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Dispersion in three dimensional electrodes Gao, Lixin 2001

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DISPERSION IN THREE DIMENSIONAL ELECTRODES by Lixin Gao .A.Sc (Chemical Engineering), Harbin Engineering University, Harbin, China, 1991 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF M A S T E R OF APPLIED SCIENCE in THE F A C U L T Y OF G R A D U A T E STUDIES DEPARTMENT OF C H E M I C A L ENGINEERING We accept this as confirming to the reajjjred standard THE UNIVERSITY OF BRITISH C O L U M B I A December, 2000 © Lixin Gao, 2000 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 written permission. The University of British Columbia Vancouver, Canada Department DE-6 (2/88) Abstract ABSTRACT Dispersion of mass is a measure of the deviation of transportation of fluid in a reactor from ideal reactor behavior (perfect mixing or plug flow) caused by the combined effects of diffusion, convection and migration. Axial dispersion is always undesirable because it reduces the driving force of the reaction and therefore causes a lower level of conversion. On the other hand, transverse dispersion is often a desirable feature since good transverse mixing will reduce the transverse concentration and temperature gradients and hence improve the selectivity of a thermochemical reactor. Transverse dispersion of mass is of more importance in a three-dimensional flow-by electrochemical reactor than that in a thermochemical reactor because the potential drop is in the transverse direction and the reaction rate and selectivity are determined by the potential as well as concentration and temperature distributions. The transverse dispersion of mass is expected to have a more profound effect on the performance of a 3D electrochemical reactor due to the strong interaction among the concentration, temperature and potential distributions in the transverse direction. In the present work, the axial and transverse dispersion of mass were studied with a two-dimensional dispersion model in two types of rectangular packed bed: i) randomly packed glass beads with the average bead diameter of 2 mm and a macroscopic bed porosity of 0.41; ii) a representation of a 3D flow-by electrode - consisting of a bed of carbon felt with the carbon fibre diameter of 20 urn and a macroscopic bed porosity of 0.95. A tracer stimulation-response system was set up and axial and transverse dispersion of Abstract 0.7M CuS0 4 in a flow of 12 wt % Na 2S04 were measured in a 32cm long by 5cm wide by 2.6cm thick rectangular tested bed filled with glass beads and with carbon felt, for Reynolds number ranging respectively from 1.8 to 7.2, and from 0.008 to 0.032. Axial and transverse dispersion coefficients D a and D t were found by parameter estimation based on a pulse tracer experiment. D a and D t were selected such that they gave the least sum of squares of the differences between the measured and calculated tracer concentrations. The latter were calculated by employing a computer program written in FemLab and MatLab to solve the two-dimensional time-dependent partial differential material balance equation governing the tracer concentration distribution within the tested bed, assuming that the transverse and lateral dispersion coefficients were of the same magnitude and the tracer concentration gradients were equal in the two directions, except near the walls of the bed. dc ^ ^ d2c d2c dc — = 2Dt — +Da — -u — dt ck dy dy Traditionally, a two step experimental technique has been employed to find axial and transverse dispersion coefficients in packed beds: the axial dispersion coefficient D a is first estimated from a pulse tracer experiment with the assumption that there is no concentration gradient in the transverse direction; then the transverse dispersion coefficient D t is calculated from a step tracer measurement with the previously calculated D a . Two improvements were achieved in the present work by finding axial and transverse dispersion coefficients simultaneously from one single set of pulse tracer experiment. First, the potential of the systematic error introduced by assuming no transverse concentration gradient for calculating D a was eliminated. Second, the accuracy of the parameter estimation of D a and D t in Abstract was improved by the greater number of tracer sampling points obtained from a pulse tracer experiment technique than have been obtained from step tracer measurements. Simultaneously estimated axial and transverse dispersion coefficients and other parameters are summarized in Table 1, along with a comparison to literature values [3, 9]. Table 1: Axial and transverse dispersion parameters estimated by simultaneous solution: (The maximum standard deviation of the parameter was 50%.) Parameters Glass beads Type of bed Carbon felt Liquid flow rate (ml min ') 30 - 120 30 - 120 Superficial velocity (m s" ') 3.8- 15.4 3.8-15.4 Re* 1.8-7.2 0.008 - 0.032 ' D a O O ^ m V ) 6-21 1 -6 Pea 0.35 0.007 P^a, literature 0.4 not available Das/udp 1.3 160 D as/ud p > literature 1.5 not available D t C l O ^ m V ) 0.5-7.6 2 - 9 Pe t 7-1 0.004 Pet, literature - 3 0 - - 1 0 not available Re* was based on diameter of beads or fibres. Axial dispersion parameters were also estimated from the variances of two tracer concentration curves measured at two points, which were at the same horizontal but different vertical positions within the tested bed, assuming that the lateral and transverse dispersion effects were negligible. The results are summarized in Table 2: - iv -Abstract Table 2: Axial dispersion parameters estimated by neglecting transverse dispersion: (the maximum standard deviation of the parameters was 20%) Type of bed Liquid flow rate Re D a Pe a Das/udp (ml min"1) (10 -«m 2 s"1) glass beads 30- 120 1.8-7.2 6-24 0.30 1.3 carbon felt 30- 120 0.008-0.032 1.5-6.8 0.006 160 It seems that the axial dispersion coefficient was slightly overestimated when the effects of transverse and lateral dispersion were neglected. The axial and transverse dispersion coefficients were not affected by repacking of the beds and the axial dispersion coefficient was uniform throughout the entire bed. The axial dispersion parameters for glass beads bed agree with the literature values. The transverse Peclet number of the glass beads bed was only one-fourth that of the literature values and this may be caused by the different reactor configuration (rectangular) of the bed tested in the present work compared with that of the previous investigations (cylindrical). No comparison data were found in the literature for dispersion parameters in a packed bed with similar characteristics to carbon felt in terms of bed porosity and bed material dimension. TABLE OF CONTENTS ABSTRACT TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES NOMENCLATURE ACKNOWLEDGEMENT Chapter 1: INTRODUCTION Chapter 2: BACKGROUD AND LITERATURE REVIEW 2.1 Flow Patterns in Packed-Bed Chemical Reactors 2.1.1 Ideal and real flow 2.1.2 Real flow modelling 2.1.2.1 Tanks-in-series model 2.1.2.2 Dispersion model 2.2 General Mathematical Treatment for the Dispersion Model 2.2.1 Axial dispersion 2.2.2. Transverse dispersion 2.3 Special Considerations for Electrochemical Reactors 2.3.1 Electrochemical vs. thermochemical reactors 2.3.2 Three-dimensional electrodes 2.4 Literature Review - v i -Table of Contents 2.4.1 Dispersion in thermochemical reactors 35 2.4.1.1 Axial dispersion 38 2.4.1.2 Transverse dispersion 43 2.4.2 Dispersion in electrochemical reactors 46 Chapter 3: OBJECTIVE OF THE PRESENT WORK 49 Chapter 4: EXPERIMENTAL APPARATUS AND PROCEDURES 50 4.1 Experimental Apparatus 50 4.2 Experimental Procedures 59 4.2.1 Scope of the work 59 4.2.2 Residence time distribution measurement 60 4.2.2.1. Preparation and calibration of copper wires 60 4.2.2.2. Pre-treatment of carbon felt 61 4.2.2.3. Packing of the test section 63 4.2.2.4. In-situ copper wire potential measurements 63 4.2.3 Data analysis procedure 64 4.2.3.1 Calculation of C u 2 + concentration 64 4.2.3.2 Preliminary calculation of axial dispersion coefficient D a 64 4.2.3.3 Methodology for simultaneous calculation of axial and transverse dispersion coefficients D a and D t 65 Chapter 5: RESULTS AND DISCUSSIONS 71 5.1 Setting up the Tracer Stimulation-Response Measuring System 72 5.1.1 Effect of fluid distributor 72 - vii -Table of Contents 5.1.2 Tracer and main fluid selection 74 5.1.2.1 Tracer selection 74 5.1.2.2 Main fluid selection 74 5.1.3 Calibration of copper wire potential vs. [Cu2 +] 75 5.1.3.1 Effect of flow rate on copper wire potential 75 5.1.3.2 Effect of different main fluid on copper wire potential 75 5.1.3.3 Effect of the copper wire surface condition on copper wire potential 79 5.2 Axial Dispersion Coefficient Estimated by Neglecting Transverse Dispersion 86 5.2.1 Randomly packed glass beads bed 87 5.2.1.1 Effect of flow rate on axial dispersion 87 5.2.1.2 Uniformity of D a in the entire bed 89 5.2.1.3 Effect of re-packing of glass beads bed on axial dispersion coefficient 89 5.2.2 Carbon felt bed 92 5.2.3. Comparison of axial dispersion in glass beads and carbon felt beds 95 5.3 Simultaneous Calculation of Axial and Transverse Dispersion Coefficients 98 5.3.1 Determination of the upper and lower boundary of the rectangular regime 98 - viii -Table of Contents 5.3.2 Determination of axial and transverse dispersion coefficients D a and D t 98 5.3.3 Samples of measured and calculated tracer concentration curves 104 5.3.4 Axial and transverse dispersion coefficients D a and D t for glass beads and carbon felt bed 107 5.3.4.1 Glass beads bed 107 5.3.4.2Carbonfelt bed 111 5.4 Comparison of Axial Dispersion with and without Considering Transverse Dispersion 115 5.5 Effect of Neglecting Axial Dispersion Coefficient on Transverse Dispersion Coefficient Estimation 115 Chapter 6: CONCLUSIONS AND RECOMMENDATIONS 117 6.1 Conclusions 117 6.2 Recommendations 118 References 120 Appendices 124 1. Fluid Distributor Calculation 124 2. Slope of the Copper Wire Potential vs. log [Cu2 +] Calibration Curve Calculated from Nernst Equation 128 3. CuS0 4 and N a 2 S 0 4 Aqueous Solution Density Data 129 4. Sample Calculation of Axial Dispersion Coefficient by Using One-Shot Technique, Neglecting Lateral and Transverse Concentration Gradient 131 - ix -5. Sample Calculation of Flow Velocity, Re, D a/ud p, D as/ udp, and Pe a 6. A Computer Program for Solving Axial and Transverse Dispersion Coefficients Simultaneously List of Tables LIST OF TABLES Axial and transverse dispersion parameters estimated by simultaneous solutions iv Axial dispersion parameters estimated by neglecting transverse dispersion v Properties of glass beads and carbon felt beds 59 D a /uL, Pea, Das/ udp.and D a at different flow rates in glass beads bed 87 D a /uL, Pea, Das/ udf; and D a at different flow rates in carbon felt bed 92 Sum of squares of differences between measured and calculated tracer concentrations for wire 1 and 8 for D a and Dti ranging from 10"3-25 to lO'Ws"1 99 Table 5.4 Sum of squares of differences between measured and calculated tracer concentrations for wire 1 and 8 with D a ranging from 2-22 x 10"6 and Dti ranging from 4-24x 10"6 m 2 s"1 100 Table 1 Table 2 Table 4.1 Table 5.1 Table 5.2 Table 5.3 - xi -List of Figures LIST OF FIGURES Figure 2.1 Properties of E and F curves for various flows 10 Figure 2.2 E curves for misbehaving plug flow reactors 11 Figure 2.3 E and F curves for the tanks-in-series model 13 Figure 2.4 Illustrative flow velocity profile for dispersion model 14 Figure 2.5 Mass balance for material A over a differential volume in a packed bed. Similar expressions can be made for y- and z-directions as for x-direction 16 Figure 2.6 Spreading of pulse tracer due to axial dispersion 18 Figure 2.7 Tracer response for closed vessels with large deviations from plug flow 22 Figure 2.8 Tracer response for open vessels with large deviations from plug flow 23 Figure 2.9 Step tracer response curves in open vessels 24 Figure 2.10 Configuration of three-dimensional electrodes 32 Figure 2.11 Axial Peclet number Pe a for liquids in random beds of sphere 42 Figure 2.12 Experimental findings on dispersion of fluids flowing with mean axial velocity u in packed beds 42 Figure 2.13 Radial Peclet numbers Per for gases and liquids in random beds of spheres 45 Figure 2.14 Variation of the fluid-mechanical Peclet group for radial dispersion with Reynolds number 45 - x i i -List of Figures Figure 4.1 A photograph of the experimental apparatus 51 Figure 4.2 Experimental arrangements for measuring axial and transverse dispersion 52 Figure 4.3 Details of the test section 55 Figure 4.4 Details of the tracer injection syringe 56 Figure 4.5 Details of the wires in one probe 57 Figure 4.6 Measurement of copper wire potential outside the test section with a salt bridge 62 Figure 4.7 Algorithm of the program for solving D a and D t simultaneously 69 Figure 4.8 Rectangular regime for solving Equation (28) 70 Figure 5.1 Comparison of fluid patterns in the test section filled with glass beads with and without fluid distributor: wire potential vs. time with 0.7M CuS0 4 pulse injection 73 Figure 5.2 Effect of flow rate on copper wire potential 77 Figure 5.3 Relationships of copper wire potential and copper ion concentration in three main fluids 78 Figure 5.4 Relationships of wire potential and copper ion concentration for active and "deactivated" wires 80 Figure 5.5 Relationships of wire potential and copper ion concentration within and without carbon felt. (In 12 wt % Na2S0 4, pH = 2 solutions) 82 Figure 5.6 Wire potential in 12 wt % N a 2 S 0 4 with 1M [Cu 2 +], pH = 2, - xiii -List of Figures within a carbon felt in the beaker 83 Figure 5.7 Slopes of calibration curves for all the wires in several runs 85 Figure 5.8 Relationship between D as/ud p, Pe a and Re for glass beads bed 88 Figure 5.9 D a /uL i . 7 and D a /uL 4 . 7 at different flow rates for glass beads bed 90 Figure 5.10 Comparison of Da/uL vs. Re of different packings of glass beads bed 91 Figure 5.11 Relationship between axial dispersion coefficient D a and Reynolds number Re for carbon felt bed 93 Figure 5.12 Relationship between axial Peclet number Pe a and Reynolds number Re for carbon felt bed 94 Figure 5.13 Comparison of D a for glass beads and carbon felt beds 97 Figure 5.14 Sum of squares of differences between measured and calculated tracer concentrations for wire 1 and 8 as a function of D a and Dtl 102 Figure 5.15 Sum of squares of differences between measured and calculated tracer concentrations for wire 1 and 8 with D a and D t i ranging from2-22 x 10"6 and 4-24 x IO - 6 m 2 s"1 103 Figure 5.16 Measured and calculated tracer concentrations for wire 1, 7, and 8. (glass beads bed, 90 ml min"1, D a = 14 x 10"6 and D t = 4 x 10"6 m 2 s"1) 105 Figure 5.17 Measured and calculated tracer concentrations for wire 1, 7, and 8. (carbon felt bed, 30 ml min"1, D a = 1 x 10"6 and D t = 2 x 10"6 m 2 s"1) 106 Figure 5.18 Axial and transverse dispersion coefficients at different flow rates for glass beads bed 109 Figure 5.19 Axial and transverse dispersion Peclet numbers at different flow - xiv -List of Figures rates for glass beads bed Figure 5.20 Axial and transverse dispersion coefficients at different flow rates for carbon felt bed Figure 5.21 Axial and transverse dispersion Peclet numbers at different flow rates for glass beads bed Figure 5.22 Comparison of axial dispersion coefficient calculated with and without considering lateral and transverse dispersion effect for glass beads and carbon felt bed Figure A-1 Density of CuS04 and N a 2 S 0 4 solution at 20 °C Figure A-2 Copper wire potential as a function of time. (Run 04282a: glass beads bed, 0.5ml 0.7M CuS0 4 with 0.01 wt % Trypan blue, flow rate = 30 ml/min.) Figure A-3 Copper concentration as a function of time 110 113 114 116 130 133 134 - xv -Nomenclature N O M E N C L A T U R E a Specific surface area of 3D electrode {m2 m"3} c Copper ion concentration {kmol m"3} c e Experimental measured tracer concentration {kmol m"3} Ct Calculated tracer concentration {kmol m"3} CRS Concentration of reduced species at the electrode surface {kmol m"3} cos Concentration of oxidised species at the electrode surface {kmol m" } D m Molecular diffusion coefficient {m2 s"1} D a Axial dispersion coefficient {m2 s"1} D r Radial dispersion coefficient {m2 s"1} Di Lateral dispersion coefficient {m2 s"1} D t Transverse dispersion coefficient {m 2 s"1} Dti Sum of transverse and lateral dispersion coefficients (= Di + D t) {m2 s"1} D x Dispersion coefficient in the x-direction {m2 s"1} D y Dispersion coefficient in the ^-direction {m2 s"1} D z Dispersion coefficient in the z-direction {m2 s"1} dp Diameter of the glass beads {m} df Diameter of the carbon fibre {m} E Operating electrode potential {volt} E e Equilibrium electrode potential {volt} AEe° Standard equilibrium electrode potential {volt} A E e Equilibrium cell potential {volt} - xvi -Nomenclature F Faraday constant (=96480 kC kmoi"1} AG Free energy change (kJ) j Local Faradaic current density (kA m"2} k a Anodic reaction rate constant (m s"1} k c Cathodic reaction rate constant {ms"1} L Length of the reactor (m) m Anodic reaction order m' Cathodic reaction order N Number of tanks in tanks-in-series model n Electron stoichiometry coefficient Pe a Axial Peclet number (=udp/Da) Pet Transverse Peclet number (=udp/Dt) Q Activity quotient of reaction (=ao/aR) Rrate Reaction rate, =f (i), {kmol m"3 s"1} Re Reynolds number based upon bead or fiber diameter and interstitial velocity (= udpp/u) r Radius {m} S Sum of squares of differences between measured and calculated tracer concentration Sc Schmidt number (= p/p D m ) t Time {s} t Residence time in the vessel (s) u Interstitial fluid velocity (m s"1} x Co-ordinates - xvii -Nomenclature y Co-ordinates z Co-ordinates dx Small change in the x-direction dy Small change in the y-direction dz Small change in the z-direction a Anodic charge transfer coefficient K e Effective electrolyte conductivity (mho m"1} K m Effective electrode matrix conductivity (mho m"1} s Bed porosity p Fluid density {kg m"3} p Fluid viscosity (m 2 kg"1 s"1} V R Stoichiometric coefficient for reduced species vo Stoichiometric coefficient for oxidised species a 2 Variance {s2} o~e2 Dimensionless variance 4> Potential field (volt) - xviii -Acknowledgement A C K N O W L E D G M E N T I sincerely appreciate Professor Colin W. Oloman for his guidance, encouragement, support and patience throughout this work. Special thanks go to Professor Jeoff Kelsall for his insightful advice concerning the electrolyte and tracer selection at the early stage of the present work. My deep thanks also go to the staff of Chemical Engineering Department, the help from Helsa, Lori, Shelagh, Alex, Horace, Qi, Peter, Robert, and Chris, was essential to the completion of the present study. I want to take this opportunity to express my gratitude to my husband James J. Wang for his love, encouragement and the in depth discussions in the process of this project. - xix -Chapter 1: Introduction Chapter 1 INTRODUCTION Mixing is important in chemical reactors. Conventionally, mixing in the direction parallel to the main flow is called axial mixing, and that in directions perpendicular to the main flow is called radial mixing in a cylindrical reactor or transverse and lateral mixing in a rectangular reactor. Axial mixing usually lowers the level of conversion in a reactor due to the degradation of concentration and temperature driving forces caused by axial mixing; while, radial (transverse, lateral) mixing, often improves the selectivity in thermochemical reactors by reducing undesirable radial (transverse, lateral) temperature and concentration gradients. Dispersion of mass in the radial (transverse, lateral) direction may play a more profound role in electrochemical reactors due to the interaction among concentration, temperature, and potential distributions, which determine the selectivity of a process in such reactors. But an attempt to increase the radial (transverse, lateral) mixing may also increase the axial mixing. A good knowledge of the inter-relationship is necessary for reactor design. In the consequent treatment of this thesis, the transverse and lateral dispersion coefficients (D t and Di) were assumed equal to D t when a rectangular reactor was studied in a simplified two-dimensional model; and the term transverse dispersion coefficient D t was also used in place of radial dispersion coefficient (D r) in a cylindrical reactor in order to simplify the nomenclature of the text. The degree of mixing in packed-bed reactors depends upon the mechanisms of flow, upon the molecular diffusivity of the fluid constituents, and upon the geometry of the packing and the - 1 -Chapter 1: Introduction vessel [1]. The flow pattern in packed-bed reactors is not too greatly deviant from plug flow, which can be described by a tanks-in-series or a dispersion model. The tanks-in-series model describes the effect of backmixing by setting up a series of N ideal stirred tanks along the reactor and the effect is quantified by assigning an appropriate N to represent the residence time distribution (R.T.D.) data. The axial and transverse dispersion coefficients (D a and D t) are defined in a dispersion model to describe the degree of mixing in different