chemical reactor modeling multiphase reactive flows

Chemical Reactor Modeling Hugo A Jakobsen Chemical Reactor Modeling Multiphase Reactive Flows Prof.Dr Hugo A Jakobsen Norwegian Univ of Science & Technology Dept of Chem Engineering N-7491 Trondheim Norway hugo.jakobsen@chemeng.ntnu.no ISBN: 978-3-540-25197-2 e-ISBN: 978-3-540-68622-4 Library of Congress Control Number: 2008924079 c 2008 Springer-Verlag Berlin Heidelberg This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer Violations are liable to prosecution under the German Copyright Law The use of general descriptive names, registered names, trademarks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use Cover design: eStudio Calamar, Girona, Spain Printed on acid-free paper springer.com To Sara Preface This book is based on lectures regularly taught in the fourth and fifth years graduate courses in transport phenomena and chemical reactor modeling, and in a post graduate course in modern reactor modeling at the Norwegian University of Science and Technology, Department of Chemical Engineering, Trondheim, Norway The objective of the book is to present the fundamentals of the single-fluid and multi-fluid models for the analysis of single- and multiphase reactive flows in chemical reactors with a chemical reactor engineering rather than mathematical bias Organized into 12 chapters, it combines theoretical aspects and practical applications and covers some of the recent research in several areas of chemical reactor engineering This book contains a survey of the modern literature in the field of chemical reactor modeling I hope this book can serve as a guide for my future Ph.D students, as well as other interested scientists, to get a thorough introduction to this field of research without spending too much of their invaluable time searching for and reading a large number of books and papers Comments on the contents of the book: In chap a survey of the elements of transport phenomena for single phase multicomponent mixtures is given This theory serves as basis for the development of most chemical engineering models as well as the multiphase flow concepts to be presented in the following chapters The first part of the chapter considers laminar single phase flows for multicomponent mixtures In the second part of the chapter the governing equations are applied to turbulent flows Chapter contains a summary of the basic concepts of kinetic theory of dilute and dense gases This theory serves as basis for the development of the continuum scale conservation equations by averaging the governing equations determining the discrete molecular scale phenomena This method is an alternative to, or rather both a verification and an extension of, the continuum approach described in chap These kinetic theory concepts also determine the basis for a group of models used describing granular flows, further outlined in chap A pedagogical advice basically for the students intending VIII Preface to obtain their very first overview of the content of reactor modeling on the graduate level may concentrate on the continuum formulations first and, if strictly needed, go back to the chapters that are dealing with kinetic theory (i.e., chaps and 4) after they feel confident with the continuum modeling concepts Chapter contains a survey of a large number of books and journal papers dealing with the basic theory of multi-fluid flow modeling Emphasis is placed on applying the multi-fluid model framework to describe reactive flows This is perhaps the main contribution in this book, as there exist no textbook on reactive multiphase flow modeling intended for reactor engineers In the more advanced textbooks the basic multicomponent multiphase theory is introduced in a rather mathematical context, thus there is a need for a less demanding presentation easily accessible for chemical reaction engineering students Chapter contains a summary of the basic theory of granular flow These concepts have been adopted describing particulate flows in fluidized bed reactors The theory was primarily used for dense bed reactors, but modified closures of this type have been employed for more dilute flows as well Compared to the continuum theory presented in the third chapter, the granular theory is considered more complex The main purpose of introducing this theory, in the context of reactor modeling, is to improve the description of the particle (e.g., catalyst) transport and distribution in the reactor system In chap an outline of the basic theory of the required