: (aT) 'x-O The overall heat transfer coefficient is given by: Ko 2^« /^™ KA where the subscripts o and i denote outer and inner surface, respectively and: he = convective heat transfer coefficient = thermal conductivity of the tube material Dti . Dto = the inside and outside diameters of the tube A; , Ao = the inside and outside surface areas of the tube Ai^ = the log mean area of the heat transfer surface The inside and outside convective heat transfer coefficients depend on the geometry of the heat exchanger, the flow velocity, the specific heat, the viscosity and the thermal conductivity of the fluids and the tube material. Except for viscosity these heat transfer coefficients increase with increases in the values of the other two cited fluid properties which. in turn, are functions of temperature. In Appendix II, a calculation method is outlined to give a "theoretical" overall coefficient, based on established equations for inside and outside heat transfer coefficients. Equation (2-1) can be expressed as: Q ^ ^^im ^ Temperature Driving Force Total Heat Transfer Resistance (2-4) The denominator in equation (2-4) represents the overall thermal resistance of a clean heat exchanger. In order to calculate the overall thermal resistance of a fouled heat exchanger, the unsteady state fouling resistances are arbitrarily added to the steady state heat transfer resistances, expressed by: ± , ± , ^D,-I>.)\ , ^ , « V ! , (2-5) v., K. 2k,A,, h„A, " A, where Rf is called the fouling resistance and the subscript d denotes the dirty (or fouled) heat exchanger. Combining equations (2-3) and (2-5) yields: 1 1 ^ ^îiK (2-6) The overall clean heat transfer coefficient for a constant heat flux heat exchanger can be assumed to be independent of the fouling process and constant over time assuming that the steam side coefficient is constant over time, i.e. the steam temperature rise with time which is used to keep the heat flux constant has negligible effect on the steam side coefficient. The surface roughness effect caused by the fouled layer is not corrected for, and is therefore incorporated into the fouling resistance. In the present case of using building steam in the DPHE, there is practically little fouling on the steam side. Hence, /?. = /? ,= J _ - J _ (2-7) od oc In practice, if one starts with a clean heat exchanger at time=0, the overall heat transfer coefficient decreases with time. Hence, the fouling resistance increases with time, as expressed by: = - ^ U . , (2-8) ^od ^oc In practice, the rate of the heat flow varies slightly from time to time, and since condensation is occurring inside the tube, the inside heat transfer coefficient and hence the overall heat transfer coefficient will vary. This is not generally a problem for pure sensible heat transfer where the overall heat transfer coefficient does not vary significantly with the heat transfer rate. The assumption of constant clean overall heat transfer coefficient can cause some errors in the computation of the fouling resistance. As shown in Table II-1 (Appendix II), the 'theoretical' overall heat transfer coefficients, each one calculated for a particular velocity and rate of heat transfer, agree moderately well with the measured values. The variation of the clean heat transfer coefficient with respect to the rate of heat transfer was determined experimentally by carrying out various clean runs at fixed velocities and varying steam temperatures. A straight line was fitted to the versus Q data using linear regression methods and the slope of the straight line was substituted in the following equation to give the variation of the rate of heat transfer versus time. (U ),. , = (f/ ), „ + ( — ) A ( 2 (2-9) ^ oc 'Umemt y^oc-'UmemO V ' ^ where: AO = 0,^.., - Q,^,, (2-10) This calculation permits Rf(t) to be determined if Q and hence changes slightly during the experiment. The thermal resistance of a cylindrical deposit having and as inside and outside diameters is: ^ ^ i^fo-^^o)Ao ^ ^ (2.11) 2^/Am ^/Am where Xf is the thickness of the cylindrical deposit and the log mean area is given by: 2x7 (2-12) If Xf<^ Do, Ato, can be approximated to A„. /?, « J (2-13) For a mass deposition rate of foulant onto the tube, the fouling resistance build-up can be expressed by differentiation of equation (2-13): (2-14) de k^dd p^kjA^ p^kj. where pf is the bulk density, m^^ and w„e, are the net rate of mass deposition in kg/s and the net flux of mass deposition in kg/m^.s, respectively, and A^ is area of deposition on the heat transfer surface. The net rate of deposition in the Kem-Seaton model (11,12) is expressed as the difference between a deposition term and a removal term as given by the following equation: The removal term is expressed as a function of fluid shear stress, T, the deposit thickness, Xf, and a constant representing the mechanical strength of the adhering material, M . The net growth velocity of the layer formation can, therefore, be expressed by: dXf _ w x^T (2-16) From (2-14) and (2-16), we obtain: ^ = - ^ ^ _w_ _ (2-17) dd pykjr Mkj, pjrkj. M Integration with initial condition ^ =0, Rf=0 and boundary condition as ^ -* oo , Rf ^ Rf yields: = R; (1 - e^') (2-^^^ where 6^ is called the time constant and is given by: 9 = — (2-19) T and asymptotic resistance, Rf*, is expressed by: R; = (2-20) Substituting (2-19) and (2-20) in (2-17) yields: (2-21) de e^ e^ In the case of low velocity runs and for very adherent scale deposits, e^ becomes very large and the removal term becomes insignificant resulting in a linear behaviour given by: 2.4 Types of Fouling Curves The fouling curve under a given set of conditions is usually studied by recording the experimental fouling resistance as a function of time. The most commonly obtained shapes of the curves are linear, asymptotic and falling rate as shown in Figure 2-1. The general shape of the curve can be rationalised in terms of equation (2-15). The linear curve represents the case where the removal rate is negligible relative to the deposition process and the deposition process is constant w.r.t. time, or when the difference in the deposition and removal process is constant over time. The asymptotic curve arises when rh^^ becomes zero. This can result when the net rate decreases with time, because of either decrease in the deposition rate or increase in removal rate or both, until the deposition rate equals the removal rate. The falling rate curve arises from an intermediate condition between these two extremes, i.e. deposition rate decreases with zero or negligible removal, or a constant deposition rate and dR (2-22) FIGURE 2-1 TYPES OF FOULING CURVES 19 an increasing removal rate. 2.5 Chemistry of Calcium Phosphates:Equilibrium Considerations The calcium-phosphorus aqueous system involves some 13 ionic species (12). The distribution of these species is a function of temperature, pH and the total calcium and phosphate concentrations. It is of interest to calculate the concentrations of these species in the aqueous system. The following sets of relationships should be taken into account in solutions from which calcium phosphates precipitate: (i) Phosphate species and water equilibria: ^ H^PO I + H* (2-23) H^PO 4" ^ HPO I' + W K 2 (2-24) HPO I' ^ + POl 3- K 3 (2-25) H^O ^ W + OH- (2-26) (ii) Ion-pair equilibria: Ca^' + OH- ^ CaOH* K4 (2-27) Cà"* + H^PO 4- ^ CaH^PO : K 5 (2-28) Ca^* + HPO \- T± CaHPO l (2-29) Cû^^  + PO 4 " ^ CaPO ; K 7 (2-30) The concentration of all different species can be calculated using the following equations: (i) Equation for total calcium species: = (Ca'^) + (CaOH^ + (CaH^PO I) ^2-31) + (CaHPO 4") + (CaPO I) (ii) Equation for total orthophosphate species: T,^, = (H,PO t) + (H^PO 4 ) + (HPO l) + (PO D + (2-32) (CaH^POl) + (CaHPO I) + (CaPO I) where the brackets represent the concentration in moles/liter and Tc, and Tp^, are the total calcium and phosphate concentrations obtained by an analysis of equilibrium solutions. (iii) Equation for elecro-neutrality: (iv) Equilibrium relationships: •fo = (HPO ]-) fu (Hp) " (CaOW) 1 „ (Ca'*)(OH-) (CaH,PO I) 1 (Ca^+XH^PO;) Id {CaHPOD 1 — 1^. (CaPO ;) (Ca^'XPOl-) fofr pH = - l og [ (HVJ (2-42) where K 1 - K 7 are thermodynamic equilibrium constants. Let Kj-K, be the corresponding equilibrium constants based on activities. Activity coefficients of monovalent, divalent and trivalent ions are denoted respectively by f^ ,, fp and fj. The ions having equal charges are assumed to have the same activity coefficients. Undissociated molecules are assumed to have an activity coefficient of unity. The activity coefficients of various ionic species in solution of ionic strength I are calculated from the Debye-Huckel-Davies equation (13): -log f,=Az -r ^ -0.31] =Az] F(I) (2-43) for I < 0.1 mol/L and where fj is the activity coefficient of a species having a charge i and A is a temperature dependent collection of constants. In their interionic theory, Debye and Huckel calculated the electrical contribution to the free energy, and the mean activity coefficient of the ions on a mole fraction scale was given by the limiting law (13,14): log/= -\z,z_\Al 4T - 0.37] (2-44) V/ + 1 where: (2^^^^)^ (2-45) 2.3026 (10A:er)' At 25°C, the value of A equals 0.509. Hence, for the monovalent ion: -log/„ = 0.51 X F(I) (2-46) For the divalent ion: -log/^ = 2.04 X F(I) (2-47) For the trivalent ion: -log/^ = 4.59 X F(I) (2-48) The ionic strength of a solution is expressed by: / = ^ 5 : z ? M , (2-49) where M; is the molality of i * component and Z; is the charge. In the present case, artificially hardened water is prepared by dissolving technical grade KH2PO4 and CaCl2.2H20. Hence, the ionic strength of the solution can be expressed by: / = 4(Ca^*) + 9(P0 I') + (CaOH*) + (CaH^PO I) + (CaPO 4") + (2-50) (H^POl) + 4iHPOl') + (K*) + (CI) + (//^) + (OH) The ionic equilibrium is described by the equations (2-31) - (2-41), (2-46) - (2-48) and (2-50). The above 15 equations contain the following 28 variables: Tc. ; (K^) ; (Ca^^) ; Tpo4 ; (CI") ; (H3PO4) ; (H^POT) ; (HPC^") ; (PC^-) ; (CaH^PO:) ; (CaHPOJ) ; (CaPO^) ; (CaOH+) ; (H+) ; (OH"); K, , , K3 , K4 , K , , , K„ , F(I) In an experiment at a given temperature the four concentrations Tc,, Tpo4, K"^, CI' and the pH are known. From the literature and Kj to K, are known. Therefore the system of equations can be solved to give all the unknown species concentrations. However, the solution to the above set of equations can also be obtained by using a trial and error method to estimate the ionic strength as given below. The ionic strength of the solution depends on the concentration of various species and vice versa. An iterative procedure is therefore used. As an initial approximation: 21 = 9T,„, + 4r^+ (K^) + (CI-) + (H*) + (OH) (2-51) This value of I can be used in equations (2-46) - (2-48) to find different activity coefficients. Al l concentrations are calculated using the mass action relationships as derived below: Subtracting (2-32) from (2-31) gives: Tea - T,o4 = iCa'') + {CaOW) - (H,PO,) - ( H , P O ( 2 - 5 2 ) - (HPOl) - (POl) Eliminating (CaOH+), ( H 2 P O J ) , (HPOj-) and (PO^) from equation (2-50) and by using equations (2-33) to (2-38) gives: Tea - T,o4 = (CafV_^ -JH,PO,) (HO {Wf iWf or: (Cû^O - C,'(H,PO,) = q (2-54) where: C = i .[ 1 + - f l - + f l ^ + 5 5 ^ 1 (2-55) 1 + K,(OH-) (H*) (Wf (Wf and: q = : '^ "^ ^ (2-56) [1 + UOH-)] From equations (2-31) and (2-38) to (2-41), it follows that: Tea = iCa'') + K,(Ca'*XOH-) + K,{Ca'*)- ^^^^^^ - C ^ - ) (2-57) or: Tea = (Cû^*)[C3 + CIH^PO,)] (2-58) where: C3 = 1 + KIOH) (2-59) and: ^ ^ ^ /S:,/:,/^, ^ ^7^A2^3 (2-60) (H^) (H^)^ {Wf Solving equations (2-54) and (2-58) simultaneously gives the following quadratic equation in (H3PO4), denoted by x: + C5JC + = 0 (2-61) where: = (2-62) and: Ca (2-63) which on solution gives: (2-64) Once (H3PO4) is known, the concentrations of the various calcium and phosphate species can be calculated using equations (2-34) to (2-41). These values are then substituted in equation (2-50) to get a new value of I. This new value is then used to calculate a new set of activity coefficients. This cycle is repeated until the difference in two successive values of I is less than 0.01 % . Since the value of I no longer remains an unknown the electoneutrality equation (2-33) can be used as a check on the calculated values. The calculation of the solution concentrations thus involves solving the equations described above. The input requirements are the specifications of total calcium and total orthophosphate concentrations, pH, temperature and concentrations of the chloride and potassium ionic species present. 2.6 Distribution of Various Ionic Forms of Phosphate Species In order to estimate the distribution of various ionic forms of phosphate species, it is necessary to understand the chemistry of dissociation of the phosphoric acid. Because the mathematics of a polyprotic acid such as phosphoric acid is quite involved, a simplified approach will be considered. Phosphoric acid can dissociate in three steps as represented by equations (2-23) to (2-25), the dissociation constants for the three successive reactions being Kj, K 2 and K 3 . The activity coefficients are assumed to be unity. Depending on the pH of the solution, the polyprotic acid may exist mainly in its undissociated form or one or more of its ionic forms. The distribution as a function of pH and at a constant temperature ( 25 °C in the present case) can easily be determined using equations (2-34) to (2-36) and a total mass balance on phosphate given by equation (2-69). If Tpo4 and pH (or concentration of H"^ ) are known, there are four equations in four unknowns. Let «j denote the ratio of the concentration of the species i to the total analytical concentration and the subscripts 0,1,2 and 3 denote the fractions present as H3PO4, H 2 P O 4 , HPO|~ and PC^~ , respectively. Solving the above four equations yields: a = [1 + _ ^ + 5 5 + 5 5 5 ] ' ' (2-65) *2 - "3 a. = a. (2-66) K. a, = « 3 . (2-67) The mass balance equation can also be expressed as: (2-69) Figure 2-2 is a graph of the various fractions as a function of pH. It may be observed that at no range of pH, the phosphoric acid is present in more than two species. Hence the mathematics gets simplified in this case, mainly because the successive ionization constants are such that K i ^ K j ^ K j . The various fractions can be easily obtained from this graph by drawing a vertical line at any pH. This graph has four regions from left to right belonging to H3PO4 , H 2 P O 4 , HPOl" , PC^", respectively. The fraction of a vertical line falling in that region is the fraction of phosphoric acid existing as that species. Thus at pH = 7.0, the solution would contain roughly 55% of the total phosphate as H2PO4 and 45% as HPC^" Another useful and popular representation is a logarithmic concentration diagram or a pC-pH diagram. Such a diagram for 0.01 molar phosphoric acid (or any of its salts) is drawn in Figure 2-3. It has a similar appearance as the superimposed species distribution diagram. However, it can be seen that at very low concentrations of the phosphate ions, the lines abruptly change their slopes from +1 to +2 . For the crystallization process, the extent of supersaturation is of interest. Once the concentration of all the species is calculated the supersaturation potentials (S) and the degree of FIGURE 2-2 DISTRIBUTION OF VARIOUS PHOSPHATE SPECIES AS A FUNCTION OF pH AT 25 DEGREE CELSIUS T ""1 * I ' I ' I '—1—^—I—*—I—I—r pH FIGURE 2-3 LOGARITHMIC CONCENTRATION DIAGRAM FOR THE PHOSPHATE SPECIES ASSUMING 0.1 M PHOSPHORIC ACID supersaturation of various calcium phosphates, namely dicalcium phosphate dihydrate (DCPD), tricalcium phosphate (TCP), octacalcium phosphate (OCP), and hydroxyapatite (HAP) are calculated as follows: SocPD = iCa'*)(HPO 1-) - % (2-70) / 2 S^^, = (Ca'*)\PO l-f - 3EL (2-71) 7 2 / 3 S^c, = iCa'*)\POT)\H*) - -IfEL_ (2-72) / 2 / 3 / 1 SHAP = iCa'*y (PO IfiOH-) - (2-73) / 2 / 3 / 1 OCP is sometimes written as CagH2(P04)3.5H20. It may be noted that since activity coefficients are fractions, neglecting them can cause considerable computational errors. Similarly, if iCa^*)\PO D 3-s2 TCP K sp2 Afl (2-75) OCP K. sp3 (2-76) i8 HAP K. sp4 Aflf, (2-77) Values of the solubility products at 25°C are given below (15): K p^, = 1.05xl0-^xe^^^-'''r(« = 2.49x10-' (mol/L)^ K,p2 = 1.15x10-2^  (mol/L)* K,p3 = 1.25x10^' (mol/L)' K,p4 = 1.8x10-^ » (mol/L)^ Note that the temperature dependencies are generally not established except for K,pi. According to Brown (17), "the application of solubility principles even to a simple ternary system Ca(OH)2-H2P04-H20 is far more complex than is usually thought". There are many factors contributing to the situation. Orthophosphoric acid is both a weak acid and a polybasic acid. The presence of impurities and other metal ions, such as magnesium, formation of a large number of ion-pair complexes, and the existence of various solid forms (molecular composition) in which calcium phosphates can precipitate will complicate matters. According to NancoUas (15), the precipitation process depends on pH, supersaturation, ionic strength, temperature and the nature and extent of the solid phases that are already present. Even if the solubility product of a particular phase is exceeded, the precipitation kinetics may vary depending upon the above factors. Metastable phases may persist for very long time, with no precipitation occurring (15). The various calcium phosphate phases can undergo considerable transformations over time (18). In cooling water systems, hydroxyapatite (abbreviated as HAP), Ca((P04)0H (a dense white solid) is the form of salt that is commonly identified. But it is not this phase that is initially precipitated. The phase that precipitates initially is popularly known as amorphous calcium phosphate, abbreviated as ACP. This amorphous phase has a broad and diffuse X-Ray diffraction pattern. The lack of crystallinity may be explained due to water of crystallinity present. Some authors (15,17,19) have proposed CagH2(P04)6.5H20 (Octacalcium Phosphate), abbreviated as OCP to be the precursor phase. This amorphous phase transforms to crystalline hydroxyapatite over time. 2.7 Solid Forms of Calcium Phosphates There are five solid calcium phosphates that are relatively stable in aqueous systems. The order of decreasing solubilities of these salts are: Ca/P CaHP04.2H20 dicalcium phosphate dihydrate (DCPD or brushite) 1.00 CaHP04 anhydrous dicalcium phosphate (ADCP or monetite) 1.00 Ca8H2(P04)6.5H20 octacalcium phosphate (OCP) 1.33 Ca3(P04)2 tricalcium phosphate (TCP) 1.50 Ca5(P04)3(OH) hydroxyapatite (HAP) 1.67 The solubilities of various calcium phosphates at 25°C are plotted as a function of pH in Figure 2-4. It can be seen that HAP becomes more soluble than DCPD if the solution pH is below 4.3. Such points, known as singular points, depend on the concentration of other components. Below pH 4.8, for example, the most stable salt is CaHP04. Further reducing the pH to 4.3 leads to DCPD being more soluble than HAP. Under extremely acid conditions, solubilities of two monocalcium phosphates namely Ca(H2P04)2 and Ca(H2P04)2.2H20 may become less than that of DCPD. Following is a brief description of calcium phosphates (17): Hydroxyapatite appears most commonly in boilers using hard water containing phosphate ions or when soluble sodium phosphates are used to precipitate any residual hardness in the boiler water during a process called internal treatment. This form of calcium phosphate is the principal constituent in tooth, bone and many naturally occurring minerals, including rock phosphate. It has a well defined structure that can be found in literature (17). Octacalcium phosphate has considerable similarity (in structure) with hydroxyapatite. Hence, it has been proposed to be a precursor to hydroxyapatite. Hydroxyapatite is considered to form through three mechanisms: (a) OCP is precipitated first by hydrolysis of CaHP04.2H20 or by titration of a calcium solution with a phosphate solution at room temperature and neutral pH. After this, OCP can be hydrolysed