directions in the reactor. The larger value of the coefficient, the larger the degree of mixing. These two models are almost identical. Mixing in two types of packed beds was studied with the dispersion model in the present work. Extensive work has been done for packed-bed thermochemical reactors (TCR's) since the 1950s, and a relatively comprehensive set of experimental axial and transverse dispersion data is available [2,3]. However, there are few such data available for electrochemical reactors (ECR's). The dispersion effect has simply been neglected in most of the electrochemical reactor modeling work [4]. Some authors used dispersion coefficients roughly estimated from the knowledge of TCR's, though these experimental data and the mathematical treatment might not be suitable to ECR's due to the difference between the two categories of reactor. Electrochemical reactions are heterogeneous reactions that take place at the interface between the electrode (an electronic conductor) and the electrolyte (an ionic conductor). There are two distinguishable features of electrochemical processes make them advantageous over thermochemical processes: the massless reagent electron and the double reaction driving forces -thermal and electrical energy. The usage of a massless reagent makes electrochemical processes Chapter 1: Introduction more environmental friendly, which is a critical issue in today's world. The application of electrical energy makes it possible for a reaction with unfavourable thermal equilibrium to be taken to high conversion at ambient temperature. For an electrolytic cell, the free energy for the overall chemical change is related to the equilibrium cell potential by the equation [5]: A G = -10"3 nFAEe where n is the electron stoichiometry coefficient, F is Faraday constant (96480 kC kmof1), and A E e is the equilibrium cell potential (Volt). Therefore, a few volts potential change will give a free energy change of hundreds of kJ mol"1. However, these advantages of electrochemical processes are balanced by the complexity of electrochemical reactions and electrochemical reactor design. In addition to the mass and energy balances which need to be taken into account as in a thermochemical reactor, a third factor, the voltage balance, must be considered simultaneously with the two balances above. The consequent strong interaction among the concentration of the electroactive species, temperature and potential distributions creates difficulties in the design of electrochemical reactors. Three-dimensional electrodes have caused great interest and have been under fast development over recent years due to the potential of high space-time yield provided by their high specific surface areas. These electrodes are particularly useful for reactions that have inherently low rate, where the high specific area can make them an economically viable proposition. As a general guide, if operating real current densities are less than approximately 300-400 Am"2 then three-dimensional electrodes should be considered as alternatives to two-dimensional systems [4]. Three-dimensional electrodes are analogous to heterogeneous catalytic thermochemical Chapter 1: Introduction systems. Transverse dispersion of mass is of more importance in electrochemical reactors than in thermochemical reactors, especially in a 3D flow-by electrode where the potential drop is also in the transverse direction. The conversion in an electrochemical reaction under kinetic control would increase from 0.56 to near 1 when the transverse dispersion coefficient increases from 10"7 to 10"5 m 2 s"1 in a 3D flow-by electrode [31]. Unfortunately, there are few data available on transverse dispersion coefficients in electrochemical reactors. One of the prime objectives of the present work is to measure the transverse dispersion in a rectangular packed-bed of carbon felt, which is a representative of 3D flow-by electrode. Two factors point to the possibility that direct application of dispersion coefficients obtained from thermochemical reactors to electrochemical reactors might be inappropriate and that specific investigation of flow patterns in three-dimensional electrodes is necessary: First, different packed-bed materials with quite different properties are used in TCR's and ECR's. For example, a typical packed-bed thermochemical reactor is packed with particles with diameters in the range of several millimetres and the macroscopic bed porosity is about 0.4. These properties are quite different from those of the electrode materials widely used in ECR's, such as carbon felt, whose macroscopic porosity is about 0.9 - 0.95, and the diameter of the carbon fibre is about 20pm. Second, TCR's and ECR's have different geometries. Due to the special current distribution, potential distribution and energy consumption concerns, parallel plate electrodes with a gap of a few millimeters are widely employed in ECR's to provide the essential potential gradient and give simple and easily controlled potential profile in the reactor. Accordingly, the Chapter 1: Introduction partial differential equation describing the dispersion phenomena in an electrochemical reactor will be in rectangular instead of the cylindrical co-ordinate system which is almost exclusively used in thermochemical reactors [2,3,6,7]. As a result, the mathematical treatments of the two categories of reactors are different. The traditional approach for finding the dispersion coefficients in the literature is that D a is first estimated from pulse tracer experiments assuming that there is no concentration gradient in the transverse direction, and D t is then solved from a stationary partial differential equation with this pre-calculated D a and the tracer concentration measurements at different transverse positions in the reactor obtained from a step tracer injection experiment [3]. This two-step procedure is employed due to the mathematical difficulty in solving D t from a pulse tracer experiment and to the fact that D a is not sensitive to step tracer measurement. However, the assumption of no transverse dispersion for calculating D a inevitably brings error into the result. This weakness was overcome in the present work by investigating D a and D t simultaneously from one single pulse tracer experiments and with the aid of newly developed computer techniques: axial and transverse dispersion coefficients were obtained simultaneously by solving a time-dependent partial differential equation. In addition to the ability to calculate axial and transverse dispersion coefficients simultaneously, the parameter estimation accuracy was improved by employing pulse instead of step tracer measurements. The number of sampling points composing the measured and calculated tracer concentration curves, which affects the accuracy of the parameter estimation, is the product of the number of tracer sensors and the number of samples taken by each sensor. The number of tracer sensors that can be mounted in the transverse direction in a reactor is - 5 -Chapter 1: Introduction constrained by the dimension of the cross section. Usually only 5-6 sensors can be mounted. Only one tracer concentration sample (the steady state tracer concentration) will be obtained by each sensor when a step tracer technique is employed; however, several tracer concentration samples can be taken by each sensor when a pulse tracer is employed (the tracer concentration can be sampled at several time intervals, not only at the steady state), e.g., hundreds of tracer samples were taken by each sensor in each run in the present work. By this means the total number of sampling points on tracer concentration curves will increase significantly and a better parameter estimate will be obtained when the pulse tracer technique is employed. In conclusion, the scope of the present work included • Developing a tracer stimulation-response system for single phase liquid flow in a rectangular packed-bed; • Setting up a test section with related tracer experimental facilities; • Measuring the tracer concentration profiles in a pulse tracer experiment at different axial and transverse positions within two types of packed-beds (glass beads and carbon felt) at different liquid flow conditions; • Developing a computer program which can be further embedded into a complete rectangular packed-bed model to calculate axial and transverse dispersion coefficients simultaneously; • Investigating the relationship between dispersion parameters and liquid flow rate. Chapter 2 Background and Literature Review Chapter 2 BACKGROUND and LITERATURE REVIEW 2.1 Flow Patterns in Packed-Bed Chemical Reactors 2.1.1 Ideal and Real Flow When a fluid flows through a vessel at a constant rate, one of two ideal flows is commonly assumed for the purpose of calculation. (i) Perfectly mixed flow: the fluid in the vessel is completely mixed so that its properties are uniform and identical with those of the outgoing stream. This assumption is frequently made for calculations on stirred reactors or blenders. (ii) Plug-flow: elements of fluid which enter the vessel at the same moment move through it with constant and equal velocity on parallel paths, and leave at the same moment. This type of behavior is normally assumed when considering flow through heat exchangers, catalytic reactors, packed-beds, etc. However, flow in real equipment always deviates from these ideals. For example, even with the best designed fluid distributor and flow channel, perfect plug flow will never be obtained with Newtonian fluids. There will always be some longitudinal mixing due to viscous effect and molecular or eddy-diffusion [8]. Elements of fluid taking different routes through the reactor may take different lengths of time to pass through the vessel and such differences determine the conversion and product distribution of the process. The distribution of the residence times for the stream of fluid leaving Chapter 2 Background and Literature the vessel is called the exit age distribution E, or the residence time distribution R.T.D. of fluid [9]. By definition, So the fraction of exit stream of age between / and t + dt is E dt, and the fraction younger Measurement of the R.T.D. is carried out by tracer techniques, whereby an inert substance, with some measurable physical property, is introduced into the fluid at the vessel inlet. Elements of fluid then pick up this tracer and follow associated paths through the vessel. The concentration of the tracer in the stream is measured at several sampling positions within the vessel at known time intervals. The measurement of this "concentration" by, for example, electrical conductivity, light adsorption, etc., gives the tracer R.T.D. in the vessel. There are four major forms of tracer signal used, pulse, step, periodic, and random. The first two are easy to use in terms of experimental design and data analysis, and therefore are commonly used. (i) Pulse tracer injection experiment: M units of tracer are "instantaneously" introduced into a fluid entering a vessel of volume V m 3 at v m3/s, and the concentration-time curve of the tracer leaving the vessel is recorded. This is the so-called CPui s e curve. The E curve is then established by dividing the concentration reading by the area under the C p u i s e curve. (ii) Step tracer injection experiment, at time t = 0 the fluid flowing through a vessel of volume V m 3 at v m3/s is switched from an ordinary fluid to a fluid with tracer concentration C m a x , and the outlet tracer concentration C s t e p is measured versus t. This is called the C s t e p curve, and Chapter 2 Background and Literature the dimensionless form of C s t e p curve is the F curve, which is obtained by having the tracer concentration rise from zero to unity. dF According to the definitions of E and F curves, it is obvious that — =E. 5 dt The properties of E and F curves for ideal and non-ideal flows are shown in Figure 2 .1 . Deviations from ideal flow in real equipment are unavoidable due to channeling of fluid, recycling of fluids, or creation of stagnant regions in the vessel. Therefore, compartment models are developed in an effort to quantitatively describe real reactors by dividing the total volume V of the vessel into three parts: plug flow region (Vp), mixed flow region (Vm), and dead or stagnant region (Vd) [9]. Accordingly, the total flow through the vessel can be considered as: active flow that flows through the plug and mixed flow regions (va), bypass flow (vb), and recycle flow (v r). By comparing the E curve or F curve for the real vessel with the theoretical curves for various combinations of compartments, a model that best fits the real vessel can be developed. This model can also be used to diagnose the flow pattern in the vessel, i.e., to pinpoint faulty flow and suggest causes. Two examples of E curves for misbehaving plug flow reactors are given in Figure 2.2. Chapter 2 Background and Literature Plug flow Mixed flow Arbitrary flow V Figure 2.1 Properties of E and F curves for various flows. - 1 0 -Chapter 2 Background and Literature C (b) Multiple decaying peaks at regular intervals indicate strong internal recirculation. Figure 2.2 E curves for misbehaving plug flow reactors. -11 -Chapter 2 Background and Literature 2.1.2 Real Flow Modeling Studies show that flow in packed bed reactors approaches plug flow with small deviation [1]. Mathematical representations of flow with not too large a deviation from plug flow commonly use the tanks-in-series model or the dispersion model. They are roughly equivalent -both exhibit flow characteristics somewhere between those of plug flow and ideally mixed flow, and often give identical results for all practical purposes. 2.1.2.1 Tanks-in-Series Model The tanks-in-series model is simple and can be used with any kinetics. It can be extended without too much difficulty to any arrangement of compartments, with or without recycle. It visualizes the effect of backmixing as setting up a series of N ideal stirred tanks along the reactor such that the outlet concentration of the (i-lth) tank is the inlet concentration for the ith tank. The flow is quantified by assigning an appropriate N to represent the R . T . D . data. The pulse response and E curve; step response and F curve are show in Figure 2.3. Plug flow is approached when N = oo, and it is mixed flow when N = 1. -12 -Chapter 2 Background and Literature Chapter 2 Background and Literature 2.1.2.2 Dispersion Model On the other hand, the dispersion model has the advantage in that all correlations for flow in real reactors invariably use this model. Consider plug flow of a fluid, on top of which is superimposed some degree of backmixing, the magnitude of which is independent of position within the vessel. This condition implies that there exist no stagnant pockets and no gross bypassing or short-circuiting of fluid in the vessel. This is called the dispersed plug flow model, or simply the dispersion model. Figure 2.4 shows the condition visualized. The velocity profile in ideal plug flow is flat, while it fluctuates in dispersed plug flow due to different flow velocities and due to molecular and turbulent diffusion [9]. (a) ideal plug flow (b) dispersed plug flow Figure 2.4 Illustrative flow velocity profile for dispersion model. The mechanism of dispersion in packed beds is studied in terms of the principal modes of dispersion, molecular diffusion, and convection. When the velocity of fluid in a packed bed is sufficiently low, dispersion of material relative to the main flow by convection is small compared - 1 4 -Chapter 2 Background and Literature to that due to molecular diffusion. The contribution of convection to the total dispersion increases as the Reynolds number of flow increases and dispersion is dominated by convection when the Reynolds number is sufficiently high (e.g., Re > 10"4 for liquid flow). 2.2 General Mathematical Treatment for the Dispersion Model Dispersion due to convection and to molecular diffusion is described as a stochastic process and obeys Fick's law of diffusion. Mass balance for material " A " over a differential volume in the reactor as shown in Figure 2.5, assuming no production or consumption of " A " by reaction, i.e., no "source term", gives, A ;„ - A out= Accumulation of A 3c 3^c 3c 3c dydz[uxc + Dx (— + dx)] - dydz[ux (c + —dx) + D—dx] + ox ax ox ax 3c d2c dc 3c dxdz[u c + £ > ( — + —Tdy)]-dxdz[u (c + —dy) + D—dy]+ (1) dy dy 3c 0^ c 0c 3cc 3c dxdy[u2c + DZ (— + — - dz)] - dxdy[uz (c H dz) + D2 dz] = — dxdydz dz dz dz dz dt Rearrangement of Equation (1) leads to dc _ d2c ^ d2c _ d2c dc dc dc — = D X — - + D V — T + DZ—T--UX—-UV (2) & dx2 y dy2 2 dz2 x dx y dy z dz W A similar expression can be written for cylindrical configuration, a ^ d2c D* d , dc. a a — = D,—T+—'- ( r — ) - « _ u. — (3) a dx r or a dx a * D r is replaced by D t - 15 -Chapter 2 Background and Literature Figure 2.5 Mass balance for material " A " over a differential volume in a packed bed. Similar expressions can be written for y-and z- directions as those shown for the x-direction. -16-Chapter 2 Background and Literature The initial value and the boundary conditions of the above partial differential equations are determined by the tracer injection technique, flow condition of the vessel and the sampling positions in the vessel. There are two types of boundary conditions. One is called open boundary conditions, where the flow is undisturbed as it passes the entrance and exit boundaries. The other is called closed boundary conditions, where it is plug flow outside the vessel up to the boundaries. This gives rise to four combinations of boundary conditions, closed-closed, open-open, and mixed (closed-open and open-closed). These dispersion coefficients are usually divided into two groups according to their directions parallel or perpendicular to the direction of the main flow. When the coordinates are chosen in such a way that the main flow is parallel to _y-direction, then D y is called axial or longitudinal dispersion coefficient D a , which characterizes the degree of backmixing during flow; while D x and D 2 are called transverse and lateral dispersion coefficients D t and D] (or D r , radial dispersion coefficient, in cylindrical configuration), respectively, which characterize the degree of mixing in the directions perpendicular to the main flow. (As stated earlier in Chapter 1, D t and Di will be assumed equal and D t will be used in place of D r). The quantities of the two groups may be quite different in magnitude. Axial and transverse dispersion coefficients agree at low Reynolds numbers (<10"4 for liquids) because dispersion is due to molecular diffusion and the transverse and the axial structures of the bed are similar. At high Reynolds numbers (>10"4 for liquids) where convective dispersion dominates the values are different because axial dispersion is primarily caused by differences in fluid velocity in flow channels while transverse dispersion is primarily caused by deviations in the flow path caused by the bed materials [10]. -17-Chapter 2 Background and Literature It is a common exercise that axial and transverse dispersion are treated separately due to their different mechanisms and roles in the reactor behavior combined with the mathematical difficulty to solve axial and transverse dispersion coefficients simultaneously, 2 . 2 . 