University of Southern California Dissertations and Theses

A structural model for gas-solid reactions with a moving boundary

(USC Thesis Other)

A structural model for gas-solid reactions with a moving boundary

PDF

Download

Open document

Flip pages

Download a page range

Download transcript

Copy asset link

Request this asset

Transcript (if available)

Content A STR U C TU R A L M O D E L F O R GAS-SOLID REACTIONS W ITH A M O VIN G B O U N D A R Y by K. Shankar A Thesis Presented to the FACU LTY O F TH E S C H O O L O F ENGINEERING UNIVERSITY O F S O U TH E R N CALIFORNIA In Partial Fulfillm ent of the Requirements fc5r the Degree M A S TE R O F SCIENCE IN CHEM ICAL ENGINEERING M A Y 1982 UMI Number: EP41808 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. UMI EP41808 Published by ProQuest LLC (2014). Copyright in the Dissertation held by the Author. Dissertation Publishing Microform Edition © ProQuest LLC. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 4 8 1 0 6 -1 3 4 6 Ch J Z K s y x T h is thesis, w ritte n by K. Shankar u n d e r the guidance o/h i s F a c u lty C om m ittee and a p p ro ve d by a ll its m em bers, has been presented to and accepted by the S choo l of E n g in e e rin g in p a rtia l fu lfillm e n t o f the re quirem ents f o r the degree o f MASJ£R..Q£..S£IENC£.JW CHEM ICAL.. ENGINEERING.. D ate. April 23, 1982 F a cu lty C om m ittee sJt zQt i S. s 2 > . Chairman ......... To my loving parents . . . . ii ACKNOWLEDGEMENTS The author appreciates the generous advice provided by his adviser Prof. Yanis Yortsos throughout the course of this study. The numerous sessions of stimulating analysis and discussion helped, to a considerable extent, in channelizing the directions of this research e ffo rt. F in ally, the author wishes to acknowledge the Department of Chemical Engineering, U S C for th eir support throughout his program of study. K.S. TABLE OF CONTENTS P A G E DEDICATION-------------------------- ii ACKNOWLEDGEMENTS....................- - ........................................... i i i LIST O F FIGURES - - - - - ............................... v ABSTRACT................ vi C H A PTER I . INTRODUCTION-------------- 1 I I . A SU R VEY O F SINGLE-PARTICLE M O D E L S F O R GAS-SOLID REACTIONS ------------------------- 6 I I I . FO RM ULATIO N O F A SINGLE-PORE M O D E L - -------------- 21 IV. ANALYTICAL SOLUTIONS TO TH E SINGLE-PORE M O D E L - 30 V. RESULTS A N D DISCUSSION.........- - .............................. 42 VI. CONCLUSIONS A N D R EC O M M E N D A TIO N S F O R FU TU R E WORK------------------------------------------------ 57 N O M E N C LA T U R E ------------------------------ —' -------------------------— 60 REFERENCES-------------------- - - .......................... - 63 APPENDICES A. Conditions for Pore-Mouth Plugging - - - - - - 66 B. Details of the Hodograph Transformation - - - - 68 C. Solutions to the Slab-Model - - - - - - - - - - 69 iv LIST O F FIGURES FIGURE P A G E 1. Strategy for the Analysis of Gas-Solid Reaction Systems - - - - - - - - - - - — - - - - - - 3 2. Schematic Representation of a Single-Particle Reaction - - - - - - - - - - - - - - — - - 7 3. Unreacted Shrinking Core Model for Non-Porous Particles - - - - - - - - - - - - - - - - - - 1 1 4. Geometry of the Single-Pore System - - - - - - - - (a) Before Reaction. (b) During Reaction : case a > 1 (c) During Reaction : case a <1 23 5. Concentration Profiles for Gas A within the Pore - 43 6. Enlargement of Pore-Radius with Time : case a '< 1 - 44 7. Shrinkage of Pore-Radius with Time : : case a > 1 - 45 8. Pore-Pluggincffor Low In itia l Porosity : ' x. case a > 1 - - - - - — _ _ _ _ _ _ _ _ _ _ 47 9. Propagation of the Reaction Zone within the Pore - 48 10. Variation of the Pore-Mouth Concentration Gradient of Gas A with Time - - - - - - - - - - - 49 11. Typical Plot of Conversion vs. Time’ - - - - - - - 50 12. Plots of Conversion vs. Time for Large Pore- Diffusional Resistances - - - - - - - - - - - 5 1 13. Plots of Conversion v?. Time for Very Fast Reactions - - - - — - - - - - - - - - - - - 52 V ABSTRACT In many gas-solid reactions, the porous solid undergoes structu ral changes due to differences in the molar volumes of the solid reactant and product. A survey of single-particle models which high lig h t these structural changes was undertaken. Based on the survey, an analytical single-pore model was developed for these structural reactions. The model accounts for the influence of pore diffusion, product layer diffusion, firs t-o rd e r surface reaction and changes in the pore-geometry. Using a new solution technique, exact-analytical solutions were derived for the equivalent case of an isolated semi in fin ite pore. For large pore diffusional resistances or fast re actions, these solutions simulate the behavior of the fin ite-po re model very w ell. The phenomena of pore-mouth plugging and incomplete conversion for low in itia l porosities were analyzed using this model. For su ffic ie n tly large in itia l porosities, the corresponding case of a moving reaction zone and complete conversion was qonsidered. A sim ilar analysis was carried out for the case of slab-geometry within the p article. F in ally, the possibility of extending the analytical single-pore model to the more general cases of non-linear, non- isothermal kinetics was discussed. vi C H A PTER I INTRODUCTION . Systems involving gas-solid, non-catalytic reactions are rele vant to a broad range of chemical process operations. Typical examples [1,2] of gas-solid reactions include the combustion and gasification of solid fuels, reduction of metal oxides, desulfurization of flue gases, incineration of solid wastes.etc. The need for synthetic fuels has recently spurred considerable interest in a variety of gas-solid reactions involving combustion and gasification of coal and other fo s s il-fu e ls . Reactions such as the reduction of iron oxides to m etallic iron, roasting of copper and zinc sulfides and other reactions play an important role in the overall analysis and design of metallurgy operations. Abatement processes for sulfur dioxide emissions from fo ssil-fu el power plants also involve gas-solid reactions. The com m on characteristic of a ll these in d u strially important reactions is the direct participation of the solid phase in the over a ll reaction. As the reaction proceeds, the structure of the solid material changes continuously and this d ire c tly affects the rate of the reaction. This is contrasted to hetewgePeoU's^cei^lyltTc reactions where, barring any structurally dependent deactivation or poisoning, the structure of the catalyst essentially remains unaffected by the reaction. Thus the inherently 'unsteady1 nature of gas-solid reactions introduces a number of complexities which are absent in co m m on hetero geneous catalytic systems. 1 The approach used in analyzing such processes is sim ilar to that employed in the analysis and design of process systems in heterogeneous catalysis. Based on this approach, the analysis begins by studying the reaction between an isolated solid particle and the reactant gas. To fa c ilita te the development of a tractable mathematical model for * the p a rtic le , various assumptions may have to be m ade for the sub units composing the p article such as pores, grains,etc. The various steps involved in the reaction such as diffusion of the gaseous reactant and product, surface chemical reaction and structural changes associated with the p article are then analyzed systematically and combined into an overall reaction rate for the p a rtic le . The inform ation from these structure-based single p article models is incorpora ted into a macroscopic model for the analysis and design of gas-solid reactors. Together with the development of mathematical models for the reactor, physical data for the reaction system are needed in order to assign numerical values for the parameters of the model. This information m ay be obtained from laboratory experiments on gas-solid ractions and from relevant c rite ria for scale-up to large- scale industrial reactors. The strategy for the analysis of gas-solid reactions is shown, schematically, in Figure 1. Different types of reactors may be used in various reactions. S om e typical reactor configurations [3] include packed bed reactors, moving bed reactors, fluidized bed reactors, etc. In translating models for single-particles to reactor models fo r m ulti-p article assemblies, various types of problems have to be faced. The physical 2 Experimental data and scale-up c rite ria Simulator to analyze indus tr ia l reactors Particle Model for overall reaction rate in a p article Macroscopic model for reactor as a multi - p article assembly Microscopic model for sub-units of p a rtic le , such as pores, grains, etc. _____________ Figure !• Strategy for the Analysis of Gas-solid Reaction Systems* nature of the assembly affects many aspects of the system including flu id -flo w , dispersion, interaction effects between p articles, etc. The reaction time between .the p article and the gas m ay not be constant due to a distribution of reactor residence times for particles and gas. Also, the composition of the reactant gas varies sp atially over the reactor. I t follows from the above discussion that m u lti-p article reactor systems are a great deal more complex, in general, than sin g le -p a rti cle systems. These phenomena relating to non-idealities of flu id flow through a reactor have been analyzed [4 ], in great d e ta il, in other contexts. A singular aspect of gas-solid reactions is the structural changes associated with the solid phase. Thus, i t is important to develop a single-particle model which can highlight effectively the changes associated with p article conversion. The present study of gas-solid reactions is confined to the analysis of sing le-particle systems. In Chapter I I , a comparative study of the various models for single-particle systems is presented. While the salient features of various models proposed are examined c r itic a lly , no attempt is made to explain the detailed mathematical formulation of each model. In Chapter I I I , a single-pore model is developed for gas-solid reactions. In this model, the reacting p article is idealized by a set of p a ra lle l, cylindrical pores. The model accounts for pore and product layer diffusion of the reactant gas, surface chemical reaction and changes in pore geometry with time. 4 Since exact-analytical solutions are not known at present, asymptotic solutions to the single-pore model are obtained in Chapter IV for the important cases of large pore diffusional resistance or fast chemical reaction. These solutions are exact for the case of tru ly in fin ite pores. In Chapter V, the resulting solutions for the single-pore model are discussed. The sen sitivity of the system to various parameters is examined and a regime for v a lid ity of the asymptotic solutions is established. The conclusions based on the single pore model are highlighted in Chapter VI. The applications of such micro-models to overall reactor configurations are discussed and areas for future work are id en tified . 