closure laws and constitutive equations is provided The first section presents the closures related to averaged of products (i.e., the analogous to turbulence type of closures) The following sections describe models for the interfacial transport phenomena occurring in multiphase reactive systems An overview of the important models for the different forces acting on a single particle, bubble or droplet is given Model modifications due to swarm or cluster effects are discussed The standard theories for interfacial heat and mass transfer are examined In the last section the literature controversy originating from the fact that with the present level of knowledge, there is no general mathematical theory available to determine whether the 3D multi-fluid model is well posed as an initial-boundary value problem, is examined In chap the derivation of the classical reactor models is examined starting out from the microscopic heat and species mass balances In chemical reactor engineering the idealized models like the plug flow reactor (PFR) - and continuous stirred tank reactor (CSTR) models are well known from basic courses in chemical reaction engineering For non-ideal flows the dispersion models (DMs) are frequently used These standard models are deduced from the microscopic heat and species mass balances employing a cross-sectional area averaging procedure Similar, but not identical, models can be obtained by simplifying the governing microscopic transport equations In chap a brief summary of the agitation and fluid mixing technology is given The main emphasis is placed on examining the modern strategies used Preface IX to model the momentum transfer from the impeller to the fluid The methods are sketched and the basic equations are listed A few model simulation examples are presented In chap the basic bubble column constructions and the principles of operation of these reactors are described The classical models for two- and three phase simple bubble column reactors are defined based on heat and species mass balances The state of the art on fluid dynamic modeling of bubble column reactors is then summarized including a few simulations of reactive flows In chap an outline of the basic theory of the population balance equation is provided Three different modeling frameworks are defined, the macroscopic formulation, the microscopic continuum - and kinetic theory formulations The macroscopic model is formulated directly on the macroscopic scales, enabling a suitable framework for practical engineering calculations In this framework a simple and inaccurate numerical discretization scheme has become an integrated part of most closure laws Since the numerical discretization schemes cannot be split from the physical closure laws in a trivial manner, the more popular closures for bubble coalescence and breakage rates are discussed in this chapter as well The more rigorous microscopic formulations are presented and future reactor analysis should preferably be based on these concepts, enabling more accurate closure laws to be formulated and more optimized solution methods to be used The status on population balance modeling of bubble coalescence and breakage phenomena is summarized Chapter 10 contains a literature survey of the basic fluidized bed reactor designs, principles of operation and modeling The classical two- and three phase fluidized bed models for bubbling beds are defined based on heat and species mass balances The fluid dynamic models are based on kinetic theory of granular flow A reactive flow simulation of a particular sorption enhanced steam reforming process is assessed In chap 11 an overview of the basic designs, principles of operation, and modeling of fixed packed bed reactors is presented The basic theory is applied to describe the performance of particular chemical processes operated in fixed packed bed reactors That is, porous media reactive flow model simulations of particular packed bed sorption enhanced steam reforming processes are assessed In chap 12 a group of finite volume solution algorithms for solving the multi-fluid model equations is described The basic single phase finite volume method solution strategies, spatial discretization schemes, and ODE solution methods in time are examined The selected multiphase algorithms are extended versions of the single-phase SIMPLE-like algorithms However, alternative algorithms can be found in the literature Some of these methods are briefly outlined in this chapter Moreover, several numerical methods for