1 Axial Dispersion The axial dispersion coefficient is often estimated with the assumption that there is no dispersion effect in the transverse direction. The effect of axial dispersion can be visualized as such that an ideal pulse of tracer is introduced into the fluid entering a vessel and the pulse spreads as the fluid passes through the vessel as shown in figure 2.6. c c+(dc/dx)d u > u x=0 x x+d x=L Figure 2.6 Spreading of pulse tracer due to axial dispersion -18-Chapter 2 Background and Literature With neglecting transverse concentration gradient, Equation (2) and (3) will be simplified to dc d2c dc — = Da — - — u — (4) dt dx dx In dimensionless form where z = (ut + x)/L and 6= t/t = tu/L [11], * = ( 5 t ) ^ £ - . * (5) dO uL dz2 dz where the dimensionless group DJuL is called the axial dispersion number. When DJuL, is small (< 0.01), the extent of axial dispersion is very small, there is little effect exerted by the boundary conditions and the E curve is a symmetrical Gaussian curve. There are a number of ways to evaluate DJuL from an experimental curve: by evaluating its variance, by measuring its maximum height or its width at the point of inflection, or by finding the width which includes 68% of the area. For large deviation from plug flow, DJuL > 0.01, the pulse response is broad and it passes the measurement point slowly enough that it changes shape, which gives a nonsymmetrical E curve with a long tail and the dimensionless residence time is larger than 1. In this case, DJuL is usually evaluated by using Equation (7), (8), or (9) according to the corresponding boundary conditions. The reciprocal of DJuL, which is called the axial Peclet number Pea, is a more popularly used parameter that describes the extent of axial dispersion. A larger axial Peclet number means a smaller degree of axial dispersion. Axial dispersion is negligible when Pea --> co, and hence it is plug flow; while axial dispersion is infinite when Pe a --> 0, and hence it is ideal mixed flow. 19 Chapter 2 Background and Literature The shapes of E curve under various boundary conditions are similar to each other. Two examples are given in Figure 2.7 and 2.8. Analytical solution is available to equation (5) for pulse tracer injection with open-open boundary conditions [9], i r (\-ef , exp[—v~ ] Zjx(DJuL) F L MP Jul) (6). and ACT' r i A KPe.J r 1 A + 8 (7) There is no analytical solution for closed-closed or mixed b.c.'s, but the E and F curves under these conditions can be obtained by solving Equation (5) numerically. For closed-closed b.c.'s: ACT 2 = 2 ' 1 w (8) and for closed-open or open-closed b.c.'s r i ~N Aal = 2| r 1 > + 3 (9) where a 2 , called variance, a measure of the spread of the curve, can be calculated as follows Jo (t-tfcdt jQ rcdt jcdt jcdt 5>,A/, 2 C A (10) •20 Chapter 2 Background and Literature Here t is the mean time of passage or mean residence time, which can be calculated as ftcdt YtcAt Vcdt 2 > A , Jo cr 2 < = (12) For an ideal pulse input, o~,2„ = 0, so 2 Aa2 =cre2 - a 9 2 =cx92 - 0 = o - / (13) The additive property of variances shown by Aris [12] make it possible to use any sloppy one-shot tracer injection instead of an ideal pulse tracer input to calculate Pea, e'°ut e"" (A*)2 (tout-tinf ( H ) DJuL (= 1/Pea) can also be found by step input experiment and F curve. The output F curve is S-shaped. An example is given in Figure 2.9. DJuL (= 1/Pea) can be estimated by finding the slope of the F(9)~ 0 curve at the mean residence time, 6= 9 , d{F{6))\ dO 1 I U L (15) 9-9 2VflD, This method is simple, but not very accurate. Generally speaking, the pulse injection experiment is preferred because the E curve is more sensitive, while the F curve is a smooth good-looking curve, which could hide real effects, or in other words, it is less sensitive to the tracer response change. -21 -Chapter 2 Background and Literature 0 0.5 1.0 1.5 2.0 Figure 2.7 Tracer response for closed vessels with large deviations from plug flow. (From Levenspiel, 1999) -22-Chapter 2 Background and Literature Chapter 2 Background and Literature (a) Step response curves for small deviation from plug flow. (b) Step response curves for large deviation from plug flow in closed vessels. Figure 2.9 Examples of step tracer response curves. (From Levenspiel, 1999) -24 -Chapter 2 Background and Literature 2.2.2 Transverse Dispersion It is unavoidable to solve the differential equation governing dispersion in packed beds in order to find the transverse dispersion coefficient D t . The value of D t can be estimated based on the previous knowledge of axial dispersion coefficient D a or calculated simultaneously with D a by using a curve fitting routine program when enough points of sample can be taken in the vessel. The program seeks the minimum of S: where c e is the experimental measured tracer concentration, Ct is the calculated tracer concentration from Equation (2) or (3), and n is the number of the sampling points. In order to avoid the mathematical difficulty in solving the time-dependent two- (three-) dimensional partial differential Equation (2) or (3) analytically, it is a common technique for transverse dispersion coefficient to be solved from a step tracer experiment at steady state. However, the weakness of this approach is that axial and transverse dispersion coefficients, D a and D t , cannot be obtained simultaneously because D a is insensitive to step tracer response. The differential equation for transverse and axial dispersion without chemical reaction at steady state is: for rectangular system, n s = Tk-ct] (16) i=i (17) -25 -Chapter 2 Background and Literature and for cylindrical configuration, _ d2c D* d , dc. dc dc n , s Dx—r+— (r—)-ux ut — = 0 (18) dx r or dr ox dr * D r is replaced by D t Analytical solution to Equation (18) was studied by several researchers. For example, Klinkenberg [13] gave an analytical solution to Equation (18) with the following boundary conditions: — = 0, for r = 0, and r = R; dr c(0,0) = c0-S(r,x); c = 0, for x = - oo. c _^J0(Knr/R)exp[(0.5 + q)ux/Da] j2K v^) where q = j0.25 + DaDtK2J(!i2R2) and K n is the nth positive root to the Bessel function Ji ( K „ ) = 0. Here, c is the tracer concentration, c 0 is the tracer concentration at the exit, r is the radial position within the bed, and R is the radius of the bed. In 1996, Aral and Liao [14] presented analytical solutions for a two-dimensional rectangular coordinate system with both pulse and step tracer injection. -26 -Chapter 2 Background and Literature The general dimensionless dispersion equation analyzed in their study was defined for a steady and uniform two-dimensional velocity field with first-order decay function and time-dependent dispersion coefficients. R ^ = Da(t)^+Dl(t)^-u^-v^-pj?c + q(x,y,t) ax oy a ~°s/ax2 '~ty'' dy2 (20) with c (x,y,0) = f (x,y) and -oo < x, y < co where, R' is the retardation coefficient; x, y are co-ordinates; u, v are the steady state velocity in the x and y direction, respectively; D a and D t are axial and transverse dispersion coefficients; c is the tracer concentration; and p is the first-order decay coefficient. In the case that an instantaneous point tracer source is injected into this field with zero initial concentration distribution, then q(x,y,t)=Ms(x)S{y)S{t) and c(x,y,0) = f(x,y) = 0 where M is the mass injection rate, n is the porosity of the bed, and 8( ) is the Dirac delta function. The solution to equation (20) is: f / \ cM cix, y,t)= exp a = ylDJDt pj u x / R + a v y 1 R 4DJ R (21) >27< Chapter 2 Background and Literature Similarly, for a continuous point source with zero initial concentration distribution, c(x,y,0) = f(x,y) = 0 and q(x, y,t) = 5(x)5(y), c(x v A- f S£° &J-u(t-t N ( ^ - ^ - ^ O ) ) 2 + ( ^ - ^ - ^ O ) ) 2 ^ C ^ ~ l 4xR<(a(t)-a(t0)fXP{ ^ Q <a(t)-a{ta)) (22) \dtn J where v ' Jo /?' R For the purpose of finding dispersion coefficients in chemical packed bed reactors, R' = 1 and p = 0. The major constraints of this method are that only the strict ideal pulse tracer input can be handled and it can only be handled in two-dimensional. On the other hand, the partial differential equations (2) and (3) can be solved numerically by using the finite element method. With the rapid progress in computer application, it is possible to overcome the difficulty in solving time-dependent two-dimensional partial differential equations with the help of one of the newly developed software kits such as MatLab PDE Toolbox and Femlab. Even a three-dimensional time-dependent partial differential equation can be tackled with the newest version of FemLab which was released in October, 2000. 2.3 Special Considerations for Electrochemical Reactors 2.3.1 Electrochemical vs. Thermochemical Reactors An electrochemical reactor is alike to a heterogeneous catalytical thermochemical reactor in that the reaction occurs at the catalyst (electrode)-process fluid interface. But there are some 28 Chapter 2 Background and Literature notable differences that need also to be recognized, e.g., there are two outstanding differences in a comparison of an electrolysis cell with a gas-phase catalysis reactor [4]. (i) Mass transport processes in the liquid phase are always very slow in comparison with those in the gas phase. This will cause both mass transport limitations on the rate of chemical changes in the reactor and the creation of reaction layers at the solid-fluid interface to be much more troublesome. (ii) While, in both types of reactor, it is desirable to maximize the surface area per unit volume, the designer of an electrolytic cell has to be concerned with the problem of non-uniform current and potential distributions. The electrochemical reaction rate is a function of potential as well as electroactive species concentration and temperature. The electrochemical reaction will only occur if there is an appropriate local potential difference between the electrode surface and the electrolyte (the electrode potential) and the reaction rate is an exponential function of electrode potential. The geometric arrangement of anode and cathode is thus a key factor. Unlike in thermochemical processes where cylindrical configuration is the norm, two or a set of two plate electrodes with gap of a few millimeters is the most commonly used configuration in electrochemical processes due to potential distribution and energy consumption considerations. A two-dimensional partial differential equation is adequate to describe the dispersion process in a cylindrical thermochemical reactor, while a three-dimensional model is needed for the rectangular configuration of an electrochemical reactor. The space-time yield is one of most valuable statements of reactor performance. The space-time yield of an electrochemical reactor depends on the concentration of electroactive -29 -Chapter 2 Background and Literature species, the mass transport regime and the current efficiency. It is directly proportional to the effective current density and to the electrode surface area. In comparison with other chemical reactors (particularly catalytical heterogeneous reactors), electrolytic cells have poor space-time yield; the expectation for many chemical reactors is a space-time yield of 0.2-1 kg h"11"1 while that, for example, for a typical copper electrowinning cell is only 0.08 kg h"1 l " 1 [4]. As a result, many electrochemical researchers have sought to increase this figure of merit by revolutionary changes in cell design, e.g., the introduction of three-dimensional electrodes. 2.3.2 Three-Dimensional Electrodes An increasingly diverse range of porous three-dimensional electrodes is available including stacked meshes, beds of granules, reticulate and microporous felts or cloths. Such materials offer particularly high values of electroactive area per unit reactor volume as well as a moderate increase in the mass transport coefficient. The result is a significantly increased space-time yield, compared to two-dimensional electrodes. The high porosity of 3D electrodes may also facilitate the release of gaseous products and reduce weight of the whole reactor. These characteristics may be utilized in various ways including allowing a compact reactor for a given duty; the capability of providing a high fractional conversion per pass given a sufficiently large residence time; and the ability to maintain a reasonable production rate and current efficiency when processing solution contains low levels of electroactive species. Porous, three-dimensional electrodes have been widely used in metal removal from dilute process liquors and in inorganic and organic electro synthesis. Many battery and fuel cell electrodes utilize an active material, which is a porous, three-dimensional matrix, while miniature -30 -Chapter 2 Background and Literature porous electrodes have found use in electrochemical detector systems for high-pressure liquid chromatography analysis. There are, in particular, two extreme configurations of three-dimensional electrodes with respect to the directions of electrolyte flow and current flow as shown in Figure 2.10. A flow -through electrode in one of its simplest forms consists of a bed of electroactive spherical particles through which electrolyte flows without channeling. Current is fed into the bed by a current feeder positioned at the downstream end and is collected by a counter electrode positioned upstream of the bed. This counter electrode may be free in solution or be separated from the reactant stream by a suitable diaphragm. A major problem with flow - through electrode configuration is that the attainment of a high conversion and a uniform potential distribution are generally incompatible. In addition the requirement of a long electrode in the direction of current flow causes large voltage loss in the electrode. The flow-by configuration, whereby the electrolyte flow is perpendicular to the current flow and the dimension in the direction of electrolyte flow is large and that in current flow is small, may be used in order to overcome these difficulties. Another advantage of the flow-by configuration is that we can adjust potential distribution and hydrodynamic conditions "independently" and thereby "optimize" performance. The flow-by electrode is also more amenable to scale-up. -31 -Chapter 2 Background and Literature Or 1" Electrolyte flow direction t Porous anode Current flow Three-dimensional cathode Porous cathode feeder / bed support (a) flow-through configuration Cathode feeder Anolyte flow direction Three-dimensional cathode (usually rectangular) Bed support Catholyte flow (b) flow-by configuration Figure 2.10 Configuration of three-dimensional electrodes (only the cathode is shown as three dimensional) -32-Chapter 2 Background and Literature The need to design a reactor with a high active surface area per unit volume is a more difficult challenge to electrochemical engineers than to thermochemical engineers. In the case of the electrochemical reactor the rate is determined by concentration, temperature and potential, so performance should have a more complex dependence on mass and heat dispersion than observed in thermochemical processes. The complex interaction among the electroactive species concentration, temperature and potential in an electrochemical reactor employing 3D electrode can be illustrated by the several equations coupled in the modeling of such a reactor [15]: i) Mass balance equation: *L = D ^ c - U V c - ^ ^ - R r a t e dt RT V ra,e ii) Bulter-Volmer equation: • = nFkacmRS ccnFE _ nFkccm0S (\-ct)nFE va K RT v„ K RT ) iii) Potential distribution in a 3D electrode d2E r 1 1 ^ -aj 1 1 \Ke KmJ dx1 iv) Faraday equation j = — Fr * v a specific surface area of 3D electrode (m2 m"3) c concentration of the electroactive species (kmol m"3) c R S concentration of reduced and species at the electrode surface (kmol m"3) Cos concentration of species at the electrode surface (kmol m"3) -33 -Chapter 2 Background and Literature D dispersion coefficients (m2 s"1) E operating electrode potential (volt) F the Faraday constant (98480 kC kmol"1) j local Faradaic current density (kA m"2) k a anodic reaction rate constant (m s"1) kc cathodic reaction rate constant (m s"1) m anodic reaction order (-) m' cathodic reaction order (-) n electron stoichiometric coefficient (-) Rrate reaction rate, =f (i), (kmol m"3 s"1) r* specific reaction rate (kmol m"2 s"1) U electrolyte flow velocity (m s"1) x distance through electrode parallel to the current (m) z charge number of species (-) a anodic charge transfer coefficient (-) iQ effective electrolyte conductivity (mho m"1) K m effective electrode matrix conductivity (mho m"1) Vo stoichiometric coefficient for oxidised species (-) V R stoichiometric coefficient for reduced species (-) v reactant stoichiometric coefficient <|> Potential field (volt) - 3 4 -Chapter 2 Background and Literature Three-dimensional electrodes generally give rise to more problems with potential and current distributions than their two-dimensional counterparts. This is primarily due to the anisotropy of porous- and packed-bed electrodes, with respect to electrode conductivity, electrolyte flow and concentration of the electroactive species. It is remarkably difficult in practice to obtain a high surface area value whilst maintaining a uniform reaction rate over the entire electrode surface, e.g., porous- and packed-bed electrodes often suffer from poor potential, and current, distributions. Often, the adoption of a three-dimensional electrode involves a trade-off between increased electroactive area and diminished selectivity. This situation may be improved with a better understanding of the current, potential, and electroactive species concentration profile (including dispersion effects) within an electrochemical reactor. 2.4 Literature Review 2.4.1 Dispersion in Thermochemical Reactors Dispersion in thermochemical reactors received extensive investigation in the 1960's and 1970's. Investigation reports were given for single-phase gas and liquid flow in packed beds and fluidized bed. A wide range of experimental conditions covered by various workers showed the effect of the principal variables, and some approximate relationships between the Peclet group and the Reynolds and Schmidt groups were presented. A number of models for the dispersion process have been put forward. One of the earliest was that of Kramers and Alberda [15] who proposed the mixed-cell model that the packed bed was viewed as a series of well-mixed tanks. The transverse dispersion was not taken into account in their model. Deans and Lapidus [16] extended the above stirred-tanks model to include radial dispersion by suggesting a hybrid arrangement of tanks with two outlets for each tank so that the -35 -Chapter 2 Background and Literature magnitude of D t can be included by specifying the flow rate through each outlet. However, the computation of this method is very tedious. The simplicity of boundary conditions for the stirred-tank model is its greatest commendation. One drawback of the tanks-in-series model comes from the fact that a tracer particle entering a void one particle diameter long will have zero probability of leaving the void between the time of entry and a time interval of at least 0.1 dp/u (dp is the diameter of the particle and u is the interstitial fluid velocity) corresponding to a minimum time of residence of the fast velocity stream. This finding is at variance with the predictions from the stirred - tank model which gives an exponential decrease of the probability density with the particle most likely to leave immediately after entry [3]. Gunn in 1969 [1] first proposed the dispersion model in packed beds based on a probability theory; which considered the mechanics of dispersion under single and combined influence of fluid mechanics and molecular diffusion. Some other investigators like Gantang Chen [11] derived the same differential equation (Equations (1)) from a different approach by making a material balance over the reactor with the assumption that dispersion is a diffusion-like process that obeys Fick's law. The proper form of boundary conditions has also received a great deal of attention. Danckwerts [8], Wehner and Wilhelm [17], and Pearson [18] divided the reactor into three regions: the reactor or tested bed itself, an entrance region of infinite extent, and an exit region of infinite extent following the reactor. Danckwerts [8] initially proposed that the dispersion coefficients of the inlet and outlet regions can be neglected in comparison with the dispersion coefficient of the reactor itself so that he chose the entry and exit coefficients to be zero. Wehner and Wilhelm [17] worked on introducing the characteristics of the inlet and outlet sections by -36-Chapter 2 Background and Literature means of assigning appropriate dispersion coefficients to them. However, the incapability of treating the flank section and the extremely cumbersome of this set of boundary conditions made it rarely used [3]. The application of the Wehner-Wilhelm boundary conditions to the measurement of dispersion characteristics has been considered by van der Laan [19], Bischoff and Levenspiel [20] and Gunn [3]. Van der Lan showed that if the first and second moments of a pulse are measured after dispersion the axial dispersion coefficient could be expressed in terms of the second moment by using the solution to the Laplace-transformed equations and boundary conditions without the necessity of finding the inverse. It is recommended by Gunn [3] that for diffusive dispersion, the boundary conditions to Equation (1) are given by: dc N = uc-Da — at the inlet dx dc — = 0 at the outlet dx Although the conditions only become correct at very low Reynolds numbers so that the asymptotic Gaussian distribution may be set up in very short length of packing. On the other hand, when convective dispersion dominates, the boundary conditions would be: dc N -uc-Da — at the inlet dx C = C(eq) aS X — > C O Where c is the tracer concentration, N is the flux of tracer, u is the fluid interstitial velocity, D is the molecular diffusion coefficient and D a is the axial dispersion coefficient. -37-Chapter 2 Background and Literature Analytical solution to a two dimensional version of Equation (1) in cylindrical coordinates system at steady state without chemical reactions was given by several authors like Roemer, Dranoff, Smith [21]. In 1996, Aral and Liao [14] presented a set of analytical solutions to a two-dimensional convective-dispersion equation with time dependent dispersion coefficients. These solutions can be employed in both step and pulse tracer injections. However, their method can handle only ideal tracer input. 