5 CH A PTER I I A S U R V E Y O F SINGLE PARTICLE M O D E L S F O R GAS-SOLID REACTIONS 2.1 The Principles of Single-Particle Reactions A single reacting particle represents the smallest unit of interaction in a gas-solid reaction system. A general gas-solid reaction is of the type a A (g) + b B (s) - » • c C (g) + d D (s) Typical examples [5] of practical systems which m ay be represented by the above reaction would include Fe2 O g (s) + 3 C O (g) -* • 2 Fe (s) + 3 C02 (g) C aO (s) + S02 (g) + 1/2 02 (g) - » ■ C a S04 Is) M g O (s.) + C£2 (g) + M g C^2 (s) + 1/2 02 (g) C (s) + 02 (g) - » • C02 (g) A schematic representation of the phenomena involved in such single particle reactions is shown in Figure 2. The overall process can be broken down into the following steps: Mass Transfer and Reaction 1. Gas-solid mass transfer from the bulk of the reactant gas stream to the surface of the p article. 2. a) Diffusive transport of the gaseous reactant through the pores of the solid matrix which could be a mixture of solid reactant and solid product, b) Adsorption of the gaseous reactant on the surface o f the solid matrix. 6 Bulk Conditions : Temperature (T ) Concentration of A (C^Q) Concentration of C (CCq) SURFACE REACTION PARTICLE SURFACE SOLID REACTANT HEAT C O N D U C TIO N / C O N VE C TIO N BULK GAS GAS-PHASE HEAT TRANSFER REACTANT GAS C^o PDRE STRUCTURE STRUCTURAL CHANGES DUE TO REACTION SOLID PRODUCT S-PHASE MASS TRANSFER DIFFUSION AND BULK TRANSPORT OF A & C THROUGH PORES PRODUCT GAS ~ DIFFUSION AND BULK TRANSPORT OF A 1 C THROUGH PRODUCT LAYER 'C O GAS-PHASE MASS TRANSFER Figure 2. Schematic Representation of a Single-Particle Reaction. c) Diffusive transport of reactant gas through the solid product 1ayer. d) Chemical reaction at the reaction interface between solid product and reactant, e) Diffusive transport of product gas through the solid product layer. f) Desorption of the product gas from the surface of the solid matrix. g) Diffusive transport of the gaseous product through the pores of the solid matrix to the surface of the p a rtic le . 3. Mass transfer of the gaseous product from the surface of the par tic le to the bulk of the gas-phase. Heat Transfer For exothermic or endothermic reactions, the m ass transfer and reaction steps w ill be accompanied, by: 1. Convective (and possible radiative) heat transfer between the gas stream and the surface of the solid p a rtic le , and 2. Conductive and convective heat transfer through the pore spaces and the solid matrix. Structural Changes In som e cases [1 ], d is tin c t changes occur in the structure of the solid matrix due to substantial differences between the molar volumes of the solid product and reactant. Thus, changes in the pore or grain structure of the solid matrix and other effects such as sintering could have a marked effect on the overall conversion of the solid reactant. 8 Id eally, a ll the above-mentioned effects should be included for a proper representation of the reaction within the p a rtic le . But this would lead to a complex model which could be d iffic u lt to solve. At present, most models for such reactions involve som e simplifying assumptions which either 'lump1 som e of the-reaction steps together or neglect them altogether. S om e of the reaction steps such as mass and heat transfer between phases are understood quite well whereas the phenomena of pore diffusion and reaction, structural changes, etc. within the p article are understood to a m uch lesser extent. The present study is mainly confined to the study of models which account for structural changes due to reaction. 2.2 Reactions in Non-Porous Solids These models can be used in the cases where the void fraction of the solid particle is low so that i t is essentially non-porous. The two types of models for such reactions are: A. Gasification type model The simplest gas-solid reaction is o f the form a A (g) + b B (s) -> c C (g) In this reaction, no solid product is formed and the reactant particle gradually shrinks as the reaction proceeds. Typical examples [6 ,7 ] of such a reaction include the complete combustion and gasification of coal, the decomposition of solids into gases, fluorination and chlorination of metals, etc. The system may be studied [8 ,9 ] under two lim iting mechanisms : chemical reaction controlled and film - resistance controlled cases. In both these cases, the ash layer is 9 absent and does not contribute any resistance to the overall reaction. W hen chemical reaction controls the process, the to tal rate of reaction is proportional to the available surface of the unreacted core. Film resistance at the surface of a particle is dependent on numerous factors such as the rela tiv e velocity between p article and gas, size of the p a rtic le , flu id properties, etc. These phenomena have been correlated [1] for various contacting methods such as packed bed, moving bed, fluidized beds, etc. B. Unreacted shrinking core (URC) model with ash formation [3] In this model, the reaction occurs fir s t at the outer skin of the p artic le . The reaction plane moves into the unreacted core gradually, and leaves behind completely reacted solid and in e rt solid. The process is shown schematically in Figure 3. The overall size of the particle m ay or m ay not change, depending on the rela tiv e densities of the solid product and reactant. In general, the overall rate of the reaction is governed by the combined resistances due to film mass transfer, ash-layer diffusion and surface chemical reaction. Comparisons of the kinetic and diffusional control models for such reactions have been presented in many investigations [10,,11]. Transi tions between the two mechanisms are shown to be a function of a Thiele type diffusion modulus. By analogy to catalyst p articles, a time-varying effectiveness factor is defined for the reaction. I t can be shown [1 ,3 ] for such models that the overall resistance to reaction m ay be approximated by a series combination of three in d iv i dual resistances which account for film , ash-layer diffusion and 10 t =0 unreacted core t = T unreacted solid ash/product layer completely reacted solid ash Figure 3. Unreacted Shrinking Core (URC) Model for Non-Porous Particles. 11 reaction control regimes of operation,respectively. I f the reaction is non-isothermal, temperature gradients m ay exist in the p a rtic le . Here, the rate of the reaction varies over the entire p a rtic le. In such cases, the mass balances for the reactant are coupled to heat balances over the reacting p article. The re a c ti v ity , s ta b ility and effectiveness factors in such systems have been examined by various workers[8, 9,12,13]. Exothermic reactions may present the interesting and often important problems of m u ltip lic ity of steady states and thermal in s ta b ility . This problem has been dis cussed by Denbigh [10] and other workers [14,15 ] . For exothermic reactions controlled by diffusion through the product layer, the m axim um temperature rise in the particle can be calculated as a function of the various reaction parameters [1, 13]. High temperature rises in a particle could cause sintering and other structural changes [16] during the course of the reaction. The shrinking core model does not provide a re a lis tic descrip-i tion of reactions occurring in very porous solids. In such cases, the reaction is distributed throughout the solid and there is no sharp reaction fro nt. Models for such structure-based reactions are dis cussed in the next section. 2.3 Reactions in Porous Solids In many gas-solid reactions, the "reactivity" of the solid phase is higher due to the fin e ly dispersed, porous structure of the solid reactant. In such cases, diffusive transport of the gaseous reactant takes place within the pore spaces of the solid. The high 12 specific surface area of the solid provides a large surface for the reaction. Reactions within porous solids are thus characterized by the combined mechanisms of in te rs titia l pore diffusion and chemical reaction spread over the entire p a rtic le , in contrast to the shrinking core model which describes the reaction phenomena by a sharp-front macroscopic model. Numerous models have been developed for such reactions. The pore and grain models have been frequently used to represent such structure-based reactions. These models describe the progress of the reaction in terms of characteristic pore/grain sizes, gas phase and solid phase d iffu s iv itie s , porosity and an in trin s ic surface reaction rate constant, which is unconstrained by any empirical structural dependence.. In this section, a comparative analysis of various struc tural models for gas-solid reactions is undertaken. Most of the survey w ill be related to a description of various pore/grain models for gas-sol id reactions. Grain-Models for Gas-Solid Reactions In this model, the solid body is idealized as a series of uniformly sized spherical grains [17] arranged in concentric spheres around the center of the p a rtic le . The gaseous reactant diffuses through the interstices between the grains and then reacts with individual grains according to the "shrinking core" model described before. Beginning from the surface of the p a rtic le , consecutive rows of grains react with the gas in progressively lower concentra tions. The gas diffuses through the interstices according to a Fick's!' Law diffusion type mechanism. The most serious lim itation. 