ࡱ> ~  "6u{|xWXY\{ n R S s } rh^e}z%&'cV D E F G H I J #$$$%%%%%%%h)i)* *+++ + + + +.J/K/000000000006G bjbjَ ~}] 4pj*ABBB/[ szXZZZFLL8$\E a~/W/[a~a~\BBcpa~B BXa~X05L l XBHgrgE 8WASHINGTON UNIVERSITY SEVER INSTITUTE OF TECHNOLOGY ____________________________________________________________________ fillin "Type the title of the dissertation (for example, A Study of Corn Futures on the Commodities Exchange)." \* upperPERFORMANCE STUDIES OF TRICKLE BED REACTORS By Mohan R. Khadilkar Prepared under the direction of Prof. M. P. Dudukovic and Prof. M. H. Al-Dahhan ______________________________________________________________________ A dissertation presented to the Sever Institute of Technology Washington University in partial fulfillment of the requirements for the degree of DOCTOR OF SCIENCE August, 1998 Saint Louis, Missouri, USA WASHINGTON UNIVERSITY SEVER INSTITUTE OF TECHNOLOGY ____________________________________________________ ABSTRACT _____________________________________________________________________ fillin "Type the title of the dissertation (for example, A Study of Corn Futures on the Commodities Exchange)." \* upperPERFORMANCE STUDIES OF TRICKLE BED REACTORS by Mohan R. Khadilkar __________________________________________________________ ADVISORS: Prof. M. P. Dudukovic and Prof. M. H. Al-Dahhan ___________________________________________________ August, 1998 Saint Louis, Missouri, USA _______________________________________ A thorough understanding of the interaction between kinetics, transport, and hydrodynamics in trickle bed reactors under different reaction and operating conditions is necessary to design, scale-up, and operate them to achieve the best performance. In this study, systematic experimental and theoretical investigation has been carried out to study their performance in different modes of operation and reaction conditions to improve understanding of the factors governing scale-up and performance. The first part of this study has been focused on comparison of performance of down-flow (trickle bed reactors-TBR) and up-flow reactors (packed bubble columns-PBC) without and with fines to assess their applicability as test reactors for scale-up and scale-down studies for different reaction systems (gas and liquid reactant limited). This has been accomplished by experimentation on hydrogenation of SYMBOL 97 \f "Symbol"-methylstyrene to cumene in hexane solvent over 2.5% Pd on alumina extrudate catalyst as a test reaction, and, by comparing the predictions of existing models for both modes of operation with each other and with the data. It has been shown that trickle bed performs better than up-flow reactor at low pressure, gas limited conditions due to ready access of the gas to the incompletely externally wetted catalyst, whereas up-flow reactor performs better at high pressure, liquid reactant limited conditions due to completely wetted catalyst. Comparison of the two reactors at different pressures, liquid reactant feed concentrations, and gas flow rates has been presented, and differences in performance explained on the basis of the observed shift from gas limitation to liquid limitation. Experiments in beds diluted with fines have been shown to yield identical performance in both up-flow and down-flow modes of operation under both gas and liquid limited conditions corroborating the fact that hydrodynamics and kinetics can be de-coupled by using fines. It has been also shown that the advantage of upflow or downflow depends on whether liquid or gas reactant is rate limiting, and that a single criterion for identifying the limiting reactant can explain most of the data reported in the literature on these two modes of operation. Comparison of the experimental observations and the predictions of the reactor scale and pellet scale models available in the literature have been made with reference to the experimental data. The second part of this study has been devoted to investigating the performance of trickle bed reactors under unsteady state liquid flow modulation (periodic operation) for gas and liquid limited reactions. Periodic operation under gas limited conditions has been shown to ensure completely internally wetted catalyst pellets, direct access of gaseous reactant to the catalyst sites, replenishment of catalyst with liquid reactant, periodic removal of products by fresh liquid, and quenching of a predetermined rise in temperature. Under liquid limited conditions, catalyst wetting and liquid supply to the particles are important, and, periodic operation has been shown to reduce and eliminate liquid maldistribution, ensuring a completely irrigated bed and quenching developing hotspots. Exploitation of the opportunity of alternately and systematically supplying the liquid and gaseous reactants to the catalyst during and after the liquid pulse, respectively, has been shown to result in performance different from that obtained under steady state conditions. Rigorous modeling of the interphase transport of mass and energy based on the Maxwell-Stefan approach at the reactor and catalyst level has been used to simulate the processes occurring under unsteady state conditions for a general multi-component system. The influence of partial wetting of the catalyst due to flow modulation as well as the volatilization of the solvent has been considered in performance prediction. Several strategies such as liquid on-off and liquid high-low flow modulation have been simulated. The effect of interphase transport on the hydrodynamics has also been accounted in the solution of holdup and velocity profiles for unsteady state operation. Results for several cycle times and amplitudes have been discussed with reference to reactor performance for a test hydrogenation reaction. The effect of key parameters such as extent of gas/liquid limitation, total cycle period, cycle split, liquid mass velocity, and liquid solid contacting have been investigated experimentally to demonstrate the cause-effect relationships in unsteady state operation. TABLE OF CONTENTS Page  TOC \o "1-3" LIST OF ILLUSTRATIONS  PAGEREF _Toc417212199 \h ix LIST OF TABLES  PAGEREF _Toc417212200 \h xii CHAPTER 1. INTRODUCTION  PAGEREF _Toc417212201 \h 1 1.1 Motivation  PAGEREF _Toc417212202 \h 3 1.1.1 Comparison of Down-flow (Trickle Bed Reactor-TBR) and Up-flow (Packed Bubble Column-PBC) Reactors  PAGEREF _Toc417212203 \h 3 1.1.2 Unsteady State Operation of Trickle Bed Reactors  PAGEREF _Toc417212204 \h 5 1.2 Objectives  PAGEREF _Toc417212205 \h 10 1.2.1 Comparison of Down-flow (TBR) and Up-flow (PBC) Performance  PAGEREF _Toc417212206 \h 10 1.2.2 Unsteady State Operation of Trickle Bed Reactors  PAGEREF _Toc417212207 \h 11 CHAPTER 2. BACKGROUND  PAGEREF _Toc417212208 \h 13 2.1 Laboratory Reactors Performance Comparison and Scaleup Issues  PAGEREF _Toc417212209 \h 13 2.1.1 Literature on Performance Comparison  PAGEREF _Toc417212210 \h 13 2.1.2 Criterion for Gas and Liquid Reactant Limitation  PAGEREF _Toc417212211 \h 14 2.2 Literature on Unsteady State Operation of Trickle Bed Reactors  PAGEREF _Toc417212212 \h 18 2.2.1 Strategies for Unsteady State Operation  PAGEREF _Toc417212213 \h 21 2.3 Review of Models for TBR Performance  PAGEREF _Toc417212214 \h 30 2.3.1 Steady State Models  PAGEREF _Toc417212215 \h 30 2.3.2 Unsteady State Models for Trickle Bed Reactors  PAGEREF _Toc417212216 \h 37 2.4 Modeling Multicomponent Effects  PAGEREF _Toc417212217 \h 40 CHAPTER 3. EXPERIMENTAL FACILITY  PAGEREF _Toc417212218 \h 43 3.1 High Pressure Trickle Bed Setup  PAGEREF _Toc417212219 \h 43 3.1.1 Reactor and Distributors for Upflow and Downflow  PAGEREF _Toc417212220 \h 44 3.1.2 Gas-Liquid Separator and Level Control  PAGEREF _Toc417212221 \h 45 3.1.4 Liquid and Gas Delivery System  PAGEREF _Toc417212222 \h 46 3.1.5 Data Acquisition and Analysis  PAGEREF _Toc417212223 \h 51 3.2 Operating Procedures and Conditions  PAGEREF _Toc417212224 \h 52 3.2.1 Steady State TBR-PBC Comparison Experiments  PAGEREF _Toc417212225 \h 52 3.2.2 Bed Dilution and Experiments with Fines  PAGEREF _Toc417212226 \h 54 3.2.3 Unsteady State Experiments  PAGEREF _Toc417212227 \h 56 CHAPTER 4. EXPERIMENTAL RESULTS  PAGEREF _Toc417212228 \h 60 4.1 Steady State Experiments on Trickle Bed Reactor and Packed Bubble Column  PAGEREF _Toc417212229 \h 60 4.1.1 Effect of Reactant Limitation on Comparative Performance of TBR and PBC  PAGEREF _Toc417212230 \h 60 4.1.2 Effect of Reactor Pressure on Individual Mode of Operation  PAGEREF _Toc417212231 \h 63 4.1.3 Effect of Feed Concentration of a-methylstyrene on Individual Mode of Operation  PAGEREF _Toc417212232 \h 64 4.1.4 Effect of Pressure and Feed Concentration on Comparative Performance of TBR and PBC in Transition from Gas to Liquid Limited Conditions  PAGEREF _Toc417212233 \h 66 4.1.5 Effect of Gas Velocity and Liquid-Solid Contacting Efficiency  PAGEREF _Toc417212234 \h 70 4.2 Comparison of Down-flow (TBR) and Up-flow (PBC) Reactors with Fines  PAGEREF _Toc417212235 \h 73 4.2.1 Effect of Pressure in Diluted Bed on Individual Mode of Operation  PAGEREF _Toc417212236 \h 76 4.2.2 Effect of Feed Concentration in Diluted Bed on Individual Mode of Operation  PAGEREF _Toc417212237 \h 76 4.3 Unsteady State Experiments in TBR  PAGEREF _Toc417212238 \h 79 4.3.1 Performance Comparison for Liquid Flow Modulation under Gas and Liquid Limited Conditions  PAGEREF _Toc417212239 \h 79 4.3.2 Effect of Modulation Parameters (Cycle Time and Cycle Split) on Unsteady State TBR Performance  PAGEREF _Toc417212240 \h 81 4.3.3 Effect of Amplitude (Liquid Flow and Feed Concentration ) on Unsteady State TBR Performance  PAGEREF _Toc417212241 \h 83 4.3.4 Effect of Liquid Reactant Concentration and Pressure on Performance  PAGEREF _Toc417212242 \h 85 4.3.5 Effect of Cycling Frequency on Unsteady State Performance  PAGEREF _Toc417212243 \h 87 4.3.6 Effect of Base-Peak Flow Modulation on Performance  PAGEREF _Toc417212244 \h 90 CHAPTER 5. MODELING OF TRICKLE BED REACTORS  PAGEREF _Toc417212245 \h 92 5.1 Evaluation of Steady State Models for TBR and PBC  PAGEREF _Toc417212246 \h 92 5.2 Unsteady State Model for Performance of Trickle Bed Reactors in Periodic Operation  PAGEREF _Toc417212247 \h 103 5.2.1 Reactor Scale Transport Model and Simulation  PAGEREF _Toc417212248 \h 106 5.2.2 Flow Model Equations  PAGEREF _Toc417212249 \h 110 5.2.3 Multicomponent Transport at the Interface  PAGEREF _Toc417212250 \h 116 5.2.4 Catalyst Level Rigorous and Apparent Rate Solution  PAGEREF _Toc417212251 \h 119 CHAPTER 6. CONCLUSIONS  PAGEREF _Toc417212252 \h 138 6.1 Recommendations for Future Work  PAGEREF _Toc417212253 \h 140 CHAPTER 7.NOMENCLATURE  PAGEREF _Toc417212254 \h 141 CHAPTER 8. REFERENCES  PAGEREF _Toc417212255 \h 144 APPENDIX A. Slurry Experiments: Intrinsic Rate at High Pressure  PAGEREF _Toc417212256 \h 149 APPENDIX B. Correlations used in Model Evaluation  PAGEREF _Toc417212257 \h 152 APPENDIX C. Flow Charts for the Unsteady State Simulation Algorithm  PAGEREF _Toc417212258 \h 153 APPENDIX D Maxwell-Stefan Equations for Multicomponent Transport  PAGEREF _Toc417212259 \h 154 APPENDIX E Evaluation of Parameters for Unsteady State Model  PAGEREF _Toc417212260 \h 159 APPENDIX F. Experimental Data from Steady and Unsteady Experiments  PAGEREF _Toc417212261 \h 162 VITA  PAGEREF _Toc417212262 \h 163  LIST OF ILLUSTRATIONS Figure Page  TOC \c "Figure 2." Figure 2. 1 Time Averaged SO2 Oxidation Rates of Haure et al. (1990)  PAGEREF _Toc417212263 \h 24 Figure 2. 2 Experimental and Predicted Temperature Profiles of Haure et al. (1990)  PAGEREF _Toc417212264 \h 24 Figure 2. 3 Enhancement in Periodic Operation Observed by Lange et al. (1993)  PAGEREF _Toc417212265 \h 24  TOC \c "Figure 3."  Figure 3. 1 Reactor and Gas-Liquid Separator  PAGEREF _Toc417212267 \h 48 Figure 3. 2 Down-flow and Up-flow Distributor  PAGEREF _Toc417212268 \h 49 Figure 3. 3 Experimental Setup for Unsteady State Flow Modulation Experiments  PAGEREF _Toc417212269 \h 50 Figure 3. 4 Data Acquisition System  PAGEREF _Toc417212270 \h 51 Figure 3. 5 Basket Reactor Catalyst Stability Test.  PAGEREF _Toc417212271 \h 54  TOC \c "Figure 4."  Figure 4. 1 Trickle Bed and Up-flow Performance at CBi=7.8%(v/v) and Ug =4.4 cm/s at 30 psig.  PAGEREF _Toc417212284 \h 62 Figure 4. 2 Comparison of Down-flow and Up-flow Performance at CBi=3.1%(v/v) at 200 psig.  PAGEREF _Toc417212285 \h 62 Figure 4. 3 Effect of Pressure at Low a-methylstyrene Feed Concentration on Upflow Reactor Performance.  PAGEREF _Toc417212286 \h 67 Figure 4. 4 Effect of Pressure at Low a-methylstyrene Feed Concentration (3.1% v/v) on Downflow Performance.  PAGEREF _Toc417212287 \h 67 Figure 4. 5 Effect of Pressure at Higher a-methylstyrene Feed Concentration on Downflow Performance.  PAGEREF _Toc417212288 \h 68 Figure 4. 6 Effect of a-methylstyrene Feed Concentration at 100 psig on Upflow Performance.  PAGEREF _Toc417212289 \h 69 Figure 4. 7 Effect of a-methylstyrene Feed Concentration at 100 psig on Downflow Performance.  PAGEREF _Toc417212290 \h 69 Figure 4. 8 Effect of a-methylstyrene Feed Concentration at 200 psig on Downflow Performance.  PAGEREF _Toc417212291 \h 70 Figure 4. 9 Effect of a-methylstyrene Feed Concentration at 200 psig on Upflow Performance.  PAGEREF _Toc417212292 \h 70 Figure 4. 10 Effect of Gas Velocity on Reactor Performance at 100 psig.  PAGEREF _Toc417212293 \h 72 Figure 4. 11 Pressure Drop in Downflow and Upflow Reactors and Contacting Efficiency for Downflow Reactor at 30 and 200 psig.  PAGEREF _Toc417212294 \h 72 Figure 4. 12 Effect of Fines on Low Pressure Down-flow Versus Up-flow Performance  PAGEREF _Toc417212295 \h 75 Figure 4. 13 Effect of Fines on High Pressure Down-flow Versus Up-flow Performance  PAGEREF _Toc417212296 \h 75 Figure 4. 14 Effect of a-methylstyrene Feed Concentration at Different Pressures on Performance of Downflow with Fines.  PAGEREF _Toc417212297 \h 78 Figure 4. 15 Effect of a-methylstyrene Feed Concentration at Different Pressures on Performance of Upflow with Fines.  PAGEREF _Toc417212298 \h 78 Figure 4. 16 Comparison of Steady and Unsteady State Performance under Liquid and Gas Limited Conditions  PAGEREF _Toc417212299 \h 80 Figure 4. 17 Comparison of Steady and Unsteady State Performance under Liquid and Gas Limited Conditions  PAGEREF _Toc417212300 \h 80 Figure 4. 18 Effect of (a) Cycle Split and (b) Total Cycle Period on Unsteady State Performance under Gas Limited Conditions  PAGEREF _Toc417212301 \h 82 Figure 4. 19 Effect of (a) Cycle Split and (b) Total Cycle Period on Unsteady State Performance under Gas Limited Conditions  PAGEREF _Toc417212302 \h 82 Figure 4. 20 Effect of Liquid Mass Velocity on Unsteady State Performance under Gas Limited Conditions  PAGEREF _Toc417212303 \h 84 Figure 4. 21 Effect of Liquid Reactant feed Concentration (a) and Operating Pressure (b) on Unsteady State Performance  PAGEREF _Toc417212304 \h 86 Figure 4. 22 Effect of Liquid Reactant feed Concentration (a) and Operating Pressure (b) on Unsteady State Performance  PAGEREF _Toc417212305 \h 86 Figure 4. 23 Effect of Cycling Frequency on Unsteady State Performance  PAGEREF _Toc417212306 \h 89 Figure 4. 24 Unsteady State Performance with BASE-PEAK Flow Modulation under Liquid Limited Conditions  PAGEREF _Toc417212307 \h 91 Figure 4. 25 Effect of Liquid Mass Velocity on Steady State Liquid-Solid Contacting Efficiency  PAGEREF _Toc417212308 \h 91  TOC \c "Figure 5."  Figure 5. 1 Upflow and Downflow Performance at Low Pressure (gas limited condition): Experimental data and model predictions  PAGEREF _Toc417212796 \h 100 Figure 5. 2 Upflow and Downflow Performance at High Pressure (liquid limited condition): Experimental data and model predictions  PAGEREF _Toc417212797 \h 100 Figure 5. 3 Effect of Feed Concentration on Performance (Downflow)  PAGEREF _Toc417212798 \h 101 Figure 5. 4 Effect of Feed Concentration on Performance (Upflow)  PAGEREF _Toc417212799 \h 101 Figure 5. 5 Estimates of volumetric mass transfer coefficients in the range of operation from published correlations.  PAGEREF _Toc417212800 \h 102 Figure 5. 6 Phenomena Occurring in Trickle Bed under Periodic Operation  PAGEREF _Toc417212801 \h 104 Figure 5. 7 Representation of the Catalyst Level Solution  PAGEREF _Toc417212802 \h 127 Figure 5. 8: Transient Alpha-methylstyrene ((-MS) Concentration Profiles at Different Axial Locations  PAGEREF _Toc417212803 \h 132 Figure 5. 9 Transient Alpha-methylstyrene Concentration Profile Development with Time (shown in seconds in the legend table).  PAGEREF _Toc417212804 \h 132 Figure 5. 10 Axial Profiles of Cumene Concentration at Different Simulation Times (shown in seconds in the legend table)  PAGEREF _Toc417212805 \h 133 Figure 5. 11 Transient Hydrogen Concentration Profiles at Different Axial Locations  PAGEREF _Toc417212806 \h 133 Figure 5. 12 Transient Liquid Holdup Profiles at Different Axial Locations in Periodic Flow  PAGEREF _Toc417212807 \h 134 Figure 5. 13 Transient Liquid Velocity Profiles at Different Axial Locations in Periodic Flow  PAGEREF _Toc417212808 \h 135 Figure 5. 14 Cumene Concentration Profiles during Periodic Flow Modulation  PAGEREF _Toc417212809 \h 136 Figure 5. 15 Intra-catalyst Hydrogen Concentration Profiles during Flow Modulation for a Previously Externally Wetted Catalyst Pellet at Different Axial Locations.  PAGEREF _Toc417212810 \h 137 Figure 5. 16 Intra-Catalyst Alpha-methylstyrene Concentration Profiles during Flow Modulation for a Previously Externally Dry Catalyst Pellet at Different Axial Locations.  PAGEREF _Toc417212811 \h 137  TOC \c "Figure A."  Figure A. 1 Slurry conversion versus time at different pressures  PAGEREF _Toc417212328 \h 151 Figure A. 2 Comparison of the Model Fitted Alpha-methylstyrene concentrations to experimental values  PAGEREF _Toc417212329 \h 151  LIST OF TABLES Table Page  TOC \c "Table 2." Table 2. 1 Identification of the Limiting Reactant for Literature and Present Data.  PAGEREF _Toc417212387 \h 17 Table 2. 2 Literature Studies on Unsteady State Operation in Trickle Beds  PAGEREF _Toc417212388 \h 28 Table 2. 3 Review of Recent Steady State Reaction Models for Trickle Bed Reactors  PAGEREF _Toc417212389 \h 34 Table 2. 4 Review of Unsteady State Models for Trickle Bed Reactors  PAGEREF _Toc417212390 \h 39  TOC \c "Table 3."  Table 3. 1 Catalyst and Reactor Properties for Steady State Experiments  PAGEREF _Toc417212406 \h 55 Table 3. 2 Range of Operating Conditions for Steady State Experiments  PAGEREF _Toc417212407 \h 55 Table 3. 3 Catalyst and Reactor Properties for Unsteady State Conditions  PAGEREF _Toc417212408 \h 59 Table 3. 4 Reaction and Operating Conditions for Unsteady State Experiments  PAGEREF _Toc417212409 \h 59  TOC \c "Table 5."  Table 5. 1 Governing Equations for El-Hisnawi (1982) Model  PAGEREF _Toc417212826 \h 97 Table 5. 2 Governing Equations for Beaudry (1987) Model  PAGEREF _Toc417212827 \h 98 Table 5. 3 Typical Equation Vector for the Stefan-Maxwell Solution of Gas-Liquid Interface  PAGEREF _Toc417212828 \h 118 Table 5. 4 Typical Stefan Maxwell Equation Vector for a Half Wetted Pellet  PAGEREF _Toc417212829 \h 122 Table 5. 5 Energy Flux Equations for Gas-Liquid, Liquid-Solid, and Gas-Solid Interfaces  PAGEREF _Toc417212830 \h 124 Table 5. 6 Equation Set for Single Pellet Model  PAGEREF _Toc417212831 \h 126 Table 5. 7 Catalyst Level Equations for Rigorous Three Pellet Model  PAGEREF _Toc417212832 \h 128 Table 5. 8 List of Model Variables and Equations  PAGEREF _Toc417212833 \h 129  TOC \c "Table A."  Table A. 1 Rate Constants Obtained from Slurry Data at Different Pressures  PAGEREF _Toc417212865 \h 150  CHAPTER 1. INTRODUCTION Trickle bed reactors are packed beds of catalyst with cocurrent down-flow of gas and liquid reactants, which are generally operated in the trickle flow regime (i.e. the flowing gas is the continuous phase and liquid flows as rivulets and films over the catalyst particles). Trickle beds are extensively used in refining and other petroleum treatment processes. In fact, tonnage wise, they are the most used reactors in the entire chemical and related industries (1.6 billion metric tonnes annual processing capacity (Al-Dahhan et al. (1997)), with millions of dollars invested in design, set-up and operation of these type of reactors. Trickle bed reactors have several advantages over other type of multiphase reactors such as: plug flow like flow pattern, high catalyst loading per unit volume of liquid, low energy dissipation, and greater flexibility with respect to production rates and operating conditions used. Despite these advantages, trickle bed reactors have not found applications to their fullest potential due to difficulties associated with their design and uncertainty in the scale-up strategies used for their commercial application. These difficulties are introduced by incomplete catalyst wetting and flow maldistribution in industrial reactors which are not predicted by laboratory scale experiments (Saroha and Nigam, 1996). In order to expand the horizon of applications of these reactors, it is necessary to understand all the relevant complex phenomena, on the macro, meso and micro scale, that affect their operation. Scale-up strategies for trickle bed reactors to date have been considered an art and have not been developed beyond the realm of hydrodesulphurization and to some extent, hydrotreating. The proper choice of laboratory reactors for testing of catalysts and feedstocks, in order to scale up or scale down, has not been dealt with comprehensively. This results in commercial trickle bed reactors which are either grossly over-designed or perform poorly below design criteria. At the same time, a reliable method for scale-down in investigating new catalysts and feedstocks is needed for rapid selection of optimal processing conditions and for cost effective performance. Thus rationalization of scale-up procedures is needed. Since the investment (to the order of millions of dollars) has already been made, it may be worthwhile to investigate if a strategy exists to obtain an enhancement in performance by modifying the method of operation such as unsteady state operation, and whether an optimal performance can be obtained with the existing setup by using this strategy. Any enhancement in performance of the pre-existing reactors, even by a few percent, would translate to a significant financial gain without further capital investment. Also, any small improvement in the design of new reactors can also lead to substantial savings in the future. The major goal of this study is to conduct systematic experimental and theoretical comparison of the performance of trickle bed reactors under different modes of operation. The first part focuses on comparison of performance of laboratory scale trickle bed (TBR, down-flow) and packed bubble column (PBC, up-flow) reactors, without and with fines, to ascertain their use as test reactors for scale-up and scale-down studies based on different reaction systems (gas and liquid limited). The second part focuses on studying (theoretically and experimentally) the effect of periodic operation of trickle bed reactors on their performance and the magnitude of this effect as a function of the system used (gas and liquid limited) over a wide range of operating conditions that covers from poorly irrigated to completely wetted beds. 1.1 Motivation 1.1.1 Comparison of Down-flow (Trickle Bed Reactor-TBR) and Up-flow (Packed Bubble Column-PBC) Reactors Trickle bed reactors are packed beds of catalyst over which liquid and gas reactants flow cocurrently downwards, whereas in packed bubble columns the two phases are in up-flow. Trickle bed reactors, due to the wide range of operating conditions that they can accommodate, are used extensively in industrial practice both at high pressures (e.g. hydroprocessing, etc.) and at normal pressures (e.g. bioremediation, etc.). In laboratory scale trickle beds (typically few inches in diameter) packed with the commercially used catalyst shapes and sizes (the reactor to catalyst particle diameter ratio is undesirably low), low liquid velocity is frequently used in order to match the liquid hourly space velocity (LHSV) of the commercial unit. These conditions give rise to wall effects, axial dispersion, maldistribution and incomplete catalyst wetting which are not observed to the same extent in commercial reactors. Hence, in laboratory reactors, an accurate estimate of catalyst wetting efficiency is essential to determine their performance (Dudukovic and Mills, 1986, Beaudry et al., 1987). The reaction rate over externally incompletely wetted packing can be greater or smaller than the rate observed over completely wetted packing. This depends on whether the limiting reactant is present only in the liquid phase or in both gas and liquid phases. For instance, if the reaction is liquid limited and the limiting reactant is nonvolatile, such as occurs in some hydrogenation processes, then a decrease in the catalyst-liquid contacting efficiency reduces the surface available for mass transfer between the liquid and catalyst causing a decrease in the observed reaction rate. However, if the reaction is gas limited, the gaseous reactant can easily access the catalyst pores from the externally dry areas and consequently a higher reaction rate is observed with decreased level of external catalyst wetting (Dudukovic and Mills, 1986). Thus, the difficulties of using trickle bed reactors in laboratory scale investigation for scale-up and scale-down are mainly caused by the interactions between the gas, the liquid and the solid-catalyst phases; all of these interactions being strongly dependent on the reacting system used. Hence, up-flow reactors are frequently used in laboratory scale studies for testing catalysts and alternative feedstocks for commercial trickle bed processes, since in them complete catalyst wetting is ensured and better heat transfer (due to continuous liquid phase), and higher overall liquid-solid mass transfer coefficients can be achieved. However, as will be shown in the present study, and as the diversity of literature results discussed herein indicate, the relative merit and the performance of up-flow and trickle beds is dependent on the reaction system used. Up-flow as a test reactor may not portray the trickle bed reactor performance for scale-up and scale-down for each and every reaction and operating condition. It is therefore important to investigate the comparative performance of both reactors in order to address the following important questions : a) When will up-flow outperform down-flow and vice versa? b) When can up-flow be used to produce accurate scale-up data for trickle bed operation? Another alternative for scale-up and scale-down studies that is practiced in industry is the use of trickle bed reactors diluted with fines (which are inert particles an order of magnitude smaller in size compared to the catalyst pellets). The lack of liquid spreading (due to the use of low liquid velocities) in laboratory reactors is compensated by fines which provide additional solids contact points over which liquid films flow. This improved liquid spreading helps achieve the same liquid-solid contacting in laboratory reactors as obtained in industrial units at higher liquid velocities. Fines can thus decouple the hydrodynamics and kinetics, and provide an estimate of the true catalyst performance of the industrial reactor by improving wetting and catalyst utilization in a laboratory scale unit at space velocities identical to those in industrial reactors. The diluted bed studies reported in the open literature investigated the performance of down-flow only, but did not compare it with up-flow performance (without or with fines), nor did they incorporate the impact of the reaction system (Van Klinken and Van Dongen, 1980; Carruthers and DeCamillo, 1988; Sei, 1991; Germain, 1988; and Al-Dahhan,1993). It is noteworthy to mention that the use of fines in up-flow reactors would eliminate the possibility of channeling, while still making use of the improved spreading due to dilution. Most of the studies reported in open literature deal with atmospheric pressure air-water systems an very little is available at high pressure air-water systems and very little is available at high pressure at which the transport and kinetics may be quite different and result in completely different performance as will be shown. A recent review by Al-Dahhan et al. (1997) elaborates on high pressure hydrodynamic studies and touches upon the lacunae of reaction studies at high pressure to complement them. 1.1.2 Unsteady State Operation of Trickle Bed Reactors Industrial trickle bed reactors are commonly operated at high pressures and temperatures in order to obtain desirable reaction-transport behavior in these reactors. High pressures are used to remove or reduce the extent of gas reactant limitation and reduce the extent of deposition of byproducts or poisons on the active surface of the catalyst. Higher temperatures are used to improve reaction rates and fluidity (in case of petroleum feeds). In hydrogenation applications, high temperatures give the added advantage of higher hydrogen solubility and can thus reduce operating pressures. Traditionally, trickle beds have been designed with the objective of steady state operation through a stepwise empirical approach. The development of alternative approaches for designing and operating existing and new reactors depends on a better understanding of their performance under different operating conditions. Thorough understanding and optimal design of trickle bed reactors is complicated by the presence of multiple catalyst wetting conditions induced by two-phase flow, which can affect the reactor performance depending upon whether the reaction is gas reactant limited or liquid reactant limited (Mills and Dudukovic, 1980). Typically, high liquid mass velocities and completely wetted catalyst are desirable for liquid limited reactions, whereas low liquid mass velocities and partially wetted catalyst are preferable in gas limited reactions. Due to the competition between the phases to supply reactants to the catalyst, the possibility of performance enhancement by operating under unsteady state conditions exists in these reactors. Unsteady state (periodic) operation can, in principle, yield better performance, reduce operating pressures, improve liquid distribution, and help control temperature. It has not been used industrially as an established strategy, and optimal conditions for any given reaction system are not yet available for its implementation. Unsteady state operation has been shown to yield better performance in other multiphase reactors on an industrial scale but has not been tried on industrial trickle beds due to difficulty in prediction and control of unsteady state conditions. It is also unknown as yet as to the choice of the tuning parameters which can give an optimal performance under periodic operation. Industrial implementation of unsteady (periodic) operation in trickle bed reactors will follow rigorous modeling, simulation, and laboratory scale experimental investigation on test reaction systems. Unsteady state operation can also yield a better insight into the reaction-transport phenomena occurring during steady state operation. Performance improvement in reactors, both new and existing ones can have significant economic impact due to high capital costs and large capacity, particularly in the refining industry. Due to the empirical approach in design of trickle beds without recourse to any serious modeling unsteady state operation has not been seriously considered. But the development of advanced computational tools, and hence predictive capabilities, demands rethinking of existing strategies for design and operation. The modeling effort in unsteady state trickle bed operation has also been preliminary and inconclusive (Stegasov et al., 1994) due to the large (and previously unavailable) computational effort required resulting in a lack of a generalized model or theoretical analysis of the phenomena underlying unsteady state performance. The use of periodic operation has been attempted on laboratory scale trickle beds for very few studies. With the advent of advanced control strategies it is now plausible to consider reaping benefits obtained by operating industrial trickle beds under unsteady state conditions. The use of trickle bed reactors under unsteady state or periodic conditions is motivated by different factors depending upon the reacting system used. Before getting into details of periodic operation we must address the two scenarios under which such a strategy can be employed to achieve improvement in performance. A. Gas Limited Reactions These are reactions where the limiting reactant is in the gas phase, and the performance of the trickle bed is governed by the access of this gaseous reactant to the catalyst. The access of the limiting reactant, through the catalyst areas wetted by liquid, to the catalyst particle is subject to an additional resistance due to the presence of the external liquid film, and higher wetting of the catalyst under these circumstances can be detrimental to the accessibility of the gaseous reactant to the reaction sites in the catalyst. The externally dry zones which exist at low liquid mass velocities, and the resulting incomplete external catalyst wetting, results in improved performance of the trickle bed reactor for a gas limited reaction (Beaudry et al., 1987). But for this to happen, the catalyst must be completely internally wetted and replenished with liquid phase reactant from time to time to avoid liquid limited behavior to occur in these externally dry pellets. In trickle bed reactors operated at steady state conditions, this may not occur due to the fact that the externally dry pellets may not get fresh liquid reactant frequently enough, and may remain depleted of liquid reactant indefinitely, so that any advantage gained due to easier access of gas through dry areas to the particles will be negated. This can be accentuated by the localized temperature rise and local increased evaporation of the liquid reactant, and the advantage due to partial wetting may not be seen after and initial surge in reaction rate. This temporary advantage could be sustained if the catalyst were to be doused with liquid reactant followed by supply of gaseous reactant with very low external wetting, thus facilitating easier access to the catalyst which is full of liquid reactant. If this were to be induced periodically, the maximum benefit of partial wetting could be achieved. It has also been observed in several cases that the (liquid phase) product(s) of the reaction may be the cause of decreased catalytic activity or may exhibit an inhibiting effect on the progress of reaction in catalyst pores (Haure et al., 1990). This necessitates the periodic removal of products by large amount of fresh solvent or liquid and restoration of catalytic activity. The maldistribution of the liquid may also cause local hot spots to develop in the zones where liquid may not wet the catalyst at all under steady state conditions. The reaction rates in this zone of higher temperatures may be higher than in the zone of actively wetted catalyst area, and may result in complete evaporation of the liquid reactant and very high temperatures resulting in catalyst deactivation. This can be prevented or put to productive use in periodic operation by allowing a predetermined rise in the catalyst temperature after which introduction of fresh liquid will bring the temperature down to a lower operating temperature. B. Liquid Limited Reactions Many industrial trickle bed reactors operate under high pressures, typically 10-20 MPa, at which the extent of gas limitation is no longer significant due to high liquid phase concentration of the gaseous reactant (high solubility at high pressure). In fact, they operate under liquid limited conditions at which the extent of external catalyst wetting is tied intimately to reactor performance, and the higher the wetting the better the performance. Due to the nature of the liquid flow, it tends to occur in rivulets which prefer to flow over externally pre-wetted areas rather than dry ones. This leads to parts of the catalyst external areas remaining dry or inactively wetted until there is a change in flow (and hence wetting) over that area. Poorly irrigated beds are a cause of concern in industrial reactors due to the reasons mentioned above (along with imperfect distributors and flow patterns in large reactors). The objective in an ideally wetted bed is to wet every area of the entire catalyst by film flow which would achieve maximum performance at a given flow rate. It is also desirable to eliminate developing hot spots, caused by the above mentioned phenomenon, by ensuring complete wetting, at least for some duration of time after which the liquid has a much higher number of alternative paths to flow over the catalyst. This would achieve better film-like wetting of the catalyst (even at low flow rates) resulting in higher catalyst utilization (i.e., higher conversions could be achieved in shorter bed heights thus reducing pressure drops) and eliminating hot spots at the same time. A high flow rate slug would also help remove stagnant liquid pockets by supplying fresh reactants and removing products. It can be seen from the above discussion that under both scenarios of operation it may be advantageous to consider periodic operation of trickle beds to achieve maximum performance in existing reactors as well as to set-up new reactors designed to operate under periodic operation. A comprehensive study of these phenomena is not available in literature for a wide range of operating conditions necessary to attempt industrial or even pilot scale implementation of unsteady state operation. 1.2 Objectives The main objectives of both parts of this study are outlined below. Details of the implementation of experiments and modeling are discussed in Chapter 3 and Chapter 5, respectively . 