2.4.1.1 Axial Dispersion A dimensional analysis of longitudinal dispersion in a packed bed gives[3]: where u is the fluid interstitial velocity, dp, d t is the diameter of the particle and bed, respectively, D is the molecular diffusion coefficient, L is the length of the bed, p and p are the viscosity and density of the fluid. L/dp has importance only when it is so small that changing velocity distribution affects the characteristics of dispersion. Genkoplis and co-workers [22,23,24] found from fluid-mechanical studies that the effect of L/d p was confined to a dozen layers of particles and is probably not very important. The effect of d t/dp has received more attention and there did not appear to be any convincing evidence that it is significant when the ratio is greater than 12 [3]. It has been pointed out that the velocity of fluid in a packed bed is greater in the close vicinity of the wall than in the center of the packing because of the higher porosity near the wall but in most industrial = /(• dpup ju L dt fi ' pD'd/dp -38-Chapter 2 Background and Literature applications the ratio of the diameter of the bed to the diameter of the particle is very large. Thus in view of the practical application, and because of the inherent statistical variation in dispersion coefficient in packed beds, the refinement of writing D and u as function of the radial coordinates is not usually justified for packed beds. As a rough rule it can be stated that the variation of fluid velocities and dispersion coefficients may be neglected when the value of d t/dp is greater than 12 in thermochemical reactors. Levenspiel [9] summarized that the vessel axial dispersion number D a /uL was a product of two terms — = (intensity of dispersion) (geometric factor) (D.Yd uL \ud j where d is a characteristic length = dmbe or dp, — is the reciprocal of axial Peclet number Pea, and ud — = —^—=f (fluid properties) (flow dynamics) =/(Re, Sc) ud Pea When Reynolds number is small (<10"4 for liquids), the dispersion of material will be little affected by velocity distribution, so that: D —2- = constant D and when the Reynolds number is large (>10"4 for liquids), the dispersion effect of fluid retarded near solid surfaces and accelerated in the middle of flow channels ensures that dispersion by fluid-mechanical forces predominates, and — =f(Re) D •39. Chapter 2 Background and Literature The magnitude of the Schmidt number is also an important influence so that in liquids where Sc ~ 103 the effect of molecular diffusion will disappear at Reynolds numbers that are small compared to the Reynolds numbers at which the effect of molecular diffusion disappears in gaseous systems where Sc ~ 1. Figure 2.11 shows a collection of experimental measurements of axial Peclet number Pea obtained during the flow of liquids in random packed spheres of porosities within the range 0.37 -0.43, and d t/dp within the range of 8 - 100, pulse tracer injection technique was commonly used and the effect of radial dispersion was neglected [3]. The results are presented as the dependence of axial Peclet number Pe a upon Reynolds number. When the predominant mode of dispersion is that due to molecular diffusion the dispersion coefficient is similar in magnitude to the molecular diffusivity and independent of Reynolds number. When convective dispersion dominates, axial Peclet number is independent of Reynolds number and its magnitude is about 0.4. The scatter is evident but possibly not surprising, as it has been shown by a number of workers. Pryce [10] and Dranoff, Roemer, and Smith [21] showed that repacking of the bed at the same porosity did not usually give the same dispersion characteristics. Pryce [10] re-packed a bed of spheres and measured dispersion coefficients a number of times to find that the standard deviation of the axial Peclet number was 0.15 Pe and about three times as large as the standard deviation obtained by repeating the measurements without repacking the bed. Evidently an inherent statistical scatter was generated on re-packing, which arose from the fact that although the porosity of the bed could be accurately reproduced, the fluid-mechanical characteristics were -40-Chapter 2 Background and Literature not defined by porosity alone. The relative arrangement of particles was important and that could not be reproduced in random beds. The boundary conditions chosen by the investigators corresponded to a semi-infinite bed because in most cases the bed was rather long. It is generally desirable that the dispersion of a given tracer should be measured at two points within the bed so that the signal is examined at two points. Many of the experimenters employed only one point of measurement and the form of the input pulse was taken for granted. This would bring error into the measurements. Gunn [3] pointed out that the relationships were applicable to random arrangements of sphere but in the absence of more specific information it was suggested that dispersion characteristics of beds containing non-spherical particles might also be estimated from the same relationships. However, it is remarkable to notice that Gunn and Pryce [10] found convection controlled mixing characteristics of regular cubic and rhombohedral arrangements did not obey Equation (1). The regular and repeated pattern of mixing from regular arrangements was not described by the conditions of the central limit theorem. Another parameter, D ae/udp, describing the degree of axial mixing for gas and liquid flow in packed beds was summarized by Levenspiel [9] as shown in Figure 2.12. Satisfactory empirical correlation has been proposed for gaseous system by Hiby [25] based on assumption that the total axial dispersion is the sum of the convective and the diffusive modes of motion. No similar correlation is available for liquid phase flow. 1 0.65 0.67g ReSc (23) + -41 -Chapter 2 Background and Literature C a r b e r r y * ' « ' A , , , I 1 ). t I I 1 ' ' ' I 1 1—«-J 1 — w 100 woo fit —REYNOLOS NUMBER (-U.0, pfcil 0-1 Figure 2.11 Axial Peclet number Pe, for liquids in random beds of sphere. (From Gkinn, 1968) 10 100 Re = dpup/ii 1000 2000 Figure 2.12 Experimental findings on dispersion of fluids flowing with mean axial velocity u in packed beds (From Levenspiel, 1999). -42-Chapter 2 Background and Literature 2.4.1.2 Transverse Dispersion Figure 2.13 shows the experimental relationship between transverse Peclet number Pe t and Reynolds number for both liquid and gas-phase in cylindrical packed beds. Again there was a significant scatter. The magnitude of the scatter is similar to that observed for axial dispersion. The general difficulty of obtaining reproducible results when the bed was repacked was also evident in transverse dispersion measurements. The transverse dispersion coefficient D t was measured from a step tracer experiment separated from the pulse tracer experiment from which axial dispersion coefficient D a was obtained. None of the investigators took axial dispersion into account properly; D a was taken to be equal to D t or simply neglected. This error could be decreased by using a simple transformation to introduce the previously measured value of the axial dispersion coefficient into the calculation of D t . It was estimated by Roemer, Dranoff and Smith [21] that the neglecting of axial dispersion in calculations of D t for experiments of this type can cause errors of 10 - 20% in the value of D t . Figure 2.13 shows that at low Reynolds number diffusion is the dominant mode of dispersion, while at high Reynolds numbers the transverse Peclet number approaches a value of about 10. Axial and transverse dispersion coefficients agree at low Reynolds numbers because dispersion is due to molecular diffusion and the transverse and axial structures of the bed are similar. At high Reynolds numbers, where convective dispersion dominated the values are different because axial dispersion is primarily caused by differences in fluid velocity in flow channels while transverse dispersion is primarily caused by deviations in the flow path caused by particles. -43 -Chapter 2 Background and Literature A satisfactory correlation was obtained for gaseous flow [3], 1 ! - + - ^ T (24) Pet 11 ^ReSc where s is the bed porosity and y is the tortuosity factor. The liquid-phase dispersion coefficients do not agree with this equation except for values of Re larger than 40, and this suggests that the transverse dispersion due to fluid-mechanical forces is reduced at low Reynolds numbers. Gunn [3] recommended that transverse dispersion can be predicted by adding the effects of convective and diffusional modes as shown in Equation (25): - ^ = ^ + ^ _ (25) Pet Pef rrReSc ' Here s is the bed porosity and x r is the tortuosity factor. The fluid-mechanical Peclet number of transverse dispersion, Pef, is a function of Reynolds number and gives the transverse dispersion due to the convective mode alone. The dependence of the fluid mechanical group, Pef, upon Reynolds number is shown in Figure 2.14. -44-Chapter 2 Background and Literature Figure 2.13 Transverse Peclet numbers Pet for gases and liquids in random beds of spheres. (From Gunn, 1968) ctOOp o 01 1 » 100 WOO Rt — REYKOLOS KUMBER Figure 2.14 Variation of the fluid-mechanical Peclet group for transverse dispersion with Reynolds number. (From Gunn, 1969) -45 -Chapter 2 Background and Literature 2.4.2 Dispersion in Electrochemical Reactors Dispersion has so far received much less attention in the electrochemical engineering literature, and research has been mainly directed to extending methods and principles known in the discipline of chemical reaction engineering to certain electrolyzers. Dispersion (or diffusion) of mass in 3D electrodes has been considered by several authors. The majority of this previous work was concerned with the behavior of porous flow-through electrodes. Both "radial" and axial dispersion and diffusion have been examined for this case. For instance, the axial dispersion coefficient measured in a flow-through porous electrode composed of rolled 80-mesh platinum screen is of the order of 10"6 m 2 s"1 [26]. However, it seems that the mass transport was studied only in terms of the effect of diffusion on the performance of porous flow-through electrodes by many authors, especially in the radial direction [27,28,29,30]. As we noticed that in most practical fixed-bed reactors, the mass and heat dispersion coefficients are orders of magnitudes greater than the respective true diffusion coefficient and thermal conductivity in the fluid. Analysis of mass transport in fixed-bed systems such as 3D electrodes in terms of simple diffusion and migration can be quite erroneous in the face of the overiding effect of dispersion in real systems [31]. Concerning flow-by electrodes there has been some discussion of the effect of axial (longitudinal) dispersion, but it seems generally to be assumed that axial dispersion is negligible (unless channeling occurs) and radial transport is solely by diffusion and migration [31]. It was recommended by Scott as a rule of thumb that for flow in packed beds, which is of relevance in three-dimensional electrode design, Bodenstein numbers (Bo = udp/eDa) for liquids are given approximately by Equation 26 [4]: -46-Chapter 2 Background and Literature Bo = = 0.5 Re< 10 Bo = = 2 Re> 103 s-Bo = 0.2 + 0.011 Re' 0.48 10<Re< 103 (26) These correlations came from the assessment of the significance of axial dispersion in electrochemical reactors aided by the availability of experimental data. These data are published for flow in empty tubes and packed tubes [32] as correlations of Peclet number versus Reynolds number or as the Bodenstein number versus Reynolds number, where, d p, is the characteristic tube diameter or particle diameter in packed beds, s is the porosity of the packed bed, and D a is the axial dispersion coefficient. Experimental data are also available for other flow channels, such as annuli or between rotating cylinders, but unfortunately little is known about the effect of dispersion on the popular parallel plate reactor configuration. The usual approach for this system is to take the value of dispersion coefficient determined for the Reynolds number based on the hydraulic diameter. The correlations in Equation (26) are applicable to packed beds which are not too short, i.e., L/d p>10. If narrow beds are used, say, d p /d t > 0.1 (d t is the diameter of the packed bed), then an increased flow at the wall may considerably decrease Bodenstein numbers and therefore, increase dispersion. This effect is more pronounced at low Re. In such cases, sometimes two regions, or multi-parameter models are recommended for multiphase and fluidized bed electrochemical reactor design. -47 -Chapter 2 Background and Literature Few data are available on dispersion coefficient in radial (transverse) direction, which is of particular importance in cases such as flow-by three-dimensional electrodes because of the interaction between the concentration, current and potential distributions. The usual range of the transverse Peclet number (Pet = ud/sDt*) in a packed-bed electrolyzer was predicted as 5 < Pe t < 13 [33], or the value of transverse dispersion coefficient in the order of 10"8 to 10 There appears to be some ambiguity in the definition of terms, such as the axial and transverse Peclet number, axial and transverse dispersion number, with respect to standard chemical engineering usage. Careful attention should be paid when quoting and comparing these terms from different sources. *The bed porosity s is usually not included in the Peclet number in the thermochemical engineering literatures. -48-Chapter 3: Objective Chapter 3 OBJECTIVE OF THE PRESENT WORK The present work is the first component of a project the objective of which is to obtain a comprehensive set of axial and transverse dispersion coefficients for flow-by packed-bed electrochemical reactors that will subsequently be used in a complete electrochemical reactor model. An experimental tracer stimulation-response measuring system and a mathematical methodology will be set up. The axial and transverse dispersion coefficients will be obtained simultaneously from pulse experiments in a rectangular testing section filled with glass beads and with carbon felt at the single phase liquid load range typical of that for an electrochemical reactor. -49-Chapter 4: Experimental Apparatus and Procedures Chapter 4 EXPERIMENTAL APPARATUS and PROCEDURES The apparatus employed in this work consisted of a packed bed test section, a tracer injection system, ten local potential sampling copper wires with a computer data acquisition interface, and necessary piping and fittings. The procedures included: • Copper wire calibration • Pulse tracer response measurement in a glass beads and a carbon felt bed at liquid flow rate from 30 to 120 ml min"1 • Calculation of axial and transverse dispersion coefficients The axial dispersion coefficient was first estimated from the difference of the variances of tracer input and its response curves, neglecting the effect of lateral and transverse dispersion. Then axial and transverse dispersion coefficients were calculated simultaneously by solving the governing differential equation with a computer program written in MatLab and FemLab with the assumption that the lateral and transverse dispersion coefficients were equal. 4.1 Experimental Apparatus Figure 4.1 and 4.2 respectively shows a photograph and a schematic diagram of the arrangement of the apparatus, consisting of an electrolyte reservoir, a reference electrode pot, a test section, a tracer injection system, ten copper wires grouped into four probes, and a computer interface to measure and record the copper wire potentials. -50-Chapter 4: Experimental Apparatus and Procedures Figure 4.1 A photograph of the experimental apparatus (From left to right - air pressure regulator, pressurized liquid reservoir (carboy) and reference electrode pot, packed bed test section with tracer injection syringe, data acquisition computer) - 51 -Chapter 4: Experimental Apparatus and Procedures Compressed air or nitrogen - 0 Outlet Reference electrode UB Inlet copper wires j ii E A: Electrolyte reservoir, 20L B: Reference electrode pot, IL C: Tracer injection system D: Test section E: Computer data acquisition interface Figure 4.2 Experimental arrangements for measuring axial and transverse dispersion. -52-Chapter 4: Experimental Apparatus and Procedures The electrolyte was driven by compressed air to flow through the system. The flow rate was adjusted by manipulating the air pressure and measured with a graduated cylinder and a stopwatch. The detail of the test section is shown in Figure 4.3. The tested bed (glass beads or carbon felt bed) was held between two 20" long by 5" wide by 1" thick transparent Plexiglas plates fitted with 1/4" inlet and outlet connectors and a rectangular Plexiglas spacer whose side surfaces were specially treated to be transparent. The transparent feature of this test section made it possible to visually observe the flow pattern. The tested bed (32cm long by 5cm wide by 2.6cm thick) was composed of either glass beads (2mm in diameter) or four side-by-side pieces of carbon felt (Each piece of the carbon felt was 32cm long by 5cm wide, cut from a big piece of carbon felt of 0.65cm thick). The test section must maintain a certain thickness in order to obtain a measurable tracer concentration profile in the transverse direction which is of primary interest of the present work. Therefore, four pieces of carbon felt were put together without compression to construct the carbon felt bed with a thickness of 2.6cm. The contact effect between the carbon-felt pieces was considered to be negligible. To ensure good distribution of liquid in the test section, an expansion section, a piece of perforated rubber plate, and a 2.5cm deep bed of 2mm diameter glass beads were provided at the base of the test section. This precaution was necessary because early experiments carried out without a fluid distributor had shown flow channeling at the entrance of the bed. The depth of the glass beads bed was so selected that the pressure drop through the distributor was about 30% of the total pressure drop through the whole test section. The rubber fluid distributor was placed in -53 -Chapter 4: Experimental Apparatus and Procedures and just above the elbow leading into the section with 36 holes (1mm diameter) evenly distributed on it, which were arranged into four rows and nine columns with 5 mm apart in either direction. 0.7M CuS0 4 with 0.1 wt % Trypan Blue was chosen as the tracer because it had the same density as that of the main fluid which was 12 wt% Na2S0 4 adjusted to pH = 2 by adding 98% H 2 S 0 4 . The tracer was injected by a syringe with a 22G stainless steel needle (0.020"O.D.) at approximately the same velocity as that of the main flow. The tracer injection syringe needle entered the test section at 0.5cm below the top surface of the glass beads bed fluid distributor, lcm from the tip, the needle was bent upwards against the wall of the test section, so that 0.5cm of the needle protruded into the tested bed from the bottom, as shown in Figure 4.4. Repeat experiments were performed with the syringe needle in the center of the bed and against the wall, no different tracer stimulation-response results were found. Ten copper wires were arranged into four probes inserted at four vertical positions in the test section as illustrated in Figure 4.2. Probe 4 was as the same vertical level as that of the injection point (the tip of the injection needle) and contained one wire while probes 3, 2, and 1 were inserted 10, 20, and 30cm above the injection point, respectively, and contained three wires each. The tip of wire 10 in probe 4 was 3 mm from the test section wall. The lengths of the three wires in each of the other three probes were so arranged that the tips of the three wires were at different horizontal positions, 3mm, 11mm, and 19mm from the wall of the test section, respectively, as shown in Figure 4.5. Each wire was insulated except for the cross section at the tip. By this means the potential measured by the copper wire was the potential at the particular point of the wire tip that, in turn, corresponded to the local copper ion concentration. -54-Chapter 4: Experimental Apparatus and Procedures A: inlet I: carbon felt (or glass beads) bed, 31cm x 5cm x 2.6cm B: injection syringe J: 2mm diameter glass beads bed fluid distributor, 2.5cm deep C: copper wire K: perforated rubber fluid distributor 5cm x 2.6cm x 1.5cm D: outlet L: 1" long Neoprene expansion section E: 1" long Neoprene contraction section M: probe 1 F: 14 sets of 1/4" tie bolt and nuts N: probe 2 G: Plexiglas plate O: probe 3 H: Plexiglas spacer P: probe 4 Figure 4.3 Details of the test section -55 -Chapter 4: Experimental Apparatus and Procedures Figure 4.4 Details of the tracer injection syringe -56-Chapter 4: Experimental Apparatus and Procedures A: Bare copper wire tip (diameter: 0.39mm) B: Copper wire insulator (diameter: 0.43mm ) C: Glass beads or carbon felt bed D: Probe 1, 2 or 3 E: probe 4 Figure 4.5 Details of wires in probes -57-Chapter 4: Experimental Apparatus and Procedures The reference electrode used in this project was a Rosemont Analytical Model 399 sensor that included a general-purpose hemi-bulb pH electrode and a double junction gel filled reference electrode. The double junction reference cell configuration was resistant to process solution containing ammonia, chlorine, cyanides, sulfide, or other poisoning ions. The glass electrode was housed in a molded Telzel body and sealed with Viton O-rings to guard against process leakage. The cable end of the sensor was also sealed, eliminating cable shorts caused by exposure to moisture. Constructed without the flowing electrolyte junction found in conventional reference electrodes, the problem associated with junction clogging was eliminated. In addition, this electrode did not require electrolyte replenishment or external pressurization. Thus, it eliminated the requirements for elevated head reservoirs, air set regulator, or differential pressure regulators to overcome backpressure of the sample being measured. The physical specification of this electrode was: pH measure range: 0-14 Temperature compensation: 0 - 85 °C Maximum Pressure: 790 kPA (100 psig) at 0 - 65 °C The copper wire potentials were measured against the reference electrode and recorded by an IBM-compatible 486 computer through a data acquisition program. These potentials were transformed into local copper wire concentration with copper wire potential - copper ion concentration calibration curves. -58-Chapter 4: Experimental Apparatus and Procedures 4.2 Experimental Procedures 4.2.1 Scope of the work Study was done in the test section filled with two kinds of packed beds - glass beads and carbon felt. Experimental conditions were: The test section dimensions: 60cm (length) x 5cm (width) x 2.6cm (thickness) Temperature: Inlet Pressure: Flow rate: Reynolds number. Main fluid density: Main fluid viscosity: Tracer density: Tracer viscosity: room temperature -10 kPa (gauge) 30 to 120 ml min"1 1.8 to 7.2 (glass beads bed) 0.008 to 0.03 (carbon felt bed) 1.12 x 103 kgm"3 1.2 x 10" 3kgmV 1.11 x 103kgm"3 1.4 x lO^kgm-'s Characteristics of the beds are summarized in Table 4.1 below: Table 4.1 Properties of glass beads and carbon felt beds: Glass beads bed Carbon felt bed Diameter of the glass bead or carbon fiber 2mm 20um Porosity 0.41 0.95 dp/dt 0.077 7.7xl0"4 -59-Chapter 4: Experimental Apparatus and Procedures dp: Glass beads or carbon felt diameter d t: Test section thickness. 