to hydroxyapatite by boiling or by treating with an alkaline solution. If the crystals are large, hydrolysis may be difficult because of coating with hydroxyapatite and a mixed composition may be obtained. (b) HAP can also be directly precipitated without an intermediate phase (OCP), for example, in lakes and rivers. However, the kinetics of such a process would be extremely slow. (c) A combination of the above processes occurs. For example, a unit-cell-thick precipitation layer on a crystal of HAP subsequently hydrolyses to a double-unit-cell-thick layer of HAP. Such a mechanism was proposed because of structural similarities between OCP and HAP. 2.8 Kinetics of Crystallisation of Calcium Phosphates The chemistry of calcium phosphates in general and HAP in particular is of significant interest because of its relevance to many diverse fields such as water geochemistry, sedimentation mineralogy, fertilizer and food technology, osteology, urology, dental sciences and last but not least, heat exchanger fouling (23). The stoichiometry of the initial precipitate obtained by mixing solutions containing orthophosphate ions never corresponds to HAP, which is the final crystalline phase obtained in many applications. The molar Ca/P ratio of the initial amorphous precipitate is always less than the value of 1.66 for HAP (22), and usually has a value of about 1.45, which corresponds to that of TCP (12,22). This phase lacks a crystalline order and hence, its molecular structure and solubility products are not defined. There is a great deal of uncertainty as to the forms of calcium phosphate precipitating as a function of the variables of the system in which the precipitation is occurring. From Figure 2-4, the position of singular points ( at which two or more phases coexist with each other ) depends on pH and the ionic strength of the solution. The apparent solubility products for a particular phase may differ significantly if it gets coated with a more acid calcium phosphate, resulting in misleading interpretation of the solubility behavior of the phosphate phase. Thermodynamic analysis cannot predict whether, in a reasonable time, precipitation of a particular phase will occur or not. The kinetic factors become important in determining which phases are actually precipitating as a function of the variables of the system. The kinetics of calcium phosphate crystal growth has been studied by NancoUas in detail using constant composition techniques, by injecting well characterised seeds of DCPD or HAP and observing the concentration change with time. These methods are claimed by the author to be highly reproducible, in contrast with the method in which heterogenous nucleation occurs on the impurities, which results in less precise measurements. While DCPD, OCP and ACP are proposed to be precursors to the formation of thermodynamically stable HAP, their transformation process into HAP is also little understood (17,22). At the pH value below about 6.0, as can be seen from Figure 2-4, DCPD and HAP are the only thermodynamically stable phases present. The kinetics of precipitation of DCPD using the seeded growth technique was studied by NancoUas using a metastable solution (22). The following rate expression was found to fit the experimental data: f ! ^ = -kstf (2-78) de where k is the specific reaction rate constant, s is a term proportional to the number of DCPD growth sites and N is the number of moles of DCPD to be precipitated before the equilibrium is reached. The kinetic expressions for the precipitation of other phosphate phases such as OCP and HAP are not given by the author. In order to demonstrate the uncertainty in the nature of the initially precipitating phases, the author carried out four sets of experiments at pH 7.4 and 37°C. At this pH all four types of calcium phosphates can precipitate. At supersaturation corresponding to level c in Figure 2-5 a solid with Ca/P ratio of 1.45 was obtained. This solid was definitely not TCP. When the supersaturation of the metastable solution was increased from the level c to level b keeping everything else constant, the Ca/P ratio of the initial precipitate was obtained as 1.33. The fact that OCP was precipitating was confirmed by x-ray and infra-red techniques. At very high supersaturation corresponding to level a in Figure 2-5, DCPD was produced initially. At low supersaturation level d, HAP is precipitated giving the Ca/P ratio of 1.66, which indicates that HAP can precipitate directly at low supersaturation without the need for the precursor formation. The above experiments clearly indicate that the levels of supersaturation determine the nature of the initially precipitating phosphate phases. In another paper by NancoUas, the kinetics of precipitation of DCPD using the same CoHPO^ -SHjO CoHPO« Co,Hj(PO^)j-SHjO P-Ca,(PO^)j COjCPO^JjOH FIGURE 2-4 SOLUBILITIES OF VARIOUS PHOSPHATES AS A FUNCTION OF pH AT 25°C (17) 16 mMCa 1-2 08 0-4 DCPD - b 00 HAP Mitmimu 200 400 Time (min) 600 FIGURE 2-5 TOTAL CALCIUM VERSUS TIME IN GROWTH OF HAP CRYSTALS AT pH 7.4, 37°C (15) techniques was described by the following equation (21): ^ dT Ca = ks[iCa'*)(HPO ;) - 5 ] f M (2-79) where Wj and w are the weights of the seed material initially and at any instant of time. The overall rate of crystallisation was considered to be mainly surface reaction controlled. A considerable change in the stirring rate in a batch reaction vessel resulted in negligible changes in the reaction rate constant. In an earlier paper, NancoUas used the following expression to interpret the DCPD kinetics at similar experimental conditions using the same techniques (20). The following expression was fitted to the experimental data: where the symbols have their usual meanings as above. The rate was found to be markedly dependent on the rate of stirring, indicating a strongly diffusion controlled process. In none of the papers, however, were the explicit values of the rate constants specified. It is surprising that different empirical expressions fit so well the experimental data for the same kinetics. At pH values of less than 5.6, the DCPD kinetics have been studied because it is the only dm = ks[(Ca^*)^iHPOiy - (2-80) M phosphate precipitating. As the pH is increased to 7.4 (the physiological value), the crystallisation of calcium phosphates is much more complicated. There are discontinuities in the growth curves, which is very different from the growth curve of the DCPD. In a typical experiment by NancoUas, ACP was found to precipitate as confirmed by scanning electron micrography. As proposed by the author, the discontinuities were probably due to superposition of a number of reactions in parallel, each one taking place at a different rate. The concentration of seed material also plays an important role in the precipitation kinetics. In another study done by NancoUas (22), a solution supersaturated w.r.t. both DCPD and HAP, the addition of small amounts of HAP seed crystals resulted in the crystallisation of DCPD . At high concentrations of seed crystals, DCPD was not formed. The explanation offered was that at high concentration of chemicals, the calcium and phosphate concentrations fell too steeply to prevent the heterogenous nucleation of DCPD. The concentration of Mg has an important part in the precipitation kinetics of calcium phosphates (18). In the presence of Mg and at low pH, ACP and DCPD are precipitated. The initial phases are unstable and undergo further transformations. The transformation of ACP to HAP is influenced by the amount of Mg added. Magnesium acts as a crystal poison and inhibits nucleation. Even in the presence of Mg, DCPD nucleates at any pH between 5.5-8.0 . Sometimes DCPD can coexist with OCP in a limited range of pH. Metastable phases can remain unchanged for a significant time period. The evolution process is affected by the concentration of Mg. The crystal inhibitors are supposed to act through the following mechanisms (22): 1) The concentrations of one of the potentially scale forming species can decrease if the additive forms a stable chemical complex. 2) Adsorption of the additive takes place at the preferential sites of growth, leading to a reduction in the growth process and affecting the crystal structure significantly. 3) The additive may change the ionic strength of the solution, which in turn affects the solubility of the precipitating phase. In practice, the concentrations of the additives is considerably less than that which would affect the ionic strength. The organic phosphonates and polyphosphates are very effective crystal growth inhibitors even at a concentration as low as 5 x 10"* mol/L. The remarkable effect of these additives at a concentration of only 0.5 ppm in inhibiting the growth of calcite crystals was demonstrated by NancoUas (22). The reaction was almost 100 % inhibited. At an additive concentration of about 1 ppm, less than 1 % of the seed surface was covered by the additive. This suggests that only a tiny fraction of the growth units actively participate in the crystallisation process. By studying the kinetics of calcite growth in the presence of an inhibitor, it was shown by NancoUas that the rate constant is significantly affected by the presence of the inhibitors. 2.9 Related Studies in the Area of Phosphate Fouling In the area of phosphate fouling, relatively less work has been done as compared with other kinds of precipitation fouling such as calcium carbonate and calcium sulphate fouling. The previous work (3,4,9,23) is mainly concerned with the performance of various organic and polyphosphate corrosion inhibitors. As mentioned earlier, the chemistry of calcium phosphate precipitation is very complicated and unpredictable and hence a fundamental approach to develop theoretical models is lacking in all of the above cited references. Knudsen et al (9) studied the effects of various variables e.g. pH, velocity and surface temperature on the hard water fouling process both in the presence and absence of polyphosphate and phosphonate corrosion inhibitors with the purpose of establishing the overall effect of the inhibitors. The fouling process was found to be strongly dependent on these variables. The deposit obtained by the authors was described to be some form of calcium phosphate, but the chemical or physical structure was not mentioned. No attempt was made to explain the results or model the fouling process. The authors mainly concentrated on monitoring the performance of adding HEDP to the cooling water systems resulting in reduced corrosion but simultaneously giving rise to phosphate fouling. In related work (23) by Barton et al, the rate of deposition of DCPD on a heat transfer surface was studied at one fixed set of conditions . The experimental results of the rate of DCPD deposition were not reproducible, varying by a factor of about 10 for the same set of conditions. A mathematical model was proposed to predict the rates of deposition or the rate of fouling under this particular set of conditions. However, a fundamental approach was lacking and the reaction constant for the deposition process was derived by equating the calculated rate of deposition to the observed rate of deposition. The theoretical model as such is supposed to be valid over the limited set of conditions specified in the paper. The wide fluctuations in the observed rate of deposition were attributed to variations in density and porosity of the deposit, resulting in varying values of thermal conductivity. The values of thermal conductivity or density were not specified in their paper. Humphris (4) studied the problem of phosphate scaling in power station condensers. He concluded that lowering the pH of the cooling water was an effective way to prevent the phosphate fouling. He assumed that the deposits were CaHP04 , but there was no evidence to support this assumption. A nomogram was developed to predict the pH at which the calcium phosphate would precipitate, given the calcium and phosphate content of the water. By adding a required amount of an acid, the actual pH of the water was manipulated to a value just below the extracted value from the monograph. Parry et al (3) studied a similar problem of calcium phosphate scaling in power station condensers. The operating solution adopted was to lower the pH of the circulating water from 8.6 to 7.5, which resulted in a 97% reduction in scaling. Addition of one of thirty available commercial additives, namely a polycarboxylic acid, was chosen for a plant test. Addition of about 2 ppm of the additive reduced the scaling by about 96%. The advantage of using the additive rather than acid treatment using sulphuric acid is that overdosing would not damage the steel and concrete structures of the cooling water circuits. Further details of the above studies are given in the "Results and Discussion" section. § 3. EXPERIMENTAL APPARATUS 3.1 Water Flow Loop The apparatus was originally constructed by Sheikholeslami (32) and was substantially modified for the current work. The process flow diagram is shown in Figure 3-1. Water is pumped from the storage tank by means of a Cole-Parmer model K-7084-20 centrifugal pump coupled with a totally enclosed fan cooled (TEFC) 3/4 HP motor. The suction header is 51 mm reinforced PVC pipe. The rest of the piping is also made up of rigid PVC to avoid corrosion build-up in the system and has an ID of 38.1 mm. The flow of the water from the pump is controlled by a manual valve and is metered via either a Brooks model R-12M-25-4 rotameter or a magnetic flow meter (the calibration curves can be found in the Appendix I). The test fluid flows upwards through the annular space of the double pipe heat exchanger. The heat exchanger is 1.4 metres long and consists of two concentric tubes made of stainless steel. The shell is flanged at both ends to the stainless steel headers. In between the steel headers and the heat exchanger shell, there are two short pyrex glass pipe sections having the same ID as the shell, which allow for the visual observation of the fouling process. Pressure drop is measured by manometers over a length of 0.6 m. Results are discussed in Appendix II-6. The artificially hardened water is heated by steam flowing downwards through the inner CA Compressed Air CP Centrifugal Pump CC Cooling Coil(s) CW Cooling Water D Drain(s) DL Datalogger F Filter HE Heat Exchanger KT KOH Tank(s) L Level Gage MF Magnetic Flowmeter M Manometer P Pressure Gage PC PH Controller RM Rotameter R Pressure Reducer S Sampling Point SS Steam Supply ST Salt Make-up Tank TK Supply Tank TC Thermocouple FIGURE 3-1 PROCESS FLOW DIAGRAM OF THE APPARATUS tube. The water enters and leaves the heat exchanger in a direction perpendicular to the axis of the heat exchanger. The heated test fluid returns to the supply tank where it is cooled by exchanging heat with building supply water which flows inside a helical coil made-up of stainless steel. The test fluid is filtered to remove any particulates formed due to bulk precipitation process as well as any chunks of the deposits from the fouled tube as fouling progresses. The filter can be bypassed, if desired. The filter housing has a Mikro-Klean 25 or 50 microns cartridge which separates particulates from the test fluid. The particulates can be collected and removed at the end of a run. 3.2 pH Control System The pH for a particular run is controlled automatically except at the very start of a run where a manual control is used. One hundred c.c. of 8 mol/L KOH solution is diluted to 20 litres and is stored in tank K T l . Five litres of this solution flows by gravity into tank KT2 where it is again diluted to 20 litres. This 0.01 mol/L KOH solution is slowly fed (at the rate of about < 1 litre per minute) into the circulating water suction header, with vigorous air sparging, to ensure uniform mixing. This initial control in raising the pH is manual. A Cole-Parmer Series 7142 proportional type pH controller is used to make the final adjustments to attain the starting pH. This unit also controls the pH during the run. The feed tank for the automatic controller contains 0.04 mol/L KOH solution. The pH controller is calibrated before each run using standard buffer solutions. 3.3 Steam Supply System The steam supply comes from the building steam header. A solenoid valve is fitted upstream of the heat exchanger which closes if a power failure occurs, and has to be energised again in order to be opened again when the power supply resumes, in order to protect the equipment from getting damaged by the steam. A needle valve is used to control the steam pressure at the entrance to the heat exchanger and, hence, the temperature in the inside tube of the DPHE. The condensate is collected and removed via #60 Clark type steam trap and is rejected into the mat. The steam outlet line has a steam trap bypass line that can be used as steam condensate drain line while the experiment is in progress. Flow rate of the condensate can be measured manually by means of a receiver and a timer. 3.4 Storage and Supply Tanks The test fluid for a particular run is prepared in and recirculated from two 230-L PVC tanks. The suction header of the centrifugal pump is 51 mm ID and is connected to both tanks via 51 mm NPT bulk head fittings, also made up of PVC to avoid corrosion. Drain lines are connected to the pump suction header and to the bottom of both tanks. Al l piping, except the pump section header, is made of PVC and is 25.4 mm ID. One of the tanks has a valve connected at mid-height for sampling during a run. Both tanks are equipped with stainless steel cooling coils of 18 m length each and 12.7 mm OD. The coil diameter is about 30 cm each. A level gage fitted to one of the tanks permits determination of the volume of the water in the tanks. 3.5 Heat Exchanger A schematic diagram of the double pipe heat exchanger is shown in Figure 3-2. The 1.4 m long stainless steel heat exchanger shell is 38.1 mm in ID, and comprises of a central S.S. section of 90 cm length which is flanged at both ends to pyrex glass sections, having the same ID as the shell, to allow visual observation of the fouling process. The inner tubes are of two types, both 19.1 mm in OD, but differing in their ID. The two thicknesses were used to vary the temperature of the fouling surface. The thin one has an ID of 16.7 mm and the thick one has an ID of 13.7 mm. The materials of construction of the tubes are S.S. Type 304 and S.S. Type 321, respectively. Both tubes are plain and smooth finished. The dimensions and the thermal conductivities of the tubes are summarised in Table 3-1 . Table 3-1 SUMMARY OF THE DIMENSIONS OF TUBES Tube Type Material of I .D. O . D . Ao ^ 1 Construction (mm) (mm) (m )^ m^K/kW (W/mK) Thin SS 304 16.7 19.1 0.084 0.077 16.6 Thick SS 321 13.7 19.1 0.084 0.194 16.4 Glass Tube 3.81ID Inner S.S. Tube 1.91 O D X 1.37 ID S.S. Shell 3.81 ID Manometer Tapping Glass Tube 3.81 ID .Water Inlet 2.5 ID All dimensions are in cm. FIGURE 3-2 SCHEMATIC DIAGRAM OF THE HEAT EXCHANGER 52 3.6 Temperature Measurements The inlet and outlet temperature of water and steam are measured using sheathed J-type Iron-Constanton thermocouples. The thermocouples are connected to a switch, which, in turn, is connected to a DORIC Digitrend datalogger that can either record voltage output of the thermocouples or the temperature directly, at specified intervals of time such as one hour. The temperature is displayed using an Omega DP 41-TC panelmeter having an accuracy of 0.2°C . The thermocouples were calibrated using a Hewlett Packard model 2801 quartz thermometer using a standard ice-bath and a constant temperature bath. A photograph of the apparatus is shown in Figure 3-3. FIGURE 3-3 FRONT PHOTOGRAPH OF THE APPARATUS § 4. EXPERIMENTAL PROCEDURES 4.1 General Strategy The feed tanks were filled with tap water to a level corresponding to the desired total system volume. The pump was turned on, as was the steam and the cooling water. Al l the temperatures were monitored until steady state was achieved. A measured weight of technical grade CaCl2.2H20 was then added to the tank, accompanied with violent agitation caused by sparging of air at about 15 psi gage to promote rapid and complete mixing. After about 15 minutes, a concentrated 4-litre solution containing a known weight of KH2PO4 was added, and the mixing was continued. The pH of the resulting solution was always less than the desired pH for all the runs. Hence, the pH was raised by adding KOH solution first manually and then under automatic pH control with the vigorous sparging of air to avoid local high-pH pockets. The first water sample was then taken. Several water samples were subsequently taken and analyzed during each run, and make-up chemicals were added to keep the water analysis as constant as possible. In all cases where chemicals were added initially or as make-up doses, they were completely dissolved in water before addition to the supply tanks and they were added very slowly and with high turbulence by sparging air to avoid local supersaturation pockets and the resulting precipitation that might occur. Step-by-step procedural details for the runs are given in § 4.4 . 