1.2.1 Comparison of Down-flow (TBR) and Up-flow (PBC) Performance I. Experimental Studies The objective of this part of the study were set to be as follows To investigate the comparative performance of laboratory trickle bed (TBR) and up-flow reactors (PBC) under gas and liquid reactant limited conditions using hydrogenation of a-methylstyrene to cumene in hexane solvent over 2.5% Pd on alumina 1/16" extrudate catalyst as a test reaction. Examine the effect of operating parameters such as pressure, feed concentration liquid-solid contacting, gas velocity, etc., on upflow (PBC) and downflow (TBR) performance. To study the effect of bed dilution (with inert fines) on the comparative performance of upflow (PBC) and downflow (TBR) performance. Determine and recommend the most suitable mode of operation for scale-up and scale-down studies for trickle bed reactors under different reaction and operating conditions. Study the effect of high pressure on intrinsic (slurry) reaction rate and determine kinetic parameters based on these experiments. II. Model Predictions The objective determined for this part was Test the experimental data obtained in part 1 by comparing the model predictions of the models developed at CREL by El-Hisnawi (1982) and Beaudry et al. (1987) for trickle bed and upflow reactors. Suggest improvements in the model suggested, if necessary, for cases where both reactants could limit the reaction or if model predictions are qualitatively incorrect. 1.2.2 Unsteady State Operation of Trickle Bed Reactors The goals of this part of the study can be summarized in two sub parts as follows : II. Experimental Study of Periodic Operation The objective here is to experimentally investigate the effect of liquid flow modulation (periodic operation) on the performance of a test reaction under steady state and unsteady state conditions. To examine effect of reactant limitation i.e., gas and liquid limited conditions on the comparative performance. Examine the effect of operating pressure and feed concentration on performance under unsteady state conditions. Investigate the effect of periodic operation parameters such as total cycle period, cycle split, cycling frequency for the operating condiions under which performance enhancement is observed. Examine both ON/OFF and BASE/PEAK flow modulation under some of the conditions chosen on the basis of above results. II. Model Development and Solution The objective of this part of the study were To develop a generalized model for trickle bed reactors which is capable of simulating unsteady state behavior and capture the phenomena observed in the literature and our experiments for periodic operation. Another objective of this exercise was to quantify the enhancement in performance with respect to parameters such as cycle period, cycle split, amplitude and allowable exotherm. The investigation of the distribution of velocities and liquid holdups during periodic operation using multiphase flow codes (CFDLIB of Los Alamos), or solution of one dimensional momentum equations (for both gas and liquid phase), will be attempted to demonstrate the qualitative picture and provide a physical basis for the model. CHAPTER 2. BACKGROUND 2.1 Laboratory Reactors Performance Comparison and Scaleup Issues Goto No systematic study has been reported which compares the performance of down-flow and up-flow operation over a wide range of operating conditions, particularly reactor pressure. The few studies that are available in the open literature do not relate the observed performance to the type of reaction system used (gas-limited or liquid-limited), nor do they conclusively elucidate which is the preferred reactor for scale-up/scale-down. 2.1.1 Literature on Performance Comparison Goto and Mabuchi (1984) concluded that for atmospheric pressure oxidation of ethanol in presence of carbonate, down-flow is superior at low gas and liquid velocities but up-flow should be chosen for high gas and liquid velocities. Beaudry et al. (1987) studied atmospheric pressure hydrogenation of a-methylstyrene in liquid solvents at high liquid reactant concentrations and observed the down-flow performance to be better than up-flow except at very high conversion. Mazzarino et al. (1989) observed higher rates in up-flow than in down-flow for ethanol oxidation and attributed the observed phenomenon to better effective wetting in up-flow without considering the type of reaction system (gas or liquid limited). Liquid holdup measurements at elevated pressure using water/glycol as liquid with H2, N2, Ar, CO2 as the gas phase by Larachi et al. (1991) indicate that liquid saturation is much greater in up-flow than in downward flow at all pressures (up to 5.1 MPa). Lara Marquez et al. (1992) studied the effect of pressure on up-flow and down-flow using chemical absorption, and concluded that the interfacial area and the liquid side mass transfer coefficient increase with pressure in both cases. Goto et al. (1993) observed that down-flow is better than up-flow at atmospheric pressure (for hydration of olefins), and noted that the observed rates in down-flow were independent of gas velocity while those in up-flow were slightly dependent on it. Thus, there is no clear guidance as to which reactor will perform better for a given reaction system. An extensive study of the effect of operating conditions is necessary to understand the interplay of factors in the particular reacting system in order to explain why these reactors perform differently and whether up-flow can be used for scale-up of trickle bed reactors. This study provides the rationale behind the literature results and their conclusions. This gives us the rules by which to 'a priori' judge whether an up-flow or down-flow reactor is to be preferred for laboratory testing. 2.1.2 Criterion for Gas and Liquid Reactant Limitation The performance of up-flow and down-flow reactors depends upon the type of reaction, i.e., whether gas (reactant) limited or liquid (reactant) limited. A simple and usable criterion for establishing gas or liquid limitation is needed. In order to obtain such a criterion for the complex processes involved, a step by step comparison of the different transport processes contributing to the observed rate, as illustrated below, is required. For a typical reaction A(g) + bB(l) = Products(l), the limiting step can be identified by first comparing the estimated rates of mass transfer with the observed reaction rates. The estimated volumetric mass transfer coefficients for the system under study can be evaluated from appropriate correlations in the literature (e.g., (ka)GL from Fukushima and Kusaka (1977), and kLS from Tan and Smith (1982) or Lakota and Levec (1989) listed in Appendix C). The comparison of maximum mass transfer rates, with the experimentally observed rates, (rA)obs as per inequality (I), where CA* is the gas solubility at the conditions of interest, confirms that external gas reactant mass transfer does not limit the rate, if inequality (2.1) is satisfied. EMBED Equation  (2.1) The observed rate in the above criterion is the mean rate for the reactor evaluated from the mass balance on the system. For systems where the conversion space time relationship is highly nonlinear, criterion (I) should be applied both at the entrance and at the exit conditions of the reactor. If inequality (I) is satisfied, the limiting reactant can then be identified by further comparing the effective diffusivity terms with the observed rate. This can be achieved by evaluation of the Weisz modulus (SYMBOL 102 \f "Symbol"We= (rA)obs(VP/SX)2/(DeC)), (where DeC is the smaller of the two, DeBCBi/b or DeACA*) (which for our reaction system yielded SYMBOL 102 \f "Symbol"We > 1 (see Table 1)). In order to identify the limiting reactant in case of SYMBOL 102 \f "Symbol"We > 1, the diffusional fluxes of the two reactants should be compared (Doraiswamy and Sharma, 1984), whereas for SYMBOL 102 \f "Symbol"We < 1, it is the ratio of the liquid reactant concentration and the gas reactant dissolved concentration that counts. The ratio (SYMBOL 103 \f "Symbol" = (DeB CBi ) /b(DeA CA* )) is indicative of the relative availability of the species at the reaction site. Thus, a value of SYMBOL 103 \f "Symbol" >> 1 would imply a gaseous reactant limitation, while SYMBOL 103 \f "Symbol" << 1 indicates liquid reactant limitation for the conditions mentioned above. This criterion is relied on for analyzing our results as well as the literature data. The limiting reactant in a gas-limited reaction can enter the porous particles through both the actively and inactively wetted surfaces, but it enters at different rates (Mills and Dudukovic, 1980). Accordingly, for a gas limited reaction, the trickle bed reactor is expected to perform better due to its partially wetted catalyst over which gas reactant has an easy access to the particles than the up-flow reactor, in which the only access of the gaseous reactant to the catalyst is through the liquid film engulfing the catalyst. In a liquid-limited reaction, the liquid reactant can only enter the catalyst particle through its actively wetted surface, leaving the inactively wetted areas unutilized. Liquid limited conditions therefore result in a better performance in up-flow, where particles are completely surrounded by liquid, than in down-flow where particles may be only partially wetted. The literature data confirm the above assertion and give support to the use of the proposed criterion for identifying the limiting reactant (values of SYMBOL 102 \f "Symbol"We and SYMBOL 103 \f "Symbol" are listed in Table 1). The experimental data of Goto and Mabuchi (1984) have SYMBOL 103 \f "Symbol" values around 300 (approximate estimate using reported concentration and molecular diffusivities), which indicate clearly a gas limited behavior, and so their observation that down-flow performs better than up-flow at low liquid and gas velocities is a forgone conclusion. Beaudry et al. (1987) operated under SYMBOL 103 \f "Symbol" values ranging between 20 and 100, and again down-flow outperformed up-flow, except at very high conversion when SYMBOL 103 \f "Symbol" was lower than 3 (i.e., of order one) and liquid reactant limitations set in , as explained by Beaudry et al. (1986), in which case up-flow tends to perform close to down-flow. The range of SYMBOL 103 \f "Symbol" values (SYMBOL 103 \f "Symbol"=0.5 to 17) encountered by Mazzarino et al. (1989), show both liquid and gas limited regimes. At one set of conditions (SYMBOL 103 \f "Symbol"=0.5) liquid limitation is indicated based on our criterion and the fact that upflow outperforms down-flow can be anticipated. However Mazzarino et al. (1989) report that upflow performs better than downflow even at SYMBOL 103 \f "Symbol"=17, which should be a gas limited reaction. It may be noted that their experiments in down-flow (Tukac et al., 1986) and up-flow (Mazzarino et al., 1989), while using presumably the same catalyst, were not performed at the same time. Our experience with active metals on alumina catalyst (Mazzarino et al. used Pt on alumina) is that exactly the same state of catalyst activity is very difficult to reproduce after repeated regeneration. This sheds some doubt whether the two sets of data can be compared. In addition, at the temperature used, ethanol will not behave as a non-volatile reactant and does not satisfy the conditions of our hypothesis. Goto et al. (1993) report down-flow performance superior to up-flow for their reaction system, for which estimates of SYMBOL 103 \f "Symbol" yield values of over 8000, which is why any short circuiting of the gas via the dry areas yields higher gas transfer rates to particles and hence higher conversion in down-flow. Table 2.  SEQ Table_2. \* ARABIC 1 Identification of the Limiting Reactant for Literature and Present Data. ________________________________________________________________________ Rate(obs) mol/m3.s Vp/Sx m CA* mol/m3Cbi mol/m3DeB/DeAWeisz Modulus (SYMBOL 102 \f "Symbol"We)Gamma (SYMBOL 103 \f "Symbol")Limiting ReactantGoto (1984)1.50E-029.50E-040.9776.00E+020.5139.5314GasBeaudry (1987)1.73.24E-043.761.71E+030.2021.092GasMazzarino(1989) Extreme I*5.0E-045.58E-040.550.550.511.520.51LiquidExtreme II*3.0E-025.58E-041.6655.310.5115.317GasGoto(1993)1.0E-021.3E-0412.05.5E+042.20.0410300GasThis studyExtreme I**1.42.91E-0414.05200.23107.78.8GasExtreme II**1.12.91E-04632.37E+020.2321.40.87Liquid_______________________________________________________________________ Extreme I* : Low Liquid Reactant Feed Concentration, Atmospheric Pressure. Extreme II* : High Liquid Reactant Feed Concentration, Atmospheric Pressure. Extreme I** : High Liquid Reactant Feed Concentration, Low Pressure. Extreme II** : Low Liquid Reactant Feed Concentration, High Pressure. 2.2 Literature on Unsteady State Operation of Trickle Bed Reactors The earliest literature studies investigating unsteady state behavior in chemical systems are those of Wilhelm et al. (1966) and Ritter and Douglas (1970). They have investigated parametric pumping and frequency response of stirred tanks and heat exchangers, and, have observed higher unsteady state performance with introduction of concentration or velocity disturbances. Several studies have also looked at thermal cycling and flow reversals in fractionators, ion exchangers, and adsorbers for operating under unsteady state to overcome equilibrium limitations (Sweed and Wilhelm (1969); Gregory and Sweed (1972); Wu and Wankat (1983); Park et al., 1998). In case of trickle bed reactors however, unsteady state operation has been considered only in the past decade or so and several strategies such as modulation of flow, composition, and activity have been suggested (Silveston, 1990). Modulation of flow of gas or liquid is done to get desired ratio of liquid and gaseous reactants on the catalyst as well as to allow a controlled exotherm (Gupta, 1985; Haure, 1990; Lee et al., 1995). Modulation of composition can improve selectivity or control phase change by addition of inerts or products (Lange et al., 1994), or injecting gaseous cold shots (Yan, 1980). Modulation of activity is done usually by an extra component, which can help catalyst regeneration and prevent build up of poisons or inhibitors from the catalyst (Chanchlani et al., 1994; Haure et al., 1990). None of these has been used industrially as an established strategy due to lack of knowledge of desirable operating parameters for any given reaction system. Rigorous experimental and modeling effort is necessary to understand the phenomena underlying unsteady state operation for gas and liquid limited reactions. This study is a step in that direct and contributes in expanding the knowledge and understanding of unsteady state behavior of trickle bed reactors. Literature results on unsteady state operation in trickle bed reactors are summarized in Table 1 and only key observations are briefly discussed here. Haure et al. (1990) and Lee et al. (1995) have studied periodic flow modulation of water for SO2 oxidation to obtain concentrated sulfuric acid. They have observed an enhancement in supply of SO2 and O2 to the catalyst during the dry cycle, resulting in higher performance and temperature rise of up to 10-15 oC. They have also attempted to model unsteady state operation based on the Institute of Catalysis approach (Stegasov et al., 1994) and have been able to predict the rate enhancement with limited success. Lange et al. (1994) have experimentally investigated 1) hydrogenation of cyclohexene, and 2) the hydrogenation of alpha-methyl styrene on Pd catalysts by manipulation of liquid feed concentration and feed rate, respectively. They have used non-isothermal composition modulation for cyclohexene to control conversion and keep the reaction system from switching from a three phase system to a two phase one, and, have designed their total cycle time based on this criterion. For the case of hydrogenation of alpha-methylstyrene under isothermal conditions, the authors have observed maximum improvement at a cycle period of 8 min. at cycle split of 0.5. The observed improvement (between 2 and 15 %) has been attributed to better wetting due to the pulse (removal of stagnant liquid). The present study extends work on this system to look at the performance governing factors for a wider range of operating and reaction conditions. Chanchlani et al. (1994) have studied methanol synthesis in a fixed bed (gas-solid system) under periodic operation. They have studied composition modulation under isothermal conditions to observe improvement in performance of methanol synthesis (from CO and H2) in a packed bed reactor by taking advantage of the peculiarity of their reaction system which shows intrinsically higher rates with some CO2 present in the feed. They have observed significant improvement in some cases but also obtained performance worse than the average steady state in others. They have obtained 20-30 % improvement with cycling between H2-CO2 mixtures with 30/70 in partial cycle and 95/05 in the rest. The authors attribute the improvement due to activation of the catalyst surface structure due to composition modulation between H2 and CO2. Although this is a gas phase catalytic reaction under their operating conditions, the use of composition modulation to achieve improvement is well illustrated through their experiments. Most of the studies reported in open literature are restricted to gas limited conditions with a very limited number of data available for the few studies published. Periodic operation under gas limited conditions can ensure completely internally wetted catalyst pellets, direct access of gaseous reactant to the catalyst sites, replenishment of catalyst with liquid reactant, periodic removal of products by fresh liquid, and quenching of a predetermined rise in temperature. Under liquid limited conditions, wetting and liquid supply to the particles is crucial, and periodic operation can reduce and eliminate liquid maldistribution, ensure a completely irrigated bed, and, quench developing hotspots. Several industrial reactors are operated under liquid limited conditions at high pressure and suffer from maldistribution of the liquid reactants, which can cause externally dry or even internally completely dry catalyst pellets. At high liquid and gas mass velocities, under pulsing flow regime, a significant improvement in catalyst wetting and effective removal of hot spots has also been reported (Blok and Drinkenburg, 1982). But this is not always practical in industrial reactors due to a large pressure drop and little control over the slugging process. Periodic flow modulation with a low base flow and a periodic slug of very high liquid flow can improve catalyst utilization even at low mean liquid flows (lower pressure drop) and still achieve temperature and flow control due to artificially induced pulses (or slugs). No study has been reported in the open literature on liquid limited reactions, and translation of lab scale reactor unsteady state performance data to large reactors. Some industrial processes do employ periodic localized quenching of hot spots by injection of cold fluids at axial locations into the reactor (Yan, 1980). The investigation of periodic operation in trickle beds is a relatively unexplored area as very few studies reported in the open literature have been conducted to test different strategies with which a performance improvement can be achieved. Most of the reported investigations consist of a few experimental runs of isolated systems. They do not consider quantitatively the extent of improvement achieved in each reaction system, nor do they attribute it to a corresponding quantitative change in the manipulated variables or controlling factors. Careful examination of the experimentation with periodic operation undertaken in the literature, and the corresponding observed enhancement, leads us to classify the strategies used into three categories. 2.2.1 Strategies for Unsteady State Operation I. Flow Modulation a) Isothermal 1. Improvement in wetting and liquid distribution 2. Introduction of dry areas to improve direct access of gaseous reactants. b) Adiabatic/Non Isothermal 1. Allow a controlled increase in temperature and, hence, enhance rate of reaction as well as evaporation of external liquid film. II. Composition Modulation This involves pulsing the feed composition by adding another component which changes reactant ratios to favor one particular reaction (in case of multiple reactions). It can also be used to prevent the reaction from going to gas phase reaction by quenching with an inert component or product of the reaction to and still operate the reactor under semi-runaway conditions. This is done under typically under adiabatic conditions. III. Activity Modulation a) Enhance activity due to the pulsed component . b) Removal of product from catalyst site to prevent deactivation or regenerate catalyst sites by use of the pulsed component. Although the focus of the present study is primarily on strategy I, it is important to consider all the above in some detail. In order to understand the factors causing the actual enhancement, the literature results need to be examined in the light of gas and liquid limited reactions as introduced earlier. This understanding is crucial to the modeling to be undertaken to achieve realistic predictions for any given reaction system, as well as to obtain an optimal performance out of the complex cause-effect relationships that result in the enhancement of performance. The terminology used in periodic operation studies must be noted here, i.e. the total time of one cycle is referred to as cycle time (or period, denoted as SYMBOL 116 \f "Symbol" ) and the part when the modulation is active is referred to as the ON part (denoted by sSYMBOL 116 \f "Symbol", where s is the fraction of total time corresponding to the ON part) and the rest as the OFF part (corresponding to (1-s)SYMBOL 116 \f "Symbol"). Details of literature investigations of periodic operation are summarized in Table 2. Gas Limited Reactions As mentioned in the previous section, in case of a gaseous reactant limitation, the necessity to obtain complete internal wetting and rejuvenation of the liquid in each particle in addition to the need for externally dry areas indicates a contradictory situation for a steady state operation. However, this can be achieved using periodic operation by replenishing the liquid reactant via pulses and ensuring internal wetting. Then the liquid flow must be stopped in order to allow for the gaseous reactant to enter the catalyst pores unhindered by external liquid film, which means that the liquid feed pulse should be followed by a zero liquid feed interval. Such a strategy, strategy Ib of liquid flow modulation, has been employed by Haure et al.(1990), and Lee et al.(1995), for SO2 oxidation.  EMBED Word.Document.8  Figure 2.  SEQ Figure_2. \* ARABIC 1 Time Averaged SO2 Oxidation Rates of Haure et al. (1990)  EMBED Word.Document.8  Figure 2.  SEQ Figure_2. \* ARABIC 2 Experimental and Predicted Temperature Profiles of Haure et al. (1990)  EMBED Word.Document.8  Figure 2.  SEQ Figure_2. \* ARABIC 3 Enhancement in Periodic Operation Observed by Lange et al. (1993) They have observed, based on intrinsic kinetics, that the rate of reaction is first order with respect to the gaseous reactant (O2), and the reaction results in formation of SO3 adsorbed on the catalyst until it is washed by the pulse of water, which results in sulfuric acid formation as well as restoration of the catalytic activity (strategy IIIb). Their operating conditions and cycle times are listed in Table 2. The enhancement of the rate (Figure 1) is attributed to two factors . 1. Temperature increase (The temperature rise during OFF time of about 10-15 oC) 2. Rate of oxidation is much higher on dry catalyst than on wetted catalyst due to gaseous reactant being present at saturation concentrations on the catalyst. They have also attempted to model the reaction based on the Institute of Catalysis approach (Stegasov et al., 1994) and have not been able to successfully predict the rate enhancement that they have observed experimentally. The temperature variation (Figure 2) and its dependence on cycle split is, however, reasonably well predicted by their model (Figure 2). Lange et al. (1993) have recently attempted to experimentally investigate periodic operation of two systems: 1) the hydrogenation of cyclohexene and 2) the hydrogenation of alpha-methyl styrene on Pd catalysts by manipulation of liquid feed concentration and feed rate, respectively. They have used non-isothermal composition modulation for cyclohexene (strategy IIb) and observed partial, as well as total, evaporation of the reactant at higher feed concentration (at temperature close to the boiling point of cyclohexene) which caused the reaction to shift from a 3 phase to a 2 phase one, at which point the next pulse was introduced and reaction was quenched with the feed of lower cyclohexene concentration (or even inert cyclohexane). Since the reactor is operated adiabatically, the improvement was attributed to temperature oscillations in the reactor. For the case of hydrogenation of alpha-methylstyrene under isothermal conditions, the authors observed maximum improvement at a cycle period of 8 min. at cycle split of 0.5. Volumetric flow rate of gas and liquid is as given in Table 2, with ON/OFF as well as base flow cycling used (strategy Ia) (Figure 3). The range of improvement was between 2 and 15 %. The authors attribute the observed improvement to better wetting due to the pulse (removal of stagnant liquid). The actual improvement may be due to direct hydrogen access to the inactive liquid during the OFF part of the cycle. Chanchlani et al. (1994) have studied methanol synthesis in a fixed bed (gas-solid system) under periodic operation. The authors have studied composition modulation under isothermal conditions (strategy IIa) to observe improvement in performance of methanol synthesis (from CO and H2) in a packed bed reactor by taking advantage of the reaction system which shows intrinsically higher rates with some CO2 present in the feed. The two reactions taking place in the system are methanol synthesis and water gas shift reaction. The experimental conditions are listed in Table 2. The temperature and pressure were 225 oC and 2.86 MPa. The strategies that were employed were either pulse CO or CO2 with simultaneous reduction in hydrogen feed, or pulse the other two components keeping H2 or CO2 constant. The optimum cycle split was observed to be 0.5. They observed significant improvement in some cases but also obtained performance worse than the average steady state rates in some other cases. They obtained 20-30 % improvement with cycling between H2-CO2 mixtures with 30/70 in partial cycle and 95/05 in the rest. They also observed 20-25 % improvement with cycling with CO2 constant. The authors attribute the improvement due to activation of the catalyst surface structure due to composition modulation between H2 and CO2 (strategy IIIa). They also suggest that the particular ratios help establish stoichiometric compositions on the surface by ensuring availability of the limiting component . Although this is a gas phase catalytic reaction (not trickle bed), the use of composition modulation to achieve improvement is well illustrated through the authors experiments. Add Castellari and Haure (1995) 2.2.2 Liquid Limited Reactions This type of reactions (i.e., where the liquid reactant is the limiting or controlling reactant) are important in industrial practice particularly at high pressure operation. Periodic operation can be of interest due to the fact that industrial reactors suffer from maldistribution of the liquid reactants which can cause externally dry or completely dry catalyst pellets. This is one of the major problems in design and operation of trickle bed Table 2.  SEQ Table_2. \* ARABIC 2 Literature Studies on Unsteady State Operation in Trickle Beds Author(s)System StudiedModulation StrategyL and G flow ratesCycle Period (() and Split (s)% EnhancementHaure et al. (1990)SO2 oxidationFlow (Non Isothermal)VL=0.03-1.75 mm/s VG=1-2 cm/s(=10-80 min (s=0.1,0-0.5)30-50 %Lange et al. (1994)1. Cyclohexene hydrogenationComposition (Non isothermal)QL=80-250 ml/h Conc=5-100 %(=up to 30 min.(s=0.2-0.5)2-15 % (Temp Rise =30oC )2. (-MS hydrogenationLiquid Flow (Isothermal)QL=0-300 ml/h, QG= 20 l/h(=1-10 min (s=0.25-0.5)Chanchlani et. al (1994)Methanol SynthesisComposition (gas phase)Mixture, SVG=37 ml(STP)/g.cat/s(=2-30 min (s=0.2-0.8)20-30 % for 30:70 H2:CO2Stegasov et al. (1994)SO2 oxidationModelVL=0.1-0.5 cm/s, VG=1.7-2.5 cm/s(=10-30 min. (s=0.1-0.5)Max=80 %Lee et al. (1995)SO2 oxidationAdiabatic Flow modulationVL=0.085-0.212 cm/s, SVG=1000 h-1(=up to 60 min (s=0.02-0.1)Castellari (1996)(-MS hydrogenationNon IsothermalQL=2.27 ml/s QG=900 ml/s(=5 to 45 min (s=0.3-0.5)400 % (Temp. Rise = 35oC) Cassein et al (1997)(-MS hydrogenationNon IsothermalQL=2.27 ml/s QG=900 ml/s(=10 to 45 min (s=0.3-0.5)400 % (Temp. Rise = 35oC)  reactors since the catalyst particles that are not fully wetted by the trickling liquid result in an underutilized bed and loss of production, particularly at low liquid mass velocity. At high mass velocity, the catalyst utilization is much better compared to that at low liquid velocities with the drawback of higher pressure drop and smaller column diameters. It has been observed that at high liquid and gas mass velocities, natural pulsing occurs in trickle beds with slugs of liquid traveling intermittently with slugs of gas (Weekman and Myers, 1964). Under these conditions, a significant improvement in catalyst utilization and effective removal of hot spots has also been reported (Blok and Drinkenburg, 1982). But this is an impractical way of improving the performance of industrial reactors since the high mass rates required for natural pulsing to occur cause high pressure drops and risk instability due to little control over the slugging process. This study investigates the possibility of achieving improved catalyst utilization by forcing pulsed flow in the trickle bed reactor at flow rates lower than those at which natural pulsing occurs, thus reducing the pressure drop and still achieving control over pulse frequency and amplitude due to artificially induced pulses (or slugs). No study has been reported in the open literature on liquid limited reactions and the quantification and translation of lab scale reactor data to large reactors is yet uncertain. Some industrial processes do employ periodic localized quenching of hot spots by injection of cold fluids at axial locations into the trickle beds (Shah et al., 1976, Yan, 1980). 2.3 Review of Models for TBR Performance The following subsections briefly describe the models available in literature for trickle bed reactor performance modeling under steady and unsteady state operation. The models are reviewed by discussing characteristic features and assumptions and their validity discussed in context of our steady and unsteady state experiments as well as industrial applications. The lists presented in Table 2.3 and Table 2.4 are by no means exhaustive and reflect the major contributions in recent years (since 1980). 2.3.1 Steady State Models Steady state models that are available in the open literature for predicting trickle bed reactor performance have been developed primarily for laboratory scale test reaction systems (as listed in Table 2.3) and have several features not typically observed in industrial reactors as discussed at the conclusion of this subsection. Historically, trickle bed reactor models have considered isothermal operation and have been modeled simply by pseudo-homogeneous approach ( ) or a heterogeneous plug flow for gas and liquid phase ( ) with some models accounting for liquid flow non-uniformities and maldistribution by using an axial dispersion model ( ). Most investigations have dealt with hydrogenations or oxidations in pure or moderately concentrated organic or aqueous solutions (large excess of liquid reactant) and hence have considered zero order rate w.r.t. the liquid reactant concentration and first order w.r.t. gaseous reactant concentration ( ). Liquid reactants/solvents have been assumed to be non-volatile and gas phase assumed to be pure at constant partial pressure of the reacting gas. Thus, the primary model variables of interest have been the liquid phase concentrations of the gaseous reactant and resulting conversion of the liquid phase reactants. The key effect that has been incorporated in most recent models is that of partial wetting and transport of gaseous reactant to dry areas of the catalyst resulting in higher rates observed in most of the experimental data ( ). Some models have considered non isothermal effects and used a pseudo-homogeneous energy balance to solve for the temperature at any axial location (). Some models have considered evaporation effects by adding VLE calculations and flash units to simplified pseudo-homogeneous or equilibrium model mass balance equations on the reactor scale( ). Other variants of the reactor model include a cell model ( ), a cross-flow model ( ) , and some others based on liquid flow mal-distribution ( ) or stagnant liquid zones in the reactor ( ). Pellet scale reaction diffusion has been studied by taking reactant limitation in to account in simpler versions ( ) and in general cas by considering partial internal wetting of pellets with both gas and liquid phase reaction zones and solution o a gas-liquid interface by considering liquid imbibition, pore filing and capillary condensation ( ) . Approxiamte solutions from gas-solid ctalayst level equations have also been veridfied by numerical solution for non-linear as well as Langmuir-hinshelwood type of kinetics ( ). The level of complexity and essential features available in the models in literature are sufficient for model evaluation of the steady state experiments for comparison of trickle beds and packed bubble columns as outlined in section 1.2. These models are still far from mimicing reality in industrial hydrocracking and hydrotreating applications due to the following general shorcomings (although some of the literature models have addressed selectively some of the issues mentioned below) Most of the models consider single, simple reactions and model the reaction transport of the limiting species by keeping other concentrations constant. This is far from many industrial reactions where a large set of reactions are occuring simultaneously. Multicomponent transport between the phases has not been considered in most models. This means that net mass and energy fluxes across the interface such as that ocurring in evaporation, condensation etc., (as observed in industrial hydrocrackers) are not accounted in the model calculations. Transport coefficients for both mass and energy in most models are based on correlations which predict low flux, dilute solution under thermodynamically ideal conditions which are not observed in most organic solvents where fluxes and concentrations are far not always low and behavior far from an ideal solution. Most models for trickle beds assume no effect of transport on the hydrodynamics and assume constant phase velocities and holdups eventhough it has been seen in some laboratory scale reactions and many industrial cases that evaporation, condensation etc., can transform a completely liquid phase reaction to a completely vapor phase reaction within the bed, for which assumptions of most models in literature would be invalid. Even the state of the art multiphase codes such as CFDLIB of Los Alamos (Kashiwa et al., 1994) as well as FLUENT ( ) do not inherently handle multicomponent mass transfer and have not been used to investigate the coupling between reaction-transpoort and hydrodynamics in industrial systems (in the open literature). This forms the motivation to develop a code which can overcome some of the abovementioned effects which can be first tested on laboratory systems and then on some industrial reactors to get realistic model predictions. Table 2.  