4.2.2 Residence Time Distribution Measurement 4.2.2.1 Preparation and Calibration of Copper Wires A l l copper wires were cut with a sharp knife and immersed in 10% HNO3 for 1 minute to wash out any impurity on the cross section at the tips, then rinsed thoroughly with distilled water. Calibrations of copper wire potential against copper ion concentration were performed individually for three main fluids, De-ionized water, 12 wt% Na2S04 and deoxygenated 12 wt% Na 2 S0 4 . The pH of each solution was adjusted to 2 with 98% H2SO4. For each main fluid a series of samples containing CuS0 4 concentration from 0.001M to 1M were prepared for the purpose of calibration. Cleaned copper wires were first immersed into a beaker filled with the main fluid containing zero CuS0 4 , which was connected to the reference electrode pot through a salt bridge as shown in Figure 4.6. The salt bridge was a U-shaped glass tube filled with the main fluid and sealed with filter paper 0.5cm of both ends of the tube. The function of the salt bridge was to provide the necessary electric circuit to this system while preventing bulk mixing of the solutions in the beaker and in the reference electrode pot, therefore the environment around the reference electrode remained undisturbed and the potential of the reference electrode remained constant. Wire potentials became stable after about 10 minutes. The potentials were measured and recorded as a function of time by the data acquisition system. About 15 minutes later, all the wires were taken out of the beaker, dried with a paper towel, and moved into the second beaker containing the main fluid with 0.001M CuS0 4 . One end -60 -Chapter 4: Experimental Apparatus and Procedures end of the salt bridge was kept in the reference electrode pot, while the other was moved into the second beaker. Again the potentials were recorded for about 15 minutes. The same procedure was subsequently repeated for the main fluid with 0.01M, 0.1M, and 1M CuS0 4 . The effect of contacting between carbon felt and copper wire cross section was also investigated by calibrating the wire potential in the same manner, except that all the wires were inserted into a small piece of carbon felt which had been soaked in each solution for a period long enough for the copper ion concentration to reach equilibrium within and outside the carbon felt as shown in Figure 4.6 (b). The activity of the copper wires deteriorated with time and needed to be checked periodically by calibration. 4.2.2.2 Pre-Treatment of Carbon Felt The dry hydrophobic carbon felt cut from a large piece needed to be wetted with wetting agent Makon NF-12 of the concentration 0.025 wt% and rinsed thoroughly with distilled water before being assembled into the test section. The carbon felt needed to be kept wet each time it was taken out of the test section for repacking. -61 -Chapter 4: Experimental Apparatus and Procedures CD -a d o x> d o o -(-> o o OH CD -o O <D is Xi O i 3 (D ^ I • - H d js s CO WH d o , , x O o 3 O O <D i—' o •g CD O <D O d <D •'3 d '3 a CD -4-» CO CD CD (50 O .9 £ <D O i-i rt O 0) <i> rd u o Uc I M xs o <j U W PH' rt co rt -d d o <D co +^ co CD CD -d CD 'co H-> 3 O " r t -»-» O OH CD I-CD OH OH O O <4-C O CD u co rt CD CD -62 -Chapter 4: Experimental Apparatus and Procedures 4.2.2.3 Packing of the Test Section The calibrated copper wires were installed into the four probes that were mounted onto the Plexiglas plate (as illustrated in Figure 4.3) with Nylon compression fittings to prevent leakage. The lengths of the wires were carefully adjusted so that their tips were 0.7cm, 1.5cm, and 2.3 cm away from the surface of the plate. Al l wires were carefully marked. The other piece of Plexiglas plate, Plexiglas spacer (a piece of thin Neoprene gasket of the same shape was attached to each side of it to prevent leakage), the perforated rubber fluid distributor, and two pieces of Neoprene which defined the expanding and contraction section of the flow channel were assembled according to Figure 4.3. Glass beads were filled into the test section above the fluid distributor up to the contraction section. For the carbon felt bed, four pieces of pretreated felts were packed side by side 2.5cm above the rubber fluid distributor and the space between the felts and the distributor was filled with glass beads carefully. Then the Plexiglas plate onto which the probes were attached was put on top of the assembly and the whole section was tied up with 14 sets of 1/4" bolts, nuts and washers. The test section was held vertical with its holder and attached with necessary inlet and outlet piping. 4.2.2.4 In-Situ Copper Wire Potential Measurements The main fluid was driven through the test section at the desired flow rate by compressed air or nitrogen. The copper wire potentials were recorded as a function of time as soon as the flow system started, and tracer was injected into the test section when the wire potentials became -63 -Chapter 4: Experimental Apparatus and Procedures stable. A run was considered complete several minutes after all the tracer passed the exit of the test section and the data acquisition for this run was stopped. 4.2.3 Data Analysis Procedure 4.2.3.1 Calculation of Cu 2 + Concentration Local copper ion concentration was calculated from the copper wire potential by using the copper wire potential - [Cu 2 +] calibration curve. Since the wire potential was not sensitive to the change of [Cu 2 +] when [Cu 2 +] was less than 10"3M, the copper ion concentration corresponding to the wire potential base line was set to 10"3M. [CV +]=10 s l o p e (27) Where, V was the measured potential, Vb a s e was the base line of the measured potential, and slope was the slope of the potential - [Cu 2 +] calibration curve. 4.2.3.2 Preliminary Calculation of Axial Dispersion Coefficient D a The axial dispersion coefficient was preliminarily calculated by assuming no dispersion effects in the lateral and transverse directions. For each run, the axial Peclet number was calculated from Equation (7) with the residence times and variances of two tracer concentration curves measured by two wires at the same horizontal but different vertical positions within the tested bed. Open-open boundary conditions were selected because the injection point and all the sampling points were within the packed bed. -64-Chapter 4: Experimental Apparatus and Procedures 4.2.3.3. Methodology for Simultaneous Calculation of Axial and Transverse Dispersion Coefficients D a and D t For each experimental run, the axial and transverse dispersion coefficients were obtained by parameter estimation based on the measured and calculated tracer concentrations at two representative positions in the tested bed: the tips of wire 1 and wire 8. Tracer concentration at wire 1 was greatly affected by axial dispersion and that at wire 8 was largely affected by transverse dispersion. The measured tracer concentrations were obtained from a pulse tracer experiment and the theoretical tracer concentration distribution in a packed-bed for any set of estimated D a and D t can be calculated from the time-dependent three-dimensional partial differential Equation (2) governing the mass balance in the bed (refer to Figure 2.5). X - , y-, and z- directions are called transverse, axial and lateral direction, respectively. The dispersion coefficients in these directions, D x , D y , and D z are called transverse, axial and lateral dispersion coefficient, D t , D a , and Dj, accordingly. In the present experimental arrangement, u x = u z = 0. Equation (2) was simplified to a two-dimensional equation (Equation 28) by assuming isotropic conditions in the transverse and lateral directions - the transverse and lateral dispersion coefficients had the same values and the tracer concentration gradients in the two directions were equal, except near the walls of the bed. Tracer concentration profiles within the bed were obtained by solving Equation (28) in a time period, e.g., 540 seconds, with a 2 seconds sampling interval. Two vectors containing tracer concentrations calculated at discrete times at the tips of wire 1 and wire 8 were picked up from the solution. -65 -Chapter 4: Experimental Apparatus and Procedures (2) dc_ a a_ a So d2c a (28) Dtl = 2D, = 2D, Equation (28) was solved for several sets of D a and D t , and the set that gave the minimum sum of squares of the differences between calculated and measured tracer concentration at the tips of wire 1 and wire 8 were the sought after axial and transverse dispersion coefficients for the studied bed in that specific run. A program written in MatLab and FemLab was developed to perfprm the calculation. The algorithm of this program is shown in Figure 4.7. The partial differential Equation (28) was solved with FemLab in a rectangular regime as shown in Figure 4.8. (i) Boundary Conditions of the Rectangular Regime To solve Equation (28), the boundary conditions at the four edges of the rectangular regime must be specified. It was not difficult to specify the b.c.'s at the two sides since there were no fluxes passing through those two sides, i.e., dc _ dx (29) -66 -Chapter 4: Experimental Apparatus and Procedures However, the conditions of the lower and upper boundaries had to be determined by calculation. After a shot of tracer was injected into the test section, most of the tracer traveled upwards with the main fluid while a small amount would drop due to axial dispersion effect and eventually flow upwards again with the main fluid. Tracer concentration profiles within the bed were recorded in a period before the tracer was injected till the tracer passed all the monitoring wires. It usually took about 3 to 15 minutes for the tracer to pass all the wires after injection depending on the flow rate in a specific run. It took less time when the flow rate was higher. The distance that the tracer could reach above the injection point was proportional to time. Within the run period (3 to 15 minutes), the tracer concentration would always be zero above a certain distance from the tracer injection point, and the distance that tracer would reach below the injection point due to backmixing was also limited. Therefore, the b.c.'s at the upper and lower boundaries was c = 0 (30) The upper and lower boundary of the rectangular regime a and b were determined by trial and error with experimental data. If a and b were selected too large, computer time would be wasted on calculation on unnecessary areas. On the other hand, if a and b were too small, the zero tracer concentration condition might not be held, (ii) Treatment of the Input for Equation (28) The discrete tracer concentration data measured by wire 7 was transformed into a function of time fitted with a non-symmetrical curve, -67-Chapter 4: Experimental Apparatus and Procedures y = A exp - e x p ( - f e ^ ) ) - f e ^ ) + l v y C ' C (31) Amplitude: A; x at Max: B; Area: 2.72C This known tracer concentration input was treated as the boundary condition of a very small circle centred at the tip of wire 7, (0.003m, a + 0.1m) in Figure 4.7, with a diameter of 0.0005m considered negligible compared to the dimension of the rectangular regime. -68-Chapter 4: Experimental Apparatus and Procedures For i = 1:10, Da(i)= 10"3 x 1O^2) Forj = 1:10, Dtl(j)=10-3xl0(-J/2) 4 Known C7 = f(t) Solve Eqn. (28)=f(Da(i), Dm t) Calculated tracer cone.: fCl( i , j , t ) , fC8(i,j,t) Measured tracer cone: rCl(t), rC8(t) S(i,j)=I((fCl(i,j,t)-rCl(t)) 2 +(fC8 (i, j , t)- rC8(t))2) 4 When i=i' and j=j ' v S(i', j ')= min(S(i, j)) Figure 4.7 Algorithm of the program for solving D a and D t simultaneously. -69-Chapter 4: Experimental Apparatus and Procedures c = 0 dc/dx = 0 3mm 7T 8mih c = 0 dc/dx = 0 - Wirel Wire 7 " Wire 8 Injection point 26mm < • Figure 4.8 Rectangular regime for solving Equation (28) (Wire 7 and 8 were at the same vertical position and very close to each other. They were displayed apart in order to make the figure clear, a and b have to be determined experimentally.) •70. Chapter 5: Results and Discussions Chapter 5 RESULTS and DISCUSSIONS The objective of the present work was to establish an experimental system and to develop a mathematical methodology to measure and calculate the axial and transverse dispersion coefficients in a rectangular packed-bed. In order to achieve the proposed goal, a tracer stimulation-response measuring system was first set up and suitable main fluid and tracer were selected. Then the tracer response experiments were performed in the test section filled with randomly packed glass beads or carbon felt at different flow rates. The effect of repacking the beds was also investigated. For each experimental condition, the axial dispersion coefficient was calculated by two methods: 1) estimated from the first and second moment of the measured tracer concentration curves at two different axial positions within the bed, neglecting the effects of lateral and transverse dispersion; 2) calculated simultaneously with the transverse dispersion coefficient by solving the material balance equation in the bed. The axial dispersion coefficients obtained by the two methods were compared with each other. The transverse dispersion coefficient was calculated simultaneously with the axial dispersion coefficient by using a computer program written in MatLab and FemLab to solve the time-dependent two-dimensional partial differential equation with the assumption that the lateral and transverse dispersion coefficients were equal. - 71 -Chapter 5: Results and Discussions 5.1 Setting up the Tracer Stimulation-Response Measuring System 5.1.1 Effect of Fluid Distributor A typical tracer response in the test section filled with glass beads that had proper fluid channels but not a fluid distributor is shown as solid squares in Figure 5.1. Severe deviation from plug flow was identified by comparing with compartment models. The relatively early peak and the long tail indicated stagnant backwaters, and the multiple peaks were signs of internal recycling or channelling [9]. It was also confirmed by visual observation with the aid of blue dye in the tracer that the fluid moved much faster on the side of the test section to which the inlet connector was attached than on the other side. This poorly behaved test section was improved as shown in Figure 5.1 (the solid circles) by adding a fluid distributor - a piece of perforated rubber fluid distributor and glass beads bed under the bottom of the tested packed bed. The perforated rubber forced the fluid to divid into evenly distributed small streams and the glass beads bed above it further damped the velocity fluctuations in the cross area of the test section. Detailed calculation of pressure drop through the test section and through the fluid distributor is given in Appendix 1. - 72 -Chapter 5: Results and Discussions -200 Time (min) • with fluid distributor • without fluid distributor Figure 5.1 Comparison of fluid patterns in the test section filled with glass beads with and without fluid distributor: wire potential vs. time with 0.7M C u S 0 4 pulse injection. - 73 -Chapter 5: Results and Discussions 5.1.2 Tracer and Main Fluid Selection 5.1.2.1 Tracer Selection C U S C M was selected as the tracer in the present study due to the easy availability of copper wires and relatively accessible Cu 2 + /Cu potential. Trypan Blue was added to CuS0 4 solution to facilitate visual monitoring of the flow conditions in the test section. From 0.5M to 2 M C U S C M was tested as tracer, it was found that higher concentration tracer gave sharper response peak. However, it was also observed that concentrated CuS0 4 dropped along the test section instead of moving upwards with the main flow after injection because the density of the tracer was much higher than that of the main fluid. 0.7M CuS0 4 with 0.01 wt % Trypan Blue was selected as the tracer determined by a compromise between the density of the tracer and that of the main fluid. The concentration of the tracer was limited by the density of the main fluid. 5.1.2.2 Main Fluid Selection KC1 was first selected as the main fluid flowing through the test section because of its electrochemical advantage of equivalent cation and anion mobility. However, later experimental results showed that the tracer stimulation - response system was unstable due to the formation of copper-chloride complex ions. Then deionised water adjusted to pH = 2 with 98% H2S0 4 was selected as the main fluid. It is shown from the potential-pH equilibrium diagram for copper-water system [22] that when pH is smaller than 3 only the Cu 2 + /Cu redox pair exists in the system and the copper wire potential is a straightforward function of local copper ion concentration. But the problem with water was that the density is too small to support the CuS0 4 tracer. - 74 -Chapter 5: Results and Discussions 12 wt % Na2S0 4 adjusted to pH = 2 with 98% H 2 S O 4 was selected as the main fluid in order to increase the density of the main fluid with a relatively simple composition and a reasonable cost. Na2S0 4 and CuS0 4 aqueous solution density data are attached in Appendix 3. 5.1.3 Calibration of Copper Wire Potential vs. [Cu2+] The influences of several parameters including flow rate, different main fluid, oxygen, and copper wire surface condition on the copper wire potential were individually investigated in several sets of experiments. 