13 of such a uniform grain-size model is the assumption that structural changes within the grain may be neglected. Such changes are lik e ly to occur due to differences in molar volumes between solid reactant and product, sintering, agglomeration, etc. Sohn and Szekely [1,18] extended the uniform grain-model to a general structural model allowing for spherical and f la t p la te -lik e grans within the p a rtic le . Asymptotic and numerical solutions are obtained for the extent of reaction with a generalized Thiel e-type reaction modulus as a parameter. Again, the most c ritic a l assumption in this model is the neglect of structural changes within the grains and p article as the reaction proceeds. This model has been extended [19] to account for the effect of intragrain diffusion for a system with large grains or fast reactions. Recently, Georgakis et a l . [20] have developed a changing grain size model which accounts for density differences between the solid product and reactant. Stoichiometric considerations yield a relation between the changing grain size and the rate of the chemical reaction. The system equations are solved numerically [21] to yield the local conversions and grain sizes with time. By integrating over the p article volume, the average particle conversion and porosity m ay be obtained. W hen the molar volume of the product solid is larger than that for the reactant solid, a c ritic a l value for the in itia l porosity is established. Beyond this value, complete conver sion of the sol id reactant is possible. A critical^'pore-plugging time is an alytically calculated for the case where the in itia l 14 porosity is less than the c ritic a l value. For such systems, the conversion of the solid levels o ff asymptotically with time. This phenomenon has been observed in the reaction between limestone and sulfur dioxide in flue gas desulfurization systems. Hartman et a l. [22,23,24] have used the grain model to study this reaction. They correlated,successfully, experimental results for the limestone - SC ^ system with a grain model which accounted for the in itia l porosity, density and diffusional characteristics of limestone. Pore Models for Gas-Solid Reactions Reactions between porous solid and gases can be described by assuming that the p article is composed of pores with a distribution of sizes and shapes. Petersen [25] considered two cases : a single uniform pore and a porous cylindrical sample containing uniform cylindrical pores with random intersections. These models consider only gasification-type reactions where no solid products are formed. Comparison of the model with experimental results gave values for the effective d iffu s iv ity which were an order of magnitude lower than the corresponding bulk d iffu s iv ity at the sam e temperature and press ure. Petersen proposed that this would happen whenever the real pore system containing pores of varying sizes is replaced by an idealized system of uniform, cylindrical pores characterized by an average diameter. C hu et a l. [17,26] developed pore and p a ra lle l-p ia te models for reactions where structural changes in the solid phase were not sign ifican t. These models, however, accounted for product layer diffusional resistances. Analytical expresssions were obtained for 15 the progress of the reaction as a function of time and various physi cal and chemical parameters. E xplicit relations for the reaction "penetration" depth were derived as a function of time and the system parameters. Such models can be applied to the combustion of carbona ceous deposits on catalysts, reduction of metal oxides, etc. Ramachandran and Smith [27] studied the structural changes associated with a firs t-o rd e r reaction in a single, cylindrical pore. The model accounted for the influence of pore diffusion, diffusion through the growing product layer which builds up on the pore w alls, and surface reaction at the product-reactant interface. By considering a single pore, i t was possible to include the effects of the changes in pore structure with reaction. The extreme cases of pore-mouth closure/incomplete conversion and uniform deposition of the product/ complete conversion could be analyzed using this model. The system equations for the pore radius and concentration profiles were solved numerically to yield conversion-time profiles for the p a rtic le . Experimental measurements fo r particle parameters such as specific surface area, pore volume and porosities of the unreacted and reacted p article were used to obtain approximate values for the in itia l pore radius, length and effective d iffu s iv ity through the product layer. Based on these physical parameters and reaction rate data, the model simulated reactions such as the reduction of nickel oxide particles with carbon monoxide and the sulfation of limestone by S O2 fa ir ly wel l . Ulrichson and Mahoney[28] extended the single pore model to include the effects of bulk flow and reversible chemical reactions. 16 A variable effective d iffu s iv ity , which accounted for bulk diffusion and Knudsen diffusion [1 ], was assumed. Again, the system of coupled diffusion-reaction equations were solved numerically and the model was used to simulate the chlorination of magnesia. Chrostowski and Georgakis [29] used a sim ilar single-pore model for structure based reactions in particles. A n attempt was m ade to analyze the problem asymptotically by deriving semi-analytical solutions for som e lim iting cases. A n exact expression was obtained for the pore-plugging time as a function of the parameters of the system. A perturbation expansion for small times was developed to examine the case where the kinetic resistance controlled the overall reaction. The other asymptotic case of high pore diffusional resis t ance was analyzed by using a perturbation-collocation method of solution. This method was found to be more satisfactory than the regular collocation method, especially for high pore diffusional resistances. In typical gas-solid reactions, the solid p article consists of wide range of pore or grain sizes. The in itia l pore structure and pore size distribution of the solid reactant alters [30] the d iffu sional resistance of the gaseous reactant and the specific reaction surface area of the solid. As the reaction proceeds, the pore size distribution is continuously altered so that the diffusion and re- ' action characteristics change with time. D ifferent types of models have been proposed for taking into account the effects of changing pore and grain size distributions. 17 Hashimoto and Silveston [31] developed a model describing the development of specific surface area, volume and porosity with extent of reaction based on considerations of pore size growth, in itia tio n of new pores and coalescence of adjacent pores. Population balances were used [32,33] to follow the evolution of the pore size distribu tion with time. The p article specific reaction areas and pore volumes were formulated as moments of the pore size d istrib u tio n . The model was used to follow the course of the reaction in gasification-type ■ processes where no solid product was formed. This model was la te r extended [34] to allow for mass transfer of the gaseous reactant to the particle and in trap article.d iffu sio n al resistances. The extended model predicts, as expected, that m ass transfer lim its the reaction while in trap article diffusion shifts the reaction to the outer surface of the p article. Simons and Finson [35,36] developed a semi-empirical model to describe the structure of porous coal char. Based on experimental -•-3 data, they suggested a r_ distribution of pore number density, where r r is the radius of a pore. The 'pore-tree' concept was then incor- r porated into a general model [37] to describe diffusion transport and heterogenous chemical reaction. The gasification rate per pore tree was integrated over the semi-analytical pore size distribution to determine the overall p article gasification rate. I t was shown that the gasification process was governed by a combination of mechanisms including diffuional and kinetic resistances, pore growth and combi nation, etc. 18 Christman and Edgar [38] extended the pore size distribution evolution model to account for solid product deposition with extent of reaction. The macroscopic solid properties were obtained by integrating over the entire pore size distrib ution . The model was used to simulate the sulfation of limestone particles in an atmosphere of S O2 . Bhatia and Perlmutter [39,40] developed a random pore model for flu id -s o lid reactions which allows for arb itrary pore size dis tributions. This model u tiliz e s an adjustable structural parameter to characterize solid re a c tiv ity . I t was shown th a t, in general, the three parameters of pore volume, surface area and length are su fficien t to adequately characterize the structure for any arbitrary pore size distrib ution . The model has been used [41] to illu s tra te the effect of pore structure on the S O2 - limestone reaction. Simonsson and Lindner [42] in a recent review compared various structural models for gas-solid reactions in porous solids undergoing structural changes. They showed that a ll the previous models,though d ifferin g in th eir geometric description of the in itia l structure of the sol id, prim arily focus on the consi'deratibns of the decreasing diffusion rate in the growing product layer and the decreasing gas- solid in terfacial area and reaction surface area. They further developed a new model which regards the in itia l solid structure as an aggregate of truncated spheres in contact with each other as in the in itia l stage of sintering. This model was then used to predict the conversion-time curves for the sulfation of limestone. They 19 concluded that a semi-empirical approach for modeling the effects of the structural changes, can give essentially the sam e results with less numerical e ffo rt. 20 C H A PTER I I I FORM ULATION O F A SINGLE-PORE M O D E L A single-pore model is used to analyze the behavior of an isolated reactant particle exposed to reactant gas at a uniform ambient concentration. By idealizing the p article as a set of parallel pores, i t is possible to account for structural changes associated with the conversion of the solid in a quantitative manner. The model accounts for pore diffusion, product layer diffusion and surface chemical reaction. Depending on the physical and reaction parameters chosen, the model can simulate both pore-mouth closure/incomplete conversion, as well as uniform product deposition/complete conversion. The major assumptions associated with the development of the model are as follows : 1. The p article is idealized by a set of p a ra lle l, cylindrical pores of fin ite length (L) and in itia l radius (r^ ). 