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I; [.35 1‘30!!! - pt.va\}lllllt¢f I}... . fix-.01! «I: :4 Ms L- J mcmc Illllll E UNIVERSITY LIBRARlES l lijll llll‘l‘l‘ m \llll ll ii; 3 \(ioe‘eiimoz95 This is to certify that the dissertation entitled Characterization of a Three-Phase Magnetically Stabilized Fluidized Bed Bioreactor presented by Vicki Sue Thompson has been accepted towards fulfillment ofthe requirements for PhoDo degreein Chemical Engr. z? 72% admit.“ Ma jOr professor Date March 31: 1993 MSU is an Affirmative Action/Equal Opportunity Institution 0‘ 12771 a.._ a. .. _, ._ .. nan—m, ,‘__.. - LIBRARY Mlcmgan State University PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. DATE DUE _ DATE DUE DATE DUE l MSU Is An Affirmative AotlorVEqueI Opportunity Inditution ' cMmHJ CHARACTERIZATION OF A THREE-PHASE MAGNETICALLY STABILIZED FLUIDIZED BED BIOREACTOR By Vicki Sue Thompson A DISSERTATION Submitted to Michigan State Universit in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Chemical Engineering 1993 ABSTRACT CHARACTERIZATION OF A THREE-PHASE MAGNETICALLY STABILIZED FLUIDIZED BED BIOREACTOR By Vicki Sue Thompson Three-phase fluidized bed systems have many desirable properties such as excellent phase contacting, low pressure drops, reduced plugging, and reduced shear. However, at high gas flow rates, bubbles coalescence tends to reduce phase contacting, and increase shear and turbulence. Studies have shown that application of a magnetic field to two-phase fluidized beds can reduce or eliminate this undesirable behavior. The focus of this study was to examine the properties of a three-phase fluidized bed with an applied magnetic field. Bed regimes, gas void fraction, axial liquid mixing and gas-to- liquid mass transfer were examined in this study. Six bed regimes were identified: random, chain, chain-channel, destabilized, channel, and frozen. Bed properties varied markedly from regime to regime. Local gas void fraction was found to increase 200% in the frozen regime. The axial dispersion coefficient, a measure of liquid mixing, decreased 400% in the chain-channel regimes for low liquid velocities, and gas to liquid mass transfer increased by 30% in the chain-channel and destabilized regimes. A mathematical model was also developed to predict the effect of intraparticle tracer diffusion into the solid phase on measurement of axial mixing. The mathematical model allowed development of criteria to determine when intraparticle diffusion could be ignored relative to effects of dispersion and convection, and allowed correction of experimental measurements. ACKNOWLEDGEMENTS I would like to thank my parents, Kenneth and Carol Freund, for their support and patience through all of'this. None of this would have occurred without them. I would like to thank my husband, David N. Thompson, for his support (both emotional and technical) during my project and the writing of this dissertation. I would like to thank my advisor, Mark Worden for his guidance during this project. Finally, I would like to acknowledge the Upjohn Company and the National Institute of Health for providing me fellowship support during my work. iv TABLE OF CONTENTS page List of Tables . .............................. viii LiSt 0‘ Figures 0000000000 O O O O O O O O O O O O O O O O O O O O O O x NomenCIature O O O O ‘O O O O O O O O O O O O O O O O O O O O O O O O O O O Xiii Chapterl: Introduction ...... ................1 Chapter 11: Literature Survey ..................... 3 l. Fluidized Beds ............................. 3 2. Magnetically Stabilized Fluidized Bed (MSFB) ......... 4 1. Early Studies ............................ 4 2. Gas- Solid MSFB ................. . ......... 5 3. Liquid-Solid MSFB ........................ 9 4. Gas-Liquid-Solid MSFB ..................... 10 5. Potential Applications of the MSFB ............. 12 3. Phase Holdups ............................. 13 1. Definition and Theory ...................... 13 2. Measurement Techniques .................... 13 4. Liquid Mixing ............................. 15 1. Theory ................................ 15 2. Expenmental Methods to Measure Dispersion Coefficients ............................ l6 3. Data Analysis ........................... 17 4. Interaction of the Tracer with the Solid Phase ...... 22 5. Gas to Liquid Mass Transfer ................... 24 1. Theory ................................ 24 2. Measurement Techniques for kLa ............... 26 1. Dynamic Method ....................... 26 2. Steady State Method ..................... 28 Chapter 111: Materials and Methods . . . . . ...... . . . . . 31 l. Fluidized Bed ............................. 31 1. Construction ............................ 31 2. Liquid Phase ............................ 31 3. Gas Phase .............................. 33 4. Solid Phase ............................. 33 2. Solenoid ................................ 34 1. Design ................................ 34 2. Construction ............................ 36 3. Gas Void Fraction .......................... 36 1. Probe Construction ........................ 36 2. Calibration and Data Analysis ................. 39 4. Liquid Phase Dispersion ...................... 43 1. Probe Construction and Calibration ............. 43 2. Data Analysis ........................... 45 3. Diffusion Coefficient ...................... 48 1. Experimental Technique .................. 48 2. Data Analysis ......................... 48 3. Solute Exclusion Technique to Determine Pore Structure ............................ 50 5. Gas to Liquid Mass Transfer ................... 54 1. Experimental Procedure .................... 54 2. Data Analysis ........................... 57 Chapter IV: FluidizedBedModel . .. .. . . .. 60 '1. Derivation ...................... . ........ 6O 2. Simulation ............................... 63 3. Dimensional Analysis ........................ 65 ChapterV: Results...... ................ 66 l. Solenoid ................................ 66 2. Bed Regimes .............................. 66 3. Gas Void Fraction .......................... 73 4. Liquid Dispersion .......................... 83 1. Conductivity Probe Performance ............... 83 2. Diffusion Coefficients ..................... 83 3. Pore Volume Distribution ................... 85 4. Dispersion Coefficients ..................... 91 5. Gas-to-Liquid Mass Transfer ................... 97 vi 6. Dimensional Analysis of the Fluidized Bed Model ..... 105 Chapter VI: Discussion ....... . . . . . . . . ......... 115 1. Bed Regimes ............................. 115 2.GasVoidFraction............. ............ 116 3. Liquid Dispersion ......................... 118 1. Diffusion Coefficients .................... 120 2. Pore Volume Distributions .................. 121 4. Gas-to-Liquid Mass Transfer .................. 123 5. Dimensional Analysis of the Fluidized Bed Model ..... 124 6. Bioreactor Potential ........................ 125 7. Scale-up Considerations ..................... 126 8. Conclusions ............................. 129 9. Future Work ............................. 131 Appendices............. ................... 133 A. Derivation of Equations ..................... 133 1. Unsteady State Well Mixed Model ............. 133 2. Steady State Well Mixed Model ............... 134 3. Steady State Plug Flow Model ................ 135 4. Steady State Dispersed Plug Flow Model . . ....... 136 5. Stirred Tank in Series Model ................ 139 B. Computer Programs ........................ 141 1. Solenoid Design ......................... 141 2. Gas Void Fraction ....................... 144 3. Optimization Program ..................... 146 4. Liquid Dispersion ....................... 149 5. Mass Transfer Coefficients ................. 155 6. Diffusion Coefficients .................... 158 7. Fluidized Bed Simulations .................. 163 C. Tabulated Data ........................... 173 Bibliography . . . . . . ................ . ........ 201 vii . Table 3-1 4-1 5-1 5-3 6-1 CA 0.2 c-3 c-4 C-6 C-7 C-8 C-9 C-10 C-ll C-12 C-l3 LIST OF TABLES Title Page Molecular Probes Used in the Solute Exclusion Technique .............................. 52 Values Used in Parametric Studies .............. 64 Values of D, and a from Curve Fits of Experimental Data ........................ 88 Curve Fits for Pore Size Distributions ............ 88 Curve Fit of ’1 versus 4> Data ................. 112 Comparison of Measured and Literature Pe Numbers ............................ 119 Data from Figure 5-1 ...................... 173 Data from Figure 5-2 ...................... 175 DatafromFigure5-3.............. ........ 176 Data from Figure 5-4 ...................... 176 Data from Figure 5-5 ...................... 176 Data from Figure 5-7 ...................... 177 Data from Figure 5-8 ...................... 177 Data from Figure 5-9 ...................... 178 Data from Figure 5-10 ..................... 178 Data from Figure 5-11 ..................... 179 Data from Figure 5-12 ..................... 179 Data from Figure 5-13 ..................... 180 Data from Figure 5-14 ..................... 180 viii C-l4 C-15 C-16 C-17a C-17b C-18 C-l9 C-20 C-21 C-22 C-23 C-24 C-25 C-26 C-27 C-28 C-29 C-30 C-31 C-32 C-33 C-34 C-35 Data from Figure 5-15 ..................... 183 Data from Figure 5-16 ..................... 184 Data from Figure 5-17 ..................... 185 Data from Figure 5-l8a ..................... 186 Data from Figure 5-18b .................... 186 Data from Figure 5-19 ..................... 187 Data from Figure 5-20 ..................... 189 Data from Figure 5-21 ..................... 189 Data from Figure 5-22 ..................... 190 Data from Figure 5-23 ..................... 190 Data from Figure 5-24 ..................... 191 Data from Figure 5-25 ..................... 191 Data from Figure 5-26 ..................... 192 Data from Figure 5-27 ..................... 192 Data from Figure 5-28 ..................... 193 Data from Figure 5-29 ................. ,0 . . . 193 Data from Figure 5-30 ..................... 193 Data from Figure 5-31 ..................... 194 Data from Figure 5-32 ..................... 194 Data from Figure 5—33 ..................... 194 Data from Figure 5-34 ..................... 195 Data from Figure 5-35 ..................... 199 Data from Figure 5-36 ..................... 200 Figure 2-1 2-2 3-1 3-3 3-4 3-5 3-6 3-7 5-1 5-2 5-3 5-4 5-6 5-7 LIST OF FIGURES Title Page Phase Diagram for a Gas-Solid MSFB ............. 6 Magnetic Forces on a Bubble .................. 11 Schematic of experimental apparatus ............. 32 Solenoid parameters ....................... 35 Schematic of solenoid cross-section ............. 37 Schematic of gas void fraction probe ............. 38 Experimental Data from One Bubble ............. 41 Calibration Apparatus for Optical Probe .......... 42 Schematic of Conductivity Probe ............... 44 Schematic of Experimental Apparatus for Measuring Mass Transfer Coefficients .............. ' ..... 55 Magnetic Field Strength versus Current: Experimental and Theoretical Values ..................... 67 Axial Variation of Magnetic Field: Experimental and Theoretical Values ........................ 68 Bed Regimes For a Liquid Velocity of 5.15 cm/s ..... 70 Bed Regimes for a Liquid Velocity of 6.44 cm/s ..... 71 Bed Regimes for a Liquid Velocity of 7.71 cm/s ..... 72 Experimental Data from Optical Probe ........... 74 Gas Void Fraction Data as a Function of Magnetic Field Strength for a Liquid Velocity of 5.15 cm/s and Gas Velocities of 0.6, 1.1, and 2.0 cm/s ............. 75 5-8 5-9 5-10 5-11 5-12 5-13 5-14 5-15 5-16 5-17 5-18a 5-18b 5-19 5-20 5-21 5-22 Gas Void Fractions as a Function of Magnetic Field Strength for a Liquid Velocity of 6.44 cm/s and Gas Velocities of 0.6, 1.1, and 2.0 cm/s .......... 76 Gas Void Fractions as a Function of Magnetic Field Strength for a Liquid Velocity of 7.71 cm/s and Gas Velocities of 0.6, 1.1, and 2.0 cm/s .......... 77 95% Confidence Intervals for Gas Void Fraction Data in Figure 5-7 ........................ 78 95% Confidence Intervals for Gas Void Fraction Data in Figure 5-8 ........................ 79 95% Confidence Intervals for Gas Void Fraction Data in Figure 5-9 ........................ 80 Comparison of the Valve Technique and the Optical Probe Technique for Field Strengths of 0 and 275 Gauss .............................. 82 Experimental Data from Conductivity Probes in a Tracer Experiment ........................ 84 Experimental Diffusion Data with One and Two Parameter Fits ........................... 86 Effect of Agitation Rate on Measured Diffusion Coefficient ............................. 87 Pore Size Distributions and Curve Fits for 0, 5, and 50% by Weight Magnetite Alginate Beads . .' ..... 89 Bead Porosities for 0, 5, and 50% by Weight Magnetite Alginate Beads versus Molecular Diameter ......... 90 Percentage of Total Accessible Bead Volume Available for Diffusion versus Molecular Diameter .......... 90 Theoretical Fit of Experimental Tracer Data ........ 92 Comparison of Peclet Numbers Calculated From Dispersion Models with and without the Effect of Tracer Diffusion Included ................... 93 Peclet Numbers as a Function of Magnetic Field Strength for a Liquid Velocity of 5.15 cm/s and Gas Velocities of 0.6, 1.1, and 2.0 cm/s ....... 94 Peclet Numbers as a Function of Magnetic Field Strength for a Liquid Velocity of 6.44 cm/s and Gas Velocities of 0.6, 1.1, and 2.0 cm/s ....... 95 xi 5-23 5-24 5-25 5-26 5-27 5-28 5-29 5-30 5-31 5-32 5-33 5-34 5-35 5-36 Peclet Numbers as a Function of Magnetic Field Strength for a Liquid Velocity of 7.71 cm/s and Gas Velocities of 0.6, 1.1, and 2.0 cm/s ....... 96 95 % Confidence Intervals for Peclet Number Data in Figure 5-21 ............................. 98 95% Confidence Intervals for Peclet Number Data in Figure 5-22 ............................. 99 95% Confidence Intervals for Peclet Number Data in Figure 5-23 ............................ 100 Experimental and Theoretical Oxygen Profiles ..... 101 Mass Transfer Coefficients as a function of Magnetic Field Strength for a Liquid Velocity of 5.15 cm/s and Gas Velocities of 0.6, 1.1, and 2.0 cm/s ...... 102 Mass Transfer Coefficients as a function of Magnetic Field Strength for a Liquid Velocity of 6.44 cm/s and Gas Velocities of 0.6, 1.1, and 2.0 cm/s ...... 103 Mass Transfer Coefficients as a function of Magnetic Field Strength for a Liquid Velocity of 7.71 cm/s and Gas Velocities of 0.6, 1.1, and 2.0 cm/s ...... 104 95% Confidence for Mass Transfer Coefficients in Figure 5-28 ............................ 106 95% Confidence for Mass Transfer Coefficients in Figure 5-29 ............................ 107 95% Confidence for Mass Transfer Coefficients in Figure 5-30 ............................ 108 Correction Factor, 17, versus d as a function of Fem, ......................... 110 Correction Factor, 1;, versus o,” as a function of Pe,” ......................... 113 Effect of System Parameters on the Correction Factor, 17 ............................. 114 xii mgmpocn M once I wanna: :1: m #0 no N H rggnrrr N '9 l" NOMENCLATURE E . . n cross-sectional bed area gas-liquid interfacial area per unit liquid volume inner solenoid radius outer solenoid radius half length of solenoid amount adsorbed in the beads bead concentration experimentally measured concentration final solute concentration gas concentration concentration at gas-liquid interface initial gas concentration initial solute concentration liquid concentration concentration at liquid-gas interface liquid concentration in equilibrium with gas phase initial concentration dispersion coefficient diffusion coefficient diameter of molecule i flow rate transfer function gas flow rate acceleration due to gravity mass of water Henry’s law constant based on partial pressure in gas phase Henry’s law constant based on concentration in gas phase bed height height of gas maximum field strength axial field strength angle of light incidence angle of light refraction current adsorption rate constant film mass transfer coefficient gas phase mass transfer coefficient liquid phase mass transfer coefficient volumetric mass transfer coefficient adsorption equilibrium constant overall mass transfer coefficient partition coefficient distance between measuring points xiii Units cm cm cm cm cm g/mL g/mL g/mL g/mL g/mL g/mL g/mL g/mL g/mL g/mL g/mL g/mL cm’ls cmZ/s mL/s mL/s cm/s2 8 atm mL/g [-1 CH1 cn1 gauss gauss amps cm/s cm/s cm/s g/mL cm/s cm Ln C m,’ M M. 11 “highest no N N n N v 0: 0: P P P3 P1 :~«f-o nax.input "HF. T F U U U V V V V V W x02 2 nax.response bubble length column length nm weighted moment particle magnetization n"I moment refractive index of fluid highest moment used refractive index of probe number of turns of wire ratio of kinetic energy to magnetization energy voidage modulus optical rotation corrected optical rotation dry weight of beads pressure partial pressure of gas at electrode total system pressure gas transfer rate radial distance in beads radius of the beads cross-correlation function Laplace variable time time for bubble to traval across the probe time probe spends in gas phase time when input curve is maximum time when response curve is maximum total time superficial velocity bubble velocity interstitial velocity accessible bead volume gas volume volume of water inaccessible to a molecule of diameter Di liquid volume particle volume mass of solute mole fraction of oxygen in gas phase axial position in bed Greek Letters mama D . . ratio of liquid volume to accessible bead volume fractional increase in pressure due to weight of bed bed conductivity pure liquid conductivity xiv MU)“ U) cm/s cm/s cm/s mL mL mL mL mL 8 1'] cm [-1 [-1 ohm ohm specific mass of water in pores in- accessible to a solute of size i time delay between input and response curves root-mean-square error gas void fraction liquid void fraction bead porosity solid bed fraction an 1e between magnetic field and d1sturbance angle between flow and disturbance bead density gas density liquid density particle density solid density density of water 1/2 time for the response tail to disappear electrode response time time for bubble to travel between probes chord susceptibility differential susceptibility XV Chapter I: Introduction Industrial applications for three-phase fluidized beds are numerous, with such examples as the H-oil process for hydrogenation and hydrodesulfurization of residual oil‘, flue gas desulfurization’, the H-coal process for coal liquefaction’, biological oxidation of waste waters‘, hydrogenation of unsaturated fats’, and Fischer- Tropsch reactions’. The advantages of fluidized beds for these processes include excellent phase contacting, which eliminates the need for mechanical mixing, low pressure drops and shear, which prevent damage to fragile particles, and reduced plugging, which decreases the amount of process down time. Unfortunately, the desirable properties of three-phase fluidized beds disappear at higher gas velocities and in low density particle systems“. Under these conditions, bubbles tend to coalesce leading to increased agitation and shear, and less effective phase contacting. These problems have been alleviated in two phase fluidized beds by applying a uniform axial magnetic field to the bed. With the applied magnetic field, fluidized beds exhibit a wide range of operating conditions where bubble coalescence is eliminated, solids move through the bed in plug flow, mixing of the fluid phase is greatly reduced, and contacting efficiency between phases is increased”’. To date, few studies have been conducted on three-phase magnetized fluidized beds, and no comprehensive analysis has been carried out on the bed characteristics. In this study, the effect of an applied magnetic field on the gas void fraction, liquid mixing, and gas to liquid mass transfer properties of a three-phase system was examined. In addition, the 1 2 macroscopic bed structure, and how it changed with magnetic field strength, was characterized. This dissertation is divided into six chapters. Following the introduction, the second chapter examines the history and development of the magnetically stabilized fluidized bed (MSFB) and discusses the methods used to measure and analyze the data from gas void fraction, liquid mixing and mass transfer studies. The third chapter is devoted to describing the design and construction of the three-phase magnetically stabilized fluidized bed, the experimental methods used to measure the bed properties, and the data analysis techniques used to analyze the data. In the fourth chapter, a mathematical model of the fluidized bed is developed to analyze axial dispersion data when intraparticle diffusion into the solid phase is present. The fifth chapter presents the results of this study, and the sixth chapter discusses the significance of these results and conclusions of the study. Also included in the dissertation are three appendices which include derivation of key equations, listings of the computer programs used to analyze the data, and tables of the experimental data. Chapter 11: Literature Survey 2.1 Fluidized Beds Fluidized bed reactors offer properties that make them preferable to other types of reactors for the growth of living cells. They have lower pressure drops and fewer plugging problems than packed beds. The excellent phase contacting achieved in fluidized beds eliminates the need for mechanical mixing; thus, energy requirements are less than in stirred tank reactors. In addition, immobilizing cells on or within solid particles gives high cell densities and protects fragile cells from mechanical shear. Unfortunately, except under conditions of very low gas flow rates or in systems of large dense particles”, the gas bubbles in fluidized beds tends to coalesce, even forming large slugs in reactors having diameters less than three inches. Coalescence is particularly inherent in systems of light particles such as porous biocatalyst particles“. Coalescence and slugging greatly reduce phase contacting and may reduce the rate of oxygen transfer below that needed by the cells. Also, larger bubbles increase bed turbulence and damage to the particles and cells. Entrainment of particles and loss from the bed is also exacerbated by coalescence. Over the last decade, extensive work on gas-solid and liquid- solid fluidized bed systems has shown that application of a magnetic field can greatly reduce or even eliminate the problems mentioned above. However, the effects of magnetizing three-phase fluidized beds have not been well characterized to date. The following sections of this chapter will examine in depth the properties of gas-solid, liquid-solid and three-phase magnetized 3 4 systems, and examine the theory and measurement techniques for bed properties of phase holdup, liquid mixing and mass transfer of three- phase fluidized beds. 2.2 Magnetically Stabilized Fluidized Bed (MSFB) 2.2.1 Early Studies The first reports of an MSFB were published in the early 1960’s in the eastern European literature by Filippov’H’, who magnetized water-fluidize‘d bed of iron particles magnetized using an A.C. magnetic field. He observed that the transition from a packed to fluidized bed occurred at the same pressure drop as an unmagnetized bed. He also observed the appearance of five distinct regimes or phases in the bed properties which depended upon field strength and fluid velocity: i) packed, ii) pseudopolymerized, characterized by chaining of particles and lack of particle movement, iii) start of particle motion, iv) extensive particle mixing, and v) particle entrainment at very high fluid velocities. Kirkoand Filippov”, showed the entire bed could be levitated as a cohesive unit at very high field strengths. This phenomenon is now referred to as a frozen bed“. At approximately the same time, Herschlern'” observed that a fluidized bed of permanently magnetized particles subjected to an A.C. magnetic field was free of bubbles, slugs and channelling. Katz and Sears‘9'2° found that a fluidized bed could still be stabilized down to a solid-phase volume fraction of 20%. Shumkov and Ivanov“ reported that bed porosity became almost uniform along the length of the bed when a magnetic field was applied. Several researchers observed an increase in bubble frequency and decrease in bubble size in the MSFBZZ‘”. Stabilization 5 was also shown to increase the apparent viscosity of the bed“ and decrease the heat transfer coefficient from that of an unmagnetized fluidized bed”. More complete reviews of early studies have been 1.”. Almost withoutexception, the done by Siegell” and Liu et a above studies were conducted with time-dependent magnetic fields or fields with high gradients. In the late 1970’s researchers began to focus on uniform, D.C. magnetic fields starting the Rosensweig’s work on. gas-solid systems. 