5.1.3.1 Effect of Flow Rate on Copper Wire Potential The potentials of three copper wires at different flow rates in 12 wt % Na2S0 4, pH =2 solution were displayed in Figure 5.2. It indicated that the wire potential was independent on flow rate. 5.1.3.2 Effect of Different Main Fluid on Copper Wire Potential Calibrations of copper wire against copper ion concentration were performed in three main fluids, deionised water, 12 wt % N a 2 S 0 4 and deoxygenated 12 wt % Na2S0 4. The pH of all the solutions was adjusted to 2 with 98 % H2SO4. Four solutions containing 0.001M, 0.01M, 0.1M and 1M C u 2 + , respectively, were prepared in four beakers for each main fluid. The solutions were obtained by adding CuS0 4 into the main fluid in each beaker. Copper wires were inserted into the solutions, in turn, from the most diluted to the most concentrated of Cu 2 + . Copper wire potentials were measured and recorded against the reference electrode inserted in the reference electrode pot that was filled with the main fluid with zero copper contents and connected to the beakers with salt bridges. - 75 -Chapter 5: Results and Discussions All the wire surfaces were freshly cut and treated with -15% HNO3 for about 1 minute. Straight lines in a semi-log plot of copper wire potential vs. copper ion concentration were obtained for all the 10 wires in all the three main fluids. The lines for all the wires were very close to each other and all the slopes obeyed the Nernst equation. The relationships between wire potential and copper ion concentration for a representative wire are illustrated in Figure 5.3. No evident discrepancies in either the copper wire potentials or the relationships between copper wire potential and copper concentration were observed between the deionised water and 12 wt % Na2S04 solutions. The wire potential response to copper ion concentration was not affected by the existence of Na + , which was expected since only the concentration of C u 2 + will affect the copper wire potential under the experimental condition. Dissolved oxygen in 12 wt % Na2S0 4 solutions was removed by purging the solutions with pressurized N2 for 30 minutes. Copper wires potentials were measured with these solutions being carefully covered to prevent them from dissolving 0 2 . O2 had little effect on the relationship between copper wire potential and copper ion concentration. This could be explained by the fact that ( V O F f redox pair would not interfere with Cu 2 + /Cu redox pair under the experimental pH condition (pH = 2). Copper wire potential was not sensitive to copper ion concentration less than 10"3M for all the main fluids, so the lower limit of the measured copper ion concentration was set to 10"3M. - 76 -Chapter 5: Results and Discussions -210 -220 > E 1 -230 c CD ' O Q. -240 -250 i i l I I I L 20 40 , , I i i . . I ' i i i I i ' ' ' ' ' 1 1 1 1 60 80 100 120 140 Flow rate (ml min"1) • wire 2 A wire 9 wire 8 Figure 5.2 Effect of flow rate on copper wire potential. - 77 -Chapter 5: Results and Discussions -140 -160 -180 •B -200 c O °- -220 i -240 -260 -4 i i i i i i i i i I i i i i I — i — i — i — i — -3 -2 -1 log [Cu 2 +] 0 Deionised water 12wt% N a 2 S 0 4 Deoxygenated 12 wt% N a 2 S 0 4 Figure 5.3 Relationship between copper wire potential and copper ion concentration in three main fluids. - 78 -Chapter 5: Results and Discussions 5.1.3.3 Effect of the Copper Wire Surface Condition on Copper Wire Potential (i) Aging of the wire surface The freshness of the cross section of copper wire tip was a key factor in the copper ion concentration measurement. Figure 5.4 shows potentials of wires 7 days after being inserted into 12 wt% Na2SC>4, pH = 2 solution. Wires 7 and 9 did not give the normal potential response to copper ion concentration change - the slopes of calibration curves for wires 7 and 9 were much smaller than those of the normal wires such as wire 2 and 6. It was found out later under a microscope that copper inside wires 7 and 9 were about 1mm eaten into the insulator caused by the corrosion of the electrolyte solution to the copper. This is called aging of the wire. The large amount of stagnant water kept in the tip of the wire insulator prohibited the copper in the wires from monitoring the copper ion concentration change. In other words, the wires were deactivated with aging. - 79 -Chapter 5: Results and Discussions • wire 2 * wire 7 log [Cu 2 +] wire 6 wire 9 Figure 5.4 Relationships of wire potential and copper ion concentration for active and "deactivated" wires. - 80 -Chapter 5: Results and Discussions (ii) Effect of Contacting between Copper Wire and Carbon Felt on Wire Potential A small piece of carbon felt was put into each of the beakers containing 12 wt% Na2SC>4, pH = 2, with different C U S C M concentrations. The effect of contact between copper wire and carbon felt on wire potential was studied by performing calibration procedure with the copper wires inserted in the carbon felt. Straight calibration lines were obtained as shown in Figure 5.5. It is interesting to notice that the calibration curve obtained with the wire inserted into carbon felt parallelly moved about 70mv upwards comparing to that obtained without the existence of the carbon felt. One possible explanation for these higher values of copper wire potential could be that the existence of the carbon felt lowered the electric resistance between the reference electrode and the wires. Another possible explanation is that the surface condition of the wires were changed by contacting with the carbon felt, in other words, the recorded potentials were given by a carbon electrode. The second explanation seems to be more plausible. Wire potential oscillated between two values when a wire was moved within the carbon felt from time to time as shown in Figure 5.6. One value was higher than normal potential; the other was the same as that measured with no carbon felt inside the beaker. It seems that the latter was given by the copper wire and the former was given by the combination of the copper wire and carbon felt. - 81 -Chapter 5: Results and Discussions 0 • Without carbon felt • Within carbon felt Figure 5.5 Relationships between wire potential and copper ion concentration within and without carbon felt (In 12 wt % Na 2 S0 4 , pH = 2 solutions). -l82 -Chapter 5: Results and Discussions i l i i i i l i i i i I i i i i I i—i—i—i—1—i—i—i—i—I 5 10 15 20 25 30 Time (min) Wire potential in 12 wt % N a 2 S 0 4 with 1M [Ci f ], pH = 2, within a carbon felt in the beaker. - 83 -Chapter 5: Results and Discussions The second explanation was further confirmed by later experimental finding that when copper wires with newly cut surface were inserted into a carbon felt bed, all the wires gave identical potential and had no response to the tracer injection. There was no individual tracer response because all the copper wires touched the carbon felt and formed one single "big" electrode. This created a difficult situation for tracer experiment for the test section filled with carbon felt bed. Newly cut copper wires could not be directly used in a carbon felt bed as in glass beads bed. The wires must be left in the main fluid for a period of time in order to have the copper corroded by the electrolyte and backed into the insulator to prevent it from touching the carbon felt. On the other hand, the wires would be deactivated if the copper were over corroded as illustrated in Figure 5.4. There was no systematic experimental data on the period of time needed for the wire surface to reach the optimum condition. It had to be determined by trial and error for each specific run. The performances of the wires were closely monitored all the time and the wires were calibrated periodically outside the test section. The surfaces of the wires were retreated when deactivation happened, (iii) Reproducibility of Copper Wire Calibration The slopes of wire potential vs. copper ion concentration curves in 6 calibration runs are displayed in Figure 5.7. The average slopes for each wire over these runs are illustrated with solid circles. A reasonable reproducibility was obtained and the average slope for all the wires was 23.9 mV. This slope agrees with the theoretical slope calculated by Nernst equation, which is given in Appendix 2. - 84 -Chapter 5: Results and Discussions E 50 CD a 40 + CM 13 o c? 30 ••5 20 .= 10 0 a J_ 3 4 5 6 Wire No. 8 10 o Run 1 • Run 2 A Run 3 v Run 4 o Run 5 o Run 6 • Average Figure 5.7 Slopes of calibration curves for all the wires in several runs. - 85 -Chapter 5: Results and Discussions 5.2 Axial Dispersion Coefficient Estimated by Neglecting Transverse Dispersion With the assumption of negligible lateral and transverse concentration gradients, the axial dispersion coefficient was calculated from the spread of the tracer stimulation-response curves monitored by two wires positioned at same horizontal but different vertical positions within the tested bed. In order to quantitatively relate the tracer stimulation-response experimental data to the axial dispersion coefficient in a packed bed, the local copper ion concentrations were first calculated from measured local copper wire potentials by using the potential - concentration calibration curves. Then D a /uL was obtained from the information of the mean residence times and variances of these two concentration curves by using Equation (9). A sample calculation of D a /uL and axial dispersion coefficient D a is given in Appendix 4. Two axial dispersion parameters, axial Peclet number (Pea) and (Dae/udp) for glass beads bed or (Das/udf) for carbon felt bed were calculated from D a /uL and other characteristics of the beds including particle size d p (or carbon fibre diameter, df) and bed porosity s. The influences of flow rate and re-packing of the beds on axial dispersion were investigated in the present work. The flow rate in a test section is represented by Reynolds number, Re = udpp/u. (or Re = udfp/u). The degree of axial dispersion is represented by the axial Peclet number, Pe a = udp/Da (or Pe a = udf/Da), D ae/ud p (or Dae/udf), and the axial dispersion coefficient D a . - 86 -Chapter 5: Results and Discussions 5.2.1 Randomly Packed Glass Beads Bed The porosity of glass beads bed was about 0.41±0.01. It was calculated by measuring the maximum volume of water added into the interstitial spaces between glass beads contained in a known volume test section. V V test _ sec hen A sample calculation of other parameters such as interstitial flow velocity u, Reynolds number Re, D a/ud p, D as/ud p, axial Peclet number Pe a and axial dispersion coefficient D a is given in Appendix 5. 5.2.1.1 Effect of Flow Rate on Axial Dispersion Axial dispersion parameters based on the experimental data of wires 7 and 1 at different flow rates are summarized in Table 5.1, while D ae/ud p and Pe a versus Re are plotted in Figure 5.8. The distance between wires 7 and 1 was L = 0.2m. (All the wires mentioned here are referred to Figure 4.1 unless annotated otherwise). Each parameter was the average of values calculated from at least 3 runs at each flow rate without repacking the test section. The maximum standard deviation of the estimated parameters was 20%. Table 5.1 D a /uL, Pea, D ae/ udp and D a at different flow rates for glass beads bed: Flow rate Re u D a /uL D a/ud p Pe a D a s/ udp D a (ml min"1) (lO^ms"1) (10" 6mV) 30 1.8 9.6 0.033 3.3 0.30 1.32 6.3 60 3.6 19.2 0.032 3.2 0.31 1.28 12.3 90 5.4 28.8 0.033 3.3 0.30 1.32 18.4 120 7.2 38.4 0.031 3.1 0.32 1.26 23.8 - 87 -Chapter 5: Results and Discussions • - • 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 1 2 3 4 5 6 7 8 Re • D a s/ud p • P e a Figure 5.8 Relationship between D as/ud p , P e a and Re for glass beads bed. - 88 -CD CL ro Q 1.5 1.0 0.5 Chapter 5: Results and Discussions Pe a and D as/ud p were around 0.3 and 1.3, respectively, nearly constant within the range of the flow rate studied. Pe a was slightly smaller than the literature value of 0.4 for Re from 1 to 10 as shown in Figure 2.10 [3]. D as/ud p was about the same as given by Levenspiel [9], illustrated in Figure 2.11. D a was proportional to the flow rate: 6 to 24 x 10"6 m 2 s"1 for Reynolds number from 1.8 to 7.2. The fact that D a increased with flow rate indicated the contribution of convective dispersion was significant. 5.2.1.2 Uniformity of D a in the Entire Bed D a /uL calculated from two sets of wires, wires 7 and 1, and wires 7 and 4, were shown in Figure 5.9. As illustrated in Figure 4.1, wire 7, 4 and 1 were placed at the same horizontal but different vertical positions in the tested bed. The distance between wire 7 and wire 1, L 1.7, was 0.2m and that between wire 7 and wire 4, L 4.7, was 0. lm. D a /uL from wire 7 and 4 was about twice that from wire 7 and 1. Since the distance L 4.7 was half of L 1.7, D a estimated from these two sets of wires were about the same. This suggests that D a was reasonably uniform throughout the entire bed. 5.2.1.3 Effect of Re-Packing of Glass Beads Bed on Axial Dispersion Coefficient D a /uL estimated from three packings of glass beads bed are shown in Figure 5.10. There was little difference in D a /uL between different packings. This result indicated that the conditions of randomly packed glass beads bed were reproducible with consistent and careful repacking. - 89 -Chapter 5: Results and Discussions ra Q 0.10 0.08 h 0.06 0.04 0.02 0.00 • D.AJL . , .7 • D a /uL 4 . 7 Figure 5.9 D g / u L ^ and D a /uL 4 . 7 at different flow rates for glass beads bed. - 90 -Chapter 5: Results and Discussions 0.10 0.08 0.06 CO Q 0.04 h Re o Pack No. 1 • Pack No. 2 A Pack No. 3 Figure 5.10 Comparison of D a/uL vs. Re of different packings of glass beads bed. - 91 -Chapter 5: Results and Discussions 5.2.2 Carbon felt bed Axial dispersion at different flow rates was also studied in the test section filled with four pieces of carbon felt. The effect of the contact between the pieces of the carbon felt was neglected. A sample calculation of interstitial flow velocity u, Re, Da/udf, D a e/ udf, Pe a and D a is given in Appendix 5. Average values of D a /uL, Pe a and D a s / udf at each flow rate in carbon felt bed are listed in Table 5.2. Two to four runs were performed at each flow rate and the maximum standard deviation of the estimated parameters was 20%. Table 5.2 Da/uL, Pea, D a s / udf, and D a at different flow rates for carbon felt bed: Flow rate Re u D a /uL D a/ud f Pe a D a s/ udf D a (ml min"1) (lO-Vs" 1) (10 - 6 mV) 30 0.008 4.3 0.018 180 0.0056 162 1.5 60 0.016 8.6 0.016 160 0.0063 144 2.7 . 90 0.024 12.9 0.018 180 0.0056 162 4.6 120 0.032 17.2 0.020 200 0.0050 180 6.8 The axial dispersion coefficient D a increased from 1.5 to 6.8 x 10"6 m 2 s"1 when flow rate increased from 30 to 120 ml min"1 (Re from 0.008 to 0.032) as shown in Figure 5.11. It can be concluded from Figure 5.12 that Pe a was constant at about 0.006 in carbon felt bed for flow rate from 30 to 120 ml min"1. The constant Pe a indicated that convective dispersion was significant in the tested flow region. D a s/ udf was about constant around 160. - 92 -Chapter 5: Results and Discussions 0.005 0.010 0.015 0.020 0.025 0.030 0.035 Re Figure 5.11 Relationship between axial dispersion coefficient D a and Reynolds number Re for carbon felt bed. - 93 -Chapter 5: Results and Discussions CD CL 0.010 0.008 0.006 0.004 h 0.002 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 Re Figure 5.12 Relationship between axial Peclet number P e a and Reynolds number Re for carbon felt bed. - 94 -Chapter 5: Results and Discussions No dispersion data were found in the literature for packed beds with similar characteristics to the carbon felt bed investigated in the present work in terms of bed porosity and diameter of the bed particle (fibre). No similar Pe a value was found in the literature. Pe a was reported as about 0.4 for a packed bed as shown in Figure 2.10 [3]. The Bodenstein number of the present carbon felt bed of 6 x 10"3 was only about hundredth of the literature value, which was 0.5 for Re less than 10 as shown in Equation (26) [4]. The extreme low Peclet and Bodenstein number in the tested carbon felt bed was caused by the small carbon fibre diameter, which is about hundred times smaller than that of particles used in ordinary packed beds, and the high porosity of the carbon felt, which was about twice the porosity of an ordinary packed bed. No evidence of re-packing effect on axial dispersion parameters was found in the carbon felt bed and D a was found to be uniform through the entire bed by checking with two sets of monitoring wires. 5.2.3 Comparison of Axial Dispersion in Glass Beads and Carbon Felt Beds Pe a value at similar Reynolds number range (-0.01 to 0.03) for glass beads bed with porosity of 0.4 were about 0.5 (Figure 2.10) [3] which was about 100 times higher than that obtained for the carbon felt bed. The extreme low Pe a value for carbon felt bed was caused by the very small diameter of the carbon fibre. On the other hand, Dae/udf for the carbon felt bed was about 100 times larger than the literature value for glass beads bed within the same Reynolds number range. (Refer to Table 5.2 and Figure 2.11) However, the axial dispersion coefficient in the carbon felt was in the same order as that in glass beads at the same flow rate (i.e., the same superficial liquid velocity) as shown in Figure - 95 -Chapter 5: Results and Discussions 5.13. The larger axial dispersion coefficient in glass beads could be explained by its smaller bed porosity - broader spread of the tracer concentration curve would be expected because the fluid velocity fluctuation would be more severe in a bed with smaller porosity. - 96 -Chapter 5: Results and Discussions 0 30 60 90 120 Flow rate (ml min"1) 150 • Carbon felt bed • Glass beads bed Figure 5.13 Comparison of axial dispersion coefficient D a for glass beads and carbon felt bed. - 97 -Chapter 5: Results and Discussions 5.3 Simultaneous Calculation of Axial and Transverse Dispersion Coefficients The axial and transverse dispersion coefficients D a and D t were found from the best fit between the measured and calculated tracer concentrations at two representative points in the tested bed. Calculated tracer concentrations were obtained by solving Equation (28) with the assumption that the lateral and transverse dispersion coefficients were of the same magnitude. Time-dependent partial differential Equation (28) was solved with FemLab in a rectangular regime. 5.3.1 Determination of the Upper and Lower Boundary of the Rectangular Regime As discussed in Section 4.2.3.3, the boundary conditions at the upper and lower boundaries of the rectangular regime were set to be c = 0. With the experimental design in the present work, the tracer concentration would always be zero lm above the injection point, i.e., lm / velocity of the fluid > time period of the run. It was also observed that the tracer could not reach 0. lm below the injection point due to backmixing by using FemLab animation function to check the tracer concentration distribution during a typical run for each flow rate. Therefore, the dimension of the rectangular region was selected to be 1.6m x 0.0026m. 5.3.2 Determination of Axial and Transverse Dispersion Coefficients D a and Dt Determination of axial and transverse dispersion coefficients D a and D t is illustrated with a sample run 04284a (glass beads bed, flow rate: 90 ml min"1). Equation (28) was solved for several sets of D a and D t i (D ti = 2D t = 2Di) with the computer program attached in Appendix 6. The sum of squares of differences between measured and calculated tracer concentrations for wire 1 and 8 with each set of D a and D t i was - 98 -Chapter 5: Results and Discussions exported to a separate file named 'sum04284a_csv'. The sum as a function of D a and D ti is listed in Table 5.3 and displayed in Figure 5.14. Table 5 . 3 Sum of squares of differences between measured and calculated tracer 3 25 V 2 1 concentrations for wire 1 and 8 for D a and D ti ranging from 10" ' to 10" m s" . Sum (10"3) D a (m 2 s 1 ) 1 r j 3 2 5 1 o"3 5 1 Q-3.75 10"4 10-425 10"45 1 0-4.75 10"5 1 0-5.25 1 Q-5.5 1 Q-5.75 10"6 1 Q-6.25 1Q-65 1 Q-6.75 10 7 10"^ 189.0159.0129.3101.1 76.4 57.0 43.2 34.9 31.3 31.0 32.3 33.5 34.4 34.8 35.0 35.2 1 Q - 3 5 185.7156.9126.0 97.3 72.0 51.7 37.3 28.4 24.2 23.3 23.7 24.2 24.4 24.4 24.4 24.4 1 0-3.75 180.5151.1 119.8 90.8 65.4 45.3 31.1 22.6 18.7 17.7 17.7 17.6 17.2 16.9 16.7 16.6 10"4 174.4144.2112.9 82.9 57.7 38.3 25.3 17.8 14.7 14.2 14.4 14.2 14.2 13.8 13.5 13.3 1 0-4.25 167.6136.1 104.0 74.1 49.2 31.0 20.0 14.2 12.3 12.7 13.2 13.5 13.5 13.3 13.4 13.3 10"45 160.2127.0 93.9 64.0 40.4 24.1 14.8 11.6 11.3 11.7 13.4 14.1 14.4 14.8 14.7 14.7 1 0-475 152.7116.8 82.8 53.7 32.0 18.2 11.4 9.4 10.6 13.1 14.6 15.4 16.1 15.9 15.8 15.7 10"5 145.9106.3 71.6 43.8 24.6 13.7 9.3 9.4 11.8 14.8 16.8 17.3 17.0 16.7 16.4 16.3 1 0-525 140.7 96.9 61.3 35.3 18.8 10.7 8.8 10.5 13.8 16.9 18.4 18.4 17.8 17.4 17.1 17.1 1 0 - 5 5 137.7 89.8 53.5 28.9 14.8 9.3 9.4 12.4 15.9 18.6 19.6 19.2 18.6 18.1 17.9 17.9 1 Q-5.75 136.6 85.5 48.2 24.7 12.5 9.0 10.7 14.3 17.6 19.7 20.3 19.9 19.2 18.7 18.5 18.5 10"6 135.9 82.7 44.9 22.2 11.3 9.3 12.2 16.0 18.7 20.3 20.7 20.3 19.6 19.1 18.9 18.9 1 Q-6.25 135.6 81.0 42.9 20.7 10.8 9.8 13.4 17.1 19.3 20.6 21.0 20.6 19.9 19.2 18.8 18.6 1 0-6.5 135.4 80.1 41.8 19.9 10.6 10.2 14.2 17.8 19.5 20.7 21.2 20.7 20.0 19.1 18.2 17.6 1 Q-6.75 135.3 79.5 41.1 19.4 10.5 10.5 14.8 18.2 19.7 20.8 21.3 20.8 20.0 18.8 17.5 16.5 10"7 135.3 79.2 40.8 19.2 10.4 10.7 15.1 18.4 19.7 20.8 21.3 20.8 19.9 18.6 16.9 15.8 - 9 9 -Chapter 5: Results and Discussions It was found from Table 5.3 that D a = 10" 4 7 5 and D t i = 10"5 2 5 m 2 s"1, which corresponded to axial dispersion coefficient D a = 17 x 10"6 and transverse dispersion coefficient D t = 10"5 2 5 / 2 = 3 x 10"6 m 2 s"1, gave the best fit between the measured and calculated tracer concentrations. The same procedure was also performed for a narrower range of D a and Dti and the result was shown in Table 5.4 and Figure 5.15. The minimum sum was obtained with D a = 14 x 10"6 and D t = 4 x 10"6 m 2 s"1 (D ti = 8 x 10"6 m 2 s'1), respectively. Table 5.4 Sum of squares of differences between measured and calculated tracer concentrations for wire 1 and 8 with D a ranging from 2 - 22 x 10"6 and D t i ranging from4-24xl0"6m2s"1. Sum (10°) D a ( lO^mV) 4 8 Dti(10"6 12 m1 s"1) 16 20 24 2 6.09 4.65 3.28 2.30 1.71 1.48 6 3.90 1.83 1.15 1.28 1.8 2.81 10 2.51 0.951 1.12 2.08 3.44 5.04 14 1.65 0.727 1.57 3.11 4.99 7.05 18 1.12 0.776 2.1 4.13 6.39 8.77 22 0.789 0.952 2.73 5.06 7.62 10.3 However, it was noticed from Figure 5.14 and 5.15 that the best fit between the measured and calculated tracer concentrations under the present experimental conditions was found in a small region rather than at a distinguishable point defined by D a and D t . In other words, in -100 -Chapter 5: Results and Discussions contrast to D a = 14 x 10"6 and D t = 4 x 10"6 m 2 s"1, it is more suitable to argue that for the tested glass beads bed with a flow rate of 90 ml min"1, axial and transverse dispersion coefficients were in the ranges of D a = 10.1-31.6 x 10"6 (D a = I O " 4 7 5 * 0 2 5 ) and Dtl = 3.1-10.1 x 10"6 (D ti = 10" 5 2 5 ± 0 2 5 ) m 2 s"1 (marked as the bold figures in Table 5.3). In all the runs of both glass beads and carbon felt bed, the sums of squares of the differences between the measured and calculated tracer concentrations were in the range of 10"5 to 10" with D a and D t i ranging from 10"" to 10" m s" ; the best fitted axial and transverse dispersion coefficients were found in a small region (lO* 0 2 5 ) rather than at a sharp point, and the sums were about 10"5 in