2. Each pore is in it ia lly surrounded by an annular reactant solid of radius x*- 3. The chemical reaction is irreversible and f ir s t order with respect to the gas and occurs at a sharp interface between the reactant and product solid. 4. Isothermal conditions prevail throughout the system. 5. Diffusion of gas in the pore and solid product layer occurs under quasi-steady state conditions [43,44]. 21 6 . Diffusion of gas in the radial direction within the pore is negligible compared to diffusion in the longitudinal direction. 7. Bulk flow and Knudsen diffusion are negligible. 8 . Effective d iffu s iv itie s for diffusion of the gas A through the pore and the product layer are constant. 9. There is no axial diffusion of A in the product layer. 10. External mass transfer resistances for the gas A are neglected. The general reaction is considered to be of the form : The geometry of the reacting single-pore system is depicted in Figure 4. The case a < 1 corresponds to a reaction-where the stoichiometric molar volume of the solid product (D) is less than that for the solid reactant (B). In such a case, the pore structure of the reacting solid opens up with time and complete conversion is possible. For a > 1, the stoirliio metric molar volume of the solid product is greater than that for the reactant so that the pore radius shrinks with time. Steady State Pore Diffusion Neglecting bulk flow, the fluxes of species A in the x *- direction and r * - direction are given respectively by a A (g) + b B (s) -> c C (g) + d D (s) (D * (2 ) and * (3) 22 (a) r = o (c) i * * 0 0 C « | AO GASA Reactant solid x*= 0 Product solid (D) r # ^7-*^— GAS A Reactant solid (6) Product solid (D) GASA -Reactant solid (B) Figure 4, Geometry of the Single-Pore System. (a) Before Reaction. (b) During Reaction : case a > 1. (c) During Reaction : case a < T. 23 where CD, D g) and (CA > C^J denote the d iffu s iv itie s and concentrations of gas A in the pore and product layer, respectively. Invoking assumptions (1 ), (.5), (6 ) , (7 ), and (8 ) , the transport equation for diffusion along the x *- direction within the pore is (ri N AxJ = 2 r*NAr* J y 1 (4) Using assumption (9) for steady-state diffusion in the product layer, 2 > K Ar‘ |v * ■ 2 *•« c b 1 . ‘ 2 Substituting equations 2 and 4a into 4 yields: (4a) a r * ax* [r i 2 ac-ft ax* • { 2 K r, D 'Ar = 0 (5) r 2 Equation (5) is supplemented by the following boundary conditions 'A ac A ax1 * C A o ’ x* = 0 ; x* = L (5a) (5b) Steady-State Product Layer Diffusion Sim ilarly,the equation for diffusion of gas A in the r*- direction becomes a r* ac r* D Ar = 0 along with * * * C „._ = C A ; r* = r., 'Ar (6 ) (6a) 24 r * * \ " D e 3 r* = K C Ar ' ’ r* = r2 Conservation of Mass The rate at which the reactant/product interface recedes into the solid annulus surrounding the pore can be related to the surface chaemical reaction by stoichiometry: ’2. _ I d I I "D 3r„ 1 * W [ - (7) ■5^*“ " | a | | - 7 ( I - - “ v ' J K C Ar Equation (7) is supplemented by the in itia l condition t* = 0 , r*2 = rQ (7a) Note that pD is the density of porous solid product D. To relate the radius of the reactant annulus to that of the product layer, w e consider the following mass balance: At time t * , for an element Ax* along the axis of the pore, the number(Am) of moles of reactant B that have been consumed is given by 2 p M (r * - r^) (Ax*) (1 - eB) = A m (8 ) B S im ilarly, for solid product D U ) (r* 2 - r * 2[) (Ax*) (1 - sD) (-jjS-) = (9) 2 1 D Eliminating 'Am1 between equations (8 ) and (9) f - ro> t’ - (1 - eD > ^ W or *2 9 *2 r*i = a rQ + (1 - a) rg . (1 1 ) 25 d VD (1 - ep) H e re ,a is a p aram eter = — ( T ^ i - ) ( 12) 'D J and (7B, VD) are the 's k ele ta l' molar volumes of (B, D). Equations (5 ), (6 ) , (7) and (11) completely specify the system. Thus, * 'fC ^ “ jk T "ft * ^ in principle, the four unknowns (C^, C^r , r^, rg) can be determined. Then, the important quantities of in terest, such as the local conver sion n (x *, t* ) = 2 X - (13) and the average conversion associated with the entire pore L L 'o T ) (t* ) J T) ( X * , t* ) d x* (14) can be calculated. To proceed, the system is rendered dimensionless by using the natural characteristics of the system: l> V D e C Ao* The new dimensionless variables are as follows : x = x*/L Ti V V ro r z = V r © x = x * / r 0 C A = V CAo C Ar = C Ar/CA o t = t* De 1 1 d i [ MD 1 o 3= O 11 r ? U ) , PD 1 1 " eD . (15) (16) (17) (18) (19) ( 20) (21) Using the above 'dimensionless.,variables, the dimensionless form of.,-s 26 the . system equations.are i t 4 M - <! r 2 C Ar2 " 0 (2 2 ) subject to B.C : C A 1 X = 0 (2 2 a) and X A . . '.3x 0 ; x = 1 (2 2 b) Also, 3 ’ r 3CAr ‘ = 0 (23) 3r L J subject to.'B ;C„ : C Ar = C'A * r := r ] (23a) 9C Ar _ 3r - B 1 C Ar ; r = r2 (23b) 3r 2 3t Bi C Ar2 (24) subject to I .C : r2 = 1 ; t = 0 (24a) F inally, r 2 = a + (1 - a) To (25) Here, the parameter 0 is a Thiele-type diffusion modulus defined by : 02 = 2 K L2 o 0 I t expresses the ratio of the characteristic time for pore diffusion along the length of the pore to the characteristic time for the surface chemical reaction. For a specific reaction at a fixed temperature 27 (constant K), small values of 0 imply very low pore diffusional resistances (S m all/L /r^). In contrast, large values of 0 indicate that pore diffusion controls the overall process, fo r a specific reaction. The Biot modulus (Bi) is defined as : K r 31 " ' V and expresses the ratio of the characteristic time for diffusion :.-.v through the solid product in the radial direction to the characteristic time for the surface chemical reaction. For specific pore geometries (constant r Q) and diffusion characteristics of the gas A, small values of Bi imply a very slow rate of chemical reaction compared to product layer diffusion. Large values of Bi indicate a very fast chemical reaction. I t may be noted that the Thiele modulus and the Biot modulus for the system are related by : D 02 = 2 Bi r o e D (26) The chemical parameter a reflects the rela tiv e differences in molar volumes of the product solid and the reactant solid and is defined by: a d ?D (1 -.£ „) b VB (! - eD ) Values of a < 1 correspond to systems where reaction leads to a more 'open* pore-structure. The lim iting case of a-> 0 corresponds to a gasification-type reaction where no solid product is formed. O n the other hand, values of a > 1 correspond to systems where the pores radius continuously shrinks with reaction due to the deposition of a high molar-volume product solid. 28 Estimation of In itia l Pore Radius (rQ) and Effective Length (L) Since the pore is assumed to be c y lin d ric a l, the average pore radius for a monodisperse particle can be estimated [27] from the equation ' • ■ w 0 , 1 where e , p and S are the in itia l porosity, density and specific r y surface area of the p a rtic le , respectively. I f the porosity of the single-pore system at t = 0 is the sam e as that for the unreacted p a rtic le , then the in itia l radius of the solid reactant annulus can be calculated from the equations ; 2 tt r . e = 0 (28) *2 TT X * or - £ L = _ I _ (29) r o The effective pore length (L) determines the significance of d iffu r sional gradients within the single pore. I t is logical to choose i t in such a way that the original p article and the single-pore system have the sam e diffusional characteristics at t = 0. Based on these arguments [27,29], the effective pore length is given by L = — 5— (30) 3^ o where R = Radius of the p article. 29 C H A PTER IV ANALYTICAL SOLUTIONS TO TH E SINGLE-PORE M O D E L In this chapter, the.asymptotics of the single-pore model are examined for the cases of._ iv/ery large pore diffusional resistances. A n analytical solution is obtained for the variation of pore-mouth radius with time. This solution is valid for a ll possible combinations of the system parameters. Exact solutions are also obtained to the single-pore model for the case of an in fin ite pore. I t is numerically shown that these solutions simulate the behavior of the fin ite -le n g th pore very well for moderately large pore-diffusional resistances. The solution is extended to the moving boundary case to account for the propagation of the reaction zone into the pore. In developing a solution to equations (22) - (25) w e f ir s t observe that equation (23) for diffusion through the porous product layer can be solved to yield : Thus, the concentration of A at the reaction interface is calculated as C 'A Bi r 2 C Ar, 1 + Bi r9 L n 2 r. 2 (32) This allows elimination of C^r and ^ between equations (22), (25) and (32) to yield: 30 3x L - £ * ] ^ C A T T r j T subject to B.C : C A = 1 ; x = 0 3C A and 3x 3r, = 0 x = 1 (33) (33a) (33 b) 3t Bi (1 - a) C ; r-, g ( r ] ) subject to I.C At t = 0 , r-. = 1 where n ^ _ f /2 + I r - aj Bi Ln r ^ - a Lr^ (1 - a> (34) (34a) (35) Solution for Pore-Mouth Radius ( r ^ ) At the pore-mouth, is id en tically equal to 1 and equation (34) is d irec tly integrated to give : 1/2 Bi r 10 ~ a 1 - a 1 4" n “"ay u L n r 2 i r jL i 10 - a - a Ln 10 ” a „r^Q (1 - a). . 1 - a . (36) For the case a > . 1, the pore^plugging time can be obtained by setting r-jQ = 0 in equation (36). I t can be shown [Cf. Appendix A] that pore- plugging and incomplete conversion w ill occur when the in itia l porosity 31 ( ■ £ ■ ) of the p article is less than a c ritic a l porosity given by : - O ' ' “ T 1 < 3 7 > Equation (36) provides the leading term in a perturbation solution for C A and r p valid for very small pore-diffusional resistance. Higher order terms can be obtained by carrying out a Regular Perturbation Analysis for the system in terms of the small parameter 0. Exact Solutions to the In fin ite Pore Model For large values of the pore diffusion resistance 0, the fin ite pore can be e ffec tive ly replaced by an in fin ite pore. Then the boundary condition (33b) is replaced by : 0 ; x - * ■ °° (33c) The problem defined by equations (33) and,(34) in the sem i-infinite domain admits an exact solution. F irs t, a hodograph transformation of the variables is made by interchanging the roles of x and r (Appendix B) Variables : Independent Dependent Old System : (x, t) (CA > r-j) N ew System : (r , t) (C^, x) The transformed system in the new variables is governed by the equa tions : J - 9 | r 2 (9 C A/9 r ] ) , 0 2 c A (8 x/Sr-j) 3r-j L 1 (a x /3r-j) ~ g (r-,) Subject to B.C. : (38) C A = 1 ; r-j = r 1 Q(t) (38a) 32 "A 0 * r l = 1 and ax -Bi (1 - a) C A ax at Subject to B.C.: r ] g (r-,) x = 0 r, = r 1Q (t) r, - 1 (38b) (39) (39a) (39b) Guided by a preliminary, asymptotic analysis for large 0 using the Liouville-Green approximation [ 4 5 ] , the solution is assumed to be in the form : r l x = /" f (r) d r (40) where f ( r ) is , as yet, unspecified. Notice that as r-j -> 1; x Therefore, f (r-j) C O as ^ + 1. Then 3x 3r 1 and Sx = f (r-j) = - f (f'm ) (41) at ‘ io 1 v,io where the dot denotes tim e-derivatives. Substituting equations (4 1 )'and (42) into (38) and (39), w e get (42) 1 f (r-|) 8 r r r i f A . 