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SIMULTANEOUS HIGHER ORDER REACTIONS AND DIFFUSION IN A SLAB By DAVID KOOPMAN A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1992 This dissertation is dedicated to my parents and sister without whose support and encouragement it would never have been possible. ACKNOWLEDGEMENTS I would like to gratefully acknowledge the assistance and inspiration provided by Dr. Hong H. Lee both through his constant assistance in this endeavor and through his textbook, Heterogeneous Reactor Design. I would also like to express my appreciation to Rutherford Aris, who collected and reviewed many of the early accomplishments in this area of research. Without the accomplishments of those who have gone before, this work would not have been possible. TABLE OF CONTENTS ACKNOWLEDGEMENTS .................................. iii KEY TO SYMBOLS ..................................... vi ABSTRACT .......................................... ix CHAPTER 1 AN INTRODUCTION TO THE PROBLEMS OF SIMULTANEOUS REACTION AND DIFFUSION IN A SLAB WITH A REVIEW OF PREVIOUS RESULTS ..... 1 Introduction ....................................... 1 General Problem Statement ............................. 4 The ZerothOrder Reaction ............................. 8 The FirstOrder Problem .............................. 10 Effectiveness Factors ................................. 17 The Generalized Thiele Modulus ......................... 22 Shape Factor Normalizations ............................ 26 Summary and Goals ................................. 27 CHAPTER 2 CONVENTIONAL SECONDORDER REACTION SYSTEMS ........... 30 Overview of SecondOrder Reaction and Diffusion .............. 30 Governing Equations and Dimensionless Groups ................ 34 Solution of the Dirichlet Problem ....................... 45 The Two Fundamental Parameters ..................... 52 The Weierstrass PeFunction, .p .......................... 66 Concentration Profiles From the Analytical Solution ............. 75 FirstOrder Subcases of the General Problem .................. 78 Internal Effectiveness Factors of SecondOrder Reactions .......... 84 Concluding Remarks ................................. 104 CHAPTER 3 CONVENTIONAL THIRDORDER REACTION SYSTEMS ............ 110 Introduction to ThirdOrder Reaction and Diffusion ............. 110 ThreeParameter Equations and Concentration Profiles ............ 111 Restrictions on the Feasible Parameter Space .................. 126 Internal Effectiveness Factors of ThirdOrder Reactions ........... 132 Concluding Remarks ................................ ..... 144 CHAPTER 4 EXTENSIONS TO OTHER REACTION AND DIFFUSION PROBLEMS .... 146 Introduction ........................................ 146 The Robin's Problem ................................ 148 The Single Species Robin's Problem ....................... 155 Autocatalytic Reactions .............................. 164 Multiple Reaction Systems ............................. 171 Concluding Remarks .................................. 182 CHAPTER 5 RECOMMENDATIONS AND FUTURE WORK ...... Recommendations ..................... Future W ork ......................... APPENDIX NUMERICAL METHODS AND COMPUTER PROGRAMS ...... . 185 ......185 ......186 . 189 . .. .. 189 REFERENCE LIST ................... ................... 208 BIOGRAPHICAL SKETCH ................................ 211 :\::: KEY TO SYMBOLS a coefficients of the single species secondorder kinetic expressions A coefficients of the Taylor series expansion of q Bi Biot number for mass transfer c' normalized dimensionless concentration c" normalized dimensionless concentration c dimensional concentration C expansion coefficients of the Weierstrass elliptic Pefunction D effective diffusivity g polynomial invariants for the Weierstrass Pefunction k reaction rate constants L characteristic length of the slab (i.e. half width) n order of a power law rate expression p coordinate system identifier for slab, cylinder or sphere p Weierstrass elliptic Pefunction q derivative of a concentration with respect to a length scale variable r kinetic rate expression x dimensional position variable z dimensionless position variable, x/L Greek Letters a constant in the integration of q, also dummy integration variable S arbitrary constant in the development of Z(0)/c, v stoichiometric coefficient t Thiele, or firstorder, or generalized, modulus ?7 effectiveness factor I' secondorder modulus a transformation variable during integration 7 transformation variable during integration X thirdorder modulus Subscripts A specific chemical species B specific chemical species C specific chemical species eq equilibrium value f forward direction G generalized, or normalized, modulus i property appropriate to generic chemical species, I int internal j,k properties appropriate to generic chemical species, J and K l,m properties appropriate to generic chemical species, L and M m+ associated with the surface of repeated equilibrium roots based on the positive square root m associated with the surface of repeated equilibrium roots based on the negative square root o at the midplane, x = 0 or z = 0 P specific chemical species; or related to product formation in the rate r Q specific chemical species r reverse direction R specific chemical species s at the surface of the slab medium, x = L or z = 1 1st firstorder Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy SIMULTANEOUS