4.2 Calcium Content of the Solutions Hardness, or calcium content, is measured using the EDTA Titrimetric Method (24,25). When EDTA ( ethylenediaminetetraacetic acid or its salts ) is added to water containing both Ca+^ and Mg^"^ ions, it preferentially combines with Ca^"^. Hence, calcium can be determined by making pH so high that Mg"^ "^  is precipitated as its hydroxide. An indicator changes colour when all the Ca^"^ has reacted with EDTA. The Erichrome Blue Black indicator is prepared by grinding together in a mortar 200 mg powdered dye and 100 g solid NaCl to 40 to 50 mesh. Standard EDTA titrant of 0.01 mol/L is prepared by weighing and dissolving 3.723 g analytical reagent-grade EDTA in 1000 mL distilled water. Standard EDTA titrant is equivalent to 400.8 ng Ca/1.00 mL Because of the high pH used, the sample should be titrated immediately after addition of alkali and indicator. Hard waters with alkalinity higher than 300 mg CaCOj/L are analyzed by taking a sample less than 50 c.c. and diluting to 50 c.c. To carry out the analysis, about 2.0 c.c. of 1.0 mol/L NaOH is added to bring the pH to between 12 and 13. Then 0.1 to 0.2 g of the indicator is added and the sample is stirred thoroughly. The EDTA titrant is added slowly, preferably dropwise, with a continuous shaking of the flask until the colour changes from red through purple to bluish purple to a pure blue with no trace of bluish-purple tint. The calcium hardness can be calculated using the following expression: mg ^ AxBx\,m (4.1) L mL sample where: A = mL titrant for sample B = mg CaCOj equivalent to 1.00 mL. EDTA at the calcium indicator end point 4.3 Phosphate Content of Solutions Total orthophosphate is measured using the Vanadomolybdophosphoric acid Colorimetric Method (24,25) . In a dilute solution containing orthophosphate ions, ammonium molybdate reacts under acidic conditions to form molybdophosphoric acid. In the presence of vanadium, yellow vanadomolybdophosphoric acid is formed. The intensity of the yellow colour is proportional to the phosphate concentration. Vandate-molybdate reagent is made by mixing two solutions: one made by dissolving 25 g ammonium molybdate, (NH4)6Mo7024.4H20), in 300 mL distilled water, and the other made by dissolving 1.25 g ammonium metavandate, NH4VO3, by heating to boiling in 300 mL distilled water. This solution is cooled to room temperature and 300 mL of concentrated HCl is added. After mixing both solutions, a dilution to one liter is done by adding distilled water. The sample to be analyzed for phosphate, 35 mL or less, containing less than 1.0 mg phosphorus is put into a 50-mL volumetric flask. 10 mL of vandate-molybdate reagent is added and the sample is diluted to the mark. The reference or the blank solution is prepared by using distilled water instead of the sample. After at least 10 minutes, the absorbance is measured using a spectrophotometer at 470 nm wavelength and 1 cm path length. A calibration curve is plotted by diluting a standard phosphate solution and measuring absorbance versus known concentration of the phosphate. The calibration curve. Figure 1-2, is shown in Appendix I. 4.4 Step-by-step Procedure for a Scaling Run (1) Weigh the required amounts of KH2PO4 and CaCl2.2H20 and dissolve them in 4 L water, separately. (2) Fill the supply tanks to an appropriate level corresponding to a known volume, typically about 350 L. (3) Start the pump and manually adjust the flow rate to the desired value. (4) Open the steam valve and manually adjust the steam pressure and hence, the temperature to the desired value. (5) Open the cooling water flow, in the cooling coil, inside the supply tanks to its maximum value. (6) Turn on the temperature panelmeter to observe the various temperatures as a function of time. (7) Calibrate the pH controller and adjust the set point at the desired value. (8) After the steady state has been reached, as made evident by constant temperature readings, add the CaCl2.2H20 solution with continuous sparging with compressed air at about 10 psi gage pressure. Wait for about 15 minutes and then add the solution of KH2PO4 slowly, using a plastic funnel passing through the hole in the centre of the top lids of both the tanks. Sparge with air for at least 15 minutes. (9) Feed the KOH solution manually by opening a valve using tank #2. Keep the flow rate below about IL/min. (10) Reduce the pressure of the sparging air to about 2 psi gage After the pH of the test fluid has reached within ± 1 units of the set point pH, commission the pH controller. (11) Turn on the datalogger. Set the clock and program the period and output of recording sequence. (12) Register the time when the pH of the test fluid becomes equal to the pH set point. (13) Keep a watch on all the temperatures and the test fluid rotameter reading. Note down the temperature rise of the water, since this corresponds to a particular value of heat duty of the exchanger. Note down the manometric height differential. (14) Try to keep the heat flux constant by controlling the steam pressure, since as the fouling proceeds, heat transfer rate will decrease and the outlet temperature of the water will fall. (15) Take a water sample after every 12-24 hours and analyze it for total orthophosphate and total calcium using the procedures outlined above. If there is a significant decrease in the concentration of either calcium or phosphate ions, or both, over time, add make-up chemicals to keep the water chemistry nearly constant. The chemicals are added after draining out 20 litres of the test fluid and slowly adding the make-up into the suction line of the pump, accompanied with air sparging at 10 psi gage pressure. The amounts of chemicals to be added are calculated using the following expressions: wc = ( ^ ( [ ^ ( 0 ^ ] w c ( 0 ) . ^ § | x w ; , ( 0 ) } (4-2) ^ e p(0) ^ V p(0) ^ ^ where: wc, wp = weights of calcium and phosphate salts to be added C(0), P(0) = concentrations of calcium and phosphate at the start of a run C{9), P(ô) = concentrations of calcium and phosphate at time d 0 + \ = time (from the start ) of the addition of the salts in hours V = total recirculating volume in liters (16) Take water samples after the addition process is complete and analyze them for the calcium and phosphate contents. (17) Feed the data into the computer program and calculate the heat flux, the overall heat transfer coefficient and the fouling resistance as a function of time for each reading. (18) After there is no longer an appreciable change in the fouling resistance, or the steam pressure reaches its maximum value, terminate the run. (19) Shut off the steam to the tube by manually closing the needle valve and by cutting off the power supply to the solenoid valve. (20) Turn off the pump and the cooling water supply to the cooling coil. Drain the supply tanks through both bottom and side exits and let the system cool down. (21) Turn off the pH controller and immerse the submersible electrode in water to prevent the drying of the tip. (21) Shut off the datalogger and collect the recorded output. (22) Disconnect the steam connection at the top of the heat exchanger and the steam trap connection at the bottom. Unscrew the bolts of the flanges of the heat exchanger and unscrew the thermocouple connection. Remove the tube from the heat exchanger. Inspect the layer on the tube for its hardness, uniformity and removability. Measure the thickness of the tube as a function of length using a micrometer screw gage. Scrape off the scale with a spatula and weigh it and store it for future analysis. (24) Have the calcium and phosphate content of the scale determined by the inductive coupled plasma (ICP) analysis at a commercial laboratory. Measure the bulk density using a specific gravity bottle and examine the scale under the optical microscope for its structure. (25) Replace the tube in the exchanger. Connect the steam line at the top and the steam trap at the bottom. Tighten the flange at the top and the fittings at the bottom. Connect the thermocouples and insulate the steam line again. (26) Clean the heat exchanger and the rest of the equipment using the method described below. (27) Prepare for another run. Follow steps 1 to 26. 4.5 Cleaning of the heat exchanger After cracking off and scraping the deposit from the tube, the tube was further cleaned by using steel wool and occasionally, fine emery cloth. It was then further chemically cleaned in a dilute HCl solution in a batch process using a plastic column about 1.5 m high and 4 cm. in diameter The tube was then wiped off and put back in the heat exchanger. Water was filled to about one-third of the tank's height and about 1 L of concentrated HCl solution was added to the tanks with mild air sparging. The pump was started and the discharge valve was wide open to allow rapid cleaning of the tube and other internals of the system. Tap water was added to the tank to top it up and the drain was opened to purge out the remains of the deposit and HCl. The water was allowed to recirculate for about hour. The acidic water was then allowed to drain using the drain valves with air sparging at 10-15 kg/cm^gage to drain any particulates from the system which would otherwise settle at the bottom of the tanks. After the draining, the tanks were refilled with tap water, the pump primed and started again and the draining procedure repeated. Water samples were drawn and tested for the presence of any dissolved solids using a Cole-Parmer TDS meter, which had a minimum count of 10 mg/L. When the TDS meter showed zero, the cleaning process was judged to be complete. s 5. RESULTS AND DISCUSSION 5.1 General Overview Artificially hardened water was prepared by adding 50 grams each of Technical Grade KH2PO4 and CaClj.HjO to about 380 L tap water. The molar concentrations of the total calcium and orthophosphate were thus kept approximately equal at values of about 90 ppm as CaCOj and 92 ppm as PO4. If one of these species was fed in much lower amount than the other, its depletion over the course of time would necessitate very frequent additions to keep water quality parameters constant, besides causing many absolute errors in the measurements of concentration. These chemicals were added after making sure that a steady state had been reached, which previous experimentation in clean runs had shown could be reached in about 1-2 hours. It took about one-half to one hour after adding the chemicals to reach the set pH. The fouling run was considered to be started at this point. The nucleation on the tube was normally visible within a few hours of the start, through the pyrex glass section of the heat exchanger. However, sometimes it occurred almost instantly. The nucleation time was not considered to be of any practical significance over the fouling runs which extended from 15 to 100 hours. The on-line filter was used for all medium velocity and low velocity runs. For high velocity runs, the filter was bypassed frequently, as clogging of the filter caused a considerable drop in the flow rate. In case the rotameter was in line, the rotameter setting was also periodically monitored as the flow was maintained as constant as possible by adjusting a manual valve. In the first 12 runs, the presence of suspended solids in the water samples was observed visually after storing the samples for 2-3 days. For the subsequent runs, all the water samples were filtered using a Gelman 47 mm Filter through 0.45 /im filter papers and no bulk precipitation in the water samples could be observed after that for a few days. The water samples were withdrawn and analysed for total calcium, total phosphate and pH. As the run progressed, if a considerable decrease in concentration was observed, make-up salts were added to keep the water quality parameters constant as described in section 4.4. The pH was maintained constant by the automatic pH controller. A typical plot of the water analysis with time is shown in Figure 5-1. Water analysis data are tabulated in Appendix II-9. 5.2 Variation of the Clean Overall Heat Transfer Coefficient with the Heat Transfer Rate The variation of the clean overall heat transfer coefficient with the heat transfer rate was determined experimentally by carrying out runs at different velocities ( 0.19, 0.47, 0.93 m/s) and by varying the steam temperature. The thick tube was used for all these runs. As was expected, the clean overall heat transfer coefficient always increased with the steam temperature. Increasing the steam temperature increased the outside wall temperature and hence the outside heat transfer coefficient increased, leading to an increase in the clean overall heat transfer coefficient. For identical conditions, the clean overall heat transfer coefficient for the thin tube was greater than that for the thick tube because of higher thermal resistance for the thick tube. The slopes of the linear plots of clean overall heat transfer coefficient versus the heat transfer rate, obtained by the least squares fit method are shown in Table 5-1. Average values of the slopes were used in equation (2-9) to correct Uœ for the variations in the heat transfer rate. TABLE 5-1 SLOPES OF VERSUS Q CURVES Run V Range of Q Slope of Uoc vs Q Line (m/s) (kW) (l/m^K) LV 0.19 5.6-10.1 0.04954 1 M V 0.47 8.9-13.7 0.04439 1 HV 0.93 12.6-17.6 0.01931 LVR 0.19 6.1-10.1 0.04560 MVR 0.47 9.7-14.0 0.05200 HVR 0.93 12.9-17.2 0.00721 R = repeat Run L = low velocity M = medium velocity H = high velocity The effect of the variations in the heat transfer rate on the experimental overall heat transfer coefficient is shown in Figure 5-2 through Figure 5-7. It can be noted that at higher velocities, the variation of the overall heat transfer coefficient with the heat transfer rate is relatively less pronounced than at lower velocities. It was observed that the variation in the heat transfer rate due to the fouling process did not affect the clean overall heat transfer coefficient and, therefore, the calculated fouling resistance appreciably in any of the runs. In all subsequent analysis, the term fouling resistance refers to the "corrected" fouling resistance, calculated by using the clean overall heat transfer value which corresponded to the measured heat flow, as indicated below. 5.3 Fouling Resistance Versus Time The datalogger was programmed to record the temperatures every hour. As the fouling progressed, the outlet temperature of the water would tend to decrease. The steam temperature was therefore raised by manually opening the 12 mm needle valve on the steam line gradually, in order to maintain a constant heat flux with time. From the results of section 5.3 it is evident that fluctuations in Q would produce fluctuations in the clean overall heat transfer coefficient. Thus in evaluating the fouling resistance, it is necessary to ensure that during the dirty overall coefficient measurement at time t, the heat flow is the same as that used for the clean coefficient i.e.: '-'od ^oc This Rf value is referred to as the corrected fouling resistance since it uses clean overall heat transfer coefficient corrected to Q = Q, which may differ slightly from Qo. The uncorrected fouling resistance was also calculated for each run as follows: '-'od '-'oc Figure 5-8 compares the corrected and uncorrected fouling resistance versus time curves. The scatter which is probably due to the fluctuations in the heat transfer rate appears to be reduced with the correction, hence in all subsequent results the corrected fouling resistance is used. The "corrected" and "uncorrected" fouling resistances were calculated for each run. All three types of behaviours i.e. linear, falling rate and asymptotic were observed. Typical curves are shown in Figures 5-9 to 5-11. The initial fouling rates and the conditions of the runs are summarised in Table 5-2. The concentrations of important ionic species and the activity coefficients are summarised in Table 5-3. The initial fouling rate was determined by applying a linear regression to the initial part of the curve that appeared to be a straight line. The water chemistry was assumed to be constant during this initial part of the run. As well, for this initial period, the removal processes were expected to have the least effect on the net fouling process. In some runs, an initial negative Rf was observed, which is presumably due to roughness effects. 