9ri L f i ri) L 3ri »2 ca 9 ('•1) (43) and 33 * 1 0 f <r1 0 > . ' Bi (1 - a) C A f (r-j) r r g ( r ,) (44) Eliminating between (43) and (44) yields an ordinary d iffe re n tia l equation for f(r -j) : 2 d r jf r l d F r 1 g ( r1 > 1 \ f (r-j) d r- 1 L f (r-j) 0 r, (45) Integrating once 2 . i a r r i 9 (ri ) r t - f(n ] ) d r-, [ f (r-,) 1 2 v’ l (rf - 1) (46) Euqation (46) is .fu rth e r integrated by introducing the variable r, g (r-,) w(ri> V - r r p ^ 1 - (47) which transforms (46) to 2 W d W dr 0 9 (r-,) 2 Equation (48) is integrated to give rs /■ (r f - 1 ) (48) (r - 1) d r (49) r1 0 ( t) where W Q = W (r-jq(t ) ). But from the definition of W and B.C. (38b) w e have r-, = 1 , W = 0 (50) Using (49) and (50) gives r.1 0 (t) w ? = 0 2 / -SL .M (r2 , 1 ) d 1 r o (51) 34 which further implies that where W 2 (r-j) = 02 I ( r ,) I (r,) - f (r2 - 1) d r (52) (53) 1 Taking the upper and lower sign to denote a > 1 and a < 1, respectively w e fin a lly get W ( r ,) = ± 0 I 1 / 2 (r-,) (54) and from (47) f (r-j) = t ^ 9 (r-,) 0 I 1/2 (r-j) (55) Substituting (55) in (40) yields : x = + j - t a j V t ) \ , \ (r) « r (56) V ' (r) From (44) and (55) the concentration p ro file for A is obtained: i _ . . r 1 i 1/2 Car, L r r ^ r J "A 'Ao > • “ H F Then the gradient at the pore-mouth is obtained 3Ch x = 0 (57) 3 x r a C a/3 r1 1 L 3x/sr-| J = + 9 ( r T0 - 1} (58) 2 r2 I ^ 2(r ) * r 10 1 i r 1 0 ; r l = r1 0 ( t) For large pore diffusional resistances, the conversion for the fin ite - 35 pore system can be approximated from the solution to the infinite-p ore model. From equation (14), r? - 1 Lim n (t) z d x I (1 - a )(X2 - 1 ) 0 > > 1 , fin ite pore o 1 (1 - a ) ( x Z - 1) J <ri 11 3rl d ^ r10(t) or 5 (t) » ---------------- — f rJr^JLLaJr): d r (59) 0 (1 - a)(x, - 1 ) J I 1/2 (r) r 10(t) The results presented in equations (56) through.(59) are valid up to a certain time t Q, where the solid reactant B is completely depleted at the pore-mouth. Beyond this time, the reaction zone propagates into the pore and a moving boundary problem has to be solved for diffusion and reaction within the pore. For the case a > 1, reactant depletion w ill occur at the pore mouth before the pore plugs whenever. s ' - ^ H r1 <60> (see appendix A). Moving Boundary Analysis This analysis is valid for the cases ( i) a < 1 ; V t > t and 36 ( i i ) a > 1 ; V t > t Q a £o > — ~ For any time t > t Q, the pore can be separated into two zones ’ (Figures 6 and 7 ). Zone I - In this zone, the solid reactant is completely converted and the concentration p ro file fo r gas A can be described by a quasi-steady-state diffusion equation without any reaction-sink.term . Zone I I - In this zone, both diffusion and reaction take place and the system is governed by the original diffusion-reaction equations. The moving reaction boundary [<5(t)] between the two zones is governed by the appropriately matched condi tions for the concentration and gradient of the gas A at the boundary. In Zone I; 0 < x < 6 ( t ) , the steady-state diffusion of A is governed by : 5)2 C A D r-S - = 0 3x subjct to B.C. : C° = 1 ; x - 0 The solution to (61), subject to (61a) is: C° = C ,(t) X + 1 where C |(t) is , as yet, unspecified. In Zone I I ; 6 ( t) s x < the steady-state diffusion with chemical reaction is described by : 37 (61) (61a) (62) 1 2 0 C A/ 3^i ) Ox/Sr-j) 3r-j 1 (Bx/3^) subject to the B.C. 0 2 C A g (r,) C fl + 0 ; x (r-, 1) (63) (63a) At the moving boundary, continuity of concentration and flux dictates r-, = X ca = c; 3C A ■ . ac A 3x 3x (64a) (64b) where X is given by : X2 = a; + (1 - a) x2 Finally, 3x 3t -Bi (1 - a) C A 3X r x 9 ( ^ ) 3 r-, B.C. r-j = X ; x = 6 ( t) r-, = 1 * x = o o B y analogy to the previous analysis for t < t Q, the solution is (65) (66) (6 6 a) (6 6 b) assumed to be of the form : x = s (t) + f (r ) d r (67) where 6 (t) and f ( r ) are, as yet, unspecified. Then, 3 'x arn = f (r’-j) (68) 38 and 3x 3t (69) where 6 ( t) represents the velocity of the moving front. Using equations (63) through (69) a sim ilar analysis is carried out, to yield: r. x = 6 (t) + r a (r) I1/2 (r) d r and C A = + 0 act) I1/2 (r-,) Bi (1 - a) Then 9C^ (3C^/3r-|) 3x “ Ox/ar-j) -r 6( t) 2 Bi (1 - a) f 2 - V The matching condition (63a) yields constant C -j (t) : C -, (t) = - A § (t) where A is a positive constant defined by i1 A = 02 (A2 - 1) 2 Bi A2 (1 - a) Also, the matching condition (64a) yields an O D E for. s (t) : [A 6 ( t ) ,+ B] 6 ( t) = 1 subject to I.C . : At t = t Q ; 6 ( t ) = 0 where B is a positive constant defined by : (70) (71) (72) (73) (74) (74a) 39 „ _ 7 0 l l / Z (A) B ‘ + Bi (1 - a) Equation (74) can be integrated to give : A 6 ( t) + 2 B <5(t) - 2 (t - t Q) = 0 Thus, the reaction front moves according to the equation i 6 (t) - [B2 + 2 A (t - tn ) ] 1/2 - B (75) (76) (77) implying a front velocity of the form 5 (t) = 1 [B2 + 2 A ( t - t 0 )]1/2 (78) Thus, the concentration of gas A and the pore radius at any position in the reaction zone can be e x p lic itly determined by equations (70), (71), (77) and (78). The pore-mouth concentration gradient is given by 3C A 9x = - A S(t) (79) Fin ally, the conversion of the solid reactant can be expressed in terms of the pore-mouth gas flux as follows : t 3C, n (t) 2 Bi 02 (x2 - 1) 10 3x d t 1 (80) x = 0 For times t > t Q, equation (80) becomes — / + \ 2: — j_ 2 Bi A 0 (t) q + — «------- p ------ 0 (x - 1 ) ( [B2 + 2 A (t - t 0 )]1/2 - b}(81) where n is the conversion obtained up to the c ritic a l time t w hen v U 40 the reactant solid is completely depleted at the pore-mouth, and can be obtained from equation (59) by setting r-jQ (t) = A. I t should be again emphasized that equation (81) is valid for the fin ite pore only for cases of very large pore diffusional resistance or fast chemical reaction. 41 CH A PTER V RESULTS A N D DISCUSSION The results from the computer simulations of the analytical solutions obtained in Chapter IV are presented here. The sen sitivity of the system to various parameters is discussed. Based on the results a regime fo r v a lid ity of the solutions is established. Standard lib ra ry subroutines [46] were used to evaluate the double quadratures and roots ( i^ q) of a non-linear equation needed in equations (56) through (81). All the simulations were performed on a D E C 1090 computer at the Engineering Computer Laboratory, USC, using single precision arithm etic. RESU LTS The concentration profiles for gas A inside the pore are shown in Figure 5 for various times. The parameters chosen for the simu lation were : a = 0.5 , 0 = 10.0? Bi = 0.T, eQ = 0.25. For the moving boundary solution, the two zones of quasi-steady state diffusion (Zone I) and diffusion with reaction (Zone I I ) are d e li neated. Figure 6 shows the enlargement of pore radius with time at various depths within the pore for the case cr< 1. The parameters chosen here are the sam e as in Figure 5. For the corresponding case a > 1 , the shrinkage of the pore radius with time is depicted in Figure 7. To highlight the moving reaction zone, here, the parameters chosenwere : a = 1.5, 0 = 10,0, Bi = 0.1, eQ = 0.4. In contrast 42 Computer Simulation t = 10 t = IOO t =200 t =300 zone V T zj Figure 5. Concentration Profiles for G as A Within the Pore. Parameters : a = 0.5 , c b = 10.0 , Bi = 0.1 , sQ = 0.25. 43 3.0 Computer Simulation t = IO O f = 2 0 0 / E / H I zone! t = IO 2.0 1 .0 w 1. 0 2.0 zone I 3.0 1 . 0 0.8 0 . 6 0.4 0.2 x Figure 6. Enlargement of Pore-Radius with Time : case a <1. Parameters : a = 0.5 , ^ = 10.0 , Bi = 0.1 , eQ = 0.25. 44 Computer Simulation 0.5 1=1350 1=550 1 =150 t =4 0.5 zone L «i 1 . 0 1 . 0 0 . 6 0. 8 0.2 0.4 0 x Figure 7. Shrinkage of Pore-Radius with Time : case a > 1. Parameters : a = 1.5 , = 10.0 , Bi = 0.1 , eQ = 0.4. to Figure 7, the case of pore-mouth plugging for low in itia l porosity (e0 = 0.2) is illu s tra te d in Figure 8. The movement of the reaction front into the pore is shown in Figure 9 for the two cases of a < 1 and a > 1 , respectively. To ensure complete conversion of the solid reactant, a value eQ = 0 .4 is chosen for the case a > 1; while a value eQ = 0 .2 5 suffices for the case a < 1. The values chosen for the other'parameters were : 0 = 10.0, Bi = 0 .1 . Figure 10 shows the changes in the concentration gradient of gas A at the pore-mouth with time. The parameters for the two cases a < 1 and a > 1 are the sam e as in Figure 9. For the asymptotic cases of large pore diffusional resistances or fast reactions, the reactant conversion for the fin ite pore m ay be approximated from the solutions to the sem i-infinite pore model. Typical plots of conversion vs. time,' are shown in Figure 11 for the cases a < 1 and a > 1. For these simulations, the system para meters were assigned the values : 0 = 10 and Bi = 0.1. The effect of large pore diffusional resistances (large 0) on the conversion vs. time behavior is shown in Figure 12 for 0 = 20.0, Bi = 0.1. The other asymptotic case of fast reactions (large B i) was also investigated and the corresponding conversion vs. time^ profiles are shown in Figure 13. * DISCUSSION From the extensive range of simulations investigated for various combinations of parameters, i t is evident that the analytical solutions 45 V . - 0 * =7.8 Computer Simulation t p =8.14 0 . 8 1 . 0 Figure 8. Pore-Plugging for Low In itia l Porosity ; case a > 1. Parameters : a = 1.5 , c b = 10.0 , Bi = 0.1 , eQ = 0.2 47 Comouter Simulation c0 =o.4 « s I 5 0.6 Parameters : $ = 10.0 1000 1250 1500 500 750 - e * C O t Figure 9. Propagation of the Reaction Zone within the Pore. 30 Computer Simulation 25 < y 10 o t = 0.5, co = 025 150 200 25C 50 1 0 0 Figure 10. Variation of the Pore-Mouth Concentration Gradient of Gas A with Time. Parameters Computer Simulation « =0.5 €o=0.25 0.8 0.6 0 4 Parameters: 0 = 10.0 0.2 0 250 500 750 1000 1250 1500 t c n Fiaure 11. Typical Plots of Conversion vs,Tirne . I Computer Simulation 0.8 Parameters : 0 = 20.0 0 IOOO 2000 3000 40 0 0 5000 6000 1 c n Figure 12. Plots of Conversion vs. Time for Large Pore-Diffusional Resistances. Computer Simulation 0.8 0. 