SEQ Table_2. \* ARABIC 3 Review of Recent Steady State Reaction Models for Trickle Bed Reactors AuthorsModel FeaturesReaction SystemRemarksEl-Hisnawi et al. (1982)Heterogeneous plug flow, effect of contacting efficiency, Rate enhancement, gas limited conditions(-methylstyrene hydrogenationEffect of external incomplete contacting on rate enhancement shownMills and Dudukovic (1984)Heterogeneous plug flow, isothermal, partial wetting effect(-methylstyrene hydrogenationComparison of earlier models shownCollins et al. (1984)Isothermal homogeneous model with gas-liquid at equilibriumHydrodesulfurization of benzothiopheneEffect of solvent vaporization consideredSicardi (1984)Model for volatile reactant Developed correlation for higher rate due to volaile on partially wetted catalystBerruti et al. (1984)2 models developed Two zone model (wet and dry) Single zone model (wet)SO2 oxidation on carbonGood prediction of experimental rate and liquid velocity effectRing and Missen (1984)Multizone external wetting model for catalyst pelletHydrodesulfurization of dibenzothiopheneBetter predictions than 2 zone model shown and effet of liquid velocity on overall effectiveness predicted.Herskowitz (1985)Plug flow, isothermal, partial wettingXylose hydrogenationOnly gas limited conditions consideredChu and Ng (1986)Dispersion model for trickling and pulsing flowsDispersion coefficient from Monte Carlo usedTable 2.3 (contd..)Beaudry et al. (1987)Pellet Scale diffusional effects (multiple wetting conditions)(-methylstyrene hydrogenationEffect of poor wetting and liquid limitation at low liquid flow consideredLaVopa and Satterfield (1988)Alternate CSTRs and flash units Completely wetted catalystHydrodeoxygenation of dibenzofuran and hydrogenation of n-butylbenzeneVolatility effects considered, but wetting changes not accountedLemcoff et al. (1988)Pellet scale effectiveness facor modelsEthanol OxidationNumerical Validation of approximate solutions for effectivenessFunk et al. (1990)Computer generated 2-D bed with flow distribution, intraparticle effectsEffect of flow mal-distribution effects and bed featuresKheshgi et al. (1992)Pseudo-homogeneous model, non isothermal effects, bed drying Cyclohexene hydrogenationHystereis and multiplicity due to dry and wet operation predictedYang and Li (1992)Multizone reaction model with stochastic flow distributionHydro-dewaxingEffect of feed temperature and flow well predictedRuzicka and Hanika (1993)Vertical string of pellets, decoupled hydrodynamics from mass and heat transfer Cyclohexene hydrogenationThree possible steady state examined, a wetted, dry and mixed phase reaction consideredHarold and Watson (1993)Pellet scale equations considering partial filling, imbibition, condensation and evaporation in catalyst pores.Hydrazine decomposition, Alpha-methylstyrenePartial internall wetting, nonisothermal effects and hysteresis predictedBrad Sims et al. (1994)Isothermal, two-region cell model with partial wettingDecomposition of hydrogen peroxideFlow regime and liquid velocity effects well predictedSundmachar and Hoffmann (1994)Multicomponent distillation model, acid catalyzed reaction on packingsMTBE synthesisSolid-liquid and Gas-liquid transport effects examinedTable 2.3 (contd..)Hekmat and Vortmeyer (1994)Differential mass balance modelAerobic Oxiadation of ethanol and alkylated benzenesBoth gas (oxygen) limitattion and substrate limitation well predictedTsamatsoulis and Papayannakos (1995)Cross-flow model with parameters obtained from step response testsHydrotreatingToppinen et al. (1996)Plug flow, non-isothermal, multicomponent transport Toluene hydrogenationComparison of multicompnent and effective (lumped) transport consideredKorsten and Hoffmann (1996)Film theory based model, wetting effects, LH kineticsHydrodesulfurization of vacuum gas oilEffect of wetting on pilot and industrial performance well predictedValerius et al. (1996)Multi-pellet and cylindrical shell model3-hydroxypropanol hydrogenationComparison of pilot reactor data with bed effectiveness from the 2 modelsRajashekharam et al. (1998)Plug flow, non-isothermal, wetted, dry, and stagnant zonesHydrogenation of 2,4 dinitrotolueneTemperature rise, hysteresis behavior well predicted 2.3.2 Unsteady State Models for Trickle Bed Reactors The earliest unsteady state modeling approach considered uses a plug flow equilibrium model for predictiong the hotspot formation and movement during startup of atrickle bed and investigation of the introduction of a gas /liquid quench steam at an axial location to quench the hotspot (Yan, 1990; Warna and Salmi, 1996). Mass transfer terms are considered in extensions of these models to predict periodic variation of temperature and concentration (Haure et al., 1990; Stegasov et al., 1994). Spatial variation terms have been dropped in some variations of this model to study time variation of mass transfer coefficients and enhancement in rates and selectivity for model reaction system (Wu et al., 1995) and by addition of catalyst wetting effects under fully externally wetted, externally dry and fully internally dry during periodic operation (Gabarain et al., 1997). This is done primarily to reduce computational complexity. Activity modulation has also been incorporated in recent transient models for optimizing cycle period in periodic operation (Yamada and Goto, 1997). These models capture the time variation of the temperature well but lack in simulating the concentration effects, multicomponent effects, and ignore the coupling between mass transfer and hydrodynamics due to their assumption of a specified variation of holdup and phase velocities with time and space. Since the existing models have been applied to laboratory reactors, the necessity of incorporating all the factor mentiuoned in section 2.3.1 and developing codes for real industrial reaction systems has not been felt. This development has also been limited by the lack of computational power sufficient to handle realistic industrial systems. Table 2.  SEQ Table_2. \* ARABIC 4 Review of Unsteady State Models for Trickle Bed Reactors AuthorsModel FeaturesReaction SystemRemarksYan (1980)Plug flow, gas-liquid equilibrium, apparent rate on catalystHydrocrackingTemperature profiles with and without quenching predictedHaure et al. (1990)Non-isothermal plug flow model with liquid ON-OFF, mass transfer controlling the rateSO2 OxidationThermal waves and temperature rise compared with experimental observationsStegasov et al. (1994)Non-isothermal plug flow with liquid ON-OFF, mass transfer controlSO2 OxidationTemperature and concentration profiles predictedWu et al. (1995)Local transient model with multiple reaction, mass transfer variation with high frequencyThree model systems of multiple series and parallel reactionsRate enhancement with varying frequency examined for natural pulsingWarna and Salmi (1996)Parabolic PDEs for gas and liquid phases, pseudo-transient simulation to steady stateToluene hydrogenation, SO2 OxidationReactor dynamics from start-up to steady state studiedYamada and Goto (1997)Hydrogenation to amino acidOptimal cycle period for product removal and catalyst regenaration calculatedGabarain et al. (1997)Three zone model, completely wetted, externally dry and internally dry pelletsHydrogenation of (-methylstyreneVariation of rate enhancement with cycle period predicted 2.4 Modeling Multicomponent Effects The assumptions and simplifications of most of the steady and unsteady state models presented in the previous section are not valid for most real industrial reactors as mentioned therein, especially for cases where the effect of multiple components transferring acroos can influence the overall reaction-transport ocurring in the system. Several phenomena that are observed in industrial reactions cannot be explained by convenstional models used on laboratory scale model test reactions. Krishna (1994) has presented a clear analysis of three cases where the multicomponent intercation effects observed in simple experiments are incapable of being explained by single component approaches. Phenomena such as reverse diffusion, where the diffusive flux of a component can occur against its own concentration gradient, or osmotic diffusion, where diffusion can occur despite a zero concentration gradient, and, mass transfer barrier where a zero flux of a component is observed despite a non-zero gradient have been demonstrated elegantly to demonstrate the importance of multicomponent models. This section presents the basic equations used in multicomponent modeling of the transport with and across phases required for model equations developed in chapter 5. The use of Stefan-Maxwell models are discussed briefly as a background for their implementation in the model equations presented later. For high pressure gases and liquids the transport of the basic equation for transport or rate of change of moementum for species of type i can be written as product of average momentum exchange per collision and the number of collisions as  EMBED Equation.3  (2.  SEQ 2. \* ARABIC 1) This can be rewritten in terms of the absolute fluxes Ni (using Ni=ciui) or in terms of the relative fluxes Ji (using Ji=Ni-xiNt) as  EMBED Equation.3  (2.  SEQ 2. \* ARABIC 2) Due to the restriction imposed by Gibbs-Duhem restriction on summation of the total driving forces to be zero, it is convenient to convert the RHS of Equation 2.1 above equations to a n-1 term equation and RHS of Equation 2.2 to a (n-1x n-1) matrix form. The RHS of Equation 2.1 can be written by applying the Gibbs-Duhem condition as  EMBED Equation.3  (2.  SEQ 2. \* ARABIC 3)  EMBED Equation.3  (2.  SEQ 2. \* ARABIC 4) The RHS of Equation 2.2 can be written in a matrix form as  EMBED Equation.3  (2.  SEQ 2. \* ARABIC 5) The combination of equations 2.4 and 2.5 forms the basis of the equation vector used to obtain the fluxes for transport within a phase. The equation that is obtained is  EMBED Equation.3  (2.  SEQ 2. \* ARABIC 6) This equation implies that only n-1 equations are available for the n unknown fluxes. At the interface between the three phases (gas, liquid, and solid), similar equations can be written for the mass transfer fluxes. The extra condition required in this case can in principle be obtained by a solution of the momentum equation for the interfacial region. In practice, however, the real hydrodyamics is replaced by a film model in many engineering applications due to its simplcity. The film model however stipulates a stagnant film on both sides of the interface, implying that Nt be automatically zero. But this approach has been used to model non-equilibrium separation by using other conditions as a replacement for the simplified hydrodynamics. This condition is called as the bootstrap condition (Taylor and Krishna, 1993) and formulated using a generalized flux equation as  EMBED Equation.3  (2.  SEQ 2. \* ARABIC 7) This can be converted to a matrix form in order to be combined with equation 2.6. This matrix represented by ( is obtained by converting the condition given in equation 2.7  EMBED Equation.3  (2.  SEQ 2. \* ARABIC 8)  EMBED Equation.3  (2.  SEQ 2. \* ARABIC 9) The equation 2.6 can then be rewritten interms of mass transfer coefficient and the bootstrap matrix as  EMBED Equation.3  (2.  SEQ 2. \* ARABIC 10) Here the mass transfer coeficient [ko] is the high flux mass transfer coeffcient determined from the low flux coefficient [k] by applying a flux correction as shown in Appendix D. Several conditons have been suggested in distillation and absorption litrature for using as the condition in equation 2.7 depending upon the knowledge of the physics of the particular problem. Some of the special cases are shown below Case I : Equimolar Counterdiffusion (No net bulk flux) e.g., Multicomponent Distillation a) Nt = 0 ( Ni = Ji or (i = (n (2.  SEQ 2. \* ARABIC 11) b)  EMBED Equation.2  (2.  SEQ 2. \* ARABIC 12) Case II : Stefan Diffusion (Inert Component flux =0 ) e.g., Evaporation of vapor into an inert, Condensation in presence of non-condensibles, Absorption of a gas from a stream of containing inert component Nn = 0 ( (i = 0, (n ( 0 (2.  SEQ 2. \* ARABIC 13) Case III : Stoichiometric Diffusion (Reaction Controlled) e.g., Heterogeneous reactions, Condensation of mixtures Ni = zi Nt ( (i =( - 1/ zi , (j = 1 (2.  SEQ 2. \* ARABIC 14) Case IV : Simultaneous Energy and Mass Energy Balance e.g., Simultaneous evaporation and absorption (Cases where other bootstraps cannot be applied). The bootstrap basis equation EMBED Equation  (2.  SEQ 2. \* ARABIC 15)  EMBED Equation.2  (2.  SEQ 2. \* ARABIC 16)  EMBED Equation.2  and  EMBED Equation.2  (2.  SEQ 2. \* ARABIC 17) Case V : No Net Volumetric Flux e.g., Transport to constant Volumes (Catalyst Pellets, Bubbles etc.)  EMBED Equation.2  ,  EMBED Equation.2  (2.  SEQ 2. \* ARABIC 18) CHAPTER 3. EXPERIMENTAL FACILITY 3.1 High Pressure Trickle Bed Setup Our first objective was to experimentally investigate the comparative performance (under identical external operating conditions) of trickle bed (TBR) and up-flow (PBC) reactors. This involved collecting data in up-flow and down-flow reactors for a hydrogenation reaction over a range of pressures and liquid mass velocities in order to create the data bank required for comparison of the two reactors. Identical experimental investigation in the presence of fines was also outlined in the objectives. Both these objectives required a versatile reactor setup in which the reaction could be carried out with the same batch of acatalyst in both trickle bed and packed bubble column configuration without removing and reactivating the catalyst bed. This necessitated the modification of several components of the high pressure trickle bed reactor facility previously setup by Al-Dahhan (1993) for high pressure hydrodynamic studies. These are discussed in the next section. Several modifications were also done to the system for the unsteady state operaion conducted in the second part of the study and are presented subsequently. 3.1.1 Reactor and Distributors for Upflow and Downflow Two reactors were designed for the two parts of this study, each consisting of a 60 cm long stainless steel tube with a 2.54 cm outer diameter and 2 mm wall thickness enough to withstand 500 psig pressure. One of the reactors was designed to run at near isothermal conditions by an extrenal cooling jacket (5.08 cm outer diameter), while the other was designed for adiabatic operation and the cooling jacket was substituted by a fiber glass insulation with 3 ports for temperature and pressure drop measurements. Connections between the reactor and distributor sections are made with flanges on bothe the bottom and the top of the reactor. Liquid and gas are fed to the reactor through inch stainless steel lines euipped with three way valves to allow for both down-flow and up-flow configurations by merely swithching the valve positions. For trickle bed operation, the valves V1, V2, V3, and V4 were all switched to position #1, while for packed bubble colum (upflow) operation, all of them are switched to position #2 (as shown in Figure 3.1). This allows switching between the two modes of operation without discharging and enables performance comparison at identical catalyst activity. The gas and liquid distribution is critical in downflow operation, while gas distribution is important in upflow operation. The downflow distributor consists of 21 stainlesss steel tubes (1.6 mm inside diameter) to distribute the liquid phase through a perforated plate welded inside the distributor (Figure 3.2). The gas phase enters the space surrounding the tubes and leaves through 3.55 mm holes around the liquid tubes as well as through the extra holes (0.76 mm in diameter) provide in between. The gas distributor at the bottom of the reactor is used for upflow operation and is designed in the form of a U-shaped stainless steel box welded to a inch stainless steel line. On top of the box are 7 holes (0.2 mm in diameter) to distribute the gas uniformly in the reactor (Figure 3.2). The liquid line for this distributor is ajust a inch stainless steel tube concentric to the gas delivery line and feeds the liquid all around the gas distributo box. The distributor was tested using a clear acrylic reactor to ensure that no channeling of gas occurs during up-flow operation. 3.1.2 Gas-Liquid Separator and Level Control The gas-liquid separator was designed and constructed from a 7 inch diameter optically clear acrylic rod sealed at both ends by stainless steel flanges (Figure 3.3). The top flange consists of the liquid level measurement device at the centre with connections for the gas-liquid mixture feed to the separator and gas exit from the separator on the sides, while the bottom flange has a liquid exit line with a sampling line. The funnel shape originally chosen by Al-Dahhan (1993) in order to minimize the liquid volume is used in this case as a reservoir for the paddle of a liquid level measurement device (Dynatrol, E-103C), which measures vibrations in the liquid and transmits the liquid level signal to a controller (Omega 7600). The controller signal is then converted to a pneumatic signal using and I/P transducer and then used to operate a pneumatic valve controlling the liquid leaving the separator. Liquid level control in the separator is crucial to maintaining a stable pressure in the reactor as well as to prevent hydrogen leakage from the liquid exit line. A stainless steel mesh demister is also placed in the top flange to trap the mist from the effluent gas stream. 3.1.4 Liquid and Gas Delivery System The liquid phase is delivered to the reactor by a set of two high pressure metering pumps either to a high pressure feed tank or directly to a small tank used to damp the pulsations and then sent on to the reactor. The liquid feed rate is controlled by the pump head setting and also by a recycle line from the small damper tank back to the pump inlet. Direct liquid supply from the pump is used in steady state experiments, but this is not suitable for unsteady state experiments. A pulsation free liquid supply is obtained by using the high pressure feed tank which is independently pressurized by nitrogen supplied from a high pressure cylinder. The liquid supplied either from the high pressure tank or from the damper is sent through two sets of parallel lines through needle valves to a combination of three way and solenoid valves which are used to switch between steady flow and ON/OFF or BASE/PEAK flow required for experiments operated under unsteady state conditions. The solenoid valves are controlled by an electronic timer to which the cycleperiod and split data are supplied. The liquid outlet stream coming from the gas liquid separator is split into a sampling stream and a waste stream. The sampling stream is connected to a needle valve for withdrawal of samples analysed by the gas chromatograph (Gow Mac Model 550). The wast stream is passed through two high pressure liquid regulators used to drop the liquid pressure from the operating pressure to atmospheric. The liquid is collected in a 110 litre stainless steel wate tank. Liquid flow measurment is also done by a rotameter placed before the the waste tank or manually by collecting the exit liquid over several time intervals. Good level control is important in obtaining reproducible flow rate data. The gas delivery system consists of high pressure hydrogen gas cylinders connected to the gas delivery line via a pressure regulator. Before the gas is delivered to the reactor, it is bubbled through two small tanks (similar to the damper tank) full of solvent in order to prevent excessive evaporation of solvent in the catalyst bed. The gas is then supplied to the distributor via a needle valve used to control its feed rate. The gas outlet stream from the gas-liquid separator passes through a series of pressure regulators to drop the pressure from the reactor pressure to a pressure of 8 psig at which it flows through the rotameter which calibrated to detemone the flow rate at that pressure. This is then corrected to calculate the actual superfircial velocity at the operating pressure. The exit gas from this arrangement is then vented to the atmosphere via a copper vent tube connected to the exit hood in the laboratory. Gas pressure indicators and safety relief valves are mounted at several locations to prevent pressure buildup in the gas delivery and exit stream. EMBED MSDraw \* mergeformat Figure 3.  SEQ Figure_3. \* ARABIC 1 Reactor and Gas-Liquid Separator EMBED MSDraw \* mergeformat Figure 3.  SEQ Figure_3. \* ARABIC 2 Down-flow and Up-flow Distributor EMBED Word.Picture.8 Figure 3.  SEQ Figure_3. \* ARABIC 3 Experimental Setup for Unsteady State Flow Modulation Experiments 3.1.5 Data Acquisition and Analysis Pressure drop measurments were conducted using a high pressure differential pressure transducer (Viatran Model 374, 0-20 WCD), the output of which is connected to the terminal block of a multiplexed data acquisition unit (National Instruments SCXI-1100) which in turn is connected to a data acquisition board (National Instruments-AT MIO-16) of a PC Computer (Dell 486) as shown in Figure 3.4. Transducer outputs from the thermocouples (type-K) is also connected similarly to the SCXI-1100 unit. The software Labview (National Instruments V4.0) is utilized to develop the required virtual instruments (VIs) for both temperature and pressure drop measurements.  EMBED Word.Picture.8  Figure 3.  SEQ Figure_3. \* ARABIC 4 Data Acquisition System The concentration data obtained from the gas chromatograph (Gow Mac Model 550) is also sent to the same computer using a data acquisition board and Peaksimple-II software which can give the peaks of the species and the area percent values which are then converted to concentrations using a predetmined calibration curve. 3.2 Operating Procedures and Conditions 3.2.1 Steady State TBR-PBC Comparison Experiments Catalyst (2.5% Pd on alumina extrudate (properties as listed in Table 3.1)) supplied by Engelhard was packed to a height of 27.5 cm with glass beads (3 mm diameter) packed on both sides to a total reactor length of 59 cm. The bed was activated by first preheating it in a chromatographic-type oven at 140 oC for 4-5 hours with a low flow rate of nitrogen passed through the bed to remove any moisture present in the catalyst. The bed was cooled and activated by reducing it with a low flow of hydrogen at near room temperature for 6-8 hours. The bed was then stored in solvent (hexane) under a nitrogen atmosphere to maintain the activity and keep the catalyst wetted at all times. SYMBOL 97 \f "Symbol"-methylstyrene (Eastman Chemicals) was pre-purified by passing over a bed of alumina to remove the polymerization inhibitor. This was then diluted in hexane (ACS grade, 99.9% purity) and was used as the liquid phase delivered to the reactor by a precalibrated high pressure pump or the pressurized feed tank as mentioned in Section 3.1.4. Pure hydrogen (prepurified, analytical grade) bubbled through a hexane saturator (to prevent evaporation of hexane in the reactor) was used as the gas phase. The reactor was kept isothermal at room temperature by using a cooling jacket (24oC with maximum 2oC temperature rise) easily due to dilute liquid reactant. At these conditions very little loss of liquid can occur due to volatilization. Typically a high liquid flow rate was chosen for the first experiment and then the flow was reduced for subsequent experiments and so on. This was done to eliminate the possibility of hysteresis seen when flow rates are increased from low to high flow rates. Liquid samples were drawn from the gas-liquid separator after steady state was reached at each liquid flow rate (about 20-25 minutes, after which no further change in exit concentration of SYMBOL 97 \f "Symbol"-methylstyrene was observed). The samples were analyzed on a gas chromatograph (Gow Mac Series 550, with thermal conductivity detector) from which the conversion of a-methylstyrene was determined. Pressure drop across the bed was measured using a differential pressure transducer as shown in Figure 3.1. Experiments were conducted at a fixed gas flow rate by varying liquid flow rate for each mode of operation. The stability of the catalyst over the duration of the experiments was verified by taking intermittent samples at identical operating conditions. The reproducibility of the data was observed to be within 3%. The range of operating conditions investigated in this study is presented in Table 3.2. In the present study the operating conditions were selected (Table 3.2) on the basis of three considerations: i) liquid mass velocities were chosen to cover a range of wetting conditions from partially wetted to completely wetted, ii) liquid feed concentrations and pressures were set to encompass both gas and liquid limited conditions (SYMBOL 103 \f "Symbol" values range from 0.7 to 10), iii) liquid reactant was diluted with solvent (n-hexane) to keep the system isothermal. Prior to the above mentioned experimentation, stirred basket reaction tests were performed to establish a reproducible activation procedure and to test the stability of the catalyst before undertaking the packed bed reactor studies. Preheating in an inert (Nitrogen) atmosphere to 140oC followed by activation with Hydrogen at reaction temperature (~24oC) yielded consistent activity for over 40 hours of operation (after an initial 9 hours of stabilization runs). EMBED ExcelChart \s \* mergeformat Figure 3.  SEQ Figure_3. \* ARABIC 5 Basket Reactor Catalyst Stability Test. 3.2.2 Bed Dilution and Experiments with Fines Upon completion of the entire set of experiments without fines, the catalyst was discharged and repacked with inert fines (Silicon Carbide from Washington Mills) using a reproducible packing procedure developed in at CREL by Al-Dahhan et al. (1995). Several methods (such as the dry and wet method) have been suggested in literaure with different proportions of fines to catalyst ratios and have been shown by Al-Dahhan et al. (1995) to give different and somewhat non-reproducible holdup and pressure drop characteristics. The procedure developed at CREL and followed in the present experiments was to use the dry method of packing and the volume of the fines added was calculated on the basis of the bed void volume in the undiluted condition. The bed was packed in two portions (halves) in the present study by packing the catalyst to half the bed height first and then packing the predetermined volume of fines. These were packed in the intersticies of the catalyst by using a vibrating decvice on the bed until the top layer of catalys is visible. This was then followed by the second half of the catalyst which was similarly packed and reactor vibrated until the top layer of catalyst ws visible. The increase in the overall volume of the diluted bed over the undiluted bed was kept to with 2 % to ensure that the structure of the original bed is not disturbed by the addition of fines. Experimental runs similar to those done previously were conducted now with the diluted bed to study the effect of fines on the performance of down-flow and up-flow modes of operation. Table 3.  SEQ Table_3. \* ARABIC 1 Catalyst and Reactor Properties for Steady State Experiments Catalyst Properties Reactor Properties Active metal2.5% Pd.Total length50 cm Catalyst support Alumina  Catalyst length 27.5 cm  Packing shapeCylinderDiameter2.2 cm Packing dimensions (cm) 0.13 x 0.56        Fines   Surface area (m2 / g) 184  Material Silicon Carbide  Total pore volume (cm3 / g) 0.481  Diameter 0.02 cm  Pellet density rP (g / cm3) 1.222   True pellet density rt (g / cm3) 2.965     Pellet porosity eP 0.5878      Table 3.  SEQ Table_3. \* ARABIC 2 Range of Operating Conditions for Steady State Experiments Temperature24oC Pressure30 - 200 psig (3 -15 atm.) Concentration of a-methylstyrene3.1 - 7.8 %(v/v) (230-600 mol/m3) Superficial Gas Velocity (Mass Velocity)3.8 - 14.4 cm/s (3.3x10-3-12.8x10-3 kg/m2 s) Superficial Liquid Velocity(Mass Velocity)0.09-0.5 cm/s (0.63 - 3.85 kg/m2 s)  3.2.3 Unsteady State Experiments The reaction system chosen for this study was hydrogenation of alpha-methylstyrene to iso-propyl benzene (cumene) which is the same system as was used for experiments in section 3.1. The high pressure packed bed reactor facility described in section 3.1 was used in this study. The liquid delivery system was modified for unsteady state experiments by adding a set of solenoid valves and a timer (as shown in Figure 3.5) to obtain liquid ON-OFF flow, liquid BASE-PEAK flow, and steady liquid flow as desired (presented in Figure 3.5). The catalyst used for these experiments, 0.5% Pd on alumina spheres (different from that used in the earlier steady state experiments) was packed to a height of 26 cm (with glass beads on both sides to a total height of 59 cm) and reduced in situ since this reactor did not have an external jacket it was easy to pre-heat, cool and activate in situ. The reaction was run in this activated bed for several hours at steadys state until a constant catalyst activity is obtained. Since the activity varied slightly between runs, steady state runs were performed before and after each set of unsteady state runs to ensure reproducibility of the catalyst activity in each set. (-methylstyrene (99.9% purity and prepurified over alumina to remove polymerization inhibitor) in hexane (ACS grade, 99.9% purity) was used as the liquid phase. Pure hydrogen (pre-purified, industrial grade) was used as the gas phase. The reactor was operated under adiabatic conditions. Liquid samples were drawn from the gas-liquid separator after steady state was reached at each liquid flow rate. The samples were analyzed by gas chromatography (Gow Mac Series 550, with thermal conductivity detector) from which the steady state conversion of a-methylstyrene was determined. Unsteady state conversion was determined by evaluating concentration of a liquid sample collected over multiple cycles to get the flow average concentration. The reproducibility of the data was observed to be within 3%. The ranges of operating conditions investigated are presented in Table 3.4. The feed concentrations and operating pressures were chosen so as to examine both gas limited and liquid limited conditions (as evaluated approximately by the ( criterion; ( < 1=> liquid limited and ( > 1 => gas limited, (Khadilkar et al., 1996)). The liquid mass velocities were chosen so as to cover partial to complete external wetting of the catalyst. Both liquid ON-OFF and BASE-PEAK flow modulation were studied over a range of liquid mass velocities for each set of experiments as illustrated in Figure 3.6.  EMBED Word.Picture.8  Figure 3.6 Schematic of Flow Modulation: Connections and Cycling Strategy Table 3.  SEQ Table_3. \* ARABIC 3 Catalyst and Reactor Properties for Unsteady State Conditions  Table 3.  SEQ Table_3. \* ARABIC 4 Reaction and Operating Conditions for Unsteady State Experiments Superficial Liquid Mass Velocity 0.05-2.5 kg/m2s Superficial Gas Mass Velocity 3.3x10-3-15x10-3 kg/m2s Operating Pressure30 -200 psig (3-15 atm) Feed Concentration2.5 - 30 % (200-2400 mol/m3) Feed Temperature20-25 oC Cycle time, t (Total Period)5-500 s Cycle split, s (ON Flow Fraction) 0.1-0.6 Max. Allowed temperature rise 25 oC CHAPTER 4. EXPERIMENTAL RESULTS 4.1 Steady State Experiments on Trickle Bed Reactor and Packed Bubble Column 4.1.1 Effect of Reactant Limitation on Comparative Performance of TBR and PBC Comparison of the two reactors is achieved by studying the conversion at identical volumetric nominal space times (defined as reactor length/ superficial liquid velocity) and identical reactant feed concentration. This is the proper scale-up variable, (space time = 3600/LHSV) when the beds for upflow and downflow are identically packed (i.e., bed voidage = constant) and the reaction rate is based per unit volume of the catalyst. At low pressure (30 psig) and high feed concentration of a-methylstyrene (CBi= 7.8 %v/v), the reaction is gas limited (SYMBOL 103 \f "Symbol" = 8.8). In this case downflow performs better than upflow reactor as shown in Figure 3(a). This is due to the nature of the hydrogenation reactions which are typically hydrogen (gas reactant) limited at low pressure (at or just above atmospheric) and high a-methylstyrene concentrations (Beaudry et al., 1987). It is obvious that this is due to low hydrogen solubility at these pressures. In downflow mode of operation, the catalyst particles are not fully wetted at the liquid flow rates used (Figure 4 shows contacting efficiency calculated using the correlation of Al-Dahhan and Dudukovic (1995)). This facilitates the access of the gas reactant to the pores of the catalyst on the externally dry parts, and reduces the extent of gas limitation compared to fully wetted pellets in upflow reactor. The result is a higher conversion in downflow than in the upflow mode of operation. In case of upflow, since the catalyst is almost completely wetted, the access of gaseous reactant to the catalyst site is limited to that through liquid film only. This provides an additional resistance for the gaseous reactant (especially at high space time i.e., low liquid flow rate) and results in conversion lower than that obtained in downflow. This effect is more prominent at higher liquid reactant feed concentrations, due to the larger extent of gas limitation at such conditions (higher SYMBOL 103 \f "Symbol" values). As liquid mass velocity increases (space time decreases), the downflow performance approaches that of upflow, due to catalyst wetting efficiency approaching that of upflow (contacting efficiency approaches 1 as seen in Figure 4). As the reactor pressure increases and the feed concentration of a-methylstyrene decreases, the value of SYMBOL 103 \f "Symbol" decreases and the reaction approaches liquid limited behavior as postulated earlier. This is reflected in a complete reversal in performance at higher pressures and at low a-methylstyrene concentration (Figure 3(b)), where the performance of upflow becomes better than downflow. This is because under these conditions the catalyst in downflow is still partially wetted (since at the operating gas velocities and gas densities (hydrogen), high pressure only slightly improves wetting in downflow (Figure 4 based on Al-Dahhan and Dudukovic, 1995) while catalyst is fully wetted in upflow. In a liquid limited reaction, the conversion will be governed by the degree of catalyst wetting, and since upflow has higher wetting (100 %) than downflow, it will outperform downflow. As the liquid mass velocity increases, and the contacting efficiency of downflow approaches 100 %, the performance of the two reactors approaches each other, as evident in Figure 3(b) at low space times. Thus, as pressure is increased from 30 to 200 psig, and feed concentration of SYMBOL 97 \f "Symbol"-methylstyrene is decreased from 7.8% to 3.1%(v/v), the reaction is transformed from a gas-limited (SYMBOL 103 \f "Symbol"=8.8) to a liquid-limited regime (SYMBOL 103 \f "Symbol"=0.8). The criterion (SYMBOL 103 \f "Symbol") is dependent on two factors (apart from the diffusivity ratio), pressure (hydrogen solubility) and feed concentration of the liquid reactant (a-methylstyrene). Further insight into the gas and liquid limitation can be obtained by investigating these two contributions individually for the set of operating conditions examined. EMBED ExcelChart \s \* mergeformat Figure 4.  SEQ Figure_4. \* ARABIC 1 Trickle Bed and Up-flow Performance at CBi=7.8%(v/v) and Ug =4.4 cm/s at 30 psig. EMBED ExcelChart \s \* mergeformat Figure 4.  SEQ Figure_4. \* ARABIC 2 Comparison of Down-flow and Up-flow Performance at CBi=3.1%(v/v) at 200 psig. 4.1.2 Effect of Reactor Pressure on Individual Mode of Operation As reactor pressure increases, the performance of both upflow and downflow improves due to increase in gas solubility, which both helps the rate of transport to the wetted catalyst (in both modes) and improves the driving force for gas to catalyst mass transfer to the inactively wetted catalyst in the downflow mode. At low feed concentration of the liquid reactant (a-methylstyrene (3.1%v/v)) and at high pressure (>100 psig), the reaction becomes liquid reactant limited (or liquid reactant affected) as can be seen from Figures 5(a) and 5(b) where no further enhancement is observed when pressure is increased from 100 to 200 psig (where SYMBOL 103 \f "Symbol" drops from 1.5 at 100 psig to 0.8 at 200 psig). This means that any further increase in the reactor pressure and hence liquid phase hydrogen concentration, will have minimal effect since hydrogen is not the limiting reactant anymore. To confirm the above observation the reaction was studied at higher feed concentration of a-methylstyrene (4.8%v/v) in order to determine whether gas limited behavior is observed at higher SYMBOL 103 \f "Symbol" values. The performance indeed improves when pressure is increased from 100 to 200 psig (Figures 6) implying that the reaction is not completely liquid limited at this feed concentration at 100 psig operating pressure (SYMBOL 103 \f "Symbol"=2.44). It becomes liquid limited at pressures above 200 psig (SYMBOL 103 \f "Symbol"=1.3) at this feed concentration, whereas it is liquid limited at lower a-methylstyrene concentration (3.1%v/v) even at lower pressures as noted previously in Figures 5((a) and (b)). Both upflow and downflow conversion increases with increasing pressure, primarily due to increase in solubility of the gaseous reactant as the pressure increases. A significant improvement in performance (conversion) occurs when pressure is changed from 30 to 100 psig as compared to the change in conversion when pressure changes from 100 to 200 psig. This confirms that the effect of pressure diminishes when liquid limitation is approached (as SYMBOL 103 \f "Symbol" approaches 1.0 from above (Figure 6)). Experimental pressure drop measurements were also made for both modes of operation during the reaction runs. The data obtained (shown in Figure 4) indicates higher pressure drops for upflow at both ends of the pressure range covered (30 and 200 psig) than for downflow, which is in agreement with expectation and the pressure drop data reported in the literature. The higher downflow performance (conversion) at 30 psig, despite lower pressure drop, confirms that poor contacting does yield better conversion due to reaction being gas limited, which seems contrary to the notion that higher transport always involves higher pressure drop (which is observed to be true here in case of liquid limited reaction at 200 psig). 4.1.3 Effect of Feed Concentration of a-methylstyrene on Individual Mode of Operation Atmospheric pressure hydrogenation of SYMBOL 97 \f "Symbol"-methylstyrene has been known to be a zeroth order reaction with respect to SYMBOL 97 \f "Symbol"-methylstyrene and first order with respect to hydrogen (Beaudry et. al, 1986). Our observations confirm that at 30 psig as well as at 100 psig, the reaction is zero order with respect to a-methylstyrene as shown in Figures 7(a) and 7(b) for upflow and downflow, respectively. An inverse proportionality of conversion with liquid reactant feed concentration (typical of zero order behavior) is observed especially at higher liquid flow rates (lower space times). At lower liquid flow rates, at 100 psig the zero order dependence appears to vanish and a first order dependence (due to a-methylstyrene transport or intrinsic rate limitations), i.e., conversion independent of feed concentration, is observed. This shift in feed concentration dependence is confirmed by data at higher pressure (200 psig, Figures 8(a) and 8(b)). When liquid limitation is observed there is no effect of feed concentration on the conversion in either mode of operation, as can be seen in Figure 8(a) and 8(b). This is a consequence of the liquid (reactant) transport or intrinsic rate limitation which shows up as a first order dependence making conversion independent of feed concentration. The effect of gas velocity on reactor performance was also examined for both upflow and downflow reactors. A significant effect was not observed in the range of the gas velocities studied (3.8-14.4 cm/s, i.e., gas Reynolds number in the range of 6-25) on either downflow or upflow performances at all the concentrations tested. This is in agreement with the observations of Goto et. al (1993). Comparison of the two reactors is achieved by studying the conversion at identical volumetric nominal space times (defined as reactor length/ superficial liquid velocity) and identical reactant feed concentration. This is the proper scale-up variable, (space time = 3600/LHSV) when the beds for up-flow and down-flow are identically packed (i.e. bed voidage = constant) and the reaction rate is based on per unit volume of the catalyst. At low pressure (1 atm.) and high feed concentration of a-methylstyrene (CBi= 7.8 %v/v), the reaction is gas limited (SYMBOL 103 \f "Symbol" = 8.8). In this case, the down-flow reactor performs better than the up-flow one as shown in Figure 6. This is due to the nature of the hydrogenation reactions which are typically hydrogen (gas reactant) limited at low pressure (at or just above atmospheric) and high a-methylstyrene concentrations (Beaudry et al., 1987). It is obvious that this is due to low hydrogen solubility at these pressures. In down-flow mode of operation, the catalyst particles are not fully wetted at the liquid flow rates used (Figure 8 shows contacting efficiency calculated using the correlation of Al-Dahhan et al. (1994)). This facilitates the access of the gas reactant to the pores of the catalyst on the externally dry parts and reduces the extent of gas limitation compared to fully wetted pellets in the up-flow reactor. The result is a higher conversion in down-flow than in the up-flow mode of operation. In case of up-flow, since the catalyst is almost completely wetted, the access of gaseous reactant to the catalyst sites is limited to that through liquid film only. This provides an additional resistance for the gaseous reactant and results in conversion lower than that obtained in down-flow. This effect is more prominent at higher liquid reactant feed concentrations due to the larger extent of gas limitation at such conditions (higher SYMBOL 103 \f "Symbol" values). As liquid mass velocity increases (space time decreases) the down-flow performance approaches that of up-flow due to catalyst wetting efficiency approaching that of up-flow (Contacting efficiency approaches 1 as seen in Figure 8). 