HIGHER ORDER REACTIONS AND DIFFUSION IN A SLAB By David Koopman December 1992 Chairperson: Dr. Hong H. Lee Major Department: Chemical Engineering Analytical solutions are presented for a broad class of simultaneous steadystate, isothermal secondorder and thirdorder reactions and diffusion in a slablike medium with constant diffusivities. All intrinsically thirdorder reversible and irreversible kinetic schemes can be unified into a single threeparameter problem, while similar secondorder kinetic schemes can be unified into a single twoparameter problem. Analytical solutions for dimensionless concentration profiles, internal effectiveness factors, etc. are determined as functions of these dimensionless parameters: a firstorder, or Thiele, modulus, ?,,, a secondorder modulus, I, and a thirdorder modulus, X, for thirdorder reactions only. The analytical solutions employ the Weierstrass elliptic Pefunction, p, to characterize the spatial dependence of the concentration profile. The analysis provides semiquantitative conditions for both diffusion free and diffusion limited behavior. The modulus, 9, is especially adjustable by varying physical parameters. Other limiting interrelationships between the three moduli are developed that restrict the size of the feasible solution space for conventional secondorder and thirdorder reactions. Existing approximation methods are only partially validated. Errors exceeding 20% are quite possible. Improvements to some methods are suggested by the analysis. Autocatalytic secondorder and thirdorder reactions do not conform to established norms for conventional reversible and irreversible reactions. Internal effectiveness factors well in excess of unity are observed. The analytical solution methods developed here are valid for either the Dirichlet or Robin's boundary condition at the surface. These methods apply to varying degrees for selected multiple reaction systems, such as parallel reactions of different orders and Van der Vusse kinetics. As an overall consequence of these developments, the number of elementary chemical reactions possessing an analytical solution for simultaneous reaction and diffusion has increased tremendously. CHAPTER 1 AN INTRODUCTION TO THE PROBLEMS OF SIMULTANEOUS REACTION AND DIFFUSION IN A SLAB WITH A REVIEW OF PREVIOUS RESULTS Introduction The study of the theory of isothermal, coupled, steadystate reaction and diffusion problems dates back to the time of E. W. Thiele and his seminal paper: "Relation Between Catalytic Activity and Size of Particle". The concepts of a modulus, characteristic of the reacting species, and of a ratio of actual achieved rate to maximum possible external rate of a chemical reaction were presented there. Such moduli bear Thiele's name today, while the reaction rate ratios are referred to as internal effectiveness factors. Thiele derived solutions for the case of a firstorder irreversible single species reaction with constant diffusivity and a Dirichlet surface boundary condition for both a sphere and an infinite slab of finite thickness. Also derived was an expression for the internal effectiveness factor of a single species irreversible secondorder reaction in a slab. More will be said about this result later. The fifty years following Thiele's work saw the addition of analytical solutions for single reactions with zerothorder kinetics, with Robin's boundary conditions, in cylindrical coordinates, and a thorough analysis of the firstorder problem with multiple reactions, as well as geometries with two or three characteristic length scales such as the 2 rectangular parallelepiped. It was recognized by Aris (1957, 1975) that use of the proper characteristic length scale brought the results for different geometries into asymptotic agreement. It was further recognized simultaneously by Bischoff (1965), Aris (1965), and Petersen (1965) that the use of a specially normalized, or generalized, Thiele modulus brought the internal effectiveness factor results for arbitrary kinetics into asymptotic agreement at both large and small modulus magnitudes. This work was collected and summarized by Rutherford Aris in 1975. Some of the emphasis in this field shifted from analytical to numerical solutions with the proliferation of high speed computers in the 1960's. The first studies on the behavior of LangmuirHinshelwood kinetics coupled with diffusion began to appear; see Aris (1975) or Satterfield (1970) for summaries. Problems with complex pore networks, nonisothermal reactions, variable diffusivities, reactions with volume changes, etc. were analyzed by numerical methods. However, after all of this effort, published analytical solutions for the concentration profiles of single reaction systems, either reversible or irreversible, existed only for the following rate laws: (1) r, = k,, c, > 0 rf = 0, c, = 0 (2) rf = kc1 (3) rf = k1c, k2 restrictions on k2 may apply (4) rf = kc, k2C2 (5) r, = kc6 where the result published for (5) was only for the concentration at the slab midplane. This work derives analytical solutions for 25 elementary kinetic rate laws for irreversible 3 and reversible thirdorder and secondorder reaction systems. It also explains how to derive analytical solutions for five secondorder and thirdorder reactions reversible with a zerothorder reaction. Additional solutions for several simple parallel reactions with secondorder and/or thirdorder terms are also