6 0.4 Parameters : 0 - 10.0 0.2 150 1 0 . 0 12.5 2.5 5.0 7.5 t Figure 13. Plots of Conversion vs. Time for Very Fast Reactions. simulate the fin ite-p o re model well for moderately Targe pore d iffu sional resistances or fast reactions. The concentration of gas A decreases monotonically [figure 5] within the pore. For times greater than t Q, the concentration p ro file can be separated into two d istin ct regions. In zone I , the p ro file is lin e a r, as expected for the case of steady state diffusion without reaction. In zone I I , the concen tration drops to zero steeply due to the combined effects of diffusion and reaction. As complete conversion is approached, the reaction 'penetration depth1 reaches the end of the pore. In such a situation, i the concentration of gas A at the end of the pore deviates slig h tly from zero, implying that the solution is not valid for times close to complete conversion. For the case of a < 1» the model predicts a more 'open' pore structure as the reaction proceeds [figure 6 ]. For t > t Q, the pore radius attains a constant value \ [given by equation (65)] in Zone I; while in Zone I I i t decreases monotonically due to progressively lower rates of reaction within the pore. Again, the solution seems to be valid up to time close to complete conversion. For the case a> 1 , the analytical model is able to predict a shrinking pore radius as the reaction proceeds [figure 7 ]. I t is seen that the moving boundary analysis can predict complete conversion of the solid reactant for s u ffic ie n tly large in itia l porosities of the p article [c rite ria in appendix A]. This is contrasted to the case depicted in Figure 8 where the pore-mouth plugs before complete conversion due to the low in itia l porosity of the p article. Thus, in itia l p article preparation should ensure s u ffic ie n tly large poro- 53 si tie s so that complete conversions can be obtained. The position of the moving reaction boundary can be e x p lic itly determined at various times [figure 9 ]. The reaction zone penetrates into the pore with a faster velocity for the case a < 1 than fo r the case a > 1, a ll other parameters being the same. Thus complete conversion can be attained in a shorter time for the case a < 1. For the case a < 1, the model predicts a monotonically decreasing concentration gradient for the gas A at the pore mouth [Figure 10]. This is to be expected since the pore radius continuously increases with time; so that the reaction zone penetrates deeply into the pore due to enhanced pore-mouth fluxes for gas A. This is in contrast to the case a > 1, where the pore-mouth gradient increases steeply for times less than t Q and decreases gradually for times greater than t .' U p to the time t Q where solid reactant B gets depleted at the pore- mouth, the pore-mouth radius shrinks rapidly with time. Thus the gradient at the pore-mouth w ill increase sharply for t < t Q. Beyond O L ™ 1 the time t Q, the reaction zone moves into the pore, provided eQ > In such cases, concentration gradients in Zone I are expected to be smaller due to pure diffusion without any reaction sink terms. As the reaction proceeds towards complete conversion, steady-state diffusion in Zone I dominates the system almost e n tirely. This would imply that the pore-mouth gradients would rapidly fla tte n out at times approaching complete conversion. From typical simulations for the solid reactant conversion with time [Figure 11], i t is seen that the sem i-infinite pore solutions provide good approximations to the fin ite pore conversion, p articu lar- 54 ly at large values of the pore diffusional resistance (0 > 10) or for very fast reactions (Bi > 1 .0 ). The time required to attain a given conversion increases as a increases. Most of the solid reactant is converted only a fte r the reaction interface begins to recede into the pore. Thus, a moving boundary model, which is valid for time periods almost up to complete conversion, seems to provide a better representation of structure-based gas-solid reactions.. O n increasing the pore-diffusional resistance (large 0 : Figure 12), i t is seen that the time required to achieve a given conversion increases. For a specified reaction at a constant temperature, higher conversions may be obtained at smaller times by ensuring that p article preparation provides small, highly porous particles. For very fast reactions, the sem i-infinite pore solutions yield excellent results since the reaction is confined to a narrow zone near the pore entrance for most times. The end of the pore 'observes' an appreciable concentration gradient of gas A only at times close to complete conversion. As expected, fast reactions reduce the time required to attain a given conversion. The cases a < 1 and a> 1 are illu s tra te d for intermediate and fast reactions in Figure 11 and Figure 13 respectively. From the above discussion, i t is seen that the exact-analytical solutions to the sem i-infinite pore model provide an excellent representation of the fin ite pore model fo r times not too close to complete conversion. I t is expected that the sem i-infinite pore solution would deviate from the true fin ite -p o re solution at large 55 times approaching complete conversion, due to the proximity of the reaction zone to the end of the pore. In p articu lar, the solutions provide good results for the useful asymptotic cases of large pore diffusional resistances (0 > 10) or fast reactions (Bi > 1 .0 ). A sim ilar approach can be used to evolve a slab-model for gas- solid reactions. A completely analogous set of exact solutions have been derived for the case of a sem i-infinite slab. The details of the solutions to the slab-model are shown in Appendix C. I t is expected that these solutions w ill also provide a good representation of the fin ite -s la b model for large diffusional resistances or fast reactions. 56 C H A PTER VI CO NCLUSIO NS A N D R EC O M M EN D A TIO N S F O R FU TU R E W O R K A single-pore model has been formulated for gas-solid reactions involving structural chagges due to significant differences in the molar volumes of the solid reactant and product. The model accounts for the influences of pore diffusion, product layer diffusion, f ir s t - order surface reaction and changes in the solid geometry. By using a new solution technique, exact-analytical solutions have been derived for the equivalent case of a sem i-infinite single-pore model. For large pore diffusional resistances (0 > TO) or fast reactions (Bi > 1 ), these solutions simulate the behavior of the fin ite pore model very w ell. The phenomena of pore-mouth plugging and incomplete conversion for low in itia l porosities were analyzed using this model. For s u ffic ie n tly large i n i t i a l . porosities, the corresponding case of a moving reaction zone and complete conversion was considered. A sim ilar analysis was carried out for the case of slab geometry within the p artic le .. In many gas-solid reactions, bulk flow of the gaseous reactant m ay be important due to substantial concentrations of the reactant gas or deviations from tru ly equimolal counter-diffusion. Also, Knudsen diffusion of the gas may augment bulk diffusion fo r the case where the characteristic pore sizes of the particle are of the sam e orders of magnitude as the mean free path of the reactant gas molecules. 57 The analytical technique for the single-pore model can be extended to account for the effects of bulk flow and Knudsen diffusion of the reactant gas. In som e cases, the kinetics of the gas-solid reaction cannot be idealized by a one-step, firs t-o rd e r chemical reaction. Also, heat effects may play an important role in the case of strongly exothermic or endothermic gas-solid reactions. Such non-linear kinetic behavior, accompanied by heat effects, is especially important in fuel combust ion and gasification-type systems [ 2,8 ]• The applications of the analytical single-pore model to gas-solid reactions involving non lin e a r, non-isothermal kinetics need to be investigated. The single-pore model may also, be modified to allow for the transient behavior of the concentration and temperature profiles within the p a rtic le . These unsteady state conditions may occur during the start-up and shut-down of reactors involving non-isothermal gas-solid reactions. In such cases, i t is expected that the transient behavior of the entire reactant p article m ay be approximated from a s ta b ility analysis of the single-pore model. The non-linear, tim e-invariant single-pore system may be rendered lin ear and tim e-variant by perturbing the system about the transient conditions prevailing at the pore-mouth. In practice, reactant particles participating in gas-solid reactions are composed of pores with a wide distribution of sizes and shapes. Thus, the analytical single-pore model should be extended to allow fo r a distribution of pore sizes within the p a rtic le . The analytical solutions from the single-pore model should be used in 58 conjunction with an evolution-type pore size distribution model employed by Hashimoto, Perlmutter et a l . [31,41] to yield a macroscopic model for diffusive transport and reaction within the p a rtic le . Such evolutionary models for gas-solid reactions within a particle can be used to predict time-varying 'effectiveness factors' for the overall reaction rate within the p article. The information from these generalized structural models for sing le-p article reactions may be incorporated into the overall conservation equations for gas-solid reactors. Subsequently, the analysis for a specific gas-solid reaction may be carried out exclu sively at the macroscopic reactor le v e l, without considering the detailed structural effects in single particles. Thus, the advantage of a hierarchial approach to analyzing structure-based gas-solid reactions lie s in the essential decoupling which is achieved between the conservation relationships for the sing le-particle and the reactor. In perspective, i t is noted that a detailed analysis of the structure-based reaction phenomena at the microscopic level within a single reactant particle is an essential prerequisite to understand ing .the macroscopic transport phenomena involved in gas-solid reactors. Thus, a rigorous model which accounts for a ll the effects associated with the reaction within a single p article needs to be developed prior to an overall analysis of gas-solid reactors. 