4.1.4 Effect of Pressure and Feed Concentration on Comparative Performance of TBR and PBC in Transition from Gas to Liquid Limited Conditions As the reactor pressure increases and the feed concentration of a-methylstyrene decreases, the value of SYMBOL 103 \f "Symbol" decreases and the reaction approaches liquid limited behavior as mentioned earlier. This is reflected in a complete reversal in performance at higher pressures at low a-methylstyrene concentration (Figure 7) where the performance of up-flow becomes better than down-flow. This is because under these conditions the catalyst in down-flow is still partially wetted (since at the operating gas velocities and gas densities (hydrogen), high pressure only slightly improves wetting in down-flow (Figure 8) (Al-Dahhan and Dudukovic, 1995)) while we could assume fully wetted catalyst in up-flow. In a liquid limited reaction, the conversion will be governed by the degree of external catalyst wetting, and since up-flow has higher wetting (100 %) than down-flow it will outperform down-flow. As the liquid mass velocity increases, and the contacting efficiency of down-flow approaches 100 %, the performance of the two reactors approaches each other as evident in Figure 7. Thus, as pressure is increased from 30 to 200 psig, and feed concentration of SYMBOL 97 \f "Symbol"-methylstyrene is decreased from 7.8% to 3.1%(v/v), the reaction is transformed from a gas-limited (SYMBOL 103 \f "Symbol"=8.8) to a liquid-limited regime (SYMBOL 103 \f "Symbol"=0.8). The criterion (SYMBOL 103 \f "Symbol") is dependent on two factors (apart from the diffusivity ratio), i.e. pressure (hydrogen solubility) and feed concentration of the liquid reactant (a-methylstyrene). EMBED ExcelChart \s \* mergeformat Figure 4.  SEQ Figure_4. \* ARABIC 3 Effect of Pressure at Low a-methylstyrene Feed Concentration on Upflow Reactor Performance. EMBED ExcelChart \s \* mergeformat Figure 4.  SEQ Figure_4. \* ARABIC 4 Effect of Pressure at Low a-methylstyrene Feed Concentration (3.1% v/v) on Downflow Performance. EMBED ExcelChart \s \* mergeformat Figure 4.  SEQ Figure_4. \* ARABIC 5 Effect of Pressure at Higher a-methylstyrene Feed Concentration on Downflow Performance. EMBED ExcelChart \s \* mergeformat Figure 4.  SEQ Figure_4. \* ARABIC 6 Effect of a-methylstyrene Feed Concentration at 100 psig on Upflow Performance. EMBED ExcelChart \s \* mergeformat Figure 4.  SEQ Figure_4. \* ARABIC 7 Effect of a-methylstyrene Feed Concentration at 100 psig on Downflow Performance. EMBED ExcelChart \s \* mergeformat Figure 4.  SEQ Figure_4. \* ARABIC 8 Effect of a-methylstyrene Feed Concentration at 200 psig on Downflow Performance. EMBED ExcelChart \s \* mergeformat Figure 4.  SEQ Figure_4. \* ARABIC 9 Effect of a-methylstyrene Feed Concentration at 200 psig on Upflow Performance. 4.1.5 Effect of Gas Velocity and Liquid-Solid Contacting Efficiency Effect of gas velocity on reactor Performance : There was no significant effect of gas velocity on either downflow or upflow performances at all the concentrations tested (as shown in the sample plot in Figure 19 for upflow performance) . There was a very slight variation in upflow performance(Figure 19 lower curve) which may be attributed to increase in holdup and kla but not significant enough at higher pressures and other concentrations (within experimental error) to be further investigated .  EMBED Excel.Sheet.8  Figure 4.  SEQ Figure_4. \* ARABIC 10 Effect of Gas Velocity on Reactor Performance at 100 psig. EMBED ExcelChart \* mergeformat \s Figure 4.  SEQ Figure_4. \* ARABIC 11 Pressure Drop in Downflow and Upflow Reactors and Contacting Efficiency for Downflow Reactor at 30 and 200 psig. 4.2 Comparison of Down-flow (TBR) and Up-flow (PBC) Reactors with Fines These two contributions to the shift in gas to liquid limitation have been investigated in detail by performing experimental runs at three different concentrations and pressures, and results are presented by Khadilkar et al.(1996) and Wu et al. (1996). Fines (nonporous inert particles, order of magnitude smaller than catalyst pellets packed only in the voids of the catalyst) are used to investigate the performance of the two modes of operation using the same reaction in an attempt to demonstrate the decoupling of hydrodynamic and kinetic effects. A way to establish this decoupling is to use the upflow and downflow modes, which are intrinsically hydrodynamically different (as discussed earlier), and asses whether fines can indeed yield the "true" kinetic behavior (more properly called "apparent" rates on catalyst pellets of interest, i.e., rates unmasked by external transport resistances and hydrodynamic effects). The two extreme cases discussed before, i.e., gas limitation (downflow performance better than upflow, Figure 3(a)), and liquid limitation (upflow performance better than downflow, Figure 3(b)) are now conducted in the presence of fines. Figures 9(a) and 9(b) show the performance of both reactors when the bed is diluted with fines. It can be seen by comparing Figure 9(a) with Figure 3(a) and Figure 9(b) with Figure 3(b) that fines have eliminated the disparities between the two modes of operation even in the extreme cases of reactant limitation. This is primarily due to the fact that fines improve liquid spreading considerably and achieve comparable (and almost complete) wetting in both modes of operation. It must be noted that Figures 3(a) and 9(a), or Figures 3(b) and 9(b), could not be directly superimposed due to slightly different catalyst activity obtained after repacking the bed with fines and catalyst and reactivating it. Nevertheless, fines have successfully decoupled the hydrodynamics and apparent kinetics, and the data with fines reflect the kinetics in the packed bed under "ideal" liquid distribution conditions. It can be observed in Figure 9(a) that at low liquid flow rate and low pressure (gas limited reaction), a trickle bed with fines still performs slightly better than upflow with fines, which indicates that the degree of wetting is still not complete in downflow resulting in some direct exposure of the internally wetted but externally dry catalyst to the gas . This may be due to the fact that at low liquid flow rate even with fines , the catalyst is not completely externally wetted (Al-Dahhan and Dudukovic, 1995). At high pressure (liquid limited reaction) Figure 9(b) reveals identical performance of both reactors where complete wetting is achieved in both modes. Since we studied the impact of the two factors, pressure and feed concentration on the performance without fines, the same study was conducted for the bed diluted with fines. EMBED ExcelChart \s \* mergeformat Figure 4.  SEQ Figure_4. \* ARABIC 12 Effect of Fines on Low Pressure Down-flow Versus Up-flow Performance EMBED ExcelChart \s \* mergeformat Figure 4.  SEQ Figure_4. \* ARABIC 13 Effect of Fines on High Pressure Down-flow Versus Up-flow Performance 4.2.1 Effect of Pressure in Diluted Bed on Individual Mode of Operation The effect of pressure on the performance of both modes of operation in beds with fines is illustrated in Figures 10(a) and 10(b). At higher pressure the performance of both upflow and downflow is better than that at low pressure. This observation and the reasoning behind is also consistent with the data obtained without fines. 4.2.2 Effect of Feed Concentration in Diluted Bed on Individual Mode of Operation At 30 psig, the conversion is higher at lower feed concentration of a methyl styrene (lower 2 curves for downflow (Figure 10(a)) and upflow (Figure 10(b)). At higher reactor pressure, there is no effect of feed concentration (upper 2 curves, Figure 10(a) and 10(b)). This was also observed for the reactors without fines and explained on the basis of liquid limitation in the previous section. The fact that it is observed with fines confirms the feed concentration dependence (of performance) in case of gas and liquid limited reaction. Fines (inert particles, order of magnitude smaller than catalyst pellets, packed only in the voids of the catalyst) are used to investigate the performance of the two modes of operation using the same reaction in an attempt to demonstrate the decoupling of hydrodynamic and kinetic effects. A way to establish this is to use the up-flow and down-flow modes, which are intrinsically hydrodynamically different (as discussed earlier), to see if fines can indeed yield "true" kinetic behavior (i.e. rates unmasked by external transport resistances and hydrodynamic effects) in order to establish their usage as a preferred technique for scale-up and scale-down studies. The two extreme cases discussed before i.e., gas limitation (down-flow performance better than up-flow, Figure 6) and liquid limitation (up-flow performance better than down-flow, Figure 7) are now conducted in the presence of fines (Khadilkar et al., 1995; Wu et al., 1996). Figures 9 and 10 show the performance of both reactors when the bed is diluted with fines. It can be seen by comparing Figure 9 with Figure 6, and Figure 10 with Figure 7, that fines have eliminated the disparities between the two modes of operation even in the extreme cases of reactant limitation. This is primarily due to the fact that fines improve liquid spreading considerably and achieve comparable (and almost complete) wetting in both modes of operation. It must be noted that Figures 6 and 9, or Figures 7 and 10, could not be directly superimposed due to slightly different catalyst activity obtained after repacking the bed with fines and catalyst and reactivating it. Nevertheless, fines have successfully decoupled the hydrodynamics and the data with fines reflect the kinetics (including internal diffusional effects) in the packed bed under "ideal" liquid distribution conditions. It can be observed in Figure 9 that at low liquid flow rate and low pressure (gas limited reaction), trickle bed performs slightly better than up-flow which indicates that the degree of wetting is still not complete resulting in some direct exposure of the internally wetted but externally dry catalyst to the gas. This may be due to the fact that at low liquid flow rate, even with fines, the catalyst is not completely wetted (Al-Dahhan and Dudukovic, 1995). At high pressure (liquid limited reaction) Figure 10 reveals identical performance of both reactors where complete wetting is achieved in both modes. EMBED ExcelChart \s \* mergeformat Figure 4.  SEQ Figure_4. \* ARABIC 14 Effect of a-methylstyrene Feed Concentration at Different Pressures on Performance of Downflow with Fines. EMBED ExcelChart \s \* mergeformat Figure 4.  SEQ Figure_4. \* ARABIC 15 Effect of a-methylstyrene Feed Concentration at Different Pressures on Performance of Upflow with Fines. 4.3 Unsteady State Experiments in TBR 4.3.1 Performance Comparison for Liquid Flow Modulation under Gas and Liquid Limited Conditions Performance comparisons were done under gas and liquid limited conditions by evaluating the flow averaged conversion under steady and unsteady state conditions for a total cycle time of 60 s and a cycle split of 0.5. Under near liquid-limited conditions (i.e., high pressure and low feed concentrations) no enhancement is observed with ON/OFF modulation except at very low liquid mass velocities (high mean space times (= VR/QL)). Under these conditions the bed is poorly irrigated and the disadvantage due to liquid maldistribution can be overcome by high flow rate liquid (Figure 2a, ( < 4). At lower space times, performance enhancement is not seen under laboratory conditions due to the small reactor diameter and a very good distributor. The conditions investigated in the present experiments correspond to fairly high liquid hourly space velocities (LHSV) in comparison with industrial trickle beds, where this maldistribution effect may be seen to be more pronounced. LHSV varied from 7 to 36 in our experiments as compared to 1.5 to 10 used in industrial reactors. In case of gas limited conditions (i.e., at low operating pressures and high feed concentration), it can be seen that unsteady state performance (conversion) is significantly higher than that under steady state conditions at all space times (Figure 2b, ( ~ 25). This case illustrates the conditions of a liquid reactant full catalyst and enhanced supply of the gaseous reactant leading to better performance. This enhancement improves as the extent of partial wetting is increased as seen at higher space times (lower liquid mass velocities). The trend of the unsteady state data at high space time indicates that by reducing the catalyst wetting and increasing the gaseous reactant supply, further enhancement could be obtained. A small exothermic contribution is also observed during unsteady state operation here with maximum bed temperatures reaching ~ 6 oC higher than steady state temperatures. EMBED Excel.Sheet.8 Figure 4.  SEQ Figure_4. \* ARABIC 16 Comparison of Steady and Unsteady State Performance under Liquid and Gas Limited Conditions EMBED Excel.Chart.8 \s Figure 4.  SEQ Figure_4. \* ARABIC 17 Comparison of Steady and Unsteady State Performance under Liquid and Gas Limited Conditions 4.3.2 Effect of Modulation Parameters (Cycle Time and Cycle Split) on Unsteady State TBR Performance To explore whether further performance enhancement is achievable by increasing gaseous reactant supply to the catalyst, a constant mean flow was chosen and cycle split (() was varied at one of the values in Figure 2b under gas limited conditions (i.e., at low pressure and high liquid reactant feed concentration). It can be seen that further enhancement is indeed possible as cycle split is lowered from steady state (( = 1) to a split of ( = 0.25, the performance improved by 60% over steady state at the same mean liquid mass velocity (Figure 3a). This improvement continues up to the point where liquid limitation sets in at very low cycle split (where the liquid will be completely consumed in a time duration less than the OFF time of the cycle) beyond which the performance will be controlled by liquid reactant supply. This implies that performance improvement can be maximized by operating at a given liquid mass velocity and total cycle period. At the cycle split value of 0.33, where performance enhancement was significant (in Figure 3a), the total cycle period influence was investigated. The performance enhancement is seen to increase with total cycle period up to a point after which it drops to near steady state values. Similar maxima was observed by both Lange et al. (1994) and Haure et al. (1990) for different reaction systems. Due to the competition between gas reactant starvation at the lowest cycle periods and liquid reactant starvation at the higher ones, there exits a feasibility envelope in which the enhancement can be optimized (Figure 3b). The location of maxima is seen to be dependent upon the liquid reactant concentration as shown by higher values observed by Lange et al. (1994) (~ 7.5 min) for much higher reactant feed concentrations (~ 50 % v/v) for alpha-methylstyrene hydrogenation under similar operating conditions. This could be quantified by an effective ( parameter under dynamic conditions, which reflects both the effect of reactant limitation and transient variation of liquid mass velocity. EMBED Excel.Chart.8 \s Figure 4.  SEQ Figure_4. \* ARABIC 18 Effect of (a) Cycle Split and (b) Total Cycle Period on Unsteady State Performance under Gas Limited Conditions  EMBED Excel.Sheet.8  Figure 4.  SEQ Figure_4. \* ARABIC 19 Effect of (a) Cycle Split and (b) Total Cycle Period on Unsteady State Performance under Gas Limited Conditions 4.3.3 Effect of Amplitude (Liquid Flow and Feed Concentration ) on Unsteady State TBR Performance The feasibility envelope (region where performance enhancement is possible as shown in Figure 3b) is strongly dependent on the relative supply of gaseous reactants to the liquid reactants. To explore whether the feasibility region can be altered with changing mass velocity, the liquid mass velocity was reduced and performance examined at two mass velocities at a cycle split of 0.25. A significantly higher improvement is observed by lowering liquid mass velocity. At a mass velocity of 0.137 kg/m2s, a similar feasibility envelope is seen as in Figure 3b which ends in degradation of performance to below steady state at higher cycle periods due to depletion of the liquid reactants. The lower liquid mass velocity allows more time for liquid reactant supply (higher mean space times) to the catalyst. This is reflected in the shift in the liquid starvation to even higher total cycle periods (Figure 4), while the gas starvation side remains unchanged except for higher enhancements at lower mass velocity similar to that observed in Figure 2b. EMBED Excel.Sheet.8 Figure 4.  SEQ Figure_4. \* ARABIC 20 Effect of Liquid Mass Velocity on Unsteady State Performance under Gas Limited Conditions 4.3.4 Effect of Liquid Reactant Concentration and Pressure on Performance Two key parameters, which decide the extent of gas or liquid reactant limitation are liquid reactant feed concentration and operating pressure. The effect of liquid reactant feed concentration was examined under gas limited conditions by evaluating enhancement at different cycle splits. With increase in gas limitation due to higher liquid reactant feed concentration, we would expect higher enhancement but this is not observed in Figure 5a. Since the absolute value of the conversion at higher feed concentrations is lower (due to gas reactant limitation), the enhancement seen is not as high even if lower mean mass velocity is used. Mass velocity used at the higher feed concentration was 0.1 kg/m2s as compared to 0.24 kg/m2s at the lower concentration. The effect of operating pressure was examined at constant gas velocity of 5.4 cm/s and liquid mass velocity of 0.085 kg/m2s. Under gas limited conditions, both steady and unsteady performance improves with increase in pressure as expected (due to enhanced solubility at elevated pressures). This enhancement should diminish as liquid limited conditions are approached at further higher pressures especially at high liquid mass velocities where the bed irrigation is complete (under laboratory conditions). In the present experiment the liquid mass velocity used is fairly low hence the performance enhancement is seen even at the highest pressure studied (Figure 5b). EMBED Excel.Sheet.8 Figure 4.  SEQ Figure_4. \* ARABIC 21 Effect of Liquid Reactant feed Concentration (a) and Operating Pressure (b) on Unsteady State Performance EMBED Excel.Sheet.8 Figure 4.  SEQ Figure_4. \* ARABIC 22 Effect of Liquid Reactant feed Concentration (a) and Operating Pressure (b) on Unsteady State Performance 4.3.5 Effect of Cycling Frequency on Unsteady State Performance Liquid ON/OFF flow modulation can be considered as square wave cycling about the mean flow for the case with a cycle split of 0.5. Conversion under periodic conditions can then be used to examine the dominant time scales affected by induced flow modulation (IFM) by looking at the frequency (() dependence of flow averaged conversion. Both Figures 6a and 6b show performance enhancement as a function of the IFM frequency with similar trends seen at different pressures, feed concentrations, and even for a case of non-square wave pulsing (s = 0.2). The performance in both cases shows degeneration of the enhancement at low frequencies tending to the steady state operation at zero frequency. But as frequency is increased, the conversion reaches a clear maximum improvement point. Ritter and Douglas (1970) observed similar frequency dependence (in dynamic experiments on stirred tanks) and have attributed the maximum to correspond with the resonance frequency of the rate controlling process. All transport processes typically have a natural frequency corresponding to their characteristic time scale. Gas-liquid transport in trickle beds corresponds to 0.2 to 0.8 Hz (at low pressures), liquid-solid transport corresponds to 0.5 to 2 Hz, whereas catalyst level processes correspond to much lower frequency depending upon intrinsic reaction and diffusion rates in pellets. Typical industrial reactions in trickle bed reactors have a frequency of 0.01 to 0.1 Hz (Wu et al., 1995). Natural pulsing occurs in trickle beds at high liquid flows and displays frequencies of 1 to 10 Hz, at which external transport is significantly improved (Blok and Drinkenberg, 1982; Wu et al., 1995). The present IFM frequencies are much lower than that observed under natural pulsing. The frequency dependence of the performance (Figure 6a and 6b) shows that the highest influence of IFM can be observed at low frequencies (( ~0.1 Hz) at which catalyst level processes could be predominantly affected to obtain the observed enhancement. Some effect on external transport processes can also be seen (in Figures 6a and 6b) with some enhancement observed corresponding to their natural frequencies. This opens up the possibility of selectivity enhancement and control for complex reaction schemes by controlling reactant supply by the proper choice of the IFM frequency for the desired reactant (Wu et al., 1995). The low enhancement seen at both ends (( (0 and ( (() can be explained on the basis of the frequency analysis similar to that done by Lee and Bailey (1974) and Park et al. (1998). The very low IFM frequency (( (0) corresponds to a equilibrium state or pseudo steady state, where the reaction transport processes have time to catch up with the modulated variable (flow in this case) and the overall system behaves as a combination of discrete steady states. On the other hand at high IFM frequency (( ((), the input fluctuations are so rapid that none of the reaction-transport processes in the system can respond to the IFM and any gain in performance due to the modulated variable is again not feasible. EMBED Excel.Sheet.8 Figure 4.  SEQ Figure_4. \* ARABIC 23 Effect of Cycling Frequency on Unsteady State Performance EMBED Excel.Sheet.8 Figure  STYLEREF 1 \s 0.22 Effect of Cycling Frequency on Unsteady State Performance 4.3.6 Effect of Base-Peak Flow Modulation on Performance For the case of liquid limited conditions (i.e., at high pressure and low feed concentrations) the use of complete absence of liquid during the OFF part of the cycle is not recommended (Lange et al., 1994). A low base flow similar to the mean operating flow can be used with a periodic high flow slug introduced to improve liquid distribution and open up multiple flow pathways for the liquid to flow during the rest of the cycle. The cycle split (() here is the fraction of the cycle period for which the high flow rate slug is on (typically chosen to be very short). Tests were conducted at a cycle split of 0.1 and cycle times varying from 30 to 200 s at an operating pressure of 150 psig and low liquid reactant feed concentration to ensure liquid limited conditions. This strategy is shown to yield some improvement over steady state performance (Figure 7, ( < 1.2), although this is not as high as observed under gas limitation (maximum enhancement observed here was 12 % as against 60 % in case of gas limited conditions). This is primarily due to intrinsically better flow distribution in small laboratory reactors, which would not be the case in typical pilot or industrial reactors where much higher enhancement can be anticipated. EMBED Excel.Sheet.8 Figure 4.  SEQ Figure_4. \* ARABIC 24 Unsteady State Performance with BASE-PEAK Flow Modulation under Liquid Limited Conditions  EMBED Excel.Sheet.8  Figure 4.  SEQ Figure_4. \* ARABIC 25 Effect of Liquid Mass Velocity on Steady State Liquid-Solid Contacting Efficiency CHAPTER 5. MODELING OF TRICKLE BED REACTORS 5.1 Evaluation of Steady State Models for TBR and PBC The qualitative analysis and arguments made in the discussion of the experimental comparison in section 4.1 on the basis of reactant limitation, liquid-solid contacting, and effect of pressure on kinetics needed to be verified by compariosn with model predictions of some of the existing models. A history of the model development of trickle bed reactors was presented in Chapter 2 ad salient features of each presented in Table 2.3. Based on the discussion therein, two modles were chosen to compare predictions to experimental data. The key parameters of distinction between downflow and upflow are contacting efficiency and hence the solution of partially wetted pellet performance for downflow and fully wetted pellets in performance for upflow. The effect of liquid mass velocity and gas-liquid and liquid-solid transport in both reactors needs to be modleled correctly. Hence, two models, one with reactor scale equations (El-Hisnawi, 1982) and other with pellet scale equations (Beaudry, 1987) both developed at CREL. The intrinsic kinetics required for this were obtained earlier as discussed in section 4.3. Reactor Scale Model (El-Hisnawi et al., 1982) El-Hisnawi et al. (1982) model was originally developed for low pressure rickle bed reactor to account for rate enhancement due to externally inactively wetted areas. Analytical solutions derived for first order kinetics fr the equations aat low pressure (as shown in Table 5.1). Since A is the limiting reactant, its surface concentration is solved for and rate evaluated and substituted in the plug flow equation for concentration of B to obtain conversion of B at each velocity specified. At high pressure, the kinetics were observed to be non-linear and surface concentration of B was solved from the non-linear equation given in Table 5.1 to get the rate equation and then solved numerically to get the concentration profile of B and conversion at each space time. The pellet effectiveness factor can be determined from the thiele modulus but was used here as a fitting parameter at one superfircial velocity and used for all other cases. This was done due to the uncertainity in the catalyst activity (rate constant) and the effective diffusivity evaluateion at different pressure. The liquid-solid contacting effciency was deterined at kow pressure by correlations developed by El-Hisnawi (1981) and at high pressure using the correlation of Al-Dahhan (1993). The upflow reactor was assumed to have completely wetted catalyst in all cases. Gas-liquid and liquid-solid mass transfer coefficients were determined for downflow by correlations for gas-liquid mass transfer coefficient from Fukushima and Kusaka (1977), for liquidsolid mass transfer coefficient from Tan and Smith (1980),and gas-solid mass transfer coefficient from Dwiwedi and Upadhyay (1977). For upflow prediction, the gas-liquid mas transfer coefficient was obtained by correlation by Reiss (1967) and liquid-solid mass transfer coefficient by Spechhia (1978). The variation of the mass transfer coefficients calculated from the above correlations with space time is shown in Figure 5.5. The predictions of El-Hisnawi model at low pressure (gas limited) compare well with the downflow experimental data as shown in Figure 5.1, whereas overpredicts the upflow data slightly. At high pressure liquid limited conditions, however, El-Hisnawi model predictions compare well with the experimental data as shown in Figure 5.2. Pellet scale model (Beaudry et al., 1987). The Beaudry model considers the catalyst pellets in the form of infinte slabs with two sides expoesed to either gas or liquid on both sides or a half-wetted pellet exposed to gas and liquid on either side. At low pressure (gas limited conditions), the gaseous reactant supplied from both sides of the pellet depleted to almost zero within a short distance depending upon the extent of the limitation. Hence the solution of pellet effectiveness considered both the dry and wetted side for the half wetted pellet , and solution of the completely wetted pellet for the gas limited case (As shown Table 5.2). For the completely dry pellet the effectiveness for this case was zero since no liquid reactant could be suplied to this pellet. The analytical solutions to this case for the first order kinetics are available in Beaudry (1987) and were used to obtain the oevrall effectiveness factor as a weited average of the containg and the effectiveness of each type of pellet. At high pressure, under liquid limited conditions the solution is much more complicated due to the non-linear reaction rate which demands the solution of the reactioon diffusion equations for the externally wetted pellets on both sids and one on only one side. Here, the value of ( is the point where the liquid reactant depletes completely and is the boundary for the liquid reactant concentration soluiton. This solkution needs to be done at each point in the reactor to get a local effectiveness factor corresponding to the local concentration of the liquid reactant. Instead of doing this as a coupled system of equation both on the pellet and reactor scale, the pellet scale equations were solved at different extrenal concentrations and then fitted as a polynomial of effectivenss as a fucntion of surface concentration. This polynomial is then used to solve the reactort scale equations numericall to obtian conversion at each space time. Although this approach does not require any fitting parameters as needed in the El-Hisnawi model, the rate constant was similarly fitted to match the conversion at one spacetime and used to compare with the experimental data at all other space times. As can be seen from Figure 3 and 4, this model predicts the observed data for down flow at low pressure and at high pressure well, but not so well for up-flow especially at low pressure and high feed concentration. The reason may be due to mass transfer correlations used which may predict a lower performance (than observed experimentally) at high space times in the upflow operating mode. This model has the drawback of not considering pellet scale phenomena and has to be supplied with apparent kinetics (pellet effectiveness factor) as an input (or a fitting parameter). The Beaudry (1987) model predictions are also shown on the Figure 5.1 and 5.2 for low and high pressure respectively. As can be seen, Beaudry's model predicts downflow performance exactly as El-Hisnawi's model does, but under-predicts upflow performance at higher space times (low liquid velocities) due to some effect of estimation of mass transfer can be seen at high space times due to the correlations used. For high pressure, on the other hand, Beaudry's models predicts experimental data quite well both for downflow and upflow, since the effect of mass transfer is not seen as much as that at low pressure. The effect of the feed concentration of both the models was also examined for both downflow and upflow reactors as shown in Figures 5.3 and 5.4 respectively. As mentioned earlier in the discussion, the downflow predictions are almost identical for both models for downflow, at which the model predictions agree with experimental data. The predictions for the upflow experimental data for both modles differs slightly diue to the effect of mass transfer coefficient for both the models, as seen in Figure 5.4. In both cases, the inverse relationship of liquid feed concentration with conversion seen in the experiments is predicted correctly in both the cases. The performance of upflow and downflow is reaction system dependent i. e., whether the reaction is gas or liquid limited under the conditions of investigation. The laboratory reactors are operated in the range of partially to fully wetted catalyst to demonstrate that the influence of wetting can be detrimental or beneficial, depending upon the reactant limitation. Models that account for these two effects can predict the performance over the entire range of operating conditions. The intrinsic kinetics of the reaction studied is different at different pressures and hence it is recommended to study slurry kinetics at the operating pressure before any scale up or modeling is attempted. A rate expression with different rate constants at different pressures can be used to predict the trickle bed reactor data well. It must be mentioned that the reactor scale model failed to predict experimental data at 100 psig when the reaction is neither gas limited nor liquid limited because the model assumptions were for the extreme conditions of one reactant being limiting. Pellet scale modified model should be able to cover all cases. The predictions of the reactor scale and pellet scale models are satisfactory for current conditions although there is a need for high pressure correlation for mass transfer coefficient and interfacial area in order to predict performance with greater certainty, especially in cases where the rate is affected significantly by external mass transfer. Table 5.  SEQ Table_5. \* ARABIC 1 Governing Equations for El-Hisnawi (1982) Model Original model equations (gas limited conditions) EMBED Equation  EMBED Equation  and EMBED Equation  EMBED Equation  Boundary conditions: EMBED Equation  Equilibrium feed EMBED Equation  Non-Equilibrium feed and EMBED Equation  Model equations at high pressure (liquid limited conditions) (boundary conditions are the same as at low pressure) embed Equation  embed Equation  embed Equation  (for liquid reactant limitation) Table 5.  SEQ Table_5. \* ARABIC 2 Governing Equations for Beaudry (1987) Model Pellet Scale Equations: a) Low Pressure (Gas Reactant Limited) with rate first order in A.  embed Equation  Boundary conditions: EMBED Equation  (=0 for m<1) EMBED Equation  EMBED Equation  (=0 for m<1) EMBED Equation  EMBED Equation  EMBED Equation  EMBED Equation  EMBED Equation  where: EMBED Equation  EMBED Equation  b) High Pressure (Liquid Reactant (diffusional) Limitation for Langmuir Hinshelwood rate form) embed Equation  Boundary conditions: EMBED Equation EMBED Equation EMBED Equation EMBED Equation EMBED Equation For both cases a) and b) above, the reactant conversion in general is given by embed Equation  where embed Equation  (at low pressure) embed Equation  (at high pressure) and overall effectiveness factor is given by embed Equation  EMBED ExcelChart \s \* mergeformat Figure 5.  SEQ Figure_5. \* ARABIC 1 Upflow and Downflow Performance at Low Pressure (gas limited condition): Experimental data and model predictions EMBED ExcelChart \s \* mergeformat Figure 5.  SEQ Figure_5. \* ARABIC 2 Upflow and Downflow Performance at High Pressure (liquid limited condition): Experimental data and model predictions EMBED ExcelChart \s \* mergeformat Figure 5.  SEQ Figure_5. \* ARABIC 3 Effect of Feed Concentration on Performance (Downflow) EMBED ExcelChart \s \* mergeformat Figure 5.  SEQ Figure_5. \* ARABIC 4 Effect of Feed Concentration on Performance (Upflow) EMBED ExcelChart \s \* mergeformat Figure 5.  SEQ Figure_5. \* ARABIC 5 Estimates of volumetric mass transfer coefficients in the range of operation from published correlations. 5.2 Unsteady State Model for Performance of Trickle Bed Reactors in Periodic Operation This section discusses the development of a generalized model which can account for phenomena occurring in experimentation discussed in section 4.2 to evaluate performance under unsteady state operation on a test reaction under gas and liquid limited conditions. The following steps can help understand the phenomena ocurring in the experiments conducted on unsteady state flow modulation presented in section 4.2. These also confirm the recent explanation of Gabarain et al. (1997) and their modeling efforts for periodic flow modulation in terms of different zones from completely externally wetted pellets, partially externally wetted pellets and completely externally and internally dry pellets. A typical cycle period can be considered in terms of the steps outlined below. 1. Supply of liquid reactants by a slug of liquid to the catalyst with almost complete wetting when the slug passes (Figure 5.6(a)). 2. This is followed by a liquid OFF (or a low liquid flow) period (Figure 5.6 (b) and (c)) in which supply of gaseous reactants occurs directly to the catalyst surface, and, hence, reaction occurs in the catalyst. 3. The may be allowed to occur isothermally or non-isothermally. Temperature rises and, hence, higher rates are achieved during the OFF part of the cycle (Figure 13 (c)). The effect of increased temperature affects the evaporation of liquid reactant. In case of a gas reactant limited reactions, no liquid flow is allowed in the OFF part of the cycle(Figure 5.6 (c)). In case of liquid reactant limited reaction a low flow of liquid is used in the OFF (referred to as section 4.2) part following the slug of higher liquid flow rate (Figure 5.6 (b)) This is followed by re-introduction of another slug of liquid which quenches the heat and replenishes the liquid reactant (Figure 13 (a)). EMBED MSDraw \* mergeformat Figure 5.  SEQ Figure_5. \* ARABIC 6 Phenomena Occurring in Trickle Bed under Periodic Operation The phenomena occurring in periodic operation as described above can be summarized and converted to the required equations as follows 1. Dynamic variation in the gas and liquid phase species concentrations is obtained by solving species mass balance equations for the gas and liquid phase. 2. Gas-liquid and liquid-solid mass transfer is modeled by the Maxwell-Stefan equations for multicomponent transfer. 