2.2.2 Gas-Solid MSFB The first comprehensive examination of gas-solid MSFB was conducted by Rosensweig, who was awarded two patents for the magnetically stabilized bed7'3°. In his first paper“, he described the response of a gas fluidized bed of nickel particles with and without an applied field. Without a field, gas in excess of the minimum fluidization velocity passed through the bed in the form of bubbles and slugs reducing contacting efficiency. When a uniform, axial D.C. field was applied bubbles did not form, and the bed was free of agitation or solids movement. The bed exhibited characteristics of a liquid: light objects floated and dense objects sank; also the suspension flowed from an orifice easily like a liquid. As the gas velocity increased, this stable regime was found to eventually give way to bubbling. Figure 2-1 shows a phase diagram of a magnetized fluidized bed. The transition from the stable phase to the unstable was described by Rosensweig"'9 as sharp and reproducible. He also found that the magnetic field had no effect on the minimum fluidization velocity, and that the pressure drop increased linearly with gas velocity until the pressure drop equalled the weight of the Unstabilized Stabilized Fluid Velocity Field Strength Figure 2-1 Phase Diagram for a Gas-Solid MSFB. 7 bed divided by the cross-sectional area. At this point the bed was fluidized, and the pressure drop remained independent of the flow rate until the bed became unstable. The onset of instability was characterized by fluctuations in pressure drop. Rosensweig also demonstrated the effects of field orientation on stabilization‘, showing that the transition to bubbling behavior occurred at much lower field strengths for a horizontal field than a vertical one. He demonstrated solids transport without solids mixing by placing bands of colored solids in the bed and removing them sequentially from the bed intact’. Later studies by Rosensweig9 found radial dispersion of the gas to be greatly reduced in a magnetized system, and comparable to that of a packed bed. Heat transfer coefficients were found to be higher in the MSFB than in a , packed bed indicating better gas-solid contacting. Rosensweig established criteria for gas-solid bed stability"l based on examination of the equations of motion combined with magnetic stresses as shown in Equations 2-1 to 2-5 N_N,>1 Unstable (2-1) N.N,=l Marginally Stable (2-2) ~_N,<1 Stable (2-3) where N“, the ratio of kinetic to magnetostatic energy, and N,, the voidage modulus, are defined below: N Ag (2-4) I M2 4'u(3-2e‘,)2 ' c081). 1v,- {1+(1-e xo-(l-e (Ari ]cos’0—— (2-5) eiu-e‘) ‘) ‘) J 00326 In these equations, p, is the particle density (g/mL), U is the superficial gas velocity (cm/s), M is the particle magnetization (gauss), e, is the void fraction of the bed, x, is the chord susceptibility, )2, is the differential susceptibility, A is the angle between the magnetic field direction and the disturbance, and 0 is the angle between the flow direction and the disturbance. This analysis also showed that magnetism produces an "elastic like restoring force"31 against the growth of bubbles. Lee32 examined rheological properties of the MSFB, and found that the MSFB is visco-plastic in behavior. Yield stresses measured in the bed were found to increase with increasing field strength. Solids settled more slowly in a magnetically stabilized bed than an unmagnetized system. Arnaldos et at.” looked at the effects of mixtures of magnetic and non- magnetic particles and determined that stabilization could be achieved with as low as 10% magnetic material present; however, the range of magnetic field strengths where bed stabilization occurrred was small at percentages lower than 30%. ArnaldOs and Casal" found that mass transfer of water from humid air was more rapid in a MSFB than either an unmagnetized bed or a packed bed. 2.2.3 Liquid-Solid MSFB The first recent studies of liquid-solid MSFB were performed in 1987 by Siegell” who used water-fluidized beds of steel and composite spheres. He noted four hydrodynamic regimes: packed, stable, roll-cell, and random. The stable regime is similar to that described for gas-solid systems and is characterized by no particle movement. The roll-cell regime is recognized by ”gulf streaming", characterized by independent movement of areas of the bed, and the random regime is characterized by random solids movement. The flow regimes were controlled by the field strength, liquid velocity, and, to a lesser extent, the liquid distributor. Siegell also found the pressure drop profile to be independent of magnetic field strength. Axial dispersion in the liquid-solid MSFB was found to be almost identical to that of a packed bed. Graves and Goetz“ also measured axial dispersion in a liquid-solid MSFB and found results comparable to Siegell. Siegell observed hysteresis in measurements of the bed fraction and bed pressure drop. With increasing liquid flow rate, the bed pressure drop increased linearly until fluidization was achieved and levelled off. As the flow rate was decreased, however, the bed pressure drop was lower than that measured at the same flow rate when the flow rate was increasing. The void fraction was consistently higher for decreasing flow rates than for the identical flow rate when the flow rate was increased. Burns and Graves” determined that liquid radial dispersion was much less in the MSFB than in an unmagnetized bed, but still 2-3 times higher than in a packed bed. — 10 2.2.4 Gas-Liquid-Solid MSFB There are few studies of gas-liquid-solid MSFB in the literature. The first, by Hu and Wu” appeared in 1987. Their system consisted of polyacrylamide gel beads containing Mn-Zn ferrite and fluidized by water and air. They reported bed regimes similar to those of Siegell. At low field strength, bed properties typical of an unmagnetized system were observed. At higher strengths, chains of beads formed and aligned with the field lines. At very high field strengths, the bed coalesced into a single mass. Only a slight variation (1%) in void fraction was found with field strengths from 0-500 gauss. Field strength also had little or no effect on radial distributions of gas holdup; however, there was a pronounced increase in solids holdup with the magnetic field due to bed contraction. Kwauk et a1.” studied bubble size in a three phase MSFB of iron particles fluidized by water and air and found a pronounced decrease in bubble size with increased field strength. They proposed a mechanism for bubble break up, shown schematically in Figure 2-2, in which field lines are ”flexed” by a gas bubble. This "flexing" creates a magnetic force in the direction to restore the field lines to their original shape (i.e. , toward the center of the bubble). This force, in addition to the gravity and inertial forces of particles above the bubble, destabilize the bubble roof and lead to bubble disintegration. From a force balance on the bubble roof, they were able to calculate the maximum stable bubble size for each field strength. These values agreed quite well with the experimentally determined values. 11 Field Lines Bubble Figurei2-2 Magnetic Forces on a Bubble 12 2.2.5 Potential Applications of the MSFB The capability of continuously contacting a dispersed solid phase with a fluid phase without mixing of the solid phase has led to several applications of the MSFB. One such application is filtration of fine particles. An MSFB using magnetite particles of 250-400 11111 in diameter showed 99% efficiency in removal of fly-ash particles greater than 4 um and 95% efficiency for particles less than 2.1 um“. Other uses are the magnetic distributor-downcomer (MDD) , and the magnetic valve for solids (MVS)‘°'“. The MDD controls solids flow in a multistage process by turning on the current to a solenoid to freeze the solids in place and turning off the current to allow solids flow at prescribed intervals. The MVS is an on-off valve for solids flow in a pipe where the solids are frozen into place when the current in on and can move freely when it is off. A circulating magnetically stabilized bed reactor (CMSBR) has been proposed by Pirkle“. This reactor would approximate plug flow of both the solid and fluid phases, and computer simulations have shown that temperature profiles could be produced that would eliminate the need for heat exchangers or interstage cooling of reactants. The CMSBR could provide temperature profiles that would optimize reactor conversion in equilibrium-limited, exothermic reactions, and for highly exothermic reactions, the CMSBR is much less sensitive to temperature runaway than conventional reactors. Other uses for the MSFB are continuous adsorptive separation using molecular sieves“"‘; continuous adsorptive separation for gas purification involving ethylene, cryogenic-plant gas, natural gas, and flue gas"; and adsorptive separations of hydrocarbon mixtures“. 13 The liquid-solid MSFB has been demonstrated to be very effective in chromatographic separations‘9'“. The MSFB has also been demonstrated to work effectively as a bioreactor33'”. 2.3 Phase Holdups 2.3.1 Definition and Theory Volume fraction, also called holdup, is defined as the fraction of bed volume taken up by a particular phase. By definition, the phase holdups in a fluidized bed sum to one as shown in Equation 2-6 e,+sl+e‘=l (245} where e, is the solid fraction, 51. is the liquid holdup, and e, is the gas holdup. Solid holdup can be easily calculated from the bed height, H (cm), and volume of solid particles present, V, (mL), as shown in Equation 2-7 V 2.5;: (24) where A is the cross-sectional area of the bed (cm’). 2.3.2 Measurement Techniques Measurement of the static pressure profile of the column is commonly used to determine phase holdups. The differential force balance shown in Equation 2-8’3 relates the static pressure gradient (dP/dz), with units of g/cm2- s’, to the phase holdups _d_P.= L 2-8 dz (93‘:+9L3L+prec)gc ( ) where p, is the solid density (g/mL), pL is the liquid density (g/mL), 14 p, is the gas density, g is the acceleration due to gravity (cm/32), and gc is the unit-conversion constant. By solving Equations 2-6 to 2-8 simultaneously, the three phase holdups can be calculated. Another method to measure gas holdup involves simultaneously shutting off the gas and liquid flow rates (the valve technique), and then measuring the distance which the liquid falls in cm (H,) as the gas bubble escape. The gas holdup is then calculated by Equation 2—9’“4 .551 (2-9) 3. where H, is the total column height (cm). Tracer techniques” can be used to calculate the interstitial velocity of the gas or liquid phases between two points. The gas or liquid holdup is then calculated by Equation 2-10 .11 e U, p (2-10) where U, is the interstitial velocity (cm/s), and U is the superficial velocity of the gas or liquid phase calculated by Equation 2-11 =5 (2.11) where F is the volumetric flow rate (mL/s) of the gas or liquid phase. Conductivity can be used to measure liquid holdup”. It has been found that the liquid holdup is proportional to the bed conductivity as shown in Equation 2-12 =1— (2.12) where 7 is the bed conductivity (ohm) and 7, is the conductivity of the pure liquid (ohm). Several optical probes have been described""""‘9 that give different responses to gas and liquids. Gas holdup can then be calculated by . =5; (2.13) ‘ t where t, (s) is the length of time the probe detects gas and t is the total time (s). i 2.4 Liquid Mixing 2.4.1 Theory Liquid mixing studies often use one of three common mixing models: well mixed, plug flow, and axially dispersed plug flow. A mass balance on a continuous-flow, well mixed liquid phase is shown in Equation 2-14‘so Refer—€35 (2.14) where C, is the inlet concentration of solute in the liquid phase (g/mL), CL is the effluent concentration of solute in the liquid phase (g/mL), F is the liquid flow rate (mL/s), and t is time (s). The plug flow model is shown in Equation 2-15“0 16 BCL GCL (2.15) where z is the axial position in the column (cm). . The axial dispersion model is commonly used to describe liquid mixing in fluidized beds. This model is shown in Equation 2-1661 ac =0 filyfl (2-16) 737 “522 67. where D“ is the axial dispersion coefficient (cm’ls). This model is based on the assumption that fluid moves in plug flow with a dispersive mixing mechanism superimposed on it. This dispersive mixing mechanism is analogous to diffusion, but occurs on a macroscopic scale. The validity of this model has been challenged by Alvarez-Cuenca and Nerenberg“2 who divided the column into mixing zones, the grid and bulk, where the grid zone was characterized by complete mixing, and the bulk zone exhibited plug flow. 2.4.2 Experimental Methods to Measure Dispersion Coefficients The axial dispersion coefficient can be measured by monitoring the progression of a tracer as it is carried through the bed by the fluid phase. Two types of tracer methods are commonly used: the one-shot method and the imperfect pulse method. In the one-shot method, a mathematically describable tracer profile is injected into the column, and its shape is measured at a position downstream. In the imperfect pulse method, an arbitrary tracer profile is injected into the column and then measured at two axial positions in the 17 column. The imperfect pulse method is easier experimentally, since producing a perfect mathematical pulse is difficult, but data analysis is more difficult. Commonly used tracers for both methods are salts“, dyes“, acids or bases", isotopes“, and heat“. 2.4.3 Data Analysis Wakao and Kaguei“ reviewed several techniques for determining dispersion coefficients from tracer concentration (C,,,,) versus time data, and calculated errors associated with each method. In the moment method, the first and second moments of the input and response curves (M1 and M2, respectively) are calculated, and the dispersion coefficient and the interstitial velocity are then determined using Equation 2-17 and 2-1861 Mf—M’af? (2-17) 20 . M,"-(Ml 2-M,’-(M,’)1= ”‘3‘," (2-18) where the superscripts I and II refer to the input and response curves respectively, and L is the distance between the two measuring points (cm). The n“I moment of the tracer data is defined by Equation 2-19“ '6 '4: M iii (2.19) . EC“ The shortcoming of this method is that the tail portion of the curves are weighted heavily by the t“ factor especially for the higher l8 moments. Thus, measurement errors in the tail portion of the curve are magnified. The weighted moment method reduces this sensitivity to tailing errors by using a weighting factor exp(-st) that approaches zero both at short and long times. The nlh weighted moment is then defined by Equation 2-20“ .C ”e "d: :J0 “"t (2.20) f. Cw The transfer function, F(s), of Equation 2-16 describes the relationship between the input and response curves in the Laplace domain. The transfer function is related to the moments as shown in Equations 2-21 and 2-22 0U %.I=F(s) (2-21) mo ELELLG). (2.22) "'0‘” mo" H" where F(s) and F’(s)/F(s) are defined in Equations 2-23 and 2-24 1 F(s) =u-p[fl{1 ..(1 +4253] 2] (2-23) 20“ (12 -1 £9). = -_L.(1 +431?) 2 (2-24) F(s) U The difficulty with this method lies in choosing the optimal s value 19 to use. The optimal s value may also be different for each moment calculated. Equation 2-25 gives suggested values of s to use‘59 3. "m- (2.25) t t AtD ~ max m+ max rayon“- where 1111.1." is the higher order moment used in the parameter estimation, MD is the time delay between the input and response curves, and twain”, and tm,,,,,,,,, are the times when the input and response curves have maximum values, respectively. The transfer-function-fitting method used Equation 2-26 or 2- 27“. -flntF(s)1}“=%suntF Probe Ports §\\\\\\\\\\\\\\\\\\\\\\\ Ring 1 Spam" '1 ‘\ Calming l Region Air Inlet Water Inlet Figure 3-1 Schematic of experimental apparatus. 33 3.1.3 Gas Phase Compressed air was metered into the column with a rotameter. It was sparged through a stainless steel ring 3.5 cm in diameter with 0.2 mm holes spaced at 5 mm intervals around the ring. Gas flow rates used in this study (measured at 25°C and a one atmosphere pressure) were 11.8, 21.6, and 39.2 cm3/s. 3.1.4 Solid Phase The solid phase consisted of 4 mm composite beads made from a calcium alginate and powdered magnetite (Fe304). The beads were produced by mixing equal weights of two percent by weight, aqueous, low viscosity sodium alginate (Sigma, St. Louis MO) and magnetite powder (Aldrich, Milwaukee WI). The resulting mixture was dripped through a 3 mm tube into a solution of 0.2 M calcium chloride and allowed to harden for 48 hours. To prevent leakage of magnetite, the beads were then coated with an additional layer of alginate. The coating procedure involved blotting exceSs water from the beads’ surfaces. then shaking the beads in a plastic bag containing sodium alginate powder. After shaking off excess powder, the beads were placed on wax paper for 15 minutes to allow partial dissolution of the sodium alginate powder with water from bead pores. The beads were then cured in 0.2 M calcium chloride for 48 hours, and washed twenty times with a volume of RO water equal to twice the bead volume to remove the excess calcium chloride before being stored in R0 water. 34 3.2 Solenoid 3.2.1 Design The design equations for a finite solenoid are shown in Equations 3-1 to 3-3“. 3 H 0 =11 —"—— (34) 1(1’ ) 0 (az+zz)3’2 _NI 1 _ -m “(envy/2] F( . )-— (3-3) 2 is the axial position (cm), -H,(z,0) (gauss) is the field strength along the center line of the solenoid, H, (gauss) is the maximum field strength, F(d,l3) is the field factor, N is the number of turns of wire, I (amps) is the current, the parameters a,, a, and B are defined in Figure 3-2 and a (cm) is the average of a, and a2. The sclenoid length and diameter, length of wire, the current, and power comsumption had to be specified in the design. The solenoid length was limited to approximately 80 cm, due to the size of the fluidized bed column. The solenoid inner diameter had to be at least 6.5 cm to accomodate the fluidized bed. A maximum field strength of 300 gauss was chosen based on previous results”. Then, based on simulations using Equations 3-1 to 3-3, values for the current, power consumption, and wire length were chosen to be 5 amps, 350 watts, and 1000 meters respectively. The computer programs used in these simulations are listed in Appendix B. These chosen values were somewhat arbitrary, since several combinations of current and wire 35 32 a, is the inner radius of the solenoid a2 is the outer radius including wire 2b is the length of the solenoid a = azla, B = b/a, Solenoid Wire . Figure 3-2 Solenoid parameters 36 length would have given a field strength of 300 gauss. When examining commercially available power supplies, one was found capable of producing 5 amps and 350 watts, so that particular set of parameters was chosen. The solenoid was designed with an internal water jacket as described below, to remove excess heat. 3.2.2 Construction A cross-section of the solenoid is shown schematically in Figure 3-3. The solenoid consisted of 1000 m of 17 gauge coated copper wire (MWS Wire Industries, Westlake Village CA) wrapped around an 82 cm long copper cylinder having a 10 cm outer diameter and 0.25 cm wall thickness. Inside this cylinder was a second, concentric copper cylinder having a 7.5 cm outer diameter and wall thickness of 0.25 cm. The cylinders were held in place with two square Teflon endpieces that were secured by four aluminum rods, one in each corner of the Teflon piece. Fittings were mounted into the Teflon endpieces to allow flow of cooling water thrOugh the annular region between the two cylinders. Power for the solenoid was provided by a 100 volt, 5 amp D.C. power supply (Kepco, Flushing NY). This unit provided magnetic field strengths from 0- 300 gauss, as measured by a Gaussmeter (Magnetic Instrumentation Inc., Indianapolis IN). Tap water at approximately 10°C was pumped through the solenoid to remove the heat generated. 3.3 Gas Void Fraction 3.3.1 Probe construction Gas void fraction was measured using a fiber-optic probe‘3'“, shown schematically in Figure 3-4. The probe was constructed by bending an acrylic optical fiber (Edmund Scientific, Barrington NJ) Water Outlet 4.. Teflon endpieces 37 \\‘ ..... .......... .... ........ ......... ...- ......... .... ......... ......... ..... ......... ........ ......... .‘. . .... ..... ........ .:._.‘.;. ..... ......... Zvfi-Z ..... n ...... n ------ aaaaaa ..... a ..... \\\\\\\\\\\\\\\\\\\\\\\\§ ...... ........ ..... v0.- ........ ......... ......... ......... .,.,...;. ..... .... ......... ........ ....... ....... .... ..... ......... ..... ..... .... .... .... ..... ......... ......... .... ......... ....... ....... ......... ..... .aa. ......... ‘‘‘‘‘‘‘ ..... . o ........ ., ...... . .. ....... ......... .... ........ ..... ....... .;.:. .;. I' t' ..... ..... . e ....... . Etta. fiégég : '13 S' :2. .fifiéfi- .’ Copper Wire Annular Water Jacket luminum Rods Copper Cylinders Figure 3-3 Schematic of solenoid cross-section. 38 Photodetector \ \ > a Laser ll: Probe' In Air Photodetector v/ .\ Laser / ln Probe' in Water Figure 3-4 Schematic of gas void fraction probe 39 250 microns in diameter into a U shape and pulling the ends through a stainless steel tube 570 microns in diameter and 2.5 cm long until approximately 1 mm of the U remained outside the tube. The probe tip was then ground with 1 micron grit aluminum oxide paper to form a 90° included angle. Monochromatic light from a HeNe laser (Newport Corporation, Fountain Valley CA) was focused onto one end of the fiber with a microscope lens. A silicon photodetector (Newport Corporation) measured the light from the emitted from the other end. The operation of the probe is based on Snell’s Law“, which can be applied at the interface between the probe and the fluid: It sin(i) =no sin(i,) (3-4) where n, is the refractive index of the probe, i, is the angle of incidence, n is the refractive index of the medium surrounding the probe, and i is the angle of refraction. For the fiber used, n,= 1.495, and for the probe geometry shown in Figure 3-4, i, = 45°. Thus, when n is less than 1.06, the light rays should be reflected back into the probe, and when n is greater than 1.06, the light should be reflected out of the probe. Water and air have refractive indices of 1.33 and 1.00 respectively, and can thus be discriminated by the probe. 3.3.2 Calibration and Data Analysis The probe tip was centered radially at an axial position 45 cm above the gas sparger in a horizontal position. The photodetector output was sampled by a data acquisition system (Labtech Notebook, Wilmington VA) at 200 Hz for 45 seconds for each void fraction 40 determination, and five replicates were measured for each set of conditions. The gas void fraction was calculated as the time the probe spent in the gas phase divided by the total time. The time in the gas phase was taken to be the sum of the peak widths. However, as shown in Figure 3-5, some of the peaks are triangular, and the correct width to use (W, cm) is not clear. The appropriate width was determined by calibrating the probe. Figure 3-6 shows the calibration apparatus. Liquid was metered into a 3 mm inner diameter vertical tube with two optical fiber probes spaced 6.5 cm apart, and a bubble was injected at the bottom of the tube. Due to the small diameter of the tube, the bubble filled the entire cross- section, so the probe could not miss the bubble. A 35 mm camera was used to take a picture of each bubble, and a ruler was placed next to the tube to provide a scale for the bubble length. The cross- correlation function of the two signals was calculated as follows:"2 R,=%.fo’x(oy(z+a)dx as where R,,(r) is the cross-correlation function, x is the signal from the first probe, y is the signal from the second probe, and T (sec) is the total time. The value of 1 that maximizes R,,, designated 1,“, is the time the bubble requires to travel between the two probes. The bubble velocity, U3 (cm/s), was then calculated from Equation 3-6. v.=f—°- (3-6, 41 25* A 2.0- ”(F :‘i O 1.5- 3 <1) P 07 g 1.0- O W > A 0.5- H N ___ l v v / 0.0 Illlfl§llltllitiltlrj 26.60 26.62 26.64 26.66 26.68 26.70 Time (seconds) ‘ H % Peak Height: p _ N x 100 Figure 3-5 Experimental Data from One Bubble 42 fl ‘ Optical L . Fiber 65 mm '3 Probes O \ Y \ Bubble v T Water lnlet * Not Drawn to Scale ' Figure 3-6 Calibration Apparatus for Optical Probe 43 The bubble length, L, (cm), was measured from the photo of the bubble, and the time the bubble required to travel across the probe, t, (sec), was calculated from this length and the bubble velocity using Equation 3-7. -_-_ (3.7) The appropriate width of the probe signal was then determined, as a percent of the peak height as shown in Figure 3-5. A value of 10% was found to be optimal for all the liquid velocities used in this study. The computer program used to calculate void fraction from experimental data is listed in Appendix B. Measurements of gas void fraction using the valve technique (Section 2.3.2) were also conducted to determine how the local gas void fraction compared with the averaged gas void fraction. 