derived, as well as solutions for about 17 types of autocatalytic behavior undergoing net secondorder or thirdorder kinetics. These analytical solutions are restricted to assumptions of constant diffusivity, infinite slabs, and isothermal behavior; however, both the Dirichlet and Robin's problems are successfully analyzed. Past studies of problems of simultaneous reaction and diffusion are now a part of traditional chemical engineering. Advances in this area tend to inherit some of this established relevance. What other reasons are there to study such problems? First and foremost, simultaneous reaction and diffusion problems have considerable industrial significance due to the importance of supported catalysts in many commercial processes. Secondly, while this problem was first identified over 50 years ago, there are only a handful of single reaction kinetics which have an analytical solution in some geometry. Methods were developed for approximating behavior in arbitrary catalyst geometries once a solution was found in one geometry. This was an inducement to create a library of analytical solutions in some geometry that has never been fully exploited. Thirdly, certain types of theoretical analyses require an analytical solution as a starting point. For example, when the reaction occurring is exothermic, there is a potential for the reaction and diffusion problem to possess multiple steady states. One type of analysis which assumes an isothermal pellet with a temperature gradient confined to an external 4 boundary layer is often valid. The analysis of the multiplicity of steady states as a function of the problem data has been done for such firstorder reaction problems, Burghardt and Berezowski (1990). It is now possible to extend these analyses to second order and thirdorder problems. Finally, the behavior of many problems can be approximately preserved through linearizations of the variables. The analysis of the linearized problem is generally easier to perform. Nonlinear behavior is, nevertheless, best studied directly. Elementary reactions involving autocatalysis are an excellent example. They will be seen to exhibit behavior well outside the range found in linear problems of simultaneous reaction and diffusion. General Problem Statement The governing equation for onedimensional, isothermal, steadystate, simultaneous reaction and diffusion in an infinite slab of finite thickness for the concentration of species, i, ci, takes the form d2'i D = vr, 1.1 dx2 where rp is the rate of product formation, Di is the effective diffusivity of species i, x is the length scale variable measured from the slab midplane to the surface, and vi is the stoichiometric coefficient of i in the reaction described by rp. Two geometries for the problem in a slab are illustrated in Figure 1.1, parts a and b. Identical external conditions are assumed when there are two exposed surfaces in order to give a symmetric Exposed Slab Surface (Porous) (a) Sealed x=O 0 0 / 0 o10 0 \ / / 0 S 0 0 0 0 v 0 0 0 0 4 4 (b) x=L Open Cylindrical Pores S Sealed Walls x=L x=O x=L (d) Figure 1.1 The Geometries for the Simultaneous Reaction and Diffusion Problem. (a) the sealed slab, (b) the symmetric slab, (c) the sealed cylindrical pore, and (d) the symmetric cylindrical pore. Exposed .. Slab Surface (Porous) w X 6 problem about the slab midplane. The slab medium is infinite in the y and z directions. Two additional geometries, or interpretations, are shown in Figure 1.1, parts c and d. The cylindrical pore interpretation requires an assumption to neglect radial concentration gradients, which, for surface driven heterogeneous reactions amounts to making the kinetics homogeneous within the pore as well. The analysis to follow for secondorder and thirdorder kinetics requires identical conditions at +L, where L is the slab half width. The imposition of net gradients across the 2L wide systems gives a slightly altered final equation to integrate, but analytical solutions may still be possible; see Ince (1927). The boundary conditions for the Dirichlet problem are ci = cs at x = L 1.2 = 0 at x = 0. 1.3 dx The boundary conditions for the Robin's problem are ki(if is) = Di at x = L 1.4 dc d = 0 at x = 0 1.5 dx where C is the concentration of i in the free stream fluid away from the surface, and kgi is the mass transfer coefficient for species i from the free stream fluid to the slab surface. The following additional result is very important for systems where rp depends on the concentration of more than one species Did2 D d2c _ = r 1.6 v, dx2 vj dx2 Integration of eq. (1.6) from x = 0 to x gives D, dc, D_ d 1.7 vi dx vj dx using the boundary condition at x = 0 common to the Dirichlet and Robin's problems. Integration of eq. (1.7) from x to x = L gives Di D. .