5.4 Reproducibility of the Results Some runs were repeated to ascertain the degree of reproducibility. In most cases, a reasonable reproducibility could be observed. In fouling experiments, a perfect reproducibility is hard to achieve because of limited direct and indirect control over the large number of the variables involved and experimental errors associated with their measurements. Where the extent of fouling is large, measurement errors are much less significant; by contrast a change in the dirty overall heat transfer coefficient of a few percent due to fouling is difficult to measure accurately. Runs 3 and 4 were done at almost identical conditions of heavy fouling yielding Rf values of about 0.35 m^K/kW after 100 hours (Appendix IV). As can be seen from the Table 5-2, the initial fouling rate of Run 4 is only 9% more than that of Run 3. Runs 14 and 19 were done at r • I 1 I I 1 1 1 1 r 5 6 7 8 9 10 Q ( k W ) I ' I ' 1 ' 1 ' \ • \ > 1—=1 8 9 10 11 12 13 14 Q ( k W ) 2.16 - I 1 1 1 1 1 1 r I • I ^ I ' I ' I ' I ' I 12 13 14 15 16 17 18 Q ( k W ) Q ( k W ) 2.11 -2.10-2.09-2.08 -2.07-2.06 Velocity = 0.93 m/s r 14 "T ' 1 ' 1 ' 15 16 17 18 Q ( k W ) 13 Time ( h ) FIGURE 5-8 RUN 3 CORRECTED A N D UNCORRECTED FOULING RESISTANCES VS. Time ( h ) • I l l I I I I » I ' ' ' ' I ' ' ' " I ' • • • I I ' ' ' ' I pH'7.5 V'0.47 m/s Re*9290 Q*10.3 kW Tca«90 mg/L Tpo4»92 mg/L T,-42 C I I 1 1 11 1 1 40 45 50 Time ( h ) • • • I pH«7.0 V«0.47 m/s Re*10602 Q ' U . S k W Tca*205.4 mg/L Tpo4* 108.1 mg/L T/65 C T T T — r 10 "T" 15 - i — I — r 20 25 Time ( h ) T A B L E 5-2 S U M M A R Y O F T H E RESULTS Run pH Vel (m/s) Re. Q.V. (kW) (ppm) Tp04 (ppm) t ; (•C) ( d R / d « ) « (m'KAcWh) Comments 2 7.5 0.47 8950 9.0 90 92 52 0.0077 3 7.5 0.47 9117 9.8 90 92 38 0.0053 Restarted after power failed at t = 1 hour 4 7.5 0.47 9195 10.9 90 92 38 0.0058 5 7.0 0.93 18,792 12.3 90 92 23 0.0074 6 7.5 0.2 3612 6.5 90 92 62 0.0027 7 7.0 0.93 19,777 13.9 90 92 44 0.0016 8 7.0 0.47 10,120 9.6 90 92 58 0.0030 9 8.0 0.2 3884 6.1 90 92 69 0.0096 Deposit very sticky and loose 10 8.0 0.93 19968 9.1 90 92 28 0.0692 Fouling very rapid, lasted about 3-4 hrs 11 7.8 0.93 19142 8.4 90 92 28 0.0183 12 7.3 0.47 10204 8.7 90 92 60 0.0060 13 6.5 0.47 9729 8.1 90 92 49 -5.6E-5 Negligible fouling observed in 24 hrs 14 6.8 0.47 9856 9.0 90 92 49 -.0009 Negligible fouling observed in 24 hrs 15 7.0 0.93 19,807 12.7 90 92 30 Fouling Rate could not be specified 16 7.5 0.93 19,946 11.0 90 92 43 0.0177 17 7.5 0.93 19,467 12.5 90 92 21 0.023 18 7.0 0.19 4028 5.7 90 92 71 0.00127 19 6.8 0.47 9,213 9.2 92 94 52 -0.0008 Negligible fouling observed in 24 hrs 20 7.5 0.47 10,183 9.7 90 92 87 0.033 Run pH Vel (m/s) (kW) (ppm) (ppm) (»C) ( R f ) « Comments 21 7.5 0.47 9290 10.3 90 92 42 0.0120 22 7.5 0.47 9509 10.4 90 92 50 0.0168 23 7.0 0.47 10,602 11.3 200 103 65 0.032 24 7.0 0.47 10,622 9.6 90 184 56 .0237 25 7.0 0.47 10,577 9.9 90 138 51 0.009 26 7.0 0.47 10,954 10.2 135 92 57 0.0198 27 7.0 0.47 10,367 9.5 225 92 55 0.030 28 7.0 0.47 10,307 9.5 90 230 54 0.028 29 7.5 0.47 10,290 10.0 100 103 63 0.0188 30 7.0 0.47 10,415 11.3 100 103 60 0.0027 31 7.0 0.47 10,748 11.2 93 96 72 0.0007 Negligible fouling observed 32 7.0 0.47 10,457 9.4 400 103 77 0.0307 Deposit very sticky and porous 33 8.0 0.47 10627 10.0 100 103 61 0.0478 * calculated ( see Appendix II ) T A B L E 5-3 S U M M A R Y O F T H E CONCENTRAT IONS A N D ACT IV I TY C O E F H C I E N T S FOR A L L RUNS Run pH Ca (mol/L) H jPOj (mol/L) HPOJ (mol/L) P O Î (mol/L) (mol/L) O H (mol/L) fa ft 2 7.5 .76 X 10 ' .21 X 10 ' .62 X 10' .22 X 10-' .34 X 10 ' .28 X 10^ .93 .73 .50 3 7.5 .76 X 10 ' .21 X 10 ' .62 X l a ' .22 X 10 ' .34 X 10 ' .28 X 10^ .93 .74 .50 4 7.5 .76 X 10 ' .21 X 10 ' .62 X 10' .22 X 10-' .34 X 10 ' .28 X 10^ .93 .74 .50 5 7.0 .80 X 10 ' .45 X 10 ' .42 X 10' .45 X 10^ .11 X 10-* .89 X 10 ' .93 .74 .51 6 7.5 .77 X 10 ' .21 X 10 ' .61 X 10' .22 X 10 ' .34 X 10 ' .28 X 10^ .93 .74 .50 7 7.0 .80 X 10 ' .45 X 10 ' .40 X 10' .42 X 10^ .11 X 10-« .13 X 10^ .93 .75 .52 8 7.0 .80 X 10 ' .45 X 10 ' .42 X 10' .45 X 10^ .11 X 10-* .89 X 10 ' .93 .74 .51 9 8.0 .71 X 10 ' .76 X 10^ .71 X 10' .80 X 10 ' .11 X 10 ' .90 X 10^ .92 .72 .48 10 8.0 .71 X 10 ' .75 X 10-» .69 X l a ' .75 X 10 ' .11 X 10 ' .13 X 10-* .92 .73 .45 u 7.8 .73 X 10 ' .12 X 10 ' .69 X 10' .48 X 10 ' .17 X 10 ' .56 X 10^ .92 .73 .49 12 7.3 .77 X 10 ' .30 X 10 ' .53 X l a ' .11 X 10 ' .54 X 10 ' .25 X 10^ .93 .75 .52 13 6.5 .84 X 10 ' .71 X 10 ' .20 X l O ' .68 X 10-' .34 X 10-* .28 X 1 0 ' .93 .75 .53 14 6.8 .81 X 10 ' .56 X 10 ' .32 X 10-' .22 X 10-« .17 X 10-* .56 X 10 ' .93 .75 .52 15 7.0 .80 X 10 ' .46 X 10 ' .41 X 10' .45 X 10^ .11 X 10-* .90 x l O ' .93 .75 .52 16 7.5 .76 X 10 ' .21 X 10 ' .62 X l O ' .22 X 10 ' .34 X 10 ' .28 X 10^ .93 .74 .50 17 7.5 .76 X 10 ' .21 X 10 ' .62 X l a ' .22 X 10 ' .34 X 10 ' .28 X 10-* .93 .74 .50 18 7.0 .80 X 10 ' .45 X 10 ' .42 X 10-' .45 X 10^ .11 X 10^ .89 X 1 0 ' .93 .74 .51 19 6.8 .82 X 10 ' .56 X 10 ' .32 X l a ' .22 X 10-» .17 X 10^ .56 X 1 0 ' .93 .75 .52 20 7.5 .78 X 10 ' .21 X 10 ' .62 X l a ' .22 X 10 ' .34 X 10 ' .28 X 10^ .93 .73 .50 23 7.0 .19 X 10-2 .45 X 10 ' .43 X 10' .52 X 10^ .11 X 10^ .91 X 10 ' .91 .69 .43 24 7.0 .75 X 10 ' .91 X 10 ' .85 X l a ' .98 X 10-» .11 X 10-* .90 X 10 ' .92 .71 .43 25 7.0 .78 X 10 ' .68 X 10 ' .63 X 10' .71 X 10-» .11 X 10-* .9 X 10 ' .92 .73 .43 26 7.0 .12 X 10-2 .43 X 10 ' .40 X l O ' .46 X 10^ .11 X 10^ .9 X 10 ' .92 .72 .47 27 7.0 .21 X 10^ .39 X 10 ' .38 X 10' .47 X 10^ .11 X 10^ .90 X 10 ' .91 .68 .41 28 7.0 .72 X 10 ' .11 X 102 .11 X 102 .13 X 10 ' .11 X 10^ .91 X 10 ' .91 .70 .45 29 7.5 .80 X 10 ' .23 X 10 ' .68 X 10' .24 X 10 ' .34 X 10 ' .28 X 10^ .92 .73 .49 30 7.0 .91 X 10 ' .50 X 10 ' .46 X l O ' .51 X 10^ .11 X 10^ .89 X 10 ' .93 .73 .50 31 7.0 .91 X 10 ' .50 X 10 ' .46 X 10-' .51 X 10^ .11 X 10-* .89 X 10 ' .93 .73 .50 32 7.0 .38 X 10-2 .38 X 10 ' .4 X 10' .55 X 10-» .11 X 10-* .93 X 10 ' .89 .62 .34 33 8.0 .81 X 10 ' .82 X 10-^  .78 X Iff' .89 X 10 ' .11 X 10 ' .9 X 10^ .92 .71 .47 identical conditions and virtually no fouling could be observed in about 24 hours. Run 28 differs from Run 8 only with respect to the concentrations of the calcium and the phosphate, but has virtually identical results, the initial fouling rate of Run 28 being about 7% less than that of Run 8. There were, however, some cases where the reproducibility was poorer, e.g. Runs 21 and 22, where the former showed a fouling rate 40% greater than the other, probably due to experimental errors. The wide range of fouling rates measured ( 0.07 - 0.003 m^K/kW.h ) with considerable accuracy indicates the quality of measurements. Rates below about 10"^  m^K/kW.h or fouling resistances below about 0.03 m^K/kW were subject to large error. All other measurements related to the water chemistry and calibration procedures were repeated until reproducible results of satisfactory accuracy were obtained. For the clean runs, the reproducibility can be seen in Table 5-1, where the slopes for the repeat runs are within 17% of the original values, except for the runs at highest velocity. 5.5 Explanation for the Type of the Fouling Behavior The traditional explanation of asymptotic fouling is the Kem-Seaton (10) model which assumes that as the fouling proceeds, the thickness of the fouled deposit increases and the removal term increases until it becomes equal to the deposition term and the net rate of fouling becomes zero. According to this model, all fouling resistance versus time curves should ultimately become asymptotic. However, an interesting observation was made in the present work that in some cases the fouling curve tended to become asymptotic as the calculated outside tube wall temperature approached 100°C. As well, Barton et. al. (23) reported that asyptotic behaviour occurred at measured wall temperatures above 100°C. This leads to a possibility that the water contained in the pores of the fouled deposit comes in contact with the wall, gets converted into steam continuously and escapes outwardly i.e. away from the tube wall. The deposition of the phosphate salt on the heat exchanger surface occurs after a set of cation(s) and anion(s) combine and nucleate at a particular spot. This subcooled boiling would either assist in removal processes already taking place or tend to reduce transport to the surface and hence reduce the net rate of fouling. It may however be noted that in the case where the heat transfer flux is not constant, the falling surface temperature or the decreasing species concentration can also give rise to a falling rate or even asymptotic behavior. Table 5-4 categorises the different fouling curves obtained and the data suggest that asymptotic behavior may occur when the calculated outside temperature exceeds about 90°C. Further experimentation with a measured wall temperature is essential to confirm this hypothesis. 5.6 Effect of the pH on the Initial Fouling Rate The pH determines the ionic distribution of the different phosphate and calcium species as described in section 2.6, and since these species participate in the fouling process, it significantly influences the fouling process. As seen in Figures 5-12 and 5-13, the effect of pH on the initial fouling rate is very pronounced at both velocities. On plotting the initial fouling rate versus the pH on a log linear scale, as shown in Figures 5-14 and 5-15, a linear relationship is obtained for both velocities which suggests that, keeping everything else constant: T A B L E 5-4 SHAPES O F T H E E X P E R I M E N T A L F O U L I N G C U R V E S 1 Run Type of Fouling Behavior Remarks | 2 Falling Rate At the end, T^ = 6 4 ' C 3 Falling Rate At the end, T „ = 82'>C 4 Asymptotic At t = 80 h. T , „ = 84''C 5 Falling Rate At the end, T^ = 63''C Linear At the end, T „ = 61 " C 7 Linear At the end, T « = 39 ' 'C 8 Falling Rate At the end, T « = 65«C 9 Falling Rate At the end, T „ = 6 9 ' C 10 Asymptotic At t = 7 h, T „ = 97 ' 'C 11 Negligible At the end T „ = 105''C 12 Negligible At the end T „ = 66°C 13 Negligible At the end T „ = 490C •4 Negligible At the end T „ = 47 ' 'C 15 Negligible At the end T „ = 36''C 16 Asymptotic At t = 20 h, T „ = 92' 'C 17 Asymptotic At t = 20 h, T^ = 78«C 18 Negligible At the end T „ = 76"'C 19 Negligible At the end T „ = 500C 20 Asyptotic At t = 20 h, T^ = 105-C 21 Falling Rate At the end T „ = 81 °C 22 Falling Rate At the end T , „ = 78°C 23 Asymptotic At t = 15 h, T^ = 9 6 ' C Asymptotic A t t = 15 h, T , „ = 8 r C 25 Falling Rate At the end, T „ = 6 4 ' C 26 Falling Rate At the end, T „ = 81 " C 27 Asymptotic At t = 15 h, T , „ = 86''C 28 Falling Rate At the end, T^ = SO^C 29 Asymptotic At t = 10 h, T „ = 83 " C 30 Negligible At the end, T „ = 73°C 31 Negligible At theend, T , „ = 71 " C 32 Asymptotic At t = 12 h, T^ = 97 " C 33 Asymptotic At t = 10 h, T^ = 93 - C 0.08 0.06 H I ' ' ' ' V«0.93 m/s Tca-90 mg/L Tpo4*92 mg/L T « S I C s^,avg. -^^  T—I—I—r Run 10 J3 « 0.04 bù 1 'a 0.02 H Run 11 Run 5 Run? 0.00 ' I ' 7.0 " T -7.2 — T 7.4 T—r 7.6 I ' ' ' • I 7.8 8.0 pH J T 0.050 H 0.045 H 0.040 H ' ' ' I ' ' V'0.47 m/s Tca*90 mg/L Tpo4*92 mg/L T..avg.-57 C I • ' ' ' I • ' I ' • • ' I ' • ' ' I ' • ' Run 33b h 0.035 H J3 0.030 H 0.025 H ^ ;S 0.020 s o : l 0.015 a Run 29 0.010 H 0.005 H 0.000 H Run 13 R u n g Run 14 Run 12 I I I I I I I I I I I I I I I 6.6 6.8 7.0 I I I I I I 11 6.4 7.2 pH ' ' I ' 7.4 -i-i-I-f 7.6 7.8 r 8.0 Logio { Initial Fouling Rate } = a.pH + b where a and b are constants. It follows from above that: Initial Fouling Rate oc (OH")' At V = 0.47 m/s a = 1.562, while at v = 0.93 m/s a = 1.557. Such a drastic effect of pH on the fouling rate has also been observed by other authors (3,4,9). In the power station condensers using water containing dissolved orthophosphate of the order of about 10 ppm where the fouling process occurs in relatively much longer time periods, a pH reduction of about 1 unit resulted in variation of the rate of fouling by about an order of magnitude. Figures 5-14 and 5-15 show increases in rate of a factor of 16 and 43 respectively as pH increases from 7 to 8. This similarly strong effect of pH on the fouling rate at low concentrations of phosphate and calcium of about 10 ppm and 100 ppm, respectively (i.e. practical situations), and at relatively higher concentrations in the present case indicates that pH plays a most critical role in the fouling phenomena. As mentioned in section 2-6, pH affects the distribution of ionic species and also the concentration of hydroxyl ions. It can be seen from Figure 2-3 that at a fixed total phosphate content increasing pH increases the concentration of both PC^~ and 0H~ ions drastically, while the concentration of HPC^" does not change much from pH 7 to 8. The strong effect of pH in this range indicates that either OH~ or P0 | " or both are playing an important role in the fouling process. Figure 5-21 shows the data on pH effect (total phosphate content fixed) expressed as fouling rate versus calculated P0|~ concentration. The linearity of plot indicates strongly that it is the PC^" ion concentration which dictates the fouling rate, and hence the deposit should contain TCP, OCP or HAP rather than DCPD. 5.7 Effect of the Velocity on the Initial Fouling Rate The velocity has an effect on both deposition and removal processes and hence its effect on the net fouling rate (or the measured fouling rate) is somewhat complicated. Runs were carried out at three different annular velocities of 0.19, 0.47 and 0.93 m/s. The average bulk and film Reynolds numbers were calculated at the average bulk and average film temperatures, respectively, and covered the range of about 3,500 to 20,000 and 7,000 to 30,000 respectively. The variation of the initial fouling rate with velocity is shown in the Figure 5-16. At pH 8.0 and 7.5, a consistent rise in the initial fouling rate with the velocity is observed. This suggests that in the range of velocity tested transport processes are important in determining initial fouling rates. However, more data are needed at higher velocities to assess the effect of the velocity on the fouling process. In a similar study done by Knudsen (9) at higher velocities in the range of 0.9 -2.44 m/s, it was found that the fouling rate over the first 100 to 200 hours decreased with increases in the velocity. Since at infinite velocity, the deposition would be prevented and at very low velocities the fouling resistance is non-zero, the velocity vs. fouling rate curve should have a maximum point at some velocity, say v ^ . Below this velocity, the fouling rate should increase with the velocity and above that, the opposing trend should be followed. It should be noted that the initial rates of the present work extend to significant Rf values, and hence are not restricted to the events involved in the initiation of fouling ( Figure 5-9). The effect of the velocity on the fouling rate can be explained as follows. At a particular scale-water interface temperature and composition of the water, the rate of fouling is the net difference between the deposition and the removal process. In a simplified approach, where surface integration is neglected, the deposition term is directly proportional to the mass transfer coefficient ko, which is almost directly proportional to the velocity. The removal processes are functions of shear stress, which is proportional to the square of the velocity. If the removal process is also assumed to be independent of the thickness of the deposit, the fouling rate as function of velocity alone can be expressed by the following simplified expression: / ( V ) =k,v - k^v^ (5-3) where k, and kj are constants that depend on the water chemistry and the deposit characteristics, respectively. On differentiating the above expression, the velocity at which the fouling is maximum can be derived as follows: V = A (5-4) 2k. It can be easily noted that the velocity at which the fouling becomes zero is exactly twice "HT I I 1 1 1 1 1 1 1 1 0.075 -0.070 -i 0.065 -j 0.060 z 0.055 -, 0.050 0.045 -^  0.040 i 0.035 -, 0.030 -i 0.025 -• pH = 7 5 • T = 32C • pH = = 53C pH = 7.0;T,,3,g. = 58C Run 33 Run 10 Run 16 ^ Run 9 Run 6 , - -Run 18 Run 2 Run 8 Run 7 I I I I I I I I I I I I I I I I I I I I I I I I I I I 0.2 0.3 0.4 0.5 0.6 Velocity (m/s) 0.7 I I I I T-r-p 0.8 T 0.9 1.0 30 ' ' • • I ' • • • I ' • ' • I ' • • • I ' ' ' • I ' ' ' • I • • ' ' I • ' • ' I ' • ' • I ' ' • ' 0 1 2 3 4 5 6 7 8' 9 10 Velocity FIGURE 5-17 QUALITATIVE PRESENTATION OF T H E VARIATION OF FOULING R A T E WITH VELOCITY v ^ . It can be concluded that if the deposition term is much greater than the removal term or the ratio of kj to kj is very large, v ^ will increase with kj/kj and the rate will increase with respect to the velocity. This is true of diffusion-controlled deposition where the deposits loosely adhere to the heat transfer surface. In the case where the ratio is not very large, i.e. the deposition term is not much greater than the removal term, the fouling rate may decrease with increasing velocity. This is shown qualitatively in Figure 5-17 by considering four arbitrary cases in which the removal term is the same, but the deposition term varies. An arguement can also be made that in the absence of removal processes, and where both mass transfer and surface integration affect the deposition process, that the initial fouling rate may either increase or decrease with velocity. 5.8 Effect of the Surface Temperature on the Initial Fouling Rate The surface temperatures were calculated on the basis of the one-dimensional steady state heat conduction equation. Since a fairly good agreement between the calculated and the experimental clean overall heat transfer coefficient was found (Appendix II), the calculated surface temperatures were regarded as a good representation of the actual surface temperatures. If the fouling phenomena was purely diffusion controlled in the ranges of variables studied, the surface temperature would have little effect on the deposition rate. The surface temperature effects the fouling process mainly in the following two ways: 1) It affects the crystallization rate constant which is often found to follow an Arrhenius type relationship. 2) It affects the properties of the fluid in contact with the heat transfer surface, namely viscosity, thermal conductivity, specific heat and density of the fluid and hence affects the convective mass transfer coefficient. The effect of the surface temperature on the initial rate of fouling can be seen from Figure 5-18, where ln(Rfi) is plotted against T ' . A near linear relationship is obtained. The slope of the straight line obtained by applying a linear regression is 4139. The activation energy of the precipitation reaction occurring on the solid-fluid interface calculated from the value of the slope is 8.2 kcal/mol, which is between the values of 10.5 kcal/mol and 4.5 kcal/mol for the precipitation of DCPD reported by NancoUas (21). From Tables 5-2 and 5-7, comparing Runs 3 and 20, it can be seen that increasing surface temperature from 38°C to 87°C increases the calculated diffusive mass transfer coefficient kp by about 11 % only, but the measured initial fouling rate increases by about 6 times, which indicates that the surface reaction rather than mass transfer effectively controls the fouling process in this range of temperature. 5.9 Effect of Concentrations of Species on the Initial Fouling Rate Since the potentially scale forming species i.e. calcium and orthophosphate ions have to diffuse from the bulk fluid and then react at the heat transfer surface, their bulk concentrations FIGURE 5-18 ARRHENIUS PLOT OF INITIAL FOULING R A T E WITH A B S O L U T E T E M P E R A T U R E have a direct bearing on the rate of fouling. This is evident in Figure 5-19 and 5-20, where the initial rate of fouling is plotted against the concentrations of the total phosphate and calcium ions at fixed pH. It can be noted that increasing the calcium concentration increased the initial fouling rate in a near linear relationship. However, increasing the total phosphate concentration appeared to increase the initial fouling rate more markedly. A log-log plot of initial fouling rate versus phosphate concentration yielded a straight line with a slope of about 2.6. Hence, it was concluded that keeping all other variables constant: It is interesting to note that Ferguson et. al (31) in their study to model the kinetics of calcium phosphate precipitation at various concentration levels of carbonate, formulated the following expression from purely empirical observations: (5-6) dCpo4 _ . 4.1 2.7 where C's denote the concentrations in millimol/L and B is in hours. The tests were done at slightly alkaline pH as in the present study. The kinetic exponent 2.7 indicated a strong sensitivity of the reaction rate to the phosphate concentration. 5-10 Characteristics of the Deposit After a run was completed, the equipment was shut down and the water drained. The deposit, if it appeared very sticky, was 'baked' for a few minutes before pulling the tube out. 1 " 1 ' 1 r - n — ' — \ — ^ — \ — ' — I — i — \ — ^ — I — . — , — I — I 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 Tea X 10^  (mol/L) 0.001 H 1 — I — I 1 , 0.001 0-002 0.003 Tpo4 ( mol/L ) FIGURE 5-20 VARIATION OF INITIAL FOULING R A T E WITH T O T A L PHOSPHATE CONCENTRATION T ' 1 ' 1 ' 1 ' 1 ' 1 0 20 40 60 80 100 (PO^-) X 10^  mol/L FIGURE 5-21 E F F E C T OF C A L C U L A T E D (POl) C O N T E N T ON INITIAL FOULING R A T E A T pH 6.5 TO 8.0 It was observed that the thickness of the scale increased with the distance from the water entrance. The deposits were, in most cases, cream colored, hard and brittle. They were porous and compressible, and generally had very weak adhesion with the surface of the tube. Deposits could be easily scraped off with a spatula, leaving little spots of residue behind on the tube. In other cases, however, the deposits were wet, sticky and tightly adherent to the wall of the tube. The deposits were sent to an external laboratory for the determination of the chemical composition. The ICP analysis gave results that are shown in Table 5-5. The samples were finally powdered, homogenised and dissolved in a solvent before analysis. It can be seen that the Ca/P molar ratio varies considerably from one sample to the other. However, the reproducibility of this method is poor, as is evident from the analysis for Run 5, where the Ca/P ratio is 11 % higher in the original analysis than in the repeat. Comparing Tables 5-5 and 5-6 suggests the following: 1) DCPD is not the only phase precipitating in any of the runs listed in Table 5-5. 