3. Variation of holdup with time and axial position, during and after the pulse of liquid/gas, is obtained by solving the continuity equations for the liquid and the gas phase. 4. Variation of interstitial velocity with time and axial position is accounted for by solving the liquid and gas phase momentum balance equations. 5. Periodic boundary (temporal) conditions are incorporated by changing the conditions of the inlet variables at partial and full cycle (sSYMBOL 116 \f "Symbol" and SYMBOL 116 \f "Symbol"). 6. Allowance for variation of parameters such as contacting efficiency and mass transfer coefficients with axial position and time is made by dynamic evaluation of the parameters at each time instant at all the axial locations. 7. Temperature rise during the OFF pulse part, and the ensuing evaporation, is accounted for by simultaneous solution of gas-liquid heat transfer equations (along with multicomponent mass transfer equations). Different temperature for solid, liquid and gas phase are also modeled to maintain rigor. 8. Allowance for the reaction to continue due to static holdup during the OFF pulse part, and for the exchange of species during both parts of the cycle, is made by inserting catalyst level accumulation terms (and Maxwell-Stefan diffusion flux terms) for reactant and product species. Scope : The preliminary model equations proposed here are based on the assumptions as mentioned below and will be refined based on preliminary results to impose/relax some of the conditions as required. A list of the number of variables and corresponding equations necessary to solve the model is given in Table 5.6. It must be emphasized here that the use of Maxwell-Stefan formulation is made in order to account for concentration effects, thermodynamic non-idealities and bulk transport of heat and mass across the interfaces as discussed in section 2.4. Assumptions : 1. The density of liquid phase remains constant. 2. Variation of temperature, concentration, velocity and holdup in radial direction is negligible as compared to axial direction. 3. All the parameter values are equal to the cross sectionally averaged values and vary only with axial location and time. These are determined at every time instant for the different axial location and used to solve for pertinent variables at the next time instant. 4. The heat of reaction is released only to the solid and then transferred to other phases through interphase heat transfer. 5. No temperature gradients exist inside the catalyst pellet. 6. The catalyst pellets are modeled as slabs of three types: fully externally wetted, half wetted and fully dry. 5.2.1 Reactor Scale Transport Model and Simulation Bulk phase equations for species and energy for gas, liquid and solid Equations for the bulk phase species concentrations can be written in terms of the convection, accumulation and interphase fluxes as given below. The expression and models for the fluxes are discussed in detail in the next section and are the crucial factors as far as solution of these equations is concerned. Liquid Phase Species Balance Equations The generalized 1-D equation for flowing liquid phase species concentrations can be written as a balance of convection, interphase (gas-liquid, liquid-solid) mass transfer and net accumulation terms (nc-1 equations) (where nc is the number of species). EMBED Equation  (5.  SEQ 5. \* ARABIC 1) Gas Phase Species Balance Equations This is written as a balance between convection and interphase transport to get accumulation term for any given species. Thus, nc-1 equations can be written (for a total of nc species) as: EMBED Equation  (5.  SEQ 5. \* ARABIC 2) The catalyst level species balance equations are discussed in a separate subsection due to several strategies used for their formulation and solution. Energy Balance Equations A three temperature model with catalyst, liquid and gas having different temperatures Gas Phase Balance : This incorporates convection, liquid to gas phase heat transfer (conductive and due to evaporation), heat loss to ambient and accumulation written in terms of gas temperature. All other contributions (such as due to viscous dissipation, pressure effect and mechanical work etc.) are not considered in the equations. The terms for heat loss to ambient have been put in for sake of generality, but solution will primarily focus on adiabatic conditions where these are omitted. EMBED Equation  (5.  SEQ 5. \* ARABIC 3) where  EMBED Equation.3  (5.  SEQ 5. \* ARABIC 4) Liquid Phase Balance : Similarly, writing balances for liquid phase in terms of convection, liquid to gas heat transfer, solid to liquid heat transfer, heat loss to ambient and accumulation, we have EMBED Equation  (5.  SEQ 5. \* ARABIC 5) where  EMBED Equation.3  (5.  SEQ 5. \* ARABIC 6) Catalyst (Solid Phase) Balance : Catalyst is assumed to be fully internally wetted and to have no internal temperature gradients. Axial temperature variation is obtained by EMBED Equation  (5.  SEQ 5. \* ARABIC 7) where  EMBED Equation.3  (5.  SEQ 5. \* ARABIC 8) The interphase mass and heat transfer fluxes are evaluated as discussed later in this chapter. The species enthalpy shown in equations 5.4, 5.6, and 5.8 are obtained as a combination of heat of formation and a Cp(T term which gives the phase temperature. The equations 5.1, 5.2, 5.3, 5.7, and 5.7 are solved at each time at axial location by marching in space using a semi-implicit predictor step which is used as the guess value for the fully implicit solver used to obtain solution of the source terms for interphase fluxes, compositions and temperatures. These are then added to the vector of guess values used for the solution of the non-linear equations at the interface which is done by using a globally convergent multivariable newton solver with a line search algorithm incorporated into it. All the concentrations, temperatures and other scalar variables are defined at the cell centers of a staggered grid employed here for convenience in solution of pressure and continuity in the flow solver discussed in the next section. Although it is convenient in many cases to non-dimensionalize the equations using characteristic values of each variable to obtain the familiar dimensionless groups, it was found to be inappropriate in this case due to two reasons 1) large variations in the variables during each periodic cycle yielding no single charateristic value for non-dimensionalization, and 2) The transport coeffcients differ for each pair of components and are written in matrix form, which would result in a matrix of dimensionless Sherwood or Stanton numbers resulting again in no single dimensionless grop to analyze effects. However, non-dimensionalizing the variables was deemed necessary for the successful usage of the numerical routines used. This was done by considering the guess vector at each point to be the charateristic set of values of the variables and was used to non-dimensionalize the variables before the newton solver was called. This dynamic non-dimensionalization improved the effectivness of the non-linear solver considerably. 5.2.2 Flow Model Equations Continuity Equation for the flowing liquid and gas The liquid present in the reactor is divided into flowing liquid and stagnant (intra-catalyst) liquid for the sake of convenience in modeling. The continuity equation for the flowing liquid and gas can be given in terms of the accumulation and convection terms balanced by the total mass transferred to and from the other phase (written in terms of interphase fluxes for gas-liquid, liquid-solid and gas-solid equations, each discussed in the subsequent section). EMBED Equation  (5.  SEQ 5. \* ARABIC 9) EMBED Equation  (5.  SEQ 5. \* ARABIC 10) The momentum equations for liquid and gas can be written in terms of accumulation, convection terms on the left hand side and the gravity, pressure, drag due to the packed phase, gas-liquid interphase momentum exchange (with exchange coefficient K) and momentum gain due to mass exchange (assuming it is added at the interfacial velocities for liquid and gas phase, uIIL and uIIG respectively) terms on the right hand side as : EMBED Equation  (5.  SEQ 5. \* ARABIC 11) EMBED Equation  (5.  SEQ 5. \* ARABIC 12) Expanding and simplifying using continuity equations with the assumption of identical velocity of bulk and interface in each of the phases EMBED Equation  (5.  SEQ 5. \* ARABIC 13) EMBED Equation  (5.  SEQ 5. \* ARABIC 14) Here K is the momentum exchange coefficient between the gas and liquid (suitably defined or used as a fitting parameter) and the drag term is obtained from Holub's three zone analysis (Holub, 1990) for different flow conditions as follows (with zone selection based on the local value of liquid holdup obtained from the continuity equation and Reynolds numbers calculated at each axial location denoted by subscripts R, F, and D for rivulet, film and dry zones respectively) Film zone: This occurs during the liquid pulse beyond the entrance zone and at the beginning of the liquid OFF cycle (or if there is a small base flow during OFF cycle) when both liquid and gas present (1.2SYMBOL 101 \f "Symbol"sL <SYMBOL 101 \f "Symbol"L < 0.8SYMBOL 101 \f "Symbol") (assuming that flow is not in the high interaction regime). EMBED Equation  (5.  SEQ 5. \* ARABIC 15) EMBED Equation  (5.  SEQ 5. \* ARABIC 16) Solution of Continuity and Momentum Balance Equations The method of solution of the continuity and momentum equations is the modified MAC method (Harlow and Welch, 1965) which is a semi-implicit method, i.e., it is implicit in pressure and interface exchange terms and explicit in all other terms such as gravity, drag and convective terms. A staggered grid is used for scalars such as pressure and phase holdups for which values are evaluated at the midpoint of the nodes, whereas vectors such as velocity, drag, etc. are evaluated at the node points themselves (Patankar, 1980, Versteeg and Malalsekhara, 1996). A simplification made here for executing this code is that both the phases are incompressible (which discussed in the subsequent discussion section). The algorithm thus consists of three main steps, Evaluation of the explicit terms to obtain an intermediate velocity Uil* and Uig* (which does not satisfy continuity). Substituting these to evaluate pressure at the next instant by solution of poisson equation for pressure). Evaluating the velocities and holdups at the new instant according to the newly evaluated pressure (which ensures that continuity is satisfied). The calculations at each of the above steps are as described below (here n represents the current time values and n+1 represents the next time instant values). Usually inlet superficial velocities are known for both the phases, but interstitial velocities are needed for the reactor inlet. These are calculated by estimating liquid holdup using Larachi et al. (1990) correlation as  EMBED Equation.3  (5.  SEQ 5. \* ARABIC 17) The superficial velocities are then corrected to obtain inlet interstitial velocities. Step 1. Evaluation of intermediate velocities from explicit equations. The momentum equation is evaluated explicitly (at n th time instant) to obtain an estimate of the velocities at the next time (n+1). Since these velocities will not satisfy continuity, they are considered only as intermediate value for further calculations. The phase holdups used here are also the n th level values to be updated later.  EMBED Equation.2 (5.  SEQ 5. \* ARABIC 18)  EMBED Equation.2 (5.  SEQ 5. \* ARABIC 19) Step 2. Similarly we can write implicit equations for the velocities at the next time instant with all the terms evaluated implicitly. Subtracting above equations from the implicit equations, we can get a relationship between the intermediate values and the actual velocity values (at the n+1 th time step).  EMBED Equation.2  (5.  SEQ 5. \* ARABIC 20)  EMBED Equation.2  (5.  SEQ 5. \* ARABIC 21) In order to calculate the pressure at the next time step, we need to eliminate the velocities of gas and liquid at the next time instant (n+1). This is done by substituting the above equations into the continuity equations (and eliminating the time derivatives by addition) to obtain the Poisson equation for pressure. This is done differently in two cases as follows: a) Constant Density (Volume Fraction Addition) Method: This is done in two stages, first by simplifying the RHS of above equations to eliminate non-primary intermediate variables (u*iG in liquid equation and u*iL in gas equation), to get  EMBED Equation.2  (5.  SEQ 5. \* ARABIC 22)  EMBED Equation.2  (5.  SEQ 5. \* ARABIC 23) where  EMBED Equation.2  (5.  SEQ 5. \* ARABIC 24) and  EMBED Equation.2  (5.  SEQ 5. \* ARABIC 25)  EMBED Equation.2  (5.  SEQ 5. \* ARABIC 26) Substituting these in the continuity equation and eliminating n+1 level velocities of gas and liquid, we have  EMBED Equation.2 where  EMBED Equation.2  (5.  SEQ 5. \* ARABIC 27) This is the familiar Poisson equation for pressure, written here for the pressure difference in time ( EMBED Equation.2 ). This has to be solved for the entire z domain and can be arranged in the form of a tri-diagonal matrix of equations (when written for all z). The boundary conditions for pressure used are that the pressure gradients in time ((p) just outside the inlet and the exit are zero, which is true based on the physics of the problem. It must be noted here that the pressure values are evaluated at the center of the grid over which velocity is calculated. The tri-diagonal system is then solved efficiently by the Thomas algorithm. b) Variable Density Correction (Rigorous Algorithm): Here equation 10a and 10b are used to substitute in the continuity equation by pre-multiplying them with the corresponding density and phase holdups. Thus Equation 12 is written on elimination of the time derivatives of holdup (which get eliminated on summation of the continuity equation) as  EMBED Equation.2  (5.  SEQ 5. \* ARABIC 28) where  EMBED Equation.2  and  EMBED Equation.2 . (5.  SEQ 5. \* ARABIC 29) Here the derivatives of density can be neglected for simplicity. Step 3. The pressure obtained from the solution of the above set of equations is used to update the velocities (using the above equations for the n+1 th time instant). These velocities are in turn used to update holdup at the next instant using the continuity equations. EMBED Equation  (5.  SEQ 5. \* ARABIC 30) EMBED Equation  (5.  SEQ 5. \* ARABIC 31) This procedure for pressure calculation and velocity correction is continued until velocity and pressure converge. Adaptive time setpping is used in the flow solution due to the explicit part of the procedure. Solution for holdup and velocty over the entire reactor length calculated using a full and two half time steps are compared to check for convergence as shown in the flow sheet in Appendix B. 5.2.3 Multicomponent Transport at the Interface Interphase Mass and Energy Transport using Maxwell-Stefan equations The interphase fluxes for the three transport processes must be rigorously modeled for periodic operation due to the fact that assumptions for conventional isothermal, steady state dilute solution transport of single species do not apply in this case. Since flow rates are changing with time and axial position by a large magnitude, any assumption of simple (or equilibrium) mass or heat transfer controlling steps will not hold for the entire simulation. Hence the use of Stefan-Maxwell equations to model interphase fluxes is made in the simulations. The important factor in modeling Stefan-Maxwell processes is that an extra equation connecting the relative flux equations to the total (or some reference) flux is necessary. This reference relation (referred to as bootstrap) is discussed for each process after the individual solution sections. Solution of Stefan-Maxwell Equations at the Gas-Liquid Interface: The gas-liquid flux can be written for each component as follows:  EMBED Equation.2  (5.  SEQ 5. \* ARABIC 32) where EMBED Equation  is the liquid side mass transfer coefficient matrix, EMBED Equation  is the bootstrap matrix and EMBED Equation . The (x thus denotes the mean heat of vaporization of the liquid. As can be seen here, the interface energy balance equation is used as the reference (bootstrap) relation, which affects the transport fluxes of all the species through the EMBED Equation  matrix. The (q value is the net conductive heat flux at the interface and given as  EMBED Equation.2 . Similar equations can be written for the gas side using EMBED Equation , the gas side mass transfer coefficient matrix as well as the gas side bootstrap matrix based on (y (the mean heat of vaporization based on gas side mole fraction). The actual calculation of the bootstrap matrix is given in the next section. It has been assumed here that the interface reaches equilibrium instantaneously, and that there is no accumulation at the interface from which one can write EMBED Equation  (which is used in further reference to the gas-liquid flux) The equilibrium at the gas-liquid interface also provides with relations for interface concentrations given for each species as EMBED Equation  (5.  SEQ 5. \* ARABIC 33) The number of unknown variables in these relations is nc fluxes, nc interphase mole fractions for liquid and vapor interface each and interface temperature (total = 3nc+1). The number of equations available for solution are the liquid and gas phase absolute flux equations (nc-1 each) , the equilibrium relations at the interface (nc), the mole fraction summations (to unity) of the gas and liquid interface compositions (2) and the energy balance relation to obtain gas-liquid interface temperature. These are coupled simultaneous non-linear equations, which are expressed as a combined vector of the functions to be solved as shown below. Table 5.  SEQ Table_5. \* ARABIC 3 Typical Equation Vector for the Stefan-Maxwell Solution of Gas-Liquid Interface  EMBED Equation.2  The solution of this matrix of non-linear equations is accomplished by a modified multi-variable Newtons method for which the Jacobian is evaluated numerically and partial variable correction is taken each iteration to improve convergence. 5.2.4 Catalyst Level Rigorous and Apparent Rate Solution Combined Solution of Stefan-Maxwell Equations for Liquid-solid and Gas-Solid Interface The liquid solid transport equations can be formulated in the Stefan Maxwell scheme by writing flux equations of the general form EMBED Equation  (5.  SEQ 5. \* ARABIC 34) The total number of unknowns for this set of equations is 2nc+1 (of which n are the liquid solid fluxes, nc are the liquid-solid interface compositions and 1 liquid solid temperature). The number of independent equations that can be written for the liquid solid system are only 2nc (of which nc-1 are the mass transfer flux equations on the liquid side, nc-1 are the catalyst wetted side fluxes from the reaction diffusion equations, 1 equations from the interface mole fraction summation and 1 from the energy flux equation). This needs another relation to solve the set of equations. Similar equations should also be written for the (externally) dry side of the catalyst. The unknowns in this case are 3nc+1(of which n are gas solid fluxes, 2nc are gas side and solid side interface concentrations and one gas solid interface temperature). The number of independent equations available to us are also 3nc+1 (nc-1 flux equations on the gas side, nc-1 catalyst side flux equations from reaction diffusion equations, nc equilibrium relations between the gas and solid (wetted) interface compositions, 1 equation each for the interface mole fraction summation on the gas and solid side, and 1 equation for the energy flux). The two sets of equations are coupled through the catalyst level reaction diffusion equations because they incorporate the boundary conditions on the pellet. Hence have coupled bootstrap condition in order to make the entire set of equations consistent. This can be obtained by satisfying mass balance of the inert component or more rigorously the total mass balance as given by  EMBED Equation.2  (5.  SEQ 5. \* ARABIC 35) or more accurately as  EMBED Equation.2  (5.  SEQ 5. \* ARABIC 36) This completes the set of equations that are necessary to solve for all the transport processes on the liquid-solid and gas-solid interfaces (5nc+2 total equations). This is in addition to the catalyst level equations (which are (nc-1)x(ncat-1)) and the bulk phase species and temperature equations (2nc+3). The system of equations for just the liquid-solid and gas-solid equations combined can be represented in Table 5.4. Table 5.  SEQ Table_5. \* ARABIC 4 Typical Stefan Maxwell Equation Vector for a Half Wetted Pellet  EMBED Equation.2  The solution of this system involves Jacobian evaluation (or iteration) followed by matrix inversion. Inspection of the origin of some of these variables such as intra-catalyst concentrations shows that the catalyst flux equations also need to be solved simultaneously with the above (since neither the flux nor the interface concentrations at the next time step are known until the Stefan Maxwell solution is converged). Energy Flux Relations for Interphase Transport by Stefan Maxwell Approach For gas-liquid, liquid-solid and gas-solid transport, the interphase energy flux equations can be written based on the Maxwell-Stefan formulation (as described in section 2.3 ) as a combination of the conductive (without mass exchange) and bulk transport (such as with evaporatioon, condenstaion etc., ) flux for bulk to interface for each phase and then for continuity of the flux at the interface as shown in Table for energy transport flux from bulk vapor to interface Table 5.  SEQ Table_5. \* ARABIC 5 Energy Flux Equations for Gas-Liquid, Liquid-Solid, and Gas-Solid Interfaces ___________________________________________________________________ Gas-Liquid Interface EMBED Equation  EMBED Equation  Interface Equilibrium EMBED Equation  Liquid-Solid Interface EMBED Equation  EMBED Equation  Interface Equilibrium EMBED Equation  Gas-Solid Interface EMBED Equation  EMBED Equation  Interface Equilibrium EMBED Equation  ___________________________________________________________________ Solid Phase Species Balance Equations Catalyst level equations need to be solved at each axial location separately at each time and are typically of the reaction-diffusion type. The catalyst pellets are considered to be in the form of infinite slabs with the two sides exposed to the flowing phases similar to those assumed by Beaudry et al. (1987). Three approaches are considered as alternatives to solve the catalyst level equations and are discussed with pros and cons of each approach. The general equation in all the three types is as given below EMBED Equation  (5.  SEQ 5. \* ARABIC 37) where x is the intra-catalyst spatial dimension i) Rigorous Single Half Wetted Pellet Approach This formulation consider a single type of pellet which is half-wetted i.e., wetted on one side and dry on the other (Figure 5.6(a)). For the intra-catalyst calculations, the flux can be expressed in terms of the stefan maxwell diffusion matrix as a product of the inverse of the diffusion matrix and the non-ideality matrix ([B]-1[(] as discussed in section 2.3). But this is a function of the concentration and location thus complicating the solution of the reaction-diffusion equations. This is simplified by using and average diffusivity matrix or by evaluating the diffusivity matrix at each location. This simplification is still not as unrealistic as the usage of the film model for the extracatalyst equation as discussed in section 2.3. This leads to the set of equations (written for nc-1 species) in terms of central finite difference for the spatial derivatives is as shown in Table 5.5. The disadvantages of this approach is that 1) it assumes a half wetted pellet at all times and axial locations, which is not the case in the system of our interest, where the flow modulation causes all three cases i.e., fully externally wetted, half wetted and fully externally dry pellets, 2) this approach assumes constant wetting conditions and is typically suitable for steady state simulation, and, 3) the total number of equations is (nc-1)*(ncat-1)is very large to be solved at each axial location and each time instant due to the use of finite difference where the ncat required to capture all species concentration profiles is quite large. Table 5.  SEQ Table_5. \* ARABIC 6 Equation Set for Single Pellet Model ________________________________________________________________________  EMBED Equation.2 ________________________________________________________________________ ii) Apparent Rate Three Pellet Form This approach considers the possible wetting conditions for a slab type of catalyst, i.e., it considers three types of pellets which can have a both its side wetted, a half wetted pellet, and a pellet with both sides externally dry (Figure 5.6 (b)). But this necessitates the use of three times the number of equations and a very large computational effort. This is bypassed by using apparent rates with locally calculated effectiveness factors and assuming that the reaction occurs either at the interface or in the bulk phase of the catalyst. This assumpiotn is realistic only at very high thiele modulus (surface rate) and very low thiele modulus (bulk rate), but is not suitable for the intermediate case or for a general reaction case. But it is used due to its advantage of possibilty of multiple wetting conditions, as well as allowing for accumulation of the non-limiting reactant which can then be consumed during some part of the transient cycle. The governing equations for this case are given in general as  EMBED Equation.3  (5.  SEQ 5. \* ARABIC 38)  EMBED Word.Picture.8  Figure 5.  SEQ Figure_5. \* ARABIC 7 Representation of the Catalyst Level Solution iii) Rigorous Three Pellet Form (Collocation Solution) The disadvantages of both the approaches discussed above i.e., large computaional requirement for rigorous three pellet model (approach ii) and the lack of modeling multiple wetting (in approach i) are overcome by using a collocation solution (using a Jacobi polynomial) for the spatial derivative which requires less number of computational points and hence less number of equations to be solved. This approach does not have the shortcomings of the approach ii, but does require extra effort in solution of the catalyst level equations, particulary since both the multicomponent and the colocation solution require solution of a diffusivity ([(][B]-1[(]) and derivative matrix ([Bcol]. The set of equations thus can be written as shown in Table 5.7 Table 5.  SEQ Table_5. \* ARABIC 7 Catalyst Level Equations for Rigorous Three Pellet Model _______________________________________________________________  EMBED Equation.2  _______________________________________________________________ Table 5.  SEQ Table_5. \* ARABIC 8 List of Model Variables and Equations (component i, c = total number of components) VariableSymbol# of varsEquation Basis# of EquationsEqn. # Gas phase concentration CGi c-1 Gas phase mass balance c-1 6  Liquid phase concentration CLi c-1 Liquid phase mass balance c-1 4  Mass transfer of i from gas to interface NVi c Gas phase multicomponent transferc-17 Mass transfer of i from interface to liquid NLi c Liquid phase multicomponent transfer c-1 5  Interface vapor mole fraction of i yIi c V-L Interface flux balance c 8  Interface liquid mole fraction of i xIi c V-L Interface equilibriumc9 Gas temperature TG 1 Gas phase energy balance 1 21  Liquid temperature TL 1 Liquid phase energy balance 1 22  Interface (G-L) temperature TI 1 V-L Interface energy equilibrium 1 25  vapor to interface energy transfer rateEV1Gas to V-L interface heat transfer123 liquid to interface energy transfer rate EL 1 Liquid to V-L interface heat transfer 1 24     V-L interface vapor mole fraction sum 1 11     V-L interface liquid mole fraction sum110 Gas Holdup SYMBOL 101 \f "Symbol"G 1 Gas phase continuity 1 1  Liquid Holdup SYMBOL 101 \f "Symbol"L 1 Liquid phase continuity 1 2  Velocity of Gas UG 1 Gas phase momentum 1 18  Velocity of Liquid UL 1 Liquid phase momentum117 Concentration of i in catalyst NiCP c Reaction diffusion equations c 16  Liquid-Solid transfer rate NiLS c Liquid solid mass transfer c-1 12  Liquid-Solid interface concentration xiLS c Equilibrium equations on catalyst surfacec     L-S interface mole fraction sum 1   Temperature of Catalyst TS 1 Solid phase energy balance 1 29  Gas Density SYMBOL 114 \f "Symbol"G 1 Gas EOS 1 3  Total Variables  9c+9 Total Equations 9c+9    DISCUSSION OF RESULTS A four component system is considered in the simulation of hydrogenation of Alpha-methylstyrene to Cumene in n-hexane solvent. The conditions of the bed, catalyst activity, rate form, flow rates etc. are as in section 4.3. The transient simulation of the apparent rate three pellet approach (approach ii) yields a set of equations ranging from 92 to 116 depending upon the wetting conditions which are solved using an implicit finite difference approach with simultaneous solution of the non-linear system of equations resulting from the Maxwell-Stefan equations using either the multivariable Newtons method with linear search as discussed earlier in this chapter. For the pseudo-transient simulation to steady state, the velocity and holdup of each phase is supplied to the code, and concentration, flux, and temperature profiles are simulated. Liquid inlet flow at the feed concentration and temperature is specified and the reaction transport equations are solved at each time at all the axial locations to follow the development of concentration profiles in the gas, bulk liquid, and catalyst phase until steady state is achieved. Figures 2 and 3 show the alpha-methylstyrene concentration profile development at different locations in the reactor with time. As can be seen in the Figures 2 and 3 the alpha-methylstyrene concentration reaches steady state relatively early compared to the cumene concentration profiles which are shown (Figure 4). Since this is the case of gas-limited conditions with zero order rate with respect to alpha-methylstyrene, linear concentration profiles for steady state profiles are observed as expected. Figures 4 and 5 show the buildup of liquid phase cumene and hydrogen concentration at different axial positions in time. The liquid phase hydrogen concentration is seen to build to saturation at the exit where low alpha-methylstyrene concentration is present, whereas at the inlet it is consumed faster than transferred from the gas phase and results in much lower concentrations as indicated by the lower curves in Figure 5.  EMBED Excel.Chart.5 \s  Figure 5.  SEQ Figure_5. \* ARABIC 8: Transient Alpha-methylstyrene ((-MS) Concentration Profiles at Different Axial Locations  EMBED Excel.Chart.5 \s  Figure 5.  SEQ Figure_5. \* ARABIC 9 Transient Alpha-methylstyrene Concentration Profile Development with Time (shown in seconds in the legend table).  EMBED Excel.Chart.5 \s  Figure 5.  SEQ Figure_5. \* ARABIC 10 Axial Profiles of Cumene Concentration at Different Simulation Times (shown in seconds in the legend table)  EMBED Excel.Chart.5 \s  Figure 5.  SEQ Figure_5. \* ARABIC 11 Transient Hydrogen Concentration Profiles at Different Axial Locations Flow Modulation : Velocity and Holdup Simulation A typical set of simulated velocity and holdup profiles for a total cycle time of 80 seconds and LIQUID ON time of 15 seconds are shown in Figure 5.12 and 5.13. The liquid flow rate used is 42 ml/min (1.4 kg/m2s) during LIQUID ON time and 2 ml/min (0.067 kg/m2s) during the rest of the cycle with gas flow of 400 cc/min (0.0192 kg/m2s). The inlet pulse is seen to be sharp with significant spreading as the liquid pulse travels down the reactor. This is observed even with an order of magnitude reduction in spatial and temporal step size and corroborates experimental observations.  EMBED Excel.Chart.5 \s  Figure 5.  SEQ Figure_5. \* ARABIC 12 Transient Liquid Holdup Profiles at Different Axial Locations in Periodic Flow  EMBED Excel.Chart.5 \s  Figure 5.  SEQ Figure_5. \* ARABIC 13 Transient Liquid Velocity Profiles at Different Axial Locations in Periodic Flow Periodic Flow: Reaction-Transport Simulation This involves the solution of the transient reaction transport equations (described in section 5._) when coupled with the flow simulation (described in section 5._) for reaching a steady state and then with periodic flow modulation. The coupling between evaporative fluxes and their influence on holdup is solved in an explicit manner at each step for the momentum equations. The converged velocity and holdup profile is then used to do the rest of the reaction transport calculations implicitly. Reaction rates at steady state and time averaged rates under periodic flow conditions are then compared to evaluate performance under unsteady state conditions. For a test case with liquid flow cycling for 20 seconds each with liquid ON flow mass velocity of 1.01 kg/m2s and OFF (Base) flow mass velocity of 0.05 kg/m2s with a constant gas mass velocity of 0.017 kg/m2s, typical product (cumene) concentration in the bulk liquid phase is shown in Figure 8. It can be observed that the access of gas and liquid and hence the rate of product (cumene) formation is enhanced during the periodic flow modulation cycle. Figures 9 and 10 show intra-catalyst concentration of hydrogen (in a previously wetted pellet with very small reaction rate) and concentration of alpha-methylstyrene in a previously dry pellet during the entire 40 second cycle. This demonstrates the access (to the catalyst) of gaseous reactant during the OFF cycle and that of the liquid reactant during the ON cycle which can cause enhancement in overall reactor performance.  EMBED Excel.Chart.5 \s  Figure 5.  SEQ Figure_5. \* ARABIC 14 Cumene Concentration Profiles during Periodic Flow Modulation  EMBED Excel.Chart.5 \s  Figure 5.  SEQ Figure_5. \* ARABIC 15 Intra-catalyst Hydrogen Concentration Profiles during Flow Modulation for a Previously Externally Wetted Catalyst Pellet at Different Axial Locations.  EMBED Excel.Chart.5 \s  Figure 5.  SEQ Figure_5. \* ARABIC 16 Intra-Catalyst Alpha-methylstyrene Concentration Profiles during Flow Modulation for a Previously Externally Dry Catalyst Pellet at Different Axial Locations. CHAPTER 6. CONCLUSIONS Performance (conversion) comparison of downflow and upflow modes of operation of laboratory packed bed reactors over a range of identical feed and pressure conditions yields two conclusions. Gas limited conditions imply better downflow performance (due to easier access of the gas to the partially externally wetted catalyst areas in downflow), while liquid limited conditions result in better upflow performance over downflow. Conversion is seen to improve with pressure in the gas limited domain and remains fairly constant once liquid limitation sets in. Zero order behavior with respect to liquid reactant feed concentration is exhibited at the gas limited end, and first order behavior with respect to the liquid reactant (either transport or kinetic) at the liquid limited end. Gas velocity has no significant effect on the performance of either mode of operation in the range of velocities examined. Studies of the downflow and upflow mode of operation in bed diluted with fines under identical conditions show that fines do indeed neutralize the differences between the two modes of operation for both gas and liquid limited conditions and successfully decouple hydrodynamics (flow pattern and wetting) and "apparent" kinetics. This implies that the performance of the diluted bed is not dependent on the reactant limitation and flow mode used. This conclusion is important in establishing the use of fines in laboratory scale reactors as an effective and viable scale-up tool possibly to be preferred to upflow reactors. While one can argue that upflow reactors also provide rates on completely wetted particles and, hence, to some extent decouple the kinetics and hydrodynamics, they do so only at the expense of increased liquid holdup which can alter the relative amount of homogeneous and heterogeneous reactions occurring in more complex systems and should be avoided. In addition, upflow can lead to more dispersion and flow non-idealities than encountered in beds with fines. Finally, holdup and flow regimes in diluted beds are closer to those in trickle beds than in upflow. Further understanding and quantitative comparison of the performance with and without fines, under upflow and downflow conditions, requires detailed modeling which is currently under investigation. The performance of upflow and downflow is reaction system dependent i. e., whether the reaction is gas or liquid limited under the conditions of investigation. The laboratory reactors are operated in the range of partially to fully wetted catalyst to demonstrate that the influence of wetting can be detrimental or beneficial, depending upon the reactant limitation. Models that account for these two effects can predict the performance over the entire range of operating conditions. The intrinsic kinetics of the reaction studied is different at different pressures and hence it is recommended to study slurry kinetics at the operating pressure before any scale up or modeling is attempted. A rate expression with different rate constants at different pressures can be used to predict the trickle bed reactor data well. The predictions of the reactor scale and pellet scale models are satisfactory for current conditions although there is a need for high pressure correlation for mass transfer coefficient and interfacial area in order to predict performance with greater certainty, especially in cases where the rate is affected significantly by external mass transfer. Unsteady state liquid flow modulation (periodic operation) has been investigated in this study using hydrogenation of alpha-methylstyrene to cumene in a hexane solvent over 0.5% Pd on alumina spheres as a test reaction for gas and liquid limited conditions. Performance enhancement has been observed for a wide range of operating conditions with ON/OFF flow modulation under gas reactant limitation, but seen to be strongly dependent upon the extent of catalyst wetting under liquid limited conditions and observed with BASE/PEAK flow modulation. The enhancement feasibility has been shown to be dependent upon the extent of reactant limitation and mean liquid mass velocity. Performance enhancement has shown to be dependent upon the induced flow modulation (IFM) frequency and discussed in relation to the natural frequency of the performance governing process. Further experiments are necessary to investigate this effect to obtain insight into the cause of enhancement under a wide range of operating conditions, particularly when the rate controlling processes are different. Test simulation results show possibility of describing dynamic access of gaseous and liquid reactants and has to be tested for the entire set of experimental conditions presented here and in literature. 