( c,), = "(1.8 v v. for the Dirichlet or Robin's problem. Eq. (1.8) can be used to eliminate the cj in rp in favor of a single ci. Eqs. (1.8) and (1.4) can be combined to eliminate all but one of the unknown surface concentrations in the Robin's problem boundary condition; however, this is not necessarily desirable. (As an alternative the surface concentrations could be carried along as implicit parameters or replaced with the internal effectiveness factor.) This point will be examined further later in this chapter as well as in Chapter Four. Relationships such as eq. (1.8) permit expressing the in terms of one c,, either explicitly for the Dirichlet problem, or implicitly through the unknown c,'s for the Robin's problem. The single species reaction rate expression that results is valid within 8 the slab only, since that is the range of eq. (1.8). The resulting rate expression does not give the bulk fluid reaction rate correctly except for special cases. The ZerothOrder Reaction A zerothorder reaction is one that is independent of concentration so long as reactant remains present. This gives a two part governing equation Dd2 k > 0 dx2 0 e = 0 The solution for the Dirichlet problem is S kL2 c, 2 2D() L] 2Dc, 1.10 or, for kL2/2D, > 1, is i k L2 x* kLL2D C  = = 0O x x * *1 2Dc x* = L 1 s 1 kL x* x s L 1.11 1.12 1.13 Solutions of this type are somewhat awkward because of the transition point, x', at which both c and dc/dx are zero, superseding the usual boundary condition at x = 0. These solutions do suggest likely forms for a dimensionless concentration and length, as well 9 as one additional dimensionless group, kL2/2Dc,, which is the square of the Thiele modulus for a zerothorder reaction, 4k. The above solutions could be written c = $k(1 2) Z i 1.14 where S kL2 x c c 1.15 z C 2Dc,' L' C, or, for 4k > 1, c = (1 2) 2(0 Q)(1 z) z* _z < 1 1.16 c =0 Osz z* 1.17 I1.18 Z = i ). It should be noted that the Thiele modulus contains the surface concentration. This is typical of all but firstorder reaction systems, and complicates the transition from the Dirichlet to the Robin's boundary condition problem. The solution to that zerothorder problem will not be needed in what follows, but zerothorder Dirichlet problem results will occasionally be drawn on for comparison purposes. This review also serves to remind the reader of the possible complications that a zerothorder reaction could bring to a reversible reaction problem where the opposing reaction is not zerothorder. Generally, a real species should be associated with the zerothorder reaction, and safeguards should be taken to see that the concentration of this species is never less than zero. The FirstOrder Problem Firstorder kinetics must be discussed in this work, since secondorder and third order kinetics were found to behave identically to firstorder kinetics for some special cases or under asymptotic conditions. Somewhat more involved expressions for the dimensionless concentration will be developed for secondorder and thirdorder kinetics than are normally used for the firstorder problem. The firstorder kinetics problem review will be made using a new dimensionless concentration that was found to be the most generally useful c' (z) = 1.19 viL2 rp where rp, is the rate of product formation evaluated at the surface of the slab, and z is x/L. Note that if the reaction is in a state of equilibrium at the surface, this equation would call for division by zero. This dimensionless concentration in eq. (1.19) is a universal, or species independent, dimensionless concentration, since (i ci Di ( j) 1.20 Vi v. for the symmetric Dirichlet and Robin's problems discussed here. Using this dimensionless concentration here will familiarize the reader with some of its attributes. 11 Throughout this work, reactant species will be taken as A, B,... and product species will be taken as P, R,... The "forward" reaction rate constant, kf, will be associated with reactants, and the "reverse" reaction rate constant, k,, will be associated with products. Whether the forward reaction rate is actually larger than the reverse reaction rate will be irrelevant. Consider the reversible firstorder reaction A + P. The governing equations are d2k 1.21 DA d2= krp + k 1.21 dx2 D, 2 = k, kfA 122 dx2 Taking z = x/L along with eqs. (1.19) and (1.20) converts both eqs. (1.21) and (1.22) into d2 kfL2 krL2 c,. 1.23 dx2 DA D Setting either k, or kf to zero recovers the irreversible reaction problem for A or P respectively. The minus one term in eq. (1.23) does not vanish for irreversible reactions. The definition of c" in eq. (1.23), however, alters through rp, changing when the problem is made irreversible. The formulation in eq. (1.23) retains the boundary condition at x = 0 as dc " dz dz z =0 1.24 and has the second boundary condition c" =0 z=1 1.25 because of the definition of c" in terms of (ci, c). Achieving a zero valued surface concentration will give essential flexibility when trying to limit the parameter space dimension for secondorder and thirdorder reaction systems. Rewrite eq. (1.23) as 1.26 d2C,, 2 dc c 1. dx = c The square of the firstorder, or Thiele, modulus, 1s,,, is recognized as the sum of the squares of the two Thiele moduli for the irreversible forward reaction, A , and the irreversible reverse reaction, P , given by 1.27 OA = L fA f3 The stoichiometric coefficients were assumed to be + 1, but this restriction can be lifted in which case OP = L . \ p Lt = vP ')p VA A 1.28 Expressions that possess the feature of a stoichiometric coefficient weighted sum over all species, such as this one for the coefficient of c", will be found to persist for second order and thirdorder reaction systems. The solution of eq. (1.26) is straight forward by conventional techniques for linear secondorder differential equations with constant coefficients and gives cosh(cQO,,) cosh(l,,z) c"(z) = 1.29 l acosh( ,t) There is a less conventional technique for solving eq. (1.26), that begins by multiplying by dc"/dz and integrating from z = 0 to z that yields ,2 2 1dc" = s(c"2 "(02) (c" c"()) 1.30 2 dz ) 2 where c"(0) is an as yet unknown value of the solution. From eq. (1.30) one can derive the following dcz = 2 c"(0) 1 ,c"(0)2. 