2) A number of phases may be precipitating simultaneously giving rise to molar Ca/P ratios that are different from the Ca/P ratios for DCPD, OCP' or ACP, the precursors to the formation of HAP. 3) The transformation of the initially precipitated phases to the final phase (HAP) has not taken place, since the Ca/P ratio is less than 1.66. The deposit analysis is consistent with the interpretation of pH effects through Figure T A B L E 5-5 ICP ANALYS IS O F T H E DEPOSIT S A M P L E S Sample % Ca (by mass) % P (by mass) Mass ratio of Ca to P II Molar Ratio of Cato P Run 2 31.78 20.29 1.566 1.214 Run 2R 31.85 19.54 1.630 1.263 Run 5 (repeat) 32.29 19.39 1.665 1.291 Run 3 32.52 18.88 1.722 1.335 Run 4 33.98 19.03 1.786 1.384 Run 5 33.54 18.50 1.810 1.404 Run 6 34.03 18.70 1.820 1.410 Run 7 33.64 18.30 1.840 1.425 1 Run l A 25.17 13.12 1.920 1.487 Run4A 33.92 18.50 1.830 1.421 Run 5A 34.05 18.45 1.850 1.430 T A B L E 5-6 Ca/P M O L A R RATIOS FOR VARIOUS C A L C I U M PHOSPHATES Name Chemical Formula Ca/P Molar Ratio dicalcium phosphate dihydrate CaHP0^.2Hi0 1.00 anhydrous dicalcium phosphate CaHP04 1.00 octacalcium phosphate Ca,H2(P04)s.5Hj0 1.33 tricalcium phosphate Ca3(P0,)j 1.50 1 hydroxyapatite Ca5(PO,)3(H) 1.67 5-21, which indicated the key importance of PO^ concentration. 5.11 Comparison with Previous Studies Knudsen et. al (9) carried out experimental work to determine the effect of adding corrosion inhibitors to cooling water. They also studied the effects of pH, velocity, surface temperature and the material of the heating surface on the fouling rate. The conditions of their experiments were as follows: pH = 6.5 - 7.0 Velocity = 0.91 - 2.44 m/s T, = 328 - 344 K = 55 - 71 °C Tea = 650 ppm as CaCOj Tpo4 = 5 ppm as P O 4 The fouling rates obtained were of the order of about 10^ m^K/kWh. The fouling rates obtained in the present work were of the order of 10"' - 10^ ^ m^K/kWh. This can be explained by the fact that higher pH and phosphate concentrations were used in the present work. However, it is interesting to note that the trends in the variations of the fouling rates are to some extent similar. An increase in the surface temperature led to an increase in the fouling rate. The effect of the pH on the fouling rate was also remarkable, with fouling increasing drastically with a change in the pH of 0.5 units. In contrast to the present work, an increase in the velocity resulted in a decrease in the initial fouling rates presumably due to higher velocities used in their work. AU three types of fouUng behavior i.e. linear, falUng rate and asymptotic were observed in both studies. No firm conclusions on the fouling mechanisms were given regarding these trends. Barton et al (23) carried out experiments with a double pipe heat exchanger at a fixed pH of 5.6 -5.7 , so that only dicalcium phosphate dihydrate (DCPD) was precipitated. Their experimental conditions were as follows: Tea = 250 ppm as CaCOj Tpo4 = 1900 ppm as PO4 Re = 10,000 - 5,500 Q/A = 5.2 - 5.9 W/cm^ Their deposition rates varied between 0.7 and 7.0 mol/cm^.s . The fouling rates were of the order of about 0.06 m^K/kW.h, which is similar in magnitude to a few of the runs in the present work. The fouling rate became zero after the metal-scale interface temperature had exceeded 100 °C, while the solution was still supersaturated with respect to DCPD. No further increase in the fouling resistance could be observed even after the addition of reactants and increases in the surface temperature. No explanation was offered for this. This observation, which is supported by simUar observations in the present work, suggests the possibility of the steam participating in the removal processes. A rigorous mathematical model was proposed by them to predict the fouling rates. Under the very limited range of conditions ( pH= 5.6-5.7, T,= 80°C, Re = 5500-10,000) tested, it was concluded that both mass transfer and heterogenous reaction kinetics controlled the deposition process. The drawbacks of their research are as follows: 1) Their experimental measurements of the deposition rates were highly irreproducible and varied by as much as a factor of 10. 2) The density and thermal conductivity values were not specified. 3) The value of the reaction rate constant was arbitrarily obtained by adjustment so that the observed deposition rate equalled the calculated deposition rate. 4) The model does not account for the deposition kinetics of other phases of the calcium phosphates as is commonly observed in normal practice. Little comparison of this study with the present work is possible. Parry et al (3) studied the problem of phosphate scaling in power station condensers. The circulating water analysis was as follows: T c , = 400 ppm as CaCOj Tpo4 = 9 ppm as P O 4 pH = 8.4 The high pH was found to be the major factor responsible for the high rate of the fouling. On reducing the pH to about 7.00-7.5, a 97% -99% reduction in the scaling extent was observed. The rate of scaling was also found strongly dependent on the pH and the surface temperature of the condenser tubes. These observations are both supported by the present study. In a related work (4), Humphris mentioned that considerable fouling occurred in similar condensers at a pH of 7.2 and phosphate concentration of 16 ppm as P O 4 . 5.12 Calculation of the Rate of Fouling if Limited by Diffusion It is possible to make a rough estimate of the maximum possible rate of fouling assuming that it is a purely diffusion controlled process. The molar rate of the mass transfer, M^ , in such a case is given by: where C b^ and C^i are the bulk and interfacial concentrations of limiting reactant A and is the diffusive mass transfer coefficient. The value of k^ can be calculated using the following expression (30): where the fluid properties are evaluated at the average conditions between the phase surface and the bulk fluid, i.e. at the film temperature. The maximum rate will occur if Cj,^i<Cj^^ and can be neglected. Hence, it follows from equation (5-7) that: (5-7) = 0.027 v/?e-°"5c-°^*' (5-8) (5-9) where m^ is the molecular weight of the species precipitating. The maximum rate of fouling can be calculated using equation (2-22) as: The average values of density and thermal conductivity used in the above equation were obtained from this work (Tables II-2 and II-3 ) as 1470 kg/m' and 1.039 W/m.K respectively. Phosphate will be the limiting reactant irrespective of whether one of DCPD, OCP, ACP or HAP precipitates. One mole of phosphorus yields one mole of DCPD, two moles of phosphorus yield one mole of ACP, 3 moles of phosphorus yield one mole of OCP and HAP each, or 1 mole of phosphorus yields 136 g, 155 g (approx.), 164 g, 167 g of DCPD, ACP, OCP and HAP respectively. Since these numbers are quite close to each other, it can be said that one mole of phosphorus will yield approximately 156 g (average of the above numbers) of the deposit. Table 5-7 shows the calculated values of the fouling rates if limited by the diffusion. The calculated values are somewhat uncertain because of variations in scale density and thermal conductivity measurements (Appendix II). The use of average values of scale density ( 1470 kg/m )^ and thermal conductivity (1.039 W/mK) might be a contributing factor for the experimental rates to exceed the calculated values. In five runs the calculated transport limited rate exceeds the measured rate by a factor of 3 or more. This would indicate that chemical reaction limits the rate in some cases. The fact that the experimental rates are within ±50% of the transport limited calculated rates in ten runs suggests that the transport effects will be significant. Given the uncertainities involved the evidence from these calculations suggests that T A B L E 5-7 C A L C U L A T I O N O F FOUL ING RATES IF N O T L IMITED B Y S U R F A C E REACT ION Run ko ( X 10^ m/s ) Tro4 (xlO* mol/L) (dR^dt)cu. (d lVdt)^ Ratio II 2 0.314 97 0.011 0.0077 1.43 3 0.303 97 0.0107 0.0053 2.02 4 0.305 97 0.0108 0.0058 1.86 5 0.521 97 0.0188 0.0074 2.5 6 0.153 97 0.0054 0.0027 2.0 7 0.551 97 0.0195 0.0016 12.2 8 0.322 97 0.011 0.0030 3.6 9 0.156 97 0.0057 0.0096 0.59 10 0.534 97 0.0189 0.0692 0.27 11 0.529 97 0.0188 0.0183 1.03 12 0.324 97 0.0115 0.0060 1.91 13 0.313 97 0.011 -5.6E-5 Not defmed | 14 0.314 97 0.011 -.0009 Not defined 16 0.550 97 0.0195 0.0177 1.1 17 0.524 97 0.0186 0.0231 0.81 18 0.158 97 0.0056 0.001 5.6 19 0.314 97 0.011 -.0008 -20 0.350 97 0.0124 0.033 0.38 21 0.310 97 0.011 .0120 9.2 22 0.310 96 0.011 .0168 0.65 23 0.310 108 0.0123 0.032 0.38 24 0.310 193 0.022 0.0237 0.93 25 0.310 145 0.0165 0.009 1.83 ^ 0.310 97 0.011 0.0198 0.56 27 0.310 97 0.028 0.04 0.70 28 0.310 242 0.011 0.028 0.39 29 0.327 108 0.01145 0.0188 0.59 30 0.323 108 0.0118 0.0027 4.24 31 0.333 108 0.0120 0.0007 16.85 32 0.339 108 0.0116 0.0307 0.39 33 0.328 108 0.0124 0.0478 0.23 a mixed control situation exists where both transport and crystallization reaction influence the fouling rate. 5-13 Correlation for the Prediction of the Fouling Rates All the variables studied can be correlated by observing from Figures (5-13) to (5-20) that: 4139 in the following range of variables: 0.9 X 10-^  mol/L < Tea < 2.1 X 10"^  mol/L 10"^  mol/L < Tpo4 < 2.5 x 10"^  mol/L 0.2 m/s < v < 1.0 m/s 6.5 < pH < 8.0 30°C < T, < 75°C Thus by plotting Rf, versus the R.H.S. of the above equation and applying a linear regression, an empirical value of the proportionality constant, Kp, can be obtained. This value of Kp is used to predict the initial fouling rates for various Runs. From Figure 5-22, the value of the proportionality constant is obtained as 2.39 x 10^ ^ Figure 5-23 shows a plot of predicted versus experimental fouling rates. A linear regression is applied to this plot, the slope of the straight line is 1.03, indicating that this simplified correlation may be useful in predicting the initial fouling rates over the range 0.002 to 0.07 m^K/kW.h . Table 5-8 compares the values of experimental and predicted initial fouling rates. The empirical correlation is given in terms of the velocity and the measured or readily calculated water quality parameters i.e. total phosphate, calcium hardness and hydroxyl ion concentration available from the pH measurement. A more fundamental relation could perhaps be found based on Figure 5-21, however for application one would have to solve the ionic equilibrium equations for P0|~. 0.08 h 0.08 H ' I ' I ' I ' 1 ' 1 ' 1 r Predicted Fouling Rate (m^K/kW.h) FIGURE 5-23 PREDICTED VERSUS EXPERIMENTAL INITIAL FOULING R A T E T A B L E 5-8 E X P E R I M E N T A L A N D PREDICTED INITIAL F O U L I N G RATES Run Experimental Rg T c Tpo,^^ exp(-4139/TJ(OH-) ' « Predicted Rg 2 0.0077 58.18 0.0139 3 0.0053 32.79 0.0078 4 0.0058 58.18 0.0078 5 0.0074 33.04 0.00796 6 0.0027 33.10 0.0079 7 0.0016 15.84 0.0038 8 0.0030 13.91 0.003 9 0.0026 240.76 0.058 10 0.0692 226.7 0.054 11 0.0183 113.92 0.027 12 0.0060 48.3 0.0115 13 -5.6 E-5 1.83 4E-4 14 -0.0009 0.85 2E-4 16 0.0177 80.09 0.019 17 0.0231 30.05 0.007 18 0.00127 9.4 0.002 19 -0.0008 5.64 0.001 20 0.033 172.7 0.0413 21 0.0120 54.0 0.0129 22 0.0168 58.0 0.0139 23 0.032 68.0 0.016 24 0.0237 78.18 0.0187 25 0.009 30.46 0.007 26 0.0198 20.08 0.048 27 0.03 31.02 0.007 28 0.028 129.25 0.0031 29 0.0188 131.87 0.003151 30 0.0027 22.40 0.005 31 0.0007 23.10 0.0055 32 0.0307 163.85 0.039 0.0478 446.7 0.1067 s 6. CONCLUSIONS AND RECOMMENDATIONS 8.1 Conclusions The fouling of heat exchanger surfaces in the presence of calcium and orthophosphate ions was studied. The fouling process was found to be a strong function of pH, as has been observed in similar studies (3,4,9). At a fixed total phosphate content, this data could be expressed either as a linear increase in rate with the calculated PC^~ content, or a power law dependence on OH". Surface temperature also had a noticeable effect on the initial rates of fouling, which indicated that surface reaction processes were also participating in determining the overall fouling rate. The temperature dependence was consistent with the crystallisation kinetics literature. Increases in the velocity increased the initial fouling rate in a linear fashion. The concentration of phosphate and calcium species also affected fouling considerably. The phosphate effect was non-linear. In conclusion, the overall rate of fouling was found to be mass-transfer and surface reaction controlled in the region of variables tested. In the absence of accurate kinetic expressions being available and the structure and composition of the deposit not known completely, an empirical model has been presented which correlates all the variables studied in the present work. This model predicts with a reasonable degree of accuracy the initial fouling rates, which vary by a factor of about 100 in the range of variables studied. The slope and intercept of the straight line obtained by applying a linear regression to the predicted versus experimental initial fouling rate data are 1.03001+0.08153 and -0.00225, respectively. The coefficient of regression is 94.27%, indicating a significant degree of correlation between the experimental and predicted fouling rates. The clean overall heat transfer coefficient was found to be a linear function of the heat transfer rate over the range of heat transfer rates studied. The variations in the heat transfer rates due to fouling did not make much impact on the overall clean heat transfer coefficient in most cases. All three types of fouling behavior i.e. linear, falling rate and asymptotic were obtained. In the cases where the fouling became asymptotic, it was found that the calculated surface temperatures were high and approached 90-100°C. A hypothesis has been proposed linking the net fouling rate to the generation of steam at the metal-scale interface. This would have to be checked in a modified test section in which wall temperatures could be measured directly. Pressure drop results are discussed in Appendix II-6. The physical and chemical properties of the deposits as well as the adhesion to the tube varied from run to run. Measured scale densities varied from 1118.9 to 1775.4 kg/m'. A rough value of deposit thermal conductivity of 1.039 W/mK was estimated from thickness and fouling resistance measurements at the end of a number of runs. The Ca/P ratio varied over the range 1.214 to 1.487, which suggested that more than one species was precipitating. Analysis suggests that the final conversion to HAP had not taken place. 8.2 Recommendations Several modifications were carried out to improve the original set-up. However, the present apparatus and procedures should be modified in order to yield more information about the fouling process. The following modifications in the equipment are required to generate experimental data of practical importance: 1) The storage capacity of the supply tanks should be increased so that monitoring the water chemistry is not needed. At present, the monitoring of water chemistry is very time consuming and it makes specifying the concentrations of species difficult for a particular run. 2) Thermocouples should be installed to measure the wall temperature. This will help in specifying the surface temperature more realistically. The surface temperature is a very important parameter affecting almost all types of fouling and should be measured accurately. 3) A bigger pump should be installed so that an annular velocity of greater than 1 m/s is obtained. 4) The present rotameter should be replaced with one capable of measuring higher flow rates. 5) Automatic control systems to measure and control the water bulk, inlet and outlet (to the heat exchanger) temperatures should be installed to keep the heat flux constant throughout a run. The present set up needs 24-hr manual supervision. 6) A high temperature cut-off system should be installed on the steam line to protect the equipment against the following emergencies: a) Excessive leakage causing major water loss b) Accidental shut off / surging of the pump motor c) Accidental closure / failure of the cooling water Presently the equipment is protected against power failure only. Steam supply is cut off as soon as the power fails and doesn't resume automatically when the power comes back. 7) A flow control valve should be installed on the recirculating water discharge line because the online filter gets plugged resulting in a decreased flow. 8) The entire piping system should be replaced with a new one to avoid leaks that are a perpetual problem in the present equipment. On the operations side, the pH probe should be closely watched for a loss in performance and the thermocouples should be calibrated periodically on a regular basis. Further work is needed in this relatively unexplored area of phosphate fouling. Future studies should aim at widening the understanding of the various aspects of phosphate fouling; characterising the deposit, studying its structure and physical properties. Runs should be done for longer durations to generate data regarding the occurrence of linear vs. asymptotic fouling. The role of velocity in the removal process should be analyzed more thoroughly by conducting experiments at higher velocities. The possibility of the steam acting as a "removal agent" in the cases where the outside wall temperature exceeds 100 °C should be further investigated, which would lead to a better understanding of the removal process. And last but not least, analytical models should be developed in order to predict the fouling rates as a function of various variables. § 7. NOMENCLATURE Notation Description Units A temperature dependent constant Ai inside area of the tube m^  Ata. logarithmic mean surface area m^ Ao outside area of the tube m^ Af, outside area of the fouled layer m^ Afo outside area of the fouled layer m^ B, constant CrC7 constants Ci concentration of the species i mol/L C(0) concentration of the calcium at time=0 mol/L Cp specific heat kJ/kg.K specific heat of the water kJ/kg.K Cph specific heat of the steam kJ/kg.K De equivalent diameter of the annular space m Df, inside diameter of the fouled layer m Dfo outside diameter of the fouled layer m D,i inside diameter of the shell m Du inside diameter of the tube m D . outside diameter of the tube m DAB diffusivity mVs e charge on an electron F fj friction factor of the inner surface of the annular space fo friction factor for the outer surface of the annular surface fw activity coefficient of the monovalent ion fo activity coefficient of the divalent ion fT activity coefficient of the trivalent ion F volumetric flow rate m /^s g acceleration due to the gravity m/s^  G a parameter defined by the equation () hci convective inside heat transfer coefficient kW/m^.K h<.o convective outside heat transfer coefficient kW/m^.K k Boltzmann's constant k(. thermal conductivity of the water kW/m.K ko diffusive mass transfer coefficient m/s kh thermal conductivity of the steam condensate kW/m.K kf thermal conductivity of the foulant W/m.K k,„ thermal conductivity of the metal wall W/m.K K, first dissociation constant of the phosphoric acid K2 second dissociation constant of the phosphoric acid Kj third dissociation constant of the phosphoric acid K„ ionic product of the water K 4 - K 7 association constants for the reactions described by the equations (2-36) to (2-39) Kl - K 7 equilibrium constants based on activities K^p, solubility product of DCPD (mol/L)^ K,p2 solubility product of TCP (mol/L)^ K,p3 solubility product of OCP (mol/L)' K,p^ solubility product of HAP (mol/L)^ 1 distance between the manometer tappings on the heat exchanger m L length of the heat exchanger in contact with the water m m gross rate of mass deposition kg/s m^ molecular weight of species A m,. mass flow rate of the cold fluid (water) kg/s riif rate of mass removal kg/s ih„et net rate of mass deposition kg/s M a constant representing the mechanical strength of the deposit M A molar rate of mass transfer mol/L Mj molality of the component i mol/L N Avogadro's number P(0) concentration of the phosphate at time=0 mol/L Po total pressure (gravity and fluid) at the location z=0 N/m^ PL total pressure (gravity and static) at the location z = L N/m^ Q rate of heat transfer kW r radial distance from the axis of the heat exchanger m r,i inner radius of the shell m r^  inner radius of the tube m r^ outer radius of the tube m R rotameter setting RCb Reynolds number for the water bulk RCj Reynolds number at the inner surface of the annular space Re,, Reynolds number at the outer surface of the annular space RCs Reynolds number for the steam condensate RCf Reynolds number for the film Rf fouling resistance m^.K/kW.h Rfi inside fouling resistance m^.K/kW.h Rfo outside fouling resistance m^K/kW.h Rf asymptotic fouling resistance m^.K/kW.h thermal resistance of the metal wall K/kW Rf thermal resistance of the fouled layer K/kW Rft, uncorrected fouling resistance m^.K/kW.h Rfe corrected fouling resistance m^.K/kW.h Rs thermal resistance of the steam K/kW Rt , RT total thermal resistances K/kW R„ water side thermal resistance K/kW s slope of the Q vs. curve S supersaturation potential T absolute temperature K total calcium content of the solution ppm as CaCOj Tpo4 total phosphate content of the solution ppm as P O 4 Tc temperature of the cold stream (water) °C T^ b bulk temperature of the water stream °C T i^ temperature of the incoming stream of the water °C Teo temperature of the outgoing stream of the water °C Th temperature of the hot stream (steam) °C Ty temperature of the incoming stream of the steam °C Tho temperature of the outgoing stream of the steam °C Thb average bulk temperature of the steam °C T, scale water interface temperature °C T o^ metal water interface temperature °C TWP total wetted perimeter m Uo overall heat transfer coefficient based on the outside area kW/m^.K Uo,. clean overall heat transfer coefficient kW/m^.K dirty (or fouled) overall heat transfer coefficient kW/m^K V (average) velocity of the water stream m/s V recirculating volume Vr residual water left in the tanks L Vp internal volume of the piping cm^ internal volume of the specific gravity bottle cm^ V,vg average volume drained out by the pump L w mass deposited per unit area per unit time kg/m .