59 N O M E N C LA TU R E a, b, c, d Stochiometric coefficients in the chemical reation defined by equation {1). A, B, C, D Symbols for the species participating in the chemical reaction defined by equation (1 ). Bi Biot Modulus for reaction (Kr /D ). U C c ^ t ) Time-varying constant in equation (62). C A Concentration of gas A within the pore. C A o Bulk concentration of gas A at the entrance to the pore. c° A Concentration of gas A in Zone I with pure-diffusion. c„ Ar Concentration of gas A within the solid product layer. D Molecular d iffu s iv ity of gas A within the pore. D e Effective d iffu s iv ity of gas A through the solid product layer. f( r) Function defined by equation (40). g(r-) Function defined by equation (35). I ( r ) Integral defined by equation (53). K First-order surface reaction-rate constant. L Length of the pore. L(y) Function for the slab-model : defined by equation (C2). m Moles of reactant B consumed by reaction. Mi Molecular weight of species *i 1. N Ar* Flux of gas A through the solid product layer : defined by equation (3 ). ■ 60 N A x* Flux of gas A through the pore : defined by equation (2 ). r Radical coordinate within the pore and solid product layer. ro In itia l radius of the pore. 4 r l Radius of the pore at any time. r 10 Radius of the pore-mouth at any time. r2 Radius of the reactant/product interface at any time. R In itia l radius of the reactant p article. sg Specific pore-surface area of the p article. t Time. ‘ o Time for solid reactant (B) depletion at the pore-mouth. *p Pore-plugging time fo r the case a > 1. *1 'S keletal' molar volume of species ri ‘ . W fr,) Function defined by equation (47). W o Function defined by equation (51). X Axial coordinate along the pore-length. Half-spacing between the slabs at any time during the reaction for the slab model. Greek Symbols : a Chemical parameter to account for structural changes during reaction. Y Dependent parameter for the slab-model : defined by equation (CIO). « (t) Position of the moving reaction front at any time. ei Porosity of species ' i ' in porous form.' eo In itia l porosity of the reactant p article. 61 n(x,t) Local conversion associated with any point within the pore. n(t) Average conversion associated with the entire pore. A Constant defined by equation (65). pi Density of species 1i '. 0 Thiele modulus for pore-diffusion. X In itia l radius of the solid reactant annulus. V(y) Function defined by equation (C20) for the slab model. Subscripts : c C ritic a l value. A. B. C. D Species involved in the chemical reaction defined by equation (1). Superscripts : . * Dimensional variable. 62 REFERENCES 1. Szekely J ., Evans J.W. and Sohn H.Y., 'Gas-Solid Reactions', Academic Press, N ew York (1976). 2. Evaluation of Coal Gasification Technology, Part I (1972) and Part IT (1974), A d Hoc Panel of the National Academy of Engineering, Washington, D.C. 3. Levenspiel 0 ., 'Chemical Reaction Engi.neeri.ng1, Wiley, N ew York (1972). 4. Lapidus L. and Amundson N.R. (E ditors), 'Chemical Reactor Theory: A Review', Prentice H all, N ew Jersey (1978). 5. Wild R., Chem . Proc Engng. London, 1969 55. 6. Ishida M . and W e n C.Y., AIChE J. 1968, 14 311. 7. Ishida M . and W e n C.Y., C hem Engng. Sci. 1971, 2 6 _ _ 1031. 8. Arri L.E. and Amundson N.R., AIChE J. 1978 24 72. 9. W en C;Y. and W ei L .Y ., AIChE J. 1971 1 7 _ 272. 10. Cannon K.J. and Denbigh K.G., Chem . Engng. Sci. 1957 ^ 145. 11. Shen J. and Smith J.M ., Ind. Engng. Chem. Fundamentals 1965 4 293. 12. Ishida M . and W e n C.Y., Chem. Engng. Sci. 1971 26 1043. 13. Shettigar V.R. and Hughes R., Chem. Engng. J. 1972 3 ^ 93. 14. Luss D. and Am undson N.R., AIChE J. 1969 1 _ 5 194. 15. W en C.Y. and W ang S.C., Ind. Engng. Chem. 1970 6 2 ^ (8) 30. 16. Ramachandran P.A. and Smith J.M ., Chem . Engng. J. 1977 137. 17. Szekely J. and Evans J.W., C hem Engng. Sci. 1970 2 5 ^ 1091. 63 18. Sohn H.Y. and Szekely J.,Chem. Engng. Sci. 1972 2J_ 763. 19. Sohn H.Y. and Szekely J ., Chem . Engng. Sci. 1974 29_ 630. 20. Georgakis C., Chang C.W. and Szekely J ., Chem.Engng. Sci. 1979 24 1072. 21. Trinh T ., Engineer's Thesis, Department of Chemical Engineering, Massachusetts In s titu te of Technology (1978). 22. Hartman M . and Coughlin R.W., AIChE J. 1976 22 490. 23. Hartman M. and Trinka 0 ., Chem . Engng. Sci. 1980 3 5 _ 1189. 24. Hartman M. and Coughlin R.W., Ind. Engng. Chem . Proc. Des. . Develop. 1974 U 248. 25. Petersen E.E., AIChE J. 1957 3 443. 26. C hu C ., Chem . Engng. Sci . 1972 Z 7_ 367. 27. Ramachandran P.A. and Smith J.M ., AIChE J. 1977 2!3_ 353. 28. Ulrichson D.L. and Mahoney D .J., Chem . Engng. Sci. 1980 35 567. 29. Chrostowski J.W. and Georgakis C., Proceedings of the F ifth In t. Symp. on Chem . Reaction Engng. (1978) Houston, Texas. 30. Dogu T ,, Chem . Engng. J. 1981 2 1_ 213. 31. Hashimoto K. and Silveston P .L ., AIChE J. 1973 ]_9 259. 32. Himmelhlau D.M. and Bischoff K.B., 'Process Analysis and Simula tio n ', Chapter 6, Wiley, New York (1968). 33. Hulburt H.M. and Katz S ., C hem , Engng. Sci. 1964 1 _ 9 555. 34. Hashimoto K. and Silveston P.L.,AIChE J. 1973 19 268. 35. Simons G.A. and Finson M.L., Com b. Sci. and Tech. 1979 1 _ 9 217. 36. Simons G.A., Com b. Sci. and Tech. 1979 1 9 ^ 227. 37. Simons G.A., Com b. Sci. and Tech. 1979 20 107. 38. Christman P.G. and Edgar T .F ., Distributed Pore-Size Model 64 (Manuscript), Dept, of C hem Engng., University of Texas, Austin,Texas (1980). 39. Bhatia S.K. and Perlmutter D.D., AIChE J. 1980 2 6 _ 379. 40. Bhatia S.K. and Perlmutter D.D., AIChE J. 1981 2 7 ^ 247. 41. Bhatia S.K. and Perlmutter D.D., AIChE J. 1981 27 226. 42. Simonsson D. and Lindner B ., Chem . Engng. Sci. 1981 3 6 ^ 1519. 43. Bischoff K.B., Chem . Engng. Sci. 1963 1 8 ^ 711. 44. Luss D., Canad. J. Chem . Engng. 1968 4 (5 154. 45. Yortsos Y.C. and Tsotsis T .T ., Chem . Engng. Sci. 1982 3 7 _ 237. 46. S cien tific Subroutine Packages (SSP), IBM Corporation, N ew York (1970). 65 APPENDIX A CONDITIONS F O R PO R E-M O U TH PLUG G ING For the case a > 1, pore-plugging w ill occur at the time when r^ shrinks to zero at the pore-mouth. From equation (25), the corres ponding value for is given by : In the absence of diffusional resistances within the pore, the m axim um conversion is given by Then, the minimum porosity for complete conversion can be obtained as W hen pore diffusional resistances are sig n ifican t, the in itia l porosity of the p article should be greater than the value predicted by equation (A3). Setting r-jQ(t) = 0 in equation (36) gives the pore-plugging time (t ) as : n , m ax (a -°1)(1 - e0) (A2) In contrast, the time required ( t Q) for the solid reactant to deplete at the pore mouth is t 0 - - k [ * - 1] ♦ ? r r L ^ ) [A2 in ( - * ) - . a Ln x] (A5) 66 By c o m p a rin g e q u a tio n s (A 4) and (A5), i t can be shown t h a t whenever t < t (A6) o p eo > ^ (A7) Thus, a conservative condition for complete conversion is that Q L — 1 the in itia l porosity of the p article be greater than — - — , at least. 67 APPENDIX B DETAILS O F THE H O D O G R A P H TRANSFO RM ATIO N In achieving the hodograph transformation {(C A, r-j), ( x ,t ) } -> {(CA, x) , (r-,, t ) } i t is noted that 3 C A 3x 3 CA 3 r, 3 C . . A I + A . J l _ .dr, 3x 31 3x 1 or 3 C A (3 Cft/3 r i j ) 3x ' (3 x/3 r-j) (Bl) Also, 3 > 2 3Ca ' 3 r 2 3CA -j 9 r l + S 2 3 C 1 a 3 1 3 x L 1 3 x J 3 r ] L 1 9 X ’ J 3 x 3 t ’ l B 3 x 3 x or 3 r 2 9 ca _ 1 3 2 i ^ 9 ^A^9 ^ 1 ^ 3 x L 1 3X j (3 x/3 r 1) 3 r 1 [ r l ; . (3 x/3 r 1) (B2) Furthermore, the identity 3 if I 3 x 3 t gives 3x 31 31 3 r n - (3 x/31) (3 x/3 r-j) = -1 (B3) 68 APPENDIX C SOLUTIONS TO THE SLAB-M ODEL By analogy to the pore-model, the dimensionless equations for the slab-model are : 3 3 x L yl SC 1 02 C a Sx J L (y ,) , (Cl) where L (y) = l + y (y - l ) (C2) B.C. : x = 0 * C A = } (C3) x = 1 3Ca - 0 3x (C4) I t m ay be noted that '2 y-|1 is the dimensionless spacing between the slabs and 0 is the slab Thiele modulus given by 02 = K L2 (C5) Also, 3yl _ ' Bi (1 : a) C A (C6) 31 L (y ^ subject to I.C . : t = 0 ; y-| = i (C7) where the Biot modulus (Bi) and the chemical parameter (2t) are defined as R-i - K y o (C8) 69 D I D and a . = d Vp (1 - £g) b (1 - p) (C9) Y is a dependent parameter given by 7 . = Bi a 1 - a (CIO) The spacing at the slab-mouth (y^p) varies according to (G il) t ' Bi (1 - a) " Y ^ ylo ‘ + 2 (y10 " For the equivalent sem i-infinite slab-model, the Hodograph transform ation {(CA, y } ) , (x, t ) } ' { (CA, x) , (yr t) } changes the system equations to i _ r (8c* /ayi ) i = f Z ' (8x/3y-|) 3y1 ^ ^ (3 x/3y-,) with B.C.: yl = y10 (t) y l = 1 L (y-j ) C A " 1 ; C A - 0 and 3x a,t - Bi (1 - a) C L {y'j) A M . where and x = 0 at y] = y]0 (t) x = 0° at yl = 1 (Cl 2) (Cl 3) (C14) (Cl 5) ( Cl 6) (Cl 7) 70 A s before, the solution is assumed to be of the form rl I f (y) d y (CIS) y10 (t) where f (y) is , as yet, unspecified. Denoting the upper and lower signs for the cases a > 1 and a. <1 respectively, a sim ilar set of solutions m ay be obtained : f (y) f t L (y) 0 ^ 7 (y) where Then. x = + y10 (t) i- (y) d v * 1/2 (y) y The concentration p ro file is given by 1/2 ^(y-i) "A ftyio1 J (Cl 9) H y ) = 2 (1 - 2,y)(y - 1) + Y (y - 1) - 2(1 - y ) L n y (C20) (C21) (C22) while the concentration gradient for gas A at the slab-mouth is 3 -C A 0 (y10 - 1) x = o '10 < fy z (y10) (C23) For large pore diffusional resistances or fast reactions, the conver sion for the fin ite -s la b model m ay be approximated as 1 ti (t) -s 1 (y - i), l (y) d y (C 24) y io (t) * 1/2 (y) 71 where x 1S the dimensionless .thickness of the slab. The above results are valid up to the time ( t Q) where the solid reactant gets depleted at the slab-mouth. Beyond this time, the equivalent moving boundary problem can be analyzed for the slab-model. The solutions obtained are completely analogous to the pore-model. As expected, the analysis yields a moving reaction front of the form : <5(t) ~ V T (C25) The exact expression for the moving boundary is given by equation (77) in chapter IV. For the slab-model, the constants 'A' and 'B1 are defined by the equations A = ' £ > 2 V U ) (c9,s A 2 Bi (1 - a) L (A) <C26' B = * ±XL (C27) and Bi (1 - a) where primes denote derivatives with respect to y and the constant 'A1 is given by the equation A = a + (1 - a) x (C28) I t m ay be noted that x denotes the dimensionless thickness of the reactant slab at time (t) = 0 . 72