solving the population balance equation for dispersed flows are outlined Finally, several solution methods for the resulting algebraic discretization equations are mentioned X Preface The book may be used as a reference book of the multi-fluid theory, or for teaching purposes at different educational levels For example, at the graduate level, an introductory graduate course in single phase transport phenomena can be based on chap (and parts of chap 2) Suitable numerical solution methods for the governing single phase equations can be found in chapter 12 An introduction to reactor modeling can be based on chaps 6-11 The material in chapters 2,3,4,5 and the multiphase parts of chap 12 may be better suited at the post graduate level Taking these three courses in sequence, I hope the PhD students get the necessary knowledge to give future contributions in this field of science I have received a great deal of help from numerous persons, over the nearly twenty years association with this subject, in formulating and revising my views on both reactor modeling and chemical reactor engineering I would like to acknowledge the inspiring discussions I have had with the colleagues at NTNU during my work on this book I am particularly incepted to the present and former members of the staff at the Chemical Engineering Department at NTNU In addition, I wish to thank the PhD students that have taken my graduate subjects and thus read the lecture notes carefully and supplied me with constructive criticisms (among other comments) and suggestions for further improvements on the text It is fair to mention that my students, especially Dr ing Carlos A Dorao, MSc H˚ avard Lindborg, MSc Hans Kristian Rusten and MSc Cecilie Gotaas Johnsen, have contributed to this book in many ways This includes technical contributions either in a direct or indirect way, and reading parts of the draft manuscript I must also thank Associate Professor Maria Fernandino for her valuable suggestions and comments regarding chapter Finally, my thoughts are due to my wife, Jana, who strongly believes quality is better than quantity Her reviews and criticism of the contents surely improved the book Trondheim, November 2007 Hugo A Jakobsen Nomenclature Latin Letters A Hamaker constant (J) A chemical component in general reaction A empirical model parameter (−) A macroscopic surface area defining the control volume (m2 ) A model parameter (−) A shorthand notation for advective term a coefficient in the FVM discretization equation a non-linear function in PDE classification theory a stoichiometric coefficients in general reaction (−) A(t, r) generalized variable dependent on time and space catchment area (m2 ) A0 valve opening area (m2 ) A0 parameter in prescribed velocity profile in laminar boundary layer a0 theory (−) surface of phase in two phase system (m2 ) A1 parameter in prescribed velocity profile in laminar boundary layer a1 theory (−) surface of phase in two phase system (m2 ) A2 parameter in prescribed velocity profile in laminar boundary layer a2 theory (−) parameter in prescribed velocity profile in laminar boundary layer a3 theory (−) aC (x, r; x , r , Y, t) coalescence frequency or the fraction of particle pairs of states (x, r) and (x , r ) that coalesce per unit time (1/s) heat exchange surface of reactor (m2 ) Ah interface area (m2 ) AI interfacial area density denoting the interface area per unit volume aI (m2 /m3 ) coefficient in generic algebraic equation in TDMA outline constants in MWR approximation of the solution C.4 The 2D Axi-Symmetric Bubble Column Model 1229 aN = Dn + max[−Cn , 0] aS = Ds + max[Cs , 0] aE = De + max[−Ce , 0] aW = Dw + max[Cw , 0] SC,m + a0 kl,P P b= m ΔV aP = (αl ρl )P + aN + aS + aE + aW + Cn − Cs + Ce − Cw − Δt SP,q ΔV q (C.410) To avoid negative coefficients, the relation for the coefficient aP can be modified using the continuity equation, as shown for the liquid phase velocity equations The alternative a∗ and b∗ coefficients are defined by: P ν b∗ = −mC2 kl,P + SC,m + a0 kl,P P m ΔV (αl ρl )0 + aN + aS + aE + aW + mC1 − a∗ = P P Δt (C.411) SP,q,l ΔV q The convective and diffusive fluxes are approximated in the following way: Cn = An (αl ρl vr )n = An (αN ρN + αP ρP )vr,N Cs = As (αl ρl vr )s = As (αP ρP + αS ρS )vr,P Ce = Ae (αl ρl vz )e = Ae (αE ρE + αP ρP )vz,E Cw = Aw (αl ρl vz )w = Aw (αP ρP + αW ρW )vz,P An Γn Dn = δrN P As Γs Ds = δrP S Ae Γe De = δzEP Aw Γw Dw = δzP W μl,ef f Γn = ( )n = (αN μN + αN μN ) σk 2σk μl,ef f )s = (αP μP + αS μS ) Γs = ( σk 2σk μl,ef f )w = (αW μW + αP μP ) Γw = ( σk 2σk μl,ef f Γe = ( )e = (αP μP + αW μW ) σk 2σk (C.412) 1230 C Trondheim Bubble Column Model C.4.9 The Turbulent Kinetic Energy Dissipation Rate The equation for the turbulent energy dissipation rate is discretized in accordance with the generalized equation in sect C.4.3, with ψ = and phase k = l The source term The source terms in the equation for the turbulent kinetic energy dissipation rate are implemented through the source terms SP and SC in the following way: αl (C1 (Pk + Pb ) − C2 ρl ε) dV dt = [αl (C1 (Pk + Pb ) − C2 ρl ε)]P ΔV Δt k k ΔtΔV (C.413) where SC,1 ΔV Δt =[αl C1 Pk + αl C1 Pb ]P ΔV Δt k k ε Sp,1 ΔV Δt =[αl ρl C2 ]P ΔV Δt k (C.414) (C.415) The production terms Pk and Pb are defined in section C.4.8 Algebraic discretization equation After dividing all the terms by Δt, the discretized equation can be written on the standard algebraic form: aP εl,P = aN εl,N + aS εl,S + aE εl,E + aW εl,W + bεl (C.416) in which the coefficients are defined by: aN = Dn + max[−Cn , 0] aS = Ds + max[Cs , 0] aE = De + max[−Ce , 0] aW = Dw + max[Cw , 0] SC,m + a0 kl,P P b= m ΔV aP = (αl ρl )P + aN + aS + aE + aW + Cn − Cs + Ce − Cw − Δt SP,q ΔV q (C.417) To avoid negative coefficients, the relation for the coefficient aP can be modified using the continuity equation, as shown for the liquid phase velocity equations The alternative a∗ and b∗ coefficients are defined by: P C.4 The 2D Axi-Symmetric Bubble Column Model b∗ = −mC2 εν + l,P SC,m + a0 ε0 P l,P m a∗ = P 1231 ΔV (αl ρl )0 + aN + aS + aE + aW + mC1 − P Δt (C.418) SP,q,l ΔV q The convective and diffusive fluxes are approximated in the following way: Cn = An (αl ρl vr )n = An (αN ρN + αP ρP )vr,N Cs = As (αl ρl vr )s = As (αP ρP + αS ρS )vr,P Ce = Ae (αl ρl vz )e = Ae (αE ρE + αP ρP )vz,E Cw = Aw (αl ρl vz )w = Aw (αP ρP + αW ρW )vz,P An Γn Dn = δrN P As Γs Ds = δrP S Ae Γe De = δzEP Aw Γw Dw = δzP W μl,ef f Γn = ( )n = (αN μN + αN μN ) σ 2σ μl,ef f )s = (αP μP + αS μS ) Γs = ( σ 2σ μl,ef f )w = (αW μW + αP μP ) Γw = ( σ 2σ μl,ef f Γe = ( )e = (αP μP + αW μW ) σ 2σ (C.419) C.4.10 The Volume fraction The gas volume fraction is calculated from the continuity equation for phase k which is discretized by the scheme proposed by Spalding [20] The continuity equation for phase k (k = l, g) is derived in appendix C: ∂ ∂ ∂ (αk ρk ) + (rαk ρk vk,r ) + (αk ρk vk,z ) = ∂t r ∂r ∂z μk,t ∂αk ∂ μk,t ∂αk ∂ (r )+ ( )+S r ∂r σαk ,t ∂r ∂z σαk ,t ∂z In the FEM, this equation is integrated in time and over a grid cell volume The resulting terms are then approximated in accordance with the approach 1232 C Trondheim Bubble Column Model presented for the generalized equation The derivatives of the volume fraction in the diffusive terms are approximated by central differences and for the convection terms the upwind scheme is employed The discretized liquid phase continuity equation (k = l) can then be expressed as: αl,P ρl ΔV + (max[Cn , 0] + Dn ) + (max[−Cs , 0] + Ds ) Δt + (max[Ce , 0] + De ) + (max[−Cw , 0] + Dw ) = (αl ρl )o ΔV P + αl,N (max[−Cn , 0] + Dn ) + αl,S (max[Cs , 0] + Ds ) Δt + αl,E (max[−Ce , 0] + De ) + αl,W (max[Cw , 0] + Dw ) + SΔV (C.420) where Cn = An Fn = An (ρl vl )n = An vr,N (ρl,P + ρl,N ) Cs = As Fs = As (ρl vl )s = As vr,P (ρl,P + ρl,S ) Ce = Ae Fe = Ae (ρl wl )e = Ae wl,E (ρl,P + ρl,E ) Cw = Aw Fw = Aw (ρl wl )w = An wl,P (ρl,P + ρl,W ) (C.421) and An Γn δrN P As Γs = δrP S Ae Γe = δzEP Aw Γw = δzP W μl,t μl,t,P μl,t,N =( )n = + σαl,t σαl,t σαl,t Dn = Ds De Dw Γn Γs = ( μl,t μl,t,P μl,t,S )s = + σαl,t σαl,t σαl,t Γw = ( μl,t μl,t,P μl,t,W )w = + σαl,t σαl,t σαl,t Γe = ( μl,t μl,t,P μl,t,E )e = + σαl,t σαl,t σαl,t (C.422) C.4 The 2D Axi-Symmetric Bubble Column Model 1233 For convenience two new variables m and Sl,1 are introduced, hence the equation can be written in a more compact form: ml,i,out − αl,P αl,i,in ml,i,in − Sl,1 = Rl = i (C.423) i where ml,i,out = i ρl ΔV + (max[Cn , 0] + Dn ) + (max[−Cs , 0] + Ds ) Δt + (max[Ce , 0] + De ) + (max[−Cw , 0] + Dw ) αl,i,in ml,i,in = αl,N (max[−Cn , 0] + Dn ) + αl,S (max[Cs , 0] + Ds ) i + αl,E (max[−Ce , 0] + De ) + αl,W (max[Cw , 0] + Dw ) Sl,1 = (αl ρl )o ΔV P Δt (C.424) A similar equation can be obtained for the gas phase as well If both equations are solved for αP yields: αl,P + αg,P = i αl,i,in ml,i,in + Sl + i ml,i,out i αg,i,in mg,i,in + Sg =1 i mg,i,out (C.425) With minor manipulation of the equation, we get: ml,i,out = i ( i αl,i,in ml,i,in + Sl ) i mg,i,out + ( i αg,i,in mg,i,in + Sg ) i mg,i,out i ml,i,out (C.426) This relation is used