1.31 dz )z.d 14 The dimensionless surface concentration gradient depends only on the Thiele modulus and dimensionless midplane concentration. Formulae of this type will recur in Chapters Two and Three for secondorder and thirdorder kinetics respectively. The integration of eq. (1.30) can be continued by conventional methods after separation of variables, ultimately leading to the expression for c"(z) obtained above. Since c" was taken as zero at the surface, it might be thought that c"(0) is unconstrained in magnitude. This is not the case, however, for firstorder reversible and irreversible reactions as shown in Figure 1.2. The expression for c"(z=0) can be written as 1 1/cosh(O1st) c "(0) = 1.32 Ist with the two limiting values 1 1 lim c"(0) lim c"(0) = 1.33 ,,,o 2 1,, Is The limit at infinity is a zero of the righthand side of eq. (1.26). It corresponds to kcA(0) k,p(0) = 0, as can be shown by substituting into eq. (1.19). The righthand side of eq. (1.26) is a diffusion modified reaction rate expression, and a zero of this is a statement of positional equilibrium in the slab medium. The limit as 4~, goes to zero has physical significance as well. In eq. (1.26) take 4,,c" 1 = 1, the dimensionless 0 r surface reaction rate. Integrate twice to obtain 2 c"(z) = + az + b. 2 Applying the Dirichlet boundary conditions gives c"(z) = 1(1 2) 2 1 c"(O) = . 2 Thus c"(0) = 1/2 symbolizes a uniform reaction rate from the slab surface to the midplane. Figure 2 further suggests that 0.2 < 1,st < 5 contains the majority of the transition from one limit to the other. The Robin's boundary condition transforms to Bic d" c Biic" dc fl dz z= 1 1.36 where Bi, = kgL/D, is the Biot number for mass transfer for species i. Equivalently by eq. (1.31), 1.37 Bimc",i = 2c"(O) 4c<"(0)2. Reasons for choosing the minus sign are essentially physical. The dimensionless surface concentration, zero, is expected to lie between the dimensionless midplane concentration, 1.34 1.35 17 positive, and the dimensionless free stream concentration. Therefore the dimensionless free stream concentration is expected to be negative. A further reason for choosing the minus sign in eq. (1.31) will be given below. Insertion of the c"f expression into eq. (1.19) gives ci, for known cf. Knowledge of ci, plus c"(z) gives ci(x). Since ,,t is independent of c6,, c,, etc. for firstorder reactions, the final solution for the Robin's problem is immediately obtained from the Dirichlet problem in its present form. Some of the useful features of the c" formulation are that its midplane concentration is bounded even though its surface concentration is zero, that the choice of roots in equations like eq. (1.31) are easily made, and that it is independent of the species selected initially to be preserved in rp for the derivation. While these features are not essential in studying firstorder kinetics, they do expedite the analysis for secondorder and thirdorder reaction systems discussed in Chapters Two and Three. Effectiveness Factors Effectiveness factors are commonly used to compare the average rate of reaction actually achieved to that possible in the absence of concentration gradients. Three types are encountered: the internal effectiveness factor, r,,, the external effectiveness factor, fext, and the overall effectiveness factor, overa, = tinotex Only two need be considered at a time, and these will be Ier,,, and int. The internal effectiveness factor gives the ratio of the average reaction rate within the slab relative to the reaction rate at the exposed surface, Lee (1985) (1L) fr,(x)dx Di, x  'Iinrt =Lr rP, L r'Ps 1.38 for the slab problem. The dimensionless problem, such as eq. (1.26), is still a slab problem, with D = L = 1, so 1.39 fr"p(z)d 1"z = I t'int ) r"(z ) 1 r p(z=1) r"p(z=1) where r"p is the dimensionless rate expression obtained after applying eqs. (1.19) and (1.20) to rp. Using eq. (1.31) with eq. (1.26) gives int = 2 2c"(O) c"(0) 1.40 for the firstorder problem, again showing the need for the negative sign of the square root to obtain positive, or meaningful, internal effectiveness factors when using c". Substitution of eq. (1.32) for c"(0) gives the more familiar result Stanh(O,,) 1.41 (D lst valid for both reversible and irreversible firstorder reactions with the definition of 4, given in eq. (1.28). For the irreversible zerothorder reaction discussed earlier the equivalent result is TIint = 1 k 1 1.42 1 Sin 1 1.43 S= L 1.44 Both the zerothorder and firstorder results share asymptotic behavior of unity for small Thiele moduli values of ,k or s, and of 1/1k or 1/4w,t for large Thiele moduli values. This will be discussed in the next section. The overall effectiveness factor is identical to the internal effectiveness factor for the case of no external mass transfer resistance. When the Robin's boundary condition is required, the overall effectiveness factor is defined rp( c^ ^c...) r (CP=0) 1.45 "overall = is' c int = ext int = ? ", lint S(dc"/dz) 1.46 overall = rP(c =c) The subscript on c"f, the dimensionless free stream concentration of i, is required, since the relationships used to eliminate j from the reaction rate expression to form r"p are valid only within the slab and not in the free stream fluid. To use the last part of eq. (1.45) or