^s wc weight of calcium salt make-up g wp weight of phosphate salt make-up g W, weight of the empty specific gravity bottle g W2 weight of the empty specific gravity bottle + saturated water g W 3 weight of the empty specific gravity bottle -I- sample g W 4 weight of the empty specific gravity bottle + sample g -I- saturated water W 5 weight of the empty specific gravity bottle -I- pure water g Wp rate of the condensation of the steam Xf deposit thickness mm Xfjvg average thickness of the deposit mm z axial distance from the bottom of the heat exchanger m Zj charge on the species i (i) concentration of the species i mol/L Dimensionless Groups Pr Prandtl number, Cp/x/k Nu Nusselt number, hD/k Re Reynolds number, p\D/n Sc Schmidt number, fx/pD^^ Greek Letters Description ttj part of total phosphate present as species i /Sj degree of supersaturation r rate of weight condensation of the steam per unit kg/m.s inside perimeter of the tube e dielectric constant A difference between two values of a variable Ah height differential of the manometer cm AT,,„ log mean temperature difference °C K ratio of the outer diameter of the tube and the inner diameter of the shell T constant=3.1415.... 6 time h 6^ time constant h 6^ delay time h X constant X, heat of vaporisation ( or condensation ) of the steam kJ/kg fi viscosity N.s/m^ viscosity of the water N.s/m^ jUh viscosity of the steam condensate N.s/m^ Pe density of the water kg/m^ Ph density of the steam condensate kg/m^ Pf density of fouled deposit kg/m^ Pm density of the manometric fluid kg.m^ T„ liquid shear stress at any location (r,z) N/m^ Subscripts Description d dirty or fouled avg average b bulk i inside in incoming 0 outside out outgoing m metal max maximum Abbreviations Description ACP amorphous calcium phosphate ADCP anhydrous dicalcium phosphate C centigrade DCPD dicalcium phosphate dihydrate DPHE double pipe heat exchanger g, mg gram, milligram h hour HP horsepower HAP hydroxyapatite HTRI Heat Transfer Research Incorporated ICP Inductively Coupled Plasma ID inside diameter kW kilowatt K Kelvin L, mL liter, milliliter m, cm, mm meter, centimeter, millimeter, nanometer M molar min minute OCP octacalcium phosphate OD outside diameter PVC polyvinyl chloride s second SS stainless steel TCP tricalcium phosphate TDS total dissolved solids TEMA Tubular Exchangers Manufacturers Association TWP total wetted perimeter § 8. REFERENCES 1) Epstein, N.,"Fouling in Heat Exchangers". Proceedings of sixth International Heat Transfer Conference. Vol.6, p. 235-253, August 1978. 2) Hasson, D., "Precipitation Fouling". Fouling of Heat Transfer Equipment. Somerscales, E. and Knudsen, J.G. (Editors), Hemisphere Publishing Corporation, p. 527-568, 1981. 3) Parry, D.J., Hawthorn, D. and Rantell, A.,"Fouling of Power Station Condensers within the Midlands Region of the C.E.G.B.". Fouling of Heat Transfer Equipment. Somerscales, E. and Knudsen, J.G., (Editors), Hemisphere Publishing Corporation, p. 569-585, 1981 4) Humphris, T.H., "The Control of Phosphate Scaling in the Power Station Condensers", Central Engineering Research Laboratories, Fouling of Heat Transfer Equipment. Somerscales, E. and Knudsen, J.G. (Editors), Hemisphere Publishing Corporation, 1981 5) Strauss, S.D. and Puckorius, P.R.,"Cooling Water Treatment for control of Scaling, Fouling and Corrosion", Power. June 1984, p. S5-S24. 6) Chenoweth, J.M.,"Final Report of the HTRI/TEMA Joint Committee to review the Fouling Section of the TEMA Standards". Heat Transfer Engineering. Vol. 11, No. 1, p. 73-107, 1990 7) Thackery, P.A.,"The Cost of Fouling in Heat Exchangers Plant". Proceedings of the Conference "Fouling- Science or Art".Surrey. Guildford, England, March 27-28, 1979 8) Knudsen, J.G.,"Coping with Cooling Water Fouling in the Heat Exchangers", Heat Transfer Philadelphia 1989, A.I.Ch.E. Symposium Series 269. Volume 85, 1989 9) Knudsen, J.G., et. al.,"Fouling Characteristics of Cooling Tower Water Containing Phosphate Corrosion Inhibitors", Proceedings of the 1987 ASME-JSME Thermal Engineering Joint Conference. Honolulu, p. 109-115, 1987. 10) Kern, D.G. and Seaton, R.E.,"A theoretical Analysis of Thermal Surface Fouling". British Chemical Engineering. Vol. 4, p.258. May 1959 11) Kern, D.G. and Seaton R.E.," Surface Fouling How to Calculate Limits", Chemical Engineering Progress. Vol. 55, p. 71, June 1959 12) Varsanik, R.G.," The Nature and Control of Calcium Orthophosphate Deposition in Cooling Water Systems", Materials Performance. December 1975, p. 16-22. 13) Davies, C.W.," Ion Association ", Butterworths, London, 1962 14) NancoUas, G.H. ," Interactions in Electrolyte Solutions ", Elsevier, Amsterdam, 1966 15) NancoUas, G.H., Amzad, Z. and Koutsoukos, P.,"Calcium Phosphates - Speciation, Solubility and Kinetic Considerations", Chemical Modelling in Aqueous Systems, E.A. Jenne ( Ed.) ACS Symposium Series, Vol. 93, 1979. 16) Butler, J .N.," Solubility and pH Calculations ", Addison-Wesley PubUshing Company, Inc., Second Edition, 1973 17) Brown, W.E., Solubilities of Phosphates and Other Sparingly Compounds", Environmental Phosphorous Handbook. Griffith E.J., Beeton A. , Spencer J .M. , MitcheU D.T. (Eds. ), John Wiley, N .Y. 18) Abbona, F.," Crystallization of Calcium and Magnesium Phosphates from Solutions of Low Concentrations ", Journal of Crystal Growth ", Vol. 104, 1990, p. 661-671 19) Cowan, J.C. and Weintritt, D.J.," Water Formed Scale Deposits ", Gulf Publications, Texas, 1976 20) NancoUas, G.H. ," Kinetics of Crystal Growth from Solutions", Journal of Crystal Growth. Volume 3, No. 4, 1968, p. 335-339 21) Marshall, R.W. and NancoUas, G.H. ," The Kinetics of Crystal Growth of Dicalcium Phosphate Dihydrate", The Journal of Physical Chemistry. Volume 73, No. 11, November 1969 22) NancoUas, G.H. ," The Growth of Crystals in Solution", Advances in Colloid and Interface Science. Volume 10, 1979, p. 215-252 23) Barton, K.P. , Chapman, T.W., and Lund, D.," Rate of Precipitation of Calcium Phosphate on Heated Surfaces", Biotechnology Progress. Volume 1, No . l , March 1985, p.39-45 24) "Standard Methods of Chemical Analysis". Welcher, F.J. (Ed.), D. Van Nostrand Corporation Inc., Princeton, N.J. , Sixth Edition. Vol. No. 2, 1963, p. 2399 25) " Standard Methods for the Examination of Water and the Wastewater". Twelfth Edition, APHA, AWWA, WPCF, 1965. 26) Perry, R.H. and Green, D.," Perry's Chemical Engineer's Handbook ", Sixth Edition, Mc Graw Hill Publications, 1984 27) Wiegand, J.H.," Annular Heat Transfer Coefficient for the Turbulent Flow ", Transactions A.I.Ch.E.. Vol. 41, 1945, p. 147 28) Bird, R.B., Stewart, W.E. and Lightfoot, W.E.," Transport Phenomena ", John Wiley and Sons, New York, 1960 29) Incropera, F.P. and Dewitt, P.P. . "Fundamentals of Heat and Mass Transfer". Second Edition, John Wiley and Sons, New York, 1985 30) Treybal, R.E., " Mass Transfer Operations", Third Edition, Mc Graw Hill Publications, 1980 31) Ferguson, J.F., Jenkins, D. and Eastman, J.," Calcium Phosphate Precipitation at Slightly Alkaline pH", WPCF Journal, Vol.45, No.4, April 1973 32) Sheikholeslami, R., " Tube Material and Augmented Surface Effects in Heat Exchanger Scaling", M.A.Sc. Thesis, 1984 APPENDIX I. CALIBRATION CURVES 1.1 Calibration of the Rotameter The calibration of the rotameter was done by measuring the time taken by the water to fill a 60-litre tank. The volumetric flow rates were obtained for the rotameter readings corresponding to 10, 20, 30, 50, 80 and 100 . The calibration procedure was repeated several times to ensure accuracy. A plot was then drawn and a linear regression was applied to fit a straight line through the points. The calibration curve is shown in Figure I- l . 1.2 Calibration Curve for the Phosphate Determination A standard phosphate solution was prepared by dissolving 219.5 mg reagent grade Potassium Dihydrogen Phosphate in one litre distilled water. The solution is diluted to various extents. 35 mL of a solution is mixed with 10 mL of vandate-molybdate reagent and 5 mL of distilled water to give a final volume of 50 mL. After about 15 minutes, the absorbance was measured using a spectrophotometer at 470 nm wavelength and 1-cm optical path length. A linear regression was applied to yield a straight line shown in Figure 1-2. 0 10 20 30 40 50 60 70 80 90 100 110 Rotameter Reading FIGURE I-1 ROTAMETER CALIBRATION CURVE APPENDIX n . SAMPLE CALCULATIONS II. 1 Calculation of the Rate of the Heat Transfer The rate of heat transfer at any instant of time was calculated using: Q = m^C^iATX (II-l) where: m„ = mass flow rate Cpc = specific heat at the average bulk temperature of the water ATc = Rise in the temperature of the water and m is related to the volumetric flow rate as follows: m=pXF (n-2) The datalogger was programmed to give the output temperatures for J-type thermocouples. The volumetric flow rate was determined by noting the rotameter reading and by using the calibration curve. The density and specific heat of water at the bulk temperature were determined using the following relationships by applying a polynomial regression to the available literature data (29): p ( ^  ) = -0.3269 r^, + 1005.4 (II-3) (j^) = -0.000108 + 4.1818 (11-4) where T t^ is in degrees centigrade. II.2 Determination of the Dirty Overall Heat Transfer Coefficient The dirty overall heat transfer coefficient, based on the outside area of the inner tube of the DPHE was calculated as follows: U. 9^ (II-5) where is given by: = TTD L (II-6) o o and the log mean temperature was calculated using: AT. = CO a In-where T stands for temperature and subscripts c, h, i and o denote the cold, the hot, the incoming and the outgoing flows. The average of the inlet and outlet steam temperatures was used. 11.3 Determination of the Fouling Resistance The fouling resistance as a function of time was calculated using: = - i - - - i - (II-8) od oc where Uod is the dirty overall heat transfer coefficient at any instant of time. 11.4 Calculation of the Theoretical Overall Heat Transfer Coefficient The overall heat transfer coefficient is determined by calculating individually the inside and outside heat transfer coefficients and the thermal resistance of the metal wall, using equation (2-3). A computer program, Figure III-l, is used to facilitate the calculations. The following relationship is employed in calculating the outside heat transfer coefficient (27): (Nu\ = = (0.023)/?e "/P/-'"*(5£) (II-9) The physical properties of water are evaluated using the following expressions where T is in Kelvin (29): K ( 4 ) — r - . 1.22961 - 0.0018r^ + 3.483 x l O ' T ^ , u ( x l O ^ . M ) = 65816.344 - 418.2856T + 0.672557^, (H-ll) ms k ( x l O " ^ . ^ ) = -278.316 + 4.46287 - 0.004997^, (11-12) mK Pr^ = 460.768 - 2.8917; + 0.004587^, (H-13) The above equations are taken from pure water data (25), and do not reflect the presence of the added chemicals. The thermal resistance of the metal wall is determined using: lirkL (11-14) The inside heat transfer coefficient for the condensing steam is evaluated using the following equation for Re < 2100 (26): . K ^(CphH. _ 5.35 The Reynolds number is defined as: and: 4r (11-16) P hg = ( (11-17) W r = _ L ai-18) The physical properties are that of the condensate at the average steam temperature and are evaluated using the following expressions (25): P. (kg/m^) ^ (11-19) 0.68523 + 0.000958r'ft u. ( x l O ' . ^ ) = 4032.3786 - 17.261267, + 0.01937^ (11-20) ms C^^(xlO* - ^ ) = 6.17287 - 0.011497^ + 1.67 X 10"'71 01-21) A:^  ( xlO'* — ) = -446.29124 + 5.572897, - 0.006847'^ (11-22) mK (^) = 2306.62 + 2.31058r^  - 0.006557 ^ (11-23) The rate of mass condensation is evaluated using: (11-24) The overall heat transfer coefficient is evaluated using equation (2-3). A comparison of the experimental and theoretical clean overall heat transfer coefficients is shown in Table II-1. II.5 Determination of the Wall Temperature and the Scale Water interface Temperature The average wall temperatures and the scale-water interface temperatures throughout a run were evaluated using the steady-state, one-dimensional (radial) process of heat transfer for concentric cylinders with negligible contact thermal resistance. The following equation expresses the heat transfer: Q = K 1 c,avg (11-25) It follows from the above equation: Table II-1 Comparison of the Theoretical and Experimental Clean Overall Heat Transfer Coefficients Run V Te Q (m/s) (°C) C'C) (kW) (kW/m^K) (kW/m^K) 2 0.47 21.1 102.3 10.55 1.550 1.191 II ^ 0.47 19.6 107.0 9.38 1.278 1.208 4 0.47 22.4 119.6 12.72 1.559 1.189 5 0.93 23.2 105.3 13.28 1.927 1.376 6 0.19 20.0 102.6 7.00 1.009 0.883 0.93 26.3 124.6 13.27 1.606 1.417 8 0.47 25.9 113.1 9.36 1.278 1.242 9 0.19 21.8 102.1 6.24 0.926 0.900 10 0.93 29.0 111.6 13.90 2.008 1.390 II 11 0.93 25.0 114.6 14.60 1.939 1.377 12 0.47 25.7 107.7 8.36 1.214 1.251 13 0.47 23.1 101.9 9.04 1.366 1.222 14 0.47 24.7 103.1 9.03 1.372 1.300 15 0.93 25.1 104.6 11.62 1.741 1.409 16 0.93 26.7 124.5 13.93 1.695 1.407 17 0.93 27.9 131.5 17.90 2.058 1.361 18 0.19 24.8 102.9 5.68 0.867 0.920 19 0.47 20.9 106.0 9.04 1.265 1.219 29 0.47 27.2 101.1 10.03 1.616 1.590 30 0.47 24.3 103.6 11.21 1.682 1.557 31 0.47 25.4 111.1 11.20 1.558 1.577 32 0.47 27.1 113.9 10.19 1.400 1.610 33 0.47 29.9 111.1 12.52 1.835 1.572 T = T - Q'R, (11-26) •wo,avg h,avg ii- ^ ' where: ^"^ê^^ 1 (11-27) ' lirkL h. A.. and the temperature of the scale/water interface is then: T = T - Q'R^ (11-28) where: R^ = R^^h (11-29) The temperature gradients and the heat transfer resistances are shown qualitatively in Figure II-1. The calculated scaling temperature is subject to some uncertainty, as it relies on equation (II-5) for hi. II.6 Determination of Reynolds and Schmidt Numbers The equivalent diameter of the DPHE was calculated using the following relationship: FIGURE II-l TEMPERATURE GRADIENTS AND HEAT TRANSFER RESISTANCES Reynolds numbers, based on the bulk and film temperature were calculated using: (11-31) (11-32) where p, V and /x are the density, the velocity and the viscosity of the water and the film temperature is evaluated as: As mentioned in section 5-2, the overall heat transfer coefficient at a particular velocity is a function of the heat duty, which is, in turn, a function of the bulk steam temperature. Since it was not possible to maintain the exact same heat duty throughout the length of a run, the clean overall heat transfer coefficient was also a function of the heat duty. Therefore, in equation (2-8), the clean overall heat transfer coefficient was corrected for the variations in the heat duty as follows: These corrected values of the clean overall heat transfer coefficients were substituted in equation (2-8) to give the corrected values of the fouling resistances. (11-33) 2 UJO = UJO) ^S'[Q(t) - 0(0)] (11-34) II.7 Determination of the Chemical Composition. Density and the Thermal Conductivity of the Deposit The chemical composition of the deposit, mainly the Ca/P ratio, was determined by Acme Analytical Laboratories Limited, Vancouver using the Inductively Coupled Plasma Technique. In this technique, 200 g of the sample to be analysed is fused with 1.2 g of LiBOj and is dissolved in 100 mL of 5 % H N O 3 . The results of the analysis are presented in the Table 5-5. The Ca/P molar ratio of the deposit was variable, indicating that different phosphate phases may be present in the deposit samples. The density of the deposit was determined using a specific gravity bottle. The relationship used to calculate the density is derived below: Let: p = Density of pure water at the temperature of the experiment p, = Density of water saturated with the calcium phosphate Pf = Density of the (foulant) sample V,g = Internal Volume of the specific gravity bottle at the temperature of the experiment Wi = Weight of the empty bottle W2 = Weight of the empty bottle -I- saturated water W 3 = Weight of the empty bottle + sample W 4 = Weight of the empty bottle + saturated water + sample W 5 = Weight of empty bottle + pure water The volume of the specific gravity bottle is given as: Kg ^5 " ^ 1 (11-35) The density of the sample is given as Weight of the sample _ ^ 4 ~ ^2 (11-36) ^•'^ Volume of the sample V and the volume of the sample is calculated as: V = V - " (11-37) where the density of the saturated water is given as: W, - W. Pl = Ï7 (11-38) Combining equations (11-34) to (11-37) yields: W, - W, P^ w - w W,- W, (11-39) (— -)'[ 1 - — -] Table II-2 lists the calculated values of density of the scale for some of the runs. The average value used for calculation purposes was 1470 kg/m^. The thermal conductivity of the deposit is determined using equation, assuming steady state one-dimensional heat conduction through concentric cylinders. As an approximation, the thermal conductivity is calculated as: where Xf„g is the thickness of the scale averaged over the length of the tube and Rf is the (corrected) fouling resistance at the end of a run. Table II-3 shows the calculated values of thermal conductivity of the scale for 8 runs. The average value used for calculation purposes was I. 039 W/mK, which agrees well with the value reported in literature (3). II. 8 Calculations for the Total System Volume The total system volume was calculated by adding up the volume of the twin tanks and the internal volume of the piping. Since the volumes of the tanks were not known, these were determined by measuring the time taken to empty the tanks at a fixed flow rate of the water, which was drained instead of being recirculated back to the tanks. This was repeated thrice and an average was taken. However, the pump lost suction before all the water was drained, and the residual volume of the water was therefore determined by measuring the height and the diameter of the residual water columns left in the both tanks. The total volume was calculated by adding up all three volumes as expressed by: TABLE II-2 CALCULATION FOR THE SCALE DENSITY 1 Run w, W 3 W 4 W 5 Pf (g) (g) (g) (g) (g) (kg/m )^ 21.3215 46.8432 23.1210 48.0103 46.8432 1795.4 8 18.3385 43.6350 21.4745 45.5030 43.6350 1469.5 1 9 20.7200 45.7355 21.5220 46.2490 45.7355 1775.4 1 12 20.7035 45.9555 22.2195 46.7570 45.9555 1118.9 L 19.5940 44.8347 21.8080 45.8040 44.8347 1189.6 TABLE II-3 CALCULATED THERMAL CONDUCTIVITIES OF THE SCALE Run (mm) (m^K/kW) (W/mK) 3 0.375 0.3592 1.04 1 4 0.450 0.3227 1.39 20 0.25 0.180 1.39 23 0.30 0.246 1.22 24 0.2 0.219 0.91 25 0.1 0.109 0.92 26 0.2 0.2027 0.99 27 0.15 0.3358 0.45 V = V + V + V t avg r p 11.9 Water Analysis Calculations (11-41) Table II-5 summarises water chemistry data for various Runs. The total orthophosphate was determined by either using the calibration curve, Figure 1-2 or the following calibration equation: 7>^ ^ {Tl as PO,) = 242.19 x Absorbance,,,^ (11-42) Li The total calcium was determined by measuring the volume of EDTA consumed and using the following expression: (^ as Ca) = ^ ^"^'^ (11-43) Li O as CaCO,) = ^ ^ ^"^"^ (11-44) where: A = Volume of EDTA consumed in mL B = mg of CaCOj equivalent to 1 mL EDTA C = Volume of the sample in mL The total potassium was calculated by adding up the potassium added as KH2PO4 and as KOH, at the start of a run. Total Potassium = + + (11-45) where: M , = Moles of potassium added as K H 2 P O 4 M2 = Moles of potassium added (as KOH) from tank 1 M3 = Moles of potassium added (as KOH) from tank 1 Table II-4 summarises the water analysis data for some of the runs. 11.10 Calculations for the Pressure Drop and the Shear Stresses The pressure drop across a 0.6 m long section of the DPHE is measured by a U-tube manometer containing a Meriam Blue Fluid that is immiscible in water and has a specific gravity For an incompressible fluid flowing in an annular space, the following expression can be written (24): of 1.75. 