to substitute for the we get: ( αl,P = i αl,i,in ml,i,in + Sl ) i i ml,i,out term in (C.423), hence mg,i,out + ( i αg,i,in mg,i,in + Sl ) i mg,i,out i ml,i,out αl,i,in ml,i,in + Sl i (C.427) Algebraic discretization equation The algebraic equation that must be solved for the gas volume fraction variable can thus be written as: 1234 C Trondheim Bubble Column Model aP αl,P = aN αl,N + aS αl,S + aE αl,E + aW αl,W + Sl (C.428) in which the coefficients are defined as follows: aN = Dn + max[−Cn , 0] aS = Ds + max[Cs , 0] aE = De + max[−Ce , 0] aW = Dw + max[Cw , 0] SC = aP = (αl ρl )P ΔV + SΔV Δt ( i αl,i,in ml,i,in + Sl ) i mg,i,out + ( i αg,i,in mg,i,in + Sg ) i mg,i,out i ml,i,out (C.429) C.4.11 The Pressure-Velocity Correction Equations The pressure correction equation is derived from the liquid continuity equation and the liquid velocity correction equation formulas The SIMPLE Consistent (SIMPLEC) -approximation proposed by van Doormal and Raithby [23] is used to derive the velocity correction formulas The continuity equation for the liquid phase is given in appendix C The discretization of this equation is discussed in sect C.4.10 The discretized form of the continuity equation thus yields: [(αl ρl )P − (αl ρl )o ]ΔV P + Δt An (αl ρl vl,r )n − As (αl ρl vl,r )s + Ae (αl ρl vl,z )e − Aw (αl ρl vl,z )w = Dn (αl,N − αl,P ) − Ds (αl,P − αl,S ) + De (αl,E − αl,P ) − Dw (αl,P − αl,W ) + SΔV (C.430) The pressure correction is given by the difference between the correct pressure, p, and the guessed pressure, p∗ The velocity correction v is given by the difference between the correct velocity, v, and the guessed velocity v ∗ : p = p∗ + p u = u∗ + u (C.431) The relationship between the liquid velocity correction at the grid cell surface e and the pressure corrections, i.e., the velocity correction formula, is given as: Ae αl,e (pP − pE ) wl,e = = de (pP − pE ) (C.432) al,e − anb nb C.4 The 2D Axi-Symmetric Bubble Column Model where de = Ae αl,e al,e − al,nb 1235 (C.433) nb The correction formulas for the liquid velocity components in the other directions, as well as the corresponding correction formulas for the gas phase, can be deduced in a similar manner Algebraic discretization equation The transport equation for the pressure corrections is obtained substituting all the velocity components in (C.430) by the sum of the guessed and corrected velocities, and then substituting the velocity corrections by the corresponding pressure corrections employing the liquid phase velocity correction formulas The resulting algebraic equation to be solved is written: aP pP = aN pN + aS pS + aE pE + aW pW + b (C.434) where the coefficients are given as: aN = An (αl ρl )n dn = An dn (αl,P ρl,P + αl,N ρl,N ) aS = As (αl ρl )s ds = As ds (αl,P ρl,P + αl,S ρl,S ) aE = Ae (αl ρl )e de = Ae de (αl,P ρl,P + αl,E ρl,E ) aW = Aw (αl ρl )w dw = Aw dw (αl,P ρl,P + αl,W ρl,W ) aP = aN + aS + aE + aW b= ((αl ρl )P − (αl ρl )o )ΔV P Δt ∗ ∗ ∗ ∗ − An (αl ρl vl,r )n + As (αl ρl vl,r )s − Ae (αl ρl vl,z )e + Aw (αl ρl vl,z )w + Dn (αl,N − αl,P ) − Ds (αl,P − αl,S ) + De (αl,E − αl,P ) − Dw (αl,P − αl,W ) + SC ΔV (C.435) References Anderson DA, Tannehill JC, Pletcher RH (1984) Computational Fluid Mechanics and Heat Transfer Hemisphere, New York Aris R (1962) Vectors, Tensors, and the Basic Equations of Fluid Mechanics Dover, Inc., New York Bird RB, Stewart WE, Lightfoot EN (1960) Transport phenomena John Wiley & Sons, New York Bird RB, Stewart WE, Lightfoot EN (2002) Transport phenomena Second Edition, John Wiley & Sons, New York Boisson N, Malin MR (1996) Numerical Prediction of two-phase flow in bubble columns Int J Numer Meth Fluids 23:1289-1310 Borisenko AI, Tarapov IE (1979) Vector and tensor analysis with applications Translated and edited by Silverman RA, Dover Publication Inc, New York Clift R, Grace JR, Weber ME (1978) Bubble Drops, and Particles Academic Press, New York Gosman AD, Ideriah FJK (1976) TEACH-T: a general computer program for two dimensional turbulent recirculating flows London: Mechanical Engineering Department, Imperial College Grienberger J (1992) Untersuchung und Modellierung von Blasensăulen a Dr ing Thesis, Der Technischen Fakultăt der Universităt Erlangen-Nărnberg, a a u Germany 10 Irgens F (1982) Kontinuumsmekanikk Del III: Tensoranalyse Tapir, Trondheim 11 Irgens F (2001) Kontinuumsmekanikk Institutt for mekanikk, thermo- og fluiddynamikk, Norges teknisk- naturvitenskapelige universitet, Trondheim 12 Jakobsen HA (1993) On the modelling and simulation of bubble column reactors