3.4 Liquid Phase Dispersion Liquid-phase dispersion coefficients were determined using the imperfect pulse method. In this method a pulse of tracer is injected at the bottom of the column, and its progress is followed by probes at two different axial positions in‘the column. For this study, a calcium chloride tracer and conductivity probes were used. 3.4.1 Probe Construction and Calibration The conductivity probes were constructed by placing a 1 mm diameter brass rod inside a 2 mm I.D. brass tube with a 0.5 mm thick plastic non-conducting spacer in between, as shown in Figure 3-7. The tip of the probe was ground flat with emery paper. Wires were soldered to the inner rod and the outer cylinder and then connected 44 Brass Tube Plastic Spacer Brass , Rod Figure 3-7 Schematic of Conductivity Probe 45 to a six pin plug compatible with the conductivity meter (Cole Parmer, Chicago IL). These probes were placed at axial positions 30 and 60 cm above the gas sparger and connected to conductivity meters. Because ground loops formed in the column, both signals had to be isolated using a voltage isolator (Metrabyte, Taunton MA). The conductance of each probe was calibrated using known concentrations of aqueous calcium chloride. The conductance versus concentration curves were curve fit using linear regression and generally had correlation coefficients of 0.999. The calibration curves changed as the probes aged, making it necessary to repeat the calibration every few days. For each experiment, 2 mL of 100 g/L calcium chloride was injected at the bottom of the column. The output of each conductivity meter was recorded by the data acquisition system Labtech Notebook at 20 Hz. Five replicates were done for each set of conditions. 3.4.2 Data Analysis An unsteady-state mass balance on the salt tracer in the liquid phase is shown in Equation 3-8 along with the appropriate boundary and initial conditions“ a: "‘ azz r52" t= z=0 CL=C0 (3.3) z= CL=0 t>0 z=o° CL=0 where C, (g/L) is the tracer concentration in the liquid phase, t (sec) is time, D“ (cm’ls) is the dispersion coefficient, x (cm) is the axial 46 position, and U, (cm/s) is the interstitial liquid velocity. This model assumes that the salt tracer exists only in the liquid phase, that the salt concentration is uniform in the radial direction, and that there is no adsorption or diffusion of tracer into the solid phase of the system. However, in the system studied, the solid phase is highly porous, and diffusion into the bead pores is likely. To account for this effect Equation 3-8 must be modified to include tracer diffusion. An unsteady-state mass balance on tracer in the liquid phase that includes disappearance of tracer by diffusion is shown in Equation 3- 9. This equation is coupled to the unsteady state mass balance on tracer in the solid phase shown in Equation 3-10“ ac,_ a’c, vac, 3(1-eL-e‘)De ac __ O}, B , , (3-9) a: " 322 az eLR ar )rsR where e, is the liquid void fraction, 5, is the gas void fraction, e, is the bead porosity, D, (cm’ls) is the diffusion coefficient, C3 (g/L) is the concentration in the solid phase, R (cm) is the bead radius and r (cm) is the radial position in the bead“. The boundary and initial conditions are the same as those listed for Equation 3-8. —-—'-=D __ 6C BC . 1 ° [rt—’1 a: 1'2 6r 6r t=0 c, =0 r=R c, -rrpc, (3'1“) x8 r=0 -— =0 6r The partition coefficient, K,, of the tracer between the liquid and solid phases, was assumed to be 1.0 in this study. The assumptions 47 made in deriving Equations 3-9 and 3-10 were that film mass-transfer resistance was negligible, transport into the beads occurred only by Fickian diffusion, (i.e. convection into the beads was negligible) and no adsorption of tracer onto the beads occurred. Equations 3-8, and 3-9 and 3-10 were solved using the Laplace transforms shown in Equations 3-11 and 3-12 and 3-13, respectively“ " H '8& fl[1—(1¢.‘£".)m] I up! =6 2 ”‘2 F(s)- ° I (3-11) " '3 f. «a .. 'c” «at -'—‘a<[1--‘—D=o-q)]-;) F(s)- ° ""e =e ’ ”3 (3-Hx) Q l _ 1;ng 'dt q= 3sep(1 fife?) [coth(Rse')-1] (343) eLR D. where C‘,,,, (g/L) and C",,,, (g/L) are the experimental tracer curves from the first and second probes, respectively, and Pe, the Peclet number based on the bead radius (R) is defined as RU,/D,,. A computer program developed by Wakao and Kaguei“l coupled with the optimization program PATERN”, was used to determine the optimal values of the dispersion coefficient and the interstitial liquid velocity for each set of tracer data. The program of Wakao and Kaguei converted the experimental tracer curves into Fourier series and calculated a predicted tracer curve using the Fourier series of the data from the first probe as an input function and the above transfer functions. The optimization program, PATERN”, was then 48 used to vary the dispersion coefficient and the interstitial velocity until the mean square error between the predicted and experimental curves was minimized. These programs are listed in Appendix B. 3.4.3 Diffusion Coefficient 3.4.3.1 Experimental Technique An experimental technique was developed to measure the effective diffusion coefficient for use in Equations 3-9 and 3-10. A 250 mL beaker was filled with 100 mL of 10 g/L calcium chloride. Fifty mL of beads were measured by displacement in a graduated cylinder, and the excess water was removed by blotting the beads with paper towels. The beads were placed in the calcium chloride solution and agitated on a gyratory shaker at 200 rpm for one hour. The liquid conductivity was measured using a conductivity probe and sampled by Labtech Notebook once a minute. The conductivity probe was calibrated in concentrations of l, 5, 8, 10, and 12 g/L calcium chloride while agitating at 200 rpm. Several experiments were done at rpm values from 0-300 to ensure that film mass transfer was not controlling. 3.4.3.2 Data Analysis The unsteady state mass balance on the solid phase assuming that both film mass transfer and adsorption were negligible is given in Equation 3-10. However, different boundary and initial conditions are appropriate for this system“'”: t=0 CB=O x1. _ x5 ’= V1.3,“ Ape—,7 in (3-14) 6CD r= — =0 Br 49 where A, (cm’) is the surface area of the beads. The solution of Equation 3-10 with the boundary and initial conditions given in Equation 3-14 yields Equation 345“”. ( -D,q:f) Rsin( W C fan“): 6‘1”" " 71 ‘3'“) a 1+“ '1 9+9a+qfu2 rsin(q,,) The variable a is defined in Equation 3-16, V ¢= L (3-16) 1(an where V, (mL) is the liquid volume in the beaker, VB (mL) is the bead volume accessible to tracer. The n“l root of Equation 3-17 is q... 34 tanam= " (3-1‘7) 3+aq: If experimental conditions are such that the liquid film around the beads is negligible then the concentration at the surface of the bead is equal to the concentration in the bulk liquid. For that situation at r=R, C3 in Equation 3-15 can be substituted with CL, and Equation 3- 18 results”-“. 41,43: c fofiaz: ”W" " 1 (“8’ L 1+“ '1 9+9cz+qzat2 PATERN was used to vary the the effective diffusion coefficient until the mean square error between the theoretical and experimental 50 curves was minimized. Five replicates were measured to determine the effective diffusion coefficient. The program used to calculate diffusion coefficients from experimental data is listed in Appendix B. 3.4.3.3 Solute Exclusion Technique to Determine Pore Structure Alginate beads consist of a three dimensional structure of crosslinked B-D-mannuronic and a-L-guluronic acids“. The cross links define water-filled pores having a distribution of pore diameters. The solute exclusion technique of Stone and Scallan’7'” was used to determine this distribution of pore diameters. Because a molecule can only enter pores larger than the molecule’s diameter, all pores smaller than this diameter will be inaccessible to the molecule. Thus, by using a range of molecules with different diameters, it is possible to measure the fraction of pores that is inaccessible to a given molecule size and construct a pore size distribution. In the technique, a known mass of aqueous solute solution is equilibrated with a known mass of porous beads. The change in concentration of the diffusing solute is measured by optical rotation, and the volume of inaccessible pores is calculated from a solute balance before and after contacting, as shown in Equation 3-19” C 6, p=(w+q) —w—‘ (3-19) C! where 6, is the specific mass of pores inaccessible to a solute of size i (g of inaccessible water/g of beads), w (g) is the mass of solute solution added, q (g) is the mass of water in the beads, p (g) is the 51 dry weight of the beads, C, (g/mL) is the initial concentration of solute, and C, (g/mL) is the final concentration after equilibration corrected for minor water soluble compounds in the beads. The volume of inaccessible pores per unit volume can be calculated as shown in Equation 3-20 ”5,2: (3-19) where V, (mL of inaccessible water/mL of beads) is the specific volume of pores inaccessible to a solute of size i, p,V is the density of water, and p; is the bead density. Nine linear dextrans were used in this study, ranging in molecular weight from 6000 to 2 million glmole, in addition to three carbohydrates ranging from 180 to 504 glmole. Linear dextrans have been shown to behave as hydrated spheres in aqueous solution“; this allows a direct correlation between solute diameter and pore width. Diameters of the carbohydrates and dextrans in solution are calculated from their diffusion coefficients according to the Stokes- Einstein formula”; values used here have been taken from Weimer and Weston90 and are listed with their molecular weights in Table 3-1. Solute concentrations of 2% were prepared by dissolving 2 g of anhydrous solute in 100 mL of distilled water. A mass of beads with a volume of approximately 10 mL was weighed into a 20 mL polyethylene scintillation bottle with leakproof caps, and solute solution was added until the bottle was full. Three replicates were used per solute. Three bottles containing only distilled water and 52 Table 3-1 Molecular Probes Used in the Solute Exclusion Technique“9 Estimated Diameter‘ Solute Vendor Mw' Mw/M,‘ Glucose Sigma 180 1.0 8 Fructose Sigma 180 1.0 8 Maltose Fisher 342 1.0 10 Raffinose Sigma 504 1.0 12 Dextran 6K Fluka 6,000 N.A.” 38 Dextran T10 Pharamacia 10,500 1.74 51 Dextran 15-20K Polysciences 17,500' N.A. 61 Dextran T40 Pharmacia 38,800 1.60 90 Dextran T70 Pharmacia 72,200 1.88 118 Dextran 200-300K Polysciences 2.5 X 10" N.A. 204 Dextran T500 Pharmacia 5 .07 x105 2.16 270 Dextran T2000 Pharmacia 2.0 X 10° N.A. 560 a Obtained from vendor’s lot analysis except where indicated by (*), where M. is calculated as the midpoint of the molecular weight range reported. b N.A. = Not Available c Probe diameters estimated using the values given in the or work by Stone and Scallan 87,88 and from Weimer and Weston . '(ginal 9 53 beads were also prepared as blanks, to determine the effects of water soluble materials that diffuse out of the beads on the measured dextran concentration. The bottles were allowed to equilibrate overnight with occasional shaking. Ten mL ofthe liquid were withdrawn, filtered with a 0.45 pm acrodisc, and placed into a 10 cm polarimeter sample tube of an Autopol 111 Automatic Polarimeter (Rudolph Research, Flanders NJ). The remaining liquid was removed, and the bottles, still containing beads, were filled with distilled water and allowed to sit for 30 minutes. The water was replaced with fresh distilled water and this process was repeated 9 times to remove most of the solute from the beads. The beads were then dried to constant weight at 105°C and their dry weight recorded. The optical rotations measurements were corrected for water soluble materials originally contained in the beads by subtracting out the averaged optical rotation of the three blanks as shown in Equation 3-21. 0 of, win—pl)“ (3-21) where O,,, is the corrected optical rotation reading, 0, is the optical rotation reading of the sample, p (g) is the dry weight of the sample, (O,/p),.,,,, is the averaged optical rotation readings of the blanks divided by the dry weight of the blanks. The solute molecules were assumed to be spheres in solution“; thus a pore that was inaccessible to a solute was assumed to have a diameter smaller than the diameter of the spherical solute molecule. Pore size distributions were 54 measured for beads containing only alginate, beads containing 5% by weight magnetite, and beads containing 50% by weight magnetite. The pore size distributions for the 0 and 5% beads were empirically fit to Equation 3-22 a 8 =——"—-—z max-04 (3-22) ‘ hear" where x is defined as log,o[D] - log,o[4], D (A) is defined as the molecular diameter, and a0, a,, a2, a3, and a, are empirical constants. Water has a diameter of 4 A, and it is assumed that all pores in the material are accessible to water; therefore, the value of 5, corresponding to 4 A is zero. The logarithm of 4 A was subtracted from the logarithm of the molecule diameter to give the variable x a lower bound of zero. The pore size distributions for the 50% beads were fit to Equation 3-23 a,- a" 30:24:, (3.23) 1414'.” . +x The forms of Equations 3-22 and 3-23 were chosen solely on their ability to fit the experimental data. The values of the constants were determined using Peakfit (Jandel Scientific, San Rafael CA). 3.5 Gas To Liquid Mass-Transfer 3.5.1 Experimental Procedure Figure 3-8 shows the experimental apparatus for the mass-transfer experiment. The 55 gallon holding tank contained reverse-osmosis water that was continually sparged with nitrogen gas until the oxygen concentration was approximately 0-4 % of the air saturation value. 55 Oxy en l Stripping Column Solenoid 7 / //////////////////////////////////////////I//. ///////////////////////////////////////////////// ’/ Z Figure 3-8 Schematic of Experimental Apparatus for Measuring Mass Transfer Coefficients 56 The oxygen concentration in the tank was continuously measured with a dissolved oxygen probe (YSI, Yellowsprings OH). Water was pumped from this tank, through a rotameter, and into the column, where it was sparged with air. Water leaving the top of the column was gravity fed to the top of a glass stripping column 5 cm in diameter and 137 cm long, where it was countercurrently contacted with nitrogen sparged through a glass frit. Water exiting this column was pumped back into the holding tank. The combination of nitrogen sparging in both the stripping column and the holding tank was sufficient to keep the dissolved oxygen concentration in the holding tank within 5% of the desired value. Water at 0°C was pumped through a copper cooling coil submerged in the holding tank to remove heat generated by the pumps. The temperature in the holding tank was controlled at 25°C :l; 0.2°C using this method. The temperature was measured using a thermocouple (Cole Parmer, Chicago IL). . The oxygen concentration in the column was measured with a Clark style dissolved oxygen microprobe (Diamond General, Ann Arbor MI) with a tip diameter of 2 mm connected to a chemical microsensor (Diamond General, Ann Arbor MI). The probe was calibrated in 25°C RO water. The water was sparged with air through a glass frit for 15 minutes, at which time the reading on the microsensor was adjusted to 100%. The water was then sparged with nitrogen for 15 minutes, and the reading was adjusted to 0%. This procedure was repeated until the microsensor gave the correct reading after sparging 15 minutes with either air or nitrogen. The probe was then placed in the column at a position 5 cm above the 57 sparger. The output of the microsensor was sampled by Labtech Notebook for two minutes at a rate of 1 Hz. This process was repeated for positions 15, 30, 45, 60, and 75 cm above the sparger. Three replicates were performed for each set of conditions studied. 3.5.2 Data Analysis The steady state method was used to calculate the gas-to-liquid mass-transfer coefficient (kLa). A steady-state mass balance on the liquid in the column is shown in Equation 3-2479. 1 dZCL dCL _ s ‘-c :0 (3-24) Pe dzz dz + '(CL 1) where Pe and St are defined as U L k 1. Fe = 1' St=( La) (3'25) D are]. ”I. where CL (mg/L) is the concentration of oxygen in the water, C,‘ (mg/L) is the concentration of oxygen in the water that would be in equilibrium with the oxygen in the gas phase, 2 (axial position divided by the column length) is the axial position, UL (cm/s) is the superficial liquid velocity, D“ (cm’ls) is the liquid dispersion coefficient, 5,, is the liquid void fraction, L (cm) is the column length, and kLa (8“) is the mass transfer coefficient. Equation 3-24 was derived assuming that axial liquid mixing in the column can be described as a dispersed plug flow model and that the mole fraction of oxygen in the gas phase does not change appreciably across the column. The value of CL° can be defined as follows from Henry’s law”: c;=__2 (3-26) where P (atm) is the total pressure at a given pesition in the column, x02 (mole Oz/mole gas) is the mole fraction of oxygen in the gas phase, and H (atm mole Ozlmole gas) is the Henry’s law constant for oxygen in water. The pressure in the column varies as described in Equation 3-27"”"’0 P-P,{1+a(1-z)] aJwaoleup 8 8L (3'27) 3),. where p, is the solid density (kg/m3), p, is the liquid density (kg/m3), p, is the gas density (kg/m3), e, is the solid fraction, g (m/s’) is the acceleration due to gravity, and PT (Pa) is the pressure at the top of the column. Combining Equations 3—24 through 3-27 yields the 6 following differential equation” 1 dzCL dCL ——-—-StCL=-St(a+bz) Pe dzz dz (3-28) a=P ,sz(1+¢) b= 'P 1‘02“ H H with the following boundary conditions: dC z=0 c,=c,+i—': Pe dz (3-29) (10, 2:1 ——-=O 59 where C, (mg/L) is the inlet oxygen concentration. The analytical solution to Equation 3-28 is given by 3-30 to 3-3579 CL=A1e"z+A2e"‘+a-£+bz - where P: 48: 712:7[1 :1;\ 1+—] P: A, _ (Brze " -br,) N (-Br,e " +br2) 2 N B=[a-C,-§]Pe-b 2 2 N =rl e" -r2 e" (3-30) (3-31) (3-32) (3-33) (3-34) (3-35) The experimental data were fit using PATERN to find the value of kLa which minimized the mean square error between the experimental data and the theoretical curve given by Equations 3-30 through 3-35. The computer program used to calculate the mass-transfer coefficient is listed in Appendix B. Chapter IV: Fluidized Bed Model Determination of dispersion coefficients using the conventional dispersion model (Equation 3-8) is much more convenient than using the model including intraparticle diffusion of the tracer (Equations 3-9 and 3-10). However, there are no established criteria with which to judge whether intraparticle diffusion can be neglected. Additionally, when intraparticle diffusion is significant, it would be convenient to be able to correct the apparent dispersion coefficient determined using Equation 3-8 for the effect of diffusion. For these reasons, a mathematical model of a fluidized bed was developed that could predict the effect of diffusion on the apparent axial dispersion coefficient. The modelling results were used to calculate a dispersion correction factor (17), defined in Equation 4-1, that is analogous to the catalyst effectiveness factor. 1: =_mz . (41) D where Dan, is the apparent dispersion coefficient calculated from Equation 3-8, and D“ is the true dispersion coefficient calculated from Equations 3-9 and 3-10. Development of the model is described below. 4.1 Derivation The stirred tanks in series model have been widely used to describe fluid mixing in flow systems. In this model, the effluent from the first tank is assumed to be the feed to the second tank, and so forth, for all the tanks in series. By increasing the number of tanks, the system behavior asymptotically approaches plug flow. At 60 61 the other extreme, a single stirred tank describes a perfectly mixed system. Equation 4-2 is an unsteady-state mass balance on the liquid phase of a stirred tank containing gas, liquid and solid phases. This equation is derived in Appendix A.5 6C 1- - V 6C UA(Cl-CI)=VL 61% ep( 8‘ $.52 (4-2) at. where A (cm’) is the cross-sectional area of the stirred tank; C, (g/L) is the inlet concentration of tracer to the tank; U (cm/s) is the superficial liquid velocity; CL (g/L) is concentration of tracer in the tank; VL (mL) is the tank volume; 5,, e,_, and e, are the gas volume fraction, liquid volume fraction and the particle porosity, respectively; and CB (g/L) is the concentration of tracer in the beads. The term ac,/at coupled Equation 4-2 to the unsteady state mass balance on tracer concentration in the solid phase given in Equation 3-10. The boundary and initial conditions given in Equation 4-3 are appropriate for this case. t=0 C,=O r=R .=CL (4-3) 6C, r=0 — =0 6r A solution to Equation 3-10 when C, is constant with time was developed by Carslaw and .laeger". By applying Duhamel’s theorem to this solution for C, as a function of time and radial position in the bead, an expression can be obtained for CB as a function of time”: 62 sin("_’".) -D.n’x’(t-l) 0'0“) =ZD‘ZL‘ ”1).“? +136}! R: d}. (4-4) where A is a dummy variable of integration. The concentration in the beads, C3, must be averaged over the bead volume as shown in Equation 4-5 _ 3 3 2 c,,(:).R3fo c,(r,:)r dr (4.5) Combining Equation 4-4 and 4-5 and changing the order of integration and summation yields Equation 4-6”. 60 -D,N'x1(:-1) ( t 2 4—6) C’=?2£ 2 d j; CLC R d1 Taking the derivative of C,, with respect to time, and integrating by parts gives Equation 44". (+7) -D}t3tz(t-A) glaz- ‘__£¢ T41 61 R2 "'1 o 61 Combining Equations 4-2 and 4-7, taking the Laplace transform and rearranging yields Equation 4-8 — -1 fl his». Vl-el’u -°'-'°5)6D¢£:_1 __3 (4-8) C: ”A UAe 2R2 63 where C, and C, are the Laplace transforms of C, and C,, respectively. 4.2 Simulation For each simulation, the input concentration to tank one was a pulse described by a gamma density function" shown in Equation 4- 9. The gamma density function was chosen due to its resemblance to a real pulse. ‘=_£_ta-le 4t F(a) (49) F(a) =fo'u "‘e “"du The values of h and a were arbitrarily chosen as 5 and 15, respectively. The output from tank one was then calculated from Equation 4-8 using the computer program developed by Wakao and Kaguei“. The output of the first tank, C,, then became the input to the second tank and so on through each of the n tanks in series. The output of the tanks corresponding to axial positions of 30 and 90 cm were saved and used to calculate dispersion coefficients for each simulation using Equation 3-8 and Equations 3-9 and 3-10. A computer program capable of predicting the effects of both axial dispersion and solid phase diffusion was developed to calculate the correction factor as a function of the system variables and is listed in Appendix B. Table 4-1 shows the parameter values used to simulate the dispersion experiments. Table 4-1 Values Used in Parametric Studies Column Length, L, 1 m Column Diameter, (1, 5 cm Superficial Liquid Velocity, U 3.0-10.0 cm/s Bead Radius, R 0.25-4.0 cm Bead Porosity, e, 0.05-0.98 Li uid Fraction, 6, 0.3- .99 Di fusion Coefficient, D, 0.1 x 10‘9-5.0 x 10'9 mzls Number of Stirred Tanks, N 10-1000 65 4.3 Dimensional Analysis Dimensional analysis is frequently applied in engineering analysis because it minimizes the number of independent variables needed to describe the system and gives insight into the physical processes that control the systems’s behavior. The system variables included in the governing material balance equations were the dispersion coefficient, diffusion coefficient, liquid velocity, gas volume fraction, liquid volume fraction, bead void fraction, and. bead radius. The Buckingham Pi method” was used to determine the important dimensionless groups of these parameters. All possible dimensionless permutations of these variables were examined, and the one that best correlated the computed correction factor results is 2 _ _ 2 ¢_D‘Dme,(l e, eg— (440) 11le2 Another important dimensionless group is the Peclet number defined in Equation 4-11 Pe= (4-11) where LC (cm) is the column length. Chapter V: Results 5.1 Solenoid The magnetic field strength was measured as a function of current at the center of the column with a Gaussmeter. The results are shown in Figure 5-1 along with the theoretical values predicted by Equations 3-1 to 3-3. The experimental values agree quite well with the theoretical prediction. Figure 5-2 shows the predicted and experimental axial variation of the magnetic field at the radial center of the solenoid. The magnetic field was almost constant over the central 80% of the solenoid length and dropped off rapidly to one half the maximum field strength at the ends of the solenoid. Measurements of the magnetic field were also conducted while the magnetite impregnated beads were in the column, and there was no measurable difference in the field strength from the empty column case. Solenoid heat generation was investigated by measuring the temperature of the cooling water as it entered and exited the solenoid. For currents above 1.5 amps there was a noticeable rise in the cooling water temperature. A water flow rate of 650 mL/s at 10°C was adequate to keep the outside surface of the solenoid at room temperature for the highest current (5 amps) used in the study. 