0 = 1 d(rrj (11-46) r dr Integrating the above expression yields: TABLE n-4 WATER ANALYSIS DATA Run Sample Time Tpo4 pH Comments if (h) (ppm CaCOj) (ppm PO<) 2 1 24 70 7.5 No salt make-up | 3 1 1 98 91 7.5 Make-up after 27 and 49 hrs 2 21 78 79 7.5 3 27.5 94 88 7.5 4 46 72 81 7.5 5 49 88 78 7.5 6 93 60 67 7.5 4 1 2.5 88 83 7.5 Make-up after 25 and 50 hrs 2 23 58 68 7.5 3 30 78 81 7.5 4 48 60 72 7.5 5 1 0.5 90 93 7.5 Make-up after 24 hrs 2 23 68 77 7.5 3 30 80 86 7.5 4 47 66 77 7.5 6 1 22 54 68 7.5 No salt make-up 2R - 1 0 98 92 7.5 Make-up after 18 hrs 2 15.5 75 64 7.5 9 1 0.5 68 72 7.4 No salt make-up 2 52 26 52 7.4 10 1 3 32 52 7.0 No salt make-up 11 1 0 26 52 6.5 No salt make-up 2 16 10 28 37 12 1 0.5 66 75 7.00 No salt make-up 2 23 26 47 7.15 3 55 20 49 7.25 T A B L E W A T E R ANALYSIS D A T A (CONTD. ) Run Sample Time T c Tpo4 pH Comments # (h) (ppm CaCOj) (ppm P O J 13 1 0 90 87 6.4 No fouling observed 14 1 0 93 No salt make-up 1 2 26 88 15 1 0.5 94 95 7.2 No salt make-up 1 2 20 82 88 7.2 18 1 0.5 92 88 7.0 No salt make-up 2 7.5 84 85 7.0 3 70 50 65 7.0 23 1 0.5 190 101 7.0 2 4.5 182 - 7.0 3 6.5 178 92 7.0 4 20 150 69 7.0 24 1 0.5 92 165 7.0 No salt make-up 2 7.0 82 158 7.0 3 21 64 152 7.0 25 1 0.5 92 137 7.0 No salt make-up 2 19 72 127 7.0 26 1 0.5 136 95 7.0 No salt make-up 2 16 114 81 7.0 27 1 0.5 210 91 7.0 No salt make-up 2 6 204 86 7.0 3 26 190 74 7.0 28 1 0.5 88 7.0 No salt make-up 2 24 66 7.0 T = (— + _L (11-47) In order to determine constant B, , it can be said that the maximum in the velocity profile lies at some plane r=XR, where: B, = -(P, - (\Rf 2L (11-48) hence, iPo-PÙR rr 2L R r " R r (11-49) where: X = 1 -2 1 n l (11-50) The maximum of r„ is at r = /cR. Hence, I - r 1 -'rrz\r..R - '^oVX- j-J (11-51) 2K In — K II. 10.1 Calculation of Theoretical AP A force balance on the fluid flowing in the heat exchanger gives: ( P , - P,)ir(rl - ri) = 2 T L ( I p v ^ ) [ r , , / „ - r^f,] (11-52) where: p = (11-53) and fj and are the friction factors for the outside surface of the tube and the inside surface of the shell, both assumed to be hydraulically smooth. Their values are given by Blausius expression (24) : fi - ^ (11-54) and: cb 11.10.2 Calculation of Experimental AP An expression for experimental AP can easily be derived in terms of the height differential of the manometric fluid and is provided below: ^Pe = (P« - P)S^h (11-56) where Ah is the difference in the heights of the manometric fluid. Figures II-2 and II-3 show the effect of velocity on calculated and experimental pressure • Experimental # Theoretical 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 Velocity ( m/s ) FIGURE n-2 EFFECT OF THE VELOCITY ON THE CALCULATED AND THE EXPERIMENTAL PRESSURE DROP ACROSS THE DPHE •^0 J • I ' I ' I • I • I • I • I • I • I • I • I • I • I • I I I I 7.5 -0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 Velocity ( m/s ) FIGURE II-3 E F F E C T OF T H E V E L O C I T Y ON T H E C A L C U L A T E D SHEAR STRESS drop and shear stress for the clean condition. 11.10 Numerical Sample Calculation Using Run 2 Run 2 was carried out at the following conditions, at a rotameter setting of 50 . At time=1 h, the temperature readings were: T,i = 16.8 °C = 22.4 °C = (107.2 + 106.9) / 2 = 107.05 °C = 380.21 K T,b = (22.4 + 16.8) / 2 = 19.6 °C = 292.76 K Applying equations (II-3), (I-l) and (II-4): Pb =998.78 kg/m' F = 40.13 X 10-^  mVs Cp = 4.1797 kJ/kg°C Using equation (II-1): Q = (40.13 x 10-^ ) (998.78) (4.1797) (22.4 - 16.8) = 9.38 kW AT,^ = (22.4 - 16.8) / In {(107.05 - 22.4) / (107.05 - 16.8)} = 87.42 °C A„ = X (19.1 X 10-^ ) (1.4) = 8.4 X 10"^  m^ Hence, the overall heat transfer coefficient can be calculated using equation (II-5): = 9.38 / {(0.084) (87.4)} = 1.278 kW/m^K Similarly, at t = 10.0 h: U«, = 1.141 kW/m^K Q = 8.88 kW Assuming the clean overall heat transfer coefficient to be independent of the heat transfer rate, the uncorrected fouling resistance can be calculated as: (Rf)„ = 1 / 1.141 - 1 / 1.278 = 0.09395 kW/m^K However, the clean overall heat transfer coefficient depends on the rate of heat transfer as shown in Figures (5-2) to (5-7). The average slope of the clean overall heat transfer coefficient versus heat transfer straight line, at this velocity, is 0.048 . From equation (2-9): U« = 1.278 + 0.048 (8.88 - 9.38) = 1.254 kW/m^K Hence, the corrected fouling resistance, denoted by , is calculated as: Rfe = 1/1.141 - 1/1.262 = 0.079 kW/m^K The physical properties of water are evaluated at the bulk temperature using equations (II-10) to (11-13) as: = 1006.2 X 10** Ns/m^ K, = 600.282 X lO*^  kW/mK Pr = 7.09 Re, = (19.0 X 10-3 ) (0.47) (998.78) / 1006.2 x lO"*^  = 8864.3 Hence, the outside heat transfer coefficient is calculated using equation (II-9): h„ = (600.3 X lO-**) / 19.0 X lO'^ ) (0.023) (8864)°-« (7.09)°* (1.99)°"* = 3.124 kW/m^K The physical properties for the steam condensate are evaluated using equations (11-19) to (11-23): hf = 2238.82 kJ/kg Pb = 953.7 kg/m' fit, = 260.02 kg/m.s Cph = 4.2182 kJ/kg°C Kb = 683.7 X 10-* kW/m.K The mass rate of the condensation of the steam is calculated using equation (11-24): Wp = 9.38 / 2238.8 = 4.1897 x lO'^  kg/s The parameter T, defined by equation (11-18) is evaluated as follows: r = 4.1897 X 10 -3 / {(3.1415) (13.7 x IQ-^ )} = 9.73 x 10-^  kg/m.s The Reynolds number is given according to equation (11-16): RCb = 4 X 9.73 X 10-^  / 260.0 x 10"^  = 1498 < 2100 ( laminar regime ) Hence, the parameter G defmed by equation (11-17) is calculated as: G = [(4.1897 X lOy (953.7)2 (9.8) / {(29.6) (13.7 x lO-')^ (260. x 10"^ )}]* = 476.85 The Prandtl number is given below: Pr = (4.2182) (260. x W ) / (683.7 x 10"^  ) = 1.60 By plugging the above numerical values in equation (11-15), the value of the inside heat transfer coefficient is obtained as: hi = (4.2182) (476.85) (5.35) / {(1.60) (1497.6)} = 4.479 kW/m^K The thermal resistance of the metal wall is calculated using equation (11-14): R„ = In (19.1/13.7) / {(2.0) (3.1415) (16.1 x 10"^ ) (1.4)} = 2.3464 K/kW By substituting the above values in equation (2-3), the clean overall heat transfer coefficient is obtained as follows: 1 / Uo = 1 / {4.479 (13.7/19.1)} + 1 / 3.124 + {(2.3464) (6.025 x IQr^)} or, U,, = 1.207 kW/m^K The following calculation sums up the thermal resistances of steam and the metal wall, both in series, at time = 1 h: R, = 1/ {(4.479) (6.025 x 10"^ )} + 2.3464 = 6.0526 K/kW The outside wall temperature can be calculated according to the equation (11-26): T„ = 107.05 - (9.38) (6.05257) = 50.2 °C The total thermal resistance at time = 10 h is made up of the resistances offered by steam, metal and water, all in series and can be calculated by adding them up: RT = 2.3464 -I- 1 / {(4.633) (6.025 x lO^)} + 0.079/0.084 = 6.869 K/kW Therefore, the scale-water interface temperature is calculated according to equation (11-28): T, = 112.8 - (8.88) (6.869) = 51.8 °C and the wall temperature is calculated as: T , = 112.8 - (8.88) (5.929) = 60.2 °C The Reynolds number, at the film temperature, is evaluated as below: RCf = (19.0 X 10-3) (0.47) (994.4) / 545.4 x 10'* = 16281.2 The following Table II-5 summarises the experiments done to calculate the volume of the water pumped before the suction was lost: TABLE II-5 DETERMINATION OF THE TOTAL SYSTEM VOLUME Rotameter Reading Volumetric Flow Rate (mVs) Time taken to Empty the Tanks (s) Volume (x 10 ^ m,) 46 37.14 X 10-5 1178 437 56 44.27 X 10' 967 428 64 51.50 x 10-5 855 432 Hence, V„g is calculated as: V,,g = ( 440 + 428 + 437 ) / 3.0 = 435 liters When the pump lost suction, there was 5 cm high water column left at the bottom of both the tanks, having upper and lower diameters of 48.3 cm and 45.7 cm, respectively. Hence the residual volume is calculated as: V , = 2 X 3.1415 X 46.5^ / 4 x 51 cm' « 18 1 The piping is essentially made up of about 12.2 m of 2.54 cm ID and 6.1 m of 3.81 cm ID. Hence the internal volume of the piping is calculated as: Vp = 3.1415 ( 0.0127^ x 12.2 + 0.0191^ x 6.1 ) m' « 13 1 Hence the total system volume is calculated as: V = 435 + 18 + 13 = 466 1 Water Analysis 10 ml of vandate-molybdate reagent was added to 35 ml of sample # 1 for run 3 and finally diluted to 50 ml. The absorbance at 470 nm and 1 cm path length was measured using spectrophotometer as being 0.3741. From Figure 1-2: Tpo4 (as mg/1 PO4) = 90.6 50 ml of the sample # 1 for run 3 was titrated against EDTA. The volume of EDTA consumed was 4.9 cc. Hence, according to the equation (11-43), the total Ca-hardness is: Tea (as mg/1 Ca) = 4.9 X 1 X 400.8 / 50 = 39.3 The total potassium concentration, (K), for the run 3 was calculated by summing up the potassium added as KH2PO4 and as KOH, at the start of a run as follows: (K) = (50 / 136.1 -H 20 X 0.01 -H 11 x 0.58 x .04) / 380 = 2.1646 x 10^ mol/1 Pressure Drop The pressure drop across a 0.6 m length of the heat exchanger is evaluated by calculating the friction factors using the Blausius equation: fj = 0.0791 / (16281)'^ = 7.003 x lO'^  f„ = 0.0791 / ( 8 8 6 4 ) = 8.152 x 10^ AP, = (2) (0.6) (0.5) (996.6) (0.47)^ • [{(.00955) (7.005 x lO"') + (.0191) (8.15 x lO^)} / {(.0191)^ - (.00955)^)}] AP, = 107.6 N/m^ The experimental pressure drop is calculated using equation (11-56): AP, = (1750 - 1000 ) (9.81) (1.7 x la^) = 124.95 N/m^ The experimental pressure drops as a function of velocity are listed in the following Table II-6: TABLE II-6 EXPERIMENTAL PRESSURE DROP AT VARIOUS VELOCITIES Rotameter Reading Velocity Manometric Ah AP (m/s) (cm.) (N/m^) 20 0.1926 0.4 29.93 50 0.4704 1.8 132.44 80 0.7481 3.6 264.87 100 0.9333 5.2 382.59 APPENDIX i n COMPUTER PROGRAMS The following pages list six computer programs that were used to process the experimental data. The first computer program, Figure III-l, calculates, as functions of time, the heat transfer rate, the log mean temperature difference, the dirty overall heat transfer coefficients, the uncorrected and corrected fouling resistances; the wall, scale surface, water bulk, steam bulk temperatures and their averages and standard deviation over the entire length of the run. Also calculated are the bulk and the Reynolds numbers over the entire length of a run. The second computer program. Figure III-2, calculates the concentrations of the various phosphate and calcium ionic species as functions of pH and temperature. The activity coefficients and the ionic strength of the solution are calculated using an iterative procedure. The output for the programs III-l and III-2 are listed only for Run 2. For all other runs the output of the computer programs III-l and III-2 has been bound in a supplemental volume to the thesis which is available from Dr. A.P. Watkinson. The third computer program. Figure III-3, calculates the distribution of the phosphate species as a function of pH neglecting the ion pair complexing and assuming all the activity coefficients to be unity. The fourth one. Figure III-4, calculates the theoretical inside and the outside heat transfer coefficients and, therefore, the theoretical overall heat transfer coefficient as functions of the temperatures, geometry, velocity and the heat flux. The fifth one. Figure III-5, calculates, at a particular velocity, the theoretical and the experimental pressure drops across the heat exchanger and the theoretical shear stress as functions of temperatures and the geometry of the heat exchanger. The sixth one calculates the theoretical and experimental pressure drops and the shear stress as functions of the velocity, geometry and the temperature under "no steam" conditions. FIGURE m-1 COMPUTER PROGRAM TO CALCULATE THE EXPERIMENTAL FOULING RESISTANCES AS A FUNCTION OF TIME C THIS PROGRAM CALCULATES THE EXPERIMENTAL FOULING RESISTANCE C AS A FUNCTION OF TIME C Tl AND T2 DENOTE THE OUTLET AND INLET TEMPERATURES OF WATER IN C DEGREE C, T3 AND T4 DENOTE THE INLET AND OUTLET TEMPERATURES C OF THE STEAM, LMTD DENOTES LOG MEAN TEMPERATURE DIFFERENCE. C POSTSCRIPTS C AND H DENOTE WATER AND STEAM RESPECTIVELY. P, CP, C H, U, K DENOTE DENSITY, SPECIFIC HEAT, HEAT OF VAPORISATION, C THERMAL CONDUCTIVITY HAVING UNITS OF kg/m**3, kJ/kgK, kJ/kg, C kW/mK, RESPECTIVELY. TC AND TH DENOTE THE AVG BULK TEMPERATURES C TS DENOTES THE CALCULATED SCALE WATER INTERFACE TEMPERATURE. C RFC AND RFU DENOTE THE CORRECTED AND UNCORRECTED FOULING C RESISTANCES HAVING UNITS OF m**2K/kW. TRE DENOTES THE TOTAL C THERMAL RESISTANCE. UOC AND UOD DENOTE THE OVERALL CLEAN AND C DIRTY HEAT TRANSFER COEFFICIENTS HAVING UNITS kW/m**2K C M DENOTES THE SLOPE OF THE Q VS. UOC STRAIGHT LINE. REAL LMTD,M,KH,NREC,NREF,KS INTEGER RUN DIMENSION TIME<134),T1(134) ,T2(134),T3(134) ,T4(134) ,DT(134) DIMENSION LMTD(134),PC(134),CPC(134) ,Q(134),UOD(134),TWALL(134) DIMENSION RFC(134),RFU(134) ,TS(134),TC(134),TRE(134),TH(134) DIMENSION UOC(134),PH(134),HF(134),UH(134),CPH(134),KH(134) DIMENSION WH(134),T(134) ,G(134) ,PRH(134),HS(134),HI(134) SUMTC=0.0 SUMTH=0.0 SUMQA=0.0 SUMTS=0.0 C F IS THE FLOW RATE OF WATER IN ra**3/s C N IS THE NUMBER OF THR OBSERVATIONS FOR THE RUN READ(5,1) RUN,F,M,N 1 FORMAT(I3,E8.2,F7.6,I3) 2 FORMAT(F7.4,2F4.1,2F5.1) DO 10 1=1,N READ(5,2) TIME(I),T1(I) ,T2(I),T3(I),T4(I) TH(I)'=(T3(I)+T4(I)) /2.0 TC(I) = (Tl(I)+T2(I))/2.0 CPC(I)=(-0.000108)*TC(I)+4.181817 PC(I) = (-0.326893)*TC(I) +1005.445464 DT(I)=T1(I)-T2(I) SUMTC=SUMTC+TC(I) SUMTH=SUMTH+TH(I) LMTD(I)=DT(I)/LOG((TH(I) -T2(I))/(TH(I)-Tl(I))) Q(I) = (DT(I)*PC(I)*CPC(I) *F) UOD(I) =Q(I)/(0.084*LMTD(I)) U0C(I)=U0D(1)+M*(Q(I)-Q(1)) RFU(I)=1./U0D(I)-1,/U0D(1) RFC(I)=1./UOD(I)-1./UOC(I) C TO CALCULATE THEORETICAL STEAM SIDE THERMAL RESISTANCE, HI PH(I)=1000./(0.68523+0.000958*(TH(I)+273.)) HF(I)=2306.62+2.31058*(TH(I) +273.) -0.00655*(TH(I)+273.)**2 UH (I) = (4032.3786-17.26126*(TH(I) +273.) +0.0193*(TH(I)+273.)**2)/ *1000000. CPH(I)=6.17287-0.01149*(TH(I)+273.)+1.67E-05*(TH(I)+273.)**2 KH(I) = (-446.29124+5.57289*(273.+TH(I) ) -0.00684*(273.+TH(I))**2)/ *1000000. WH(I)=Q(I)/HF(I) T(I)=WH(I) /(3.1415*{13.7/1000.) ) G(I) = ((WH(I)**2*PH(I)**2*9.81) /(29.6*(13.7/1000.)**2*UH(I)))**(0 *.3333333) PRH(I) = (CPH(I)*UH{I) /KH(I) ) HI(I)=(({CPH(I)*G{I))/PRH(I))«5.35/(4.0*T(I)/UH(I))) AI=3.1415*1.4*(13.7/1000.) A0=3.1415*1.4*(19.1/1000.) C TO CALCULATE TOTAL THERMAL RESISTANCE,TRE, IN DEG K/JtW RM=LOG(19.1/13.7)/(2.0*3.1415*16.1/1000.*!.4) TRE(I)=RM+RFC(I)/0.084+1./(HI(I) *AI) TS(I)=TH(I)-Q(I)*TRE(I) SUMTS=SUMTS+TS(I) SUMQA^SUMQA+Qd) SWRE«RM+1./(HI(I)*AI) TWALL(I) «=TH(I)-SWRE*Q(I) 10 CONTINUE WRITE(6,3) 3 FORMAT (' DT TC TH LMTD TIME Q UOD RFC RFU TS *TWALL HI •) 4 F0RMAT(F4.1,1X,F4.1,1X,F5.1,1X,F5.1,1X,F4.1,1X,F5.2,1X,F5.3, *1X,F6.4,1X,F5.3,1X,F5.1,1X,F5.1,1X,F5.3) 2=0.0 DO 7 1=1,N WRITE(6,4) DT(I),TC(I),TH(I),LMTD(I),TIME(I), *Q(I),U0D(I),RFC(I),RFO(I) ,TS(I) ,TWALL(I),HI(I) Z=Z+1. 7 CONTINUE AVGTC=SUMTC/Z AVGTH=SUMTH/2 AVGQA=SUMQA/Z AVGTS=SOMTS/Z SWA=0.0 SFA=0.0 SBA=0.0 DO 8 1=1,N SWA=SWA+(TC(I)-AVGTC)**2 SFA=SFA+(Q(I)-AVGQA) **2 SBA=SBA+(TS(I)-AVGTS)**2 8 CONTINUE SDTW=SQRT(SWA)/Z SDQA=SQRT(SFA)/Z SDTS=SQRT(SBA) /Z WRITE(6,40) 40 FORMAT(//,' SDTW SDQA SDTS ') WRITE(6,20) SDTW,SDQA,SDTS 20 F0RMAT(F8.4,3X,F7.4,2X,F7.4) WRITE(6,5) 5 F0RMAT(//,' TC(avg) TH(avg.) Q(avg.) TS(avg.) ') 30 FORMAT(2X,F4.1,6X,F5.1,4X,F4.1,5X,F5.1) WRITE(6,30) AVGTC,AVGTH,AVGQA,AVGTS C TO CALCULATE THE FILM AND BULK REYNOLDS NUMBER DENOTED BY C NREF AND NREC, RESPECTIVELY UC=(65816.34444-418.2856*(AVGTC+273.)+0.67255*(AVGTC+273.)**2) /I *000000. PB=1000./1.002 D E = ( 3 8 . 1 - 1 9 . 1 ) / l O O O . V = F / ( ( 3 . 1 4 1 5 / 4 . ) * ( ( 3 8 . 1 / 1 0 0 0 . ) * * 2 - ( 1 9 . 1 / 1 0 0 0 . ) * * 2 ) ) N R E C = D E * V * P B / U C T F = ( A V G T S + A V G T C ) I I . U F A = ( 4 0 3 2 . 3 7 8 6 - 1 7 . 2 6 1 2 6 * ( T F + 2 7 3 . ) + 0 . 0 1 9 3 * ( T F + 2 7 3 . ) * * 2 ) / l O O O O O O . P F = 1 0 0 0 . 0 / ( l . + ( T F + 2 7 3 . - 2 8 5 . ) * . 0 0 0 2 ) N R E F = D E * V * P F / U F A W R I T E ( 6 , 5 0 ) NREF ,NREC 50 F O R M A T ( / / , ' N R E F = ' , F 8 . 1 , 7 X , ' N R E C = ' , F 8 . 1 ) STOP END FIGURE m - 2 COMPUTER PROGRAM TO CALCULATE THE CONCENTRATION OF VARIOUS SPECIES AND THE ACTIVITY COEFFICIENTS PROGRAM BPREP C THIS PROGRAM C A L C U L A T E S CONCENTRATIONS AND A C T I V I T Y C O E F F I C I E N T S C AT THE BULK TEMPRATURE AND G I V E N P H , TCA AND TP04 I M P L I C I T R E A L ( A - Z ) REAL K 1 , K 2 , K 3 , K 4 , K 5 , K 6 , K 7 , K W , I , I I , K , K S P C A L L CONCENTRATIONS ARE IN m o l / L . Z I S THE N O , OF I T ERAT IONS T P 0 4 = 9 6 . 6 7 E - 5 TCA ^ e S . SE -S PH=7.50 CL= 1 . 9 3 E - 3 K = 4 . 3 3 5 E - 3 EPS=-00001 Z=0 .0 C TC IS THE AVERAGE WATER BULK T E M P E R A T U R E , ! I S T H E AVERAGE C STEAM TEMPERATURE I N D E G . K TH=52 .+273 .16 T C = 2 0 . + 2 7 3 . 1 6 C I N I T I A L APPROXIMATION FOR THE I O N I C STRENGTH 1 = 0 . 5 * ( 9 . « T P 0 4 + 4 . * T C A + K + C L + 1 0 . * * ( - 1 . * P H ) ) C C A L C U L A T E A C T I V I T Y C O E F F I C I E N T S AT THE F I L M TEMPERATURE C US ING EXTENDED FORM OF THE D E B Y E - H : U C K E L EQUAT ION C PROPOSED BY DAVIES FOR I < 0 . 2 M O L E / L I T E R 100 F M = 1 0 . 0 * * ( - 2 6 2 0 . 7 / T C * * 1 . 5 * ( S Q R T ( I ) / ( 1 + S Q R T ( I ) ) - 0 . 3 * I ) ) F D = 1 0 . 0 * * { - 1 0 4 8 2 . 8 / T C * * 1 . 5 * ( S Q R T ( I ) / ( 1 + S Q R T ( I ) ) - 0 . 3 * I ) ) F T = 1 0 . 0 * * ( - 2 3 5 8 6 . 3 / T C * * 1 . 5 * ( S Q R T ( I ) / ( 1 + S Q R T ( I ) ) - 0 . 3 * I ) ) C C A L C U L A T E VARIOUS EQUIL IBR IUM CONSTANTS AT THE BULK TEMPERATURE K l = 1 0 . * * ( 4 2 3 . 2 3 / T C - 3 . 5 9 0 3 ) / ( F M * * 2 ) K 2 = 1 0 . * * ( 1 4 1 . 0 8 / T C - 7 . 6 1 7 ) / F D K 3 = 1 0 . * * ( 2 1 1 . 6 2 / T C - 1 2 . 8 4 5 ) * F D / ( F M * F T ) K 4 = 1 0 . * * ( - 2 8 2 . 1 5 6 / T C + 2 . 2 9 3 5 ) * F D K 5 = 1 0 . * * ( - 7 7 5 . 9 3 / T C + 4 . 0 2 2 ) * F D K 6 = 1 0 . * * ( - 9 8 7 . 5 / T C + 6 . 0 4 7 ) * ( F D * * 2 ) K 7 = 1 0 . « * ( - 6 3 4 . 8 5 / T C + 8 . 5 8 5 ) * F D * F T / F M K W = 1 0 . * * ( - 2 6 0 9 . 9 / T C - 5 . 1 8 ) / ( F M * * 2 ) C C A L C U L A T E THE H+ & O H - CONCENTRATIONS IN m o l / L H = 1 0 . * * ( - 1 . * P H ) / F M OH=KW/H C C A L C U L A T E VARIOUS CONSTANTS C 1 , C 2 , C 3 , C 4 , C 5 , C 6 C l = ( 1 . + (K l /H) + ( K 1 * K 2 / ( H * * 2 ) ) + ( K 1 * K 2 * K 3 / H * * 3 ) ) / ( 1 . + K 4 * 0 H ) C 2 = ( T C A - T P 0 4 ) / ( l . + ( K 4 * 0 H ) ) C 3 = l . + ( K 4 * O H ) C 4 = ( K 5 * K 1 / H ) + ( K 6 * K 1 * K 2 / H * * 2 ) + ( K 7 * K 1 * K 2 * K 3 / H * * 3 ) C 5 = ( ( C 2 * C 4 ) + ( C 3 * C 1 ) ) / ( C 1 * C 4 ) C 6 = ( ( C 2 * C 3 ) - T C A ) / ( C 1 * C 4 ) H 3 P O 4 = 0 . 5 * ( S Q R T ( ( C 5 * * 2 ) - ( 4 . 