to use eq. (1.46) requires more than just considerable care in writing r"p. The overall effectiveness factor remains species independent, because the numerical values of the r"p(c"r) are the same for each i, even though the expression for r"p in the free stream must be modified for each species. This ultimately is due to the species' concentrations being interrelated through ratios of mass transfer coefficients instead of ratios of diffusivities between the free stream and the slab surface. This will be illustrated for the firstorder Robin's problem. For the irreversible firstorder problem, note that Bi,, c" cAdz tin 1.47 21 and, since there is no second species to confuse the issue, r"p(c"A) = st c", 1, both inside and outside the slab, so Bi.A c f 1.48 Tioverall = 2 v,, 1 stc fA or 1 st 1 s 1 1.49 overall BiA BimA C"A BmA 1 int This sum of resistances format is discussed in Aris, p. 107 (1975). For the reversible firstorder reaction, the expression, r"p(c"fA) = 4t c"f 1, does not express the dimensionless reaction rate in the free stream nor does r"p(c"m) = st c"C 1, since both expressions contain no information about the relative rates of external mass transfer. Both expressions implicitly contain the effect of external mass transfer only for the selected species. The primary use of eqs. (1.45) and (1.46) in this work will be for single species reactions. A general expression for reactions with more than one species requires a new derivation beginning with the definition of the overall effectiveness factor in terms of a ratio of the dimensional reaction rate expressions, rather than the dimensionless reaction rate expressions in eq. (1.45). This will be discussed in Chapter Four. The Generalized Thiele Modulus In the above section, it was noted that the zerothorder and firstorder internal effectiveness factors were both asymptotic to unity and 1/4. This was not entirely accidental. The factor of two in V permits this match. In 1965 the three independent authors, Aris, Bischoff, and Petersen, developed defining relations for an arbitrary reaction rate expression that would produce a generalized Thiele modulus, to, such that ,t would be asymptotic to unity and 1/4,, Lr (e,) 1.50 2 f D,(a) rp(a) da see Lee (1985), where rp has been rendered in terms of a single species through relations like eq. (1.20) to permit integration. The lower integration limit, .q, is zero for an irreversible reaction, and a sensibly chosen zero of r,(c) = 0 for a reversible reaction, usually the one closest to c,. The two moduli derived so far, 4, and 41,, are also generalized Thiele moduli for the zerothorder and firstorder reaction respectively. Secondorder and thirdorder power law kinetics for irreversible single species reactions have been investigated in the context of traditional and generalized Thiele moduli as part of studies on general nth order kinetics Dd2 = ken. 1.51 dx2 It should be noted that the stoichiometric coefficient is neglected in writing the righthand side. This gives rise to no problem for single species reactions where it can be assumed to be absorbed into the rate constant, k, although it is a little careless. Because this work deals with many multiple species reactions, the stoichiometric coefficients will always be indicated except when discussing earlier work. Making eq. (1.51) dimensionless with the slab halfwidth, L, and surface concentration, c,, leads to the traditional moduli 2 L kc= D 1.52 D The generalized moduli are very similar in form, needing only an appropriate scaling term to produce the desired asymptotic behavior 2 (in+1)L2k 1) 1.53 The generalized moduli for n = 0,1,2,3 are given by 2 L2k Sn = 0 1.54 2De, 2 L2k SG n = 1 1.55 D 2 3L2kec 1.56 o = n = 2 2D 2 2L2k2, 1.57 = D S n = 3 D These four irreversible reaction systems will be used throughout as a basis for comparison with results obtained later for secondorder and thirdorder reaction systems. Plots of %, versus to for these values of n date to Bischoff (1965) who determined the curves for n = 2 and n = 3 by numerical integration of the reactiondiffusion equation. Figure 3 is a plot of this type generated from analytical solutions derived as part of this work. The figure is in general agreement with Bischoff's plot as well as one in Aris' book (1975), but these tend to lose the feature that the curve for n = 2 is closer to the curve for n = 3 then to the curve for n = 1 over the full range of ,o. In the context of the generalized modulus one can say that, at fixed 4 a reaction giving results similar to those for firstorder is less efficient than irreversible zerothorder reactions but more efficient than irreversible secondorder or thirdorder power law reactions. In other words, every physical single reaction system, either irreversible or reversible, has a location on Figure 3, to and %t, which can be compared to nth order behavior. This is perhaps the only setting in which diverse reaction kinetics can be readily compared. There is one further note concerning Figure 3. It is common to refer to the region 4,o < 3 as the region of negligible diffusion effects, r, 1, and to the region 4, > 4 as the region of diffusion control, i.e. insensitive to the rate expression S e so 0 I I 2 I SO o 2 0 I ^0 C. m E cj 4. *  o H 26 form. It is not common, and also not wise, to base these distinctions upon c"(0), since it may be possible in some systems to have c"(0) small and %, large. The