5.2 Bed Regimes The bed structure was observed at two positions in the column: at the center of the column, where the field strength was maximum (up to 300 gauss) and at the top of the solenoid, where the field strength was one half of maximum. The field of view at the center was limited to a square centimeter allowed by the periscope. Although observations at the top of the solenoid were limited to field 66 3007 N O ‘3 100‘ Field ‘Strength (Gauss) Figure 5-1 — Theoretical O Experimental Magnetic Field Strength versus Current: Experimental and Theoretical Values ‘ Current (amps) 68 Field Strength (Gauss) 8 l - O Experimental — Theoretical O I j I l I I I I I I I I I I I l 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 Dimensionless position Figure 5- 2 Axial Variation of Magnetic Field: Experimental and Theoretical Values 69 strengths of 150 gauss or less, more accurate observations were possible because of the wider field of view. The bed regimes observed over the range of 0-300 gauss and gas velocities from 0.6 to 2.0 cm/s are shown in Figures 5-3 to 5-5 for liquid velocities of 5.15, 6.44, and 7.71 cm/s. The experiments were performed by setting the liquid and gas flow rates and then increasing the magnetic field strength from 0 to 300 gauss. When the experiments were repeated while decreasing the magnetic field strength, no hysteresis was observed. Six basic bed regimes were observed: random, chain, chain- channel, destabilized, channel, and frozen. The random regime, observed at the lowest field strength, was characterized by solid particles in completely random motion. Bubble size ranged from 1 mm diameter to small slugs up to 2 cm in diameter. Random bed behavior was typical of a non-magnetized bed. In the chain regime, the solid particles formed chains 3-4 particles in length as the magnetic field strength induced significant magnetic dipoles in the particles. In the chain-channel regime, small chains coalesced into a three-dimensional mesh structure that contained voids or channels. These structures were not permanent, but shifted periodically. In the destabilized regime, channels were too narrow to accomodate bubble slugs. As a result these slugs often broke up the three dimensional meshes into small clumps of 10-20 particles. Within each clump, particles were arranged in a rosette structure, where the particle in the center was surrounded by six particles. In the channel regime, the bubble slugs could no longer break up the mesh, and the channels described previously remained fixed. Bubbles rose through 70 Liquid Velocity = 5.15 cm/s 2.0 =‘ R=Rondom . R [ C C 0 CN F C C=Choin A 1'8.) CC=Choin—Chonnel 8 1 D=Destabilized C3) 1'6‘ CN=Channel 8 ‘ F=Frozen .C 1.4“ ’5. c g 1.2~ m 0 '2 1.0- .9 - J LL 0.8= 0.6 I# I I I I I r Ifi jj I I I I I I T I I 0 50 ‘l 00 1 50 200 250 .300 Gas Velocity (cm/s) Figure 5-3 Bed Regimes For a Liquid Velocity of 5.15 cm/s 71 R=Rondom C=Choin CC=Choin-Chonnel D=Destabilized CN=ChanneI F=Frozen Gas Velocity (cm/s) I I I I I T T 0 so 100 150‘ ' ' '260' 250 .300 Field Strength (Gauss) Figure 5-4 Bed Regimes for a Liquid Velocity of‘6.44 cm/s 72 Liquid Velocity = 7.71 cm/s 2.0 <‘ R=Rondom . R A C C 0 ON C C=Choin 1'8" CC=Choin—Channel ’0? D=Destabilized E 1'6- CN=Channel U ‘ F=Frozen V 1.4- >. :‘L'.’ . o 2 1.2- (D > 1 4 8' 1.0- O 0.8- 0.5- ....H.......,, p..- 0 50 100 1 50 200 250 300 Field Strength (Gauss) Figure 5-5 Bed Regimes for a Liquid Velocity of7.71 cm/s ' 73 these channels, but the slugs observed in the previous structures were absent. Significant numbers of bubbles were also observed to travel next to the column wall. The only solid motion observed was particle vibration. In the frozen regime, even the particle vibrations stopped. The entire bed behaved as a single mass that was fluidized by the liquid and gas in the column. The observed bubble sizes did not change in the chain, chain- channel and destabilized regimes from those observed in the random regime. In the channel and frozen regimes, no bubble slugs were observed, indicating that they may have been broken up by the bed structure. In addition, at a liquid velocity of 5.15 cm/s and gas velocities of 0.6 and 1.1 cm/s, small bubbles (1-2 mm in diameter) became entrapped in the three-dimensional mesh structure of the chain-channel regime and were held in place for several seconds. This behavior was observed to a lesser extent for a liquid velocity of 6.44 cm/s and a gas velocity of 0.6 cm/s and not at all at the highest liquid velocity. 5.3 Gas Void Fraction Figure 5-6 shows a sample of the optical probe’s output. Each peak represents a bubble contacting the probe tip. Calibration of the probe was described in Section 3.3.2. The percentage of peak height as defined in Figure 3-6 was found to be 10%. The effects of magnetic field strength and gas velocity on gas void fraction are shown in Figures 5-7 to 5-9 for liquid velocities of 5.15, 6.44, and 7.71 cm/s, respectively. These same data are shown with their 95% confidence levels in Figures 5-10 to 5-12. Figure 5-7 shows that for a liquid velocity of 5.15 cm/s, gas void fraction 74 2.5a 1.51 l 1.04 Voltage (Volts) 0.5- 1 0 l O 20 30 40 50 Time (seconds) Figure 5-6 Experimental Data from Optical Probe 75 Liquid Velocity = 5.15 cm/s 0.3 G-(-) Gas Velocity = 0.6 cm/s B-EJ Gas Velocity = 1.1 cm/s a A—A Cos Velocity = 2.0 cm/s J c: O 2.3 0.2- U C L. L1. .9 ‘ l O > Z (I) 0.1% . A o I o I: - a e - 0,0 ttfiilvvitliriIlttttlTvrrjleT— O 50 100 i 50 200 250 300 Field Strength (Gauss) - Figure 5- 7 Gas Void Fraction Data as a Function of Magnetic Field Strength for a Liquid Velocity of 5.15 cm/s and Gas Velocities of0.6, 1.1, and 2. 0 cm/s 76 Liquid Velocity = 6.44 cm/s 0.3 o—e Gas Velocity = 0.6 cm/s Ei-El Gas Velocity = 1.1 cm/s - A—A Gas Velocity = 2.0 cm/s Gas Void Fraction 000 I r V T l I F T I l T V I I I I I I I I I l U I I I I I T I 0 50 l 00 1 50 200 250 300 Field Strength (Gauss) ' Figure 5- 8 Gas Void Fractions as a Function of Magnetic Field Strength for a Liquid Velocity of 6. 44 cm/s and Gas Velocities of0.6, 1.1, and 2. 0 cm/s Gas Void Fraction 77 Liquid Velocity = 7.71 cm/s 0.30 9-0 Gas Velocity = 0.6 cm/s B—E] Gas Velocity = 1.1 cm/s « A—A Gas Velocity = 2.0 cm/s 0.20- I 0.101 Eu 8 (J o o‘:\ /g V 0-00 '*"‘1""I'T"I's" "'ii'HTfi 0 50 100 1 50 200 250 300 field Strength (Gauss) ' Figure 5-9 Gas Void Fractions as a Function of Magnetic Field Strength for a Liquid Velocity of 7.71 chs and Gas Velocities of 0.6, 1.1, and 2.0 cm/s 78 O 3 Liquid Velocity = 5.15 cm/s G—O Gas Velocity = 0.6 cm/s G-El Gas Velocity = 1.1 cm/s . A—A Gas Velocity = 2.0 cm/s Gas Void Fraction rr' MW, 0 so 100 150 200 250 3150 Field Strength (Gauss) . Figure 5-10 95% Confidence Intervals for Gas Void Fraction Data in ' Figure 5-7 - 79 Liquid Velocity = 6.44 cm/s 0.3 9-6 003 Velocity = 0.6 cm/s ‘ [38 Gas Velocity = 1.1 cm/s ~ A—A Gos Velocity = 2.0 cm/s c: O '43 O.2~ _._.. U _— O C: 3\ .12 i — :1 O > —— _— cn o 115——\l _— C —— L.) 0 —— —— —— — __ _LN—fld—JE _=.__ j— I‘\ ‘1'- ._<:_ L __ #L. 0.0 IIIllttvvlttttltvrilIVIllit1 O 50 100 150 200 250 Field Strength (Gouss) - Figure 5-11 95% Confidence Intervals for Gas Void Fraction Data in ° Figure 5-8 ' ' 80 Liquid Velocity = 7.71 cm/s 0.30 G—O Gos Velocity = 0.6 cm/s B-E] Gos Velocity = 1.1 cm/s « A—A Gos Velocity = 2.0 cm/s C O '43 0.20— O O L LL. :9 .4 O _ o __ 4— L. #5- r ‘- l l. 0.00 j I V I l I I I V I r T r I l 1 I T I l I I 1 I I T Y Y I I 0 50 1 00 150 200 250 .300 Field Strength (Gouss)' Figure 5-12 95% Confidence Intervals for Gas Void Fraction Data in ~ Figure 5-9 - ' 81 increases with field strength for the two higher gas velocities, but decreases slightly for the lowest gas velocity. The gas void fraction for 1.1 cm/s is higher than the void fraction for 2.0 cm/s at the highest field strength of 300 gauss. Both the decrease in void fraction at the lowest gas velocity and the increase in void fraction at the higher two gas velocities occur in the frozen regime. The 95% confidence intervals indicate much greater variation in the data at high field strengths. Figure 5-8 shows similar trends for a liquid flow rate of 6.44 cm/s. The gas void fraction increases for gas velocities of 1.1 and 2.0 cm/s and decreases for a gas velocity of 0.6 cm/s. Also, the scatter in the data increases with magnetic field strength. The gas void fraction is higher for a gas velocity of 1.1 cm/s at a field strength of 200 gauss than for a gas velocity of 2.0 cm/s. However, for this liquid velocity, the increase in void fraction for the two higher gas velocities occurs in the channel regime rather than the frozen regime. The decrease in gas void fraction at the lowest gas velocity does occur in the frozen regime though. Figure 5-9 shows that for a gas velocity of 2.0 cm/s there is a slight increase in void fraction with magnetic field strength but no effect of magnetic field strength for gas velocities of 1.1 and 0.6 cm/s. In this case, the confidence intervals are unaffected by field strength. A comparison of the gas void fraction measured with the valve technique (average measurement) to the void fraction measured with the optical probe (local measurement) is shown in Figure 5-13 for a magnetic field strength of 0 and 275 gauss. There is a strong 82 A 0.20- G) 3 I [/51 — O gauss .g l/ \\ --- 275 gauss _: [,1’ \ 0 Gas velocity = 0.6 cm/s 8 0'16: [/I \ Cl Cos velocity = 1.1 cm/s l— g/’ ‘\ A Gas Velocity = 2.0 cm/s Q) 3 . o 0.12- > . V C d O 2.3 0.081 U . U h L'— < E 0.04- O . > 8 . O 0.00 . I ' I ' fi ' I . fl 0.01 0.02 0.03 0.04 . 0.05 0.06 Cos Void Fraction (Optical Probe) Figure 5-13 Comparison of the Valve Technique and the Optical Probe Technique for Field Strengths Of 0 and 275 Gauss 83 correlation between the two methods for zero magnetic field, while there is essentially no correlation at the high field strength. Since the high field strength corresponds to the frozen regime, this finding suggests that local void fraction measurements do not accurately reflect the average void fraction when there are preferential channels for bubble migration. 5.4 Liquid Dispersion 5.4.1 Conductivity Probe Performance Care was taken to make the two conductivity probes as identical as possible, and the initial calibration curves (probe output voltage versus tracer concentration) for the probes were found to be identical. In addition, the calibration curves were linear over the range from O to 10 g/L with a correlation coefficient of 0.999. Figure 5-14 shows the output from both probes during a typical tracer run. After several days of operation, the probe surfaces corroded significantly, and a loss of signal was noticed. The probes were regenerated by grinding off the corrosion. However, this procedure affected the probe response, so it was necessary to recalibrate the probes after regeneration. 5.4.2 Diffusion Coefficients Diffusion coefficients were determined as described in Section 3.4.3. Equation 3-14 was fit to the experimentally measured diffusion data by both a one and two parameter fit. In the one parameter fit, the diffusion coefficient was varied until the mean square error between the model and the experimental data was minimized, and in the two parameter fit, both the diffusion coefficient and the parameter a were optimized simultaneously. As 0.25— 0.20~ 0.15- 0.10- Voltage (Volts) 0.05- 0.00 Figure 5-14 — Probe 1 ----- Probe 2 I I I . 2 4 6 8 10 12 14 16 18 Time (seconds) Experimental Data from Conductivity Probes in a Tracer Experiment - ' 85 shown in Figure 5-15, the two parameter approach gave a much better fit. As described in Chapter 6, two parameters are needed to account for the volume of the beads obstructed by the magnetite. One assumption made in developing the model used to calculate diffusion coefficients was that film mass transfer resistance was negligible. An experiment was conducted to determine the conditions under which this assumption was valid. Figure 5-16 shows the effect of agitation rate on the apparent diffusion coefficient. The measured coefficient increased with agitation rate up to about 100 rpm. Table 5-1 shows the values of the diffusion coefficient and the parameter a that best fit Equation 3-14 to the experimental data for alginate beads containing 0, 5, and 50% by weight magnetite. The effective diffusion coefficient decreased with increasing magnetite concentration, while the parameter or increased with increasing magnetite concentration. 5.4.3 Pore Volume Distributions Figure 5-17 shows the void volume distributions of alginate beads containing 0, 5, and 50% by weight magnetite. Table 5-2 lists the empirical constants used to fit Equations 3-22 and 3-23 to these data. The asymptotic values of the curves, which correspond to the total void volume of the solid, decrease as the percentage of magnetite increases. From these data, the bead porosity may be determined as a function of the tracer diameter by finding the difference between the asymptotic value of the curve and the excluded void volume for the particular molecular diameter. Figure 5-18a shows a plot of the bead porosity as a function of the diameter of the diffusing molecule. The O, 5, and 50% magnetite beads have 86 12.0- 0 Experimental Data — Two Parameter Fit l --- One Parameter Fit 11.0- 10.0- Concentration (g/L) 9.0- 8-0 I I I I I I I I I I I I . 0.0 10.0 20.0 30.0 40.0 50.0 60.0 Time (minutes) Figure 5-15 Experimental Diffusion Data with One and Two Parameter Fits 87 6.0- e/‘0 T 5.0— O X ”a? 4.J \ o N E 3 Q) O 3.0“ e 2.0litvlfit11j',j,l"j‘l"" 'jr‘. 0 50 100 150 200 250 300 rpm Figure 5-16 Effect of Agitation Rate an Measured Diffusion ' Coefficient - 88 Table 5-1 Values of D, and a from Curve Fits of Experimental Data % Magnetite Diffusion Coefficient (cmz/s) oz 0 5.40 X 10" 2.23 5 5.41x10" 2.26 50 4.30 x 10“ 2.55 Table 5-2 Curve Fits for Pore Size Distributions % Magnetite A0 A1 A2 A3 A4 M.S.E. 0 0.94 3.33 1.82 0.015 0.03 0.000588 5 0.82 3.59 1.71 0.030 0.02 0.000216 50 0.50 3.56 1.52 0.059 0.015 0.000175 89 1.01 A O 13 o E 08 O ' s ' D a) 1 '1 E - , 3 0.6 . T) A 5 04‘ 0. 1‘ A 8 .= .0 O 2- l‘ 1 ‘ ‘3 '02? A 50% Magnetite X Lq/. ‘ Cl 5% Magnetite LU ’ O 0% Magnetite - 000 ' I I I V l I I U I I . l 0.0 100.0 200.0 300.0 400.0 500.0 600.0 Diameter (Angstroms) Figure 5-17 Pore Size Distributions and Curve Fits for 0, 5, and 50% by Weight Magnetite Alginate Beads a.) Bead Porosity be) Percent of Accessible Pore Volume (73) Figure 5-18 90 1.01 . 0-0 07.‘ Magnetite i ‘ B-E] 5% Magnetite " M 507; Magnetite 0.8" . V l V I Y I I V - o 100 200 300 400 550 600 Diameter (Angstroms) 100 G-O 0% Magnetite G-El 5% Magnetite A-A 50% Magnetite 80- . 601 .40- 20* 0 a ' l r I r l ' I 0 100 200 300 400 500 600 Diameter (Angstroms) a). Bead Porosities for 0, 5, and 50% by Weight Magnetite Alginate Beads versus Molecular Diameter b). Percentage of Tot-al'Accessible Bead Volume Available for Diffusion versus Molecular Diameter 91 approximately 93%, 83%, and 50%, respectively of their volume accessible to molecules with diameters of 10 A (the approximate diameter of the calcium chloride tracer in solution as described in Chapter 6). Figure 5-18b shows a plot of the percentage of total accessible bead volume as a function of molecular diameter of the diffusing molecule. The percentage of total accessible bead volume is calculated by dividing the bead porosity by the asymptotic value of the curve. 5.4.4 Dispersion Coefficients Predicted tracer profiles were calculated as described in Section 3.4.2 by fitting both the dispersion coefficient and the interstitial velocity to the experimental data. The predicted and experimental tracer concentration curves are compared in Figure 5- 19 for a typical run. Figure 5-20 compares the Peclet numbers (UiR/D“) calculated both neglecting intraparticle diffusion (Equation 3-8) and including it (Equations 3-9 and 3-10). Peclet numbers calculated including the effect of diffusion are higher (thus dispersion coefficients are lower) than when diffusion is ignored, indicating that diffusion contributes to spreading of the tracer curves. Figures 5-21 to 5-23 show the effects of magnetic field strength and gas velocity on the measured Peclet number for liquid velocities of 5.15, 6.44, and 7.71 cm/s, respectively. For the lowest liquid velocity (Figure 5-21), there was a sharp increase in Peclet number at 100 gauss for a gas velocity of 0.6 cm/s, and smaller increases at 150 gauss for a gas velocity of 1.1 cm/s and at 75 gauss for a gas velocity of 2.0 cm/s. The Peclet number curves peaked at the boundary of the chain-channel and destabilized regimes for gas 92 0.16-1 . — Experimental Data 014— -- Theoretical Fit 0.12“ 0.10— 008'- Voltage (Volts) 0.06- ] 1' I I l I l U I 1 0 4 8 12 16 20 Time (Seconds) Figure 5-19 Theoretical Fit of Experimental Tracer Data 0.08- 0.06- Peclet Number 0.04-- 93 A—A Liquid Velocity = 7.71 cm/s - B-El Liquid Velocity = 6. 44 cm/s Model w/o diffusion - - - Model w/ diffusion 0.02 I l 1 fl 1 ' 0.4- 0.5 0.8 1.0 1.2 Gas Velocity (cm/s) ' l 1.8 2.0 2.2 Figure 5- 20 Comparison of Peclet Numbers Calculated From Dispersion Models with and without the Effect of Tracer Diffusion Included 94 0 8 Liquid Velocity = 5.15 cm/s 9-0 0.6 cm/s " ‘ [El-El 1.1 cm/s A—A 2.0 cm/s Gas Velocity 0.6— 63 “D n E c 3 Z 0.4e E a a Q a 0.2" 7 Q A A S 2 *‘ - 0.0 I ' ' j T ' ' I I l I V I I U I I l I I I U I I I I U 0 50 1 00 1 50 200 250 300 Field Strength (Gauss) Figure 5- 21 Peclet Numbers as a Function of Magnetic Field Strength for a Lilquid Velocity of 5.15 cm/s and Gas Velocities of 0.6,1. ,and 2. O cm/s 95 Liquid Velocity = 6.44 cm/s . 0-0 0.6 cm/s 070q 9'9 1.1 CM/S ‘ A—A 2.0 cm/s Gas Velocity Peclet Number 0 2‘3 1 0000 I ' ' I U ' I I I I I I I I I I I I T I I I I I I I I I I I o 0 50 1 00 1 50 200 250 300 Field Strength (Gauss) Figure 5- 22 Peclet Numbers as a Function of Magnetic Field Strength for a Liquid Velocity of 6. 44 cm/s and Gas Velocities of O. 6, 1.1, and2. 0cm/s 96 Liquid Velocity = 7.71 cm/s 0.80 . G-O 0.6 cm/s 070_ W 1.1 cm/s . A—A 2.0 cm/s Gas Velocity 0.60- L c: 8 0.50- E 3 '1 Z 0.40— E q 0 0.30- G) 0- . 0.00 l I I1 I 1 I I I I I I T I T I l I I I. I l I I I I I I I I I A O 50 1 00 1 50 200 250 300 Field Strength (Gauss) Figure 5- 23 Peclet Numbers as a Function of Magnetic Field Strength for a Liquid Velocity of 7. 71 cm/s and Gas Velocities of 0. 6, 1.1, and2. 0cm/s 97 velocities of 0.6 and 2.0 cmls, and in the channel regime for a gas velocity of 1.1 cm/s. Figures 5-22 and 5—23 show that neither magnetic field strength nor gas velocity significantly affected Peclet number for the two higher liquid velocities- Figures 5-24 to 5-26, which show the 95% confidence intervals for the data shown in Figures 5-21 to 5-23 respectively, indicate that reproducibility was least for field strengths between 50 and 150 gauss, where the decrease in mixing was observed. Regardless, the increases in Peclet number for gas velocities of 0.6 and 1.1 cmls, in Figure 5-21 were shown to be statistically valid using a two level analysis of variance". Thus, magnetization can significantly reduce axial mixing in three-phase fluidized beds. 5.5 Gas to Liquid Mass Transfer The agreement between the predicted and experimental oxygen concentration profiles was excellent, as is shown in Figure 5-27. A one parameter fit (the overall mass transfer coefficient, kLa) of Equations 3-36 to 31-41 was used along with the dispersion coefficient data from Section 5-4. Figures 5-28 to 5-30 show the effect of magnetic field strength and gas velocity on the overall mass transfer coefficient for liquid velocities of 5.15, 6.44, and 7.71 cmls, respectively. Figure 5-28 shows that kLa peaked at approximately 75 gauss for all gas velocities, but the peak was most pronounced for the highest gas velocity. All of the peaks occurred in the chain- channel regime. As shown in Figure 5-29, there was no effect of magnetic field strength on kLa for the intermediate liquid velocity. However, Figure 5-30 shows that there was again an effect of field strength at the highest liquid velocity for gas velocities of 1.1 and 98 Liauid Velocity = 5.15 cm/s 1.2 ‘ 0-0 0.6 cm/s B-El 1.1 cm/s 1'0- A—A 2.0 cm/s Gas Velocity 0.8-4 0.6- Peclet Number 0 50 100 1 50 200 260 300 Field Strength (Gauss) ' Figure 5-24 95 96 Confidence Intervals for Peclet Number Data in ‘ Figure 5-21 - ' 99 Liquid Velocity = 6.44 cm/s 9-0 0.6 cm/s 0,7~ 133-El 1.1 cm/s - A—A 2.0 cm/s 0-5‘ Gas Velocity j Peclet Number I I I I I I I I I j .,...., W“... 100 150 200 250 300 Field Strength (Gouss)‘ Figure 5-25 95 96 Confidence Intervals for Peclet Number Data in ‘ Figure 5-22 ' 100 0 8 Liquid Velocity = 7.71 cm/s . G—O 0.6 cm/s 0.7_ G-El 1.1 cm/s . G-Q 2.0 cm/s Gas Velocity 0.6-4 0.5- Peclet Number O .1; l 0.2-i JR‘ :: 0.1=j% _gfe V _LE:_J.._9\ Jig —.— __ __ \o 0.0....,.e.-,-...,....,...._... 0 so 100 150 200 250 300 Field Strength (Gauss) - Figure 5- 26 95 96 Confidence Intervals for Peclet Number Data in Figure 5- 23 101 0 Experimental Data ‘ —- Theoretical Flt 0.6- 02 Concentration (mg/L) 0.211‘1'1'1'1'1'1'1'1'1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Axial Position (Dimensionless) Figure 5-27 Experimental and Theoretical Oxygen Profiles 102 Liquid Velocity = 5.15 cm/s 0.06 ‘ 005-: Z 5 0042' ,_ . J I 3 3L 0.035 O u g i > 0.022 ‘ . Gas Velocity 0'01? A-A 2.0 cm/s I [El-B 1.1 cm/s 0000 ‘ GI? IO.I6 lch/Is I I l I—I h l I I I I l I I I j l I I I I . 0 50 1 00 1 50 200 250 300 Field Strength (Gauss) Figure 5-28 Mass Transfer Coefficients as a function of Magnetic - Field Strength for a Liquid-Velocity 0f 5.15 cm/s and Gas Velocities of 0.6, 1.1, and 2.0 cm/s 103 Liquid Velocity = 6.44 cm/s 0.06 0.05- O.04-£5\A_____,5 at A A l ‘ 3 c“; 0.031;} B W .9 X Gas Velocity 00“ A—A 2.0 cm s ‘ [El-El 1.1 cm/s 0.00 9?.°'§,°'?/?..,....,...... .......s O 50 100 1 50 200 250 300 Field Strength (Gauss) Figure 5- 29 Mass Transfer Coefficients as a function of Magnetic Field Strength for a Liquid Velocity Of 6. 44 cm/s and Gas Velocities of 0.6, 1.1, and 2. 0 cm/s 104 Liquid Velocity = 7.71 cm/s 0.06 l 0.05— T 4 _J21 5 ~ 3 2 X t V ,3 Gas Velocity 0'01- A—A 2.0 cm s 1 i3~El 1.1 cm/s 0.00 9'1).°°§f'?/?..,....,...._...,....' 0 50 1 00 150 200 250 300 Field Strength (Gauss) Figure 5-30 Mass Transfer Coefficients as a function of Magnetic Field Strength for a Liquid Velocity of 7. 71 cm/s and Gas Velocities of0.6, 1.1, and 2. O cm/s 105 2.0 cm/s, although the effect was not as pronounced as those in Figure 5-28. The maximum kLa values occurred in the chain-channel and the destabilized regimes for this liquid velocity. Figures 5-31 to 5-33, which show the 95% confidence intervals for these data, indicate that the reproducibility of the oxygen mass transfer coefficients was much greater than for the gas void fraction and liquid dispersion measurements. 5.6 Dimensional Analysis of the Fluidized Bed Model Dimensional analysis was used to correlate the model results using dimensionless groups of the important variables of the system. These variables are L, the distance between measuring points; R, the particle radius; U, the superficial liquid velocity; D,, the diffusion coefficient; and D“, the dispersion coefficient. The particle porosity (e,), and the liquid void fraction (eL) were also included, but they are already dimensionless. Since it was not known what dimensionless groups would be important, all possible types of dimensionless groups were determined. The resulting dimensionless groups are listed in Equation 5-1 fiflfiflflL—Uflfl (5-1) I.Da D“ D. D“ D, a, It was hypothesized that all values of the parameters could be graphically correlated as a function of two independent dimensionless groups. One criterion for the dimensionless groups was that they contain all the above variables, with the exception of L, because the liquid dispersion coefficient should not be a function of the distance between the measuring points. The variable L was 106 Liquid Velocity = 5.15 cm/s 0.06 1 —i_ 005‘ #:\l ' _— 0.042 —— F‘ __ __ 3; 0031; —— _— o __ _ 2 \— 9’4): 00 _ — Gas Velocity 0'01 A—A 2.0 cm/s 1 B-El 1.1 cm/s 0.00 G‘OOFET/f..,....,....,......... 0 50 100 l 50 200 250 300 Field Strength (Gauss) Figure 5-31 95% Confidence for Mass Transfer Coefficients in Figure 5- 28 107 Liquid Velocity = 6.44 cm/s 0.06 0.05- 0'0415‘\\—1— A I _— /\ 41E; 3 I ____ ___. .9 .x .__. Gas Velocity 0'01 A-A 2.0 cm/s 1 B-EJ 1.1 cm/s 0.00 GI-Ie IO.I6 lch/Is I I I I I I I I I I Ifir l I I I I I I j I I i 0 50 1 00 1 50 200 250 300 Field Strength (Gauss) Figure 5-32 95 % Confidence for Mass Transfer Coefficients in 1 Figure 5-29 - ' 108 Liquid Velocity = 7.71 cm/s 0.06 0.05- A T 3} 0.03f J 2 _— x "L _’>_ Gas Velocity 0'01 A—A 2.0 cm s 1 B—El 1.1 cm/s 0.00 9'909ET45 ...