using a two-fluid model Dr ing thesis, the Norwegian Institute of Technology, Trondheim, Norway 13 Johansen ST, Boysan F (1988) Fluid Dynamics in Bubble Stirred Ladles: Part Mathematical Modelling Met Trans B 19:755-764 14 Launder BE, Spalding DB (1972) Mathematical Models of Turbulence Academic Press, London 15 Malvern LE (1969) Introduction to the Mechanics of a Continuum Medium Prentice-Hall Inc, Englewood Cliffs 16 Patankar SV (1980) Numerical heat transfer and fluid flow Hemisphere publishing corporation, New York 1238 References 17 Slattery JC (1972) Momentum, Energy, and Mass Transfer in Continua McGraw-Hill Book Company, New York 18 Spalding DB (1977) The calculation of free-convection phenomena in gas-liquid mixtures ICHMT seminar 1976 In:Turbulent Buoyant Convection, Hemisphere, Washington, pp 569-586 19 Spalding DB (1980) Numerical computation of multiphase fluid flow and heat transfer In: Taylor C, Morgan K (Eds) Recent Advances in Numerical Methods in Fluids, Pineridge Press, pp 139-167 20 Spalding DB (1981) IPSA 1981: New Developments and Computed Results Report HTS/81/2, Imperial College of Science and Technology, London 21 Svendsen HF, Jakobsen HA, Torvik R (1992) Local Flow Structures in Internal Loop and Bubble Column Reactors Chem Eng Sci 47(13-14):3297-3304 22 Torvik R, Svendsen HF (1990) Modeling of slurry reactors - a fundamental approach Chem Eng Sci 45(8):2325-2332 23 van Doormal JP, Raithby GD (1984) Enhancement of the SIMPLE Method for Predicting Incompressible Fluid Flows Numer Heat Transfer 7:147-163 24 Zapryanov Z, Tabakova S (1999) Dynamics of bubble, drops and rigid particles Kluwer academic publishers, Dordrecht Index activity coefficient, 674 affinity, thermodynamic forces, 64 agitation, 679 algebraic-slip mixture model, 467 angular momentum balance, 67 apse-line, 232 averaging, 394 area averaging, 86, 90, 93, 473 ensemble averaging, 118, 429 Maxwellian, 211, 246, 249 statistics, 118 time averaging, 118, 419 time- after volume averaging, 441 volume averaging, 118, 397 balance equations, balance laws, balance principle, 12 BBGKY-hierarchy, 207 Bernoulli equation, 82 billiard ball model, 208 blending, 679 Boltzmann equation, 218, 245, 246 Boltzmann stosszahlansatz, 223 breakage probability, 832 bubble wall friction force, 796 bulk expansion coefficient, 69 Capillary number, 573 Carnot cycle, 191 Cauchy equation, 250 centrifugal force, 195 centripetal force, 195 chemical reaction engineering, CRE, 336 chemical reaction equilibrium, 666 Chilton-Colburn relation, 633 classical thermodynamics, 36 closure law constitutive, 543 topological, 543 transfer, 543 coalescence time, 822 coefficient of restitution, 228 collision cylinder, 244 collision frequency, 243 collision time, 823 complete differential, 54 compressible flow, fluid, concentration diffusion, 20 configuration space, 203 conservative forces, 45 constitutive equations, 7, 543 continuous stirred tank reactor model, CSTR, 337 continuous surface force, CSF, 352 continuum hypothesis, 319 continuum mechanics, continuum surface stress, CSS, 352 control volume arbitrary Lagrangian-Eulerian (ALE), 10 Eulerian, 10 Lagrangian, 10 1240 Index material, 10 control volume approach, 8, 11 Coriolis force, 195 curvature mean curvature of surface, 348 principal curvatures of a surface, 349 principal radii of curvature, 349 curvilinear coordinate systems, 1158 Damkăhler number, 708 o Danckwerts boundary conditions, 665, 769, 905, 912 Darcy friction factor, 479 Darcy-Weisbach equation, 698 degrees of freedom, 197 diffusion barrier, 271 diffusion mixture model, 469 diffusion velocity, 263 dilute gas, 192, 318 dirac delta function, 350 Dirichlet boundary conditions, 994 dispersed flows, 339 dispersion reactor models, 337 axial, 98 heterogeneous, 484, 957 homogeneous, 957 pseudo-homogeneous, 485 distribution function, 190, 210 drift-flux model, 472 Dufour eect, 42, 266 dusty gas model, 274 Eătvăs number, 572 o o eddy viscosity hypothesis, 545 effective swept volume rate, 816 elastic collision, 209 embedded interface method, EI, 344, 362 ensemble, 203 Enskog equation, 246, 248, 323 Enskog expansion method, 256 equations of transfer, 191 equipartition theorem, 217 ergodic hypothesis, 119, 189 Euler equations, 215, 258 Eulerian-Eulerian models, 340 Eulerian-Lagrangian models, 340 excess property, 372 extent of reaction, 56, 670 Fanning friction factor, 86, 92, 479 Fick’s law of viscosity, 597 fluid mechanics, fluidization, 867 CPV model, 920 drift velocity, 919, 927 flow regimes, 869 Geldart classification of