asymptotic behavior with respect to 4,G is guaranteed, although anything is possible between the asymptotes. This is amply demonstrated in the literature for n < 0 power law kinetics, for nonisothermal exothermic reactions, and even certain types of LangmuirHinshelwood kinetics. Results for most LangmuirHinshelwood kinetic expressions, however, lie between the curves for n = 0 and n = 2 in Figure 3. Shape Factor Normalizations Zerothorder and firstorder solutions have been derived in cylindrical and spherical coordinates, as well as some twodimensional and threedimensional geometries. Previous workers have found that a characteristic length choice of Vp/SCx, the particle volume divided by the external surface area, brings firstorder internal effectiveness factor solutions for slab, cylinder and sphere into approximate agreement, i.e. about as close as the curves for n = 1,2,3 in Figure 3. There is excellent asymptotic agreement and approximate agreement between the asymptotes. A more sophisticated renormalization was proposed by Miller and Lee, (1983). The potential of shape factor normalization has been somewhat neglected, since analytical solutions existed for zeroth order and firstorder reactions in many geometries, making shape factor normalization unnecessary, while there were virtually no other analytical solutions in any geometry. This work presents a very large number of new analytical solutions in the slab geometry for which there are no corresponding solutions in any other geometry. An opportunity 27 exists to extend these new solutions to other geometries at least approximately using the established principles of shape factor normalization. Summary and Goals Ideal catalysts would always have high internal effectiveness factors. A new catalyst of arbitrary geometry can be compared to an equivalent slab through shape factor normalization. The internal effectiveness factor of the slab can be calculated numerically, estimated from a graph such as Figure 3, or calculated from an analytical solution. Such procedures are facilitated by having a large library of analytical solutions available, but this has not been the case. Analytical solutions for the internal effectiveness factor exist for n = 0 and n = 1, as well as one for n = 1 in terms of Dawson's integral, Aris (1975), plus Thiele's solution for n = 2 power law kinetics which is n2D 3L2k o) 1.58 where Co = c(x=0) is obtained from the solution of either L2 kF sn 3' 41 (s eo)J es 1.59 D 3 F .sin sin(15) SO N4 V/le,/0 C for c,/co < 1 + V3, or 2 I2K(15" ) F sin o) sin(15*) 1.60 SD 4 1 + / for c,/c > 1 + V3, where co is the dimensional midplane concentration, F is the elliptic integral of the first kind, and K is the complete elliptic integral of the first kind. Thiele's original notation was modified to conform with that used throughout this text. Thiele did not include an expression for c(x), which presumably would be similar to eqs. (1.59) and (1.60) with c(x) replacing c0 in one or more places, plus an appearance by x perhaps replacing L. Eqs. (1.59) and (1.60) are awkward in appearance, require two different elliptic integrals to evaluate, are each valid over only a specified, but implicit range, and are algebraically implicit in solution character since c0 is an unknown value of the solution which is needed in eq. (1.58) to determine t,. The extension of this solution method to other secondorder kinetic systems may be possible, but has not been studied by others as far as is known by the author. It may well be that this result discouraged other engineers from attempting to solve more complicated secondorder kinetic reaction and diffusion problems. Nevertheless, one can hardly expect to solve a nonlinear second order differential equation without at least having to evaluate one nonlinear algebraic expression to satisfy the boundary conditions. The following chapters will deal with the solution of the two problems d2 = azc2 + a, + a0 1.61 dx2 and d2c a3 + a2 + al + ao 1.62 dx2 derived from net secondorder and thirdorder reversible and irreversible elementary reaction rate laws using the interrelationships between species developed earlier and using both the Dirichlet boundary conditions and the Robin's boundary conditions. Analytical solutions for the concentration profiles will be derived that are valid throughout the allowable parameter ranges without using elliptic integrals. The solution procedure will require determination of the midplane concentration in an implicit equation to satisfy one boundary condition. This single implicit step will complete the concentration profile equation as well as giving the internal effectiveness factor, which is a simple function of the Thiele moduli and the midplane concentration. Thus the new solutions will be more compact and easier to compute than Thiele's solution for n = 2 power law kinetics, as well as covering a much broader range of kinetic diversity. CHAPTER 2 CONVENTIONAL SECONDORDER REACTION SYSTEMS Overview of SecondOrder Reaction and Diffusion Limited work exists in the literature on the analysis of isothermal, coupled steady state secondorder reaction and diffusion problems. Several specific types of kinetics and physical configurations have been studied, and the results are summarized nicely by Aris (1975). This chapter will accomplish several interrelated goals. The conventional secondorder reactions, those which are