-..,....,.fi.fi.... 0 50 100 1 50 200 250 300 Field Strength (Gauss) Figure 5-33 95% Confidence for Mass Transfer Coefficients in Figure 5- 30 109 still considered important in the analysis, since it was required to non-dimensionalize the variable R. Examination of the dimensionless groups in Equation 5-1 shows that at least four of these groups are needed to include all of the variables. The number of possible permutations of the dimensionless groups in Equation 5-1 is 35. Fifteen of the 35 possibilities resulted in L not cancelling out, and were eliminated from consideration. The correction factor (n), the ratio of the true axial dispersion coefficient to the apparent one, was calculated for each of the parametric studies described in Chapter 3 and plotted versus each of the remaining 20 possibilities. Within each possibility there were additional 16 combinations since the reciprocal of each dimensionless group in Equation 5-1 could also be used. Two dimensionless groups ( and Pe) were found that could correlate all the parametric studies. Using the Peclet number based on the column length a family of curves was obtained. The definitions of 11> and the Peclet number are given in Equation 5-2. 2 “_D‘Dmepfl-epz Pa: 11. (54) 112112 D“ The correction factor 11 is plotted versus the dimensionless group 11> in Figure 5-34. The usefulness of this plot was limited since the Peclet number contains the true dispersion coefficient, which cannot be obtained without solving the mathematical model that includes the effects of intraparticle diffusion (Equations 3-9 and 3- 10). By contrast, the apparent dispersion coefficient can be simply calculated using Equation 3-8. Although an iterative method could be used to obtain an estimate of the true disperison coefficient from 110 1.011 00¢. btélA-fls‘bo" v A. e " 2 0.00“ 0‘3A AO~.~“‘§~WV~ °%A‘ “5.1‘ wt 0.8- s. «”30” 9 'a .3. q. 3. . " 06- o ‘1 '. ' i , IL? a?“ o I q as _I EG. 0004 e A 5' 0.4.. 02000 00 a a 1500 0" ' 1 A 1000 ‘ ' - 500 ' _ II200 A 0.2 ‘ 100 g 0'50 v 20 0.0-1 ”WWW-Hum 10‘" 10‘" 10" 10“ 10‘7 10“ 10” 10“ 10" 10" o . ”Pum‘éu'fi)? Figure 5-34 Correction Factor, 11, versus 42 as a function of Pe,,“ 111 the apparent value, this method is time consuming. Thus, for simplicity, the results were cast in terms of the measurable variables, Pe,” and , which are defined in Equation 5-3. 2 1» _D‘Dmepfl-elf PC = (53) Unfortunately, the families of curves shown in Figure 5-34 do not map directly into these new coordinates (Pe,,p, ). To correlate the results, each value of '1 was tabulated as a function of I and Pe,”, thus generating a three-dimensional surface, n(, Pe,"). This surface was then curve fit using the empirical expression shown in Equation 5-4. n =ao+alx+azz+03x11arz2+asn+asfi+a¢+afz+agx2 mica/i Fe m 1000 (5-4) x=long z= Table 5-3 shows the empirical constants obtained in the fit. The form of this expression was varied by trial and error until the average deviation of the empirical fit from the data points was less than 5%. For the equation given in Equation 5-4 the average deviation was 1.5%. Isocurves showing I; as a function of 4’ for constant values of Fe were then calculated from Equation 5-3 for ‘99 different values of Pe,”. Figure 5-35 shows the family of isocurves. Figure 5-36 shows the effect of the parameters on the correction factor. Particle radius appears to have the strongest affect, and the diffusion coefficient also has a strong effect. 112 Table 5-3 Curve Fit of 11 versus d> Data Constant Value Al .09429 A2 '.6706 A3 3.327 A4 ". 1337 A, .0493 A6 1.026 A7 -9.962 A, -.00789 A9 .0717 A10 '.1760 A -2.688 113 1.0 0.811 ”an” 0.611 .1; . .. I . 'r-1 0.4-1 50. 0.2- . V 20001500 500 100 1000 200 00" . mmmmmmm 10"0 10" 10" 10" 10" 10" 10“ 10.0 10“ o . D‘DW‘:("‘8)2 R102 Figure 5-35 Correction Factor, 1;, versus 4:,” as a Function of Pe,” 114 A o 4 O . D Q a O. - 9 9 9.0 4 '0 0.8- ‘v a s ‘ 2 “L 0.7- o .I 9’ 1 v 0.6- O. Diffusion Coefficient O 5_ A Solid Fraction ' D Bead Porosity v a V Radius 0 4-. O Velocut . mm 1 0" 10" 10" 1 0" 1 0" 1 0" 1 0" 1 O" o .. ym‘fia’fi’z 12203 Figure 5-36 Effect of System Parameters on the Correction Factor, :1 Chapter VI: Discussion 6.1 Bed Regimes The unstabilized and frozen regimes observed in this study are comparable to the random and frozen regimes described by Rosensweig29'”; however, there was no regime in this study similar to the stabilized regime described by Rosensweig. His stabilized regime was characterized by both the absence of solids movement and an expanded fluidized bed. The chain regime observed in this study approached the stabilized state, but still had significant solids motion. Siegell” described an additional regime in his studies of a liquid-solid MSFB, the roll-cell regime, which was characterized by gulf streaming motion. None of the regimes found in this study resembled the roll-cell regime. Hu and Wu” reported three bed regimes in a three phase MSFB: the particulate regime that exhibits behavior of an unmagnetized fluidized bed, the chain-fluidized bed where chains of particles form, and the magnetically aggregated bed where the particles aggregate into a single unit. The counterparts of these regimes found in this study are the random, chain, and frozen regimes, respectively. However, Hu and Wu make no mention of other bed regimes or the role that the gas phase plays in the formation of bed regimes. The results of this study indicate that the gas phase stongly influences the bed behavior. As will be discussed in later sections, observed changes in key bed properties as a function of magnetic field strength can often be correlated with transitions between bed regimes. 115 116 The observable behavioral differences in the bed regimes of the MSFB suggest that it may be possible to vary the properties of the MSFB markedly by simply changing the magnetic field strength. This would be valuable for processes with rapidly changing conditions, and for systems where many different processes are run in the same reactor. 6.2 Gas Void Fraction For a liquid velocity of 5.15 cmls, the onset of the frozen regime caused a substantial increase in local gas void fraction for gas velocities of 1.1 and 2.0 cm/s, but a decrease for a gas velocity of 0.6 cm/s. There was a wide variation among the five replicate measurements for the two higher gas velocities, with replicates varying from as high as 0.33 to as low at 0.001. This trend is consistent with the observation that the bubbles channel through preferred paths of reduced resistance within the frozen bed. When one of these paths intersected the probe tip, an artificially high void fraction was observed, whereas when the probe was bypassed, an artificially low value of void fraction was obtained. In the case of the lowest gas velocity, the frozen-bed void fraction was uniformly low in value. The low gas flow rate was probably inadequate to keep the channels open against the force of the applied magnetic field. This explanation agrees with the observation that bubbles appeared to flow around the bed under these conditions. The data taken at a liquid velocity of 6.44 cm/s showed similar trends, except that the large increases in void fraction occurred in the channel regime rather than the frozen regime. This observation suggests that there is little difference between the physical behavior 117 of the channel and frozen regimes. The only observable difference between the tWo regimes was that the solid particles in the channel regime vibrated, while those in the frozen regime did not. Again, the frozen-bed void fraction was very low for the lowest gas velocity. For a liquid velocity of 7.71 cmls, the modest increase in gas holdup observed at the highest gas velocity was again associated with the channel regime. However, in this case there was apparently no effect of magnetic field strength for gas velocities of 0.6 and 1.1 cm/s. It is not clear why the void fraction data for the highest liquid flow rate were so much different than those for the lower flow rates. Qualitative observations of the MSFB revealed that the bubble size decreased as the magnetic field strength increased. Kwauk et al.” found that the bubble size in a three phase MSFB decreased exponentially with increasing field strength. As smaller bubbles exhibit lower rise velocities, this trend suggests that gas void fraction should increase with field strength. Although two and three fold increases in the local gas void fraction were observed in this study with application of the magnetic field, these results are questionable due to the point measurement technique used. Artificially high values were obtained when the probe happened to be in a bubble channel and low values were obtained otherwise. Moreover, the probe would not necessarily detect an increase in gas void fraction caused by entrapment of bubbles in the bed structure. Thus, point measurements of void fraction may not accurately reflect the radially averaged gas void fraction in the frozen and channel regimes. The comparison of values from the optical probe (point measurement) with measurements using the valve technique (average 118 measurement) shown in Figure 5-13 support this conclusion. There was a strong correlation between the valve technique and the optical probe for zero magnetic field, but little correlation at a field of 275 gauss in the frozen regime. 6.3 Liquid Dispersion There were sharp decreases in liquid mixing for the lowest liquid velocity: a four fold increase in Peclet number at a gas velocity of 0.6 cmls, and a two fold increase in Peclet number at a gas velocity of 1.1 cm/s. The scatter among replicates was largest when mixing was the least, suggesting that the phenomenon responsible for the reduction in mixing may be transient or unstable under the conditions used in this study. Neverthless, a two-level analysis of variance" of the data showed that the increases were statistically significant. It is not clear why the decrease in liquid mixing occurred in the destabilized regime for a gas velocity of 0.6 cm/s and in the channel regime for a gas velocity of 1.1 cm/s. It was surprising that the magnetic field apparently had no effect on the liquid mixing for higher liquid velocities of 6.44 and 7.71 cm/s. It is possible that the greater interparticle spacing at these velocities may not facilitate the interparticle phenomena that reduce liquid mixing. Table 6-1 compares literature Pe values (RU/D,,) to the values measured in this study. Davison" used alginate particles with a density of 1.05 to 1.15 g/mL. Kim and Kim” and Muroyama” used glass beads with a density of 2.5 g/mL. The density of the beads used in this study was 1.7 g/mL. Since it has been shown that the liquid mixing in low density particle systems is higher than that of 119 Table 6-1 Comparison of Measured and Literature Pe Numbers System Bead Pe Author Studied Re Density (g/mL) Range Davison96 Three-phase l 1.05-1.15 .002-.08 gelbeads Kim and Kim97 Three-phase 100-200 2.5 .09 glass beads Muroyama98 Three-phase 100-200 2.5 .05-.11 glass beads Present Study Three-phase 100-200 1.7 .03-. 19 magnetized gelbeads with Fe304 120 dense particle systems", it would be expected that the Peclet numbers determined in this study would be between the values measured for the low density particle system and the dense particle systems. However this was not the case; the Peclet numbers from this study are in the same range as those of the glass bead study, and in some cases are larger than those of the glass bead studies. These results reinforce the finding that application of a magnetic field can reduce the axial mixing in a three-phase fluidized bed system. It has been shown that plug flow behavior gives higher volumetric productivity for reactions exhibiting positive order kinetics or product inhibition”. These types of kinetics occur commonly in biological systems, making the MSFB a good candidate for a bioreactor system. For systems which exhibit negative order kinetics, such as substrate inhibition, the bed could be used without the magnetic field or with a field applied to only a portion of the bed, to allow more liquid mixing. 6.3.1 Diffusion Coefficients The results from the diffusion data clearly indicate that a two parameter fit to Equation 3-14 was necessary to obtain a good fit. The parameter a is defined by Equation 3-15 as the ratio of the liquid volume external to the beads to the bead volume, and it was assumed in the derivation of Equation 3-14 that the total bead volume was accessible to the diffusing molecules. However, this assumption is of questionable validity, because the metal powder occupies, and possibly blocks access to, a portion of the bead volume. The a values determined from the diffusion studies, 2.23, 2.26, and 2.55 for alginate beads containing 0, 5, and 50% by weight magnetite, 121 respectively, were significantly greater than the value of 2.0 obtained assuming the entire internal volume of the beads was accessible. These results suggest that the bead volume accessible to the diffusing molecule is less than the total bead volume. The appropriate definition of a then, should be the ratio of the liquid volume external to the beads to the accessible volume inside the beads. 6.3.2 Pore Volume Distributions Figure 5-18a shows the porosities of the alginate beads, that is, the amount of accessible bead volume per unit total bead volume, as a function of pore size. These data were used to calculate a values for the calcium chloride tracer of 2.15, 2.35, and 3.77 for alginate beads containing 0, 5, and 50% by weight magnetite, respectively. The values for the 0 and 5% by weight magnetite agree fairly well with the values obtained from diffusion data, but the value for the 50% by weight magnetite differed by 48%. One possible explanation for this discrepancy is that the solute exclusion technique used sugars or polysaccharides for the diffusing molecules, whereas the other unsteady-state diffusion experiments used an ionic compound (calcium chloride). These two classes of tracer molecules could interact differently with the alginate matrix or the magnetite in the beads. The fact that the a values agree best for the lowest magnetite concentrations suggests that calcium chloride may adsorb to the magnetite. Adsorption would increase the amount of calcium chloride taken up by the beads, causing an overprediction of the accessible pore volume, and, therefore, a smaller value of a. 122 From Figure 5-18a, the value of bead porosity, 5,, required to solve Equations 3-9 and 3-10 can be determined. The tracer used in this study was calcium chloride. Since calcium chloride dissociates in water into ions the molecular diameter of the tracer is the average of the calcium and the chloride ions. The average number of water molecules which coordinate the ions in aqueous solution is six” and spacing them evenly around the ion gives an average diameter equal to the diameter of two water molecules (diameter of 4.2 A”) plus the diameter of the ion itself (calcium ion = 2 A and chloride ion = 3.6 A”). This gives a diameter of 10.4 A for calcium and 12.0 A for chloride with an average value of 11.2 A. From Figure 5-l8a, a molecule of this size would experience a 50% porosity in alginate beads containing 50% by weight magnetite. Figure 5-17 shows that the addition of magnetite reduces the amount of accessible void volume in the beads since the asymptotic value of each curve represents the total accessible void volume of the beads. The magnetite occupies volume and obstructs pores of the beads, thus also hindering diffusion. Figure 5-18b, shows that there is a shift toward larger pores as the percentage of magnetite in the beads increases. For example, a molecule with a diameter of 10 A, would have access to have 98, 98, and 92% of the volume accessible to water in beads containing 0, 5, and 50% by weight magnetite, respectively. In contrast, a molecule with a diameter of 204 A, would have access to 12, 19, and 26% respectively in the same beads. The magnetite may interfere with the cross-linking of the alginate during bead formation, thus resulting in larger pores. 123 6.4 Gas-to-Liquid Mass Transfer It is unclear why the magnetic field affected kLa for the highest and lowest liquid velocities, but not the intermediate velocity. This behavior suggests that two different mechanisms were responsible for the increases in kLa, one of which predominated at the lowest liquid velocity, and the other predominated at the highest velocity. Since the maximum kLa values for a liquid velocity of 5.15 cm/s appeared in the chain-channel regime, the increase probably arose from bubble phenomena that are exclusive to that regime. One observation made in this regime for a liquid velocity of 5.15 cm/s and the two lower gas velocities was that small bubbles (2-3 mm diameter) became entrapped in the three-dimensional mesh of solid particles which formed in this regime. By increasing the bubble residence time, this phenomenon would increase the bubble void fraction, the interfacial area for mass transfer, and hence kLa. However, the fiber optic probe would not necessarily indicate the increase in the local void fraction. Thus, it is reasonable that no increase in local void fraction was observed under these conditions. While this observation offers an explanation for increased mass transfer at the lower gas velocities, there is none for the highest gas velocity. Further study is needed to explain increases in mass transfer for a liquid velocity of 7.71 cm/s. The MSFB would be a valuable tool for systems where the gas- to-liquid mass transfer step is rate limiting, since the gas-to-liquid mass transfer can be increased by setting the MSFB to the appropriate conditions described above. 124 6.5 Dimensional Analysis of the Fluidized Bed Model The dimensionless group can be interpreted physically in terms of two Peclet numbers, DanlRU and D,/RU. The first group would be the ratio of the dispersive transport to convective convective and the second group the ratio of diffusive transport to convective transport. When the rate of intraparticle diffusion is large compared to either dispersion or convection, the value of becomes large, and the correction factor becomes less than one. With the plot shown in Figure 5-34, it is now possible to conduct a tracer experiment in a fluidized or packed bed system, calculate the apparent dispersion coefficient using any of the methods described in Section 2.4.3, calculate and Pe,”, then graphically determine the correction factor. Multiplication of the correction factor by the observed dispersion coefficient will then give the true dispersion coefficient. This approach eliminates the need to solve Equation 3-9 and 3-10 to account for the effects of both diffusion and axial dispersion. It also shows the regimes within which the effects of diffusion may be neglected. However, there are some limitations to this model. The numerical technique used to solve the model was unstable for values of Pe less than 20 and for liquid velocities less than 5 cm/s. The IF? experimentally determined correction factors calculated from the dispersion data measurements in this study agree very well with the correction factors shown in Figure 5-34. However, these correction factors were all greater than 0.9, so the ability of the model to predict correction factors much less than 0.9 was not tested. To further verify the model predictions at higher values, highly 125 porous, small radius beads, should be used. Such experiments were attempted with highly porous alginate beads without magnetite. Unfortunately, the experimental system gave a Pe,” smaller than 20, and the model could not accurately predict the correction factor. It may thus be preferable to use small radius beads to further test the ability of the model to predict correction factors. 6.6 Bioreactor Potential Based on the results of this study, the three-phase MSFB shows potential for bioreactor application. It is possible to significantly change the bed properties of simply by adjusting the magnetic field strength. Thus, the MSFB can be customized to meet the demands of many types of biological applications. For fast growing biological systems where the rate of oxygen transfer to the liquid phase is rate limiting, the MSFB can be operated in the chain-channel regime where the gas-to-liquid mass transfer coefficient was maximized. For slow growing systems, the cell concentration maybe rate limiting. In this case, the cell concentration could be maximized by running the MSFB in the frozen regime where the packing of spherical biocatalyst particles approaches the theoretical limit. In addition, true continuous processing of the solid phase is possible by operating the MSFB in the frozen regime. Biocatalyst particles would be continuously, or periodically, loated into the top of the reactor and allowed to slowly fall through the reactor in plug flow. Spent biocatalysts would be removed at the same rate from the bottom of the reactor after they stop producing optimally, die, or otherwise exhibit undesirable behavior. Solids residence time could be carefully controlled by the rate of solids addition to maintain 126 optimal productivity of the bioreactor. This mode of operation should offer imporved performance over conventional bioreactor systems where many of the cells may be dead or in a sub-optimal physiological state. Continuous solids throughput in plug flow is also possible for two-phase MSFB, but scale up of such systems for aerobic bioprocesses would be difficult, because of the limited oxygen-carrying capacity of the aqueous medium. For reactions which exhibiting positive order kinetics and/or product inhibition, plug flow reactors maximize conversion. By operating the MSFB in the channel regime, where a 400% decrease in the dispersion coefficient was measured, axial mixing of the liquid phase can be minimized. For substrate inhibited reactions, a completely mixed system is desirable and can be easily achieved by using a recycle stream. The MSFB can thus produce a wide range of properties by simply changing the magnetic field strength, the liquid velocity, and the gas velocity. The MSFB has the capability of acting as a stirred tank by employing a recycle stream, as a packed by operating in the frozen regime, and as a conventional fluidized bed by switching off the magnetic field. In addition, the MSFB offers an excellent package of features not found together in other bioreactor configurations: such as low axial liquid mixing, efficient contacting of gas, liquid, and solid phases, low pressure drops and a low shear environment for immobilized cells. 6.7 Scale-up Considerations The two important scale-up considerations for the three-phase MSFB are the solid phase and the solenoid. The solid phase used in 127 this study consisted of 4 mm diameter calcium alginate beads containing 50% by weight magnetite. The relatively large size of the beads would probably lead to significant concentration gradients in the radial direction, leading to reduced catalyst effectiveness factors, and possibly death to cells inside a critical radius. A potential solution to this problem would be to made smaller diameter beads. In addition, the magnetite tended to leak out of the 50% magnetite beads unless a time consuming coating process was performed on the beads. This coating process would be difficult to conduct aseptically on the beads. A potential solution to this problem would be to make smaller diameter beads containing a lower concentration of magnetite. There was little leakage of magnetite from the alginate beads when only 5% magnetite was present. However, 5% magnetite was not enough for stabilization of the bed at the field strengths used in this study. Thus, there may be an optimum value between 5 and 50% by weight magnetite, where the magnetite will not leak out, but stabilization can be achieved. Higher field strengths may be required for this purpose. An additional engineering issue is the high viscosity of the magnetite-alginate suspension. Although, in principle, smaller beads can be produced by using a smaller nozzle, the use of small nozzles in making 50% magnetite beads let to plugging of the nozzle. This problem may be overcome by reducing the amount of magnetite in the beads, or better designing the gel dispersion system. The simplest method to scale up the solenoid is to increase the diameter and the length of the solenoid proportionately while 128 keeping the ratios 0: and B (as defined in Figure 3-2) the same. Reasonable criteria for solenoid scale up are that the field strength must remain constant and the ratio of power consumed/surface area available for cooling must either remain constant or decrease. Examination of the solenoid design equations“, showed that the power consumption of the solenoid was proportional to the solenoid diameter. The power/surface area ratio decreased as the solenoid diameter increased. Thus, better cooling of the solenoid will be possible for larger solenoids. One potential disadvantage to this scale up strategy is that the length of wire might become excessive for large diameter solenoid. Another scale up strategy would be to keep the length of wire constant was analyzed. The power consumption and ratio of power consumption to surface area available for cooling would remain the same. However, the trade off for this case was that very large currents must be used to obtain a constant field strength. In both of these cases, the power/volume ratio decreases as the solenoid size increases. For very large diameter solenoids, instead of keeping the ratio B (the solenoid length to diameter) constant, it may be desirable to increase the thickness of the wire layer on the solenoid (az-a1) to maintain the desired field strength and decrease the value of B. As this layer becomes thicker, a water jacket alone may not be adequate to cool the solenoid. Other more effective types of cooling have been suggested“, including the use of hollow wires where cooling water is run through the wires, placement of cooling ducts between the layers of wire, use of liquid gases such as nitrogen, and use of superconductivity. This first two suggestions are probably the 129 most cost efficient; however, these methods reduce the efficiency of the solenoid“. The use of liquid gases would be more costly, and the use of superconductivity would probably make the large scale MSFB prohibitively expensive. However, further increases in the maximum temperature at which superconductivity can be achieved could make this alternative more attractive. Another important consideration in the scale up of the MSFB is aseptic operation. Introduction of the solid phase at the top of the bed and removal from the bottom present a special problem in maintaining aseptic conditions. Aseptic conditions are especially important for a systems which will be running continuously. 