particles, 868 gulf streaming, 898 PGT model, 924 PGTDV model, 927 PT model, 921 riser, 876 transport reactor, 876 forced diffusion, 21 form drag, 556, 559 Fourier’s law, 597 free surface flow, 349 friction drag, 556, 559 front tracking method, FT, 344 fugacity, 672 fugacity coefficient, 674 Galilean transformation, 64 Galileo law, 194 Gauss’ theorem, 1130 divergence theorem, 1130 Green’s theorem, 1130 Ostrgradsky’s theorem, 1130 surface, 1132 generalized coordinates, 196, 197 generalized drag force, 554, 555, 558 generalized Eulerian transport equation, 12 generalized transport theorem, 379 generalized velocities, 197 Gibbs-Duhem equation, 295 granular flow, 503 granular material, 503 granular temperature, 505 H-theorem, 223, 252, 253 Hagen-Poiseuille law, 122 Hamiltonian variational principle, 198 heat of reaction, 58 heat transfer models, 588 heaviside function, 358 high resolution models, 340 history force, 586 Index hydraulic diameter, 92 ideal gas law, 218 ill-posed model system, 486 incompressible, flow, 4, 68 fluids, inelastic collision, 209 interface model macroscopic 2D dividing surface, 370 microscopic 3D transition region, 370 interfacial coupling, 341 four way, 341 one way, 341 two way, 341 irreversible thermodynamics, 37 transport properties, 309 isentropic, 84 isothermal compressibility, 69 jump condition formulation, 344 kinetic theory, 189 kinetic theory of dense gases, 319, 510 Knudsen number, 367 Lagrange multiplier, 669 least squares method, 996 Leibnitz theorem, 1128 Leibnitz’ theorem surface, 1131 Leibnitz’s integral rule, 1125 level set method, LS, 344, 357 lift force, 557 Liouville equations, 205 Liouville theorem, 205, 218 local equilibrium, 223 Mach number, 72 macro mixing, 707 Magnus lift force, 564 maker and cell method, MAC, 344, 345 mass transfer models, 588 lm theory, 612 Frăssling equation, 634 o laminar boundary layer theory, 618 penetration theory, 615 surface-renewal theory, 615 turbulent boundary layer theory, 624 1241 Maxwell-Stefan equations, 269 Maxwellian molecules, 191, 209 Maxwellian velocity distribution, 254 mean free path, 5, 309, 312 concept, 191 mean value theorem, 1083 mechanics, 187 chaos, 188 classical, 187 continuum, 188 dynamics, 188 fluid, 188 Hamiltonian, 194, 201 Hamiltonian integral principle, 197 kinematics, 188 kinetics, 188 Lagrangian, 194, 197 Newton, 194 Newtonian, 194 non-linear dynamics, 188 quantum, 187 solid, 188 statics, 188 statistical, 188, 203 method of manufactured solutions, 987 micro mixing, 707 mixed or Robin boundary conditions, 994 mixing, 679 mixture model, 463 molar heat of formation, 59 molecular chaos, 223 moment equations, 191 momentum balance, 25 Morton number, 572 moving bed, 867 multifluid model, 343, 391 multiphase control volume, 372 mutual diffusion, 315 Navier-Stokes equations, 262 Neumann boundary conditions, 994 Newton’s first law, 194, 195 Newton’s law of cooling, 593 Newton’s law of viscosity, 597 Newton’s second law, 194, 340, 554 Newton’s third law, 194, 195, 342 normal stresses, 214 numerical methods 1242 Index approximation function, 996 arithmetic mean values, 1067 basis function, 996 boundary-value problems, 991 boundedness, 990 central difference scheme, 1028 class method, 1077 collocation method, 996, 998 convergent, 990 deferred correction method, 1029 density-based methods, 1010 discrete method, 1077 domain decomposition parallelization method, 1107 FCT schemes, 1031 finite difference method, 993 finite volume method, 995 fractional step methods, 1010 Galerkin method, 996, 1001 Gauss-Seidel point iteration method, 1093 harmonic mean values, 1067 initial value problem, 992 initial-boundary-value problems, 992 Jacobi point iteration method, 1093 Jacobi preconditioner, 1096 Krylov subspace methods, 1095 least squares method, 1000, 1077, 1090 method of lines, 1017 method of moments, 1002, 1077 method of weighted residuals, 985, 995 multi-group method, 1077 multigrid solvers, 1102 multistep methods, 1021 numerical accuracy, 990 numerical stability, 989 ODE solution methods, 1019 orthogonal collocation, 997 predictor-corrector methods, 1021 pressure-based methods, 1010 projection methods, 1011 quadrature formulas, 1013 quadrature method of moments, 1077 QUICK scheme, 1029 Rayleigh-Ritz method, 996 Runge-Kutta methods, 1020 strong form, 1004 tau method, 996 test function, 996 trail function, 996 TVD schemes, 1032 upwind differencing scheme, 1027 von Neumann method, 989 weak form, 1004 weight function, 996 Nusselt number, 611 osmotic diffusion, 271 packed bed reactor hot spot, 