6.1 Recommendations for Future Work CHAPTER 7.NOMENCLATURE C(AMS) alpha-methylstyrene concentration C*(H2) saturation concentration of hydrogen De diffusivity (m2/s) G mass velocity of gas (kg/m2s) L mass velocity of liquid (kg/m2s) P operating pressure (psig) QL, QG volumetric flow rate of liquid, gas VL, VG superficial velocity of liquid, gas VR reactor volume, (m3) X(ss) steady state conversion X(us) unsteady state conversion ( reactant limitation criterion (=De (AMS) C(AMS) feed/ De (H2) C*(H2)) ( total cycle period, s ( cycle split flow modulation frequency, Hz aGS = Gas-solid interfacial area (m2/m3). aGL = Gas-solid interfacial area (m2/m3). a'GL = Gas-Liquid total interfacial area (m2/m3). aLS = Liquid-solid interfacial area (m2/m3). ct = Molar density (mol/m3) CA* = Concentration of gaseous reactant in liquid phase (mol/m3) CBi = Concentration of liquid reactant in liquid phase (mol/m3) CiL = Liquid phase concentration of species i (mol/m3). CiCP = Concentration of species i in the catalyst (mol/m3). CiG = Gas phase concentration of species i (mol/m3). CPmixG = Gas phase specific heat (J/kg K). CPmixL = Liquid phase specific heat (J/kg K). DeA = Effective diffusivity of gaseous reactant in the catalyst (m2/s) DeB = Effective diffusivity of liquid reactant in the catalyst (m2/s) Dei = Effective diffusivity of reactant i in the catalyst (m2/s) EV = Energy transfer flux from bulk gas to gas-liquid interface (J/ m2s). EL = Energy transfer flux from gas-liquid interface to liquid (J/ m2s). ES = Energy transfer flux from solid to liquid-solid interface (J/ m2s). ELS = Energy transfer flux from liquid-solid interface to liquid (J/ m2s). g = Gravitational acceleration (m/s2). GaL = Liquid phase Galileo number. GaG = Gas phase Galileo number. HiV = Partial molar enthalpy of component i in vapor phase (J/mol). HiL = Partial molar enthalpy of component i in liquid phase (J/mol). HiS = Partial molar enthalpy of component i in liquid phase (J/mol). hL = Gas-liquid interface to liquid heat transfer coefficient (J/ m2 s K). hLS = Solid-liquid interface to liquid heat transfer coefficient (J/ m2 s K). hS = Solid-liquid interface to solid heat transfer coefficient (J/ m2 s K). hV = Gas-liquid interface to gas heat transfer coefficient (J/ m2 s K). J = Molar diffusion flux relative to average velocity (mol/m2s). (ka)GL = Volumetric gas-liquid mass transfer coefficient ( 1/s) kLS.aLS = Volumetric liquid-solid mass transfer coefficient (1/s) [kLik] = Liquid side mass transfer coefficient (m/s). [kVik] = Liquid side mass transfer coefficient (m/s). ke = Effective thermal conductivity of catalyst (J/m s K). K = Interphase momentum transfer coefficient (kg/m3). L = Half the length of characteristic pellet (m). Mi = Molecular weight of species i (kg/kgmole). NiV = Flux of component i (vapor to gas-liquid interface) (mol/m2s). NiL = Flux of component i (gas-liquid interface to liquid) (mol/m2s). NiLS = Flux of component i (liquid to liquid-solid interface) (mol/m2s). NiS = Flux of component i (liquid-solid interface to solid) (mol/m2s). P = Pressure (N/m2). SYMBOL 68 \f "Symbol"P = Pressure drop (N/m2). q = Heat flux (J/m2s) RW = Reaction rate on fully wetted catalyst pellet (mol/m3s). RDW = Reaction rate on half wetted catalyst pellet (mol/m3s). RD = Reaction rate on fully externally dry catalyst pellet (mol/m3s). ROV = Overall reaction rate (mol/m3s). ReL = Liquid phase Reynolds number. ReG = Gas phase Reynolds number. (rA)obs = Observed rate of reaction (mol/m3s) Sx = Catalyst external surface area (m2) TCP = Catalyst temperature (K) TG = Gas temperature (K) TA = Ambient temperature (K) TL = Liquid temperature (K) TI = Gas-liquid interface temperature (K). TILS = Liquid-Solid interface temperature (K). UIL = Liquid interstitial (actual) velocity (m/s). UIG = Gas interstitial (actual) velocity (m/s). Ug = Gas velocity (m/s) Vp = Catalyst Pellet Volume ( m3) x = Mole fraction in the liquid phase. x' = Spatial coordinate in catalyst pellet (m). y = Mole fraction in the gas phase. z = Axial coordinate (m). Greek Symbols SYMBOL 98 \f "Symbol" = Bootstrap matrix in Maxwell-Stefan formulation SYMBOL 97 \f "Symbol"',SYMBOL 98 \f "Symbol"' = Order of reaction with respect to species i and j. SYMBOL 101 \f "Symbol"=SYMBOL 101 \f "Symbol"B = Bed voidage. SYMBOL 101 \f "Symbol"G = Gas holdup. SYMBOL 101 \f "Symbol"P = Porosity. SYMBOL 101 \f "Symbol"L = Liquid holdup. SYMBOL 101 \f "Symbol"sL = Static holdup of liquid. SYMBOL 110 \f "Symbol"i = Stoichiometric coefficient of component i SYMBOL 114 \f "Symbol"L = Liquid phase density (kg/m3). SYMBOL 114 \f "Symbol"G = Gas phase density (kg/m3). SYMBOL 104 \f "Symbol"CE = Contacting efficiency. SYMBOL 121 \f "Symbol"L = Dimensionless liquid phase drag. SYMBOL 121 \f "Symbol"G = Dimensionless gas phase drag. 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C., "A Modeling and Simulation Language for Combined Lumped and Distributed Parameter Systems", Submitted to Computers and Chemical Engineering, (1995). Ramachandran, P.A., and Chaudhari, R. V., Three Phase Catalytic Reactors, Gordon Breach Publishers, NY, (1983). Sei, S. T., "Scale Effects in Laboratory and Pilot plant reactors For Trickle-Flow Processes", Revue De L'Institut Francais Du Petrole, 46, 4, (1991). Shah, Y. T., Mhaskar, R. D., and Paraskos, J.A., Ind. Eng. Chem. (Proc. Des. Dev), 15, 400, (1976). Stegasov, A.N., Kirillov, V.A., and Silveston, P. L., "Modeling of Catalytic SO2 Oxidation for Continuous and Periodic Liquid Flow Through a Trickle Bed", Chem. Eng. Sci., 49, 22, p3699-3710, (1994). Tan, C. S., and Smith, J. M., "Catalyst particle Effectiveness with Unsymmetrical Boundary Conditions", Chem. Eng. Sci., 35, 1601, (1980). Taylor, R., and Krishna, R., Multicomponent Mass Transfer, John Wiley and Sons Inc., New York, (1993). Tremblay, M. D., and Luss, R., "Optimization of Non-Steady-State Operation of Reactors", Can. J. Chem. Eng., 67, 494, (1989). Tukac, V., Mazzarino, I., Baldi, G., Gianetto, A., Sicardi, S., and Specchia, V. "Conversion rates in a Laboratory Trickle-bed Reactor during the Oxidation of Ethanol" Chem. Eng. Sci., 41, 17-23, (1986). Van Klinken, J. and Van Dongen, R.H., "Catalyst Dilution for Improved performance of Laboratory Trickle-flow Reactors", Chem. Eng. Sci., 35, 59, (1980). Villadsen, J.V., and Michelsen, M. L., Solution of Differential Equation Models by Polynomial Approximation, Prentice Hall, Englewood Cliffs, New Jersey, (1978). Weekman, V.W., and Myers, J. E., AIChE J., 10, 951, (1964). Wu, Y., Khadilkar, M. R., Al-Dahhan, M. H., and Dudukovic, M. P., "Comparison of Up-flow and Down-flow Two Phase Flow Packed Bed Reactors with and without Fines : Experimental Observations", Accepted for publication to I &E. C. (Res), (1996). Yan, T. Y. Can. J. Chem. Eng, 58, 259, (1980). Al-Dahhan, M. H., and Dudukovic, M. P., 1995, Chem. Engng. Sci, 50, 2377. Al-Dahhan, M. H., Larachi, F., Dudukovic, M. P., and Laurent, A., 1997, Ind. Eng. Chem. Res., 36, 3292-3314. Blok, J., and Drinkenburg, A., 1982, Chem. Eng. J., 25, 89. Castellari, A. and Haure, P. M., 1996, AIChE J., 41, 1593-1597. Chanchlani K. G., Hudgins, R. R., and Silveston, P. L., 1994, Can. J. Chem. Eng., 72, 657-671. El-Hisnawi, A. A., Dudukovic, M. P., and Mills, P. L., 1982, Am. Chem. Soc. Symp. Ser., 196, 421. Gupta, R., 1985, U.S. Patent # 4526757. Gregory, A. R., and, Sweed, N. H., 1972, Chem. Eng. J., 4, 139-148. Haure, P.M., Bogdashev, S.M., Bunimovich, M., Stegasov, A.N., Hudgins, R. R., and Silveston P. L., 1990, Chem. Eng. Sci. 45, 2255. Khadilkar, M. R., Wu, Y., Al-Dahhan, M. H., Dudukovic, M. P. and Colakyan, M., 1996, Chem Engng. Sci. 51, 2139-2148. Khadilkar, M. R., Al-Dahhan, M. H., and Dudukovic, M. P., 1997, AIChE Annual Meeting, Los Angeles, CA. Lange,R., Hanika, J., Stradiotto, D., Hudgins, R. R., and Silveston P. L., 1996, Chem. Eng. Sci. 49, 5615-5622. Lee, J. K., Hudgins, R. R., and Silveston, P. L., 1995, Chem. Engng. Sci., 50, 2523-2530. Lee, C. K., and Bailey, J. E., 1974, Chem. Engng. Sci., 29, 1157-1163. Mills, P. L., and Dudukovic, M. P., 1980, Chem. Engng. Sci, 35, 2267. Park I. H., Petkovska, M., and Do, D. D., 1998, Chem. Engng. Sci., 53, 819-832. Ritter, A. B., and Douglas, J. M., 1970, Ind. Eng. Chem. Fundam., 9, 21-28. Saroha, A. K., and Nigam, K. D. P., Reviews in Chemical Engineering, 12, 207. Satterfield, C. N., and Roberts, G.W., 1968, AIChE J. 14, 159. Silveston, P., 1990, in Unsteady State Processes in Catalysis, 217-232. Stegasov, A.N., Kirillov, V.A., and, Silveston, P. L., 1994, Chem. Eng. Sci. 49, 22, 3699-3710. Sweed, N. H., and Wilhelm, R. H., 1969, Ind. Eng. Chem. Fundam., 8, 221-231 Wilhelm, R. H., Rice, A. W., and Bendelius, A. R., 1966, Ind. Eng. Chem. Fundam., 5, 141. Wu R., McReady, M. J., and Varma, A., 1995,Chem. Engng. Sci., 50, 3333-3344. Wu, X., and Wankat, P. C., 1983, Ind. Eng. Chem. Fundam., 22, 172-176. Yan, T. Y., 1980, Can. J. Chem. Eng., 58, 259-266. APPENDIX A. Slurry Experiments: Intrinsic Rate at High Pressure Generation of intrinsic kinetic data was necessary before any trickle bed model predictions could be attempted. Hence, the reaction rate in a slurry reactor at different pressures and over a range of initial liquid reactant concentrations was investigated. This was done in a high pressure slurry reactor using crushed catalyst (< 50 SYMBOL 109 \f "Symbol" size). Sufficient agitation was maintained to ensure no external transport limitations in the system, along with a supply of fresh hydrogen to replace whatever was consumed and, hence, maintain saturation concentration of hydrogen in the liquid phase. Slurry reactions at low pressure (30 psig) showed the expected dependence (El-Hisnawi, 1981) of zero order with respect to symbol 97 \f "Symbol" \s 12-methylstyrene concentration. A peculiar behavior, encountered sometimes only at very high pressure (>1000 psig), was observed in our high pressure experiments (max. pressure 300 psig) as shown in the Figure 15 below. The rate for a system with approximately the same liquid reactant concentration increased when pressure was raised from 30 to 100 psig but decreased when pressure was increased further from 100 to 200 and further to 300 psig. This was further augmented by the slight inhibiting effect observed at high pressure which could be attributed due to the presence of cumene, particularly at higher conversions when 8-10% (volume cumene/volume liquid solution) cumene is present in the system. This pressure dependence, and product inhibition, have been observed at pressures over 1000 psig in commercial hydrogenation and hydrodesulphurizations (Satterfield and Roberts, 1968). A mechanism based single rate form was first used to fit the data. Different single and dual site adsorption-reaction mechanisms were tried by considering a shift in the rate limiting step. No simple mechanism was capable of representing this reaction at all pressures, and since the primary objective was to study and predict trickle bed and upflow performance at the same discrete pressure values, separate fits were used at each pressure for the general Langmuir-Hinshelwood rate expression as shown below. The hydrogen concentration dependence was lumped in the numerator in the rate constant and the adsorption equilibrium constants were obtained by a constrained non-linear fit to at least 30 points of data at each pressure. The fitted parameters at different pressures are presented in Table 7 (Khadilkar et al., 1996). EMBED Equation  Table A.  SEQ Table_A. \* ARABIC 1 Rate Constants Obtained from Slurry Data at Different Pressures Pressure (psig)kvs (m3liq./m3cat./s) *(mol/m3 liq)m'-1K1K2symbol 98 \f "Symbol" \s 12300.08140001001.144.4111.4812000.0222.73x10-22.1x10-22  EMBED Excel.Sheet.8  Figure A.  SEQ Figure_A. \* ARABIC 1 Slurry conversion versus time at different pressures  EMBED Excel.Sheet.8  Figure A.  SEQ Figure_A. \* ARABIC 2 Comparison of the Model Fitted Alpha-methylstyrene concentrations to experimental values APPENDIX B. Correlations used in Model Evaluation Downflow correlations EMBED Equation  El-Hisnawi (1982) EMBED Equation  Al-Dahhan and Dudukovic (1995) EMBED Equation  Tan and Smith (1980) EMBED Equation  Fukushima and Kusaka (1977) EMBED Equation  Fukushima and Kusaka (1977) EMBED Equation  Dwivedi and Upadhyah (1977) EMBED Equation  Lakota and Levec (1989) Upflow Correlations EMBED Equation  EMBED Equation  EMBED Equation  Specchia (1978) EMBED Equation  Reiss (1967) EMBED Equation  APPENDIX C. Flow Charts for the Unsteady State Simulation Algorithm  EMBED Word.Picture.6  APPENDIX D Maxwell-Stefan Equations for Multicomponent Transport Solution of simultaneous mass and energy transfer equations: For a system with n components, the number of unknowns in the problem are 2n mole fractions at the interface (xiI and yiI), n molar fluxes (incorporating NiL=NiV=NiVL ) and an interface temperature (TI). Since energy equilibrium at the interface is also considered, it acts as a bootstrap to relate the total flux to the absolute quantities without needing any further assumptions (of inert flux being zero or stoichiometric fluxes etc.). Thus the total number of variables are 3n+1. The number of equations relating these variables are a) n-1 rate equations for the liquid phase given by EMBED Equation  (D.  SEQ D. \* ARABIC 1) where EMBED Equation  (in matrix form, (n-1)x(n-1)) (D.  SEQ D. \* ARABIC 2) and SYMBOL 76 \f "Symbol"k=(SYMBOL 108 \f "Symbol"k-SYMBOL 108 \f "Symbol"n)/SYMBOL 108 \f "Symbol"x , SYMBOL 68 \f "Symbol"q=h.L(TI-TL)-h.V(TG-TI) (D.  SEQ D. \* ARABIC 3) ( EMBED Equation  and EMBED Equation , EMBED Equation  ) (D.  SEQ D. \* ARABIC 4) b) n-1 rate equations for the vapor(gas) phase given by EMBED Equation  (D.  SEQ D. \* ARABIC 5) where EMBED Equation  (in matrix form, (n-1)x(n-1)) (D.  SEQ D. \* ARABIC 6) c) n equilibrium relations EMBED Equation  (D.  SEQ D. \* ARABIC 7) d) Interfacial energy balance EMBED Equation  (D.  SEQ D. \* ARABIC 8) e) Two more equations are necessary to complete the set and are given by mole fraction sum for both gas and liquid compositions at the interface. EMBED Equation  (D.  SEQ D. \* ARABIC 9) EMBED Equation  (D.  SEQ D. \* ARABIC 10) This completes the set of equations required to solve for all the fluxes, compositions and temperatures. The mass and heat transfer coefficients are the corrected values at the fluxes and can be revised based on this flux calculation until converged values are obtained. Details of iteration for finite flux transfer coefficents can be found in Taylor and Krishna (1993). The Bootstrap matrix for interphase transport Bootstrap matrix for gas transport is evaluated based on interphase energy flux for the gas -liquid interface  EMBED Equation.2  (D.  SEQ D. \* ARABIC 11) where  EMBED Equation.2  (D.  SEQ D. \* ARABIC 12) and  EMBED Equation.2  ,  EMBED Equation.2  (D.  SEQ D. \* ARABIC 13) For the gas-solid interface similar equations are used except the enthalpy is now of the liquid in the solid (liquid at solid temperature). Bootstrap matrix for the liquid phase is also based on the energy flux term and given by  EMBED Equation.2  (D.  SEQ D. \* ARABIC 14) where  EMBED Equation.2  (D.  SEQ D. \* ARABIC 15) and  EMBED Equation.2  ,  EMBED Equation.2  (D.  SEQ D. \* ARABIC 16) Catalyst level equations have unsymmetric boundary conditions due to gas and liquid bulk flow streams on either side of partially wetted pellets and suitable boundary conditions have to be incorporated into the bootstrap matrix to ensure that mass balance is satisfied in the pellets. Several bootstraps are possible in this case such as stoichiometric fluxes, zero flux for inert component, net zero flux for zero net mole production. But these are not deemed suitable for the present problem. Hence the most likely condition would be that net volumetric flux entering on the wetted side is equal to net mass flux leaving on the dry side. This can be incorporated directly in the fully implicit beta evaluation as  EMBED Equation.2  (D.  SEQ D. \* ARABIC 17) or in general since beta is not evaluated at each intra-catalyst location, it can be represented as  EMBED Equation.2  (D.  SEQ D. \* ARABIC 18) where  EMBED Equation.2  (D.  SEQ D. \* ARABIC 19) and  EMBED Equation.2  (D.  SEQ D. \* ARABIC 20) B) The transport matrices for interphase mass and energy flux 1. For the gas phase the low flux mass transfer coefficient matrix can be calculated as  EMBED Equation.2  (D.  SEQ D. \* ARABIC 21) where  EMBED Equation.2  (for i = j ) (D.  SEQ D. \* ARABIC 22)  EMBED Equation.2  (for i ( j) (D.  SEQ D. \* ARABIC 23) Here  EMBED Equation.2  where the diffusivity coefficients are calculated from infinite dilution diffusivities (from kinetic theory) using mole fraction weighting factors. The [(] matrix is the non-ideality coefficient matrix, which is considered to be identity for the gas side. The liquid phase mass transfer coefficient matrix are calculated similarly as  EMBED Equation.2  (D.  SEQ D. \* ARABIC 24) where  EMBED Equation.2  (for i = j ) (D.  SEQ D. \* ARABIC 25)  EMBED Equation.2  (for i ( j) (D.  SEQ D. \* ARABIC 26) Here  EMBED Equation.2  where the diffusivity coefficients are calculated from infinite dilution diffusivities (using the Wilke-Chang correlation as given in Appendix E) using mole fraction weighting factors. The [(] matrix is the non-ideality coefficient matrix, which is considered to be identity for the gas side. C. High flux correction High flux correction is important for the gas side mass and heat transfer coefficients and has been incorporated as below. Determine [(] matrix based on the calculated fluxes  EMBED Equation.2  (D.  SEQ D. \* ARABIC 27)  EMBED Equation.2  (D.  SEQ D. \* ARABIC 28) correct the low flux mass transfer coefficient matrix by the correction as  EMBED Equation.2  (D.  SEQ D. \* ARABIC 29) where exp[(] is evaluated by a Pade approximation as  EMBED Equation.2  (D.  SEQ D. \* ARABIC 30) Similarly, heat transfer coefficient is also corrected by  EMBED Equation.2  (D.  SEQ D. \* ARABIC 31)  EMBED Equation.2  (D.  SEQ D. \* ARABIC 32) APPENDIX E Evaluation of Parameters for Unsteady State Model Liquid phase infinite dilution diffusivities are determined using the Wilke and Chang (1955) correlation as  EMBED Equation.3  (E.  SEQ E. \* ARABIC 1) Gas phase infinite dilution diffusivities are determined as given by Reid et al. (1987)  EMBED Equation.3  (E.  SEQ E. \* ARABIC 2) These are corrected for concentration effect using the Vignes (1966) correlation as recommended by taylor and Krishna (1993)  EMBED Equation.3  (E.  SEQ E. \* ARABIC 3) The film thickness for gas and liquid phase is determined using the Sherwoods correlation by using the particle diameter as the charatecteristic dimension as  EMBED Equation.3  (E.  SEQ E. \* ARABIC 4) The Schmidt number used in the above correlation is different for each species and can give multiple values of film thickness according to the species used in the calculation. This dilemma is resolved using either an average value of diffusivity or the diffusivity of the gas phase key reactant. Heat transfer coefficients for gas and liquid phase are determined using either the Nusselt (E.5 ) number or the Chilton-Colburn analogy (E.6) as recommended by Taylor and Krishna (1993)  EMBED Equation.3  (E.  SEQ E. \* ARABIC 5)  EMBED Equation.3  (E.  SEQ E. \* ARABIC 6) The interfacial areas required for interphase transport (gas-liquid, liquid-solid, and gas-solid) are evaluated using the following correlations. ,  EMBED Equation.3  (E.  SEQ E. \* ARABIC 7)  EMBED Equation.3  (E.  SEQ E. \* ARABIC 8)  EMBED Equation.3  (E.  SEQ E. \* ARABIC 9) The modeling of liquid phase non-ideality effects can be done by using any of the several approaches available (such as simple 2 parameter models such as the Margules, Van-Laar, or the Wilson equation or complex rigorous methods such as the UNIQUAC or UNIFAC). In the present study, the Wilsons equation was used to obtain the equilibrium constants as well as the activity matrix ([(]) as given by the equations below. For the Wilson equation, the excess energy Q is given by  EMBED Equation.3  (E.  SEQ E. \* ARABIC 10) where  EMBED Equation.3  (E.  SEQ E. \* ARABIC 11)  EMBED Equation.3  (E.  SEQ E. \* ARABIC 12) First and second derivatives of Q w.r.t. compositions i and j are then given by  EMBED Equation.3  (E.  SEQ E. \* ARABIC 13)  EMBED Equation.3  (E.  SEQ E. \* ARABIC 14) The acivity coefficients needed for calculating the equilibrium constants are then given directly by  EMBED Equation.3  (E.  SEQ E. \* ARABIC 15) The thermodynamic factors ((ij) required for the nonideality matrix are given by  EMBED Equation.3  (E.  SEQ E. \* ARABIC 16) APPENDIX F. Experimental Data from Steady and Unsteady Experiments VITA Publications Investigation of a Complex Reaction Network for Production of Amino Alcohol: Part I. Experimental Observations in a Trickle Bed Reactor, Khadilkar, M. R., Jiang, Y., Al-Dahhan, M., Dudukovic, M. P., Chou, S. K., Ahmed, G., and Kahney, R., Accepted for Publication, AIChE Journal (1997). Investigation of a Complex Reaction Network for Production of Amino Alcohol: Part II. Kinetics, Mechanism and Model Based Parameter Estimation, Jiang, Y., Khadilkar, M. R., Al-Dahhan, M., Dudukovic, M. P., Chou, S. K., Ahmed, G., and Kahney, R., Accepted for Publication, AIChE Journal (1997). Prediction of Pressure Drop and Liquid Holdup in High Pressure Trickle Bed Reactors, Al-Dahhan, M., Khadilkar, M. R., Wu. Y., and Dudukovic, M. P., Accepted for Publication, I&EC Research (1997). Comparison of Trickle-Bed and Upflow Reactor Performance at High Pressure: Model Predictions and Experimental Observations, Khadilkar, M. R., Wu. Y., Al-Dahhan, M., Dudukovic, M. H. and Colakyan, M., Chem. Eng. Sci., 51, 10, 2139 (1996). Simulation of Flow Distribution in Trickle Bed Reactors Using CFDLIB, Khadilkar, M. R. and Dudukovic, M. P, CREL Annual Report, Washington University (1996-1997) Evaluation of Trickle-bed Reactor Models for a Liquid Limited Reaction Wu, Y., Khadilkar, M. R., Al-Dahhan, M., and Dudukovic, M. P., Chem. Eng. Sci., 51, 11, 2721 (1996). Comparison of Upflow and Downflow Two Phase Flow Reactors With and Without Fines, Wu. Y., Khadilkar, M. R., Al-Dahhan, M., and Dudukovic, M. P., I&EC Research, 35, 397 (1996). Presentations Simulation of Unsteady State Operation of Trickle Bed Reactors Khadilkar, M. R., Al-Dahhan, M., and Dudukovic, M. P., Presentation 254e, AIChE Annual Meeting, Los Angeles, CA (1997). Investigation of a Complex Reaction Network in a High Pressure Trickle Bed Reactor, Khadilkar, M. R., Jiang, Y., Al-Dahhan, M., Dudukovic, M. P., Chou, S. K., Ahmed, G., and Kahney, R., Presentation 252a, AIChE Annual Meeting, Los Angeles, CA (1997). Prediction of Two Phase Flow Distribution in Two Dimensional Trickle bed Reactors Jiang, Y., Khadilkar, M. R., Al-Dahhan, M., Dudukovic, M. P., Poster Presentation 276a, AIChE Annual Meeting, Los Angeles, CA (1997). Simulation of Unsteady State Operation of Trickle Bed Reactors Khadilkar, M. R., Al-Dahhan, M., and Dudukovic, M. P., Poster Presentation TRP19, Engineering Foundation Conference, Banff, Canada (1997). Effect of Catalyst Wetting on the Performance of Trickle Bed Reactors Wu. Y., Khadilkar, M. R., Al-Dahhan, M., and Dudukovic, M. P., Second Joint AIChE/CSCE Chemical Engineering Conference, Beijing China (1997). Comparison of Trickle-Bed and Upflow Reactor Performance at High Pressure: Model Predictions and Experimental Observations, Khadilkar, M. R., Wu. Y., Al-Dahhan, M., Dudukovic, M. H. and Colakyan, M., Presentation at ISCRE-14, Brugge, Belgium (1996). Evaluation of Trickle-bed Reactor Models for a Liquid Limited Reaction Wu. Y., Khadilkar, M. R., Al-Dahhan, M., and Dudukovic, M. P., Poster Presentation at ISCRE-14, Brugge, Belgium (1996). Effect of Fines on the Performance of Downflow (trickle-bed) and Upflow (packed bubble column) Reactors, Khadilkar, M. R., Wu, Y., Al-Dahhan, M., and Dudukovic, M. P., Presentation 66e, AIChE Annual Meeting, Miami Beach, Florida (1995). 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& MrEd%&+ & MrEd%04 & MrEd%9> & MrEd%CG & MrEd%MQ & MrEd%VZ & MrEd%`d & MrEd%in & MrEd%sw & MrEd%| & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd%  & MrEd% & MrEd%! & MrEd%66 & MrEd%XBA@?>=<;:98876554443333~3|3{3z4y4x4w5v5u6t7s8r8r9q:q;p<p=p>o?o@oAoBoCoDoEpFpGpHqIqJrKrLsMtNuNvOwPxPyQzQ{Q|Q~QQQQQQQPPONMLKJIHGFEDCB & MrEd %t<zI< & MrEd %I<? & MrEd%Trqpnmlkjihgffeddcccbbbbbcccddefghhijklmnpqrsuvwxyz{|}~~}||{zyxwvusr & MrEd%Y & MrEd %kyk & MrEd %pnmlklmnprsuyy & MrEd % & MrEd6% & MrEd%W                      & MrEd % & MrEd % & MrEd%q t & MrEd%u w & MrEd%px & MrEd%px & MrEd % & MrEd% & MrEd% & MrEd% & MrEd% & MrEd %{{|}~~|{   & MrEd%PB & MrEd%PB & MrEd%FF & MrEd%KK & MrEd %8M8B4D & MrEd%*O,A & MrEd%.O0A & MrEd%)E1E & MrEd%)J1J & MrEd %FDCBBBCEFHIKOO & MrEd%wszf & MrEd%{s}f & MrEd%vj~j & MrEd%vo~o & MrEd %sfh & MrEd%sf & MrEd%sf & MrEd%jj & MrEd%oo & MrEd %jhgfffghjlmoss & MrEd%kn & MrEd%oq & MrEd%jr & MrEd%jr & MrEd %yyv & MrEd% & MrEd% & MrEd% & MrEd% & MrEd % & MrEd% & MrEd%    & MrEd% & MrEd%yyy & MrEd%}}} & MrEd% & MrEd% & MrEd%== & MrEd%bb & MrEd%bb & MrEd%((( & MrEd%^^^ & MrEd%(^ & MrEd%^ & MrEd% & MrEd% & MrEd%PP & MrEd%rr & MrEd% & MrEd %g g & MrEd% & MrEd% & MrEd % & MrEd%   & MrEd% & MrEd% & MrEd%" & MrEd%&* & MrEd%.1 & MrEd%58 & MrEd%<? & MrEd%DG & MrEd%JN & MrEd%RU & MrEd%Y\ & MrEd%`c & MrEd%hk & MrEd%or & MrEd%vy & MrEd%} & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd%  & MrEd% & MrEd% & MrEd%  & MrEd%#& & MrEd%*. & MrEd%25 & MrEd%9< & MrEd%<< & MrEd%<< & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd%(( & MrEd%(( & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd%   & MrEd% & MrEd% & MrEd%" & MrEd%&* & MrEd%.1 & MrEd%58 & MrEd%<? & MrEd%DG & MrEd%JN & MrEd%RU & MrEd%Y\ & MrEd%`c & MrEd%hk & MrEd%or & MrEd%vy & MrEd%} & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd%  & MrEd% & MrEd% & MrEd%  & MrEd%#& & MrEd%*. & MrEd%25 & MrEd%9< & MrEd%<< & MrEd%<< & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd%HH & MrEd%HH & MrEd%HH & MrEd%HH & MrEd%WW & MrEd%WW & MrEd%WW & MrEd%WW & MrEd%HH & MrEd%HH & MrEd%HH & MrEd%HH & MrEd%WW & MrEd%WW & MrEd%WW & MrEd%WW & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd%BBB & MrEd%BJ & MrEd%RB & MrEd%EZ & MrEd%bM & MrEd%Uj & MrEd%r\ & MrEd%ez & MrEd%l & MrEd%u & MrEd%| & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd%GB & MrEd%BN & MrEd%VB & MrEd%I^ & MrEd%fP & MrEd%Yn & MrEd%va & MrEd%h~ & MrEd%q & MrEd%x & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd %HWH & MrEd%SSLLS & MrEd%LK & MrEd%KR & MrEd%RK & MrEd%KR & MrEd%RK & MrEd%KR & MrEd%RK & MrEd%KR & MrEd%RK & MrEd%KR & MrEd%RK & MrEd%KR & MrEd%RK & MrEd%KR & MrEd%RK & MrEd%KR & MrEd%RK  & MrEd%KR  & MrEd%RK & MrEd%K R & MrEd%R"K' & MrEd%K.R( & MrEd%R/K4 & MrEd%K;R5 & MrEd%R<KA & MrEd%KHRC & MrEd%RJKO & MrEd%KVRP & MrEd%RWK] & MrEd%KcR] & MrEd%RdKj & MrEd%KpRk & MrEd%RrKw & MrEd%K~Ry & MrEd%RK & MrEd%KR & MrEd%RK & MrEd%KR & MrEd%RK & MrEd%KR & MrEd%RK & MrEd%NR & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd%  & MrEd% & MrEd% & MrEd%! & MrEd%"( & MrEd%.) & MrEd%05 & MrEd%<7 & MrEd%=C & MrEd%ID & MrEd%JP & MrEd%VQ & MrEd%X] & MrEd%d_ & MrEd%ek & MrEd%rl & MrEd%sx & MrEd%y & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd%EE[[E & MrEd%EE[[E & MrEd% & MrEd% & MrEd%KK & MrEd%SS & MrEd% & MrEd% & MrEd% & MrEd% & MrEd%KK & MrEd%SS & MrEd% & MrEd% & MrEd $cYYc & MrEd %cYYc & MrEd%Y & MrEd $ & MrEd % & MrEd%} & MrEd$ & MrEd% & MrEd&$ & MrEd&% & MrEd$  & MrEd%  & MrEd$  & MrEd%  & MrEd.$ & MrEd.% & MrEd"$ & MrEd"% & MrEd $ & MrEd % & MrEd$  & MrEd%  & MrEd$ & MrEd% & MrEd$ & MrEd% & MrEd$ & MrEd% & MrEd$  & MrEd%  & MrEd$ & MrEd% & MrEd$ & MrEd% & MrEd$  & MrEd%  & MrEd$ & MrEd% & MrEd$ & MrEd% & MrEd$ & MrEd% & MrEd$  & MrEd%  & MrEd$ & MrEd% & MrEd$  & MrEd%  & MrEd $lllmmmmlponnnl & MrEd %lllmmmmlponnnl & MrEd$$wutrrrstuttstuvw & MrEd$%wutrrrstuttstuvw & MrEd($uvvwwvuttuvwxxxwvu & MrEd(%uvvwwvuttuvwxxxwvu & MrEd$ }~~~}} & MrEd% }~~~}} & MrEd$ & MrEd% & MrEd$ & MrEd% & MrEd$  & MrEd%  & MrEd$  & MrEd%  & MrEd$ & MrEd% & MrEd$ & MrEd% & MrEd$  & MrEd%  & MrEd$ & MrEd% & MrEd$ & MrEd% & MrEd$  & MrEd%  & MrEd$ & MrEd% & MrEd$ & MrEd% & MrEd$ & MrEd% & MrEd $qmuq & MrEd %qmuq & MrEd%pp & MrEd $qumq & MrEd %qumq & MrEd%pp & MrEd $vyrv & MrEd %vyrv & MrEd%evv & MrEd $uryu & MrEd %uryu & MrEd%ut & MrEd $opqrsstsrrqppo & MrEd %opqrsstsrrqppo & MrEd$$ttuvwwxxxwvutsst & MrEd$%ttuvwwxxxwvutsst & MrEd$oopqppo & MrEd%oopqppo & MrEd$xxwwx & MrEd%xxwwx & MrEd$ vutsrrstuv & MrEd% vutsrrstuv & MrEd$ vwxxwwxxxwvuv & MrEd% vwxxwwxxxwvuv & MrEd$oppooo & MrEd%oppooo & MrEd$oorpo & MrEd%oorpo & MrEd$ xxxwssrrwx & MrEd% xxxwssrrwx & MrEd$rrsssr & MrEd%rrsssr & MrEd$rrsttssr & MrEd%rrsttssr & MrEd$twxxxwtt & MrEd%twxxxwtt & MrEd$twxxxwtt & MrEd%twxxxwtt & MrEd$rrsttsr & MrEd%rrsttsr & MrEd$rrsssr & MrEd%rrsssr & MrEd$ xxxwssrrwx & MrEd% xxxwssrrwx & MrEd$rrssr & MrEd%rrssr & MrEd$rrstsr & MrEd%rrstsr & MrEd$twwxxwwt & MrEd%twwxxwwt & MrEd$twxxxwtt & MrEd%twxxxwtt & MrEd$rsttssr & MrEd%rsttssr & MrEd$rrssssr & MrEd%rrssssr & MrEd $ & MrEd % & MrEd% & MrEd $ & MrEd % & MrEd%B & MrEd$22113322 & MrEd%22113322 & MrEd"$30/-,,+**+,./03 & MrEd"%30/-,,+**+,./03 & MrEd$ **+,-,,++* & MrEd% **+,-,,++* & MrEd $+,++,2333332*+ & MrEd %+,++,2333332*+ & MrEd$$*+,-/023320/-++* & MrEd$%*+,-/023320/-++* & MrEd&$*+,-/0233320/-++* & MrEd&%*+,-/0233320/-++* & MrEd$ 332..--233 & MrEd% 332..--233 & MrEd$--..- & MrEd%--..- & MrEd$--.//..- & MrEd%--.//..- & MrEd$/2332// & MrEd%/2332// & MrEd$/23332// & MrEd%/23332// & MrEd$--.//..- & MrEd%--.//..- & MrEd$--....- & MrEd%--....- & MrEd$ 332..--233 & MrEd% 332..--233 & MrEd$--..- & MrEd%--..- & MrEd$--.//..- & MrEd%--.//..- & MrEd$/23332// & MrEd%/23332// & MrEd$/23332// & MrEd%/23332// & MrEd$--.//..- & MrEd%--.//..- & MrEd$--....- & MrEd%--....- & MrEd $''1$1+'' & MrEd %''1$1+'' & MrEd%'0' & MrEd $(,%( & MrEd %(,%( & MrEd%(( & MrEd$ $$$++,,++#$ & MrEd% $$$++,,++#$ & MrEd$$$+,,+#$ & MrEd%$$+,,+#$ & MrEd&$##$%')+,,++)'%$## & MrEd&%##$%')+,,++)'%$## & MrEd$$##$%')+,,++)'%$# & MrEd$%##$%')+,,++)'%$# & MrEd$ ,,+'&&&++, & MrEd% ,,+'&&&++, & MrEd$&&&''&&& & MrEd%&&&''&&& & MrEd$ &&'(('&&& & MrEd% &&'(('&&& & MrEd$(++,,+(( & MrEd%(++,,+(( & MrEd$(++,,+(( & MrEd%(++,,+(( & MrEd$&&'(('&& & MrEd%&&'(('&& & MrEd$&&&''&&& & MrEd%&&&''&&& & MrEd$ ,,++'&&&++, & MrEd% ,,++'&&&++, & MrEd$&&&''&&& & MrEd%&&&''&&& & MrEd$&&'(('&& & MrEd%&&'(('&& & MrEd$ (++,,++(( & MrEd% (++,,++(( & MrEd$ (++,,++(( & MrEd% (++,,++(( & MrEd$ &&&'((''& & MrEd% &&&'((''& & MrEd$&&''&& & MrEd%&&''&& & MrEd $! & MrEd %! & MrEd% & MrEd $! & MrEd %! & MrEd% & MrEd$$ & MrEd$% & MrEd*$ & MrEd*% & MrEd$$ & MrEd$% & MrEd&$ & MrEd&% & MrEd$ %"##""$$$% & MrEd% %"##""$$$% & MrEd$&%$%& & MrEd%&%$%& & MrEd$&'((&&%& & MrEd%&'((&&%& & MrEd$((()&&&( & MrEd%((()&&&( & MrEd$ ,,,-)***, & MrEd% ,,,-)***, & MrEd$*+,,**)* & MrEd%*+,,**)* & MrEd$*)()* & MrEd%*)()* & MrEd$ 1.//..0001 & MrEd% 1.//..0001 & MrEd$21012 & MrEd%21012 & MrEd$23342212 & MrEd%23342212 & MrEd$44452224 & MrEd%44452224 & MrEd$ 778956667 & MrEd% 778956667 & MrEd$67776656 & MrEd%67776656 & MrEd$6543456 & MrEd%6543456 & MrEd $=nGkFr=n & MrEd %=nGkFr=n & MrEd%oEn & MrEd$kkllmmlk & MrEd%kkllmmlk & MrEd($kllmnpqqqqpponmllk & MrEd(%kllmnpqqqqpponmllk & MrEd$ oppqqqqpo & MrEd% oppqqqqpo & MrEd$ qqqqjiiiihq & MrEd% qqqqjiiiihq & MrEd&$kllmnpqqqpponmllk & MrEd&%kllmnpqqqpponmllk & MrEd$oppqqpo & MrEd%oppqqpo & MrEd$nnmmn & MrEd%nnmmn & MrEd$ mllkklmnm & MrEd% mllkklmnm & MrEd$klmmlk & MrEd%klmmlk & MrEd&$kkllmpqpqqqpnmllk & MrEd&%kkllmpqpqqqpnmllk & MrEd$ ponnnnnopp & MrEd% ponnnnnopp & MrEd$ ppqqpqqqqqpp & MrEd% ppqqpqqqqqpp & MrEd$ qqqqpllkpq & MrEd% qqqqpllkpq & MrEd$ mllmmlkkllm & MrEd% mllmmlkkllm & MrEd$ qqpijpqqq & MrEd% qqpijpqqq & MrEd$iqqqqji & MrEd%iqqqqji & MrEd $nnn & MrEd %nnn & MrEd$kklmmllk & MrEd%kklmmllk & MrEd&$kllmnpqqqpponmllk & MrEd&%kllmnpqqqpponmllk & MrEd$oppqqpo & MrEd%oppqqpo & MrEd$qqpllkpq & MrEd%qqpllkpq & MrEd$ mllmmlkklm & MrEd% mllmmlkklm & MrEd$ lkkllpqll & MrEd% lkkllpqll & MrEd$ lkklqqpll & MrEd% lkklqqpll & MrEd$ qrsttsssrqq & MrEd% qrsttsssrqq & MrEd$ qqqqjiiihq & MrEd% qqqqjiiihq & MrEd$ qqmllkqqq & MrEd% qqmllkqqq & MrEd$hijjih & MrEd%hijjih & MrEd$kklmmllk & MrEd%kklmmllk & MrEd$$kllmnpqqqponmllk & MrEd$%kllmnpqqqponmllk & MrEd$oppqqpo & MrEd%oppqqpo & MrEd$ qqqpllkpq & MrEd% qqqpllkpq & MrEd$mllmlklm & MrEd%mllmlklm & MrEd&$kllmnopqqqponmllk & MrEd&%kllmnopqqqponmllk & MrEd$$kllmnopqqponmllk & MrEd$%kllmnopqqponmllk & MrEd$kkllmllk & MrEd%kkllmllk & MrEd$$kllmnpqqqponmllk & MrEd$%kllmnpqqqponmllk & MrEd$ oppqqqqpo & MrEd% oppqqqqpo & MrEd$ hiiijqqqqph & MrEd% hiiijqqqqph & MrEd $   & MrEd %   & MrEd%v  & MrEd$ & MrEd% & MrEd$$ & MrEd$% & MrEd$ & MrEd% & MrEd$  & MrEd%  & MrEd&$ & MrEd&% & MrEd$  & MrEd%  & MrEd $ & MrEd % & MrEd$  & MrEd%  & MrEd$  & MrEd%  & MrEd$ & MrEd% & MrEd$ & MrEd% & MrEd$ & MrEd% & MrEd$ & MrEd% & MrEd$ & MrEd% & MrEd$ & MrEd% & MrEd$  & MrEd%  & MrEd$  & MrEd%  & MrEd($ & MrEd(% & MrEd$ & MrEd% & MrEd$ & MrEd% & MrEd$  & MrEd%  & MrEd$ & MrEd% & MrEd$ & MrEd% & MrEd&$ & MrEd&% & MrEd$  & MrEd%  & MrEd $ & MrEd % & MrEd$  & MrEd%  & MrEd$  & MrEd%  & MrEd$  & MrEd%  & MrEd$$ & MrEd$% & MrEd($ & MrEd(% & MrEd$  & MrEd%  & MrEd $ & MrEd % & MrEd$$ & MrEd$% & MrEd$ & MrEd% & MrEd$  & MrEd%  & MrEd$  & MrEd%  & MrEd$ & MrEd% & MrEd$ & MrEd% & MrEd$  & MrEd%  & MrEd"$ & MrEd"% & MrEd$ & MrEd% & MrEd$ & MrEd% & MrEd$  & MrEd%  & MrEd$     & MrEd%     & MrEd$        & MrEd%        & MrEd$          & MrEd%          & MrEd$   & MrEd%   & MrEd$   & MrEd%   & MrEd$       & MrEd%       & MrEd$  & MrEd%  & MrEd$  & MrEd%  & MrEd$ & MrEd% & MrEd$  & MrEd%  & MrEd$  & MrEd%  & MrEd$ & MrEd% & MrEd$ & MrEd%.Times New Roman+-2 <iGas liquid Separator 2 2WReactor- & MrEd% & MrEd% & MrEd%&MrEd ?--? Symbol5-2 .DTimes New Roman+-2 ,PTimes New Roman+-2 ' Differential 2 (Pressure 2 ' Transducer&MrEd&:-:&Times New Roman+- 2 5210 mm&MrEd 2 218 mm&MrEd  &MrEdhh&MrEdcyyc2 sClear Acrylic Rod&MrEdvv 2 Demister&MrEd 22  2 -110 mm&MrEdm~~m 2 {35 mm&MrEd <<  2 90 mm"System---''4\ g\g\$g\& MrEdCompObjXOle10Native}p_952385373\ F^g^gOle       !"#$%&'()*+,-./012345789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuVz{|}~p*  ߷Z&MrEdMicrosoft DrawZ&MrEdTimes New Roman_  &MrEd6 6 ~  &MrEd>Cm-- mC>&MrEdRx xR- & MrEd%#L# & MrEd%*- & MrEd%@#@' & MrEd%@,@1 & MrEd%@5@9 & MrEd%@>@B & MrEd%@GBK & MrEd%BPBT & MrEd%BYB] & MrEd%BbBf & MrEd%BkBo & MrEd%BtBx & MrEd%B}B & MrEd%BB & MrEd%BB & MrEd%BB & MrEd%BB & MrEd%BB & MrEd%@E & MrEd%IN & MrEd%RU & MrEd%Y^ & MrEd%be & MrEd%in & MrEd%ru & MrEd%{~ & MrEd% & MrEd% & MrEd% & MrEd% & MrEd%   & MrEd%   & MrEd%   & MrEd%   & MrEd%   & MrEd%{ v  & MrEd%r n  & MrEd%k f  & MrEd%b ^  & MrEd%Y V  & MrEd%R N  & MrEd%I E  & MrEd%B @  & MrEd%@ @ & MrEd%@@ & MrEd%@@# & MrEd%B'B, & MrEd%B0B5 & MrEd%B9B= & MrEd%6y6z & MrEd%6}6 & MrEd%66 & MrEd%66 & MrEd%66 & MrEd%66 & MrEd%66 & MrEd%66 & MrEd%66 & MrEd%66 & MrEd%66 & MrEd%66 & MrEd%66 & MrEd%66 & MrEd%66 & MrEd%66 & MrEd%66 & MrEd%66 & MrEd%66 & MrEd%66 & MrEd%66 & MrEd%66 & MrEd%66 & MrEd%66 & MrEd%66 & MrEd%66 & MrEd%66 & MrEd%66 & MrEd%66 & MrEd%66 & MrEd%66 & MrEd%66 & MrEd%66  & MrEd%6 6  & MrEd%66 & MrEd%66 & MrEd%66 & MrEd%66! & MrEd%6"6$ & MrEd%6'6( & MrEd%6+6- & MrEd%6061 & MrEd%6466 & MrEd%696: & MrEd%6=6? & MrEd%6B6C & MrEd%6F6H & MrEd%6K6L & MrEd%6O6Q & MrEd%6T6U & MrEd%yz & MrEd%} & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd%  & MrEd%   & MrEd% & MrEd% & MrEd% & MrEd%! & MrEd%"$ & MrEd%'( & MrEd%+- & MrEd%01 & MrEd%46 & MrEd%9: & MrEd%=? & MrEd%BC & MrEd%FH & MrEd%KL & MrEd%OQ & MrEd%TU & MrEd%#' & MrEd%,0 & MrEd%59 & MrEd%>B & MrEd%GK & MrEd%PT & MrEd%Y] & MrEd%bf & MrEd%ko & MrEd%tx & MrEd%| & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd%  & MrEd% & MrEd%" & MrEd%&+ & MrEd%/3 & MrEd%8<- & MrEd%L<B< & MrEd%<< & MrEd%B<<< & MrEd%H- & MrEd%GL & MrEd%PU & MrEd%Y^ & MrEd%bf & MrEd%ko & MrEd%tx & MrEd%} & MrEd% & MrEd%<D<J & MrEd%<M<Q & MrEd%<V<[ & MrEd%<_<d & MrEd%<h<m & MrEd%<q<u & MrEd%<z<~ & MrEd%<<- & MrEd%<< & MrEd%< & MrEd%## & MrEd%Y#N# & MrEd%XX & MrEd%- & MrEd%-X- & MrEd%X-X< & MrEd%X<L<- & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd%BB & MrEd%BB & MrEd%BB & MrEd%BB & MrEd%BB & MrEd%BB & MrEd%BB & MrEd%BB- & MrEd%< & MrEd% & MrEd%S<S & MrEd%SH & MrEd% & MrEd%H & MrEd%HH- & MrEd%H & MrEd%H & MrEd%H & MrEd%H & MrEd%H & MrEd%H & MrEd%H & MrEd%H & MrEd%H & MrEd%[[ & MrEd%[[ & MrEd%[[ & MrEd%kk & MrEd%kk & MrEd%kk & MrEd%X- & MrEd%X & MrEd%-- & MrEd $&+,& & MrEd %&+,& & MrEd%{l+ & MrEd%6y8y & MrEd%;y<y & MrEd%?y@y & MrEd%yy & MrEd%yy & MrEd%yy & MrEd%yz & MrEd%} & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd%  & MrEd%   & MrEd% & MrEd% & MrEd% & MrEd%! & MrEd%"$ & MrEd%'( & MrEd%+- & MrEd%01 & MrEd%46 & MrEd%9: & MrEd%=? & MrEd%BC & MrEd%FH & MrEd%KL & MrEd%OQ & MrEd%TU & MrEd%yy & MrEd%yy & MrEd%yy & MrEd%z} & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd%   & MrEd%  & MrEd% & MrEd% & MrEd% & MrEd%" & MrEd%$& & MrEd%(+ & MrEd%-/ & MrEd%14 & MrEd%68 & MrEd%:= & MrEd%?A & MrEd%CF & MrEd%GJ & MrEd%LO & MrEd%QS & MrEd%yz & MrEd%} & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd%  & MrEd%   & MrEd% & MrEd% & MrEd% & MrEd%! & MrEd%"$ & MrEd%'( & MrEd%+- & MrEd%01 & MrEd%46 & MrEd%9: & MrEd%=? & MrEd%BC & MrEd%FH & MrEd%KL & MrEd%OQ & MrEd%TU & MrEd%yy & MrEd%yy & MrEd% y y & MrEd%y| & MrEd%~ & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd%   & MrEd%   & MrEd%   & MrEd%   & MrEd%   & MrEd%   & MrEd%   & MrEd%    & MrEd%    & MrEd%   & MrEd%   & MrEd%   & MrEd%  ! & MrEd% " $ & MrEd% ' ( & MrEd% , - & MrEd% 0 1 & MrEd% 5 6 & MrEd% 9 < & MrEd% = ? & MrEd% B D & MrEd% F H & MrEd% K L & MrEd% O Q & MrEd% T U & MrEd%&y(y & MrEd%)y,y & MrEd%-y/y & MrEd%0z0} & MrEd%00 & MrEd%00 & MrEd%00 & MrEd%00 & MrEd%00 & MrEd%00 & MrEd%00 & MrEd%00 & MrEd%00 & MrEd%00 & MrEd%00 & MrEd%00 & MrEd%00 & MrEd%00 & MrEd%00 & MrEd%00 & MrEd%00 & MrEd%00 & MrEd%00 & MrEd%00 & MrEd%00 & MrEd%00 & MrEd%00 & MrEd%00 & MrEd%00 & MrEd%00 & MrEd%00 & MrEd%00 & MrEd%00 & MrEd%00 & MrEd%00 & MrEd%0 0  & MrEd%0 0 & MrEd%00 & MrEd%00 & MrEd%00 & MrEd%00" & MrEd%0$0& & MrEd%0(0+ & MrEd%0-0/ & MrEd%0104 & MrEd%0608 & MrEd%0:0= & MrEd%0?0A & MrEd%0C0F & MrEd%0G0J & MrEd%0L0O & MrEd%0Q0S & MrEd%%y%z & MrEd%%}% & MrEd%%% & MrEd%%% & MrEd%%% & MrEd%%% & MrEd%%% & MrEd%%% & MrEd%%% & MrEd%%% & MrEd%%% & MrEd%%% & MrEd%%% & MrEd%%% & MrEd%%% & MrEd%%% & MrEd%%% & MrEd%%% & MrEd%%% & MrEd%%% & MrEd%%% & MrEd%%% & MrEd%%% & MrEd%%% & MrEd%%% & MrEd%%% & MrEd%%% & MrEd%%% & MrEd%%% & MrEd%%% & MrEd%%% & MrEd%%% & MrEd%%%  & MrEd%% %  & MrEd%%% & MrEd%%% & MrEd%%% & MrEd%%%! & MrEd%%"%$ & MrEd%%'%( & MrEd%%+%- & MrEd%%0%1 & MrEd%%4%6 & MrEd%%9%: & MrEd%%=%? & MrEd%%B%C & MrEd%%F%H & MrEd%%K%L & MrEd%%O%Q & 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& MrEd%0U2U & MrEd%5U6U & MrEd%GG & MrEd%GG & MrEd%GG & MrEd%GG & MrEd%GG & MrEd%GG & MrEd%GG & MrEd% G G & MrEd%GG & MrEd%GG & MrEd%HH & MrEd%%H(H & MrEd%)H+H & MrEd%-H/H & MrEd%6H8H & MrEd%DD & MrEd%DD & MrEd%DD & MrEd%DD & MrEd%DD & MrEd%DD & MrEd%DD & MrEd% D D & MrEd%DD & MrEd%DD & MrEd%DD & MrEd%%D&D & MrEd%)D+D & MrEd%-D/D & MrEd%6D8D & MrEd%GD & MrEd%DG & MrEd%DG & MrEd%DG & MrEd%DG & MrEd% D G & MrEd%DG & MrEd%DG & MrEd%(D(F & MrEd%-D-G & MrEd%8D8G & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd%  & MrEd%   & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd%  & MrEd%   & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd%  & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd%  & MrEd%   & MrEd% & MrEd% & MrEd%~~ & MrEd%~~ & MrEd%~~ & MrEd%~~ & MrEd%|| & MrEd%|| & MrEd%|| & MrEd%|| & MrEd%|{ & MrEd%{{ & MrEd%{{ & MrEd%{{ & MrEd%{{ & MrEd%{{ & MrEd%{y & MrEd%yy & MrEd%yy & MrEd%yy  & MrEd%xv & MrEd%vv & MrEd%vv & MrEd%vv & MrEd%vv & MrEd%vu & MrEd%uu & MrEd%uu & MrEd%uu & MrEd%uu & MrEd%uu & MrEd%uu & MrEd%uu & MrEd%ss & MrEd%ss & MrEd%ss & MrEd%ss & MrEd%ss & MrEd%rr & MrEd%rr  & MrEd%r r  & MrEd%pp & MrEd%po & MrEd%oo & MrEd%oo & MrEd%oo & MrEd%oo & MrEd%nn & MrEd%nn & MrEd%nn & MrEd%nn & MrEd%nn & MrEd%nn & MrEd%ll & MrEd%ll & MrEd%ll & MrEd%ll & MrEd%ll & MrEd%kk & MrEd%kk & MrEd%kk  & MrEd%k k - & MrEd %l## & MrEd%X & MrEd%lbl & MrEd%Xcl & MrEd %clc#Y#- & MrEd%@%?