6.8 Conclusions The three-phase MSFB exhibited several bed regimes that were distinguished by different modes of interaction between the magnetized particles and the rising bubbles: random, chain, chain- channel, destabilized, channel, and frozen. The chain regime is distinguished by small chains of solids forming. The chain-channel regime is characterized by three-dimensional meshes of chains. The destabilized regime occurs when the three-dimensional meshes break into clumps. In these three regimes, a large range of bubble sizes is present, ranging from a few millimeters in diameter to several centimeters. The channel and frozen regimes are characterized by permanent channels in the solid phase through which bubbles preferentially travel. In these two regimes only small bubbles (less than 1 cm) are observed. The chain-channel, destabilized, and channel regimes do not occur in two-phase MSFB indicating the 130 complexities introduced into three-phase MSFB behavior by the gas phase. Throughout the course of this study, it has been shown that improvements in bed behavior can be obtained by the application of a magnetic field. A 200-300 percent increase in local gas void fraction was observed in the frozen and channel regimes. A 400 percent increase in Peclet number occurred in the destabilized and channel regimes, and a 30% increase in the mass transfer coefficient occurred in the chain-channel regime. In many cases, the improvements could be attributed to the interactions of gas, liquid, and solid characteristics of a particular bed regime. The increase in mass transfer occurred in the chain-channel regime for liquid velocities of 5.15 and 7.71 cm/s. The increase at 5.15 cm/s could be explained by the small bubbles becoming entrapped in the mesh-like structure of the bed, but the increase at 7.71 cm/s could not be explained. The channel and frozen regimes exhibit similar local gas void fraction properties, while the Channel and destabilized regimes exhibit similar liquid mixing properties. The MSFB properties were strongly affected by the liquid and gas velocities, as well as the magnetic field strength. These variables can thus be used to customize the bed properties for particular applications. The behavior of the three-phase MSFB is much more complicated than the corresponding two-phase systems, and care must be taken in extrapolating data from two-phase systems to three-phase. A theoretical analysis of the effect of tracer diffusion on measured dispersion coefficients was conducted. A correction factor, defined as the ratio of the true dispersion 131 coefficient to the apparent dispersion coefficient, was graphically correlated as a function of the Peclet number. With this correlation, the extent to which intraparticle tracer diffusion affects the apparent dispersion coefficient can be determined, andlthe true dispersion coefficient can be calculated, without having to solve the governing partial differential equations. Dimensional analysis indicated that Peclet numbers that express the relative rates of diffusion, dispersion and convection govern when the correction factor would need to be applied. 6.9 Future Work The large number of unexplainable results from this work shows the complexity of MSFB phenomena. The need for more detailed study of the macroscopic and microscopic bed structures to gain complete understanding of the system is clear. The original purpose of this work was to develop and optimize a three-phase MSFB for use as a bioreactor. The ability of the MSFB to continuously contact the solid, liquid and gas phases, while the solid phase passes through the reactor in a plug flow should now be exploited in the design of the new bioreactor system. Types of biological systems which would benefit from this reactor configuration are ones that have a limited cell lifetime, produce a toxic product, or store the desired product intracellularly. In addition, a high product value would be required to justify the added cost of providing the magnetic field. An example of this type of system would be production of specialty chemicals using plant cell cultures. 132 A sterilizable MSFB bioreactor is currently being designed for this purpose. Beads containing magnetite and the desired organism will be fed to the top of the reactor. The magnetic field will prevent mixing of the solids as they proceed down the. reactor and are removed from the reactor after the desired residence time. Because there will be no solids mixing, all the cells in the reactor will have nearly identical residence times. APPENDICES APPENDIX A Appendix A: Derivation of Equations A.1 Unsteady State Well Mixed Model A mass balance on a well mixed reactor volume gives the following equation in - out + generation = accumulanon a dc], (A' 1) FC, ‘ FCL + kLaVKCL ”C’) - 1.7 where F is the flow rate (mL/s), C, is the inlet concentration in the liquid (g/mL), CL is the outlet concentration in the liquid (g/mL), VT is the liquid volume (mL), kLa is the mass transfer coefficient (3"), CL° is the concentration in the liquid phase which is in equilibrium with the gas phase (g/mL), and t is time (s). Rearranging this equation into a more convenient form yields dC p F C , They—firm = VT°+kLaCL (A-z) Equations of this form have the following solution a- +h(t)y =80) ya) =1 moan-‘3 u 14 [KM (A-3) u(t) =e For Equation A-2 then h(t) F (t) C + aC' = a+_ g = L L ‘ VT VI . (A4) I} “la «4.1.x u(t)=e ‘ ”r =e ‘ ‘9 133 134 Substituting this into the solution given in equation A-3 gives «kw-‘1» (hair C p «La-"Ly CL=e y’ f e y’ ( ‘3 + LaCL‘)dt+Be y' (A-S) :- Simplifying this equation yields C F (—9—+kLaC[) F -(lt,,a+—-)t CL- VT +86 If, (A-G) F Substituting the initial condition that the concentration in the liquid is zero at time zero gives cor , ( V + LaCL) o= 7' F +3 (A-7) k +— (La V7) Finally, substituting the value for B into Equation A-6 gives the final result (— +kLaCD 1? «Lu—)3 c - T (l-e ’r) (A—8) I. (aw-‘5) VT A.2 Steady State Well Mixed Model A steady state mass balance on a well mixed reactor volume gives the following equation in - out + generation FC - FCL + Lanai-cl) 0 accumulation (A-9) II C 135 A.3 Steady State Plug Flow Model A steady state mass balance on a differential volume element with thickness A2 and cross-section area A is shown in Equation A- 10 in - out + generation = accumulation (A-10) AUCL lz-AUCLI +kLaA(Az)(C[-C,) = o 2%: dividing through by AAz UCL L'UCL | A M: +kLa(C[ -c,) =0 (A41) Z and taking the limit as A2 goes to zero gives d(UC , - dz ') +kLa(C,_ —c,) =0 (A-12) Assuming that U is not a function of axial position and rearranging to the form given in Equation A-3 dCL ha I: a (A-l3) _+-— =_L. I: U dzU" for this differential equation h(z), g(z), and u(z) are h( )_kLa 8( )_kLaC[ z ' U z ' U (A-14) 5'34: (if): u(z) =e U =e Substituting these into the solution given in Equation A-3 yields 136 ha ha . ha -(——)2 (—)zk aC -(—)z C=e U e U —" Ldz+Be U (A-15) L f U Simplifying this equation gives «51): (A 16) CL-C[+Be U Using the boundary condition that the liquid concentration at z=0 is zero gives o=C;+B (A47) Substituting this into Equation A-l6 gives the final result he , -(—)z A-l CL=CL(1 -e " ) ( 8) A.4 Steady State Dispersed Plug Flow Model A steady state mass balance on a differential volume element of thickness A2 and cross-sectional area A is shown in Equation 2-19 in - out + generation = accumulation UACL |,+.4e,_1v|z - UACLI -AeLN|,,Az + kLaA(Az)(C[—Cl) = o zsAz (A-l9) where N is the flux through the volume element due to disperison, and EL is the liquid fraction in the volume element. Dividing through by AAz gives UCI. lz-UCL |z+Az+ 8L(le-N|2+A A2 A2 U +kLa(C,_' -c,) =0 (A40) 137 and taking the limit as Az goes to zero d(UC dN . - dz U -e‘72' +k,a(c,_ -c,) =0 (A41) Assuming that the dispersive flux can be modelled by a Fickian diffusion mechanism N= .. D 12’: (A-22) and that U is not a function of z the final form of the differential equation results 42c, dz 2 dC , “a, =07; +k,a(c, -c,) =0 (A-23) The boundary conditions for this differential equation are D dC “° “Tiff“ (A-24) dC z=L __L.= dz The characteristic equation for Equation A-23 is ka n2- U n- L =0 (MS) ”L04: ”1.04: Using the quadratic formula give the following real roots 4kaD a n1_2=——U (ltJl+—-—L 2“ ‘) (A46) art. 138 The solution of Equation A-23 will then be of the following form C L'Alen“ seize-35,43 (A-27) By inspection the constant term A, must be CL'. Applying the boundary condition at z=L gives Al as a function of A2 (”1'9 [tr-"U" " (A-zs) "1 Applying the boundary condition at z=0 and substituting Equation A- 28 give an expression for A2 _ (Co‘CD ‘2‘ (A-29) -,, D - D -3305 ”Ar—""2305 w-anq-l n2 U U Multiplying the numerator and denominator by the term nle'” gives (Co-CDnle "‘1' A2= D (A-30) D -'ae”‘+-§Mle""-—§nznl¢"‘+nle"“ Grouping like terms in the denominator gives (Co-CDnlew' 42" (A-31) I Dex Dc! nle 'fi—Eru-ll-nzehfi—a—nl-l] Expanding the first term in brackets in the denominator gives BEnI—l-Dfi U (1— “an (A-32) U U ZD cL 139 Simplifying Equation A-32, it can be shown D“ U 7’12--1=--n1—.::. (A'33) and a similar expansion can be done on the second term in brackets to show that D _¢~.,.1-1=-,.2_U. (A-34) U D“ Substituting the expressions in Equation 2-33 and 234 into Equation 2-31 gives a final expression for A2 -(Co-C£)nle"" Ell-(n15 "L-nzzefil) 42' (A-35) A similar analysis can be performed to find Al (Co-CDnze'hl' BU—(nfe "L-nzze'fil‘) ‘41 ' (A-36) A.5 Stirred Tank in Series Model A mass balance on a well stirred reactor volume including the effect of tracer diffusion into the solid phase gives in - out + generation accumulation . . dC, (A-37) UACo - UACL - lossduetodtfiilston = eLVT-I A convenient description for the term due to diffusion is the negative 140 of the term describing the accumulation of tracer inside the beads, and unit analysis is used to get the appropriate units of grams/second as shown in Equation A-38 grams tracer accessible bead volume bead volume 1 total volume accessible bead volume- time bead volume total volume dC, ? 8P C: VT (A-38) where C,, is the concentration in the solid phase (g of tracer/mL of accessible bead volume), and e, is the porosity of the solid phase. Substituting this into Equation A-37 gives dC dC UA(C,-C,)-epe,Vr7‘5=eLVr—j (A—39) APPENDIX B Appendix B: Computer Programs B.l Solenoid Design #********************************* c THIS PROGRAM CALCULATES REQUIRED SYSTEM C PARAMETERS TO PRODUCE A GIVEN SPECIFICATION FOR C A GIVEN FIELD STRENGTH OR A GIVEN LENGTH OF WIRE itt¥lfl¢¥t¢t¥t¥t*************#***** REAL*8 DS,LS,DW,HO,PI,RHO,A,A1,BETA,LW REAL*8 ALPHA,N,FAB,I REAL*8 R,V,P,LAMBDA,Q,WV,WS,D2,VEL REAL*8 D,H,DELTW,DELTC,DELTF REAL*8 DELTI,DELTS,KCU,KFORM,KI,TFORM,TI INTEGER STEP PRINT*,’DO YOU WANT TO STEP OFF LENGTH OR 3 FIELD,1=LENGTH 2=FIELD’ READ*,STEP PRINT*,’INPUT DIAMETER AND LENGTH OF s SOLENOID,AND WIRE DIAMETER 3 IN CENTIMETERS’ READ*,DS,LS,DW PRINT*,’INPUT FIELD STRENGTH IN OERSTEDS’ READ*,HO PRINT*,’INPUT LENGTH OF WIRE IN METERS’ READ*,LW OPEN(2,FILE=’CURRENT’,STATUS=’NEW’) OPEN(1,FILE=’HEAT',STATUS=’NEW') IF(STEP.EQ.1) THEN WRITE(1,*),’SOLENOID LENGTH =’,LS,’CM’ WRITE(1,*),’SOLENOID DIAMETE =’,DS,’.CM’ WRITE(1,*),’WIRE DIAMETER =’,DW,’CM’ WRITE(1,*) WRITE(2,*),’SOLENOID LENGTH =’,LS,’CM’ WRITE(2,*),’SOLENOID DIAMETE =’,DS,’CM’ WRITE(2,*),’WIRE DIAMETER =’,DW,’CM’ WRITE(2,*) GOTO40 ENDIF WRITE(1,*),’SOLENOID LENGTH =’,LS,’CM’ WRITE(1,*),’SOLENOID DIAMETE =’,DS,’CM’ WRITE(1,*),’WIRE DIAMETER =’,DW,’CM’ WRITE(1,*) WRITE(2,*),’SOLENOID LENGTH =’,LS,’CM’ WRITE(2,*),’SOLENOID DIAMETER =’,DS,’CM’ WRITE(2,*),’WIRB DIAMETER =’,DW,’CM’ WRITE(2,*) 40 PI=3.141592654 RHO=1.7E-6 LAMBDA=PI/4.*((DW-.00762)/DW)**2. KCU=3.86 KFORM=.OO99822 KI = .014 141 142 TFORM= .00762 TI=.2 WRITE(1,*),’FLOW RATE WATER TEMP SOLENOID $ TEMP FIELD STRENGTH LENGTH OF WIRE’ WRITE(2,*),’ N R Lw I P V ALPHA H0’ 20 A=(DW/2.)**2.*PI A1=DS/2. BETA=(LS/2.)/Al LW=LW*100 - ALPHA=(LW*A/(2"'Al**3*PI*BETA)+l)"0.5 N=2.*Al"2.*BETA*(ALPHA-l)/A FAB=LOG((ALPHA+(ALPHA"2.+BETA**2.)**0.5) s /(1+(BETA**2.+1)“0.5)) FAB=4*PI*BETA/10.*FAB I=2*HO*A1*BETA*(ALPHA-l)/(N*FAB) R=N**2*RHO/A1*PI*(ALPHA+1) s /(2*BETA*(ALPHA-l)*LAMBDA) V=I*R P=PI*N**2*I**2*RHO*(ALPHA+1) s /(2*A1*BETA*(ALPHA-l)"LAMBDA) Q=Pl4l.86 WV=P/(Al**3*2*PI*BETA*(ALPHA"2-1)) WS=P/(PI*(DS-0.4)*LS) D2=DS-0.4 VEL=4*Q/(PI*(D2**2-6**2.)) D=DS-6.4 H=9E-3*(1+1.5E-2*19)*VEL"0.8/D**0.2 DELTW=14+10+WSIH DELTC=WV*.2"2/(2*KCU) DELTF=WS*TFORM/KFORM DELTI=WS*TI/KI DELTS = DELTC + DELTF + DELTI IF(STEP.EQ.1) THEN GOTOlO ENDIF WRITE(2,100),N,R,LW/100,1,P,V,ALPHA,HO 100 FORMAT(F6.0,3X,F6.2,3X,F71,3X, s F7.2,3X,F8.2 3x F7.2,3X .5,3 ,F5.l) ’ , , F8 x WRITE(1,300),Q,D LTW,DELTS,HO,LW/100 300 FORMAT(F7.2,9X,F8.4,9X,F8.4,1l ,F5.l,13X,F7.1) HO=HO+10 LW=LW/100. IF(HO.GT.500.) THEN GOTO30 ENDIF GOTozo 10 WRITE(2,200),N,R,LW/l 00, 200 FORMAT(F6.0,3X,F6.2 3x,F7.1, X,F7.2, s 3X,F8.2,3X,F7.2,3X,F8.5,3X,F5.l) WRITE(1,400),Q,D LTW,DELTS, O,LW/100 400 FORMAT(F7.2,9X,F8.4,9X,F8.4,11 ,F5.1,13X,F7.1) LW=LW/100. 30 143 LW =LW+50 IF(LW.GT.2000.) THEN GOTO30 ENDIF GOT020 CLOSE(2) CLOSE(1) STOP END 8*8**************$**************** C C THIS PROGRAM CALCULATES THE AXIAL FIELD PROFILE OF A SOLENOID 813*t*************************¥***** 100 200 10 REAL*8 H(600),ALPHA,BETA,HO INTEGERI OPEN(2,FILE=’STRENGTH’,STATUS=’NEW’) z= .5 PRINT*,’ENTER THE FIELD STRENGTH’ READ*,HO Pl=3.l41592654 DO 10 I=1,160 ALPHA=1.11279 BETA=30.75/4.6 BETAMI=ABS(BETA-Z/4.6) BETAPL=BETA+Z/4.6 F=LOG((ALPHA+(ALPHA**2+BETA**2)**0.5) $ /(1 +(l +BETA**2)**0.5)) F= 4"PI*BETA*F/10 FPL= LOG((ALPHA+(ALPHA**2+BETAPL**2)**O. 5) $ /(1 +(l +BETAPL**2)**0. 5)) FPL= 4*PI*BETAPL*FPL/10 FMI = LOG((ALPHA + (ALPHA**2 +BETAMI**2)**O.5) 3 /(l +(l +BETAMI**2)**O.5)) FMI=4*PI*BETAMI*FMI/10 IF((Z/4.).LE.BETA) THEN H(I)=HO*(FPL+FMI)/(2*F) WRITE(2,100), z, H(I) FORMAT(F5.1,5X,F7.2) z=z+05 ELSE H(I)=HO*(FPL-FMI)/(2*F) WRITE(2, 200) z ,H(I) FORMAT(F5. 1, 5x, F7. 2) z= 2+0. 5 ENDIF CONTINUE CLOSE(2) PRINT*,’DO YOU WANT ANOTHER TRY?, YES=1, NO=2’ READ*,TRY IF(TRY.EQ.1) THEN GOTOS ENDIF 144 STOP END B.2 Gas Void Fraction ********$*8**#***¥************* C THIS PROGRAM CALCULATES THEVOID FRACTION C FROM DATA COLLECTED FROM A OPTICAL FIBER PROBE titltt*tttttt#******#¥*¥*8***** real a, b, c, t, v, i, nse, sum, tl, inc, ltime, noise, hival real percent, void integerj, flagl, flag2, k write(*,*) ’INPUT NUMBER OF DATA FILES’ read(*,*) n write(*,*) ’INPUT LAST TIME POINT:’ read(*,"') ltime write(*,*)’INPUT THE NOISE LEVEL AND HIGH VALUE: ’ read(*,*) noise,hival write(*,*) ’INPUT THE % ABOVE NOISE TO USE:’ read(*,*) percent i = 1./100. percent = percent/100. nse = noise + (hival - noise) * percent do 30 k = l, n write(*,*) open(2,file=’ ’,status=’old’) read(2,*) a, b sum = 0.0 if (b .le. nse) then flagl = 1 else flagl = 0 endif do 10j = 1, 10000 if (a .gc. ltime) goto 20 read(2,*) t, v if (v .le. nse) then flag2 = 1 else flag2 = 0 endif if ((flagl .eq. 0) .and. (flag2 .eq. 0)) then sum = sum + i elseif(f1agl .ne. flag2) then t1 = a + (t - a)*(nse - b)/(V - b) if (flagl .gt. flag2) then inc = t - t1 else inc =t1- a endif sum = sum + inc endif a=t 145 b = v fla l = flag 2 10 continue 20 void = sum/ltime write(*,*) ’THE VOID FRACTION = ’, void close(2) 30 continue stop end ****33*****¥***********¥******** C C C THIS PROGRAM CALCULATE THE BUBBLE VELOCITY OF A BUBBLE WHICH TRAVELS BETWEEN TWO OPTICAL FIBER PROBES *t***************t***##********# real t,tau,step,up,cross,crossl,low,fli,time,corr(55,5) real keep(55,5) integer n, num, count character*20 namel, name2 write(*,*) ’INPUT TOTAL TIME, STEP SIZE, AND NUMBER S OF POINTS’ read(*,"‘) t,step,num write("',*) ’INPUT A GUESS OF TAU’ read(*,*)tau write(*,*)’INPUT THE NAME OF FILES FOR PROBE 1 AND 2’ read(*,*) name1,name2 tau = tau - 25*step do 5 k = 1,50 10 n = tau/step open(2, file=namel, status=’old’) open(3, file=name2, status=’old’) do 20 i=l,n read(3,*) time,up 20 continue 30 40 read(2,*) tl,low read(3,*) time,up cross = low‘up do 30j=l,10000 read(2,*) tl,low read(3,*) time, up fli = low*up if(time.eq.t) then cross = fli +cross goto 40 else cross = cross + 2*f1i endif continue do 50 i=l,2 if(i.eq.l) then corr(k,i) = cross/(2*num) else corr(k,i) = tau 146 endif 50 confinue close(2) close(3) tau = tau + step write(*,*) ’loop’,k 5 confinue keep(1,l)= corr(1,l) kee (1,2) = corr(l,2) do 0i=2,50 if(corr(i,l).gt.keep(l,l)) then keep(l,l) = corr(i,l) keep(l,2) = corr(i,2) endif 60 continue write(*,*)’tau =’,keep(l,2) stop end B.3 Optimization Program The remaining programs utilize an optimization called PATERN which finds the optimum value of a variable. The PATERN program will be listed here and deleted from the subsequent programs to save space. SUBROUTINE PATERN(NP,P,STEP,NRD,IO,COST) C ----- THE SIZE OF Bl,B2,T,AND S NEED ONLY BE EQUAL TO ----- THE NUMBER OF PAR DIMENSION P(lOOO),STEP(1000),Bl(100),B2(100) DIMENSION T(100),S(100) C THE FOLLOWING COMMAND ALLOWS PATERN TO USE C AN INTEGER VARIABLE AS THE THIRD PARAMETER P(3). C O NSRC =3 P(NSRC) = ifix(P(NSRC)) ----- STARTING POINT L=1 ICK=2 ITTER=0 DOS I=1,NP B1(I)=P(I) BZ(I)=P(I) T(I)=P(I) 5 S(I)=STEP(I)*10. C ----- INITIAL BOUNDARY CHECK AND COST EVALUATION CALL BOUNDS(P,IOUT) IF(IOUT.LE.0)GOT010 IF(IO.LE.0)GOT06 WRITE(*,1005) O0 6 10 c--- 11 C--- 147 WRITE(*,1000)(J,P(J),J=1,NP) RETURN CALL PROC(P,C1) IF(IO.LE.0)GOTOll WRITE(*,1001)ITTER,C1 WRITE(*,1000)(J,P(J),J=1,NP) ----BEGINNING OF PATTERN SEARCH STRATEGY DO99 INRD=1,NRD D012 I=1,NP S(I)=S(I)/10. S(NSRC) = 1.0001 C .............................................................. 20 (3-- 21 22 23 24 25 30 31 33 IF(IO.LE.0)GOT020 WRITE(*, 1003) WRITE(*,1000)(J,S(J),J=1,NP) IFAIL=0.0 ---PRETURBATION ABOUT T D0301=1,NP IC=0 P(I)=T(I)+S(I) IC=IC+1 CALL BOUNDS(P,IOUT) IF(IOUT.GT.0)GOT023 CALL PROC(P,C2) L=L+1 IF(IO.LT.3)GOTO22 WRITE(*,1002)L,C2 WRITE(*,1000)(J,P(J),J =1,NP) IF(Cl-C2)23,23,25 IF(IC.GE.2)GOT024 S(I)=-S(I) GOT021 IFAIL=IFAIL+1 P(I)=T(I) GOT030 T(I)=P(I) C1 =C2 CONTINUE IF(IFAIL.LT.NP)GOTO35 IF(ICK.EQ.2)GOTO90 IF(ICK.EQ.1)GOTO35 CALL PROC(T,C2) L=L+1 IF(IO.LT.2)GOT031 WRITE(*,1002)L,C2 WRITE(*,1000)(J,T(J),J =1,NP) IF(Cl-C2)32,34,34 ICK=1 D033 I=1,NP Bl(I)=B2(I) P(I)=B2(I) T(I)=B2(I) 148 GOT020 34 C1=C2 35 IB1=0 D039 I=1,NP 132(1)=T(I) IF(ABS(Bl(I)- 132(1)). LT. 1. 0E- -20)IB1=IB1+1 39 CONTINUE IF(IBl. EQ. NP)GOTO90 ICK=0 ITTER = ITTER +1 IF(IO.LT.2)GOTO40 WRITE(*,1001)ITTER,C1 WRITE(*,1000)(J,T(J),J=1,NP) C ----- ACCELERATION STEP 40 SJ=1.0 D045II=1,11 DO421=1,NP C T(I)=B2(l)+SJ*(B2(I)-Bl(1)) C IF(I.EQ.NSRC)T(I)=ifix(T(I)) 42 P(I)=T(I) SJ=SJ-.1 CALL BOUNDS(T,IOUT) IF(IOUT.LT.1)GOTO46 IF(II.EQ.11)ICK=1 45 CONTINUE 46 DO47I=1,NP 47 Bl(I)=B2(I) GOT020 90 DO911=1,NP 91 T(I)=132(I) 99 CONTINUE DOlOOI=l,NP 100 P(I)=T(I) COST=C1 IF(IO.LE.0)RETURN WRITE(*,1004)L,C1 WRITE(*,1000)(J,P(J),J=1,NP) RETURN 1000 FORMAT(3(35X,I7,5X,E13.6/)) 1001 FORMAT(//1X13HITERATION NO. ,15/5x,5HCOST= S ,E15.6,20X,lOHPARAMETERS) 1002 FORMAT(10X3HNO.,I4, 8X5HCOST=,E15.6) 1003 FORMAT(/1X28HSTEP SIZE FOR EACH PARAMETER ) 1004 FORMAT(1H113HANSWERS AFTER 3 ,I3,2X,23HFUNCTIONAL EVALUATIONS // s 5X5HCOST=,E15.6,20X,18HOPTIMAL PARAMETERS ) 1005 FORMAT(1H135HINITIAL PARAMETERS OUT OF s BOUNDS ) END B.4 Dispersion Coefficients 149 *****************$*1****t***** SUBROUTINE PROC FOR PROGRAM CALLED PATERN PURPOSE: TO ESTIMATE DISPERSION COEFFICIENTS BY READING IN DATA CONTAINING TIME -VS- CONCENTRATION VALUES. ****************¥**¥***fi***¥** MAIN PROGRAM COCO *OOOOO * INTEGER NDATA,NP,NPASS,IO integer ip, i, nin, nres, nterm real pm(20), cdi(200), cwork(10), toin, dtin, tores, dtres, S cdr(200), tau, p(10), step(10) common lmainl/cdi,toin,dtin common /main2/cdr,tores,dtres common /main5/nin,nres common lpbparm/pm common /main4/tau,nterm open(2,file=’file2.csv’,status=’old’) open(3,file=’filel.csv’,status=’old’) open(4,file=’file3.csv’,status=’old’) 1 = read(2,300) (pm(i), i=l,ip) write(*,*) (pm(i), i=1.ip) 300 format(8f10.0) read(3,100) toin,dtin write(*,*)toin,dtin 100 format(2f10.0) nin=0 110 format(10f8.0) 1 read(3,110,end=90) (cwork(i), i=1,10) write(*,*) (cwork(i), i=1,10) do 10i = 1, if (cwork(i) .lt. 0.0) goto 90 nin=nin+l if (nin .gt. 200) goto 90 cdi(nin)=cwork(i) write(*,*) cdi(nin) 10 continue goto l 90 nres=0 read(3,*) tores,dtres write(*,*) tores,dtres 2 read(3,110,end=91) (cwork(i), i=l,10) write(*,*) (cwork(i), i=1,10 do 20 i=1,10 if (cwork(i) .lt. 0.0) goto 91 nres=nres+1 if (nres .gt. 200) goto 91 cdr(nres)=cwork(i) write(*,*) cdr(nres) 150 20 continue oto 2 8 91 read(4,200) tau, nterm write(*,*) tau, nterm 200 format(f10.0,i10) 399 OOOOOOOOOOOOOOO call norsig call fouexp SEARCH INITIALIZATION P(l)= 0.01 p(2)= 1.0e-3 STEP(1)=0.01 step(2)=1.0e-3 NP=2 NPASS=3 IO=3 START SEARCH CALL PATERN(NP,P,STEP,NPASS,IO,COST) SEARCH COMPLETE, PRINT RESULTS PRINT 399, P(l), P(2), COST 399 FORMAT(F10.3,10X,F10.3) OPEN (UNIT=30,FILE=’FOR30.DAT’,STATUS=’NEW’) WRITE (30,399) P(l),P(2),COST FORMAT(ElO.3,5X,F6.3,5X,E10.3) STOP END THIS FILE IS A PAIR OF SUBROUTINES WRITTEN TO BE COMPATIBLE WITH THE OPTIMIZATION SUBROUTINE PATERN. THEY SIMULATE A PROCESS USING DISCRETE DIFFERENCE EQUATIONS AND COMPARE THE SIMULATION OUTPUT WITH THE ACTUAL OUTPUT (READ IN THROUGH A DATA FILE), CALCULATING AN ERROR OR ”COST" ASSOCIATED WITH THAT SIMULATION. PATERN USES THESE SUBROUTINES ITERATIVELY IN ORDER TO FIND THE OPTIMUM SET OF TRANSFER FUNCTION PARAMETERS TO FIT THE DATA. SUBROUTINE PROC(P,COST) dimension p(10) common/pbparm/pm(20) common/main3/error Pm(2)=P(1) Pm(7)=p(2) COO 0000 10 20 151 call precur cost=error RETURN END SUBROUTINE BOUNDS(P,IOUT) DIMENSION P(10), STEP(10) IOUT=0 IF(P(1).LE.0) IOUT=1 RETURN END subroutine norsig integer znin,znres common lmainl/ cdin,ztoin,zdtin common /main2/ cdres,ztores,zdtres common /main5/znin,znres common /signal/toin,dtin,nin,cin(200), $ tores,dtres,nres,cres(200) dimension cdin(200),cdres(200) nin=znin toin=ztoin dtin=zdtin nres=znres tores=ztores dtres=zdtres call normlz(dtin,nin,cdin,ain,cin) call normlz(dtres,nres,cdres,ares,cres) - return end subroutine normlz(dt,n,cd,area,cn) dimension cd(200),cn(200) area = 0.0 do 10 i=1,n area=area+cd(i) continue area=area*dt if(area.eq.0.0) then write(*,600) do 20 i=1,n cn(i)=cd(i) continue return elseif(area.lt.0.0) then write(*,610) 30 152 endif do 30 i=1,n cn(i) =cd(i)/area continue return 600 format(lh0,’area of curve is zero(returned signal is not normal Sized). ’/) 610 format(lh0,’area of curve is negative.’/) end subroutine fouexp common lmain4/ztau,znterm common /Signa1/ toin,dtin,nin,cin(200), $ tores,dtres,nres,cres(200) common/fdata/ tau,nterm,aoin,ain(200),bin(200), aores,ares(200),bres(200) tau=ztau nterm=znterm if(tau.lt.