954 runaway, 954 partial molar enthalpy, 58 partial specific enthalpy, 58 particle Reynolds number, 573 peculiar velocity, 212 perimeter, 92 phase change, interfacial momentum transfer closure, 587 phase space, 210 phase trajectory, 203 plug flow reactor model, PFR, 337 pressure diffusion, 21 pressure tensor, 214 radii of curvature, 378 reactor flow characteristics, 338 realizable models, 990 reduced mass, 227, 230, 242 reverse diffusion, 271 reversible adiabatic, 84 Reynolds number, 611 Saffman lift force, 564 scalar quantity, 1158 scattering cross section, 235, 236 self-diffusion, 315 separated flows, 339 shear stresses, 214 Sheerwood number, 611 solid angle, 233 Soret effect, 42, 266 specific chemical potentials, 62 specific enthalpy, 52 specific entropy, 39 specific molar enthalpy, 57 Index speed of sound, 72 standard drag curve, 562 Stanton number, 611 state vector, 210 statistical thermodynamics, 189 steady drag force, 556 steady flow, 68 superficial velocity, 484 surface tension, 382 static force balance, 1133 surface theorem, 379 symmetry of stress tensor, 67 system approach, temperature equation, 1143 tensor quantity, 1159 tensor transformation laws, 1157 thermal diffusion, 21 thermal radiation, 635 absorptivity, 642 blackbody, 640 emissivity, 641 gray surface, 643 incident, 639 irradiation, 639 Kirchhoff’s law, 643 Lambert’s cosine law, 641 radiosity, 640 Stefan-Boltzmann law, 641 thermodynamic pressure, 214 thermodynamics, 36, 39 entropy, 191 first law, 40 second law, 61, 63, 191 torque, 566, 687, 688 transport properties, 309 turbulence k-ε model, 139 auto-covariance, 106 autocorrelation coefficient, 106 Batchelor spectrum, 709 Boussinesq turbulent viscosity hypothesis, 626 buffer layer, 125 coherent structures, 102 cross-term stress, 169 definitions, 99 dispersion force, 796 eddy concept, 105 1243 eddy turnover time, 112 energy cascade, 106 energy dissipation rate, 110, 112 energy spectrum, 103, 115 Eulerian longitudinal integral length scale, 109 Eulerian transverse integral length scale, 109 friction velocity, 124 gradient transport hypothesis, 161, 626 homogeneous turbulence, 108 inner wall layer, 124 intensity, 120 isotropic turbulence, 108 Kolmogorov five-third law, 116 Kolmogorov hypotheses, 113 Kolmogorov microscales, 114 Kolmogorov similarity hypothesis, 114 Kolmogorov structure function, 116 Kolmogorov two-third-law, 116 Kolmogorov-Prandtl relationship, 142 Lagrangian integral time scale, 106 Lagrangian microscale, 106 large eddy simulation, LES, 161 law of the wall, 127, 628 Leonard stresses, 169 local isotropic turbulence, 113 log-law sublayer, 125 longitudinal autocorrelation function, 108 mixing length, 105 modulation, 796 one-point two-time correlation, 106 outer wall layer, 124 overlap wall region, 124 passive scalar spectra, 708 power law velocity profile, 122 Prandtl’s Mixing length model, 123 residual stresses, 169 Reynolds analogy, 629 Reynolds averaging, 104, 129 Reynolds stress models, 133 rms-velocity, 110 Smagorinsky constant, 172 Smagorinsky eddy-viscosity model, 171 standard k-ε model parameters, 143 1244 Index statistical theory, 104 sub-grid-scale, SGS, 164 Taylor’s hypothesis, 111 transverse autocorrelation function, 108 turbulent dispersion, 796 two-point correlation function, 108 universal velocity profile, 628 velocity-defect law, 127 viscous sub-layer, 125 wall functions, 150 turbulent impeller, 680 two-fluid continuity, 384 energy balance, 387 internal energy, 387 momentum balance, 385 species mass balance, 385 variable density flow, 75 vector quantity, 1158 velocity distribution function, 190 virtual mass force, 581 viscous stress tensor, 29 volume of fluid method, VOF, 344, 346 Piecewise Linear Interface Construction, PLIC, 351 Simple Line Interface Calculation, SLIC, 351 SOLA-VOF method, 349 wall lift force, 796 Weber number, 573 well-posed model system, 486 well-posedness, 485 whole field formulation, 344, 350 Wilke equation, 273 .. .Chemical Reactor Modeling Hugo A Jakobsen Chemical Reactor Modeling Multiphase Reactive Flows Prof.Dr Hugo A Jakobsen Norwegian Univ of Science... transport phenomena and chemical reactor modeling, and in a post graduate course in modern reactor modeling at the Norwegian University of Science and Technology, Department of Chemical Engineering,... single-fluid and multi-fluid models for the analysis of single- and multiphase reactive flows in chemical reactors with a chemical reactor engineering rather than mathematical bias Organized 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