% & MrEd%<%;% & MrEd%9%6% & MrEd%5%3% & MrEd%0%/% & MrEd%,%+% & MrEd%(%&% & MrEd%%%"% & MrEd% && & MrEd%&& & MrEd%&& & MrEd%&& & MrEd%&& & MrEd% & & & MrEd%&& & MrEd%'' & MrEd%'' & MrEd%'' & MrEd%'' & MrEd%'' & MrEd%'' & MrEd%'' & MrEd%'' & MrEd%'( & MrEd%@*?* & MrEd%<*;* & MrEd%9*6+ & MrEd%5+3+ & MrEd%0+/+ & MrEd%,+++ & MrEd%(+&+ & MrEd%%+"+ & MrEd% ++ & MrEd%+, & MrEd%,, & MrEd%,, & MrEd%,, & MrEd% , , & MrEd%,, & MrEd%,, & MrEd%,, & MrEd%,, & MrEd%-- & MrEd%-- & MrEd%-- & MrEd%-- & MrEd%-- & MrEd%-- & MrEd%@/?/ & MrEd%=0;0 & MrEd%9080 & MrEd%5030 & MrEd%00/0 & MrEd%,0+0 & MrEd%)0&0 & MrEd%%1#1 & MrEd% 11 & MrEd%11 & MrEd%11 & MrEd%11 & MrEd%11 & MrEd% 1 1 & MrEd%11 & MrEd%11 & MrEd%11 & MrEd%13 & MrEd%33 & MrEd%33 & MrEd%33 & MrEd%33 & MrEd%33 & MrEd%33 & MrEd%@5?5 & MrEd%<5;5 & MrEd%8565 & MrEd%5525 & MrEd%06-6 & MrEd%,6+6 & MrEd%(6&6 & MrEd%#6"6 & MrEd% 66 & MrEd%66 & MrEd%66 & MrEd%66 & MrEd%6 6 & MrEd% 6 8 & MrEd%88 & MrEd%88 & MrEd%88 & MrEd%88 & MrEd%88 & MrEd%88 & MrEd%88 & MrEd%88 & MrEd%99 & MrEd%99 & MrEd%@:?: & MrEd%<<;< & MrEd%8<6< & MrEd%5<2< & MrEd%0</< & MrEd%,<+< & MrEd%(<&< & MrEd%#<"< & MrEd% << & MrEd%<< & MrEd%<< & MrEd%== & MrEd%= = & MrEd% = = & MrEd%== & MrEd%== & MrEd%== & MrEd%== & MrEd%== & MrEd%>> & MrEd%>> & MrEd%>> & MrEd%>> & MrEd%>> & MrEd%@A?A & MrEd%<A;A & MrEd%8A6A & MrEd%5A2A & MrEd%0B/B & MrEd%,B+B & MrEd%(B&B & MrEd%#B"B & MrEd% BB & MrEd%BB & MrEd%BB & MrEd%BB & MrEd%C C & MrEd% C C & MrEd%CC & MrEd%CC & MrEd%CC & MrEd%CC & MrEd%CC & MrEd%CC & MrEd%CE & MrEd%EE & MrEd%EE & MrEd%EE & MrEd%@G?G & MrEd%<G;G & MrEd%8G6G & MrEd%5G2G & MrEd%0G-G & MrEd%,G+G & MrEd%(G&G & MrEd%#H"H & MrEd% HH & MrEd%HH & MrEd%HH & MrEd%HH & MrEd%H H & MrEd% H H & MrEd%HH & MrEd%HH & MrEd%JJ & MrEd%JJ & MrEd%JJ & MrEd%JJ & MrEd%JJ & MrEd%JJ & MrEd%KK & MrEd%KK & MrEd%@M?M & MrEd%=M;M & MrEd%9M8M & MrEd%5M3M & MrEd%0M/M & MrEd%,M+M & MrEd%)M&O & MrEd%%O#O & MrEd% OO & MrEd%OO & MrEd%OO & MrEd%OO & MrEd%PP & MrEd% P P & MrEd%PP & MrEd%PP & MrEd%PP & MrEd%PP & MrEd%PP & MrEd%PP & MrEd%PQ & MrEd%QQ & MrEd%QQ & MrEd%QQ & MrEd%@T?T & MrEd%=T;T & MrEd%9T8T & MrEd%5T3T & MrEd%0T/T & MrEd%,U+U & MrEd%)U&U & MrEd%%U#U & MrEd% UU & MrEd%UU & MrEd%UU & MrEd%UU & MrEd%UU & MrEd% U V & MrEd%VV & MrEd%VV & MrEd%VV & MrEd%VV & MrEd%VV & MrEd%VV & MrEd%VV & MrEd%VV & MrEd%VV & MrEd%VX- & MrEd%<- & MrEd%<? & MrEd%DD & MrEd %BBBF@F & MrEd%<D;D & MrEd%yy & MrEd%yy & MrEd%yy & MrEd%yy & MrEd%yy & MrEd%yy & MrEd%yy & MrEd% y y & MrEd% yy & MrEd%yy & MrEd%yy & MrEd%yy & MrEd%yy & MrEd% y"y & MrEd%%y&y & MrEd%)y+y & MrEd%-y/y & MrEd%0y3y & MrEd%5y6y & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd%   & MrEd% & MrEd% & MrEd%"% & MrEd%)- & MrEd%26 & MrEd%;= & MrEd% & MrEd% & MrEd% & MrEd%BB & MrEd%CE & MrEd%FF & MrEd%F & MrEd%jg & MrEd%dc & MrEd%a^ & MrEd%[Z & MrEd%jg & MrEd%dc & MrEd%a^ & MrEd%[Z & MrEd%jg & MrEd%dc & MrEd%a^ & MrEd%jg & MrEd%dc & MrEd%a^ & MrEd%[Z & MrEd%jg & MrEd%dc & MrEd%a^- & MrEd%<=<- & MrEd%H & MrEd%H & MrEd%H & MrEd%H & MrEd%H & MrEd%H & MrEd%H & MrEd%H & MrEd%H & MrEd%@&?& & MrEd%<';' & MrEd%9'6' & MrEd%5'3' & MrEd%0'/' & MrEd%,'+' & MrEd%('&' & MrEd%%'"' & MrEd% '' & MrEd%'( & MrEd%(( & MrEd%(( & MrEd%(( & MrEd% ( ( & MrEd%(( & MrEd%(( & MrEd%(( & MrEd%(( & MrEd%** & MrEd%** & MrEd%** & MrEd%** & MrEd%** & MrEd%** & MrEd%@,?, & MrEd%<,;, & MrEd%9,6, & MrEd%5,3, & MrEd%0-/- & MrEd%,-+- & MrEd%(-&- & MrEd%%-"- & MrEd% -- & MrEd%-- & MrEd%-- & MrEd%-- & MrEd%// & MrEd% / / & MrEd%// & MrEd%// & MrEd%// & MrEd%// & MrEd%// & MrEd%// & MrEd%/0 & MrEd%00 & MrEd%00 & MrEd%00 & MrEd%@3?3 & MrEd%=3;3 & MrEd%9383 & MrEd%5333 & MrEd%03/3 & MrEd%,3+3 & MrEd%)3&4 & MrEd%%4#4 & MrEd% 44 & MrEd%44 & MrEd%44 & MrEd%44 & MrEd%44 & MrEd% 4 4 & MrEd%44 & MrEd%55 & MrEd%55 & MrEd%55 & MrEd%55 & MrEd%55 & MrEd%56 & MrEd%66 & MrEd%66 & MrEd%66 & MrEd%@8?8 & MrEd%<8;8 & MrEd%8868 & MrEd%5828 & MrEd%09-9 & MrEd%,9+9 & MrEd%(9&9 & MrEd%#9"9 & MrEd% 99 & MrEd%99 & MrEd%99 & MrEd%99 & MrEd%: : & MrEd% : : & MrEd%:: & MrEd%:: & MrEd%:: & MrEd%:: & MrEd%<< & MrEd%<< & MrEd%<< & MrEd%<< & MrEd%<< & MrEd%<< & MrEd%@>?> & MrEd%<>;> & MrEd%8>6> & MrEd%5>2> & MrEd%0>-> & MrEd%,>+> & MrEd%(>&? & MrEd%#?"? & MrEd% ?? & MrEd%?? & MrEd%?? & MrEd%?? & MrEd%A A & MrEd% A A & MrEd%AA & MrEd%AA & MrEd%AA & MrEd%AA & MrEd%AA & MrEd%AA & MrEd%AA & MrEd%AA & MrEd%BB & MrEd%BB & MrEd%@C?C & MrEd%<E;E & MrEd%8E6E & MrEd%5E2E & MrEd%0E-E & MrEd%,E+E & MrEd%(E&F & MrEd%#F"F & MrEd% FF & MrEd%FF & MrEd%FF & MrEd%FF & MrEd%F F & MrEd% F F & MrEd%FF & MrEd%FF & MrEd%FG & MrEd%GG & MrEd%GG & MrEd%GG & MrEd%GG & MrEd%GG & MrEd%GG & MrEd%GG & MrEd%@K?K & MrEd%<K;K & MrEd%8K6K & MrEd%5K2K & MrEd%0K-K & MrEd%,K+K & MrEd%(K&K & MrEd%#K"K & MrEd% LL & MrEd%LL & MrEd%LL & MrEd%LL & MrEd%L L & MrEd% L L & MrEd%LL & MrEd%LL & MrEd%LM & MrEd%MM & MrEd%MM & MrEd%MM & MrEd%MM & MrEd%MM & MrEd%MM & MrEd%MM & MrEd%BP?P & MrEd%=Q;Q & MrEd%9Q8Q & MrEd%5Q3Q & MrEd%0Q/Q & MrEd%-Q+Q & MrEd%)Q(Q & MrEd%%Q#Q & MrEd% QQ & MrEd%QQ & MrEd%QQ & MrEd%SS & MrEd%SS & MrEd% S S & MrEd% SS & MrEd%SS & MrEd%SS & MrEd%SS & MrEd%SS & MrEd%TT & MrEd%TT & MrEd%TT & MrEd%TT & MrEd%TT & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd%  & MrEd%   & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd%  & MrEd%   & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd%  & MrEd%   & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd%  & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd%~ & MrEd%~~ & MrEd%~~ & MrEd%~~ & MrEd%~~  & MrEd%| |  & MrEd%{{ & MrEd%{{ & MrEd%{{ & MrEd%{{ & MrEd%yy & MrEd%yy & MrEd%yy & MrEd%yy & MrEd%yx & MrEd%xx & MrEd%xx & MrEd%xx & MrEd%xx & MrEd%xx & MrEd%xv & MrEd%vv & MrEd%vv & MrEd%vv & MrEd%vu & MrEd%uu  & MrEd%ss & MrEd%ss & MrEd%ss & MrEd%rr & MrEd%rr & MrEd%rr & MrEd%rr & MrEd%rp & MrEd%pp & MrEd%pp & MrEd%pp & MrEd%pp & MrEd%pp & MrEd%pp & MrEd%pp & MrEd%oo & MrEd%oo & MrEd%oo & MrEd%oo & MrEd%on  & MrEd%n n  & MrEd%jg & MrEd%dc & MrEd%a^ & MrEd%[[ & MrEd%jg & MrEd%dc & MrEd%a^ & MrEd%jh & MrEd%fc & MrEd%a_ & MrEd%][ & MrEd%jg & MrEd%dc & MrEd%a^ & MrEd%[Z & MrEd%jg & MrEd%dc & MrEd%a^ & MrEd%[Z & MrEd%jg & MrEd%dc & MrEd%a^ & MrEd%jg & MrEd%dc & MrEd%a^ & MrEd%nffnn & MrEd%3n3 & MrEd %TjjaabbT & MrEd%h & MrEd%b & MrEd% & MrEd%n;[;[nn & MrEd%AE & MrEd%IM & MrEd%OS & MrEd%W\ & MrEd%`b & MrEd%fj & MrEd%nq & MrEd%uy & MrEd%} & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd%  & MrEd% & MrEd% & MrEd%"% & MrEd%)- & MrEd%13 & MrEd%9; & MrEd%@B & MrEd%HJ & MrEd%NR & MrEd%VY & MrEd%]a & MrEd%eh & MrEd%lp & MrEd%tw & MrEd%{ & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd%TATE & MrEd%TITM & MrEd%TOTS & MrEd%TWT\ & MrEd%T`Tb & MrEd%TfTj & MrEd%TnTq & MrEd%TuTy & MrEd%T}T & MrEd%TT & MrEd%TT & MrEd%TT & MrEd%TT & MrEd%TT & MrEd%TT & MrEd%TT & MrEd%TT & MrEd%TT & MrEd%TT & MrEd%TT & MrEd%TT & MrEd%TT & MrEd%TT & MrEd%TT & MrEd%TT & MrEd%TT & MrEd%TT & MrEd%T T & MrEd%TT & MrEd%TT & MrEd%T"T% & MrEd%BB & MrEd%BB & MrEd%BB & MrEd%B!B & MrEd%%B'B & MrEd%*B,B & MrEd%0B1B & MrEd%5B8B & MrEd%:B>B & MrEd%@BCB & MrEd%EBIB & MrEd%LBNB & MrEd%RBSB & MrEd%VAX> & MrEd%A > & MrEd% l[i & MrEd% h[f & MrEd% d[b & MrEd% a[^ & MrEd% ^[\ & MrEd% Z[W & MrEd% W[S & MrEd% S[Q & MrEd% Q[N & MrEd% N[J & MrEd% J[G & MrEd% F[E & MrEd% E[A & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd%   & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd% & MrEd%   & MrEd%T%W% & MrEd%Y%]% & MrEd%`%b% & MrEd%f%h% & MrEd%k%m% & MrEd%q%r% & MrEd%v%y% & MrEd%{%~% & MrEd%%% & MrEd%%% & MrEd%%% & MrEd%%% & MrEd%%% & MrEd%%% & MrEd%%% & MrEd%%% & MrEd%%% & MrEd%%% & MrEd%%% & MrEd%%% & MrEd%%% & MrEd%T`W` & MrEd%Y`]` & MrEd%``b` & MrEd%f`h` & MrEd%k`m` & MrEd%q`r` & MrEd%v`y` & MrEd%{`~` & MrEd%`` & MrEd%`` & MrEd%`` & MrEd%`` & MrEd%`` & MrEd%`` & MrEd%`` & MrEd%`` & MrEd%`` & MrEd%`` & MrEd%`` & MrEd%`` & MrEd%`` & MrEd%T%R) & MrEd%Q-O/ & MrEd%N3N7 & MrEd%M;M> & MrEd%MBMF & MrEd%NJNM & MrEd%OROU & MrEd %QYQZR\ & MrEd%T`Td & MrEd%ThTk & MrEd%ToTs & MrEd%TwTz & MrEd%TT & MrEd%TT & MrEd%TT & MrEd%TT & MrEd%TT & MrEd%TT & MrEd%TT & MrEd%TT & MrEd%TT & MrEd%TT & MrEd%TT & MrEd%TT & MrEd%TT & MrEd%TT & MrEd%TT & MrEd%TT & MrEd%TT & MrEd%TT & MrEd%&__&& & MrEd%kk & MrEd%-ZZ-- & MrEd%BB & MrEd%&& & MrEd%&& & MrEd%&& & MrEd%&& & MrEd%&& & MrEd%&& & MrEd%'* & MrEd%/2 & MrEd%69 & MrEd%>A & MrEd%EI & MrEd%MP & MrEd%TX & MrEd%\_ & MrEd%__ & MrEd%__ & MrEd%__ & MrEd%__ & MrEd%__ & MrEd%__ & MrEd%+&&1U``Z & MrEd%&` & MrEd%BB & MrEd%1VV11 & MrEd%>> & MrEd%>> & MrEd%> > & MrEd% >> & MrEd%>> & MrEd%>> & MrEd%>> & MrEd%>> & MrEd%>> & MrEd%>> & MrEd%>> & MrEd%>> & MrEd%>> & MrEd%>> & MrEd%>> & MrEd%>> & MrEd%>> & MrEd%>> & MrEd%>> & MrEd%>> & MrEd%>> & MrEd%>> & MrEd%>> & MrEd%>> & MrEd%>> & MrEd%>> & MrEd%>> & MrEd%>> & MrEd%~>{> & MrEd%y>v> & MrEd%r>p> & MrEd%m>k> & MrEd%g>e> & MrEd%b>`> & MrEd%\>Y> & MrEd%W>T> & MrEd%R>O> & MrEd%L>I> & MrEd%G>D> & MrEd%@>>> & MrEd%;>9> & MrEd%5>3> & MrEd%0>.> & MrEd%*>'> & MrEd%%>"> & MrEd%>> & MrEd%>> & MrEd%>> & MrEd%FF & MrEd%FF & MrEd%F F & MrEd% FF & MrEd%FF & MrEd%FF & MrEd%FF & MrEd%FF & MrEd%FF & MrEd%FF & MrEd%FF & MrEd%FF & MrEd%FF & MrEd%FF & MrEd%FF & MrEd%FF & MrEd%FF & MrEd%FF & MrEd%FF & MrEd%FF & MrEd%FF & MrEd%FF & MrEd%FF & MrEd%FF & MrEd%FF & MrEd%FF & MrEd%FF & MrEd%FF & MrEd%~F{F & MrEd%yFvF & MrEd%rFpF & MrEd%mFkF & MrEd%gFeF & MrEd%bF`F & MrEd%]F]I & MrEd%[JXJ & MrEd%TJRJ & MrEd%OJMJ & MrEd%IJHJ & MrEd%DJBJ & MrEd%?J=J & MrEd%9J6J & MrEd%4J1J & MrEd%.J+J & MrEd%)J&J & MrEd%"J J & MrEd%JJ & MrEd%JJ & MrEd%JJ & MrEd%FB & MrEd%]F]B & MrEd%]>]= & MrEd $)--) & MrEd %)--) & MrEd%:- & MrEd $$)%-!-$) & MrEd %$)%-!-$) & MrEd%$:$- & MrEd $1)3-0-1) & MrEd %1)3-0-1) & MrEd%1:1- & MrEd $?)@->-?) & MrEd %?)@->-?) & MrEd%?:?- & MrEd $M)N-L-M) & MrEd %M)N-L-M) & MrEd%M:M- & MrEd%&_ & MrEd%&_ & MrEd%&_ & MrEd%&_ & MrEd%&_ & MrEd%&_ & MrEd%-Z & MrEd%-Z & MrEd%-Z & MrEd%-Z & MrEd%-Z & MrEd%-Z & MrEd%&* & MrEd%.1 & MrEd%59 & MrEd%=A & MrEd%DH & MrEd%LP & MrEd%RV & MrEd%Z_ & MrEd%&* & MrEd%.1 & MrEd%59 & MrEd%=A & MrEd%DH & MrEd%LP & MrEd%RV & MrEd%Z_ & MrEd%&* & MrEd%.1 & MrEd%59 & MrEd%=A & MrEd%DH & MrEd%LP & MrEd%RV & MrEd%Z_ & MrEd%&* & MrEd%.1 & MrEd%59 & MrEd%=A & MrEd%DH & MrEd%LP & MrEd%RV & MrEd%Z_ & MrEd%&* & MrEd%.1 & MrEd%59 & MrEd%=A & MrEd%DH & MrEd%LP & MrEd%RV & MrEd%Z_ & 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~O44TjYtemEJlCxcKkʷ:a^C ¼ԍt*tqhQXOzg~<S+1_!^E- Ҧ1"b+t,oiп)3п)A%ɞg/gn=@?x?fۯ<Fo8c4iЧ)OCDOCA҅xfK~כ~ίJ~ï: ut~*{}}: wUN4!k?Pؑba!'qb Equation dC A dzb z=1 =(2-w)Bi LS,A (C AL -C A b z=1 )( METAFILEPICT%2  .1  @ & MathType- #7p36Times New Roman-!dC!dz !Bi5!C!CTimes New Roman-!AY:!zPJ !LSP !APq!ALP!A!z3Times New Roman-!'!( !) !(!'!'!)Py Times New Roman-!,YSymbol-!=}!=Symbol-!=!-!-Y9Times New Roman-!1!1Times New Roman-!2nSymbol-!w"System-(((((((((((Dd 0<  C A}?2'X~[8`S`!X~[8`S@}"kx+DQ9w1<㍲f'4K2E?RFΟʆBv(ϝ{+}{n!  3L('! 亮 *H3;Bj_xKGV 02_\B<9*Z}(9tI5$8lȡoldP7|GXj!s! 3ʽC/_(Y䪗6J/j2}t oX&< l)^,ME霗DM:Yڊi/wc 9sxviwIX6c1io&,asfW̕ KkԘ֓]ā'fZAH>?&F?V0q3~:MqwwV(n[P D\i{'_h6 Equation dC A dzb z=0 =0؋^Ox METAFILEPICT%h2  .1   & MathType- #736Times New Roman-!dC!dzTimes New Roman-!AY:!z3Times New Roman-!'YSymbol-!=.Symbol-!=YITimes New Roman-!0/Times New Roman-!0"System-0880ppp3DdL0<  C A~@2D?*[?βa`!wD?*[?βa2 kExcdd``$D@9@, fbd02,( 1dbfaj`~ d30&dT20``Y}"Lk,a DLA"f a03?9A``bM-VK-WM A`d`ҊL )00dzB73 1Al`Of >mbdpd@l B7fH_ 61mRGiM Mab[$sX\LآǓ=,HK鳡.qD {I)$5 =`u] Equation dC B dzb z=1 =(2-w)Bi LS,B (C BL -C B b z=1 )!5%L[T METAFILEPICTd%df2  .1    & MathType- # 7 p36Times New Roman-!dC!dz !Bi%!C!CTimes New Roman+-!BY2!zPB !LSP !BPQ!BLP!B!z3Times New Roman-!'!( !)} !(!'}!'!)Pq Times New Roman+-!,YSymbol-!=U!=Symbol-!=!-!-Y1Times New Roman-!1!1Times New Roman-!2fSymbol-!w"System-Dd 0<  C AA2+ .sm][svӠC`! .sm][svӠC L"kx/A߼O-n"~JZ 'ZRZ BҴIU\t[;k&ɛ|߼%7^e# A˲8'=đԳCh .5B,CfwVsbL SӊN3b(¢;Z`STD:>eD"H 2ɽ> 5o¶1[(<%gsa9exv<mHuANh)@ҡ jg0N~VdpQao;UQ;A{!*FCY{m>,G$^mU[mjjGU cRORzL1[66סIMK=YU\=Q)myO |RܷNi Iq%3jS+d_S0vAy'B W6 Equation dC B dzb z=0 =0 METAFILEPICT%hn2  .1   & MathType- # 7 36Times New Roman-!dC!dzTimes New Roman-!BY2!z3Times New Roman-!'YSymbol-!=&Symbol-!=YATimes New Roman-!0'Times New Roman-!0"System-x@@ ?2 6DdL0<  C AB2dta,`!zdta,2 kHxcdd``$D@9@, fbd02,( 1dbfaj`~ d30&dT20``Y}"Lk,a DL"f a03?9A``bM-VK-WMcH+218\*B !?!60p&F21īč70MLpQlZ,vy@+ssp&5V4a7=(qdAK>t 81C(p*FRpeqIj.{H`m Equation` C B b z=0 =0  METAFILEPICT(42  .1   & MathType`-***@Times New Roman-!CBTimes New Roman+-!BR!z@Times New Roman-!'Symbol-!=@FSymbol-!=aTimes New Roman+-!0@GTimes New Roman-!0"System- Dd@<  C AC2J#0~Q:f`!^J#0~Q:L ,xJ@ƿ4BoՃBE(`B @s$gd oPWv+wUTBVNs,˺etm`z7BU~LLmYE;F&K{7,L͌TOCpҺ[q<"x-yﻄ5>ϨG7qFb}'9b/GӀKNRϦ::}k$vRǝm=Kygg=2b ѼPhx?dFl?\6 Equation dX B dZ=b(1-e B )h o (rate)C Bi U SL2 METAFILEPICToH(o6  .1  ` & MathType-  6Times New Roman-!dX!dZ!b# !rate!C!UcTimes New Roman+-!Bc!Bc !o)!Bi) !SLSymbol-!=!-Times New Roman-!(!){ !(h !)HTimes New Roman+-!1Symbol-!e !h"System-@@ 6 wDd<  C AD2"HٚBbj `!"HٚBb`:x?KAg$g=HD좂$B;X V&zgB3(vbm'"jgo{mcݛ!LlYW:[ y,'9^w8h0cբ9q*5*֩І(]ix|,s)4$ɾ4^%qйBŪN3SsK'}!9hO:M~'t (:NXkQwCP޸{3;,yE&2XB<=GjY6?He -\; kkJFTVou8%E:> c r,'tn+ Uw< zqt_cj Equation` rate=k vs C Ate METAFILEPICT*6*7  .1   & MathType@ Times New Roman-!rate u!k !CpTimes New Roman-!vsp!A jSymbol-!="System-Dd<  C AE2U⥛ƑهE'\1w`!)⥛ƑهE'\ xcdd``~ @bD"L1JE `x06qYjl < %! `y&F1=[0dzbr< t@Z($fȬRfv #C63x# n?ALdb(+F6+&Gd1 lЦܤ 32̱200#RpeqIj.އ2e Equation rate=k vs C A C B (1+K 1 C B +K 2 C C ) bPP*!;Xt<u METAFILEPICT%F7 , .1    & MathType-u Times New Roman-!rate!k!Cl !C!K!C !K !CcTimes New Roman+-!vsc!Ac@ !BM!BM !CjSymbol-!=!+!+Times New Roman-!( !)Times New Roman+-!1MTimes New Roman-!1M !2qB Symbol-!b"System-z0BȈ4'DIALOGxDd0<  C AF2 / T$`! / T$X PkxMO@gmh#A`AE545<xUOU]tK dwޙibd"lȋt$u9YiNr=ud^.xEs$Pn앚6@ ~2Q4/J/z9w}ݩ9#יpZnز40CSMWcaK[ 6OpT^Zf8 deNVlDgM9W]E?G߯#xR{Bg`%1攳5S8w?U?|T^d[<fzvvq/{rDd@<  C AG2-!-V%#`!-!-V%#0h+ xcdd``db``beV dX,$XĐ h 7XRYaj`T`~35[0d H@penR~^](;VdedȀ 7+v y/.n&]\A IlET'ȣ d++&10B} ? 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" $ $ % % ' ( & MrEd$^$# % & ' ( + , - . . / / / / / /" /$ /& .' ,' ,( +( (( &( %( ( ( ( ( ( ( ( ( ( ' & $ "                           !!#%% & MrEd$[                                                                                           - & MrEd%@` ? & MrEd%0 ` @ & MrEd%` @0 p & MrEd%@`  & MrEd%0 `  & MrEd%` 0  & MrEd%0  & MrEd% p0  & MrEd%0   & MrEd%00 / & MrEd% 0 0 & MrEd%0 0 ` & MrEd%0  & MrEd% 0  & MrEd%0   & MrEd% 0  & MrEd% 0 & MrEd%0 0 & MrEd% 0 & MrEd% ` 0 & MrEd%0 & MrEd% 0  & MrEd% 0 & MrEd%0 P - & MrEd$[                                                                                           2 @ +Catalyst (Internally and Externally wetted)2  Liquid Full (Holdup=Bed voidage)!2  3Catalyst (Internally wet, externally partially wet)2 0 'Liquid films (Holdup = dynamic +static)2  $Gas accesing liquid and dry catalyst2 ` * Catalyst (Internally wet, externally dry)2 #Liquid films (Holdup = only static)2 Gas Accesing dry catalystTimes New Roman-2 pLIQUID PULSE ON2 LIQUID PULSE TRANSITON ZONE2 pLIQUID PULSE OFF- & MrEd%`  & MrEd%0 ``  & MrEd%` 0 Times New RomanF-2  $Temperature, Low (=Feed Temperature)2  %Temperature, Rise (>Feed Temperature)2 %Temperature, High (>Feed Temperature) & MrEd%P0 O & MrEd% 0 P & MrEd%0 P  & MrEd%P & MrEd%P 0 & MrEd% 0 & MrEd%0 & MrEd% P Times New Roman- 2 (a) 2 (b) 2  (c)Times New RomanF-2  (Only Scenario II)2 @ (Only Scenario II)"System---''hj=u Vi~Vj     PIC LMETA CompObj*fObjInfo, & MathType "-@l PSymbol-2  (PSymbol-2 O)PSymbol- 2 j 2 W 2 we 2 jL 2  2  eTimes New Roman- 2 tk 2 C 2 z 2 % u 2 ] C 2 N 2 a 2 N 2 ?a Times New Roman9- 2 XL| 2 iL>| 2  ILJ| 2  L| 2 WiL>| 2 i> 2 GL| 2 GL| 2 i> 2 (LS|p 2 %LS|pTimes New Roman,.- 2 ( 2 )PSymbol- 2 + 2 -= 2 - & "Systemn- FMicrosoft Equation 2.0 DS Equation Equation.29q  t(e L C iL )+zu IL e L C iL ()=N iL a Ole10Native-$Equation Native 2_902837114|FcgcgOle 7GL -N iLS a LS f@iJmJiJ t(e L C iL )+zu IL e L C iL ()=N iGL a GL -N iLS a LS~PIC 8LMETA :CompObjNfObjInfoPL5!El5!E w .1   & & MathType "-@& PSymbol-2  (PSymbol-2 )PSymbol- 2 j 2 W 2 we 2 j 2 = 2 J eTimes New Roman - 2 tk 2 C 2 z 2 d u 2  C 2 N 2 a 2 YN 2 a Times New Roman - 2 CG 2 iG> 2 7 IGJ 2  G 2 iG> 2 8i> 2 cGL| 2 pGL| 2 wi> 2 GSp 2 GSpTimes New Roman - 2 ( 2 #)PSymbol- 2 + 2 = 2 - 2 - & "Systemn- FMicrosoft Equation 2.0 DS Equation Equation.29q t(e G C iG )+zu IG e G C iG ()=-N iV a GLSymbol-eiJmJ|iJ t(e G C iG )+zu Ole10NativeQEquation Native V_907708088Fcg`cgOle [IG e G C iG ()=-N iGL a GL -N iGS a GSL+&El+&E v .1  "&`"PIC  \LMETA ^CompObj  rfObjInfot & MathType "-@4PSymbol- 2 fW 2 fr 2 f+e 2  2 f 2 f' e 2  Times New Roman- 2 f( 2 f) 2 f ( 2 fZ) Times New Roman - 2 rG 2 G 2 G 2  G 2 [ IGJ 2 G 2 GL| 2 GL| 2 GSp 2 GSp 2 >GA 2 2!GATimes New Roman- 2 fH 2 tk 2 f u 2 fr H 2 k z 2 E 2 a 2 ~E 2 a 2 ME 2 q aPSymbol- 2 + 2 n= 2 - 2 E- 2 - & "Systemn- FMicrosoft Equation 2.0 DS Equation Equation.29q r G e G C PmixG T G t+e G u IG r G C PmixG Ole10Native  uEquation Native |8_953922022mF cg cgOle T G z=-E V a GL -h GA (T G -T A )) {l{h{l{h'hJtmJ`iJ (r G e G H G )t+(e G u IG H G )z=-E GL a GL -E GS a GS -E GA a GA FMicrosoft Equation 3.0 DS Equation Equation.39qxpVIh|I H G =C iG H iG / G "CompObjfObjInfoEquation Native _907708213TF cg,dgOle PIC LMETA CompObjfL%ETl%E  .1   "&! & MathType "-@pPSymbol- 2 fW 2 fr 2 fe 2  2 f 2 f e 2 f r 2  Times New Roman0- 2 f( 2 fM) 2 fJ ( 2 f) Times New RomanE- 2 L| 2 L| 2 L| 2  L| 2  ILJ| 2  L| 2 lL| 2 GL| 2 GL| 2 |LS|p 2 KSLp| 2 LA| 2  LA|Times New Roman0- 2 f~H 2 tk 2 f4 u 2 fFH 2  z 2 E 2 a 2 RE 2 a 2 E 2 aPSymbol- 2 B+ 2 )= 2 - 2 - & "Systemn-! FMicrosoft Equation 2.0 DS Equation Equation.29q r L e L C PmixL T L t+e L u IL r L C PmixL T L z=E L a GL +E LS a SL -h LAObjInfoOle10NativeEquation Native H_953922198JF,dg dg (T L -T A ),hJlmJhJ (r L e L H L )t+(e L u IL r L H L )z=E GL a GL -E LS a SL -E LA a LAqJOle CompObjfObjInfoEquation Native  FMicrosoft Equation 3.0 DS Equation Equation.39qxbI gI H L =C iL H iL / L "_953923640"F dgKdgOle PIC !$LMETA LE lE  .1  `&  & MathType "-@ PSymbol- 2 fWr 2 fe 2 F Times New Roman%- 2 cc 2 B 2 2CP 2 t LS|p 2 QLS|p 2 GSp 2 GSpTimes New Roman)- 2 fH 2 tk 2 J E 2 {a 2 E 2 <aTimes New Roman%- 2 fw( 2 f{) 2 f1PSymbol- 2 f- 2  = 2 R + 2 + & "Systemn- FMicrosoft Equation 3.0 DS Equation Equation.39q` r c C PC (1-e B )T CP t-(1-e B )k e  2 T CP z 2 =-E S a LCompObj#&fObjInfoOle10Native%'dEquation Native S -(R OV )(DH) RTime̼bI gI "(1" B )H CP "t=+E LS a LS +E GS a GS FMicrosoft Equation 3.0 DS Eq_953923701 ;*FKdgcdgOle CompObj)+fObjInfo,uation Equation.39qzI{I H G =(1" P )C PC (T"T ref )+ P C iCP H iCP "L$Equation Native _949789483j/Fcdg@}dgOle PIC .1L$  .1  ` !&  & MathType "-||]PSymbol-2 r(PSymbol-2  )|| PSymbol- 2 -r 2 e 2  2 META  CompObj02fObjInfo3Equation Native pr 2  2  e 2 m  Times New Roman P- 2 `L| 2 GL| 2 `L| 2 / ILJ| 2 /^ L| 2 `9i> 2 TdGL| 2 `MGL| 2 `i> 2 `i> 2 TLS|p 2 `LS|p 2 `e i>Times New Roman- 2 Otk 2 u 2 + z 2 -N 2 a 2 M@ 2 N 2 a 2 M@PSymbol- 2 + 2  = 2 G+ 2 -PSymbol- 2 6+ 2 6 & "Systemn- FMicrosoft Equation 3.0 DS Equation Equation.39qTTcIgI " L  L "t+" L u IL  L ()"z=+N iGL a GL M i " "N iLS a LS M i "_9497894826F`dg@KdgOle PIC 58LMETA       "$%&'()*+,-./0123456789:;<=>?@ABCDEGJKLMNOPQRSVXYZ[\]^_`abcdefghijklmnopqrstuvwxy{~L%%  .1  `!&`! & MathType "-||PSymbol-2 (PSymbol-2  )|| PSymbol- 2 -r 2 0e 2  2 r 2  2  e 2   Times New Roman=- 2 `G 2 GG 2 `G 2 / IGJ 2 / G 2 `i> 2 TGL| 2 `GL| 2 `ki> 2 `Ki> 2 TvGSp 2 `_GSp 2 `!i>Times New Roman- 2 otk 2 6 u 2  z 2 N 2  a 2 #M@ 2 ?N 2 a 2 M@PSymbol- 2 + 2  = 2 - 2 -PSymbol- 2 6 2 6= & "Systemn- FMicrosoft Equation 3.0 DS EqCompObj79fObjInfo:Equation Native p_953926700=F@Kdg;dguation Equation.39qTTlI,mI " G  G "t+" G u IG  G ()"z="N iGL a GL M i " "N iGS a GS M i " Ole  PIC <?!LMETA #CompObj>@FfL>Gd(>G @ .1  `@&`@ & MathType "-@y GZPSymbol-2 7-[ZPSymbol-2 ?]PSymbol- 2  2 W 2 ~r 2 e 2 Y  2   2 J r 2  e 2 e 2 r 2 e 2 / 2 YTimes New Roman- 2 tk 2 u 2  z 2 u 2 u 2 ag 2 P 2 z 2 wF 2 #K 2 $u 2 'u 2 +u 2 /N 2 "2a 2 ;4M@ 2 W9N 2 ;a 2 =M@ Times New Roman - 2 fL| 2 ILJ| 2 VL| 2 2 L| 2  L| 2 ILJ| 2 lILJ| 2 VL| 2 L| 2 eL| 2 YL| 2 DD 2 K Liq|>p 2 %IGJ 2 (ILJ| 2 [,ILJ| 2 0i> 2 0GL| 2 2GL| 2 5i> 2 c:i> 2 :LS|p 2 |<LS|p 2 >i>Times New Roman- 2 ( 2 ) 2  ( 2 ]) 2 8$( 2 )) 2 h?) Times New Roman - 2 ,8PSymbol- 2 + 2 I= 2 w- 2 D- 2 !+ 2 &- 2 ~*+ 2 66-PSymbol- 2 U7 2 - & "Systemn- FMicrosoft Equation 3.0 DS Equation Equation.39qObjInfoAHEquation Native I_953926747(\DF;dgdgOle ŤbI gI ""t( L u IL  L )+""z( L  L u IL u IL )= L  L g" L "P"z L "F D,Liq +K(u IG "u IL )+u ILI N iGL a GL M i "N iLS a LS M i )  "  " []LJ*J @ .PIC CFULMETA WCompObjEHzfObjInfo|1  `D&C & MathType "-@c  yEZPSymbol-2 /[ZPSymbol-2 QC]PSymbol- 2  2 W 2 hr 2 e 2   2 z  2  r 2 5 e 2 e 2 r 2 e 2  2 Times New Roman - 2 tk 2 u 2 8 z 2 u 2 qu 2 g 2 uP 2 z 2 8 F 2 #%K 2 &u 2 )u 2 -u 2 T3N 2 5a 2 7M@ 2 =N 2 o?a 2 AM@ Times New Roman)- 2 ?G 2 IGJ 2 G 2 z G 2  G 2 pIGJ 2 7IGJ 2 ^G 2 G 2 G 2 G 2  !D 2 "GaspW 2 'ILJ| 2 *IGJ 2 .IGJ 2 e4i> 2 4GL| 2 6GL| 2 ,9i> 2 >i> 2 A>GSp 2 3@GSp 2 Bi>Times New Roman - 2 ( 2 T) 2  ( 2 V) 2 =&( 2 +) Times New Roman)- 2 !,8PSymbol- 2 /+ 2 R= 2 - 2 - 2 #+ 2 (- 2 ,+ 2 =0- 2 9-PSymbol- 2 ; 2 h1 & "Systemn- FMicrosoft Equation 3.0 DS Equation Equation.39q t(r G u IG e G )+z(r G e G u IG u IG )=eOle10NativeGI}Equation Native _907517663ZRLFdgdgOle  G r G g-e G Pzb G -F D,Gas +K(u IL -u IG ))w RǒyI{I ""t( G u IG  G )+""z( G  G u IG u IG )= G  G g" G "P"z G "F D,Gas +K(u IL "u IG )+u IGI "N iGL a GL M i "N iGS a GS M i "  " []@PIC KNLMETA  CompObjMPfObjInfoL,<,  .1  ``(& ( & MathType "-ZF vEPSymbol- 2 -r 2 e 2 q 2   2 |r 2 e 2 ]  2   2 e 2 r 2 e 2  2  Times New Roman - 2 L| 2 }L| 2 ILJ| 2 hL| 2  L| 2  ILJ| 2  ILJ| 2 L| 2 L| 2 L| 2 L| 2 D 2 Liq|>p 2 V#IGJ 2 &ILJ|Times New Roman- 2 /u 2 tk 2 Y u 2  u 2  z 2 g 2 rP 2 z 2  F 2  K 2 "u 2 %uPSymbol- 2 [+ 2 h= 2 - 2 - 2 + 2 $- Times New Roman - 2 ,8Times New Roman- 2 "( 2 ') & "Systemn- FMicrosoft Equation 2.0 DS Equation Equation.29q r L e L u IL t+r L e L u IOle10NativeOQEquation Native _907517951TFdg`_dgOle L u IL z=e L r L g-e L Pzb L -F D,Liq +K(u IG -u IL )+u IL N iL a GL M i dhJhmJhJ r L e L u IL t+r L e L u IL u IL z=e L r L g-e L Pzb L -F D,Liq +K(u IG -u IL )L--  .PIC SVLMETA  CompObjUXfObjInfo1  `)&@) & MathType "-L aEPSymbol- 2 -r 2 e 2  2 I 2 r 2 b e 2   2   2 e 2 r 2 e 2  2  Times New Roman- 2 G 2 }G 2 IGJ 2 G 2  G 2  IGJ 2 IGJ 2 FG 2 G 2 G 2 G 2 D 2 GaspW 2 u$ILJ| 2 'IGJTimes New Roman9- 2 Yu 2 tk 2  u 2  u 2 ez 2 g 2 ]P 2 z 2  F 2  "K 2 #u 2 &uPSymbol- 2 + 2 := 2 - 2 - 2  + 2 %- Times New Roman- 2 ,8Times New Roman9- 2 %#( 2 () & "Systemn- FMicrosoft Equation 2.0 DS Equation Equation.29q r G e G u IG t+r G e G u IG u IG z=e G r G g-e G Pzb G -F D,Gas +K(u Ole10NativeWYEquation Native _907515838\Fdg` egOle IL -u IG )+u IG N iV a GL M i dhJhmJhJ r G e G u IG t+r G e G u IG u IG z=e G r G g-e G Pzb G -F D,Gas +K(u IL -u IG )L 44   .1    & & MathType "-  .<PIC [^LMETA CompObj]`fObjInfo      !"#$%&'()*+,-/2345679:;<?BCFHIJKLMNOPQRSTUVWXYZ[\]^_`abdghijklmnqstuvwxyz{|}~Times New Roman - 2 LF 2 jg 2 E 2 -/Ga 2 E 2 -+Ga Times New Roman - 2 D 2 GaspW 2 G 2 G 2  L| 2 HPG 2 HZF 2 G 2 H^G 2 HhF 2 G Times New Roman - 2 ,8 2 H,8 2 H,8Times New Roman - 2 Re 2 RePSymbol- 2 = 2 - - 2 9  2 =9  2 9  2   2 =  2   2 + 2 4 2 4 2 H4 2 S 2 S 2 HS 2 Ze 2 r 2  e 2 - e 2 -@ e Times New Roman - 2 3p 2 H1p 2 H2p 2 <^2p & "Systemn- FMicrosoft Equation 2.0 DS Equation Equation.29q F D,Gas =e G rOle10Native_aEquation Native D_907515837dF eg%egOle  G gY G =r G gee-e L () 3 E 1 Re G,F Ga G +E 2 Re G,F2 Ga G []]W(oJmJiJ F D,Gas =e G r G gee-e L () 3 E 1 Re G,F Ga G +E 2 Re G,F2 Ga G []L|4  .PIC cfLMETA CompObjeh.fObjInfo01   `&  & MathType "- F N xTimes New Roman - 2 LF 2 g 2 z E 2 ,Ga 2 CE 2 ,Ga Times New Roman- 2 D 2 $Liq|>p 2 L| 2 oL| 2  L| 2 IL| 2 IF 2 L| 2 IL| 2 IF 2 kL| Times New Roman - 2 ,8 2 Ih,8 2 IT,8Times New Roman- 2 Re 2 RePSymbol- 2 = 2  2 < 2  2 P  2 <P  2 P  2 + 2   2   2 I  2  2  2 I 2 0e 2 r 2 ! e 2 , e Times New Roman- 2  3p 2 I]1p 2 I42p 2 =2p & "Systemn- FMicrosoft Equation 2.0 DS Equation Equation.29q F D,Liq =e L r L gY L =r L gee L () 3 E 1 Re L,F Ga L +E 2 Re L,F2 Ga L []L`!,`Ole10Nativegi1Equation Native 8<_950446512lF%eg$>egOle = oJmJ|iJ F D,Liq =e L r L gee L () 3 E 1 Re L,F Ga L +E 2 Re L,F2 Ga L [] FMicrosoft Equation 3.0 DS Equation Equation.39qCompObjkm>fObjInfon@Equation Native A_950454133-qF$>eg$>egxII  L =0.33 B (1"10 "g 0 )L44 w .Ole DPIC psELMETA GCompObjrtcf1   @/&/ & MathType "-<<W<<!PSymbol- 2 8e 2 r 2 de 2 r 2 <e 2 r 2  2 m 2 e 2 ?r 2 e 2  2 I Times New Roman - 2  L| 2 }L| 2 iL>| 2 7L| 2 L| 2  iL>| 2  np 2 L| 2 L| 2 iL>| 2 np 2 `iL>| 2 ~np 2 L| 2 +L| 2 L| 2 q np 2 #D 2 $Liq|>p 2 #np 2 )iG> 2 )np 2 -iL>| 2 7-npTimes New Roman=- 2  u 2 6 u 2 = tk 2 u 2 u 2 +z 2 g 2 p 2  z 2 "F 2 'K 2 -)u 2 h,u Times New Roman - 2 *p 2 @$,8Times New Roman=- 2  ( 2 (( 2 .))PSymbol- 2 = 2 ! + 2 Y- 2 + 2 - 2 ~!- 2 I&+ 2 A+- 2 S D & "Systemn- FMicrosoft Equation 3.0 DS Equation Equation.39qbI gI  L  L u iL* = L ObjInfoueEquation Native f _908299709xFKGeg_egOle o L u iLn +t(" L  L u iLn "u iLn "z+ L  L g" L "p n "z"F D,Liqn +K(u iGn "u iLn ))PIC wzpLMETA rCompObjy{fObjInfo|L66 w .1   1&0 & MathType "-<<}<2 <"PSymbol- 2 8e 2 r 2 e 2 &r 2 e 2 Rr 2  2 ~ 2  e 2 zr 2 me 2 I  2   Times New Roman- 2 G 2 }G 2 iG> 2 vG 2 G 2 e iG> 2  np 2 G 2 )G 2 iG> 2 np 2 \iG> 2 znp 2 G 2 QG 2 +G 2 !np 2 $D 2 %GaspW 2 9%np 2 `+iG> 2 ~+np 2 .iG> 2 .npTimes New Roman- 2 4u 2  u 2  tk 2 u 2 u 2 <z 2  g 2 !p 2 l!z 2 "$F 2  )K 2 *u 2 -u Times New Roman- 2 *p 2 %,8Times New Roman- 2 o( 2 %*( 2 /))PSymbol- 2 j= 2  + 2 - 2 + 2 B- 2 "- 2 '- 2 ,- 2  D & "Systemn- FMicrosoft Equation 2.0 DS Equation Equation.29qԨhJdmJhJ e G r G u iG* =e G r G u iGn +Dt(-e G r G u iGn u iGn z+e G r G g-e G p n z-F D,Gasn -K(u iGn -u iGn )Equation Native _902837000F_eghegOle PIC ~L)L22  .1   .&- & MathType "-<p<PPSymbol- 2 8e META  CompObjfObjInfoEquation Native 2 r 2 Ke 2 r 2 e 2  2  Times New RomanE- 2 L| 2 L| 2 iL>| 2 3np 2 0L| 2  L| 2 ' iL>| 2 wL| 2 np 2 np 2 iG> 2 np 2 "iG> 2 G'iL>| 2 f'np 2 ;+iL>|Times New Roman- 2 Ru 2 e u 2 tk 2 p 2 (p 2 sz 2 K 2 u 2 "u 2 &u 2 y*u PSymbol- 2 +{ 2 f+{ 2 }+{ 2 '+{PSymbol- 2 = 2 u + 2 - 2 - 2 + 2  - 2 $- 2 P)- Times New Roman- 2 11p 2 1p 2 1p 2 d(1p 2 ? *p 2 "*p 2 S+*pTimes New RomanM- 2 ( 2 J( 2 ) 2 (( 2 #) 2 %( 2 6,)))PSymbol- 2  D & "Systemn- FMicrosoft Equation 2.0 DS Equation Equation.29qliJmJXiJ e L r L u iLn+1 =e L r L u iL* +Dt(-e L (p n+1 -p n )z+K((u iGn+1 -u iG* )-(u iLn+1 -u iL* )))LC3C3  .1   .&@._902836999}FhegGegOle PIC LMETA   #     !"$)6%&'(*,+-0./13254<7Q89:;=?>@CABDEFIGHJNKLMOPRVlSTUWXY[Z\_]^`abcdefghijkmnoqprstuvwxzy{| & MathType "-<<PSymbol- 2 8e 2 r 2 ue 2  r 2 e 2  2 H Times New Roman- 2 G 2 G 2 >iG> 2 ]np 2 EG 2  G 2 { iG> 2 G 2 xnp 2 np 2 niG> 2 np 2 b#iG> 2 'iL>| 2 'np 2 +iL>|Times New Roman- 2 |u 2  u 2 tk 2 p 2 p 2 z 2 K 2 u 2 "u 2 'u 2  +u PSymbol- 2 +{ 2 +{ 2  +{ 2 x(+{PSymbol- 2 G= 2  + 2 -- 2 b- 2 G- 2 w!- 2 c%- 2 )- Times New Roman- 2 [1p 2 v1p 2  1p 2 (1p 2  *p 2 z#*p 2 +*pTimes New Roman- 2 ( 2 ( 2 K) 2 (( 2 $) 2 &( 2 ,)))PSymbol- 2 'D & "Systemn- FMicrosoft Equation 2.0 DS Equation Equation.29qlhJmJiJ e G r G u iGn+1 =e G r GCompObjfObjInfoEquation Native _902836998FGeg 8eg u iG* +Dt(-e G (p n+1 -p n )z-K((u iGn+1 -u iG* )-(u iLn+1 -u iL* )))Lp Ole PIC LMETA `CompObjf    !"#$%'*+,/123456789:;=@CEFGHIJKLMNOPQRSTUWZ[\_abcdefghijklmnopqrtwxy|~  .1  @&@ & MathType "-G  4Times New Roman(- 2 4u 2 u 2 'tk 2 ep 2 z Times New Roman- 2 iL>| 2 np 2 iL>| 2  L| 2 G PSymbol- 2 +{PSymbol- 2 = 2  - 2 e - 2  + Times New Roman(- 2 1pTimes New Roman- 2 e 1 Times New Roman(- 2 *pTimes New Roman- 2 ( 2 e[ ( 2 e ) 2 ) 2 e( 2 e)PSymbol- 2 =D 2 e a 2  r 2 ea 2 +r 2 e 2 ed 2  & "Systemn- FMicrosoft Equation 2.0 DS Equation Equation.29qObjInfoEquation Native  _902836997F 8egegOle  ԴiJhmJdiJ u iLn+1 =u iL* -Dt((1-a)r L +ar G )(dp)zL\ PIC LMETA `CompObj&fObjInfo(  .1  @`&  & MathType "-q z Times New Roman- 2 4u 2 u 2 Qtk 2 ep 2 lz Times New Roman%- 2 iG> 2 np 2 iG> 2 v G 2 L| PSymbol- 2 +{PSymbol- 2 = 2 3- 2 e - 2  + Times New Roman- 2 1pTimes New Roman%- 2 e 1 Times New Roman- 2 *pTimes New Roman%- 2 ( 2 e ( 2 e ) 2 ) 2 eu( 2 e)PSymbol- 2 gD 2 e b 2  r 2 eb 2 $r 2 e 2 ed 2  & "Systemn- FMicrosoft Equation 2.0 DS Equation Equation.29qԴiJhmJdiJ u iGn+1 =u iG* -Dt((1-b)r G +br L )(dp)zL {hEquation Native )_902836996F eg egOle -PIC .LMETA 0CompObj<fObjInfo>Equation Native ?X { a .1  @ & & MathTypePPSymbol- 2 1dTimes New Roman%- 2 p 2 p 2 p Times New Roman-- 2 ^np 2 fnpPSymbol- 2 = 2 8- PSymbol- 2 +{ Times New Roman%- 2 X1p & "Systemn- FMicrosoft Equation 2.0 DS Equation Equation.29q<mJ8iJHnJ dp=p n+1 -p nL 44@ 4 * .1   & & MathTypePPSymbol- 2 `1a 2 `r 2 `e 2 ` _902836994F egegOle APIC BLMETA D\r 2 ` e 2 ` r 2 `se 2 `r 2 `Ve 2 `6r 2 `e 2 `= 2 `8+ 2 `+Times New Roman - 2 `K 2 `tk 2 `K 2 `tk Times New Roman- 2 G 2 >G 2  G 2 L G 2  L| 2 DL| 2 L| 2 'L| 2  G 2 GPSymbol- 2 `D 2 `DTimes New Roman - 2 `/k 2 ` ( 2 `Z( 2 `S) 2 `h) & "Systemn- FMicrosoft Equation 2.0 DS Equation Equation.29qCompObjVfObjInfoXEquation Native Y_902836993Feg`eghJhmJTiJ a=Kr G e G Dt/(r G e G r L e L +(r L e L +r G e G )KDt)LH4@Ole ]PIC ^LMETA `CompObjsfH4 B .1  `&  & MathTypePPSymbol- 2 `;b 2 `r 2 `Ne 2 ` r 2 ` e 2 `f r 2 ` e 2 `Tr 2 `e 2 `r 2 `ePSymbol- 2 `= 2 `+ 2 ``+Times New Roman%- 2 `K 2 `tk 2 `K 2 `tk Times New RomanI- 2 L| 2  L| 2  L| 2  L| 2 ' G 2 ~G 2 *L| 2 lL| 2 OG 2 GPSymbol- 2 `D 2 `DTimes New Roman%- 2 `a/k 2 `! ( 2 `( 2 `y) 2 `) & "Systemn- FMicrosoft Equation 2.0 DS Equation Equation.29qlJ0qJ(mJ b=Kr L e L Dt/(r L e L r ObjInfouEquation Native v_902836992F`eg`WfgOle zG e G +(r L e L +r G e G )KDt)LAD%A  .1  ;&`;0 & MathType "-< PIC {LMETA } CompObjfObjInfo< < <5<2<<@<"<^$<_)<+<0<M3<4<7<:Times New Roman - 2 (N 2 a 2 M@ 2 jN 2 a 2 M@ 2 xN 2 a 2  M@ 2 *'u 2 &z 2 -u 2 -z 2 2tk 2 "4z 2 -9p 2 8z Times New Roman- 2 :i> 2 eGL| 2 ZGL| 2 i> 2 . G 2 .z L| 2 |i> 2 GSp 2 GSp 2 Ji> 2 .G 2 i> 2 LS|p 2 LS|p 2 8"i> 2 .L| 2 &L| 2 'iL>| 2 ?-L| 2 .iL>|Times New Roman - 2 }( 2 G) 2 8%( 2 () 2 +( 2 k/) 2 7( 2 9) Times New Roman- 2 '*p 2 .*pTimes New Roman - 2  1 2  1PSymbol- 2  r 2  r 2 r 2 r 2 u$ 2 %e 2 3& 2 + 2 j,e 2 , 2 3 2 d3 2 5w 2 %7 2 o8d 2 !8 2 F - 2 (+ 2 6+ 2 $#+ 2 )+ 2 k0= 2  5 2  5 2  5 2 : 2 : 2 :PSymbol- 2 h 2 Z 2 8PSymbol- 2 1D & "Systemn- FMicrosoft Equation 2.0 DS Equation Equation.29q(hJ\mJhJ N iGL a GL M i (1r G -1r L )+N iEquation Native D_902836990F`Wfg@ )fgOle PIC LGS a GS M i r G +N iLS a LS M i r L    +(e L u iL* )z+(e L u iL* )z=Dtzw(dp)z()  @L<< < .1  @&t & MathType "-\4\H\ \ \\\\8PSymbol- 2 0w 2 e 2 :a 2 r 2 4 aMETA CompObjfObjInfoEquation Native  2  r 2  e 2 b 2 r 2 b 2 r 2 = 2 - 2 - 2 o 2 o 2 o 2   2   2   2  + 2 - 2 ]- 2 Y 2 Y 2 Y 2 B 2 B 2 B Times New Roman- 2 @L| 2 LL| 2 L G 2 @G 2 LG 2 LL|Times New Roman)- 2 01 2 1 & "Systemn- FMicrosoft Equation 2.0 DS Equation Equation.29qhJpmJLiJ w=e L 1-ar L -ar G ()+e G 1-br G -br L ()L {h { a .1  @ &_907520608JFH2fgH2fgOle PIC LMETA  & MathTypePPSymbol- 2 1dTimes New Roman%- 2 p 2 p 2 p Times New Roman-- 2 ^np 2 fnpPSymbol- 2 = 2 8- PSymbol- 2 +{ Times New Roman%- 2 X1p & "Systemn- FMicrosoft Equation 2.0 DS Equation Equation.29q<mJ8iJHnJ dp=p n+1 -p nCompObjfObjInfoEquation Native X_950474646F 9fgJfgOle PIC LMETA  CompObjfLAD%A  .1  ;&`;0 & MathType "-< < < <5<2<<@<"<^$<_)<+<0<M3<4<7<:Times New Roman -    !"#$%&'()*+,-./02569;<=>?@ABCDEFGHIJKMPQTVWXYZ[\]^_`abcdefghijklmnopruvwxz{|}~ 2 (N 2 a 2 M@ 2 jN 2 a 2 M@ 2 xN 2 a 2  M@ 2 *'u 2 &z 2 -u 2 -z 2 2tk 2 "4z 2 -9p 2 8z Times New Roman- 2 :i> 2 eGL| 2 ZGL| 2 i> 2 . G 2 .z L| 2 |i> 2 GSp 2 GSp 2 Ji> 2 .G 2 i> 2 LS|p 2 LS|p 2 8"i> 2 .L| 2 &L| 2 'iL>| 2 ?-L| 2 .iL>|Times New Roman - 2 }( 2 G) 2 8%( 2 () 2 +( 2 k/) 2 7( 2 9) Times New Roman- 2 '*p 2 .*pTimes New Roman - 2  1 2  1PSymbol- 2  r 2  r 2 r 2 r 2 u$ 2 %e 2 3& 2 + 2 j,e 2 , 2 3 2 d3 2 5w 2 %7 2 o8d 2 !8 2 F - 2 (+ 2 6+ 2 $#+ 2 )+ 2 k0= 2  5 2  5 2  5 2 : 2 : 2 :PSymbol- 2 h 2 Z 2 8PSymbol- 2 1D & "Systemn- FMicrosoft Equation 3.0 DS Equation Equation.39q(bII N iGL a GL M i (1 G "1 L )  " +N iGS a GS M i  G +N iLS a LS M i ObjInfo Equation Native  D_950474823F SRfgcfgOle  L "  " +1 L "( L  L u iL* )"z+1 G "( G  G u iG* )"z+ L "ln L "t+ G "ln G "t=t1 L ""z L "(p)"z()+1 G ""z G "(p)"z()[]L<< < .PIC LMETA CompObj1fObjInfo31  @&t & MathType "-\4\H\ \ \\\\8PSymbol- 2 0w 2 e 2 :a 2 r 2 4 a 2  r 2  e 2 b 2 r 2 b 2 r 2 = 2 - 2 - 2 o 2 o 2 o 2   2   2   2  + 2 - 2 ]- 2 Y 2 Y 2 Y 2 B 2 B 2 B Times New Roman- 2 @L| 2 LL| 2 L G 2 @G 2 LG 2 LL|Times New Roman)- 2 01 2 1 & "Systemn- FMicrosoft Equation 3.0 DS Equation Equation.39q՜bI gI  L = L  L 1" L " G ()Equation Native 4_950474965FClfg }fgOle 7PIC 8LL<< < .1  @&t & MathType "-\4\H\ \ \\\\8PSymbol- 2 0w 2 e 2 :a 2 r 2 4 aMETA :CompObjLfObjInfoNEquation Native O 2  r 2  e 2 b 2 r 2 b 2 r 2 = 2 - 2 - 2 o 2 o 2 o 2   2   2   2  + 2 - 2 ]- 2 Y 2 Y 2 Y 2 B 2 B 2 B Times New Roman- 2 @L| 2 LL| 2 L G 2 @G 2 LG 2 LL|Times New Roman)- 2 01 2 1 & "Systemn- FMicrosoft Equation 3.0 DS Equation Equation.39q՜bI gI  G = G  G 1" G " L ()L*LH* ^ .1  @&&& & MathTypePSymbol-_9497875204FfgvfgOle RPIC SLMETA U2 (PSymbol-2 S) "- PSymbol- 2 -r 2 e 2 r 2 e 2 $ r 2   2 e 2 , Times New Roman - 2 L| 2 wL| 2 tbnp 2 L| 2 L| 2 tnp 2  L| 2 LILJ| 2 LL| 2 Vi> 2 tGL| 2 nGL| 2 i> 2  i> 2 t*!LS|p 2 #LS|p 2 %i>Times New Roman(~- 2  tk 2 <u 2 ,tz 2 HN 2 a 2 M@ 2 N 2 1"a 2 >$M@ PSymbol- 2 t+{PSymbol- 2 \= 2 + 2 :+ 2 - 2 r  2 r  2 r  2 r  2 % 2 % 2 % 2 %PSymbol- 2 V` 2 V Times New Roman(~- 2 tl1pPSymbol- 2  D & "Systemn- FMicrosoft Equation 3.0 DS Equation Equation.39q r L e L t+r L u IL e L ()z=+N iL a GL M i CompObjqfObjInfosOle10NativetEquation Native yTא@J̑J  L  Ln+1 = L  Ln +t" L u IL  L ()"z+N iGL a GL M i " "N iLS a LS M i " ()_949787502Fvfg?fgOle PIC LMETA L+H+ ^ .1  @'&`' & MathTypePSymbol-2  (PSymbol-2 ) "-9pPSymbol- 2 -r 2 e 2 r 2 :e 2 c r 2 P 2 Ve 2 ,) Times New Roman- 2 G 2 wG 2 twnp 2 G 2 G 2 tnp 2 8 G 2 LSIGJ 2 LG 2 i> 2 tGL| 2 GL| 2 i> 2 {!i> 2 t!GSp 2 #GSp 2 9&i>Times New Roman- 2 * tk 2 u 2 ,z 2 N 2 >a 2 [M@ 2 m N 2 "a 2 $M@ PSymbol- 2 t+{PSymbol- 2 q= 2  + 2 - 2 `- 2   2   2   2   2 & 2 & 2 & 2 &PSymbol- 2 V 2 V Times New Roman P- 2 t1pPSymbol- 2 @ D & "Systemn- FMicrosoft Equation 3.0 DS EqCompObjfObjInfoOle10NativeOle10ItemNameuation Equation.39q r G e G t+r G u IG e G ()z=-N iV a GL M i ?DBAllEquation Native t_907525265F?fg@gfgOle PIC LXhJlmJXiJ r G e Gn+1 =r G e Gn +Dtr G u IG e G ()z-N iGL a GL M i  -N iGS a GS M i  ()PL C .1   &4 & MathTypepTimes New RomanR- 2 XN 2 fc 2 k 2  x 2 x 2 ?x 2 q Times New Roman)- 2 vi> 2 GL| 2 tl>> 2 META CompObjfObjInfoEquation Native  L| 2  i> 2 q j> 2 wj> 2 sIJ 2 vj> 2 yL| 2 i> 2 fxcPSymbol- 2 E- 2 Z- 2 -Times New RomanR- 2 {[~ 2 ][~~ 2 z ]}~ 2  ( 2 2) 2 r( 2 ) 2 /k Times New Roman)- 2  ,8PSymbol- 2 b 2 ]l 2 D & "Systemn-L FMicrosoft Equation 2.0 DS Equation Equation.29qfiJmJiJ N iGL -c tl {[b][k L ]} i,j (x jI -x jL )-x i (Dq)/l xL++  .1  &G_907525302F@gfg@fgOle PIC LMETA  & MathType`PSymbol-2 [PSymbol-2 A]Times New Roman- 2 k Times New Roman- 2 @L| & "Systemn- FMicrosoft Equation 2.0 DS Equation Equation.29qCompObjfObjInfoOle10NativedEquation Native D` k ikL []&YesM#Rf(iJmJiJ k L []LWT_907525301FfgfgOle PIC LMETA W  .1   &< & MathType`PSymbol-2 [PSymbol-2 o]PSymbol- 2 b & "Systemn- FMicrosoft Equation 2.0 DS EqCompObjfObjInfoOle10NativedEquation Native 8uation Equation.29q` k ikL []&YesM#Rf iJ|mJiJ b[] FMicrosoft Equation 2.0 Equation Equation.29q_907525455Fcfg+fgOle CompObjcOle10Native l x =l i x ii=1n Cancel IhpLWTW  .1   &<_907525588F+fgggOle PIC LMETA  & MathType`PSymbol-2 [PSymbol-2 o]PSymbol- 2 b & "Systemn- FMicrosoft Equation 2.0 DS Equation Equation.29qCompObj fObjInfoOle10Native   dEquation Native  8                ! " # $ % ' * , / 1 2 3 4 5 6 7 8 9 ; > @ C E H I J K L M N O P Q R S T U V W X Y Z [ \ ] ^ _ ` a b c d e f g j l m n o p q r s t u v w x y z { } ` k ikL []&YesM#Rf iJ|mJiJ b[]LS{ hS{  .1  @@&_908958432Fgg,ggOle  PIC   LMETA   & MathTypePPSymbol- 2 >DTimes New Roman%- 2 (q 2 h 2 T 2  T 2 i h 2 T 2 tT Times New Roman;- 2 gL| 2 QGL| 2 IJ 2  L| 2  G 2 -G 2 "GL| 2 tIJPSymbol- 2 d= 2 - 2 < - 2 G-Times New Roman;- 2 ( 2 p ) 2  ( 2 q) & "Systemn- FMicrosoft Equation 2.0 DS Equation Equation.29qCompObj fObjInfo Equation Native  _907525722F,gg@5ggؘmJiJ,nJ Dq=h L (T GLI -T L )-h G (T G -T GLI )L++  .1  &GOle  PIC  LMETA  CompObj& f & MathType`PSymbol-2 [PSymbol-2 6]Times New Roman!- 2 k Times New Roman- 2 @jG & "Systemn- FMicrosoft Equation 2.0 DS Equation Equation.29qObjInfo( Ole10Native) dEquation Native + D_907525903F@5gg@Wgg` k ikV [].1R(4iJ|mJiJ k G []L {hOle - PIC . LMETA 0 DCompObj!: f {  .1  @ &  & MathTypePTimes New Roman - 2 XN 2 BN 2 N Times New Roman- 2 vi> 2 G 2 `i> 2 L| 2 7 i> 2 b GL|PSymbol- 2 = 2 = & "Systemn- FMicrosoft Equation 2.0 DS Equation Equation.29q` N iV =N iLM#NABLfPiJmJiJ N iG =NObjInfo< Ole10Native "= dEquation Native ? l_905281147%F@Wggogg iL =N iGLL0.C FMicrosoft Equation 2.0 DS Equation Equation.29qhJ0mJ(iJ F GL (z,Ole A PIC $'B LMETA ~!CompObj&(D f(0.  .1  *+&`+0* & MathTypep Times New Romanh- 2 LF 2 z 2  tk 2  N 2 w c 2 k 2 [x 2 :x 2 x 2 eq 2 f N 2 T c 2 k 2 [x 2 :x 2 x 2 q 2 N 2 9 c 2 k 2 x 2 ux 2 x 2 '!q 2 `Q N 2 `? c 2 `k 2 `uy 2 `y 2 `y 2 `q Times New Roman - 2  GL| 2  GL| 2 $tl>> 2 $L| 2 $Hj> 2 $.j> 2 *IJ 2 $ j> 2 L| 2 $"xc 2 X GL| 2 d tl>> 2 dL| 2 dHj> 2 d.j> 2 X*IJ 2 d j> 2 XL| 2 d"xc 2 ncpc 2 t  GL| 2 tl>> 2 dL| 2 @ncpc 2 j> 2 ij> 2 t eIJ 2 Hj> 2 t KL| 2 np 2 $xc 2 GL| 2  tg>p 2 G 2 Vj> 2 Hj> 2 !V 2 Vj> 2 RIJTimes New Romanh- 2 W( 2 ,` 2 ) 2 {[~ 2 ][~~ 2 K]}~ 2 ( 2 ) 2 ( 2 7 ) 2 !/k 2 {[~ 2 ][~~ 2 (]}~ 2 ( 2 ) 2  ( 2 Z ) 2 2!/k 2 D5.` 2 5.` 2 5.` 2 y {[~ 2 ][~~ 2 ]}~ 2 ( 2 ) 2 ( 2 !) 2 "/k 2 `{[~ 2 `][~~ 2 `Y]}~ 2 `( 2 `) 2 `( 2 `i ) 2 `A!/k Times New Roman - 2 $,8 2 d,8 2 ,8 2 ,8PSymbol- 2 = 2 V - 2 - 2 - 2 3 - 2 - 2 - 2  - 2 <- 2 - 2 ` - 2 `>- 2 `- PSymbol- 2 -{ 2 )-{ 2 1-{ Times New Romanh- 2 $ 1p 2 $p1p 2 $a1p 2 d{ 2p 2 d[2p 2 do2p 2 H 1p 2 1p 2 11pp 2 X 1p 2 ~1p 2 1pPSymbol- 2 b 2 !l 2 b 2 !l 2 b 2 #l 2 `b 2 `"l 2 {D 2 D 2 = D 2 `D Times New Romanh- 2 #yc 2 g GL| 2  tg>p 2 cG 2 Vj> 2 Hj> 2 !V 2 Vj> 2 RIJ 2 '#yc 2 ]ncpc 2 QGL| 2 ] tg>p 2 ]GG 2 ]Mncpc 2 ]j> 2 ]j> 2 Q[V 2 ]j> 2 QIJ 2 ]np 2 ]$yc 2 nIJ 2 IJ 2 ;IJ 2 IJ 2 ]!ncpc 2 Q IJ 2 ]!9ncpc 2 ]!ncpc 2 Q IJ 2 "IJ 2 "IJ 2 "IJ 2 #ncpc 2 "IJ 2 $IJ 2 $IJ 2 $IJ 2 %ncpc 2 $-IJ 2 (V 2 (L|Times New Roman - 2 . N 2  c 2 k 2 uy 2 y 2 y 2 q 2 N 2  c 2 k 2 y 2 y 2 2y 2 X!q 2 =y 2 =~K 2 =x 2 -ly 2 -KK 2 -x 2 y 2 <K 2 0x 2 =#x 2 =#x 2 =#x 2 =#Fx 2 }%y 2 }%y 2 }%y 2 }%^y 2 t( q 2 t(q Times New Romanh- 2 C 2p 2 i2p 2 2p 2 ] 1p 2 ]1p 2 ]11pp 2 :1p 2 m1p 2 1p 2 2p 2 H2p 2 21pp 2 #1p 2 #2p 2 #f3p 2 %1p 2 %2p 2 %r3pTimes New Roman - 2 =#?1 2 }%W1PSymbol- 2  - 2 >- 2 - 2  - 2 x- 2 - 2 =E- 2 -- 2 - 2 =#+ 2 =#+ 2 =#"+ 2 =#6- 2 }%+ 2 }%+ 2 }%.+ 2 }%N- 2 t(- 2 t(+ PSymbol- 2 ] -{ 2 ]6-{ 2 ]b-{Times New Romanh- 2 {[~ 2 ][~~ 2 6]}~ 2 ( 2 ) 2 <( 2  ) 2 d!/k 2 5.` 2 5.` 2 5.` 2 q {[~ 2 ][~~ 2 ]}~ 2 ( 2 ) 2 ( 2 *") 2 #/k 2 m5.` 2 =#.` 2 =#Z.` 2 =#.` 2 }%.` 2 }%f.` 2 }%.` Times New Roman - 2 ,8 2 ]#,8PSymbol- 2 b 2 '"l 2 b 2 #l 2 D 2 n DTimes New Roman - 2 t(N 2 t(H 2 t(RH Times New Romanh- 2 (i> 2 'GL| 2 (i> 2 'V 2 (`i> 2 'L| 2 )i> 2 &_np 2 *s&ncpcTimes New Roman - 2 t(6( 2 t(I) Times New Romanh- 2 *J(,8PSymbol- 2 t(- 2  2 ) 2 $ 2  2  2 w 2  2 Y  2  2 ;  2  2  2  2  2 p 2  2 R 2  2 4 2  2  2  2  2 i! 2 " 2 K$ 2 % 2 -' 2 ( 2 ^% 2 )^% 2 $^% 2 ^% 2 ^% 2 w^% 2 ^% 2 Y ^% 2 ^% 2 ; ^% 2 ^% 2 ^% 2 ^% 2 ^% 2 p^% 2 ^% 2 R^% 2 ^% 2 4^% 2 ^% 2 ^% 2 ^% 2 ^% 2 i!^% 2 "^% 2 K$^% 2 %^% 2 -'^% 2 (^% 2 m)= PSymbol- 2 )S={ 2 *\'+{PSymbol- 2 ( Times New Romanh- 2 )1p 2 *&3p 2 *'1p 2 *}(1pTimes New Roman - 2 *0 & "Systemn-ObjInfo)F Equation Native G _905675146#2,FoggrggOle h t)=N 1GL -c tl {[b][k L ]} 1,j (x jI -x jL )-x 1 (Dq)/l x N 2GL -c tl {[b][k L ]} 2,j (x jI -x jL )-x 2 (Dq)/l x ...N nc-1GL -c tl {[b][k L ]} nc-1,j (x jI -x jL )-x n-11 (Dq)/l x N 1GL -c tg {[b][k G ]} 1,j (y jV -y jI )-y 1 (Dq)/l y N 2GL -c tg {[b][k G ]} 2,j (y jV -y jI )-y 2 (Dq)/l y ...N nc-1GL -c tg {[b][k G ]} nc-1,j (y jV -y jI )-y n-11 (Dq)/l y y 1I -K 1 x 1I y 2I -K 2 x 21I .y ncI -K nc x ncI x 1I +x 2I +x 3I +...x ncI -1y 1I +y 2I +y 3I +...y ncI -1q V -q L +N iGL (H iV -H iL ) i=1n  [] 3nc+1,1 =0L   .1  &@3 & MathType`PSymbol-2 ,(PSymbol-2 ])LPSymbol-PIC +.i LMETA k 8CompObj-0| fObjInfo~ 2  (LPSymbol-2 )Times New Roman- 2 N 2 Wc 2 ' k 2  x Times New Roman- 2 -LS|p 2 t> 2 LS|p 2  LS|pPSymbol- 2 ,=Times New Roman- 2 h[~ 2  ][~~ 2 ) ]~ Times New Roman- 2  .8PSymbol- 2 b 2 " D & "Systemn- FMicrosoft Equation 2.0 DS Equation Equation.29q N LS ()=c t Ole10Native/1 Equation Native  _9056858904FggggOle   [b LS ][k LS. ]Dx() changes to Equation in PP3TTRIֈhiJmJiJ N LS ()=c t [b LS ][k LS. ]Dx()Lh PIC 36 LMETA  CompObj57 fObjInfo8   .1  @&; & MathType`Times New Roman- 2 4( 2 ) 2  ( 2 j)Times New Roman- 2 N 2 ;M@ 2  N 2  M@ Times New Roman- 2 i> 2 i> 2 L| 2 Sp 2 , i> 2 i> 2 G 2 USp PSymbol- 2 %-{ 2 -{PSymbol- 2 =PSymbol- 2  2 6  & "Systemn- FMicrosoft Equation 2.0 DS Equation Equation.29q՜ fObjInfo? Equation Native  Times New Roman$]- 2 i> 2 i> 2 L| 2 Sp 2 g CE 2 i> 2 Ti> 2 VG 2 Sp 2 CE PSymbol- 2 1-{ 2 -{PSymbol- 2 # = 2 -PSymbol- 2  2  PSymbol- 2 }h 2 hTimes New RomanLa- 2 1 & "Systemn- FMicrosoft Equation 3.0 DS Equation Equation.39qbI gI (N i M i / Li ) L"S =  " (N i M i / Li ) G"S "LU1;S0/ FMicrosoft Equation 2.0 DS Equation Equation.29q_906528085*nBF`,gg@ggOle  PIC AD LMETA $9U1;S  .1  K,&,%K & MathType "-, Times New Romanp- 2 &LF 2 &kz 2 &tk 2 8 N 2 c 2 k 2  x 2 x 2 x 2 % q 2  N 2  c 2 _k 2  x 2 x 2 x 2 H q 2 , rN 2 , c 2 , Qk 2 , Sx 2 , ]x 2 , x 2 , !q 2 l N 2 lc 2 l7B 2 lOx 2  N 2 c Times New Roman- 2 &GLS|p 2 $ LS|p 2 0tl>> 2 0LL| 2 0j> 2 0j> 2 $L| 2 0j> 2 $IJ 2 0#xc 2 dd LS|p 2 ptl>> 2 p)L| 2 pj> 2 pj> 2 dL| 2 pj> 2 dIJ 2 p#xc 2 ncpc 2 LS|p 2 tl>> 2 L| 2 ncpc 2 ?j> 2 'j> 2 *L| 2 1j> 2 -IJ 2 xncpc 2 W%xc 2 0CP 2 tCP> 2 %j> 2 eLSI|pJ 2 CP 2 ytCP>Times New Romanp- 2 &( 2 & ,` 2 &) 2 `{[~ 2 n][~~ 2 ]}~ 2 g( 2 q) 2 ( 2  ) 2 !/k 2 ={[~ 2 K][~~ 2 ]}~ 2 g( 2 q) 2 ( 2 !) 2 !/k 2 P.` 2 .` 2 .` 2 , /{[~ 2 , =][~~ 2 , ]}~ 2 , ( 2 , ) 2 , l ( 2 , ") 2 , #/k 2 l {[~ 2 l][~~ 2 l-]~ 2 l[~ 2 l&]}~ 2 l( 2 l ) 2 {[~ Times New Roman- 2 0,8 2 p,8 2 ,8 2 ,8PSymbol- 2 & = 2  - 2 - 2 H- 2  - 2 - 2 H- 2 , - 2 , !- 2 , - 2 l- 2 l  2 - PSymbol- 2 r -{ 2 -{ 2 b-{ 2 -{ Times New Romanp- 2 0@ 1p 2 01p 2 0 1p 2 p+ 2p 2 p2p 2 p.2p 2 1p 2 f1p 2 1p 2  1p 2 /1p 2 L1p 2  2pPSymbol- 2 b 2 "l 2 tb 2 "l 2 , fb 2 , ]$l 2 l@b 2 .b 2 ;D 2 ^D 2 , !D 2 l+GM..Times New Romanp- 2 ][~~ 2 ]~ 2 [~ 2 ]}~ 2 ( 2  ) 2 .` 2 .` 2 .` 2 Y{[~ 2 Y][[~~~ 2 YP]~ 2 Y[~ 2 YI]}~ 2 Y5( 2 Y}) 2  {[~ 2 ][~~ 2 \]}~ 2 ( 2 ) 2 F( 2 ) 2 q /k 2  {[~ 2 ][~~ 2 \]}~ 2 ( 2 ') 2 ( 2 ) 2  /k 2 $6{[~ 2 $D][~~ 2 $]}~ 2 $( 2 $) 2 $ ( Times New Roman- 2 ,8 2 U,8 2 ^,8 2 ,8 2 (%,8Times New Romanp- 2 %B 2 `x 2 YN 2 Y c 2 YZB 2 Yx 2 N 2  c 2 k 2 {y 2 y 2 #y 2 q 2 N 2  c 2 k 2 y 2 y 2 Fy 2  q 2 $N 2 $ c 2 $Xk 2 ${y 2 $y 2 $#y 2 $?"q Times New Roman- 2 6j> 2 vLSI|pJ 2 - ncpc 2 Q CP 2 0 tCP> 2 }ncpc 2 j> 2 LSI|pJ 2 RQ GSp 2 ^ tG> 2 ^G 2 ^[j> 2 ^Oj> 2 R=G 2 ^zj> 2 RvIJ 2 ^8"yc 2 Q GSp 2  tG> 2 G 2 ~j> 2 rj> 2 `G 2 j> 2 IJ 2 ~"yc 2 (%- ncpc 2 $Q GSp 2 (%0 tG> 2 (% G 2 (%ncpc 2 (%[j> 2 (%Oj> 2 $=G 2 (%zj> 2 $vIJ 2 (%ncpc PSymbol- 2 -{ 2  -{ 2 -{ 2 g-{ 2 (% -{ 2 (%-{ 2 (%-{PSymbol- 2 1 2 Yn - 2 Y 2  - 2 ^- 2 - 2  - 2 - 2 - 2 $n - 2 $^- 2 $- Times New Romanp- 2 1p 2 H2p 2  1p 2 R1p 2 1p 2 ^ 1p 2 ^1p 2 ^1p 2 - 2p 2 2p 2 2p 2 (% 1p 2 (%1p 2 (%: 1pPSymbol- 2 G 2 YNG 2 D 2  D 2 $U!D 2 Yb 2 b 2 7!l 2 b 2 }!l 2 $mb-&>'>(ab*a/1Times New Romanp- 2 $#) 2 $#/k 2 m'*{[~ 2 m'8][~~ 2 m'N]~ 2 m'[~ 2 m'G]}~ 2 m'( 2 m') 2 **{[~ 2 *8][~~ 2 *N]~ 2 *[~ 2 *G]}~ 2 *( 2 *3) 2 71L{[~ 2 71Z][[~~~ 2 71]~ 2 71_[~ 2 71]}~ 2 71q( 2 71) 2 X3( 2 X3a) 2 H5( 2 H5) 2 7.` 2 9( 2 9') 2 ;( 2 ;D.` 2 ;.` 2 ;.` 2 ;B) 2 |=( Times New Roman- 2 ',8 2 r*,8 2 1,8PSymbol- 2 $$l 2 m'ab 2 *ab 2 71b Times New Romanp- 2 (%%yc 2 &Q CP 2 ' tCP> 2 'Fj> 2 'qGSIpJ 2 f)Q CP 2 r* tCP> 2 r*ij> 2 *GSIpJ 2 1- np 2 0Q CP 2 1 tCP> 2 1np 2 1j> 2 1*GSIpJ 2 2{IJ 2 2IJ 2 3GSp 2 4HIJ 2 4IJ 2 5&GSp 2 x9ncpc 2 l8(IJ 2 x9ncpc 2 x9)ncpc 2 l8OIJ 2 x9GSp 2 \:IJ 2 \:IJ 2 \:IJ 2 h;Dncpc 2 \:jIJ 2 h;GSp 2 <IJ 2 <IJTimes New Roman- 2 m'N 2 m' c 2 m'XB 2 m'px 2 *N 2 * c 2 *XB 2 *x 2 71N 2 710 c 2 71B 2 71)x 2 X3y 2 X3K 2 X3x 2 H5xy 2 H5_K 2 H5x 2 9Xy 2 9K 2 9x 2 ;7x 2 ; x 2 ; x 2 ;x 2 |=7x 2 |= x 2 |= x Times New Romanp- 2 ' 1p 2 &P1p 2 'm1p 2 r*- 2p 2 f)P1p 2 r*{2p 2 18 1p 2 01p 2 1&1p 2 3G1p 2 31p 2 31p 2 5"2p 2 5]2p 2 521pp 2 h;1p 2 h;2p 2 h;3p 2 =1p 2 =2p 2 =3pTimes New Roman- 2 ;x1PSymbol- 2 m' - 2 m'A 2 * - 2 *d 2 71 - 2 71 2 X3V- 2 H5#- 2 9M- 2 ;+ 2 ;+ 2 ;p+ 2 ;l- 2 |=+ 2 |=+ PSymbol- 2 &-{ 2 f)-{ 2 1 -{ 2 0m-{ 2 1-{PSymbol- 2 m'LG 2 *LG 2 71G Times New Romanp- 2 <IJ 2 =Dncpc 2 IJ 2 >IJ 2 >IJ 2 ?\ncpc 2 >IJ 2 ?GSp 2 BG 2 B"Sp 2 B<i> 2 B_i> 2 AG 2 Bi> 2 A(Sp 2 Cmi> 2 @ncpc 2 &G)L| 2 &G Sp 2 &G'i> 2 &GJi> 2 FL| 2 &Gi> 2 FSp 2 &HXi> 2 Encpc 2 Ji> 2 Ji> 2 JLSI|pJ 2 Jvi> 2 JEi> 2 JIGSIpJTimes New Roman- 2 |=x 2 l?+y 2 l? y 2 l?y 2 l?y 2 WBRq 2 WBYq 2 WB-N 2 WBPH 2 WBH 2 FRq 2 FDq 2 FN 2 F;H 2 FH 2 UJN 2 UJ]M@ 2 UJgN 2 UJM@PSymbol- 2 |=p+ 2 |=\- 2 l?+ 2 l?+ 2 l?|+ 2 l?- 2 WB5- 2 WB+ 2 WB- 2 F - 2 F+ 2 F- 2 UJ= 2 _ 2 J_ 2 /_ 2 _ 2 _ 2 _ 2 _ 2 d _ 2 _ 2 F _ 2 _ 2 (_ 2 _ 2 _ 2 {_ 2 _ 2 ]_ 2 _ 2 ?_ 2 _ 2 !_ 2 _ 2  _ 2 t!_ 2 "_ 2 V$_ 2 %_ 2 8'_ 2 (_ 2 *_ 2 +_ 2 ,_ 2 m._ 2 /_ 2 O1_ 2 2_ 2 14_ 2 5_ 2 7_ 2 8_ 2 9_ 2 f;_ 2 <_ 2 H>_ 2 ?_ 2 *A_ 2 B_ 2 D_ 2 }E_ 2 F_ 2 _H_ 2 I_ 2 M& 2 JM& 2 /M& 2 M& 2 M& PSymbol- 2 C={ 2 &H={PSymbol- 2 BC 2 G. 2 J} 2 JTimes New Romanp- 2 |=D.` 2 |=.` 2 |=.` 2 |=B) 2 l?}( 2 l?P.` 2 l?.` 2 l?.` 2 l?Z) 2 WB( 2 WB) 2 F( 2 F) 2 UJR( 2 UJ) 2 UJ( 2 UJ) 2 |=h1 2 l?1 Times New Roman- 2 ?1p 2 ?2p 2 ?3p 2 CO1p 2 &H:1pPSymbol- 2 M& 2 M& 2 d M& 2 M& 2 F M& 2 M& 2 (M& 2 M& 2 M& 2 {M& 2 M& 2 ]M& 2 M& 2 ?M& 2 M& 2 !M& 2 M& 2  M& 2 t!M& 2 "M& 2 V$M& 2 %M& 2 8'M& 2 (M& 2 *M& 2 +M& 2 ,M& 2 m.M& 2 /M& 2 O1M& 2 2M& 2 14M& 2 5M& 2 7M& 2 8M& 2 9M& 2 f;M& 2 M& 2 ?M& 2 *AM& 2 BM& 2 DM& 2 }EM& 2 FM& 2 _HM& 2 IM& 2 &*= PSymbol- 2 KF(+{ Times New Romanp- 2 K&5p 2 K(2p 2 K)1pTimes New Roman- 2 &+0 Times New Romanp- 2 K\'ncpc Times New Roman- 2 KX),8 & "Systemn-CompObjCE fObjInfoF Equation Native  _953938618lNIF`gg@_ggf hJ(mJ0iJ F GLS (z,t)=N 1LS -c tl {[b][k L ]} 1,j (x jL -x jI )-x 1 (Dq)/l x N 2LS -c tl {[b][k L ]} 2,j (x jL -x jI )-x 2 (Dq)/l x ...N nc-1LS -c tl {[b][k L ]} nc-1,j (x jL -x jI )-x nc-1 (Dq)/l x N 1CP -c tCP {[b][B] -1 [G]} 1,j (xb LSI )N 2CP -c tCP {[b][B] -1 [G]} 2,j (xb LSI )...N nc-1CP -c tCP {[b][[B] -1 [G]} nc-1,j (xb LSI )N 1GS -c tG {[b][k G ]} 1,j (y jG -y jI )-y 1 (Dq)/l y N 2GS -c tG {[b][k G ]} 2,j (y jG -y jI )-y 2 (Dq)/l y N nc-1GS -c tG {[b][k G ]} nc-1,j (y jG -y jI )-y nc-1 (Dq)/l y N 1CP -c tCP {[b][B] -1 [G]} 1,j (xb GSI )N 2CP -c tCP {[b][B] -1 [G]} 2,j (xb GSI )N n-1CP -c tCP {[b][[B] -1 [G]} n-1,j (xb GSI )(y 1I -K 1 x 1I ) GS (y 2I -K 2 x 21I ) GS .(y ncI -K nc x ncI ) GS (x 1I +x 2I +x 3I +...x ncI ) GS -1(x 1I +x 2I +x 3I +...x ncI ) LS -1(y 1I +y 2I +y 3I +...y ncI ) GS -1q G -q S +N i (H iG -H iS ) i=1nc  q L -q S +N i (H iL -H iS ) i=1nc  (N i M i ) LSI =(N i  M i ) GSI  [] 5nc+2,1 =0 FMicrosoft Equation 3.0 DS Equation Equation.39qd  .1  @ Ole  CompObjHK fObjInfo OlePres000JL                     ! 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(T GSI "T S )+N iGS H iS (T S ) i=1n " FMicrosoft Equation 3.0 DS EqCompObj fObjInfo Equation Native  h_9539327829F كhgɝhguation Equation.39qLx}II E G =E S =E GS@LnE lnE ! .Ole  PIC  LMETA  LCompObj f1  &` & MathType "-@5PSymbol- 2 cW 2 Z 2 cL 2 ( 2 ~ n PSymbol- 2  a 2 b{Times New Roman - 2 cC 2 tk 2 c N 2 x 2  kC 2 C Times New Roman%- 2  iCP> 2 &iCP> 2 T i> 2 c iCP> 2 jCP>PSymbol- 2  + 2 : =Times New Roman%- 2 'C Times New Roman - 2 .'' 2 '' & "Systemn- FMicrosoft Equation 3.0 DS Equation Equation.39q  C iCP t+N iCP x=n i kC iCPa C jCPb t=0"xC iCP =0x=+LObjInfo Ole10Native $Equation Native  _953934384FɝhghgN iCP =e P N iLS x=-LN iCP =-e P a GS (a GS +a GL )N iVd&̌bI gI "C iCP "t+"N iCP "x'= i ROle  PIC  LMETA BCompObj fLES'  FMicrosoft Equation 3.0 DS Equation Equation.39qDbI 2 7zcWc 2 + ntp> 2 7zcWc 2 +ntp> 2 nj> 2 ) "j> 2 )"zcWc 2 !ntp> 2 )'j> 2 )A(zcWc 2 'ntp> 2 )+j> 2 )d,zcWc 2 +ntp> 2 bj> 2 n2zcWc 2 b2ntp> 2  tCP> 2 BzcWc 2  ntp> 2 zcWc 2 ntp> 2 # j> 2 K"j> 2 "zcWc 2 w9"ntp> 2 'j> 2 (zcWc 2 w'ntp> 2 ,j> 2 ,zcWc 2 w+ntp> 2 j>Times New Roman;- 2 ( 2 ,` 2 ) 2 {[~ 2 ][~~ 2 ]~ 2 _[~ 2 ]}~ 2  ( 2 <.) 2 %( 2 y() 2 h{[~ 2 h][~~ 2 h ]~ 2 h{[~ 2 h]}~ 2 # ( 2 #{.) 2 t%( 2 t() Times New Roman%- 2 77,8 2 7X,8 2 nw,8 2 )b",8 2 )',8 2 ),,8 2  ,8 2 t,8 2 ,8 2 ",8 2 6(,8 2 Y,,8PSymbol- 2 = 2 - 2 - 2 $- 2 )+ 2 1/+ 2 1- 2 h - 2 #%- 2 #)+ PSymbol- 2 ++{ 2 bo-{ 2 )|#+{ 2 "+{ 2 c(+{ 2 )4--{ 2 ,+{ 2 bt3+{ 2 d+{ 2 -{ 2 #+{ 2 w #+{ 2 w(+{ 2 s--{ 2 w,+{PSymbol- 2 D 2 1 Times New Roman;- 2 7 1p 2 +I1p 2 71p 2 b1p 2 n 1p 2 )#1p 2 Q#1p 2 (1p 2 )-1p 2  -1p 2 n )2p 2 n#11p 2 b31p 2 x 2p 2 1p 2 21pp 2  1p 2 62p 2 >$1p 2 w#1p 2 w%)1p 2 -1p 2 wH-1p 2 I)2pTimes New Roman%- 2 %2 2 #/&2PSymbol- 2 b 2 b0n 2 h b 2 G 2 L&D 2 hG 2 t&DzH z_z!z0 2 hp/+ 2 k- 2 - 2 &- 2 *+ 2 u0+ 2 ` 2 ` 2 )` 2 ` 2 ` 2 |` 2 ` 2 ^ ` 2 ` 2 @ ` 2 ` 2 "` 2 ` 2 ` 2 ?7 2 ?7 2 )?7 2 ?7 2 ?7 2 |?7 2 ?7 2 ^ ?7 2 ?7 2 @ ?7 2 ?7 2 "?7 2 ?7 2 ?7 2 == PSymbol- 2 3+{ 2 ' -{ 2  +{ 2 '6-{ 2 R:-{ 2 ^-{ 2 $+{ 2 $+{ 2 )+{ 2 x.-{ 2 -+{ 2 ^^3-{ 2 R16+{ 2 8-{ 2 ;-{PSymbol- 2 4PSymbol- 2 h0n 2 b 2 1n Times New Roman;- 2 p12p 2 Y41p 2 ' 1p 2  1p 2 '1p 2 R1p 2 ^P 1p 2 C%1p 2 $1p 2 **1p 2 .1p 2 M.1p 2 ^N*2p 2 ^31p 2 R61p 2 M91p 2 $<1pTimes New Roman%- 2 4'2 2 V>0Times New Roman;- 2 h2R 2 c 2 t x 2 x 2 1dtk 2 B 2 }"x 2 (x 2 5,x 2 z(zc 2 `4R Times New Roman%- 2 2zcWc 2 3ntp> 2 ^tCP> 2 ' ncpc 2 '= zcWc 2 5 ntp> 2 'Mncpc 2 'mzcWc 2 entp> 2 ^ncpc 2 ^(!j> 2 P#j> 2 #zcWc 2 >#ntp> 2 (j> 2 )zcWc 2 (ntp> 2 -j> 2 -zcWc 2 ,ntp> 2 Rj> 2 ^u2ncpc 2 ^@5zcWc 2 R]5ntp> 2 7ncpc 2 9ncatpcp>Times New Roman;- 2 I j.` 2 j.` 2 j.` 2 {[~ 2 ][~~ 2 ]~ 2 *[~ 2 ]}~ 2 !( 2 /) 2 &( 2 )) Times New Roman%- 2 ' ,8 2 '#,8 2 ^ ,8 2 #,8 2 ;),8 2 ^-,8 2 9,8PSymbol- 2 G 2 'D & "Systemn-ObjInfo Equation Native  `_953937279F3hghgOle                         ! " # $ % & ' ( ) * + , - . / 0 1 2 3 4 5 7 8 9 : ; < > B D G H I J K L M N O P Q R S T U V W X Y Z [ \ ] ^ _ ` a b c d e f g h i j m o r s t v w x { } x 1,zcnt+1 "x 1,zcnt dt"c tCP {[][B] "1 []} 1,j (x j,zc+1nt+1 "2x j,zcnt+1 +x j,zc"1nt+1 )(zc) 2j " + 1 R zcnt+1 c tCP x 2,zcnt+1 "x 21,zcnt dt"c tCP {[][B] "1 []} 2,j (x j,zc+1nt+1 "2x j,zcnt+1 +x j,zc"1nt+1 )(zc) 2j " + 2 R zcnt+1 ...c tCP x nc"1,zcnt+1 "x nc"1,zcnt dt"c tCP {[][B] "1 []} nc"1,j (x j,zc+1nt+1 "2x j,zcnt+1 +x j,zc"1nt+1 )(zc) 2j " + nc"1 R zcnt+1 [] nc"1,ncat"1 FMicrosoft Equation 3.0 DS Equation Equation.39qCompObj fObjInfo OlePres000 nEquation Native 6 #F  .1   &` ) & MathType "-@@Times New Roman-2 02 0R2 a2 NN2 a2 e N2 dt2 x2 x2 4c Times New Romane-2 12 Ynt 2 YApp2 12 *b2 12 ,2 a2 12 j2 iGS2 j2 yGS2 vi2 A b2  12  ,2 * a2  12  j2 lLS2 j2  LS2  i2 nt2 i2 12 nt2 i 2 tCPSymbol-2 n=2 n2 -2 -2 -2 - Symbol-2 +2 9=2 P =2 q+Symbol-2 t2   & "Systemn-̐bI`hI c tCP x int+1 "x int dt"N iLS a jLSj=1a,1b " "N iGS a jGSj=1a,1b " " 1 R Appnt+1 =0  FMicrosoft Word Picture MSWordDocWord.Picture.89q_953936899 Fhghg1Table[`*CompObj= hObjInfo?  [$@$NormalmH <A@<Default Paragraph Font 2Horu{~ &%$# *+ 2Horu{~       +,/Xr$/ig~hQYir$+.i@'+ V(  (T -x #   JA9991999?70%$  C BCDEF9991999?"" T $ <`x$xHxlxx DhxxT0 tP ,  T TEH@x}   JBCDEPFX9991999?$0<THxl DhlH$),@8!  JA9991999?90%`<  Hgֳgֳ999?-   Hgֳgֳ999?;    Hgֳgֳ999? xB   <1999?   PA9991999?70%   S BCDEF9991999?"" T $ <`x$xHxlxx DhxxT0 tP ,  T TEH@   PBCDEPFX9991999?$0<THxl DhlH$),@  Tgֳgֳ999?   Tgֳgֳ999? rB  B1999?  PA9991999?70%  S BCDEF9991999?"" T $ <`x$xHxlxx DhxxT0 tP ,  T TEH@   PBCDEPFX9991999?$0<THxl DhlH$),@  PA9991999?90%  T gֳgֳ999?    T gֳgֳ999?    T gֳgֳ999?  rB  B1999?   PA9991999?70%   S BCDEF9991999?"" T $ <`x$xHxlxx DhxxT0 tP ,  T TEH@   PBCDEPFX9991999?$0<THxl DhlH$),@   PA9991999?90%   T gֳgֳ999?    T gֳgֳ999?     Tgֳgֳ 999? rB ! B1999? " PA9991999?90% # Tgֳgֳ#999?  $ Tgֳgֳ$999?  % Tgֳgֳ%999?   & Tgֳgֳ&999?  * Ngֳgֳ*999?  + Ngֳgֳ+999? B S  ? +w T*OvT ZT& "!T#w ! T" ] T!  TR T; M T.! T  T T% ~|T C C Tc u T= T T % g T  TF T + T$Vd T l Q/ T= T09 :9 TY k T T [ T : T N! T"&@٣@GTimes New Roman5Symbol3& ArialABook Antiqua"qhhh$&jh$&!~0Preferred CustomerPreferred CustomerWordDocumentrSummaryInformation({DocumentSummaryInformation8_953940023GF@hg hgG bjbjَ ]"""RRRRR ^R'~~~~~w&w&w&&&&&&&&$')r&"w&"w&w&w&&&~~q~&&&w&@~"~&6Dw&&&&&j"&~r IfRR&& L CiCP G Type III: Both Sides Externally Dry Type II: Half Wetted Type I: Both Sides Externally Wetted G L CiCP G L CiCP G L CiCP (a) Single Half Wetted Pellet (b) Three Pellet Approach !"#'(*+PQfgB*CJhnH B*CJOJQJhnH B*CJ H*OJQJhnH B*CJ OJQJhnH  jUmH*!"'(*+APQfg{$!"'(*+APQfg{ &$ N N!"# $--%za/ig~hQY( """"za+.( Oh+'0l   ( 4 @LT\dssPreferred Customerref Normal.dotuPreferred Customer2efMicrosoft Word 8.0@G@f@,Bf՜.+,D՜.+,0 hp  CRELitl   Title 6> _PID_GUIDAN{0ACC9865-D267-11D1-A5F4-008029612A14}Ole @ PIC A LMETA CompObjC fLES'  FMicrosoft Equation 3.0 DS Equation Equation.39qbI gI F CAT (z,t)=c 1,ES   .1  @@?&? & MathType "-  .  /Times New Roman;- 2 LF 2 vz 2 tk 2  c 2 1 x 2 Rx 2 dtk 2 B 2 9!x 2 &x 2 *x 2 6'zc 2 1R 2 hO c 2 1 x 2 1x 2 tdtk 2 hB 2 #x!x 2 # 'x 2 #0+x 2 tu'zc Times New Roman%- 2 CAT| 2 nN tCP> 2 7zcWc 2 + ntp> 2 7zcWc 2 +ntp> 2 nj> 2 ) "j> 2 )"zcWc 2 !ntp> 2 )'j> 2 )A(zcWc 2 'ntp> 2 )+j> 2 )d,zcWc 2 +ntp> 2 bj> 2 n2zcWc 2 b2ntp> 2  tCP> 2 BzcWc 2  ntp> 2 zcWc 2 ntp> 2 # j> 2 K"j> 2 "zcWc 2 w9"ntp> 2 'j> 2 (zcWc 2 w'ntp> 2 ,j> 2 ,zcWc 2 w+ntp> 2 j>Times New Roman;- 2 ( 2 ,` 2 ) 2 {[~ 2 ][~~ 2 ]~ 2 _[~ 2 ]}~ 2  ( 2 <.) 2 %( 2 y() 2 h{[~ 2 h][~~ 2 h ]~ 2 h{[~ 2 h]}~ 2 # ( 2 #{.) 2 t%( 2 t() Times New Roman%- 2 77,8 2 7X,8 2 nw,8 2 )b",8 2 )',8 2 ),,8 2  ,8 2 t,8 2 ,8 2 ",8 2 6(,8 2 Y,,8PSymbol- 2 = 2 - 2 - 2 $- 2 )+ 2 1/+ 2 1- 2 h - 2 #%- 2 #)+ PSymbol- 2 ++{ 2 bo-{ 2 )|#+{ 2 "+{ 2 c(+{ 2 )4--{ 2 ,+{ 2 bt3+{ 2 d+{ 2 -{ 2 #+{ 2 w #+{ 2 w(+{ 2 s--{ 2 w,+{PSymbol- 2 D 2 1 Times New Roman;- 2 7 1p 2 +I1p 2 71p 2 b1p 2 n 1p 2 )#1p 2 Q#1p 2 (1p 2 )-1p 2  -1p 2 n )2p 2 n#11p 2 b31p 2 x 2p 2 1p 2 21pp 2  1p 2 62p 2 >$1p 2 w#1p 2 w%)1p 2 -1p 2 wH-1p 2 I)2pTimes New Roman%- 2 %2 2 #/&2PSymbol- 2 b 2 b0n 2 h b 2 G 2 L&D 2 hG 2 t&DzH z_z!z0 2 hp/+ 2 k- 2 - 2 &- 2 *+ 2 u0+ 2 ` 2 ` 2 )` 2 ` 2 ` 2 |` 2 ` 2 ^ ` 2 ` 2 @ ` 2 ` 2 "` 2 ` 2 ` 2 ?7 2 ?7 2 )?7 2 ?7 2 ?7 2 |?7 2 ?7 2 ^ ?7 2 ?7 2 @ ?7 2 ?7 2 "?7 2 ?7 2 ?7 2 == PSymbol- 2 3+{ 2 ' -{ 2  +{ 2 '6-{ 2 R:-{ 2 ^-{ 2 $+{ 2 $+{ 2 )+{ 2 x.-{ 2 -+{ 2 ^^3-{ 2 R16+{ 2 8-{ 2 ;-{PSymbol- 2 4PSymbol- 2 h0n 2 b 2 1n Times New Roman;- 2 p12p 2 Y41p 2 ' 1p 2  1p 2 '1p 2 R1p 2 ^P 1p 2 C%1p 2 $1p 2 **1p 2 .1p 2 M.1p 2 ^N*2p 2 ^31p 2 R61p 2 M91p 2 $<1pTimes New Roman%- 2 4'2 2 V>0Times New Roman;- 2 h2R 2 c 2 t x 2 x 2 1dtk 2 B 2 }"x 2 (x 2 5,x 2 z(zc 2 `4R Times New Roman%- 2 2zcWc 2 3ntp> 2 ^tCP> 2 ' ncpc 2 '= zcWc 2 5 ntp> 2 'Mncpc 2 'mzcWc 2 entp> 2 ^ncpc 2 ^(!j> 2 P#j> 2 #zcWc 2 >#ntp> 2 (j> 2 )zcWc 2 (ntp> 2 -j> 2 -zcWc 2 ,ntp> 2 Rj> 2 ^u2ncpc 2 ^@5zcWc 2 R]5ntp> 2 7ncpc 2 9ncatpcp>Times New Roman;- 2 I j.` 2 j.` 2 j.` 2 {[~ 2 ][~~ 2 ]~ 2 *[~ 2 ]}~ 2 !( 2 /) 2 &( 2 )) Times New Roman%- 2 ' ,8 2 '#,8 2 ^ ,8 2 #,8 2 ;),8 2 ^-,8 2 9,8PSymbol- 2 G 2 'D & "Systemn-ObjInfoE Equation Native F  _924505428F@WhgigOle k icnt+1 "c 1,icnt dt"{[][B] "1 []} 1,ispjj=1ncol " [B col ] ic,jj c isp,jjisp=1nc"1 " " 1 R icnt+1 (ic=1,2,....ncolforspecies1,pelletI)c 2,icnt+1 "c 2,icnt dt"{[][B] "1 []} 2,ispjj=1ncol " [B col ] ic,jj c isp,jjisp=1nc"1 " " 2 R icnt+1 (ic=1,2,....ncolforspecies2,pelletI)c nc"1,icnt+1 "c nc"1,icnt dt"{[][B] "1 []} 2,ispjj=1ncol " [B col ] ic,jj c isp,jjisp=1nc"1 " " nc"1 R icnt+1 (ic=1,2,....ncolforspecies2,pelletI)...ncol*(nc"1)equationsforeachpellet[] 3(nc"1)ncolLV/#7 FMicrosoft Excel ChartBiff5Excel.Chart.59qPIC l LBook zMETA  XPRINT(      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~     !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~ =B=,A =<X@"1Arial1Arial1Arial1Arial1Arial1Times New Roman1Symbol1Times New Roman1Times New Roman"$"#,##0_);\("$"#,##0\)"$"#,##0_);[Red]\("$"#,##0\) "$"#,##0.00_);\("$"#,##0.00\)%""$"#,##0.00_);[Red]\("$"#,##0.00\)5*2_("$"* #,##0_);_("$"* \(#,##0\);_("$"* "-"_);_(@_),))_(* #,##0_);_(* \(#,##0\);_(* "-"_);_(@_)=,:_("$"* #,##0.00_);_("$"* \(#,##0.00\);_("$"* "-"??_);_(@_)4+1_(* #,##0.00_);_(* \(#,##0.00\);_(* "-"??_);_(@_)                + ) , *   Chart2 pout3  =&APage &PMNHP LaserJet 4/4M Plus PSw odXXxQ!RdCustom page 1BBCustom page 2BBCustom page 3BB"dXX??3pout3Z3FF3Q z=0Q;HFQ;7HF3    ]  4H3    ]  43    ]  43    ]  4/3    ]  4E4FF3Q z=0.1Q;SHFQ;oHF3      4E4FF3Q z=0.2Q;HFQ;HF3      4E4FF3Q z=0.3Q;HFQ;HF3     4E4FF3Q z=0.4Q;HFQ;IF3      4E4FF3Q z=0.5Q;3IFQ;OIF3      4E4FF3Q z=0.6Q;kIFQ;IF3     4E4FF3Q z=0.7Q;IFQ;IF3      4E4FF3Q z=0.8Q;IF Q;IF3      4E4FF3Q z=0.9Q;JF Q;/JF 3     4E4FF3Q z=1.0Q;KJF Q;F 3      4E4D FAD 3OPX q 3*1@#43*#4%\GY3O5Q  time, (s)'4%,$3O!&P Q a-MS Concentration, (mol/m3)'4523  43" " w 3O" v%3O&Q44$%3O&Q43    ]  4444  ????????? ? ?????????? ? ?jt?jt?jt?jt?jt?jt?jt?jt?jt? jt? jt?5^I ?5^I ?5^I ?5^I ?5^I ?5^I ?5^I ?5^I ?5^I ? 5^I ? 5^I ?rh|?rh|?rh|?rh|?rh|?rh|?rh|?rh|?rh|? rh|? rh|?+?+?+?+?+?+?+?+?+? +? +?MbX9?MbX9?MbX9?MbX9?MbX9?MbX9?MbX9?MbX9?MbX9? 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G 2 .z L| 2 |i> 2 GSp 2 GSp 2 Ji> 2 .G 2 i> 2 LS|p 2 LS|p 2 8"i> 2 .L| 2 &L| 2 'iL>| 2 ?-L| 2 .iL>|Times New Roman - 2 }( 2 G) 2 8%( 2 () 2 +( 2 k/) 2 7( 2 9) Times New Roman- 2 '*p 2 .*pTimes New Roman - 2  1 2  1PSymbol- 2  r 2  r 2 r 2 r 2 u$ 2 %e 2 3& 2 + 2 j,e 2 , 2 3 2 d3 2 5w 2 %7 2 o8d 2 !8 2 F - 2 (+ 2 6+ 2 $#+ 2 )+ 2 k0= 2  5 2  5 2  5 2 : 2 : 2 :PSymbol- 2 h 2 Z 2 8PSymbol- 2 1D & "Systemn- FMicrosoft Equation 2.0 DS Equation Equation.29q(hJ\mJhJ N iGL a GL M i (1r G -1r L )+N iGS a GS M i r G +N iLS a LS M i r L    +(e Equation Native  D_902836997&4/F`lg lgOle  PIC .1 LL u iL* )z+(e L u iL* )z=Dtzw(dp)z()  @L\   .META  `CompObj02 fObjInfo3 Equation Native  1  @`&  & MathType "-q z Times New Roman- 2 4u 2 u 2 Qtk 2 ep 2 lz Times New Roman%- 2 iG> 2 np 2 iG> 2 v G 2 L| PSymbol- 2 +{PSymbol- 2 = 2 3- 2 e - 2  + Times New Roman- 2 1pTimes New Roman%- 2 e 1 Times New Roman- 2 *pTimes New Roman%- 2 ( 2 e ( 2 e ) 2 ) 2 eu( 2 e)PSymbol- 2 gD 2 e b 2  r 2 eb 2 $r 2 e 2 ed 2  & "Systemn- FMicrosoft Equation 2.0 DS Equation Equation.29qԴiJhmJdiJ u iGn+1 =u iG* -Dt((1-b)r G +br L )(dp)zLp   ._9028369986F lg lgOle  PIC 58 LMETA  `1  @&@ & MathType "-G  4Times New Roman(- 2 4u 2 u 2 'tk 2 ep 2 z Times New Roman- 2 iL>| 2 np 2 iL>| 2  L| 2 G PSymbol- 2 +{PSymbol- 2 = 2  - 2 e - 2  + Times New Roman(- 2 1pTimes New Roman- 2 e 1 Times New Roman(- 2 *pTimes New Roman- 2 ( 2 e[ ( 2 e ) 2 ) 2 e( 2 e)PSymbol- 2 =D 2 e a 2  r 2 ea 2 +r 2 e 2 ed 2  & "Systemn- FMicrosoft Equation 2.0 DS Equation Equation.29qCompObj79 fObjInfo: Equation Native  _949787502-L?F lg@HlgԴiJhmJdiJ u iLn+1 =u iL* -Dt((1-a)r L +ar G )(dp)zL+H+ ^ .Ole  PIC <> LMETA  CompObj=A f                    ! " # $ % & ' ( ) * + , - . / 0 1 2 3 4 5 6 7 8 : = > ? @ B C D E F G J O Q R S T U V W X Y Z [ \ ] ^ _ ` a b c d e f g h i j k m p q r s t u v w z | } ~  1  @'&`' & MathTypePSymbol-2  (PSymbol-2 ) "-9pPSymbol- 2 -r 2 e 2 r 2 :e 2 c r 2 P 2 Ve 2 ,) Times New Roman- 2 G 2 wG 2 twnp 2 G 2 G 2 tnp 2 8 G 2 LSIGJ 2 LG 2 i> 2 tGL| 2 GL| 2 i> 2 {!i> 2 t!GSp 2 #GSp 2 9&i>Times New Roman- 2 * tk 2 u 2 ,z 2 N 2 >a 2 [M@ 2 m N 2 "a 2 $M@ PSymbol- 2 t+{PSymbol- 2 q= 2  + 2 - 2 `- 2   2   2   2   2 & 2 & 2 & 2 &PSymbol- 2 V 2 V Times New Roman P- 2 t1pPSymbol- 2 @ D & "Systemn- FMicrosoft Equation 3.0 DS Equation Equation.39q r G e G t+r G u IG e GObjInfo Ole10Native@B Ole10ItemNameC Equation Native  t ()z=-N iV a GL M i ?DBAllXhJlmJXiJ r G e Gn+1 =r G e Gn +Dtr G u IG e G ()z-N iGL a GL M i  -N iGS a GS M i  ()PL*LH* ^ .1  @&&& & MathTypePSymbol-_949787520FF@Hlg@HlgOle  PIC EH LMETA  2 (PSymbol-2 S) "- PSymbol- 2 -r 2 e 2 r 2 e 2 $ r 2   2 e 2 , Times New Roman - 2 L| 2 wL| 2 tbnp 2 L| 2 L| 2 tnp 2  L| 2 LILJ| 2 LL| 2 Vi> 2 tGL| 2 nGL| 2 i> 2  i> 2 t*!LS|p 2 #LS|p 2 %i>Times New Roman(~- 2  tk 2 <u 2 ,tz 2 HN 2 a 2 M@ 2 N 2 1"a 2 >$M@ PSymbol- 2 t+{PSymbol- 2 \= 2 + 2 :+ 2 - 2 r  2 r  2 r  2 r  2 % 2 % 2 % 2 %PSymbol- 2 V` 2 V Times New Roman(~- 2 tl1pPSymbol- 2  D & "Systemn- FMicrosoft Equation 3.0 DS Equation Equation.39q r L e L t+r L u IL e L ()z=+N iL a GL M i CompObjGJ9 fObjInfo; Ole10NativeIK< Equation Native A Tא@J̑J  L  Ln+1 = L  Ln +t" L u IL  L ()"z+N iGL a GL M i " "N iLS a LS M i " ()_950448951DQNF@Hlg`lgOle H CompObjMOI fObjInfoPK  FMicrosoft Equation 3.0 DS Equation Equation.39qlI܂I CL4Equation Native L $_950454133XSF`lg`lgOle M PIC RUN LMETA P CompObjTVl fObjInfoWn Equation Native o 4 w .1   @/&/ & MathType "-<<W<<!PSymbol- 2 8e 2 r 2 de 2 r 2 <e 2 r 2  2 m 2 e 2 ?r 2 e 2  2 I Times New Roman - 2  L| 2 }L| 2 iL>| 2 7L| 2 L| 2  iL>| 2  np 2 L| 2 L| 2 iL>| 2 np 2 `iL>| 2 ~np 2 L| 2 +L| 2 L| 2 q np 2 #D 2 $Liq|>p 2 #np 2 )iG> 2 )np 2 -iL>| 2 7-npTimes New Roman=- 2  u 2 6 u 2 = tk 2 u 2 u 2 +z 2 g 2 p 2  z 2 "F 2 'K 2 -)u 2 h,u Times New Roman - 2 *p 2 @$,8Times New Roman=- 2  ( 2 (( 2 .))PSymbol- 2 = 2 ! + 2 Y- 2 + 2 - 2 ~!- 2 I&+ 2 A+- 2 S D & "Systemn- FMicrosoft Equation 3.0 DS Equation Equation.39qbI gI  L  L u iL* = L  L u iLn +t(" L  L u iLn "u iLn "z+ L  L g" L "p n "z"F D,Liqn +K(u iGn "u iLn ))_950470003ZF`lg lgOle x PIC Y\y LMETA { L66 w .1   1&0 & MathType "-<<}<2 <"PSymbol- 2 8e 2 r 2 e 2 & r 2 e 2 Rr 2  2 ~ 2  e 2 zr 2 me 2 I  2   Times New Roman- 2 G 2 }G 2 iG> 2 vG 2 G 2 e iG> 2  np 2 G 2 )G 2 iG> 2 np 2 \iG> 2 znp 2 G 2 QG 2 +G 2 !np 2 $D 2 %GaspW 2 9%np 2 `+iG> 2 ~+np 2 .iG> 2 .npTimes New Roman- 2 4u 2  u 2  tk 2 u 2 u 2 <z 2  g 2 !p 2 l!z 2 "$F 2  )K 2 *u 2 -u Times New Roman- 2 *p 2 %,8Times New Roman- 2 o( 2 %*( 2 /))PSymbol- 2 j= 2  + 2 - 2 + 2 B- 2 "- 2 '- 2 ,- 2  D & "Systemn- FMicrosoft Equation 3.0 DS EqCompObj[] fObjInfo^ Equation Native  WordDocumentF=uation Equation.39qbIgI  G  G u iG* = G  G u iGn +t(" G  G u iGn "u iGn "z+ G  G g" G "p n "z"F D,Gasn "K(u iGn "u iGn ))Oh+'0 ( D P \ ht|ssPreferred Customerrefref Normal.dotuPreferreG bjbjَ (p]NNN~~~~~~8SSSFKLL$!rNSSSS8SXN8bpSN8 @pQ0G~~X Step in time Pressure Convergence Check (uIL, and uIG)  EMBED Equation.2   EMBED Equation.2   EMBED Equation.2  t=t0, dt=dorig  EMBED Equation.2   EMBED Equation.2  T=t+dt/m (m=1 for full step, m=2 for half step) EMBED Equation  EMBED Equation  Compare dt and 2*(dt/2) Step Profiles t _PID_GUIDAN{FC2DC786-B2AD-11D1-A5F3-008029612A14} FMicrosoft Equation Equation Equation9qSW`  .1    CompObjce ]OlePres000f Ole10Native _906628956iFDlglg& MathType@Times New RomanM-!(!)![ !](!)!J!c!k !x"!xTimes New Roman-!L!t!L!L !I !LSymbol-!= !-Symbol-!"System- (J L )=c tL [k L ](x I -x L )======================[>==============================================================================================>=======>=>!5-]gk`!!5-]gk&+C\xkAߛm 6uRbR۴QZR+".Qy^6Ɏӓur*eԭKE*>FKc_('](2D[5~S^  5創sx?uZ'='zaI I I?C!~_vsz_yޱ_@og@H4B;8)A}Tɢ:Hw*o#T洑R^>]x+ye;ߛaWF+ICp*IIV? 1dԗM ;kI&p56 Equation k LS =ShD mA d pv h CE ifRe G Re L p0.02NABL METAFILEPICTf1  .1  @ & MathType-`V  Times New Roman-!k!ShDv!d!if0Times New Roman-!LSc!mA);!pv)!CEcf !G) !LUSymbol-!=C!h% Times New Roman-!Re1 !Reh !. 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TbkM( į23c?p.(6QbD@\̈ LLLą-mQ|Y-,q{d5_R=,q@rH@#e`NLLJ% lav[ Equation l y =l i y ii=1n & MaIBTO METAFILEPICTgHgh!  .1   & MathType`Symbol-!l` !l Times New RomanM-!y!i!in!i!n`mTimes New Roman-!y` Symbol-!= Symbol-!=;Symbol-!> Times New RomanM-!1"System- v Dd<  C A2z[ψ"(aKj`!bz[ψ"(aK& @20xcdd`` @b1##X`=FI!#T57Lvn K7$# X^)IAl1L@(\W(Yrb&~IQl Ad++&10Bl`ɦJ?mDf&~99p<䲉. ŌX]D\b'WٜLPs%=OQcǂ e`-ES.#RpeqIj.{t\v Equation N iV =J iV -y i L k J kVk=1n-1  +y i Dql yNABL METAFILEPICT H>0  .1   & MathType-iR8Times New Roman-!N"!J{!y !J!yq!qTTimes New Roman-!ih!V!i!V+!i !kj !kn !V!k!n!i !ySymbol-!=6!-!+n Symbol-!=f !-Symbol-! Symbol-!L!D Times New Roman-!1 !1Symbol-!l"System-Dd0 <  C A2>A!%?ED&."%e`!>A!%?ED&."%:xO1N@\ &,(wL0I q͉82:8qF^K _Mzm~ym} a$wgI 9Ddx T<  C A2u 0yu<]%`!u 0yu<4 XXJPxcdd`` @b1##X`=FI!#T57Ln K7$# ,/´lo`&0Z/ZZ O(Y4. &"~ &">",'ݜ7p{PXa(a(0J$>l0BB?f#RaH&20m&8Hlnrcw⇉{ Z\IpenR~,!gY^.8B [,+KRs] V QN Equation y iI =K i x iI   & MaIBTO METAFILEPICTH4n:   .1   & MathType@8Times New RomanM-!y8!K!x Times New Roman-!i !I2!iG!it!ISymbol-!="System-....*****.77KK:)*+C..,***K55:KG*K&&Dd@<  C A2a=( kׄ=*`!5=( kׄL xcdd``bd``beV dX,$XĐ h 7XRYaj`: sB2sSRs|Aف"+#C%܅`a( Y`HAmVOf͌` 2LDɈ)afN'Fܤ;d[,OLLJ% lde66  Equation h L. (T I -T L )+N iL H iL (T L ) i=1n  =h V. (T G -T I )+N iV H iV (T G ) i=1n  METAFILEPICT'h'hN@  .1  # & MathType-@ 9 `Times New Roman+-!h`0!T`!T`/ !N`L !H`!T`!h`!T`!T`!N`!H`!!T Times New Roman|-!L!I!LA !i !L^ !i !L!Lr!i!n+!V]!GB!I!i!V!i !V;"!G!ih!n!.e!.`|Times New Roman+-!(`t!)`E!(`!)`!(` !)` !(`H#!)`Symbol-!-`%!+`f!=`r!-`!+ Symbol-!=\!=?Symbol-!!B  Times New Roman+-!1!1"System-Ddl<  C A2!9v$/xs~qd5`!\!9v$/xs~qH#7@2*xcdd``^ $D@9@, fbd02,( 1dbfab`y d30&dT20\``Y}"Lgx@,a ˒&vU&vN"$vos&0 ZZǠ H+22d}2ɀ &"~>LE܇&"9Sgvŏ`wC9SfUO(D@B BDȊ]#7L<I|+c#7vqŮ~7?6E9p.b0L< E `WGzNIo)`W].f4 ypenR~9(樲ĵQa(Eb%͜9 DuČ[sbog/Ŝ+Rį2;qU $0CKC/V\121)W2N8 Equation` x iI =1  .1R METAFILEPICTh{5  .1  @ & MathType`Times New Roman+-!x Times New Roman-!i!ISymbol-!=Symbol-!Times New Roman+-!1"System-Ddph<  C A2e ,'l.|"A<`!9 ,'l.|"6@|xcdd``fd``beV dX,$XĐ ɁRcgaM,,He`Hˋ050a: s d++&1hJiEVF E0.bI P$3Lp &$Y`{6W&0] 1n-9% Lw>Eq0d .3kG,LLJ% la7QD Equation` y iI =1  sM#R METAFILEPICT3h3{4  .1  @ & MathType`Times New Roman-!y Times New Roman-!i!ISymbol-!=Symbol-!Times New Roman-!1"System-YYO 1V1/NU U /n/Ddh<  C A2f5SV-se&BA`!:5SV-se&6@|xcdd``fd``beV dX,$XĐ Ɂi+7Ole  PIC hk LMETA  0CompObjjl fL {Xh {  .1  @ &  & MathTypepTimes New Roman0- 2 `[~ 2 `]~ Times New Roman- 2 ,8PSymbol- 2 `b 2 `d Times New Roman0- 2 ri> 2 kc 2 -G 2 sik>c 2  i> 2  kcTimes New Roman- 2 `yPSymbol- 2 `j= 2 `- 2 ` L & "Systemn- FMicrosoft Equation 2.0 DS Equation Equation.29ql0iJ|mJiJ [b i,k ] G =d ik -y i L kL {hObjInfom Equation Native  _906629219gupFmgmgOle  PIC or LMETA  CompObjqs fObjInfot  { a .1  @ &`  & MathTypepPSymbol- 2 `>L Times New Roman8- 2 Zik>c 2 ikc 2 kncpc 2  ycPSymbol- 2 `= 2 `U-Times New Roman8- 2 `( 2 `m ) 2 `= /kPSymbol- 2 `wl 2 `yl 2 ` l & "Systemn- FMicrosoft Equation 2.0 DS Equation Equation.29qEquation Native  |_906629400wF?mgg"mgOle  PIC vy L`mJiJ,nJ L ik =(l k -l nc )/l yL ; .1  `& & MathTypePSymbol- META  CompObjxz fObjInfo{Equation Native h2 7l 2 Al Times New Roman8- 2 2yc 2 i> 2 *i> 2 i>Times New Roman- 2 QyPSymbol- 2 +=PSymbol- 2 q & "Systemn- FMicrosoft Equation 2.0 DS Eq      !"#$%&'()*,/0356789:;<=>?ADGIJKLMNOPQSVY[\]^_`abcdefgilmprstuvwxyz{|}~uation Equation.29qLmJiJ,nJ l y =y i l ii L\{ h\{  ._906629490@~Fg"mg@CmgOle PIC }LMETA h1  @`&  & MathTypePPSymbol- 2 7l Times New Roman@- 2 i> 2 i> 2 i> 2 5V 2  G 2  i> 2 L| 2 fL|Times New Roman - 2 \y 2 H 2  T 2  H 2 TPSymbol- 2 = 2 j -Times New Roman - 2 o( 2 (@a 2  ) 2 (@a 2 )) & "Systemn- FMicrosoft Equation 2.0 DS Equation Equation.29qCompObjfObjInfoEquation Native _906630962F@Cmg`KmgՐ$iJmJiJ l i =y i (H iV (@T G )-H iL (@T L ))L {Dh {  .1  @ &` Ole PIC LMETA 0CompObj+f & MathTypepTimes New Roman- 2 `[~ 2 `]~ Times New Roman- 2 ,8PSymbol- 2 `b 2 `d Times New Roman- 2 ri> 2 kc 2 BL| 2 ^ik>c 2 c i> 2  kcTimes New Roman- 2 `xPSymbol- 2 `U= 2 `- 2 ` L & "Systemn- FMicrosoft Equation 2.0 DS Equation Equation.29ql0iJ|mJiJ [b i,k ] L =d ik -x i L ObjInfo-Equation Native ._906630961F TmgSvmgOle 1kL 4@ 4 a .1   &@  & MathTypePPSymbol- 2 `>L Times New Roman-PIC 2LMETA 4CompObj@fObjInfoB 2 Uik>c 2 `kc 2 [ncpc 2  xcPSymbol- 2 `= 2 `L-Times New Roman- 2 `( 2 `X ) 2 `0 /kPSymbol- 2 `nl 2 `pl 2 ` l & "Systemn- FMicrosoft Equation 2.0 DS Equation Equation.29q`iJmJiJ L ik =(l k -l nc )/l xLEquation Native C|_906630960F}mg}mgOle EPIC FLMETA HCompObjRfObjInfoTEquation Native Uh ; .1  `& & MathTypePSymbol- 2 7l 2 Hl Times New Roman1- 2 0xc 2 i> 2 ,i> 2 i>Times New RomanE- 2 PxPSymbol- 2 .=PSymbol- 2 t & "Systemn- FMicrosoft Equation 2.0 DS Equation Equation.29qLDiJ|mJiJ l x =x i l ii _906630959FmgOmgOle WPIC XLMETA ZhL{ h{  .1  @ & & MathTypePPSymbol- 2 7l Times New RomanM- 2 i> 2 i> 2 i> 2 V 2  G 2  i> 2 L| 2 .L|Times New Roman- 2 Dx 2 H 2 T 2 u H 2 pTPSymbol- 2 = 2 < -Times New RomanM- 2 O( 2 (@a 2 h ) 2 l(@a 2 )) & "Systemn-CompObjhfObjInfojEquation Native k_954024882r@F`wmg@@mg FMicrosoft Equation 2.0 DS Equation Equation.29qՐiJ|mJtiJ l i =x i (H iV (@T G )-H iL (@T L ))Lh Ole nPIC oLMETA qCompObjf  .1  @&; & MathType`Times New Roman- 2 4( 2 ) 2  ( 2 j)Times New Roman- 2 N 2 ;M@ 2  N 2  M@ Times New Roman- 2 i> 2 i> 2 L| 2 Sp 2 , i> 2 i> 2 G 2 USp PSymbol- 2 %-{ 2 -{PSymbol- 2 =PSymbol- 2  2 6  & "Systemn- FMicrosoft Equation 3.0 DS Equation Equation.39qbI gI (N i M i / Li ) L"S =  " (N i M i / Li ) G"S "ObjInfoEquation Native _906632248F`mg`mgOle PIC LMETA 4CompObjfObjInfoL {h {  .1  @ &`  & MathTypepTimes New Roman8- 2 `[~ 2 `]~ Times New Roman- 2 ,8PSymbol- 2 `b 2 `$d Times New Roman8- 2 ri> 2 kc 2 4CP 2 ik>c 2  cic> 2  kcTimes New Roman- 2 `Q xPSymbol- 2 `= 2 `- 2 ` L & "Systemn- FMicrosoft Equation 2.0 DS Equation Equation.29qp(iJmJiJ [b i,k ] CP =d ik -x ci L kL4@Equation Native _906632247F mg smgOle PIC LMETA CompObjfObjInfoEquation Native |4 ~ .1   &  & MathTypePPSymbol- 2 `>L Times New Roman@- 2 ckc 2 kc 2 M ncpc 2  mxcTimes New Roman0- 2 `gM@ 2 `M@PSymbol- 2 `o= 2 `-Times New Roman0- 2 `( 2 `J ) 2 `" /kPSymbol- 2 ` l & "Systemn- FMicrosoft Equation 2.0 DS Equation Equation.29q`iJmJiJ L k =(M k -M nc )/l mxLU U  < .1  `` &  & MathTypePSymbol- _906632246F smgmgOle PIC LMETA 2 7l Times New Roman- 2 2mxc 2 i> 2 i> 2 i>Times New Roman!- 2 x 2 @M@PSymbol- 2 =PSymbol- 2  & "Systemn- FMicrosoft Equation 2.0 DS EqCompObjfObjInfoEquation Native l_906717206Fmg,nguation Equation.29qPiJtmJ|iJ l mx =x i M ii L {h {  .Ole PIC LMETA <CompObjf1  @ &  & MathTypePTimes New Roman0- 2 [~ 2 ]~ 2 t[~ 2  ]~ 2 [~ 2  ]~Times New Romanl- 2 k 2 B Times New Roman0- 2  GPSymbol- 2 R= PSymbol- 2 -{ Times New Romanl- 2 1pPSymbol- 2  G & "Systemn- FMicrosoft Equation 2.0 DS Equation Equation.29q:DmJiJ,nJ [k] G =[ObjInfoEquation Native `_906717320F,ng.ngOle B] -1 [G]L\  .1  `` &  & MathType "-  Times New RomanPIC LMETA CompObjfObjInfol- 2 [~ 2 ]~ Times New Roman0- 2 ,8 2 ),8 2  ,8Times New Romanl- 2 B 2 `y 2 k 2 `P y 2  k Times New Roman0- 2 i> 2 [j> 2 hi> 2 i> 2 dncpc 2  kc 2  i> 2 O kc 2  kcPSymbol- 2 = 2 +PSymbol- 2   & "Systemn- FMicrosoft Equation 2.0 DS Equation Equation.29q:՜mJiJ,nJ [B i,j ]Equation Native _906717830F.ng`?ngOle PIC L=y i k i,nc +y k k i,kk LfL f  .1  `& & MathType "-META  CompObjfObjInfoEquation Native    "$%&'()*+,-./1479:;<=>?@ABCDEFGILMPRSTUVWXYZ[\]^_`acfgjlmnopqrstuvx{~ n  Times New Roman0- 2 [~ 2 ]~ 2 (( 2  ) Times New Roman- 2 ,8 2 ,8 2  ,8Times New Roman0- 2 B 2 y 2 k 2  k Times New Roman- 2 i> 2 [j> 2 i> 2 i> 2 V j> 2 K i> 2  ncpcPSymbol- 2 = 2 - 2 A -Times New Roman0- 2 jw1 2 jX 1 & "Systemn- FMicrosoft Equation 2.0 DS Equation Equation.29q:ՌPiJxmJiJ [B i,j ]=-y i (1k i,j -1k i,nc )Lgg w ._906718361F`?ng PangOle  PIC  LMETA 1   &` & MathType "-<<ATimes New Romanl- 2 [~ 2 ]~ Times New Romanl- 2 ,8 2 M,8Times New Romanl- 2 k 2 oD 2 lk Times New Romanl- 2 oi> 2 *j> 2 i> 2 j>PSymbol- 2 = & "Systemn- FMicrosoft Equation 2.0 DS Equation Equation.29q:LmJiJ,nJ [k i,j ]=D i,j lCompObjfObjInfoEquation Native h_906720008F Pang@hngOle  PIC !LMETA #<CompObj0fL {h {  .1  @ &  & MathTypePTimes New Romanl- 2 [~ 2 ]~ 2 Z[~ 2 ]~ 2 y[~ 2  ]~Times New Romann- 2 k 2 B Times New Romanl- 2 !L|PSymbol- 2 8= PSymbol- 2 -{ Times New Romann- 2 1pPSymbol- 2 G & "Systemn- FMicrosoft Equation 2.0 DS Equation Equation.29q:DiJ|mJxiJ [k] L =[B] -1 [G]L\ObjInfo2Equation Native 3`_906720007F@hngKngOle 5PIC 6LMETA 8CompObjHfObjInfoJ  .1  `` &  & MathType "-  Times New Romanl- 2 [~ 2 ]~ Times New Roman0- 2 ,8 2 ),8 2  ,8Times New Romanl- 2 B 2 `y 2 k 2 `P y 2  k Times New Roman0- 2 i> 2 [j> 2 hi> 2 i> 2 dncpc 2  kc 2  i> 2 O kc 2  kcPSymbol- 2 = 2 +PSymbol- 2   & "Systemn- FMicrosoft Equation 2.0 DS Equation Equation.29q:՜mJiJ,nJ [B i,j ]=y i k i,nc +y k k i,kk LfL Equation Native K_906720006FKng 2 [j> 2 i> 2 i> 2 V j> 2 K i> 2  ncpcPSymbol- 2 = 2 - 2 A -Times New Roman0- 2 jw1 2 jX 1 & "Systemn- FMicrosoft Equation 2.0 DS Equation Equation.29q:ՌPiJxmJiJ [B i,j ]=-y i (1k i,j -1k i,nc )_906720004F 2 *j> 2 i> 2 j>PSymbol- 2 = & "Systemn- FMicrosoft Equation 2.0 DS EqCompObjwfObjInfoyEquation Native zh_914355460F@ng`nguation Equation.29q:LmJiJ,nJ [k i,j ]=D i,j lL\  .Ole |PIC }LMETA XCompObjf1  @` &  & MathType "-"  PSymbol- 2 7F% Times New Romanm- 2 uii>> 2 i> 2 t> 2 in>p 2  kc 2  t> 2  ik>cTimes New Roman!- 2 `N 2 c 2 k 2 ` N 2 6 c 2 Y kPSymbol- 2 = 2 $+PSymbol- 2 J & "Systemn- FMicrosoft Equation 2.0 DS Equation Equation.29qObjInfoEquation Native _914355459vF`ng ogOle cҨiJ|mJhiJ F ii =N i c t k in +N k c t k ik L  .PIC LMETA CompObjfObjInfo1  ``&  & MathType "-^   PSymbol- 2 7F% Times New Roman|- 2 eii>> 2 i> 2 t> 2 ij>> 2  t> 2  in>pTimes New Roman- 2 N 2 c 2 k 2  c 2  kPSymbol- 2 ~= 2 - 2  -Times New Roman|- 2 V( 2  ) 2 f1 2 f 1 & "Systemn- FMicrosoft Equation 2.0 DS Equation Equation.29qcҜhJtmJ`iJ F ii =-N i (1c t k ij -1c t k in )L'  .Equation Native _914355458F og!ogOle PIC LMETA CompObjfObjInfoEquation Native >>>>>> > > > >> >>>>>A>>>>>>>>>>>>> >!>">$>#>%>&>'>(>)>+>*>,>->.>/>0>1>2>4>3>5>B>@>7>8>9>:>;><>=>>>?>>>D>E>F>G>H>I>J>L>M>N>O>P>Q>R>T>U>V>W>X>Y>Z>\>]>^>_>`>a>b>c>d>e>f>g>h>i>j>k>l>m>n>o>p>q>r>s>t>u>v>w>x>y>z>{>|>}>~>1  @&4 & MathTypepTimes New Roman- 2 [~ 2 ]~ 2 [~ 2 ]~ 2 Z [~2  ][exp([~~~ 2 v])~ 2 [~ 2 !]]~~Times New Roman%- 2 k 2 Ek 2 sI Times New Roman-2  highfluxp>pp>>pc2 lowflux>p>>pcPSymbol- 2 = 2 - PSymbol- 2 --{PSymbol- 2  F% 2 HF% Times New Roman- 2 1p & "Systemn- FMicrosoft Equation 2.0 DS Equation Equation.29qcҤmJhJ$nJ [k] highflux =[k] lowflux [F][exp([F])-[I]] -1L_914355457 F4)og$CogOle PIC   LMETA x  .1    &`  & MathType "-3U Times New Romank- 2 4exp[~ 2 ]~ 2 j/[~ 2 j]~ 2 jF [~ 2 j ]~ 2 j /k 2 0[~ 2 ]~ 2 E [~ 2  ]~ 2  /kPSymbol- 2 F% 2 j F% 2  F% 2 = 2 j<+ 2 <-Times New Romank- 2 jI 2 ITimes New Roman- 2 j 2 2  2 & "Systemn- FMicrosoft Equation 2.0 DS Equation Equation.29qCompObj  fObjInfoEquation Native |_914355456FJogJogc`mJhJ$nJ exp[F]=[I]+[F]/2[I]-[F]/2L) T)  T .1   & ,Ole PIC LMETA CompObjf & MathType "- PSymbol- 2 7F% Times New Roman0- 2 |htp> 2 /Bi> 2 /pip>Times New Roman- 2 $N 2 C 2 (hPSymbol- 2 =PSymbol- 2 4 & "Systemn- FMicrosoft Equation 2.0 DS Equation Equation.29qc\mJhJ$nJ F ht =N i C pi  hL{ hObjInfoEquation Native x_914355455 FSog HogOle PIC LMETA CompObjfObjInfo{ z .1  @&` & MathTypepTimes New RomanT- 2 `4h 2 `h Times New Roman-2 highfluxp>pp>>pc2 ^lowflux>p>>pc 2 Z htp> 2 \htp>PSymbol- 2 `l= 2 `(- 2 ` F% 2 `F%Times New RomanT- 2 ` /k2 `` ((exp( 2 `T) 2 `) 2 `11 & "Systemn- FMicrosoft Equation 2.0 DS Equation Equation.29qEquation Native _954027788F Hog@ogOle CompObj f   "#&)*+.12589:;>ABEHILOPSVWZ]^_befgjmnopsvy|}cҜmJhJ$nJ h highflux =h lowflux F ht /((exp(F ht )-1) FMicrosoft Equation 3.0 DS Equation Equation.39qbI gI D ij0 ObjInfo!Equation Native _954027828'$F@ogogOle  =7.4x10 "12 ( j M j ) 1/2 T j V i0.6 FMicrosoft Equation 3.0 DS Equation Equation.39qTvI4yI D ij0 CompObj#% fObjInfo&Equation Native p_954028132)FogCog=1.013x10 "2 ((M j +M j )/M i M j ) 1/2 T 1.75 P(V i1/3 +V i1/3 ) 2, FMicrosoft Equation 3.0 DS Equation Equation.39qOle CompObj(*fObjInfo+Equation Native ӸTWIZI D ij =(D ij0 ) (1"x i ) (D ji0 ) (x i ) FMicrosoft Equation 3.0 DS Equation Equation.39q_954028347"6.FCogkogOle 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