this region. -101 -Chapter 5: Results and Discussions Da (m2 s-1) Figure 5.14 Sum of squares of differences between measured and calculated tracer concentration for wire 1 and 8 as a function of D a and Da (D ti = 2D t = 2 Di) (Run 04284a: glass beads bed, 90 ml min l). - 102 -Chapter 5: Results and Discussions 12 16X10" 6 m* s"1 2 0 X 1 0 " 6 m* s"1 2 4 x 1 0 " 6 m 2 s"1 Figure 5.15 Sum of squares of differences between measured and calculated tracer concentrations for wire 1 and 8 with D a and D„ (D„ = 2D, = 2D,) from 2 -22 and 4 - 24 x 10"6 2 -1 m s . - « - D u = 4 x 1 0 * r T r V - ^ - D „ = -m- D t l = 8 x I f J 6 m 2 s _ 1 D t l = - A - D = 12 x 10"6 m 2 s"1 - » - D , 1 = - 103 -Chapter 5: Results and Discussions 5.3.3 Samples of Measured and Calculated Tracer Concentration Curves One sample of the measured and calculated tracer concentration curves is given for glass beads and carbon felt bed in Figure 5.16 and 5.17, respectively. Measured tracer concentrations by wires 1, 7 and 8 are represented by cross, triangle and circle symbols and the calculated concentrations are shown with solid lines. Reasonable fit was considered to be obtained though there was room for improvement. For example, the gap between the measured and calculated tracer concentrations could be caused by the errors related with the slope of the copper wire potential vs. copper ion concentration curve - the wire surface condition deteriorated with time and the slope of the wire potential vs. copper ion concentration curve drifted from that when the wire was calibrated, which was assumed to be constant and employed to transfer the measured copper wire potential into copper ion concentration. A better understanding and control of the copper wire surface condition as well as a finer tuning up of the test section would be helpful to improve the accuracy of the experiments. -104 -Chapter 5 Results and Discussions [Cu2+]mea, 1 [Cu2+]mea, 7 [Cu2+]mea, 8 [Cu2+]cal, 1 [Cu2+]cal, 7 [Cu2+]cal, 8 100 150 200 250 Time (s) 300 350 400 Figure 5.16 Measured and calculated tracer concentrations for wires 1, 7 and 8. (Glass beads bed, 90 ml min"', D a = 14 x 10"6 and D t = 4 x 10"6 m 2 s"1) -105 -Chapter 5 Results and Discussions x10 -3 1000 1500 Time (sec) 2000 2500 Figure 5.17 Measured and calculated tracer concentrations for wires 1, 7 and 8. (Carbon felt bed, 30 ml min' , D A = 1 x 10"6 and D T = 2 x 10"6 m 2 s 1) - 106 -Chapter 5: Results and Discussions 5.3.4 Axial and Transverse Dispersion Coefficients D a and D t for Glass Beads and Carbon Felt Bed 5.3.4.1 Glass Beads Bed Simultaneously calculated axial and transverse coefficients at different flow rates for glass beads are shown in Figure 5.18. The values are the average values for three to four runs at each flow rate. The maximum standard deviation of the parameters calculated from replicate runs was about 20%. However, as discussed in Section 5.3.2, the uncertainty of the simultaneous estimation of the axial and transverse dispersion coefficients was about 50%. Therefore, the uncertainty of the parameters was reported as 50%. The axial dispersion coefficient D a was proportional to flow rate, increasing from 6 to 21 x 10"6 m 2 s"1 for Re from 1.8 to 7.2, while the transverse dispersion coefficient D t was also increased with flow rate but at a higher rate, from 0.55 to 7.6x IO - 6 m 2 s"1 for the same Reynolds number range. Axial and transverse dispersion coefficients should agree at low Reynolds numbers (Re <10'4 for liquid) where they both are independent of Re and equal to molecular diffusivity [35] because dispersion is due to molecular diffusion and the transverse and axial structures of the bed are similar. At high Reynolds numbers where convective dispersion dominated the dispersion values are proportional to Reynolds number and axial and transverse dispersion coefficients are different because axial dispersion is primarily caused by differences of fluid velocity in flow channels, while transverse dispersion is primarily caused by deviation in the flow path caused by bed materials. The axial dispersion coefficient becomes proportional to - 107 -Chapter 5: Results and Discussions flow rate at Reynolds number as low as 10"4; on the other hand, the transverse dispersion coefficient does not start to increase proportionally with flow rate till Re exceeds 100. Within the investigated flow rate range, (l<Re<10), the axial dispersion coefficient increased steadily with flow rate, while the transverse dispersion coefficient was under the transition from molecular diffusivity to convective dispersion. Accordingly, the axial Peclet number was constant and the transverse Peclet number was neither constant nor increased with Re - it dropped with increased Re which indicated that the transverse dispersion number increased at a higher rate than the Reynolds number did. Axial and transverse Peclet numbers at different flow rates for glass beads are shown in Figure 5.19. Pe a was about constant at 0.35 while Pet dropped from 7 to 1 when flow rate increased from 30 to 120 ml min'1 (Reynolds number from 1.8 to 7.2). Both the trends of axial and transverse Peclet numbers vs. Reynolds number agree with the data available in the literature, though it seems that Pet obtained in the present work was about one-fourth of the literature value [3]. This discrepancy might come from the different configuration of the bed tested in the present work and in previous investigations. -108 -Chapter 5: Results and Discussions , , , , I . . * . I • • • ' I ' ' ' ' ' ' ' ' 1 1 1 1 1 1 I 3 4 5 6 7 8 Re Axial and transverse dispersion coefficients D a and D t at different flow rates for glass beads bed. - 109 -Chapter 5: Results and Discussions Q_ Pe„ Figure 5.19 Axial and transverse dispersion Peclet numbers P e a and Pe t at different flow rates for glass beads bed. - no -Chapter 5: Results and Discussions 5.3.4.2 Carbon Felt Bed Simultaneously calculated axial and transverse dispersion coefficients for carbon felt bed are shown in Figure 5.20. The axial and transverse Peclet numbers are shown in Figure 5.21. The values are the average of two to three sets of data, as discussed for glass beads bed, the uncertainty was 50% for all the parameters. For the flow rate ranging from 30 to 120 ml min"1 (Reynolds number from 0.008 to 0.032), both the axial and transverse dispersion coefficients for carbon felt were proportional to flow rate, and they were 1 to 6 x 10"6 m 2 s"1 and 2 to 9 x 10"6 m 2 s"1; while the axial and transverse Peclet numbers were constant at 0.007 and 0.004, respectively. The axial dispersion coefficient for the carbon felt bed was smaller than that for glass beads at the same flow rate (i.e., the same superficial flow velocity), while the transverse dispersion coefficient for carbon felt bed was about twice that for glass beads bed at the same flow rate. No comparison data are available for a packed bed with similar porosity and bed material dimension to carbon felt. The axial and transverse dispersion coefficients were in the same order (10"6 mV 1) as the values in the literature for flow-through porous electrode [26] and estimations for 3D flow-by electrode [31]. However, it is also noted that the axial dispersion coefficient obtained in the present work was smaller than that obtained in a larger 3D flow-by electrochemical reactor (50cm wide and 0.3cm thick), 10"4 to 10"3 m 2 s"1 [31]. This high dispersion number implies substantial velocity gradient (or bypassing) in the larger reactor. Extra attention should be paid when scale-up of a laboratory 3D flow-by electrode (such as the one being tested in the present work) to industrial size (up to 50cm wide and 1 to 2m high) - I l l -Chapter 5: Results and Discussions because liquid feed and out-take may be distributed less evenly across the cell width, electrode packing density (bed porosity) may vary over the cell area and pressure can cause large flat cells to bulge and promote channelling [31]. The transverse Peclet number Pet obtained in the present work was much smaller than the values predicted for an ordinary 3D flow-by electrode (5<Per<13) [33] and this discrepancy may be caused by the small fibre diameter and the large porosity of the carbon bed. As discussed earlier in the thesis, transverse dispersion of mass in a 3D flow-by electrode should act with the potential profile to affect conversion in an analogous way to the action with temperature profile in a thermochemical reactor. However, the mass dispersion in the transverse direction may have a more profound effect in a 3D flow-by electrode in cases such as for a reaction under pure kinetic control: comparing a thermochemical reaction whose activation energy is 50 kJ mol"1 with an electrode reaction where a = 0.5 (assumed independent of temperature) an increase from 0.1 to 0.2 volt has an effect on the electrode rate equivalent to raising the thermochemical process temperature from 25 to 300 °C [31]. -112 -Chapter 5: Results and Discussions 10 8 CO CD I o X5 c ca 2 h . i . J I I I I I I I I I I 1 I I I I I I I I L 0 0.005 0.010 0.015 0.020 Re • D a - D4 0.025 0.030 0.035 Figure 5.20 Axial and transverse dispersion coefficeints D a and D t at different flow rates for carbon felt bed. - 113 -Chapter 5: Results and Discussions 0.010 0.008 0.006 0.004 0.002 0.000 J—i—i—i—i—I—i i i i I i i J i i i_ 0.005 0.010 0.015 0.020 0.025 0.030 0.035 Re • P e a • Pe. Figure 5.21 Axial and transverse Peclet number Pe and P a at different 3 t flow rates for carbon felt bed. - 114 -Chapter 5: Results and Discussions 5.4 Comparison of Axial Dispersion with and without Considering Transverse Dispersion Figure 5.22 shows axial dispersion coefficient for glass beads and carbon felt calculated with and without considering lateral and transverse dispersion. It seems that the axial dispersion coefficient was slightly overestimated when the effects of transverse and lateral dispersion were neglected. However, the significance of considering lateral and transverse dispersion effect for calculating axial dispersion coefficient D a would be better understood with further investigation considering the uncertainty related to D a estimation as discussed in section 5.3.2. 5.5 Effect of Neglecting Axial Dispersion Coefficient on Transverse Dispersion Coefficient Estimation Unlike for the axial dispersion coefficient, there is no separate mathematical method, such as that using the first and second moment of the tracer input and output concentration curves to estimate D a , available to calculate transverse dispersion coefficient alone. However, the effect of inappropriate D a estimation on D t calculation can be qualitatively concluded from Figure 5.15: when D a is underestimated (smaller than its optimum value), D t wil l be overestimated; on the other hand, when D a is overestimated (larger than its optimum value), D t will be underestimated. This trend also agrees with that observed for axial dispersion coefficient calculation, D a was overestimated when it was calculated by neglecting the effect of transverse (and lateral) dispersion. -115 -Chapter 5: Results and Discussions 20 40 60 80 100 120 140 Flow rate (ml min"1) • D a without considering D| and for glass beads bed • D g considering D| and D t for glass beads bed O D a without considering D| and D t for carbon felt bed • D a considering D| and D^ for carbon felt bed Figure 5.22 Comparison of axial dispersion coefficeint with and without considering lateral and transverse dispersion effect for glass beads and carbon felt bed. - 116 -Chapter 6: Conclusions and Recommendations Chapter 6 CONCLUSIONS and RECOMMENDATIONS 6.1 Conclusions: A tracer stimulation - response system was set up and pulse tracer experiments were performed to measure axial and transverse dispersion of 0 . 7 M C U S C M in 12 wt % Na2S04 in a rectangular test section filled with glass beads and with a carbon felt packed bed at flow rates from 3 0 to 120 ml min"1 (corresponding superficial velocity was from 3 .84 to 15.3 m s"1; with Reynolds number from 1.8 to 7.2 and 0 .008 to 0 .032 for glass beads and carbon felt bed, respectively.). Within the flow range: • Axial dispersion parameters estimated neglecting the effects of lateral and transverse dispersion were: > For glass beads beds, Pea was about constant at 0.3, D as/ud p was about constant at 1.3, and D a was from 4 to 2 4 x 10"6 m 2 s"1. > For carbon felt beds, Pea was about constant at 0.006, D as/ud p was about 160, and D a was from 1.5 to 6.8 x 10 ' 6 m 2 s"\ • Axial and transverse dispersion parameters calculated simultaneously with the assumption that the transverse and lateral dispersion coefficients were of the same magnitudes were: > For the glass beads bed, D a was proportional to Re, from 6 to 21 x 10"6 m 2 s'1, Pea was about constant at 0.35, which agrees with the literature value of 0.4 [3], and D as/ud p was about constant at 1.3, which was also close to the literature value of 1.5 [9]. Dt increased with Re at a higher rate, from 0.55 to 7.6 x 10"6 m 2 s"1, accordingly, Pet dropped from 7 to - 1 1 7 -Chapter 6: Conclusions and Recommendations 1, which was about one fourth of the literature value (about 30 to 10) in the same Reynolds number range for thermochemical packed bed reactors with radius ranging approximately from 4cm to 50cm. > For the carbon felt bed, D a was 1 to 6 x 10"6 m 2 s"1, Pe a was about constant at 0.007, D t was 2 to 9 x 10"6 m 2 s"1, and Pet was about constant at 0.004. No experimental data were found in the literature for dispersion parameters in a packed bed that had similar bed characteristics (bed material diameter and macroscopic bed porosity) with carbon felt. • The uncertainty of simultaneous estimation of axial and transverse dispersion coefficients was about 50%, which may be improved by a better control of the copper wire surface condition and a finer tuning up of the test section. • D a calculated without considering transverse and lateral dispersion effects slightly overestimated axial dispersion coefficient in both the two types of packed beds. • The axial and transverse dispersion coefficients were not affected by repacking of the beds and the axial dispersion coefficient was uniform throughout the entire bed. 6.2 Recommendations for future work: The present study sets up an experimental system and calculation methodology for a series of investigations into dispersion in rectangular beds of beads and of fibre. In the present work, the dispersion coefficient in the lateral direction of the bed was assumed to be the same as that in the transverse direction so that the concentration profile could be simplified from a three-dimensional to a two-dimensional problem. A further three-dimensional study will give a complete knowledge of the concentration profile within the bed by measuring the tracer concentration profile in lateral direction as well as axial and transverse -118-Chapter 6: Conclusions and Recommendations directions. The three-dimensional modelling may then test the assumption of equal lateral and transverse dispersion. A three-dimensional partial differential equation solver is now (November, 2000) available in FemLab. It would be valuable to have a complete set of experimental data about dispersion parameters for electrochemical reactors similar to that available for thermochemical reactors. For example, experiments can be done to cover a wider range of reactor conditions - different size of reactor, other 3-D electrode materials, different electrolytes, and different fluid flow conditions; dispersion coefficients can be measured in a liquid-gas two phase flow system, which is used in H2O2 production and other electrochemical processes. Both the concentration and potential profiles for a kinetic controlled electrochemical reaction in a 3D electrode should be investigated in order to obtain a better understanding of the interaction between the concentration and potential distributions. -119-References REFERENCES 1. Gunn, D. J., "Theory of axial and radial dispersion in Packed Beds", Trans. Instn. Chem. Engrs, 47, 1969, T351-T359 2. Bernard R. A . and Wilhelm R. H. , "Turbulent diffusion in fixed beds of packed solids", Chem. Engng. Progr., 46, 1950, 233 3. Gunn, D. J., "Mixing in packed and fluidised Beds", The Chemical Engineer, June 1968, CE153-CE172 4. Scott, K., "Electrochemical Reaction Engineering", Academic Press Limited, London, 1991 5. Oloman, C. W., "Electrochemical Processing for the Pulp and Paper Industry", the Electrochemical Consultancy Pub., Romsey, 1996 6. Bischoff, K. B. and Levenspiel, O., "Fluid dispersion generalization and comparison of mathematical models - I generalization of models", Chem. Eng. Sci., 17, 1962, 245 7. Nauman, B "Chemical Reactor Design", Wiley Series in Chemical Engineering, John Wiley & Sons, New York, 1987 8. Danckwerts, P. V. , "Continuous flow systems", Chem. Eng. Sci., 2, 1953, 1 9. Levenspiel, O., "Chemical Reaction Engineering", 3rd, ed., John Wiley & Sons, New York, 1999 10. Gunn, D. J. and Pryce, C , "Dispersion in packed beds", Trans. Instn. Chem. Engrs, 47, 1969, T341-T350 120-References 11. Chen, G. T., "Chemical Reaction Engineering", 2 n d Edition, Chemical Engineering Industry Press, China, 1989 12. Aris, R., "Notes on the diffusion-type model for longitudinal mixing in flow", Chem. Eng. Sci., 9, 1959, 266 13. Klinkenberg, A. , Krajenbrink, H. J. and Lauwerier H. A., "Diffusion in a fluid moving at uniform velocity in a tube", Ind. Engng. Chem., 45, 1953, 1202 14. Aral, M . M . and Liao, B., "Analytical Solutions for Two-Dimensional Transport Equation with Time-Dependent Dispersion Coefficients", Multimedia Environmental Simulations Laboratory, Georgia Institute of Technology, Atlanta, Jan. 1996 15. Kramers, H. , and Alberda, G., "Frequency response analysis of continuous flow systems", Chem. Eng. Sci., 2, 1953, 173 16. Deans, H. A. , and Lapidus, L . "Computational model for predicting and correlating the behavior of fixed-bed reactors: I. Derivation of model for nonreactive systems ", A.I.Ch.E.Jl, 6, 1960, 656 17. Wehner, J. F., and Welhelm, R. H. , "Boundary conditions of flow reactor", Chem. Eng. Sci., 6, 1956, 89 18. Pearson, J. R. A. , "A note on the "Danckwerts" boundary conditions for continuous flow reactors", Chem. Eng. Sci., 10, 1959, 281 19. van der Lann, E. T., "Notes on the diffusion-type model for the longitudinal mixing in flow", Chem. Eng. Sci., 7, 1957, 187 20. Bischoff, K. B. and Levenspiel, O., "Advances in Chemical Engineering", Academic Press Inc., New York, 4, 1962. 121 -References 21. Romer, G., Dranoff, J. S., and Smith, J. M . , "Diffusion in packed beds at low flow rate", Ind. Engng. Chem. Fundamentals, 1_, 1962, 284. 22. Stahel, E. P., and Geankoplis, C. J., "Axial diffusion and pressure drop of liquids in porous media, A.I.Ch.E.Jl, 10, 1964, 174 23. Strang, D. A. , and Geankoplis, C. J., "Longitudinal diffusivity of liquids in packed beds", Ind. Engng. Chem., 50, 1958, 1305 24. Liles, A. W., and Geankoplis, C. J., "Axial diffusion of liquids in packed beds and end effects", A.I.Ch.E.Jl, 6, 1960, 591 25. Fliby, J. W., in Rotterburg, P. A. , (Ed.). "The interaction between fluids and particles", The Institution of Chemical Engineers, London, 1962. 26. Sioda, R.E., "Axial dispersion in flow porous electrodes", J. of Applied Electrochemistry., 7, 1977, 135 27. Ateya, B.G., "Effects of radial diffusion on the efficiency of porous flow-through electrodes", J. of Applied Electrochemistry., 10, 1980, 627 28. Ateya, B.G. , "Effect of radial diffusion on the polarization at porous flow-through electrodes", J. of Applied Electrochemistry., 13., 1983, 417 29. Fahidy, T.Z., "The effect of axial dispersion on the concentration distribution in an electrochemical reactor", J. of Applied Electrochemistry., 5, 1975, 173 30. Scott, K. , "The role of diffusion on the performance of porous electrodes", J. of Applied Electrochemistry., 13, 1983, 709 31. Oloman, C.W., unpublished manuscript, 1986 -122 -References 32. Wen, C. Y . and Fan, L.T., "Models for Flow Systems and Chemical Reactors", Marcel Dekker, New York, 1975 33. Fahidy, T.Z., "Dispersion effects in electrochemical reactors", Paper 4a, Symposium on Electrochemical Reaction Engineering - I AIChE 1985 summer National Meeting, Seattle 34. Pourbaix, M . , "Atlas of Electrochemical Equilibria in Aqueous Solution", National Association of Corrosion Engineers, Huston, 1974 35. Fahien, R. W. and Smith J. M . , "Mass transfer in packed beds", Ameri. Inst. Chem. Engng. J., I , 1955, 28 36. Hodgson and Oloman, C.W., "Pressure gradient, liquid hold-up and mass transfer in a graphite fibre bed with cocurrent upward gas-liquid flow", Chem. Eng. Eci., 54, 1999, 5777. 37. CRC Handbook of Chemistry and Physics, 76 t h edition, CRC Press, 1995 38. Perry's Chemical Engineers' Handbook, 7 t h edition, McGraw-Hill Book Company, 1995 -123 -Appendix 1 Appendix 1 Fluid Distributor Design A perforated piece of rubber was used as the fluid distributor in the test section. The dimension of the distributor, in terms of the number of the holes in it, the diameter of each hole and the height of the rubber were selected such that the pressure drop through the distributor was about 15 - 30% that of the total pressure drop through the carbon felt bed and the distributor. 