Linked assets

University of Southern California Dissertations and Theses

Conceptually similar

PDF

Local composition models

PDF

Evaluation of Simha's approach to thermal decomposition of a linear polymer system

PDF

Mechanical characterization of polyolefin blends

PDF

Mathematical analysis of bubble growth enchanced by chemical reaction

PDF

Characterization of the copolymer of styrene and butylmethacrylate

PDF

Rheological behavior of polymer composites containing crosslinked polymers

PDF

Mathematical Modelling of solid-liquid phase transfer catalysis

PDF

Kinetics of isocyanates reactions

PDF

Rheological studies of polymer solutions and filled polymer melts

PDF

Deactivation phenomena by site poisoning and pore blockage. The effects of catalyst size, pore size and pore size distribution

PDF

Bifurcation phenomena in lumped parametric Arrhenius type system with finite activation energies

PDF

Gas-liquid partition chromatography study of the infinite-dilution activity coefficients of water in TEG, PEG, glycerol, and their mixtures

PDF

Fingernail mold simulations by C-MOLD CAE software

PDF

Measurement of electrical conductivity in carbon black filled polymers

PDF

Vapor liquid equilibrium

PDF

Enthalpy relaxation in pure and filled copolymers

PDF

Synthesis and characterization of second-order nonlinear optical polystyrene beads

PDF

Irradiation effects in Ultrahigh Molecular Weight Polyethylene

PDF

A contact angle study at solid-liquid-air interface by laser goniometry and ADSA-CD computer simulation

PDF

Further elucidation of the chemical nature of Green River oil shale

Asset Metadata

Creator Shankar, K. (author)

Core Title A structural model for gas-solid reactions with a moving boundary

Degree Master of Science

Degree Program Chemical Engineering

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag engineering, chemical,OAI-PMH Harvest

Language English

Contributor Digitized by ProQuest (provenance)

Advisor Yortsos, Yanis C. (committee chair), Salovey, Ronald (committee member), Seshadri, Kal (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c20-313288

Unique identifier UC11259471

Identifier EP41808.pdf (filename),usctheses-c20-313288 (legacy record id)

Legacy Identifier EP41808.pdf

Dmrecord 313288

Document Type Thesis

Rights Shankar, K.

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the au...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus, Los Angeles, California 90089, USA