0 * C 6 ) > - C 5 ) H2P04=(K1*H3P04)/H HP04=(K1*K2*H3P04) / ( H * * 2 ) P04=(K1*K2*K3*H3P04) / ( H * * 3 ) CA=C2+(C1*H3P04) CA0H=K4*0H*CA C A H 2 P 0 = ( K 5 * K 1 * H 3 P 0 4 * C A ) / H CAHP04= . (K6*K1*K2*CA*H3P04) / (H**2) C A P 0 4 = ( K 7 * K 1 * K 2 * K 3 * C A * H 3 P 0 4 ) / ( H * * 3 ) C C A L C U L A T E THE IONIC STRENGTH OF SOLUTION USING ABOVE VALUES I I = I I -0 .5* (4 .0*CA+9 .0*PO4+CAOH+CAH2PO+CAPO4+H2PO4+4 .0*HPO4+K+CL+H *+0H) Z=Z+1. D I = A B S ( ( I - I I ) / I I ) * 1 0 0 . I F ( D I . L E . E P S ) GOTO 110 GOTO 100 110 W R I T E ( 6 , 1 2 0 ) K 1 , K 2 , K 3 , K 4 , K 5 , K 6 , K 7 , K W 120 F O R M A T C K l ' , ' K2 ' , ' K3 K4 ' , 2 X , 4 ( E 8 . 3 , 2 X * ) , / / , ' K5 ' , ' K6 K7 ' , ' KW ' , 2 X , 4 ( E 8 . 3 , 2 X ) / ) W R I T E ( 6 , 1 3 0 ) H , O H , C A , C A O H , C A H 2 P O , C A H P 0 4 , C A P 0 4 130 FORMAT( ' H ' , ' OH ' , ' CA M 0 X , 3 ( E 8 . 3 , 2 X ) / / , * ' CAOH C A H 2 P 0 ' , ' CAHP04 ' , ' C A P 0 4 ' , 4 ( E 8 . 3 , 2 X ) / ) W R I T E ( 6 , 1 4 0 ) H 3 P 0 4 , H 2 P 0 4 , H P 0 4 , P 0 4 , I , I I , 2 140 F O R M A T C H3P04 H2P04 HP04 P04 ' , 4 ( 2 X , E 8 . 3 ) , / / * • I I I 2 ' , 8 X , 2 ( 2 X , E 1 3 . 8 ) , 2 X , F 3 . 1 / ) W R I T E ( 6 , 1 5 0 ) F M , F D , F T 150 F O R M A T C FM FD F T ' , 7 X , 3 ( 2 X , F 5 . 4 ) , / ) C CHECK FOR THE ELECTONEUTRAL I TY OF THE SOLUTION SP0S>=2. * F D * C A + F M * (CA0H+CAH2P0+K+H) S N E G » = 3 . * F T * P 0 4 + 2 - * F D * H P 0 4 + F M * ( C A P 0 4 + H 2 P 0 4 + C L + 0 H ) TCA=CA+CA0H+CAH2P0+CAHP04 +CAP04 TP04=H3P04+H2P04+HP04+P04+CAH2PO+CAHP04+CAP04 C TO C A L C U L A T E THE SUPERSATURATION POTENT IAL W R T VARIOUS C S P E C I E S K S P = 1 . 0 5 E - 8 * E X P ( 9 4 3 . 4 7 / T C ) / F D * * 2 SDCPD=CA*HP04-KSP S H A P = C A * * 5 * P 0 4 * * 3 * O H - l . 8 E - 5 8 / ( F D « * 5 * F T * * 3 * F M ) S T C P = C A * * 3 * P 0 4 * * 2 - 1 . 1 5 E - 2 9 / ( F D * * 3 * F T * * 2 ) S O C P = C A * * 4 * P 0 4 * * 3 * H - l . 2 5 E - 4 7 / ( F D * * 4 * F T * * 3 * F M ) C C A L C U L A T E THE DEGREE OF SUPERSATURATION WRT VARIOUS S P E C I E S DDCPD=CA*HP04/KSP D H A P = C A * * 5 * P 0 4 * * 3 * 0 H / ( l , 8 E - 5 8 / ( F D * * 5 * F T * * 3 * F M > ) D T C P = C A * * 3 * P 0 4 * * 2 / ( 1 . 1 5 E - 2 9 / ( F D * * 3 * F T * * 2 ) ) D O C P = C A * * 4 * P 0 4 * * 3 * H / ( 1 . 2 5 E - 4 7 / ( F D * * 4 * F T * * 3 * F M ) ) W R I T E ( 6 , 1 5 1 ) D D C P D , D H A P , D T C P , D O C P 151 F O R M A T C DDCPD DHAP DTCP DOCP ' , 4 ( IX , E 9 . 2 ) / ) W R I T E ( 6 , 2 0 0 ) S D C P D , S H A P , S T C P , S O C P 200 F O R M A T C SDCPD SHAP STCP SOCP ' , 4 ( 1 X , E 9 . 2 ) / ) W R I T E ( 6 , 1 5 5 ) SPOS ,SNEG 155 FORMAT C SPOS = • , E 8 . 3 , 2 X , ' SNEG = • , E 8 . 3 / ) WRITE ( 6 , 1 1 2 ) P H , T C A , T P 0 4 F O R M A T C PH = ' , F 3 . 1 , 8 X , ' TCA = • , E 8 . 2 , 8 X , • TP04 END FIGURE III-3 COMPUTER PROGRAM TO CALCULATE THE CONCENTRATION OF VARIOUS PHOSPHATE SPECIES C TH IS PROGRAM C A L C U L A T E S THE CONCENTRATIONS OF THE PHOSPHATE C ASSUMING NO C A L C I U M PHOSPHATE ION COMPLEXING R E A L K 1 , K 2 , K 3 DATA K 1 , K 2 , K 3 / 6 . 0 9 E - 3 , 6 . 3 3 E - 8 , 4 . 2 9 E - 1 3 / C A L L CONCENTRATIONS ARE IN m o l / L C A L L A C T I V I T Y C O E F F I C I E N T S ARE ASSUMED TO BE UNITY T P 0 4 = 1 . 1 8 4 E - 2 T C A = 1 . 2 1 0 E - 2 C C A L C U L A T E THE (H+) CONCENTRATION H = 1 0 * « ( - P H ) C C A L C U L A T E THE CONCENTRATIONS OF THE VARIOUS S P E C I E S P 0 4 - T P 0 4 / ( 1 . 0 + H / K 3 + H * * 2 / ( K 3 * K 2 ) + H * * 3 / ( K 3 * K 1 * K 2 ) ) HP04=H/K3*P04 H2P04=H/K2*HP04 H 3 P 0 4 « H / K 1 * H 2 P 0 4 W R I T E ( 6 , 1 0 0 ) P 0 4 , H P 0 4 , H 2 P 0 4 , H 3 P 0 4 , P H 100 F O R M A T d X , • P04 • , E 8 . 2 , ' HP04 ' , E 8 . 2 , * • H2P04 ' , E 8 . 2 , ' H3P04 ' , E 8 . 2 , ' PH • , F 4 . 1 ) END FIGURE III-4 COMPUTER PROGRAM TO CALCULATE THE OVERALL HEAT TRANSFER COEFFICIENTS C THIS PROGRAM CALCULATES THE OVERALL THEORET ICAL HEAT TRANFER C C O E F F I C I E N T S FOR THE D P H E IN kW/ni*niK REAL K C , K H , N R E C , K , N R E H , J C INPUT VAR IABLES ARE HEAT TRANSFER R A T E ( Q ) , OUTSIDE AND INS IDE C DIAMETERS OF THE ANNULAR S P A C E ( D O , D I ) , V E L O C I T Y ( V ) AND THE C AVERAGE BULK TEMPERATURES OF STEAM AND WATER (TH, T B ) . UNITS C ARE kW, m, m/s AND DEGREE K E L V I N , R E S P E C T I V E L Y . POSTSCRIPTS C H AND C DENOTE STEAM AND WATER, R E S P E C T I V E L Y Q=09 .38 TH=273 .16+107 . T C = 2 7 3 . 1 6 + 1 9 . 6 ROT=50.0 C CALCULATE THE FLU ID PROPERT IES AT THE BULK TEMPERATURE C PRC DENOTES THE BULK PRANDTL NUMBER AND V I S C O S I T Y ( U C ) , THERMAL C CONDUCTIV I TY(KC) AND DENS I TY (PC) OF WATER AT THE BULK T E M P . C HAVE UNITS OF N m / s * s , kW/mK AND k G / m * * 3 R E S P E C T I V E L Y UC= ( 6 5 8 1 6 . 3 4 4 4 - 4 1 8 . 2 8 5 6 * T C + 0 . 6 7 2 5 5 * T C * * 2 ) / l O O O O O O . K C « ( - 2 7 8 . 3 1 5 9 + 4 . 4 6 2 8 * T C - 0 . 0 0 4 9 9 * T C * * 2 ) / 1 0 0 0 0 0 0 . P R C - 4 6 0 . 7 6 7 8 - 2 . 8 9 0 6 * T C + 0 . 0 0 4 5 8 * T C * * 2 C C A L C U L A T E THE WATER S I D E HEAT TRANSFER C O E F F I C I E N T (HO) US ING C WIEGAND CORRELATION FOR TURBULENT FLOW IN A N N U L I . DE DENOTES C EQUIVALENT DIAMETER D O = 3 8 . 1 / 1 0 0 0 . 0 D I = 1 9 . 1 / 1 0 0 0 . V=-009259*ROT+ .007398 DE=DO-DI NREC= D E * V * P C / U C H O " ( K C / D E ) * ( 0 . 0 2 3 ) * N R E C * * 0 . 8 * P R C * * 0 . 4 * ( D O / D I ) * * 0 . 4 5 C CALCULATE THE WATER S IDE C O E F F I C I E N T (HOG) USING G N I E L I N S K I C CORRELATION FOR TURBULENT FLOW IN CIRCULAR P IPES J = l . / ( ( 1 . 8 2 * 0 . 4 3 4 2 * L O G ( N R E C ) - 1 . 6 4 ) * * 2 ) HOG-( K C / D E ) * ( ( a / 8 . ) * ( N R E C - 1 0 0 0 . ) * P R C ) / ( 1 . + 1 2 , 7 * ( J / 8 . ) * * 0 . 5 * ( P R C 1 * * 0 . 6 6 6 6 - 1 . ) ) C C A L C U L A T E THE FLUID PROPERT IES AT THE STEAM T E M P . ( T H ) , T I S I N K C HFH DENOTES THE HEAT OF VAPORISAT ION IN k j / ) c g P H = 1 0 0 0 . / ( I . 2 2 9 6 1 - 0 . 0 0 1 8 * T H + 3 . 4 8 3 E - 6 * T H * * 2 ) HFH=2306 .62+2 .31058* TH - 0 . 0 0 6 5 5 * T H * * 2 U H = ( 4 0 3 2 . 3 7 8 6 - 1 7 . 2 6 1 2 6 * T H + 0 . 0 1 9 3 * T H * * 2 ) / l O O O O O O . C P H = 6 . 1 7 2 8 7 - 0 . 0 1 1 4 9 * T H + 1 . 6 7 E - 0 5 * T H * * 2 K H = ( - 4 4 6 . 2 9 1 2 4 + 5 . 5 7 2 8 9 * T H - 0 . 0 0 6 8 4 * T H * * 2 ) / l O O O O O O . WH=Q/HFH T = W H / ( 3 . 1 4 1 5 * ( 1 3 . 7 / 1 0 0 0 . ) ) NREH=4.0*T/UH G = ( ( W H * * 2 * P H * * 2 * 9 . 8 1 ) / ( 2 9 . 6 * ( 1 3 . 7 E - 0 3 ) * * 2 * U H ) ) * * ( 0 . 3 3 3 3 3 ) PRH=(CPH*UH/KH) H I = ( ( ( C P H * G ) / P R H ) * 5 . 3 5 / ( 4 . 0 * T / U H ) ) C THE FOLLOWING STEPS C A L C U L A T E RES IS TANCE OF THE METAL WALL (RM) c K DENOTES THE THERMAL CONDUCTIV I TY OF THE METAL WALL K = 1 6 . 1 / 1 0 0 0 . R M = L 0 G ( 1 9 . 1 / 1 3 . 7 ) / ( 2 . 0 * 3 . 1 4 1 5 * 1 . 4 0 * K ) A 0 = 3 . 1 4 1 5 * ( 1 9 . 1 / 1 0 0 0 . ) * 1 . 4 0 C C A L C U L A T E OVERALL HEAT TRANSFER C O E F F I C I E N T FROM THE IND IV IDUAL C HEAT TRANSFER C O E F F I C I E N T S U 0 = 1 . / ( 1 . / ( H I * ( D O / D I ) ) + R M * A 0 + 1 . / H O ) WRITE ( 6 , 1 0 0 ) H I , H 0 , N R E H , H 0 G , U 0 100 FORMAT C H I » ' , 1 X , F 7 . 4 , / / , ' H O = ' , I X , F7 . 4 , / / , "NREH = M X , 1 F 6 . 1 , / / , ' H O G - • , 1 X , F 7 . 4 , / / , ' U O = ' , 1 X , F 7 . 4 ) STOP END FIGURE III-5 COMPUTER PROGRAM TO CALCULATE THE SHEAR STRESSES AND THE PRESSURE DROPS C THIS PROGRAM CALCULATES THE REMOVAL SHEAR STRESSES AND C PRESSURE DROP ACROSS THE HEAT EXCHANGER REAL NREF,NREC,K,KD INTEGER RUN C ROT IS THE ROTAMETER READING, F IS THE VOLUMETRIC FLOW RATE IN C m**3/s, INPUT TEMPERATURES ARE IN DEGREE C. DELHE IS THE C MANOMETRIC HEIGHT DIFFERENTIAL IN cm. V IS THE VELOCITY IN m/s ROT=50. TC=19.6 TH=50.9 DELHE=5.0 TC=TC+273.16 TH=TH+273.16 F'=(0.79032*ROT+0.63147) /lOOOOO. V=4.0*F/(3.1415*((38.1/1000.)**2-(19.1/1000.)**2)) RI=(19.1/2.)/1000. RO=(38.1/2.)/1000. DE=2.«(R0-RI) C CALCULATE THE AVG. FILM TEMP. IN DEG. K TF=0.5*(TC+TH) C FIND LIQUID DENSITY (P) AND VISCOSITY (U) AS FUNCTION OF C TEMPERATURE. UNITS OF DENSITY AND VISCOSITY ARE kg/cm**3,Nm/s**2 PF=1000./ (1.+ (TF-285.)*0.0002) PC=1000./1.002 UF«(4032.3786-17.26126*TF+0.0193*TF**2)/lOOOOOO. UC"= (65816.34444-418.2856*TC+0.67255*TC**2)/1000000. C FIND BULK & FILM REYNOLDS NUMBERS & SCHMIDT NUMBER (SC) NREF=V*DE*PF/UF NREC=V*DE*PC/UC C DIFFUSION COEFFICIENT OF ALL THE IONS IS APPROXIMATED C AS 10.**-9 m**2/s AT 25 DEG. CENTIGRADE SC=UF/(PF*l.E-09) C CALCULATE MASS TRANSFER COEFFICIENT (KD) IN THE FILM C UNITS ARE m/s KD=0.023*V*NREF**(-0.17)*SC**(-0.666) C CALCULATE DELPE, THE EXPERIMENTAL PRESSURE DIFFERENCE IN N/m**2 DELPE=(1750,-1000.) *9.81*DELHE/100. C CALCULATE DELPT, THE THERETICAL PRESSURE DIFFERENCE IN N/m**2 FI=.0791/(NREF**.25) FO=.0791/(NREC**.25) DELPT=2.*0.6*0.5*PF*V**2*(RI*FI+RO*FO)/(R0**2-RI**2) c CALCULATE TRZ, SHEAR STRESSES AT THE S C A L E L IQUID INTERFACE C UNITS ARE N/ni**2 T O = D E L P T * R O / ( 2 . * 0 . 6 ) K=RI/RO T R Z = T 0 * ( K - ( 1 - - K * * 2 ) / ( 2 . * K * L O G ( 1 . / K ) ) ) WR I TE (6 ,100 ) K D , D E L P E , D E L P T , T R Z , N R E C , N R E F , F I , F O , V , U F , P F 100 FORMAT ( I X , ' KD D E L P E DELPT TRZ ' , 4 ( I X , E 9 . 3 ) ) STOP END III-6 COMPUTER PROGRAM TO CALCULATE THE SHEAR STRESS AND THE PRESSURE DROP AS A FUNCTION OF VELOCITY C THIS PROGRAM CALCULATES THE REMOVAL SHEAR STRESSES AND C PRESSURE DROP ACROSS THE HEAT EXCHANGER UNDER NO STEAM CONDITION C FOR DIFFERENT FLOW RATES PF WATER REAL NREC,K,KD INTEGER RUN C ROT IS THE ROTAMETER READING, F IS THE VOLUMETRIC FLOW RATE IN C m**3/s, INPUT TEMPERATURES ARE IN DEGREE C. DELHE IS THE C MANOMETRIC HEIGHT DIFFERENTIAL IN cm. V IS THE VELOCITY IN m/s ROT=0.0 10 ROT=ROT+5.0 TC=20.0 TC=TC+273.16 F=(0.79032*ROT+0.63147) /lOOOOO. V»4.0*F/(3.1415*((38.1/1000.) **2-(19.1/1000.)**2)) RI«=(19.1/2.)/1000. RO=(38.1/2.)/1000. DE=2.*(RO-RI) C FIND LIQUID DENSITY (P) AND VISCOSITY (U) AS FUNCTION OF C TEMPERATURE. UNITS OF DENSITY AND VISCOSITY ARE k.g/cm**3,Nm/s**2 PC=1000./1.002 UC-(65816.34444-418.2856*TC+0.67255*TC**2)/lOOOOOO. C FIND BULK & FILM REYNOLDS NUMBERS & SCHMIDT NUMBER (SC) NREC-V*DE*PC/UC C CALCULATE DELPE, THE EXPERIMENTAL PRESSURE DIFFERENCE IN N/m**2 C CALCULATE DELPT, THE THERETICAL PRESSURE DIFFERENCE IN N/m**2 C FR DENOTES THE FRICTION FACTOR FR=.0791/(NREC**.25) DELPT-4.*0.6*0.5*PC*V**2*FR/DE C CALCULATE TRZ, SHEAR STRESSES AT THE SCALE LIQUID INTERFACE C UNITS ARE N/m**2 TO=DELPT*RO/(2.*0.6) K=RI/RO TRZ-TO* (K- (1 .-K**2) / (2 .*K*LOG (1. /K) ) ) WRITE(6,100) ROT,V,NREC,DELPT,TRZ 100 FORMAT ('ROT - ',F5.1,//,'VEL = ',F6.4,//,'NREC = ',£10.3, *//,'DELPT = ',E10.3,//,'TRZ = ',E10.3,////) IF (ROT.LE.140.) GOTO 10 GOTO 200 200 STOP END APPENDIX IV FOULING CURVES The remainder of the fouling curves are shown in Figures IV- l to IV-29. The outputs of the computer programs III-l and 111-2 for Run 2 are listed in Figures IV-30 and IV-31. The actual data points, temperatures etc. for each run are bound in a supplemental volume, available from Dr. A.P. Watkinson. 0.24 0.6 0.5 H 0.4 g 0-3 1 t2 0-2 -m ' I " • " I " " ' I " " ' I " • " I • " " I " • • • I " " • I " ' • ' I " pH'7.5 V»0.47 m/s Re'9195 Q«10 .9kW Tca'90 mg/L Tpo4-92 mg/L T,'38 C • rnglXI [p " " ^ ^ ^ 0 10 20 30 40 50 60 70 80 90 100 110 Time ( h ) I I I I { I I I » 1 I T—I—r I I I I I I I I I I I I pH«7.5 V'0.93 m/s Re* 18,792 Q'12.3 kW Tca*90 mg/L Tpo4*92 mg/L T/23 C • • -r—T-50 10 I I I I I 20 ' ' I 30 T—r T -40 Time ( h ) I I I I I I " " I " " I " " I " " I " " I " " I " " I I I I I pH-7.5 V'0.19m/s Re»3612 Q'6.5 kW Tca*90 mg/L Tpo4«92 mg/L T,'62 C a • • • ° • • a • • ° • • • • • - fOi I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I 0 5 10 15 20 25 30 35 40 45 50 55 Time ( h ) 0.14-0.12-0.10 S 0.08 0.06-4> 3 m "m ^ 0.04 î P-, 0.02-0.00--0.02 -1 • 1 1 1 1 1 1 1 1 1 1 1 pH«7.0 V'0.93 m/s • Re« 19,777 Q'I3.9kW Tca*90 mg/L Tpo4*92 mg/L -T,«44C • • • • CP • • 10 15 Time ( h ) 20 25 30 0.20 0.18-0.16-0.14-0.12-0.10-o .2 0.08 3 0.06 o 0.04-0.02-0.00--0.02 ' 1 ' 1 1 1 1 1 1 1 1 1 1 1 1 1 1 pH«7.0 V'0.47 m/s Re*10,120 Q»9.6 kW Tca*90 mg/L • Tj,o4*92 mg/L — T,«58 C -X • • • -10 15 20 25 Time ( h ) 30 35 40 45 I ' ' ' ' I ' • ' • I • • ' • I ' • • • I • • ' ' I • ' ' • I ' pH«8.0 V'0.19m/s Re'3884 Q«6.1 kW Tca«90 mg/L Tpo4'92 mg/L T/69 C • I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I r I I I I I 10 20 30 40 50 60 70 Time ( h ) pH«8.0 V'0.93 m/s Re'19.968 Q'9.1 kW Tca*90 mg/L Tpo4'92 mg/L T,'28 C Time (h ) 0.16 0.14 -0.12 -^ 0.10-I ' ' " I " " I " " I " " I " ' ' I pH'7.3 V«0.47 m/s Re* 10204 m/s Q«8.7 kW Tca*90 mg/L Tpo4«92 mg/1 T/60 C " ' 'I o 0.08 m .S 0.06 1 0.04-0.02-• 0.00 r Q i I I I I I I I I I I I I I I I 1 I I I I I I I I I I I I I I I I I I I I I I I 10 15 20 25 30 35 Time (h ) 40 45 50 55 ' ' • I ' pH»6.5 V«0.47 m/s Re«9953 Q'8.8 kW Tca*90 mg/L Tpo4*92 mg/L T/49 C T — r T — r T — r T — I — r • • • • • • • • • ° ° • • • • • I g p p I—r ' ' I ' 15 ' I • ' 20 10 25 Time ( h ) • 0.14 H 0.12 H 0.10 H 1 1 1 1 1 1 1 1 1 1 1 1 1 1 pH»6.8 V»0.47 m/s Re'9856 Q'9.0 kW Tca'90 mg/L Tpo4*90mg/L T/48 C I ' 1 1 1 1 1 1 1 n 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I 0.08 H 0.06 H <u .S g 0.04 o 0.02 H 0.00 H • • • -0.02 I I I I I I I I I I M I 10 - r r r T 1 1 1 1 1 1 1 1 1 1 1 15 20 Time ( h ) 25 30 - r r r 1—f 0.14-0.12-0.10-0.08 -0.06-pH»7.0 V«0.93 m/s Re* 19,807 Q'12.7 kW Tca*90 mg/L Tpo4'92 mg/L T/30 C 0.04 H 0.02 H 0.00 H -0.02 H -0.04 H -0.06 H -0.08 T 1 1 r • — r 10 15 Time ( h ) T T T 0.8 0.7 0.6 pH-7.5 V«0.93 m/s Re« 19,946 Q'I l .OkW Tca'90 mg/L Tpo4*92 mg/L T,-43 C 0.5 • • i y .S -^^  3 o 0.4 H 0.2-^ o . H 0.0 H Time ( h ) FIGURE IV-I4 RUN 16 FOULING RESISTANCE VERSUS TIME 1.0 - . . . , . • • • 1 ' ' ' ' 1 1 1 1—I—1—1—1—1—1—I—1—i-i 0.9 -pH'7.5 V»0.93 m/s Re-19,467 -0.8 - Q'12.5 kW Tc3*90 mg/L -0.7- Tpo4»92 mg/L T/21 mg/L --"s r 0-5-o 1 0.4-G i 0 .3 - • • • • • D - _ _ _ ^ fi^ ° 0.2- -0.1 -• -0.0- • -• ' • 1 • ' ' ' 1 ' ' ' ' 1 • • • I 1 1 1 1 1 1 1 1 0 5 10 15 20 25 Time ( h ) 0.12 T 1 r T 1 1 1 1 r 0.10 H 0.08 H pH-7.0 vO.47 m/s Re'4028 Q«5.7 kW Tca*90 mg/L Tpo4'92 mg/L T/71 C 0.06 H • • 0.04 H 0.02 H 0.00 Ht • -0.02 T 20 - r 40 60 80 Time ( h ) 0.20 T — 1 — I — I — r T 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 r pH«6.8 V'0.47 m/s Re'9213 Q'9.2 kW Tca-90 mgn. Tpo4'92 mg/L T,»52 C 1 I I I I — I — I — I — I — I — I — 1 — I — I — I — I — I — I — I — I — I — 1 — H 0 5 10 15 20 Time ( h ) pH»7.5 V-0.47 m/s Re*10783 Q'9.7 kW Tc3'90 mg/L Tpo4*92 mg/L T/89 C 0.40 T T T ' ' I ' i I — r 0.35 0.30 H 0.25 H 0.20 H 0^ 0.15 3 O tu 0.10 H 0.05 H 0.00 pH«7.5 V«0.47 m/s Re»9509 Q»10.4 kW Tca'90 mg/L Tpo4'92 mg/L T/50 C Time ( h ) 0.4 H 0.3 H pH'7.00 V*0.47 m/s Re« 17,489 Q'9.6 kW Tca*90 mg/L Tpo4*138mgA. T/56 C o •2 0.2 bû G 1 o . H 0.0 H • [ŒU-QQ- OLto UULU LTGn T 5 10 -7—T—i—i—r—|—r 15 20 25 Time ( h ) 0.20 0.15 H "s o I o 0.10 H 0.05 H 0.00 • • • 1 . . . . . . . . 1 1 1 1 1 1 1 1 1 -pH'7.00 • V-0.47 m/s • Re«17,040 -• Q-9.9 kW -Tca'90 mg/L — _ Tpo4* 184 mg/L - T,'56 C -i: • ^ • ^ — * - • • r j / n 1 10 Time ( h ) 15 20 T — I — I — I — r 0.3 H 0.2 o I ° 0.1 til 0.0 H pH«7.00 vO.47 m/s Re» 10,954 Q'10.2 kW Tea'135 mg/L Tpo4*92 mg/L T,-57 C T 1 r 1 5 T 1 1 1 — 10 Time ( h ) T r 15 20 I 0.6 pH'7.0 V«0.47 m/s Re»10,367 Q*9.5 kW Tca«225 mg/L Tpo4*92 mg/L T/55 C 0.4 H <L> I •Ë o fa • • ID 0.2 H 0.0 5 10 15 Time ( h ) ' • ' I 20 25 0.5 ' • ' I ' pH«7.0 vO.47 m/s Re* 10,307 Q*9.5 kW Tca*90 mg/L Tpo4*92 mg/L T/54 C •D • • • • n • ÛCULuL-l LP -T — I — r I ' ' 10 I ' ' 15 Time ( h ) 20 ' I ' 25 0.30-0.25 -0.20-o c 0.15 -v G 1 0.10-0.05 -0.00 pH'7.5 V»0.47 m/s Re* 10,290 Q'10.0 kW Tca'lOO mg/L Tpo4*103 mg/L T,.63 C Time ( h ) 0.12 H 0.10 H 0.08 H 0.06 <U Pi S o 0.04 H 0.02 H I I I I I I I I I pH'7.0 V-0.47 m/s Re» 10,415 Q ' I l . S k W Tca'lOO mg/L Tpo4«103 mg/L T/60 C I ' ' ' ' I ' ' ' ' I 0 .00 -9 0 ' ' I ' 5 ' I ' ' ' ' I ' I • • • ' I ' • ' ' I 10 15 20 25 Time ( h ) 30 • I ' ' ' • 35 40 0.10 0.09-0.08-0.07-0.06-0.05 0.04--0.01 H -0.02 -I ' • ' • I ' I • pH'7.0 V«0.47 m/s Re« 10,748 Q-11.2kW Tca'93 mg/L Tpo4*96 mg/L T «72 C • • • • - I — I — 1 — • I 10 15 20 Time ( h ) T — r '55 <u Oi a 1 0.36-3 0.34-: 0.32-i 0.30^ 0.28 4 0.26 -E 0.24-Ê 0.22 i 0.20 0.18 0.16-i 0.14-È 0.12-^ 0.10 I 0.08 I 0.06-j 0.04 -Ê 0.02 I 0.00-D -0.02 i _ pH«7.0 V»0.47 m/s Re»10,457 Q'9.4 kW Tca»400 mg/L Tpo4'103 mg/L 10 15 Time ( h ) pH-8.0 V«0.47 m/s Re* 10,627 Q'10.0 kW Tea* 100 mg/L Tpo4'103 mg/L T «61 C Time (h ) DT TC TH LMTD T IME Q UOD RFC RFU TS TWALL HI 5 . 6 1 9 . 6 107 . ,0 87 .4 1 .0 9 .38 1 . 2 7 8 0 . 0 0 0 0 0 . 0 0 0 50 .2 5 0 . 2 4 . 4 7 7 5 . 5 20 .1 107 . ,4 87 . 3 2 . 0 9 .21 1 . 2 5 6 0 . 0 0 8 5 0 . 0 1 3 50 .9 5 1 . 9 4 . 5 0 9 5 . 4 20 .1 108 . ,3 88 .1 3 . 0 9. .05 1 . 222 0 . 0256 0 . 0 3 6 51 .2 5 4 . 0 4 . 5 4 8 5 . 4 20 .1 109 . ,1 89 . 0 4 . 0 9, .05 1 . 2 1 0 0 . 0 3 4 0 0 . 0 4 4 51 .3 5 5 . 0 4 . 5 5 9 5 . 4 2 0 . 0 109 . ,8 89 .7 5 . 0 9 .05 1 . 2 0 0 0 . 0 4 0 5 0 . 0 5 0 51 . 3 5 5 . 6 4 . 5 6 7 5 . 4 2 0 . 0 110 . .4 90 .4 6 . 0 9 .05 1 .191 0 . 0 4 7 0 0 . 0 5 7 51 .3 5 6 . 4 4 . 5 7 5 5 . 4 2 0 . 0 1 1 1 . ,1 91 .0 7 . 0 9, .05 1 . 1 8 3 0 . 0 5 2 5 0 . 0 6 3 51 .4 5 7 . 1 4 . 5 8 3 5 . 2 2 0 . 0 1 1 1 . ,7 91 .7 8 . 0 8, .71 1 .131 0 .0811 0 . 1 0 1 51 .7 6 0 . 2 4 . 6 4 9 5 . 4 20 .1 112 . ,3 92 .2 9 . 0 9, .05 1 . 1 6 8 0 . 0 6 3 7 0 . 0 7 4 51 .6 5 8 . 5 4 . 5 9 9 5 . 3 2 0 . 0 112 . .8 92 .7 1 0 . 0 8, .88 1 .141 0 . 0 7 9 0 0 . 0 9 4 51 . 7 6 0 . 1 4 . 6 3 3 5 . 3 2 0 . 0 113 . ,4 93 .4 1 1 . 0 8, .88 1 . 132 0 . 0857 0 . 1 0 1 51 .8 6 0 . 9 4 .641 5 . 5 2 0 . 1 1 1 3 . ,8 93 .8 1 2 . 0 9. .21 1 . 1 7 0 0 . 0 6 7 3 0 . 0 7 2 51 . 5 5 8 . 9 4 . 5 8 9 5 . 6 2 0 . 2 114 . 1 93 .9 1 3 . 0 9. .38 1 . 1 8 9 0 . 0 5 8 3 0 . 0 5 8 51 .5 5 8 . 0 4 . 5 6 5 5 . 3 1 9 . 9 114 . ,3 94 .4 1 4 . 0 8. .88 1 . 1 2 0 0 .0951 0 . 1 1 0 51 .8 6 1 . 8 4 . 652 5 . 6 2 0 . 0 114 . ,8 94 .7 1 5 . 0 9. .38 1 . 1 7 9 0 . 0654 0 . 0 6 5 51 .4 5 8 . 7 4 . 5 7 2 5 . 3 1 9 . 8 1 1 5 . ,0 95 .1 1 6 . 0 8. .88 1 .111 0 . 1 0 2 2 0 . 1 1 7 51 .7 6 2 . 5 4 . 6 5 9 5 . 5 1 9 . 9 1 1 5 . ,3 95 .4 1 7 . 0 9, .22 1 . 1 5 0 0 . 0 8 1 8 0 . 0 8 7 51 . 5 6 0 . 5 4 . 6 0 6 5 . 5 1 9 . 9 1 1 5 . ,6 95 .6 1 8 . 0 9. .22 1 . 1 4 8 0 . 0837 0 . 0 8 9 51 .6 6 0 . 7 4 . 6 0 9 5 . 4 2 0 . 1 116 . 1 96 .0 1 9 . 0 9, .05 1 . 122 0 . 0 9 8 5 0 . 1 0 9 51 .9 6 2 . 5 4 . 6 4 3 5 . 4 2 0 . 1 116 . 4 96 .3 2 0 . 0 9. .05 1 . 1 1 9 0 . 1 0 1 3 0 . 1 1 1 51 .9 6 2 . 9 4 . 6 4 7 5 . 3 2 0 . 0 116 . 6 96 .5 2 1 . 0 8, .88 1 . 0 9 5 0 . 1 1 5 5 0 . 1 3 1 52 .1 6 4 . 3 4 . 6 7 8 5 . 2 2 0 , 0 116 . 6 96 .6 2 2 . 0 8, .71 1 . 0 7 3 0 . 1 2 8 8 0 . 1 4 9 52 .1 6 5 . 5 4 . 7 0 8 5 . 4 2 0 . 2 1 1 7 , ,1 96 .9 2 3 , 0 9. .05 1 . 1 1 2 0 . 1 0 6 9 0 . 1 1 7 52 .1 6 3 . 6 4 . 6 5 5 5 . 4 2 0 . 1 1 1 7 . 3 97 .1 2 4 . 0 9. .05 1 . 1 0 9 0 . 1092 0 . 1 1 9 52 . 0 6 3 . 8 4 . 6 5 6 5 . 2 1 9 . 9 1 1 7 , ,3 97 .4 2 5 . 0 8. .71 1 . 0 6 5 0 . 1 3 6 5 0 . 1 5 7 52 .1 6 6 . 2 4 . 7 1 6 5 . 5 2 0 . 1 1 1 7 . 6 97 .5 2 6 . 0 9. .21 1 . 1 2 5 0 . 1 0 1 0 0 . 1 0 6 51 .8 6 2 . 9 4 . 631 5 . 4 2 0 . 1 1 1 7 , 8 97 .6 2 7 , 0 9. .05 1 . 1 0 3 0 . 1 1 3 9 0 . 1 2 4 52 .1 6 4 , 3 4 . 6 6 2 5 . 4 1 9 . 9 1 1 7 , 8 97 .9 2 8 . 0 9. .05 1 . 1 0 0 0 . 1 1 6 6 0 . 1 2 7 51 . 9 6 4 . 4 4 . 6 6 3 SDTW SDQA SDTS 0 . 0 2 1 6 0 . 0 3 4 9 0 . 0 7 6 3 T C ( a v g ) T H ( a v g . ) Q ( a v g . ) T S ( a v g . ) 2 0 . 0 1 1 3 . 8 9 . 0 5 1 . 6 NREF= 1 6 3 9 2 . 2 NREC= 8 9 5 0 . 8 Figure IV-30 Output of Computer Program III-l for Run 2 K l K2 K3 K4 . 8 3 3 E - 0 2 . 9 9 6 E - 0 7 . 1 2 0 E - 1 1 K5 K6 K7 KW .174E+02 .257E+03 .104E+07 H OH CA . 3 4 2 E - 0 7 . 2 8 2 E - 0 6 . 7 5 6 E - 0 3 CAOH CAH2P0 CAHP04 CAP04 . 3 3 6 E - 0 8 . 2 7 8 E - 0 5 . 1 2 0 E - 0 3 . 1 7 0 E - 0 4 H3P04 H2P04 I I I FM FD DDCPD DHAP SDCPD SHAP SPOS = . 3 3 1 E - 0 2 PH = 7 . 5 HP04 P04 . 8 6 7 E - 0 9 . 2 1 1 E - 0 3 . 6 1 6 E - 0 3 . 2 1 6 E - 0 7 Z . 4 9 4 1 7 3 5 4 E - 0 2 . 4 9 4 1 7 6 8 9 E - 0 2 4 . 0 FT . 9257 . 7 3 4 4 . 4 9 9 3 DTCP DOCP 0 .96E+00 0 .96E+11 0 .17E+04 0 . 3 0 E + 0 3 STCP SOCP - 0 . 2 1 E - 0 7 0 . 7 0 E - 4 5 0 . 2 0 E - 2 4 O . l l E - 4 2 SNEG •= . 2 7 7 E - 0 2 TCA = 0 . 8 9 E - 0 3 TP04 = 0 . 9 7 E - 0 3 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0058511/manifest

Comment

Related Items