(tores+dtres*float(nres))/2.0) then tau=(tores+dtres*float(nres))/2.0*1.5 write(*,610) tau endif if(nterm.lt.l.or.nterm.gt.ifix(tau/dtres+0.5)) then nterm=ifix(tauldtres+0.5) write(*,620) nterm endif if(nterm.gt.200) then nterm =200 write(*,620) nterm . endif - call foucoe(toin,dtin,nin,cin,tau,nterm,aoin,ain,bin) call foucoe(tores,dtres,nres,cres,tau,nterm,aores,ares,bres) return 610 format(lh0,’half period (tau) is replaced by’,e15.7) 620 format(lh0,’number of terms (nterm) is replaced by’,i5) 10 end subroutine foucoe(to,dt,nt,cn,tau,nterm,ao,a,b) dimension cn(200),a(200),b(200) data pi/3.141593/ ao=0.0 sq=0.0 do 10 i=1,nt ao=ao+cn(i) sq=sq+cn(i)**2 continue ao=ao*dt/tau sq=sq*dt cotest=ao/2.0 sqtest=2.0*(ao/2.0)**2*tau do 20 n=1,nterm 30 20 40 153 x=0.0 y=0.0 sv=float(n)*pi/tau do 30 i=1,nt z=sv*(to+float(i-l)*dt) x=x+cn(i)*cos(z) y=y+cn(i)*sin(z) ' continue a(n) =x*dt/tau b(n) =y*dt/tau cotest=cotest+a(n) sqtest=sqtest+(a(n)**2+b(n)**2)*tau if(mod(n,10).eq.0) then rat=sqtestlsq endif continue return end subroutine precur common lmain3/err common/signal/toin ,dtin, nin , cin(200) , tores ,dtres, nres, S cres(200) common/fdata/tau,nterm,aoin ,ain(200) ,bin(200) ,aores, $ ares(200),bres(200) common/pbparm/ pm(20) dimension aclc(200),bclc(200) call clccoe(pm,tau,nterm,at,aoin,ain,bin,aoclc,aclc,bclc) call clccur(tau,nt,aoclc,aclc,bclc,0.0,2.0*tau,dtres) x=2.0*(aores/2.0-aoclc/2.0)**2 y=2.0*(aores/2.0)**2 do 40 i=1,nt x=x+(ares(i)-aclc(i))**2+(bres(i)-bc1c(i))**2 y=y+ares(i)**2+bres(i)**2 continue err = sqrt(x/y) return end subroutine clccoe(pm,tau,nterm,nt,aoin,ain,bin,aoclc,aclc,bclc) dimension pm(20),ain(200),bin(200),aclc(200),bclc(200) data pi/3.141593/ call transf(0,0.0,pm,an,bn) aoclc=aoin*an do 10 n=1,nterm w=float(n) *pi/tau call transf(1,w,pm,an,bn) aclc(n) =ain(n)*an+bin(n)*bn bclc(n) =bin(n)*an-ain(n)*bn if(an**2+bn**2.lt.1.0e-8) then nt=n return endif 10 154 continue nt=nterm return end subroutine transf(ictl,w,pm,an,bn) dimension pm(20) real*8 dl,d2,d3,d4,d5 save d1,d2,d3,d4,d5 complex*l6 cs,cphi,cex,coth,cq,cf if(ictl.eq.0) then d1=pm(7)/(pm(l)*pm(2)) d2=pm(1)/pm(2) $13=Pm(8)*pm(3)l(pm(4)*pm(5)) 1f(pm(8).eq.0.0) then d4=0.0 else d4=pm(6)/pm(8) endif d5=pm(5) endif if(w.eq.0.0) then an= 1.0 bn=0.0 return endif cs=cmplx(0.0,w) if(d4.eq.0.0) then lcf=cdexp(1.0/(2.0"‘d1)"‘(1.0-cdsqrt(l.0+4"‘d1"‘d2"‘cs))) e se cphi=d5*cdsqrt(cs*d4) cex=cdexp(-2.0*cphi) coth=(1.0+cex)l(1.0-cex) cq=d3*(cphi*coth—l.0) cf=cdexp(l.0/(2.0*d1)*(l.0-cdsqrt(1.0+4*d1*d2*(cs+cq)))) endif an=cf bn=dimag(cf) return end subroutine clccur(tau,nt,ao,a,b,t1,t2,dt) dimension a(200),b(200),c(500),t(500) data pi/3.141593/ if(dt.eq.0) then in=1 dtw=dt else in=ifix((t2-t1)/dt+0.5) if(in.ge.0) then in=in+1 dtw=dt else 155 in=-in+l dtw=-dt endif if(in.gt.500) then in=500 write(*,610) in endif endif do 10 i=1,in t(i) =tl +float(i-1)*dtw c(i) =ao/2.0 10 continue do 20 n=1,nt sv=float(n)*pi/tau do 30 i=1,in z=sv*t(i) c(i)=c(i)+(a(n)*cos(z)+b(n)*sin(z)) 30 continue 20 continue return 610 fgrmat(1h0,’number of data(in) is replaced by’,i5/) en B.5 Mass Transfer Coefficients ******#*It*********************tt SUBROUTINE PROC FOR PROGRAM CALLED PATERN PURPOSE. TO ESTIMATE MASS TRANSFER COEFFICIENTS FROM POSITION --VS OXYGEN CONCENTRATION VALUES *******************#*****#****** MAIN PROGRAM 0000 *0000 IMPLICIT DOUBLE PRECISION(A-H,O-Z) INTEGER NDATA,NP,NPASS,IO character'20 dfile DIMENSION P(10), STEP(10) CHARACTER CONCEN'20 COMMON lparam/ c(100),z(100),dax,el,eg,es,zl,ul,zh, $ pt,ndata,cin common/expt/ conc(100) SEARCH INITIALIZATION P(l) = .01d0 p(2) =.06d0 STEP(l)=p(1)*.1d0 step(2) =p(2)*. ld0 NP=2 OOOOO np=1 NPASS=3 156 IO=3 READ IN DATA write(*,*)’enter data file name’ read(*,*) dfile . OPEN(UNIT=20,FILE=’ ’,STATUS=’OLD’) zh = .02464 write(*,*)’enter gas and solid fractions, and inlet conc(mg/L)’ read(*,*) eg,es,cin el = 1.0d0 -(eg+es) cin=cin/100.0d0*8.26d0 C write(*,*)’enter the room pressure (atm) and column 3 length (m)’ C read(*,*) pt,zl pt=1.0d0 zl=0.8500 write(*,*)’enter the axial dispersion coefficient and superficial 3 liquid velocity’ read(*,*) dax,ul do 10 i= 1,100 read(20,*) z(i),c(i) ndata = i-l if(z(i).lt.0.d0) goto 20 10 continue 000000 000 WRITE(*,*) . WRITE(*,*)’CONCENTRATION(g/L) TIME(MIN)’ 20 DO 150 J=1,NDATA WRITE(*,350)C(J),Z(J) 350 FORMAT(6X,F6.2,18X,F6.2) 150 CONTINUE START SEARCH CALL PATERN(NP,P,STEP,NPASS ,IO,COST) SEARCH COMPLETE, PRINT RESULTS PRINT 300, P(l), P(2), COST 300 FORMAT(F10.3,10X,F10.3) OPEN (UNIT=30,FILE=’FOR30.DAT’,STATUS=’NEW’) do 500 i=1,ndata WRITE (30,*)z(i),conc(i) 500 continue 300 FORMAT(EIO.3,5X,F6.3,5X,E10.3) STOP END C 157 THIS FILE IS A PAIR OF SUBROUTINES WRITTEN TO BE COMPATIBLE WITH THE OPTIMIZATION SUBROUTINE PATERN. THEY SIMULATE A PROCESS USING DISCRETE DIFFERENCE EQUATIONS AND COMPARE THE SIMULATION OUTPUT WITH THE ACTUAL OUTPUT (READ IN THROUGH A DATA FILE), CALCULATING AN ERROR OR ”COST” ASSOCIATED WITH THAT SIMULATION. PATERN USES THESE SUBROUTINES ITERATIVELY IN ORDER TO FIND THE OPTIMUM SET OF TRANSFER FUNCTION PARAMETERS TO FIT THE DATA. 00000000000000 SUBROUTINE PROC(P,COST) IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION P(10), STEP(10), CONC(100),pos(100) dimension xfinal(100), ,gfinau (2 ,100) COMMON lparam/ c(l 0), z(100), dax, e1 ,,eg es, zl,u1,zh, 3 pt, ndata, cin common lexpt/ conc common /data/ pe,st,a,b C C INITIALIZE ARRAYS AND DEFINE PARAMETERS C C zkla=P(1) u1=p(2) ERROR=0.0 TOTAL=0.0 gamma=(es*l700+el*1000+eg*l1.8)*9.80*(zl) $ /(pt*1.01325e5) C write("‘, *)’gamma’ pe=ul*zl/(dax*el) St=zkla*zl/ul a=pt*.21/zh*(1 +gamma) b =-(pt*.21/zh)*gamma C write(*,*)’pe=’,pe,’st= ’,St, ’a= ’,a,’b= ’ ,b,’cin= ’,cin capb = (a-cin-(b/st))*pe-b r1=pe/2*(l +(1+4*st/pe)**0.5) r2=pe/2*(1-(l +4*st/pe)**0.5) capn =r1**2*exp(rl)-r2**2*exp(r2) a1=(capb*r2*exp(r2)-b*rl)lcapn a2 = (-capb*rl *exp(rl) +b*r2)/capn if(a2.lt.0.0) then a2=abs(a2) C write(*,*)’capn=’,capn, ’a1=’,al,’a2=’,a2 c if(a1.lt.0.0) then C a1 =abs(al) C a1 =-exp(log(al)-capn) C else C a1 =exp(log(a1)-capn) C endif C C 158 C a2- — -exp(log(a2)-capn) C else C a2=exp(log(a2)-capn) C endif C write(*, *)’ st=’ ,,st ’pe= ’,pe, ’capn=’,capn C write(*, *)’ a=’ a’,b=’, b, ’capb=’ c,apb,’r1=’,rl,’r2=’,r2 C write(*, *) ’before call’ . c write(*,*)’a1=’,a1,’a2=’,a2,’gamma=’,gamma do 10 i=1,ndata pos(i) =z(i) conc(i) = a1*exp(rl *pos(i))+ a2*exp(r2 *pos(i)) $ +a-(b/st)+b*pos(i) err=(conc(i)-c(i))**2 error=error+err 10 continue error=(error/(ndata-l))**0.5 cost=error return END 000 SUBROUTINE BOUNDS(P,IOUT) IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION P(10), STEP(10) COMMON /param/ c(100),z(100),dax,el,eg,es,zl,ul,zh, $ pt,ndata,cin IOUT=0 C IF(P(l). LE. 0. or. p(2).le.0) IOUT=1 if(p(1). 1e. 0) iout= RETURN END B.6 Diffusion Coefficients **#*******¥******¥*************** SUBROUTINE PROC FOR PROGRAM CALLED PATERN PURPOSE: TO ESTIMATE DIFFUSION COEFFICIENTS BY READING IN DATA CONTAINING TIME -VS- CONCENTRATION VALUES. *#*****************#************ 00*00000 MAIN PROGRAM IMPLICIT DOUBLE PRECISION(A-H,O-Z) INTEGER NDATA, NP, NPASS, IO COMMON C(4000), T(4000), Q(10000), NDATA, X, NQ,R COMMON NTYPE, XB DIMENSION P(10), STEP(10) CHARACTER CONCEN*20 0 C SEARCH INITIALIZATION 159 C C READ IN DATA C 5 WRITE(*,*)’WHAT TYPE OF DIFFUSION PROBLEM DO 3 YOU WISH TO’ WRITE(*,*)’SOLVE? ENTER THE NUMBER FOR THE s DESIRED TYPE.’ WRITE(*,*) WRITE(*,*)’1. DIFFUSION FROM LIQUID INTO $ INITIALLY’ . WRITE(*,*)’ SOLUTE FREE SPHERES.’ WRITE(*,*)’2. DIFFUSION FROM SPHERES INTO $ INITIALLY’ WRITE(*,*)’ SOLUTE FREE LIQUID.’ READ(*,*) NTYPE WRITE(*,*) IF(NTYPE.EQ.1) THEN WRITE(*,*)’ENTER INITIAL LIQUID CONCENTRATION 5 IN g/L’ READ(*,*) x WRITE(*,*) ELSEIF(NTYPE.EQ.2) THEN 3 WRIT/13(*,*)’ENTER INITIAL BEAD CONCENTRATION IN E L’ READ(*,*) XB WRITE(*,*) ELSE WRITE(*,*)’YOU DID NOT ENTER A 1 OR A 2. TRY s AGAIN’ WRITE(*,*) GOT05 ENDIF WRITE(*,*)’ENTER THE NAME OF THE S TIME-CONCENTRATION FILE’ WRITE(*,*)’AND THE NUMBER OF DATA POINTS’ READ(*,*) CONCEN, NDATA WRITE(*,*) OPEN(UNIT =20,FILE = CONCEN,STATUS = ’OLD’) 00 WRITE(*,*)’ENTER THE NUMBER OF Q VALUES DESIRED’ READ(*,*) NQ WRITE(*,*) 300 000000000000000 000 160 WRITE(*,*)’ENTER THE RADIUS OF THE BEADS’ READ(*,*) R WRITE(*,"‘) WRITE(*, *) WRITE(*, *) CONCENTRATION(g/L) TIME(MIN)’ DO 150 J=1, NDATA READ(20, *)t(J), c(J) WRITE(*,350)C(J),T(J) FORMAT(6X,F6.2,18X,F6.2) CONTINUE START SEARCH CALL PATERN(NP,P,STEP,NPASS ,IO,COST) SEARCH COMPLETE, PRINT RESULTS PRINT 300, P(1), P(2), COST 300 FORMAT(F10.3,10X,F10.3) OPEN (UNIT=30,FILE=’FOR30.DAT’,STATUS =’NEW’) WRITE (30,300) P(1),COST FORMAT(ElO.3,5X,F6.3,5X,E10.3) STOP END THIS FILE IS A PAIR OF SUBROUTINES WRITTEN TO BE COMPATIBLE WITH THE OPTIMIZATION SUBROUTINE PATERN. THEY SIMULATE A PROCESS USING DISCRETE DIFFERENCE EQUATIONS AND COMPARE THE SIMULATION OUTPUT WITH THE ACTUAL OUTPUT (READ IN THROUGH A DATA FILE), CALCULATING AN ERROR OR "COST" ASSOCIATED WITH THAT SIMULATION. PATERN USES THESE SUBROUTINES ITERATIVELY IN ORDER TO FIND THE OPTIMUM SET OF TRANSFER FUNCTION PARAMETERS TO FIT THE DATA. SUBROUTINE PROC(P,COST) IMPLICIT DOUBLE PRECISION (A H, 0- -Z) COMMON C(4000), T(4000), Q(10000), NDATA, x, NQ,R COMMON NTYPE, XB DIMENSION P(10), STEP(10), CONC(100) INITIALIZE ARRAYS AND DEFINE PARAMETERS D=P(l) ALPHA = P(2) ERROR=0.0 TOTAL=0.0 C 11 161 PURPOSE: TO CALCULATE THE Q VALUES BASED ON THE ALPHA GIVEN alow= 0.0 alov31= tan(alow)-3*alow/(3+alpha*alow**2) J _ FINC = 0.001 kount = 0 9(1)= 0 0 doS k= 2,n n+q +1 up= alo ow + FINC hival = tan(up)- 3*up/(3+alpha*up**2) if(hival.1t.0.0.and.aloval.gt.0.0) j=j+1 if(aloval.lt.0.0.and.hival.gt.0.0) j=j+1 if(j.eq.2) goto 11 alow = up aloval = hival goto 2 if(kount.gt.500) then FINC = FINC/10 alow = q(k-1) + FINC aloval = tan(alow)-3*alowl(3+alpha*alow**2) J .— goto 2 endif aloval = tan(alow)-3*alow/(3+alpha*alow**2) if(aloval.lt.0.0) then flagl = 0 else flagl = 1 endif hival = tan(up)-3*up/(3+a1pha*up**2) if(hival.lt.0.0) then flagh = 0 else flagh = 1 endif if(flagl.eq.0.and.flagh.eq.l) then guess = (up - alow)/2. + alow amidval = tan(guess)-3*guess/(3+alpha*guess**2) if(abs(amidval).lt.1.0e-8) goto 100 kount = kount + l if(amidval.lt.0.0) then alow = guess goto 11 else up = guess goto 11 endif else guess = (up - alow)/2. + alow amidval = tan(guess)-3*guess/(3+alpha*guess**2) if(abs(amidval).lt.1.0e-8) goto 100 162 kount = kount + 1 if(amidval.lt.0.0) then up = guess goto 11 else alow = guess goto 11 endif endif 100 wt) = g kount = 0 alow = guess+ FINC alogal = tan(alow)-3*alow/(3+alpha*alow**2) J: continue CALCULATE COST IF(NTYPE.EQ. 1) THEN DO 20 I=1,NDATA D010 I=2,NQ+1 TERMI =(6*(1+ALPHA)*EXP(-D*Q(I) $ **2*T(J)/R"2)) TERM=TERM1/(9+9*ALPHA+Q(I) $ **2*ALPHA**2) TOTAL=TOTAL+TERM 10 CONTINUE CONC(J) = ((X*ALPHA)/(1 +ALPHA))*(1 +TOTAL) ERR=(C(J)-CONC(J))"2 ERROR=ERROR+ERR COST=(error/(ndata-2))**0.5 TOTAL=0.0 20 CONTINUE GOTO 300 ELSE DO 30 I=1,NDATA DO 40 I=2,NQ+1 TERMl =(6*(1+ALPHA)*EXP(-D*Q(I) $ **2*T(J)/R**2)) TERM=TERMl/(9+9*ALPHA+Q(I) $ **2*ALPHA"2) TOTAL=TOTAL+TERM 40 CONTINUE CONC(J)= (XB/(l+ALPHA))*(1- TOTAL) ERR= ABS((C(J)- CONC(J))/conc(j)) ERROR= ERROR+ER COST= ERROR TOTAL=0.0 30 CONTINUE ENDIF 300 RETURN END C 11888 0000' 163 00 SUBROUTINE BOUNDS(P,IOUT) IMPLICIT DOUBLE PRECISION (A-H,O-Z) COMMON C(4000),T(4000),Q(10),NDATA, x, NQ, R COMMON NTYPE, XB DIMENSION P(l), STEP(l) IOUT=0 IF(P(1).LT.0.0R.P(2).LT.0) IOUT=1 RETURN END B.7 Fluidized Bed Simulation *****************¥*****3*****it C THIS PROGRAM SIMULATES A FLUIDIZED BED USING C A STIRRED TANK IN SERIES MODEL AND PREDICTS C BED BEHAVIOR WHEN A PULSE IS APPLIED. THE C DISPERSION COEFFICIENT OF THE SIMULATED SYSTEM C IS DETERMINED IN THE SECOND HALF OF THE PROGRAM ******tttt********#*#*******#** implicit double precision (a-h,o-z) common/pbparmlpm(20) common/counter/I,nprob1,nprob2 open(1,file=’file4.csv’,status=’old’) open(2,file = ’file5 .csv’ ,status = ’old’) open(3,file=’file6.csv’,status=’old’) open(6,file=’tank4.csv’,status=’new’) open(7,file=’tank5.csv’,status=’new’) vb=0.00196d0 write(*,*)’how many tanks do you want’ read(*,*)ntank tankvol=vb/ntank write(*,*)’enter velocity,bead radius, and bead porosity’ read(*,*)u,br,ep a=.00196d0 write(*,*)’enter the gas void fraction’ read(*,*)eg write(*,*)’enter solid fraction, and diffusion coefficient’ read(*,*)es,de ch = 1.0d0-(eg-l-es) write(3,*)u write(3,*)a write(3,*)br write(3,*)ep write(3,*)tankvol write(3,"‘)es write(3,*)de write(3,*)eg nprob1=6*ntank/10 nprob2=9*ntank/10 close(3) 10 164 call norsig call fouexp call rdparm(pm) do 10 i=1,ntank call precur continue close(8) close(7) call displ stop end subroutine norsig implicit double recision (a-h,o-z) common lsignal toin,dtin,nin,cin(200) dimension cdin(5000) write(6,610) call rdsigl(toin,dtin,nin,cdin) call normlz(dtin,nin,cdin,ain,cin) call print(toin,dtin,nin,cdin,ain,cin) return 610 FORMAT(IIHO,’*** MEASURED SIGNALS w“) end subroutine rdsigl(to,dt,n,cd) implicit double precision (a-h,o-z) dimension cd(200),cwork(10) read(1,*) to, dt write(6,600) to, dt if(to.1t.0.d0) write(6,610) if(gt.le.0.0d0) write(6,620) u: do 900 i=1,10 read(1,*,end=90) cwork(i) write(6,630) cwork(i) if (cwork(i).lt.0.0d0) goto 5 900 continue 5 10 90 91 do 10 i=1, 10 n=n+1 if(cwork(i).lt.0.0d0) return if(n.gt.200) goto 91 cd(n)=cwork(i) continue goto 1 write(6,640) stop write(6,650) stop 100 format(2f10.0) 110 format(10f8.0) 600 format(_lh0,5x,’to =’,f10.5,5x,’dt =’,f10.5/) 165 610 format(lh0,’starting time (to) is negative.’/) 620 format(lh0,’time interval (dt) is not positive.’/) 630 format(lh,10f8.2) 640 format(lh0,’end of data cannot be found. (sub.rdsigl)’/) 650 fgrmat(lho,’number of data exceeds 200. (sub.rdsigl)’/) en subroutine normlz(dt,n,cd,area,cn) implicit double precision (a- h,o-z) dimension cd(200), cn(200) area =0 0d0 do 10 i=1, n area=area+cd(i) 10 continue area=area*dt if(area.eg.0.0d0) then c write( ,600) do 20 i=1,n cn(i) =cd(i) 20 continue return elseif(area.lt.0.0d0) then c write(6,610) endif do 30 i=1,n cn(i) =cd(i)/area 30 continue return 600$ format(lhO,’ area of curve is zero(returned signal is not normal ized). ’/) 610 format(lhO,’ area of curve is negative. ’l) end subroutine print(toin,dtin,nin,cdin,ain,cin) implicit double precision (a-h,o-z) dimension cdin(200),cin(200) c write(6,600) c write(6,610) toin,dtin,nin,ain c write(6,630) do 10 i=1,nin tin=toin +float(i-1)*dtin c write(6,650)i,tin,cdin(i),cin(i) 10 continue return 600 format(l/lhO,’***** input and response signals *****’/1h0,10x $,6x,’starting time’,5x,’ time interval’, 5x,’ no of data’ ,5x, ’area’) 610 format(lh0,5x,’ input’ ,4x, HO. 4, 7x ,f10. 4, 8x, i5, 5x, e13. 5) 630 format(lh0,15x,’ ======= input 3 =======’ H15x’=======response S =======’/1h, 4x, ’n’ ,2(7x, ’time reading 3 (normalized)’ )) 166 650 fgrmat(lh,15,3x,f10.4,f10.4,3x,f10.5) en subroutine fouexp implicit double precision (a-h,o-z) common Isignal/ toin,dtin,nin,cin(200) common/fdata/ tau,nterm,aoin,ain(200),bin(200) read(2,*) tau,nterm c write(6,600) tau, nterm if(tau.lt.(toin+dtin*float(nin))/2.0d0) then tau=(toin+dtin*float(nin))/2.0d0*1.5d0 c write(6,610) tau endif if(nterm.lt. 1 .or.nterm.gt.idint(tau/dtin+0.5d0)) then nterm=idint(tau/dtin+0.5d0) c write(6,620) nterm endif if(nterm.gt.200) then nterm=200 c write(6,620) nterm endif c write(6,630) call foucoe(toin,dtin,nin,cin,tau,nterm,aoin,ain,bin) c write(6,640) return 200 format(f10.0,i10) 600 format(l/lhO,’*"** fourier expansion *****’/lh0,5x,’half Speriod =’,e13.5,5x,’no of terms=’,15) 610 format(lh0,’half period (tau) is replaced by’,e15.7) 620 format(lh0,’number of terms (nterm) is replaced by’,15) 630 format(lh0,5x, ’ + + + + + conversion check (input) + + + + + ’) 640 format(lh0,5x, ’ + + + + + conversion check (response) 3 +++++Q end subroutine foucoe(to,dt,nt,cn,tau,nterm,ao,a,b) implicit double precision (a-h,o-z) dimension cn(200),a(200),b(200) data pi/3.141593d0/ ao=0.0d0 sq=0.0d0 do 10 i=1,nt ao=ao+cn(i) sq=sq+cn(i)**2 10 continue ao=ao*dt/tau sq=sq*dt c write(6,600) cotest=ao/2.0d0 sqtest=2.0d0*(ao/2.0d0)**2*tau do 20 n=1,nterm x=0.0d0 167 y=0.0d0 sv=float(n)*pi/tau do 30 i=1,nt z=sv*(to+float(i-1)*dt) x=x+cn(i)*cos(z) y=y+cn(i)*sin(z) 30 continue a(n)=x*dt/tau b(n)=y*dt/tau cotest=cotest+a(n) sqtest=sqtest+(a(n)**2+b(n)**2)"'tau if(mod(n,10).eq.0) then rat=sqtestlsq c write(6,610) n,rat,cotest endif 20 continue return 600 format(lh0,10x,’no of terms’,8x,’ratio’,7x,’value at t=0’) 610 format(lh,10x,i5,10x,f10.5,5x,f10.5) end subroutine rdparm(pm) implicit double precision (a-h,o-z) dimension pm(20) character*4 pname(8) data pnamel’u ’,’a ’,’r ’,’ep ’,’vt ’,’es ’, 5 ads 9’seg9/ ip=8 open(3,file= ’file6.csv’ ,status = ’old’) do 10 i=1,ip read(3,*) m(i) c write(6,6 0) c write(6,*) i,pname(i),pm(i) 10 continue return 600 format(l/lhO,’*****packed-bed parameters *****’/) end subroutine precur implicit double precision (a-h,o-z) common/signal/toin,dtin,nin,cin(200) common/fdata/tau,nterm,aoin,ain(200),bin(200) common/pbparm/ pm(20) common/counter/i,nprob1,nprob2 dimension aclc(200),bclc(200) open(8,file=’tank6.csv’,status=’new’) c write(6,600) call clccoe(pm,tau,nterm,nt,aoin,ain,bin,aoclc,ac1c,bclc) if(i.eq.nprob2) then write(7,*) nterm write(7,*) aoclc 168 do 30 k=l,nterm write(7,*) aclc(k) 30 continue do 40 k=l,nterm write(7,*) bclc(k) 40 continue endif if(i.eq.nprobl) then write(8,*)nterm write(8,*)aoclc do 50 k=l,nterm write(8,*)aclc(k) 50 continue do 60 k=l,nterm write(8,*)bclc(k) 60 continue endif do 20 j = 1,nterm ain(j) =ac1c(j) bin(j) =bclc(j) 20 continue aoin=aoclc return 600 format(l/lhO,’*“***calculation of response curve *****’) end subroutine clccoe(pm,tau,nterm,nt,aoin,ain,bin,aoclc,aclc,bclc) implicit double (precision (a-h,o-z) dimension pm(2 ),ain(200),bin(200),ac1c(200),bclc(200) data pi/3.141593d0/ . call transf(0,0.0d0,pm,an,bn) aoclc=aoin*an do 10 n=1,nterm w=float(n)*pi/tau call transf(l,w,pm,an,bn) aclc(n)=ain(n)*an+bin(n)*bn bclc(n)=bin(n)*an-ain(n)*bn if(an**2+bn**2.lt.1.0d-8) then nt=n return endif 10 continue nt=nterm return end subroutine transf(ictl,w,pm,an,bn) implicit double precision (a-h,o-z) dimension pm(20) save dl,d2,d3,d4,d5,el complex*16 cs,cphi,cex,coth,cq,cf,c if(ictl.eq.0) then Q69 10 C 169 d1-'-P111(1)"'Pm(2)/((1.Odo-P(PIII(<5)+ m(8)))*Pm(5)) d2=—pm(7)*pm(4)*gm(6))*6. ..0d0/((1p-0d0-(pm(6) + pmp(8)))* "2 3:?m(1)*Pm(pZ)/(()-l -)-0d0 -(Pm(6)+Pm(3)))*Pm(5)) end i if(w. eq. 0. 0d0) then an=1.0d0 bn=0.0d0 return endif cs=dcmplx(0.0d0,w) cq=dcmplx(0.0d0,0.0d0) do 10 i=l,500 sigma=3.l415927d0*real(i)/pm(3) alpha=pm(7)*sigma"2 cq=cq+cs/(cs+alpha) continue cf=dl/(cs+dl+d2*cq) an=cf bn=dimag(cf) return end MAIN PROGRAM subroutine displ implicit double precision (a-h,o-Z) INTEGER NDATA,NP,NPASS,IO integer ip, i, nin, nres, nterm common/mainl/nterm, tau common lpbparml/pm common/four/aoclcsig, aoclcres ,aclcsig, bclcsig, aclcres, bclcres dimension pm(20), cdi(400), cwork(400), cdr(400), p(10), step(10) dimension aclcsig(500), bclcsig(500), aclcres(500), bclcres(500) open(10, file=’ input4. csv’ ,status= ’old’ ) open(11,file=’tank5. csv’ ,status= ’old’ ) open(12,file=’input5.csv’,status=’old’) ope119(13,file=’tank6.csv’,status=’old’) Ip= write(*,*)’L,U,a,eb,R,ep,Dax,De,eg’ read(*.*) (Pm(i). i=1,ip) 300 format(8f10.4) 900 910 read(13,*) nterm read(13,*)aoclcsig do 900 k=l,nterm read(13,*)aclcsig(k) continue do 910 k=l, nterm read(13, *)bclcsig(k) continue read(ll, *)nterm read(1l,’)aoclcres do 920 k=l,nterm 170 read(ll,*)aclcres(k) 920 continue do 930 k= 1,nterm read(11,*)bclcres(k) 930 continue read(12,*)tau,nt C SEARCH INITIALIZATION do 10001=l,2 if(l.eq.2) pm(8)=0.0d0 P(l)= 0.08d0 p(2)=8.0d-4 STEP(1)=0.001d0 step(2)=1.0d-5 NP-2 NPASS=3 IO=3 START SEARCH CALL PATERN(NP,P, STEP,NPASS ,IO,COST) SEARCH COMPLETE, PRINT RESULTS PRINT 399, P(1), P(2), COST 399 FORMAT(F10.3,10X,F10.3) OPEN (UNIT=30,FILE=’FOR35.DAT’,STATUS=’NEW’) if(l.eq.1) write(*,*)’diffusion is not zero’ if(l.eq.2) write(*,*)’diffusion is zero’ WRITE (30,"') P(l),P(2),COST 399 FORMAT(E10.3,5X,F6.3,5X,E10.3,5x,E10.3) 1000 continue STOP END 0000 0000 THIS FILE IS A PAIR OF SUBROUTINES WRITTEN TO BE COMPATIBLE WITH THE OPTIMIZATION SUBROUTINE PATERN. THEY SIMULATE A PROCESS USING DISCRETE DIFFERENCE EQUATIONS AND COMPARE THE SIMULATION OUTPUT WITH THE ACTUAL OUTPUT (READ IN THROUGH A DATA FILE), CALCULATING AN ERROR OR "COST” ASSOCIATED WITH THAT - SIMULATION. PATERN USES THESE SUBROUTINES ITERATIVELY IN ORDER TO FIND THE OPTIMUM SET OF TRANSFER FUNCTION PARAMETERS TO FIT THE DATA. 000000000000000 SUBROUTINE PROC(P,COST) implicit double precision (a-h,o-z) dimension p(10) 000 40 171 common/pbparml/pm(20) common/main3/error Pm(2) =p(l) Pm(7) =P(2) call precurl cost=error RETURN END SUBROUTINE BOUNDS(P,IOUT) implicit double precision (a-h,o-z) DIMENSION P(10), STEP(10) IOUT=0 IF(P(1).LE.0.0) iout=1 if(p(2).le.0.0) iout=] RETURN END subroutine precurl implicit double recision (a-h,o-z) common /main3 err common/mainl/nterm,tau common/pbparml/ pm(20) common/four/aoclcsig,aoclcres,aclcsig,bclcsig,aclcres,bclcres dimension aclcsig(50 ),bclcsig(500),aclcres(500),bclcres(500) dimension aclc(400),bclc(400) call clccoel(pm,tau,nterm,aoclcsig,aclcsig,bclcsig ,aoclc,aclc,bclc) . x=2.0d0*(aoclcres/2.0d0-aoclc/2.0d0)**2 y=2.0d0*(aoclcres/2.0d0)**2 do 40 i=1,nterm x=x+(aclcres(i)-aclc(i))**2+(bclcres(i)-bclc(i))**2 y=y+aclcres(i)**2+bclcres(i)**2 continue err = sqrt(x/y) return end subroutine clccoel(pm,tau,nterm,aoin,ain,bin,aoclc,aclc,bclc) implicit double (precision (a-h,o-z) dimension pm(2 ),ain(400),bin(400),aclc(400),bclc(400) data pi/3.141593d0/ call transfl(0,0.0d0,pm,an,bn) aoclc=aoin*an do 10 n=1,nterm w=float(n)*pi/tau call transfl(1,w,pm,an,bn) aclc(n) =ain(n)*an+bin(n)*bn bclc(n)=bin(n)*an-ain(n)*bn if(an**2+bn**2.lt.1.0d-8) then 172 nt = 11 return endif 10 continue nt =nterm return end subroutine transfl(ictl,w,pm,an,bn) implicit double precision (a-h,o-z) dimension pm(20) save dl,d2,d3,d4,d5,e1 complex*16 cs,cphi,cex,coth,cq,cf if(ictl.eq.0) then e1=pm(8)*pm(6) dl=Pm(7)/(Pm(l)*Pm(2)) d2=pm(l)/pm(2) d3= 91*Pm(3)/(Pm(4)*pm(5)) if(pm(8). berg). 0. 0d0) then d4=0 else d4=pm(6)/el endif d5=pm(5) endif if(w.eq.0.0d0) then an=1.0d0 bn=0.0d0 return endif cs=dcmplx(0.0d0,w) if(d4.eq.0.0d0) then cf=cdexp(1.0d0/(2.0d0*d1)*(1.0d0- S cdsqrt(1.0d0+4*dl*d2*cs))) else cphi=d5*cdsqrt(cs*d4) cex=cdexp(-2.0d0*cphi) coth= (l. 0d0+cex)/(1.0d0-cex) cq= d3*(cphi*coth- 1. 0d0) cf= cdexp(l. 0dO/(2. 0d0*d1)*(1. 0d0-c.dsqrt(1 0d0+4*d1*d2 $ *(cs+cq)))) endif an=cf bn=dimag(cf) return end Appendix C Appendix C: Data Tables Table C-l Data from Figure 5-1 Experimental Predicted Current(Amps) Field Strength (Gauss) Field Strength (Gauss) 0 0 0 0.17 10 0.25 18 0.34 20 0.4 31 0.51 30 0.68 40 0.75 42 0.85 50 1 60 1.03 60 1.2 70 1.25 80 1.37 80 1.5 90 1.54 90 1.71 100 1.75 100 1.88 110 2 120 2.05 120 2.22 130 2.25 136 2.39 140 2.5 150 2.56 150 2.73 160 2.75 162 2.9 170 3 180 3.08 180 3.25 196 3.25 190 3.42 200 3.5 209 3.59 210 3.75 221 3.76 220 3.93 230 4 243 4.1 240 4.25 260 4.27 250 173 174 Table C-l (continued) Data from Figure 5-1 Experimental Predicted Current(Amps) Field Strength (Gauss) Field Strength (Gauss) 4.44 260 4.5 271 4.61 270 4.75 284 4.78 280 4.9 297 4.96 290 5.13 300 175 Table C-2 Data from Figure 5 =2 Dimensionless Experimental Predicted Position Field Strength (Gauss) Field Strength (Gauss) 0.000 106.00 ' 106.00 0.033 106.00 0.067 105.98 0.100 105.96 0.133 105.93 0.167 105.90 0.200 105.85 0.233 105.79 0.267 105.71 0.300 105.63 0.333 105.00 105.52 0.367 105.40 0.400 105.26 0.433 105.08 0.467 104.88 0.500 104.63 0.533 104.33 0.567 103.97 0.600 103.53 0.633 102.99 0.667 100.00 102.31 0.700 101.45 0.733 100.34 0.767 g 98.90 0.800 97.00 0.833 90.00 94.44 0.867 90.98 1.000 45.000 1.133 23.67 1.167 18.18 1.200 14.13 1.233 11.16 1.267 8.95 1.300 7.30 1.333 6.03 1.367 5.05 1.400 4.28 1.433 3.67 1.467 3.17 1.500 15.00 2.76 176 Table C-3 Data from Figure 5-3 Field Strength (Gauss) at which regimes first appear Gas Random Chain Chain—channel Destabilized Frozen Velocity ‘ (cm/s) 0. 6 10 50 60 80 l 10 1.1 10 50 75 110 180 2 20 80 110 180 270 Table C-4 Data from Figure 5-4 Field Strength (Gauss) at which regimes first appear Gas Random Chain Chain-channel Destabilized Frozen Velocity (cm/s) 0.6 10 40 90 110 150 l. 1 20 60 90 120 300 2 30 90 1 10 130 Table C-S Data from Figure 5 -5 Field Strength (Gauss) at which regimes first appear Gas Random Chain Chain-channel Destabilized Frozen Velocity (cm/s) 0.6 20 60 100 130 240 1.1 30 60 100 150 300 2 40 80 120 180 177 Table 06 Data from Figure 5-7 Field Strength Gas Void Fraction at Gas velocities of (Gauss) 0.6 cm/s 1.1 cn1/s 2.0 cm/s 0 0.0431 0.0719 0aw095 50 0.0294 75 0.0610 0.0929 100 0.0083 150 0.0223 0.0536 0.1011 200 0.0659 225 0.1481 300 0.0060 0.2410 0.1400 Table C-7 Data from Figure 5-8 Field Strength Gas Void Fraction at Gas velocities of (Gauss) 0.6 cn1/s 1.1 cm/S 2.0 cm/s 0 0.0421 0.0702 0.1001 50 0.0401 75 0.0594 0.0922 100 0.0520 150 0.0527 0.0661 0.0879 200, 0.1698 225 0.1294 300 0.0066 0.1400 0.2288 178 Table C—8 Data from Figure 5-9 Field Strength Gas Void Fraction at Gas velocities of (Gauss) 0.6 chs 1.1 cn1/s 2.0 cn1/s 0 0.0398 0.0652 0.0994 50 0.0378 75 0.0608 0.0790 100 0.0381 150 0.0335 0.0600 0.0905 200 0.0240 225 0.1087 300 0.0355 0.0558 0. 