1. Pressure drop through carbon felt bed Pressure drop through the carbon felt bed was estimated as [36] *Pca*on / e J , = l x l O - 5 x ( « X M X 5 2 ) x V 8 J xH where, APCarbon_feit pressure drop through the carbon felt bed, bar H height of the carbon felt bed, 0.32m w width of the test section, 0.05m t thickness of the test section, 0.026m p fluid viscosity, 1.2 x 10"3 kg m'V 1 (12 wf% Na 2 S0 4 , at 20 °C) u fluid superficial velocity, m s'1 3 1 V flow rate, m s" £ porosity of the carbon felt bed, 0.95 d f diameter of carbon fibre, 20pm s specific surface area of carbon felt bed, m 2 m"3 k Kozeny constant -124-Appendix 1 for fibrous beds with 0.68 < 8 < 0.96 and 16.5 < d f < 19.5 um, £ = 3.5 = 3.5 x 8.53 f 0.93 ( l - 0 . 9 ) o : :(l + 5 7 x ( l - f ) 3 ) (l + 57x( l -0 .9 ) 3 ) df 20x l0" 6 2 x l 0 5 m 2 m " 3 The experimental flow rate ranged from 30 to 150 ml min' 1 (5.1 to 25.5 x 10"7 m 3 s"1). When the flow rate was 30 ml min"1, the fluid velocity and pressure drop through the bed were: V 5.1 x l O - 7 wxt 0.05x0.026 = 3.85xl0" 4ms" 1 carbon _ felt :(2xl05)2)> f 8.53 x (1-0.95) 0.953 ,2 A x0.5 : l x l 0 - 5 x^1.2 x 10" x3.85x10"" x = 0.011 bar = 1.11 kPa Apcarbon feit was proportional to the flow rate. 2. Pressure drop through the perforated rubber Pressure drop through the perforated rubber, Apmbber can be calculated by using Fanning or Darcy Equation [37]. From CRC 5-24 APrubber = F X p For a circular pipe, the Fanning or Darcy equation is: - 1 2 5 -Appendix 1 _ 32 xf xLxq2 " 3 . 1 4 2 xgcxD5 where / = — (for flow with Re <2000, from CRC 5-28) Re Re UxDxp M U = 4xq 3.\4xD2 V q = — n L = L' + Le f Fanning friction factor gc dimensional constant, m s"2 V 3 1 volume flow rate in the reactor, m s" n number of holes in the perforated rubber q 3 1 volume rate of flow in each hole, m s" u fluid velocity in each hole, m s"1 D diameter of each hole, m L calculated length of each hole considering entrance and exit effect, m L' true length of each hole, same as the height of the rubber, m equivalent additional hole length, m p density of the fluid, kg m" viscosity of the fluid, kg m 'V 1 -126-Appendix 1 The length of the hole had to be corrected for the entrance and exit effects. In practice the exit effect is small enough to be neglected; for flow with Re<2000 and A 2 / A i < 0.2, which is the case of the present study, the loss in sudden contraction can be estimated by an equivalent additional pipe length L e given by [28] Le/D = 0.3 + 0.04Re A 2 is the reduced flow cross-section area, i.e., the sum of the cross areas of all the holes; and A i is the original flow cross-section area. Pressure drop through the perforated rubber distributor was calculated for a wide range, where, the number of the holes was from 20 to 50; the diameter of each hole was from 1 to 2.5mm; and the height of the rubber was from 1 to 2 cm. A 1 cm high rubber distributor with 36 holes of 1.5 mm diameter was selected as the fluid distributor in the present work. The pressure drop through this distributor, AP^er , 0.5kPa at 30 ml min"1, was 31% of the sum of the pressure drops through the carbon felt bed and rubber distributor. The ratio of the pressure drop through the distributor and the whole test section was independent of flow rate because pressure drop through the distributor and through the carbon felt bed were both proportional to flow rate. -127 -Appendix 2 Appendix 2 Slope of the Copper Wire Potential vs. log [Cu2+] Calibration Curve Calculated from Nernst Equation The equilibrium electrode potential is given by Nernst Equation as: E.=E:+—logwQ a t 25°C n where E°is standard electrode potential (volt), E e is equilibrium electrode potential (volt), n is the electron stoichoimetry coefficient, and Q is activity quotient of reaction (=ao/aR). In the present study, the potential measured by the copper wire was the equilibrium potential (E e) determined by the following redox pair Cu2+ +2eoCu where n = 2 and Therefore, the theoretical slope of the copper wire potential (E e) - log [Cu 2 +] curve is 0.059 n n c i r slope = 29.5mK -128-Appendix 3 Appendix 3 C11SO4 and Na 2S0 4 Density Data From Perry's Chemical Engineers' Handbook [38], Table 3 - 4 9 and 3 - 94, density of Q1SO4 and Na2SC>4 aqueous solutions at different concentrations are listed as below: (20 °C) Concentration (%) Density (x 103 kgm"3) CuSC-4 N a 2 S 0 4 1 1.0086 1.0073 2 1.0164 4 1.0401 1.0348 8 1.084 1.0724 12 1.1308 1.1109 16 1.180 1.1506 18 1.206 20 1.1915 24 1.2336 It was found from Figure A - l below that the density of 12 wt% N a 2 S 0 4 is 1.12 x 103 kg m"3, and that of 0.7M CuS0 4 is 1.11 x 103 kg m"3. They were selected as the main fluid and the tracer, respectively, due to their similar density. -129 -Appendix 3 1.25 1.20 1.15 O .4? 1.10 CO c CD Q 1.05 1.00 . . i l l I I—1—I—I—I—I—I—1—L I I I I I I I I 1—1 1 1 L. 0 10 15 20 Concentration (%) 25 30 • C u S 0 4 . N a 2 S 0 4 Figure A-1 Density of C u S 0 4 and N a 2 S 0 4 solution at 20 °C. -130-Appendix 4 Appendix 4 Sample Calculation of Axial Dispersion Coefficient by Using One-Shot Technique, Neglecting Lateral and Transverse Concentration Gradient Figure A-2 is an example of potential responses given by two wires to a one-shot tracer injection. Wire 1 is 20cm above wire 7, i.e., 20cm downstream of wire 7. Figure A-3 shows the corresponding local copper ion concentrations calculated from the measured copper wire potentials with copper wire calibration curves. The slopes of the wire potential vs. [Cu 2 +] curves for wire 1 and wire 7 were respectively 22mV and 23mV for this specific run, and the potential base line of each wire was -235 mV and - 228 mV, respectively. [cV+l =io( 22 } ' and ( K 7 - ( - 2 2 8 ) ) [cV+J^io' 2 3 Form equation (6) and (7) for discrete experimental data with uniform At,. 7 _rZ^,A/, _ J > , C 188.33 1 ~ l £ c , A r , h~{zci 0 5 0 2 Y Y C A J , Yt.C, 134 36 tn = (4;' ' ')7 = (^L-L) 7 = = 1795 7 Xc^i L c i 0 7 5 1 and a? ,( -^C-AS1 = ( ^ X - ^ = ^ ^ - 3 7 5 - 3 7 3 4 . 4 ^ 1 V £C,Af , * 1 V £ C , 1 1 0.502 -131 -Appendix 4 ^ = ( ^ ) 7 - ^ = ^ - 1 7 9 2 =1453.8 0.751 (1) Da/uL was first calculated with small deviation from plug flow assumption method, i.e. 2 ( % = A a | = uL 2 2 cr , -cr out m a , -o- 7 ACT2 (AO2 ( C - U 2 ft-',) £> 1 3734.4-1453.8 —s- = -x — = 0.030 uL 2 (375-179) 2 which is larger than the limit for small deviation assumption (D a/uL <0.01), therefore, this method was discarded. (2) Then D a /uL was calculated from the equation for large deviation from plug flow with open-open boundary conditions: + 8 \uL j ~2 ~2 °out - OT,-H 7 \ 2 (fut - t i n ) C , - ' 7 ) 3734.4-1453.8 Aol00 = — = 0.060 s2 (375-179)' Da _ - 2 + -j4 + 4 x 8 x A o - 2 0 0 _ - 2 +V4 + 4x8x0.060 = 0.027 uL 2x8 2x8 The flow rate was 30 ml min"1, the cross section area of the test section was 5cm x 2.6cm, the porosity of this glass beads bed was 0.4, and the distance L between wire 7 and 1 was 20cm. So the fluid velocity was, 3 0ml min _ 1 x 10~6 mlm "3 u - 5cm x 10 mem x 2.6cm xlO mem x 0.4 x 60s min" = 9 .6xl0" 4 ms"1 -132-Da = —2-x« x Z = 0.027 x9.6 x K T W 1 x20c/w = 5 . 2 * l O ^ m V 1 uL -190 -200 -> -210 E (— -220 o 0_ -230 -240 h -250 i i i i i . i i i 1 ) i i i i i i i i 0 60 120 180 240 300 360 420 480 Time (s) • Wire 1 • Wire 7 Figure A-2 Copper wire potential as a function of time. (Run 04282a: glass beads bed, 0.5ml 0.7M CuS0 4 with 0.01 wt % Trypan blue, flow rate = 30 ml min"1.) -133 -Appendix 4 0.035 0.030 0.025 5 0.020 O 0.015 i 0.010 1 0.005 0.000 • f 0 60 120 180 240 300 360 420 480 540 Time (s) • Wire 1 • Wire 7 Figure A-3 Copper concentration as a function of time. - 134 -Appendix 5 Appendix 5 Sample Calculation of Flow Velocity u, Re, D a/udp (or Da/udf), Das/ udp(or Das/udf), and Pea p main fluid density (1.1 x 103 kg m"3) u. main fluid viscosity (1.2x 10"3 kg m"1 s"1) w width of the test section (0.05 m) t thickness of the test section (0.026 m) L distance between the wire 1 and 7 (0.2 m) s porosity of the bed dp diameter of the glass beads (0.002 m) df diameter of the carbon fiber (0.00002 m) V flow rate (m3 s"1) u interstitial fluid velocity (m s"1) Re Reynold number (udp p/u. or udfp/u.) Pe a axial Peclet number (udp/ D a or udf/ D a ) D a axial dispersion coefficient (m2 s"1) 1. Glass Beads Bed: (Run 04282b) D a /uL = 0.036, V=5xl0- 7 V 5x l0~ 7 n r . ^ u = = = 9.6 x 10 4 m s 1 wts 0.05x0.026x0.4 udp 9.6x10^ x 0.002 x 1.1 x l O 3 , „„ Re = — — = = 1.78 p 1.2 x l O - 3 D - D ~ 1 0 . 0 3 6 x ^ = 3.6 udp uL dp 0.002 -135 -D e D — 2 - = — 2 - x s = 3.6 x 0.4 = 1.44 udp udp Pe„ = = 3.6_1 =0.28 Da = ^xuxL = 0.036 x 9.6 x 10~4 x 0.2 = 7 x 10~6 m 2 uL 2 . Carbon Felt Bed: (Run 05092b) D a /uL = 0.018, V = 5x l0 - 7 V 5xl0~ 7 u = wte 0.05x0.026x0.95 ^ . l x l O ^ m s " 1 udfp 4 . 1 x l 0 - 4 x 0.00002 x 1.1 x l O 3 Re = — — = = 0.008 u 1.2 x l O - 3 ^ = ^ x A = 0 . 0 1 8 x ^ - = 180 udf uL df 0.00002 De D udf udf ?-xs = 180x0.95 = 171 Pe. \ u d f J 180"1 =0.0056 Da = ^ x w x L = 0.018 x 4.1 x 10~4 x 0.2 = 1.5 x l O - 6 n uL -136-Appendix 6 Appendix 6 A Computer Program for Calculating D a and D t A computer program written in MatLab and FamLab was used to solve the time-dependent two-dimensional partial differential equation (28) for several sets of axial and transverse dispersion coefficients. Data set 04284a i s used as a sample for illustration. Measured tracer concentrations CI , C7, and C8 were imported from an Excel file in which the copper wire potentials measured by all the wires were stored and converted into copper ion concentrations. Attached is the script of the program. % Filename: 04284a % FEMLAB Model M-file % Generated 19-Aug-2000 15:43:21 by FEMLAB 1.1.0.302. % Measured tracer concentration by wire 1, 7 and 8 as a function of time. % T(s) C1(M) C7(M) C8(M) rawdata=[ 0 0.000901571 0.000574237 0 000945387 2 0.000933254 0.000707946 0 000963252 4 0.000831764 0.000592553 0 000902162 6 0.000831764 0.000592553 0 000902162 8 0.000851138 0.000605088 0 000990684 10 0.001035142 0.000671851 0 001129394 12 0.000785236 0.000637596 0 000963252 - 137 -Appendix 6 14 0.000988553 0.000686062 0.001161557 16 0.000881049 0.000574237 0.001009404 18 0.001071519 0.000657933 0.001108448 20 0.000988553 0.000605088 0.001077756 22 0.001035142 0.000671851 0.001108448 24 0.000933254 0.000707946 0.00091921 26 0.000881049 0.000562341 0.001009404 28 0.000851138 0.000657933 0.000902162 30 0.000933254 0.000592553 0.000990684 32 0.000803526 0.000562341 0.000902162 34 0.000831764 0.000592553 0.00091921 36 0.000785236 0.000544959 0.000963252 38 0.000901571 0.000574237 0.000990684 40 0.000831764 0.000637596 0.001009404 42 0.000831764 0.000544959 0.000990684 44 0.000831764 0.000637596 0.000963252 46 0.000988553 0.000686062 0.001161557 48 0.000933254 0.000657933 0.001129394 50 0.000954993 0.000671851 0.001077756 52 0.000785236 0.000624388 0.000990684 54 0.000901571 0.000574237 0.001057768 56 0.000785236 0.000657933 0.001009404 - 138 -Appendix 6 58 0.000954993 0.000624388 0.001108448 60 0.000803526 0.00053367 0.000963252 62 0.000785236 0.00053367 . 0.000902162 64 0.000881049 0.000574237 0.001057768 66 0.000901571 0.000624388 0.001108448 68 0.000724436 0.000522615 0.000902162 70 0.000881049 0.000637596 0.001077756 72 0.000699842 0.00053367 0.00091921 74 0.000785236 0.00053367 0.000963252 76 0.000724436 0.000522615 0.000990684 78 0.000831764 0.000562341 0.001057768 80 0.000660693 0.000522615 0.000902162 82 0.000933254 0.000671851 0.001183507 84 0.000683912 0.000544959 0.000877181 86 0.000785236 0.000671851 0.001057768 88 0.000699842 0.000605088 0.000945387 90 0.000724436 0.000637596 0.000963252 92 0.000683912 0.000544959 0.000844947 94 0.000767361 0.000637596 0.001009404 96 0.00074131 0.00050646 0.000963252 98 0.00074131 0.000671851 0.000963252 100 0.000683912 0.000544959 0.00091921 139 Appendix 6 102 0.000699842 0.00050646 0.00091921 104 0.00074131 0.000624388 0.000963252 106 0.000785236 0.000522615 0.000990684 108 0.000645654 0.00050646 0.000877181 110 0.000767361 0.000637596 0.000963252 112 0.000881049 0.000574237 0.001108448 114 0.000831764 0.00053367 0.001028478 116 0.000630957 0.00050646 0.000963252 118 0.000851138 0.000605088 0.001077756 120 0.000785236 0.00053367 0.001009404 122 0.000831764 0.000637596 0.001161557 124 0.000767361 0.00053367 0.001057768 126 0.000831764 0.000605088 0.001057768 128 0.000831764 0.000574237 0.001108448 130 0.000699842 0.000562341 0.000902162 132 0.000767361 0.000522615 0.001028478 134 0.000767361 0.000544959 0.00091921 136 0.000851138 0.000574237 0.001057768 138 0.000803526 0.000562341 0.001057768 140 0.00074131 0.00050646 0.000963252 142 0.000803526 0.000624388 0.001057768 144 0.000645654 0.000544959 0.000902162 140 Appendix 6 146 0.000699842 0.000562341 0.000963252 148 0.000803526 0.000637596 0.001057768 150 0.000803526 0.000574237 0.001028478 152 0.000831764 0.000624388 0.001028478 154 0.000724436 0.000592553 0.000963252 156 0.000724436 0.000624388 0.000945387 158 0.000851138 0.000562341 0.001108448 160 0.000699842 0.000495969 0.00091921 162 0.000767361 0.00053367 0.001028478 164 0.000785236 0.000592553 0.001108448 166 0.000699842 0.000480638 0.00091921 168 0.000645654 0.000480638 0.000860913 170 0.000831764 0.000562341 0.001108448 172 0.000785236 0.00053367 0.001009404 174 0.000699842 0.000637596 0.001028478 176 0.000660693 0.001488426. 0.001161557 178 0.00074131 0.005171736 0.001673313 180 0.00074131 0.013265611 0.00234379 182 0.000785236 0.023344352 0.003538185 184 0.000724436 0.020589053 0.003376411 186 0.000683912 0.019134547 0.003222034 188 0.00074131 0.015199111 0.003472565 - 141 -Appendix 6 190 0.000785236 0.015199111 0.003707709 192 0.000660693 0.011699891 0.003017691 194 0.000645654 0.009589991 0.002826308 196 0.000767361 0.009101038 0.003162278 198 0.000785236 0.008819715 0.002826308 200 0.00074131 0.006375958 0.002748048 202 0.000831764 0.00651083 0.002748048 204 0.000699842 0.005449588 0.00234379 206 0.000645654 0.004328761 0.001999001 208 0.000851138 0.004561324 0.002410537 210 0.000803526 0.004239091 0.002195146 212 0.000767361 0.003817844 0.002094779 214 0.00074131 0.003623188 0.002154435 216 0.000645654 0.002818383 0.001673313 218 0.000831764 0.002818383 0.001999001 220 0.000851138 0.002818383 0.002055929 222 0.000831764 0.002408897 0.001704933 224 0.000803526 0.002238721 0.00155259 226 0.000901571 0.002358996 0.001673313 228 0.000988553 0.00216952 0.001596806 230 0.001230269 0.002286077 0.00162698 232 0.00162181 0.00216952 0.001786621 142 Appendix 6 234 0.001548817 0.001687612 0.00155259 236 0.001717908 0.001873817 0.001454125 238 0.002162719 0.001778279 0.001427156 240 0.003126079 0.001913455 0.001704933 242 0.003589219 0.001778279 0.001673313 244 0.004120975 0.001687612 0.001704933 246 0.003890451 0.001488426 0.001387639 248 0.004216965 0.001488426 0.001454125 250 0.005623413 0.001442417 0.00155259 252 0.005011872 0.001519911 0.001324193 254 0.005623413 0.001272171 0.001299634 256 0.00699842 0.001519911 0.001523796 258 0.005623413 0.001299081 0.001263648 260 0.005623413 0.001272171 0.001299634 262 0.006839116 0.001442417 0.001454125 264 0.006606934 0.001488426 0.001495536 266 0.006606934 0.001442417 0.001495536 268 0.005623413 0.001368875 0.001387639 270 0.004897788 0.001182299 0.001217211 272 0.005956621 0.001340518 0.001454125 274 0.005623413 0.001299081 0.001454125 276 0.005011872 0.001122018 0.001387639 143 Appendix 6 278 0.004897788 0.001087336 0.001324193 280 0.004120975 0.001010521 0.001361904 282 0.003801894 0.001087336 0.001217211 284 0.00402717 0.001182299 0.001324193 286 0.003388442 0.001031897 0.001161557 288 0.003890451 0.001182299 0.001387639 290 0.003198895 0.000958999 0.001183507 292 0.003198895 0.001145753 0.001427156 294 0.003198895 0.001182299 0.001263648 296 0.003388442 0.001087336 0.001427156 298 0.002570396 0.000910104 0.001217211 300 0.002290868 0.000958999 0.001161557 302 0.00242661 0.000910104 0.001240212 304 0.002213095 0.000863701 0.001183507 306 0.001927525 0.000891251 0.001161557 308 0.001883649 0.000910104 0.001108448 310 0.002290868 0.000958999 0.001324193 312 0.001819701 0.000863701 0.001108448 314 0.001927525 0.001010521 0.001263648 316 0.001927525 0.000863701 0.001217211 318 0.001883649 0.00084581 0.001129394 320 0.001995262 0.000989588 0.001263648 - 144 -Appendix 322 0.001883649 0.000989588 0.001263648 324 0.00162181 0.000910104 0.001161557 326 0.001995262 0.000958999 0.001299634 328 0.001717908 0.000828289 0.001240212 330 0.001819701 0.000828289 0.001240212 332 0.001819701 0.000939133 0.001324193 334 0.001584893 0.000802686 0.001161557 336 0.001548817 0.000958999 0.001183507 338 0.00162181 0.00084581 0.001217211 340 0.001584893 0.000939133 0.001299634 342 0.00162181 0.000910104 0.001299634 344 0.001462177 0.000910104 0.001129394 346 0.00162181 0.000828289 0.001263648 348 0.001462177 0.000910104 0.001183507 350 0.001380384 0.00076176 0.001028478 352 0.001333521 0.000828289 0.001077756 354 0.001202264 0.000786058 0.001028478 356 0.001462177 0.000910104 0.001161557 358 0.001230269 0.000722921 0.000990684 360 0.001135011 0.000786058 0.001009404 ]; % Set up the time and tracer concentration vectors, and subtract the base line from the tracer - 145 -Appendix % readings. ta=rawdata(:,l); rCl=rawdata(:,2)-0.001; rC7=rawdata(:,3)-0.001; rC8=rawdata(:,4)-0.001; % Use FemLab to solve the partial differential equation (28) with several sets of axial and % transverse (lateral) dispersion coefficients: Da and Dti, where Dti=2Dt=2Di. % Da and Dn were from IO" 3 5 to IO"8 m 2s _ 1 at 10" 0- 5 m 2s _ 1 apart. for i= 1:10 for j=l:10 % FEMLAB Version clear vrsn; vrsn.name-FEMLAB 1.1'; vrsn.major=0; vrsn.build=302; fem. version=vrsn; % Space dimensions fem.sdim={'x','y1}; % Geometry % Form the rectangular regime for Equation (28): 1.6 m by 0.026m % The input circle at the tip of wire 7: diameter 0.0005 m at (0.003, 0.15) - 146 -Appendix clear s c p objs names Rl=rect2(0, 0.025999999999999999, 0, 1.6000000000000001, 0); Cl=circ2(0.0030000000000000027, 0.14999999999999999, 0.00050000000000000044, 0); objs={Rl,Cl}; names={'RlVCl'}; s.objs=objs; s.name=names; clear objs names objs={}; names={}; c.objs=objs; c.name=names; % Define the tip of wire 7, (0.003, 0.015), % Define the tip of wire 8, (0. Oil, 0.015), % Define the tip of wire 1, (0.003, 0.035), clear objs names PTl=point2(0.0030000000000000027 ,0.15049999999999999 ); PT2=point2(0.010999999999999999 ,0.14999999999999999 ); PT3=point2(0.0030000000000000001 ,0.34999999999999998 ); objs={PTl,PT2,PT3}; names={'PTl','PT2','PT3'}; p.objs=objs; -147 -p.name=names; drawstmct=struct('s',s,'c',c,'p',p); fern. draw=drawstruct; fern. geom=geomcsg(fem); % Recorded command sequence clear appl % Application mode 1 appl appl appl appl appl appl appl appl 'grade' appl appl appl appl appl appl }.mode='flpdedf2d(,,dimH,{ncM},',submode,V,stdM,MtdiffV'onn) }.dim={'c'}; } .form-coefficient'; }.usage=[l 1]; }.border='on'; }.name-df; }.var={); }.assign={'Q';,Q';'flux';,flux,;'flux_x';'flux_x';'flux '^;'flu^ ,'gradc';'gradcx';'gradcx';'gradcy';'gradcy';'n_flux';'n_flux'}; }.equ.D={{{'1.0'}}}; }.equ.Q={'1.0'}; }.equ.ind=[l 1]; }.bnd.q={'0','0','0'}; }.bnd.g={'0','0','0'} }.bnd.c={'0','0','0'} - 148 -Appendix 6 appl{l}.bnd.type={,qgOVcVc'}; appl{l}.bnd.ind=[l 2 2 1 3 3 3 3]; appl{l).init.sd={{'0'}}; appl{l}.init.ind=[l 1]; fem.appl=appl; % Initialize mesh fem.mesh=meshinit(fem,... 'Out', {'mesh'},... 'jiggle', 'mean',... 'Hcurve', 0.33333333333333331,... 'Hgrad', 1.3); % Dimension fem.dim={{'c'}}; % Boundary conditions fem.border=l; % Usage fem.usage=[l 1]; % Problem form fem.form-coefficient'; % Differentiation fem.diff='off; % Define variables - 149 -Appendix fem.variables={... 'Da', 10A(-3)*10A(-i/2),... 'D t l ' , 10A(-3)*10*(-j/2)}; % Define application mode variables fem.var={}; % Boundary conditions clear bnd bnd.q={{{'0'}},{{'0'}},{{'0'}}}; bnd.g={{{,0'}},{{10'}},{{,0'}}}; bnd.h={{{'0'}},{{T}},{{T}}}; bnd.r={{{'0'}},{{'0,}},{{'0.0228!,!exp(-exp(-(t-184)/7)-(t-184)/7+l)'}}}; bnd.var={'n_flux','-ncu 1'}; bnd.ind=[l 2 2 1 3 3 3 3]; fem.bnd=bnd; % PDE coefficients clear equ equ.da={{{T}}}; equ.c={{{'DtlVDa'}}}; equ.al={{{'0';'0'}}}; equ.ga={{{'0';'0'}}}; equ.be={{{'0';'28.8e-4,}}}; equ.a={{{'0'}}}; equ.HICO'}}}; - 150 -equ.va^i'QU^.O^/gradcxV-cxVgradcyV-cyVgradc'/sqrtCcx.^+cy.^)', . 'flux_xV-cu 1 xVflux^V-cu 1 y Vflux','sqrt(cu 1 x. A2+cu 1 y. A2)'}; equ.ind^fl 1]; fem.equ=equ; % Evaluate initial condition fern. init=asseminit(fem,... 'context','local',... 'inif, structCsd',{{{{'0,}}}},'ind',{[l 1]})); % Solve dynamic problem fem.sol=femtime(fern,... 'tlist', 0:2:360,... 'atol', 0.001,... 'rtol', 0.01,... 'jacobian','equ',... 'mass', 'full',... 'ode', 'ode 15 s',... 'odeopt', struct('InitialStep', {[]},'MaxOrder', {5},'MaxStep', {[]}),.. 'out', {'sol'/stop'},... 'stop', 'on',... 'report', 'off,... 'context','local',... 'sd', 'off,... - 151 -Appendix 'Epoint', 'gauss2',... 'Tpoint1, 'gauss2',... 'Solcomp',1); % Save current mesh restart.mesh=fem.mesh; % Form calculated concentration vectors: fC7, fC8 andfCl, concentrations at wire 7,1 and 8, % respectvely uE=postinterp(fem,fem.sol.u,fem.mesh.p); format long uE fC7=uE(:,4); fC8=uE(:,8); fCl=uE(:,6); % Calculate the sum of squares of the differences between calculated and measured tracer % concentrations at wire 1 and 8. X=(fC 1 -rC 1). *(fC 1 -rC 1 )+(fC8-rC8). *(fC8-rC8); %form the table of sum X for all Da and Dti sum(X); SumTable(i,j)r=sum(X) end end % Export SumTable to file "sum04284a_csv" csvwrite('sum04284a_csv', SumTable) -152 -

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