1255 Table C-9 Data from Figure 5-10 Gas velocities 0.6 cm/s 1.1 cn1/s 2.0 cm/s Field void 95% void 95% void 95% Strength fraction confidence fraction confidence fraction confidence (Gauss) interval interval interval 0 0.0431 0.0104 0.0719 0.0026 0.1095 0.0068 50 0.0294 0.0129 - 75 0.0610 0.0111 0.0929 0.012 100 0.0083 0.0068 150 0.0223 0.0147 0.0536 0.0221 0.1011 0.0295 200 ' 0.0659 0.0248 225 0.1481 0.0176 300 0.0060 0.0186 0.241 0.0286 0.14 0.0707 179 Table C-10 Data from Figure 5-11 Gas velocities 0.6 cm/s 1.1 chs 2.0 cm/s Field void 95 % void 95 % ' void 95 % Strength fraction confidence fraction confidence fraction confidence (Gauss) interval interval interval 0 0.0421 0.0109 0.0702 0.0075 0.1001 0.0084 50 0.0401 0.0147 75 0.0594 0.0044 0.0922 0.0167 100 0.0520 0.0268 150 0.0527 0.0223 0.0661 0.0088 0.0879 0.0181 200 0.1698 0.0244 225 0. 1294 0.0568 300 0.0066 0.0160 0.14 0.0925 0.2288 0.0798 Table C-ll Data from Figure 5-12 Gas velocities 0.6 cm/S 1.1 cn1/s 2.0 cn1/s Field void 95 % void 95 % void 95 % Strength fraction confidence fraction confidence fraction confidence (Gauss) interval interval interval 0 0.0398 0.0044 0.0652 0.0128 0.0994 0.0065 50 0.0378 0.0115 75 0.0608 0.0081 0.0790 0.0129 100 0.0381 0.0157 150 0.0335 0.0074 0.0600 0.0162 0.0905 0.0137 200 0.0240 0.0138 225 0.1087 0.0175 300 0.0355 0.0268 0.0558 0.0120 0.1255 0.0438 180 Table C-12 Data from Figure 5-13 Gas Void Fraction Measured. By Two Techniques Wham—Mans: 6, (Probe) 6, (Valve) 6, (Probe) ’ 5, (Valve) 0.0210 0.0350 0.0170 0.1400 0.0360 0.0650 0.0330 0.1930 0.0550 0.0900 0.0440 0.0060 Table 013 Data from Figure 5-14 Experimental Data from Probe 1 Time(s) Voltage Time(s) Voltage Time(s) Voltage (Volts) (Volts) (Volts) 0 0 5.3 0.10014 10.6 0.00794 0.1 0.00313 5.4 0.10014 10.7 0.00794 0.2 0.00951 5.5 0.10489 10.8 0.00632 0.3 0.01745 5.6 0.09695 10.9 0.00794 0.4 0.03496 5.7 0.08581 11 0.00951 0.5 0.0461 5.8 0.08738 11.1 0.00951 0.6 0.05879 5.9 0.089 11.2 0.00794 0.7 0.06993 6 0.09057 11.3 0.00794 0.8 0.10489 6.1 0.08262 11.4 0.00632 0.9 0.12872 6.2 0.07787 11.5 0.00475 1 0.14461 6.3 0.07149 11.6 0 1.1 0.15737 6.4 0.07631 11.7 0.00475 1.2. 0.16368 6.5 0.07312 11.8 0.00632 1.3 0.16687 6.6 0.06517 11.9 0.00632 1.4 0.16368 6.7 0.06036 12 0.00632 1.5 0.17006 6.8 0.0550 12.1 0.00475 1.6 0.18276 6.9 0.04922 12.2 0.00475 1.7 0.19389 7 0.05879 12.3 0.00313 1.8 0.19708 7.1 0.05404 12.4 0 1.9 0.19233 7.2 0.04922 14.5 0.01074 2 0.1907 7.3 0.0461 14.6 0.01074 2.1 0.19389 7.4 0.04291 14.7 0.00917 2.2 0.20184 7.5 0.04291 14.8 0.00766 2.3 0.19865 7.6 0.0556 14.9 0.00766 2.4 0.19708 7.7 0.04128 15 0.00458 2.5 0.20027 7.8 0.03177 15.1 0.00609 2.6 0.20184 7.9 0.03015 15.2 0.00609 2.7 0.20184 8 0.03015 15.3 0.00766 2.8 0.20027 8.1 0.02858 15.4 0.00609 2.9 0.20027 8.2 0.02702 15.5 0.00609 181 Table C-13 (continued) Data from Figure 5-14 Experimental Data from Probe 1 Time(s) Voltage Time(s) Voltage Time(s) Voltage (Volts) (Volts) (Volts) 3 0.20184 8.3 0.02383 15.6 0.00609 3.1 0.19865 8.4 0.02858 15.7 0.00458 3.2 0.19233 8.5 0.02858 15.8 0.00458 3.3 0.18751 8.6 0.02858 15.9 0.00301 3.4 0.18276 8.7 0.02383 16 0 3.5 0.17482 8.8 0.0222 3.6 0.17163 8.9 0.0222 3.7 0.16687 9 0.02064 3.8 0.16212 9.1 0.01908 3.9 0.15737 9.2 0.01908 4 0.15255 9.3 0.01589 4.1 0.14942 9.4 0.01426 4.2 0.13666 9.5 0.01589 4.3 0.13985 9.6 0.01426 4.4 0.11921 9.7 0.0127 4.5 0.11127 9.8 0.01113 4.6 0.10645 9.9 0.01145 4.7 0.1144 10 0.0127 4.8 0.13347 10.1 0.01113 4.9 0.11602 10.2 0.0127 5 0.10489 10.3 0.01589 5.1 0.1017 10.4 0.0127 5.2 0.1017 10.5 0.00951 Experimental Data from Probe 2 Time(s) Voltage Time(s) Voltage Time(s) Voltage (Volts) (Volts) (Volts) 2 0 7.3 0.12414 12.6 0.0184 2.1 0.00766 7.4 0.12414 12.7 0.01532 2.2 0.01224 7.5 0.12263 12.8 0.01532 2.3 0.01532 7.6 0.11497 12.9 0.00766 2.4 0.01991 7.7 0.10882 13 0.01224 2.5 0.02606 7.8 0.10267 13.1 0.01074 2.6 0.03372 7.9 0.09965 13.2 0.01074 2.7 0.04289 8 0.09808 13.3 0.01074 2.8 0.05362 8.1 0.09657 13.4 0.01532 2.9 0.06895 8.2 0.09657 13.5 0.01375 3 0.08427 8.3 0.09193 13.6 0.01532 3.1 0.09808 8.4 0.08891 13.7 0.01224 3.2 0.10731 8.5 0.08125 , 13.8 0.01074 3.3 0.11956 8.6 0.07661 13.9 0.00917 3.4 0.12872 8.7 0.07818 14 0.01532 182 Table C—l3 (continued) Data from Figure 5-14 Experimental Data from Probe 2 Time(s) Voltage Time(s) Voltage Time(s) Voltage (Volts) (Volts) (Volts) 3.5 0.13488 8.8 0.08125 0.00917 3.6 0.13795 8.9 0.08125 0.00766 3.7 0.14254 9 0.0751 0.00917 3.8 0.14103 9.1 0.07052 0.01074 3.9 0.14411 9.2 0.07052 4 0.1502 9.3 0.07661 4.1 0.15478 9.4 0.07661 4.2 0.15786 9.5 0.06744 4.3 0.15786 9.6 0.06436 4.4 0.15635 9.7 0.06436 4.5 0.15786 9.8 0.06286 4.6 0.15786 9.9 0.05978 4.7 0.15943 10 0.05212 4.8 0.15328 10.1 0.05212 4.9 0.16094 10.2 0.04747 5 0.16401 10.3 0.04446 5.1 0.16251 10.4 0.04596 5 .2 0.15478 10.5 0.04138 5 .3 0.15478 10.6 0.05362 5.4 0.15478 10.7 0.04289 5.5 0.15478 10.8 0.03981 5.6 0.1502 10.9 0.03523 5.7 0.15177 11 0.03523 5 .8 0.15328 11.1 0.03981 5.9 0.15177 11.2 0.04138 6 0.1502 11.3 0.0383 6.1 0.1471 11.4 0.0368 6.2 0.14562 11.5 0.03523 6.3 0.14411 11.6 0.02907 6.4 0.13946 11.7 0.02606 6.5 0.13337 11.8 0.02757 6.6 0.1318 11.9 0.02907 6.7 0.13029 ‘ 12 0.02757 6.8 0.12263 12.1 0.02606 6.9 0.12414 12.2 0.01991 7 0.12722 12.3 0.01683 7.1 0.13029 12.4 0.0184 7.2 0.12872 12.5 0.0184 1% Table C-l4 Data from Figure 5-15 Ca+2 Tracer Concentration (g/L) Two Parameter Fit Experimental One Parameter Fit Time (min) 38518631 876 321100999988888877 10009999%888M&888888777777777777W 99 232 386556803715 505173 63 752075319753198 999 99 8 man Benunmnmunmmwmnmmemmmwuuunmeeewnnnew nwmn 999 000000000000000000000000000000000000000000000 0L21¢101890L23AifllSQQLZlkiflTSQQLZl4561890LZ3. l111111111222222222233333333334444 184 Table 0141 (continued) Data from Figure 5- 15 Ca+2 Tracer Concentration (g/ L) Time (min) Experimental One Parameter Fit Two Parameter Fit 45.0 8.76 8.56 8.77 46.0 8.77 8.54 ' 8.77 47.0 8.76 8.53 8.77 48.0 8.77 8.51 8.77 49.0 8.74 8.50 8.77 50.0 8.77 8.49 8.77 51.0 8.77 8.47 8.77 52.0 8.76 8.46 8.77 53.0 8.76 8.45 8.77 54.0 8.74 8.43 8.77 55.0 8.76 8.42 8.77 56.0 8.74 8.41 8.77 57.0 8.74 8.40 8.77 58.0 8.74 8.39 8.77 59.0 8.76 8.38 8.77 60.0 8.74 8.37 8.77 Table C-15 Data from Figure 5-16 Agitation Rate (rpm) Diffusion Coefficient x 10‘ (cm’ls) 0 2.51 50 5.13 100 5.66 240 5.6 290 5.72 185 Table C-16 Data from Figure 5-17 Molecular Excluded Void Volume Distribution for 0% beads Dia‘rAneter Experimental Data Curve Fit Data ( ) 4 0 0.0025 10 0.0398 0.0188 12 0.0748 0.0255 38 0.1340 0.1446 51 0.2282 0.2215 61 0.2487 0.2845 90 0.4586 0.4623 118 0.5916 0.6022 204 0.8466 0.8230 270 0.9796 0.8826 560 0.8910 0.9364 Molecular Excluded Void Volume Distribution for 5 % beads Dialrkneter Experimental Data Curve Fit Data ( ) 4 0 0.0011 10 0.0741 0.0194 38 0.1388 0.1088 51 0.1533 0.1601 61 0.2340 0.2030 90 0.3202 0.3332 118 0.4403 0.4507 204 0.6746 0.6829 270 0.7872 0.7622 560 0.8341 0.8474 Molecular Excluded Void Volume Distribution for 50% beads Dia‘IKneter Experimental Data Curve Fit Data ( ) 4 0 0 10 0.0421 0.0380 38 0.0757 0.0790 51 0.1074 0.1002 61 0.1672 0.1184 90 0.1800 0.1769 118 0.2111 0.2341 204 0.3402 0.3686 270 0.4355 0.4258 560 0.5287 0.4971 186 Table C-17a Data from Figure 5-18a Pore Porosig for Beads With Various Percentage Magnetite Diameter (A) 0 5% 50% 4 0.9339 0.8464 0.4971 10 0.9176 0.8281 0.4591 12 0.9109 0.8474 0.4971 38 0.7918 0.7386 0.4181 51 0.7148 0.6873 0.3969 61 0.6519 0.6445 0.3786 90 0.4741 0.5142 0.3202 118 0.3342 0.3967 0.2630 204 0.1133 0.1646 0.1284 270 0.0538 0.0852 0.0713 560 0.0000 0.0000 0.0000 Table C-17b Data from Figure 5-18b Pore Percent of Total Pore Volume Accessible (%) Diameter (A) 0% 5 % 50% 4 100.0 100.0 100.0 8 98.7 98.3 93.8 10 98.0 97.7 92.4 12 97.3 97.0 91.4 38 84.6 87.2 84.1 51 76.3 81.1 79.8 61 69.6 76.1 76.2 90 50.6 60.7 64.4 118 35.7 46.8 52.9 204 12.1 19.4 25.8 270 5.7 10.0 14.3 560 0.0 0.0 0.0 Table C-18 Data from Figure 5-19 187 Predicted Data Time(s) Voltage Voltage Time(s) Voltage Time(s) Voltage 0 0.00088 5.2 0.14232 10.5 0.0601 - 15.8 0.00913 0.1 0.00095 5.3 0.1421 10.6 0.0580 15.9 0.00878 0.2 0.00091 5.4 0.1419 10.7 0.0559 16 0.00844 0.3 0.00077 5.5 0.14172 10.8 0.05378 16.1 0.00812 0.4 0.00053 5.6 0.14153 10.9 0.05168 16.2 0.0078 0.5 0.00021 5.7 0.14134 11 0.04962 16.3 0.0075 0.6 0.00015 5.8 0.14109 11.1 0.04764 16.4 0.0072 0.7 0.00053 5.9 0.14077 11.2 0.04576 16.5 0.0069 0.8 0.00087 6 0.14035 11.3 0.04398 16.6 0.00661 0.9 0.00114 6.1 0.1398 11.4 0.04232 16.7 0.00632 1 0.0013 6.2 0.13909 11.5 0.04078 16.8 0.00604 1.1 0.00131 6.3 0.13822 11.6 0.03936 16.9 0.00576 1.2 0.00116 6.4 0.13719 11.7 0.03805 17 0.00549 1.3 0.00088 6.5 0.13599 11.8 0.03684 17.1 0.00523 1.4 0.000 6.6 0.13464 11.9 0.03573 17.2 0.00496 1.5 0.00015 6.7 0.13315 12 0.03469 17.3 0.0047 1.6 0.00072 6.8 0.13152 12.1 0.03372 17.4 0.00444 1.7 0.00124 6.9 0.12979 12.2 0.03281 17.5 0.00417 1.8 0.0016 7 0.12797 12.3 0.03193 17.6 0.00389 1.9 0.00168 7.1 0.12606 12.4 0.03109 17.7 0.00359 2 0.00134 7.2 0.12409 12.5 0.03028 17.8 0.00329 2.1 0.00046 7.3 0.12206 12.6 0.0295 17.9 0.00297 2.2 0.00108 7.4 0.11997 12.7 0.02874 18 0.00263 2.3 0.00338 7.5 0.11783 12.8 0.02801 18.1 0.00229 2.4 0.00654 7.6 0.11563 12.9 0.02731 18.2 0.00195 2.5 0.0106 7.7 0.11338 13 0.02663 18.3 0.0016 2.6 0.0156 7.8 0.11107 13.1 0.02597 18.4 0.00127 2.7 0.02152 7.9 0.10872 13.2 0.02534 18.5 0.00095 2.8 0.0283 8 0.10632 13.3 0.02473 18.6 0.00067 2.9 0.03586 8.1 0.10389 13.4 0.02413 18.7 0.00041 3 0.04407 8.2 0.10145 13.5 0.02355 18.8 0.0002 3.1 0.05278 8.3 0.09901 13.6 0.02297 18.9 0.00003 3.2 0.06182 8.4 0.09659 13.7 0.02238 19 0.00009 3.3 0.07101 8.5 0.09423 13.8 0.02178 19.1 0.00017 3.4 0.08015 8.6 0.09193 13.9 0.02116 19.2 0.0002 3.5 0.08905 8.7 0.08978 14 0.02052 19.3 0.0002 3.6 0.09754 8.8 0.08763 14.1 0.01985 19.4 0.00016 3.7 0.10547 8.9 0.08566 14.2 0.01915 19.5 0.0001 3.8 0.11272 9 0.08382 14.3 0.01843 19.6 0.00003 3.9 0.11918 9.1 0.0821 14.4 0.01769 19.7 0.00005 4 0.1248 9.2 0.08051 14.5 0.01693 19.8 0.00012 4.1 0.12956 9.3 0.07901 14.6 0.01616 19.9 0.00018 188 Table C-18 (continued) Data from Figure 5 - 19 Experimental Data Time(s) Voltage Time(s) Voltage Time(s) Voltage 2.3 0 7.6 0.11401 12.9 0.02989 2.4 0.00426 7 .7 0.10829 13 0.02989 2.5 0.00712 7 .8 0.10257 13.1 0.02849 2.6 0.00998 7.9 0.09691 13.2 0.02849 2.7 0.01565 8 0.09831 13.3 0.02703 2.8 0.01991 8.1 0.09977 13.4 0.02703 2.9 0.01851 8.2 0.10257 13.5 0.02563 3 0.02563 8.3 0.10257 ‘ 13.6 0.02563 3.1 0.03701 8.4 0.10257 13.7 0.02563 3.2 0.05131 8.5 0.10117 13.8 0.02563 3.3 0.06556 8.6 0.09977 13.9 0.02423 3.4 0.07694 8.7 0.09119 14 0.02423 3.5 0.08692 8.8 0.08546 14. 1 0.02423 3.6 0.09691 8.9 0.08546 14.2 0.02423 3.7 0.10403 9 0.08692 14.3 0.02137 3.8 0.10829 9.1 . 0.08692 14.4 0.01991 3.9 0.11541 9.2 0.08692 14.5 0.01851 4 0.11967 9.3 0.08692 14.6 0.01138 4.1 0.11967 9.4 0.08406 14.7 0.0171 4.2 0.1268 9.5 0.0812 14.8 0.01138 4.3 0.13112 9.6 0.08546 14.9 0.00998 4.4 0.13252 9.7 0.08400 15 0.00998 4.5 0.13678 9.8 0.08546 15 . 1 0.01278 4.6 0.13824 9.9 0.0812 15 .2 0.01565 4.7 0.13538 10 0.07122 15 .3 0.01851 4.8 0.13398 10.1 0.06842 15 .4 ' 0.01991 4.9 0.13678 10.2 0.07554 15 .5 0.0171 5 0.13678 10.3 0.07122 15.6 0.01424 5 .1 0.13678 10.4 0.06696 15 .7 0.01424 5.2 0.13964 10.5 0.0627 15 .8 0.01278 5.3 0.1425 10.6 0.05844 15 .9 0.01278 5.4 0.1439 10.7 0.05558 16 0.01278 5.5 0.14676 10.8 0.05558 16.1 0.01138 5 .6 0.14536 10.9 0.05558 16.2 0.01138 5 .7 0.1439 11 0.05412 16.3 0.00998 5 .8 0.13964 11.1 0.05271 16.4 0.00852 5 .9 0.12826 11.2 0.05271 16.5 0.00712 6 0.1268 11.3 0.05131 16.6 0.00852 6.1 0.12966 11.4 0.04985 16.7 0.00712 6.2 0.12399 11.5 0.04845 16.8 0.00712 6.3 0.1268 11.6 0.04845 16.9 0.00712 6.4 0.13112 11.7 0.04699 17 0.00712 6.5 0.13252 11.8 0.04273 17.1 0.00712 6.6 0.12399 11.9 0.03847 17.2 0.00712 6.7 0.12253 12 0.03135 17.3 0.00566 6.8 0.12399 12.1 0.02703 17.4 0.00566 189 Table C-18 (continued) Data from Figure 5-19 Experimental Data Time(s) Voltage Time(s) Voltage Time(s) Voltage 6.9 0.12826 12.2 0.02703 17.5 0.00566 7 0.13112 12.3 0.02989 ' 17.6 0.00566 7.1 0.13112 12.4 0.02423 17.7 0.00566 7.2 0.12399 12.5 0.03135 17.8 0.00566 7.3 0.12253 12.6 0.02989 17.9 0.00998 7.4 0.11827 12.7 0.02989 7.5 0.11255 12.8 0.02989 Table C-19 Data from Figure 5-20 ' «_ ‘ “1‘1 "xi ‘ h. 0 ' ,- Gas U = .71 cn1/s U = 6.44 cn1/s Velocity (cm/s) Pew Pe,", Pew, Pe,", 0.6 0.1081 0.0850 0.0618 0.0544 1.1 0.0543 0.0504 0.0395 0.0347 2.0 0.0545 0.0499 0.0291 0.0267 Table C-20 Data from Figure 5-21 Field Strength Peclet Numbers at Gas Velocities of (Gauss) 0.6 cn1/s 1.1 cm/s 2.0 cn1/s 0 0.1228 0.1549 0.0651 50 0.2098 75 0.2279 0. 1834 100 0.7678 150 0.0910 0.4900 0.1837 200 0.0746 225 0.1208 300 0.1603 0.1535 0.2080 Table C-21 Data from Figure 5-22 190 Field Strength Peclet Numbers at Gas Velocities of (Gauss) 0.6 cn1/s 1.1 cm/s 2.0 cm/s 0 0.1023 0.0675 ' 0.0490 50 0.1446 75 0.1595 0.1431 100 0.1153 150 0.1252 0.1089 0.0771 200 0.1674 225 0.1106 300 0.1160 0.1109 0.1432 Table C-22 Data from Figure 5-23 Field Strength Peclet Numbers at Gas Velocities of (Gauss) 0.6 cm/s 1.1 cm/s 2.0 cn1/s 0 0.1740 0.0873 0.0900 50 0.1809 75 0.1419 0.2155 100 0.1340 150 0.1677 0.1218 0.1036 200 0.1038 g 225 0.1242 300 0.1421 0.1077 0.0713 Table C-23 Data from Figure 5-24 191 Gas velocities 0.6 cn1/s 1.1 cm/s 2.0 cm/s Field Fe 95 % Fe 95 % Pe 95 % Strength number confidence number confidence number confidence (Gauss) interval interval interval 0 0.1228 0.0386 0.1549 0.0343 0.0651 0.0243 50 0.2098 0.2482 75 0.2279 0.0574 0.1834 0.0756 100 0.7678 0.3749 150 0.0910 0.0234 0.4900 0.2201 0.1837 0.1311 200 0.0746 0.0347 225 0. 1208 0.0425 300 0.1603 0.0595 0.1535 0.0876 0.2080 0.1616 Table C-24 Data from Figure 5-25 Gas velocities 0.6 cm/s 1.1 cn1/s 2.0 cm/s Field Pe 95 % Fe 95 % Strength number confidence number confidence (Gauss) interval interval 0 0.1022 0.0230 0.0675 0.0230 50 0.1446 0.0882 75 0.1595 0.0957 100 0.1153 0.0832 150 0.1252 0.0234 0.1089 0.0354 200 0. 1674 0.3280 225 0.1106 0.0207 300 0.1160 0.0659 0.1109 0.0650 0.1432 0.0670 Pe 95 % number confidence interval 0.0490 0.0124 0. 1431 0.0938 0.0771 0.0143 Table C-25 Data from Figure 5-26 192 Gas velocities 0.6 chs 1.1 cm/S 2.0 cm/s Field void 95 % void 95 % - void 95 % Strength fraction confidence fraction confidence fraction confidence (Gauss) interval interval interval 0 0.1740 0.0126 0.0873 0.0276 0.0900 0.0309 50 0.1809 0.1436 75 0.1419 0.0534 0.2155 0.0775 100 0.1340 0.0809 150 0.1677 0.0374 0.1218 0.0665 0.1036 0.0108 200 0.1038 0.0171 225 0. 1242 0.0363 300 0.1421 0.1066 0.1077 0.0978 0.0713 0.0005 Table 026 Data from Figure 5-27 Dimensionless Aqueous 0; Concentration (mg/L) Position Predicted Experimental 0.000 0.2525 0.059 0.3825 0.3769 0.118 0.5100 ‘ 0.176 0.6348 0.6055 0.235 0.7572 0.294 0.8770 0.353 0.9945 0.9580 0.412 1.1095 0.471 1.2221 0.529 1.3324 1.3632 0.588 1.4403 0.647 1.5460 0.706 1.6494 1.6687 0.765 1.7506 0.824 1.8495 0.882 1.9454 1.9415 0.941 2.0338 1.000 2.0850 193 Table C-27 Data from Figure 5-28 Field Strength kLa Values at Gas Velocities of (Gauss) 0.6 cmls 1.1 cmls 2.0 cn1/s 0 0.0188 0.0298 0.0393 75 0.0240 0.0391 0.0527 150 0.0198 0.0328 0.0454 225 0.0223 0.0342 0.0430 300 0.0239 0.0363 0.0456 Table C-28 Data from Figure 5-29 Field Strength kLa Values at Gas Velocities of (Gauss) 0.6 cn1/s 1.1 cmls 2.0 cmls 0 0.0184 0.0289 0.0397 75 0.0183 0.0290 0.0381 150 0.0180 0.0291 0.0394 225 0.0196 0.0306 0.0396 300 0.0203 0.0317 0.0387 Table C-29 Data from Figure 5-30 Field Strength kLa Values at Gas Velocities of (Gauss) 0.6 cn1/s 1.1 cmls 2.0 cn1/s 0 0.0188 0.0296 0.0397 75 0.0195 0.0337 0.0460 150 0.0200 0.0348 0.0462 225 0.0188 0.0292 0.0389 300 0.0177 0.0278 0.0348 194 Table C-30 Data from Figure 5-31 Gas velocities 0.6 cm/s 1.1 cn1/s 2.0 cm/s Field 95% 95% 95% Strength confidence confidence confidence (Gauss) kLa interval kLa interval kLa interval 0 0.0188 0.0014 0.0298 0.0016 0.0393 0.0021 75 0.0240 0.0029 0.0391 0.0023 0.0527 0.0027 150 0.0198 0.0038 0.0328 0.0026 0.0454 0.0031 225 0.0223 0.0023 0.0342 0.0021 0.0430 0.0027 300 0.0239 0.0010 0.0363 0.0027 0.0456 0.0028 Table C-31 Data from Figure 5-32 Gas velocities 0.6 cmls 1.1 chs 2.0 cmls Field 95% 95 % 95 % Strength confidence confidence confidence (Gauss) kLa interval kLa interval kLa interval 0 0.0184 0.0018 0.0289 0.0002 0.0397 0.0004 75 0.0183 0.0004 0.0290 0.0009 0.0381 0.0012 150 0.0180 0.0010 0.0291 0.0005 0.0394 0.0009 225 0.0196 0.0012 0.0306 0.0022 0.0396 0.0011 300 0.0203 0.0032 0.0317 0.0015 0.0387 0.0032 Table C-32 Data from Figure 5-33 Gas velocities 0.6 cm/s 1.1 cmls 2.0 cmls Field 95 % 95% 95 % Strength confidence confidence confidence (Gauss) kLa interval kLa interval kLa interval 0 0.0188 0.0009 0.0296 0.0008 0.0397 0.0017 75 0.0195 0.0007 0.0337 0.0009 0.0460 0.0009 150 0.0200 0.0006 0.0348 0.0024 0.0462 0.0013 225 0.0188 0.0010 0.0292 0.0006 0.0389 0.0019 300 0.0177 0.0018 0.0278 0.0018 0.0348 0.0010 Table C-33 Data from Figure 5-34 195 PeWe = 20 Pew = 50 1) <15" Pe,“, 1) <12," Pe,", .9987 .85 8E-08 18.20 .9980 .380E-08 45 .02 .9953 .886E-07 18.28 .9935 .394E—07 44. 83 .9934 .225E—06 18.13 .9896 .999E-07 44.64 .9907 .356E—06 18.19 .9869 .159E-06 44.52 .9817 .144E—05 18.01 .9743 .643E-06 43.94 .9727 .327E—05 17.84 .9618 .147E-05 43.36 .9699 .438E—05 17.74 .9563 .197E—05 43.10 .9671 .468E-05 17.73 .9540 .210E—05 43.01 .9640 .5 87E—05 17.67 .9496 .264E—05 42. 80 .9634 .5 88E—05 17.66 .9489 .264E—05 42.77 .9554 .926E—05 17.50 .9375 .418E—05 42.25 .9520 .106E-04 17.44 .9331 .478E—05 42.04 .9513 .115E-04 17.41 .9312 .520E—05 41.95 .9490 .119E-04 17.39 .9287 .540E-05 41.86 .9471 .135E-04 17.34 .9258 .610E-05 41.71 .9436 .150E-04 17.28 .9213 .681E-05 41.51 .9388 .185E-04 17.18 .9145 .841E-05 41.18 .9359 .202E-04 17.13 .9107 .919E=OS 41.01 .9307 .244E—04 17.02 .9032 .111E-04 40.67 .9228 .311E-04 16.86 .8921 .143E-04 40.16 .9165 .372E-04 16.74 .8836 .171E—04 39.76 .9148 .397E—04 16.71 .8809 .183E-04 39.64 .9070 .470E-04 16.56 .8708 .217E-04 39.18 .9063 .502E—04 16.54 . 8689 .231E—04 39.09 . 8784 . 876E-04 16.01 . 8322 .409E—04 37.40 .8758 .104E-03 15.94 . 8253 .488E-04 37.08 .8550 .160E-03 15.54 .7953 .760E-04 35.71 .8419 .166E-03 15.31 .7829 .790E-04 35.14 .8268 .277E-03 14.98 .7532 . 134E—03 33.77 .7996 .460E-03 14.45 .7108 .228E-03 31.83 .7412 .200E-02 13.25 .5869 .111E-02 26.16 196 Table C-33 (continued) Data from Figure 5-34 PeWe = 100 Pew, = 200 n <15", Pe,", 11 (Pm, Pe," .9975 .164E—08 98.12 .9969 . 815E-09 197.76 .9916 .170E-07 97.55 .9896 . 847E-08 196.34 .9870 .431E—07 97.07 .9836 .215E-07 195.13 .9834 .686E—07 96.72 .9797 .342E-07 194.32 .9673 .279E-06 95. 13 .9597 . l40E-06 190.34 .9515 .638E-06 93.56 .9403 .321E—06 186.52 .9444 .857E—06 92.88 .9321 .432E-06 184.85 .9418 .917E—06 92.61 .9291 .462E-06 184.23 .9361 .115E-05 92.03 .9217 .582E—06 182.81 .9354 .115E—05 91.97 .9212 .583E-06 182.65 .9210 .183E—05 90.53 .9037 .928E-06 179.20 .9156 .210E-05 90.00 .8971 .106E-05 177.90 .9134 .228E—05 89.76 .8940 .116E-05 177.29 .9092 .238E-05 89.40 .8878 .121E-05 176.09 .9063 .268E—05 89.09 . 8859 . 136E—05 175.64 .9003 .300E-05 88.50 . 8779 .153E-05 174.10 .8921 .371E-05 87.67 .8689 .189E-05 172.25 .8874 .406E-05 87.20 .8629 .207E—05 171.07 . 8782 .492E—05 86.28 . 8523 .252E-05 168.94 . 8646 .633E-05 84.93 . 8359 .325E-05 165.70 . 8539 .760E-05 83 . 87 . 8233 . 392E-05 163 . 16 .8505 .813E-05 83.54 .8192 .420E—05 162.35 .8482 .850E-05 83.31 .8045 .501E-05 159.44 .8382 .968E-05 82.32 .8014 .536E-05 158.81 .8356 .104E—04 82.06 .7482 .971E—05 148.20 .7907 .185E—04 77.60 .7378 .117E'04 146.16 .7817 .221E—04 76.72 .6948 .186E-04 137.58 .7450 .349E—04 73.06 .6773 . 195E-04 134.05 .7308 . 364E—04 71 .65 . 6346 . 339E—04 125.61 .6931 .625E-04 67.92 .5747 .600E-04 113.66 .6410 . 108E-03 62.76 .3929 .352E—03 77.56 .4828 .577E-03 47.14 .3929 .352E-03 1.00 197 Table C-33 (continued) Data from Figure 5-34 Pew, = 500 Pew, = 1000 1) in", Pe," I) <1)” Pe,“, .9864 .339E-08 491.73 .9835 . 170E-08 982.31 .9796 .862E-08 488.21 .9741 .433E-08 972.46 .9732 . 137E—07 485 . 14 .9664 .690E-08 965 .24 .9478 .564E—07 472.26 .9350 .285E-07 933. 86 .9227 . 130E—06 459.80 .9046 .664E—07 903.58 .9126 .176E-06 454.84 .8943 .896E—07 892.57 .9082 .188E—06 452.72 .8883 .963E-07 886.70 .8989 .238E—06 447.93 .8771 .122E—06 875.21 . 8984 .238E—06 447.61 . 8762 . 122E—06 874.61 . 8758 .382E—06 436.48 . 8494 . 197E-06 847.71 . 8677 .439E-06 432.28 . 8400 .226E-06 838.53 . 8648 .477E—06 430.78 . 8363 .246E-06 835 .32 .8539 .564E—06 425.38 .8235 .292E-06 821.79 .8506 .503E—06 423.97 .8106 .264E-06 809.66 . 8404. .637E—06 418.76 . 8024 .333E-06 800.96 .8323 .788E—06 414.63 .7983 .410E—06 796.75 .8235 .866E-06 410.34 .7865 .453E-06 785.09 . 8117 .105E-05 404.37 .7739 .552E-06 772.47 .7913 .137E-05 394.27 .7502 .721E—06 748. 88 .7763 .166E-05 386.64 .7325 .876E-06 731.23 .7705 . l78E-05 383. 87 .7267 .941E-06 725 .41 .7521 .214E—05 374.53 .7034 .114E-05 701.61 .7492 .229E—05 373.04 .7017 . 122E—05 700.42 .6821 .424E-05 339.56 .6205 .233E=05 619.15 .6727 .509E—05 334.98 .6153 .278E-05 614.22 .6215 .827E—05 309.47 .5596 .459E—05 557.90 .5942 . 885E-05 295 .59 .5179 .507E-05 516.30 .5517 . 155E—04 274.59 .4838 . 885E-05 482.35 .4838 .284E—04 240.64 .4118 .166E-04 410.59 .2851 .193E—03 141.58 .2127 .129E—03 211.79 J- 198 Table C-33 (continued) Data from Figure 5-34 Pew, = 1500 Pe,m = 2000 n <12," Pe," 1) <1", Pe," .9941 .108E-09 1489.38 .9961 . 813E-10 1987. 80 .9806 .113E-08 1469.31 .9789 .852E-09 1955.87 .9723 .289E—08 1457.62 .9669 .218E-08 1934.56 .9619 .462E—08 1441.35 .9586 .348E-08 1915.17 .9267 .192E—07 1388.67 .9185 .145E-07 1835 .46 . 8925 .449E-07 1337.56 . 8835 .340E—07 1765 .47 . 8816 .605E-07 1321.14 .8719 .460E—07 1740.95 . 8728 .652E-07 1308.63 . 8616 .496E—07 1722.52 . 8607 . 827E—07 1289.94 . 8476 .630E—07 1694.51 . 8594 . 828E-07 1288.41 . 8476 .630E-07 1693.90 . 8299 .134E—06 1243.88 .8145 .102E-06 1627.93 . 8201 . 154E—06 1229.50 . 8035 . l 18E-06 1606.36 .8182 .168E—06 1225.39 .8027 .128E—06 1601.79 . 8012 .200E—06 1201.01 .7838 .153E-06 1566.73 .7811 .182E—06 1170.04 .7548 . l41E—06 1509.05 .7739 .230E—06 1159. 83 .7541 .217E-06 1507.41 .7733 .282E-06 1159.25 .7503 . l78E-06 1500.45 .7600 .313E-06 1138.00 .7389 .241E-06 1475.52 .7473 .381E—06 1120.43 .7265 .294E-06 1452.44 .7234 .499E—06 1082.73 .6997 . 386E-06 1398.99 .7042 .608E-06 1053.98 .6792 .472E-06 1357.98 .6971 .655E—06 1043.30 .6726 .508E—06 1344.68 .6703 . 851E—06 1003.13 .6445 .663E-06 1288.56 .6693 .798E—06 1002.71 .6431 . 623E—06 1284.72 .5789 .166E-05 866.69 .5487 .156E-05 1096.97 .5784 . 197E-05 865 .63 .5462 . 132E-05 1090.24 .5196 . 329E-05 777.61 .4900 .262E-05 977. 37 .4674 .375E-05 698.74 .4286 .306E-05 855 . 15 .4414 .646E-05 660.54 .4106 .521E-05 818.96 .3684 .124E-04 551.32 .3376 .101E-04 673.42 .1749 .105E-03 261.27 .1502 .914E—04 298.92 Table C-34 Data from Figure 5-35 Pe,", ch," 2000 1500 1000 500 200 100 50 20 .249E-04 .000 1.000 1.000 .000 1.000 - 1.000 1.000 1.000 .304E=04 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 .372E-04 1.000 1.000 .997 1.000 1.000 1.000 1.000 1.000 .454E-04 1.000 1.000 .995 1.000 1.000 1.000 1.000 1.000 .555E-04 1.000 .988 .990 1.000 1.000 1.000 1.000 1.000 .677E-04 1.000 .984 .987 1.000 1.000 1.000 1.000 1.000 .827E—04 1.000 .982 .987 1.000 1.000 1.000 1.000 1.000 .101E-03 1.000 .980 .985 1.000 1.000 1.000 1.000 .999 .123E-03 1.000 .977 .985 1.000 1.000 1.000 1.000 .999 .151E-03 .999 .977 .981 1.000 1.000 1.000 1.000 .997 .184E-03 .989 .975 .981 1.000 1.000 1.000 .999 .996 .225E-03 .976 .970 .982 1.000 1.000 1.000 .997 .995 .275E—03 .960 .962 .980 1.000 1.000 1.000 .996 .994 .335E-03 .941 .950 .974 1.000 1.000 .999 .990 .993 .410E—03 .917 .933 .964 .999 1.000 .998 .986 .992 .500E-03 .890 .913 .950 .991 1.000 .995 .983 .991 .611E-03 .857 .887 .931 .980 .997 .991 .981 .001 .912E-03 .776 .819 .878 .946 .979 .982 .978 .000 .111E-02 .727 .776 .843 .923 .967 .976 .976 .001 .166E-02 .610 .670 .754 .860 .933 .958 .970 .002 .248E-02 .465 .536 .638 .776 .885 .931 .959 .003 .370E-02 .288 .369 .490 .665 .818 .890 .938 .005 .552E-02 .078 .168 .310 .526 .731 .834 .906 .007 .823E-02 .092 .355 .619 .758 .859 .949 .101E-01 .257 .553 .713 ' .829 .933 . l23E-01 .150 .481 .661 .794 .914 .150E-01 .033 .400 .603 .754 .891 .183E—01 .312 .539 .708 .863 .224E—01 .215 .468 .656 .830 .273E-01 .110 .389 .598 .792 .334E-01 .302 .533 .748 .408E-01 .208 .461 .698 .498E-01 .104 .382 .641 .608E-01 .295 .578 .743E-01 . 199 .507 .907E-01 .095 .428 .111E+00 .342 .135E+00 .247 200 Table 035 Data from Figure 5-36 Parameter <15, 1) Pe,", Bead Radius (m) .00200 .492E-05 .8782 86.28 .00025 .577E—03 .4828 47.14 .00050 .108E-03 .6410 62.76 .00100 .221E—04 .7817 76.72 .00300 .210E-05 .9156 90.00 .00400 .115E—05 .9354 91.97 Velocig'oéomls) . 1 .238E—05 .9092 89.40 .08000 .300E-05 .9003 88.50 .06000 .406E-05 .8874 87 .20 .03000 .850E—05 .8482 83.31 Bead Porosity .98000 .760E-05 .8539 83.87 .90000 .633E-05 .8646 84.93 .70000 .371E-05 .8921 87.67 .60000 .268E-05 .9063 89.09 .50000 .183E—05 .9210 90.53 .40000 .115E-05 .9361 92.03 .30000 .638E—06 .9515 93.56 .20000 .279E-06 .9673 95. 13 . 10000 .686E-07 .9834 96.72 .05000 .170E-07 .9916 97.55 Liquid Fraction. .80000 .857E-06 .9444 92.88 .70000 .228E-05 .9134 89.76 .50000 .968E—05 .8382 82.32 .40000 .185E—04 .7907 77.60 .30000 .364E-04 .7308 71.65 .95000 .431E—07 .9870 97.07 .99000 .164E—08 .9975 98.12 Diffusion Coefficient (m2/s) 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