irreversible, reversible with another secondorder reaction, and reversible with a firstorder reaction, will be analyzed. The analysis in this chapter will be limited to the Dirichlet problem for a slab. This will lead to equations interrelating the individual species' concentrations as well as to an analysis of the minimum number of parameters needed to formulate a single general problem. This general reaction diffusion problem will be solved analytically for the dimensionless concentration profile with the aid of the Weierstrass elliptic Pefunction, p, in terms of just two parameters. Three different dimensionless concentrations will be used to facilitate different stages of the development. While this is a little awkward, each has its advantages and disadvantages. These will be discussed. The mathematical nature of the Weierstrass elliptic Pefunction will be outlined. The general nonlinear governing differential equation leads to an implicit algebraic expression to be solved to satisfy one 31 boundary condition. The dependence of the solution of this equation on the two parameters will be examined. Secondorder reactions reversible with another second order reaction possess a special feature. Under certain values of the kinetic rate constants and diffusivities, the governing differential equation will degenerate to first order kinetics with linear differential equations. This could be forced to occur in many instances by adjusting temperature, for example, without invoking any infinite diffusivities or other asymptotic limits. Finally, an analytical expression for the internal effectiveness factor will be examined as a function of the two most fundamental parameters. Burghardt's recent article (1990) still claims that there is a lack of an explicit relationship between the internal effectiveness factor of a pellet and the characteristic parameters for other than zerothorder and firstorder reactions. Burghardt uses theorems from singularity theory to divide the feasible parameter space into regions with different bifurcation diagrams, i.e. with different numbers of possible solutions, for an exothermic reversible firstorder reaction with a uniform pellet temperature different from that in the surrounding fluid. Their analysis should now be possible for selected secondorder kinetic cases. Relationships between the internal effectiveness factor and the new analytical solution will be derived here for seven classes of secondorder kinetics. Analytical solutions for secondorder kinetic expressions are generally only possible for Cartesian coordinates and then only in terms of elliptic functions or integrals. These are either sufficiently unusual or unavailable that many engineers would undertake a numerical solution of the problem instead. This need not be the case as will be shown 32 below. Furthermore, a calculated concentration profile is not required to determine the internal effectiveness factor. The internal effectiveness factor can be calculated if the midplane concentration alone is known. As will be seen below, the initial dimensionless forms of the individual species' governing equations need contain no more parameters than the number of distinct species in the elementary reaction plus one for the Dirichlet problem, i.e. up to five for the most general secondorder reversible case, A + B 4 P + R. The number of dimensionless groups contracts to just three species' dependent parameters in the first step of the new analytical solution procedure for the dimensionless concentration profile (fewer for irreversible reaction cases). The analytical solutions derived in this chapter contain an expansion in terms of the Weierstrass elliptic Pefunction that is arguably easier to use than those for the traditionally acceptable firstorder irreversible reaction in an infinite cylinder, which requires the modified Bessel function of the first kind. Some earlier analyses exist in the literature for simultaneous secondorder reactions and diffusion, Thiele (1939), Maymo (1966), Bailey (1971), Tartarelli (1970). The previous analyses have not produced much general theory about secondorder elementary reactions. These early analyses have tended to be in the spherical coordinate system of catalyst pellets and have relied heavily on numerical methods to obtain solutions. Often these authors have invoked stoichiometric surface concentrations and/or a common diffusivity for all species to reduce the dimensionality of the parameter space in order to permit graphical presentation of the results. Indeed, one recurring problem 33 in previous presentations of solution results for simultaneous diffusion and secondorder kinetics has been the confusion in identifying the minimum dimension of its parameter space. Among these earlier analyses are some with up to eight dimensional parameter spaces. The first step reduction of the governing equations taken in this chapter produces a threeparameter basis for isothermal reaction kinetics with secondorder forward reactions, and either second, first, zerothorder or no reverse reactions (nine kinetic schemes are possible, counting two with zerothorder reverse reactions). Numerical values of the three parameters depend on the species selected for the calculation. This is not a major drawback, but it can be inconvenient. All three parameters are nonzero except when, (1) a special firstorder degeneracy occurs, (2) the surface concentrations reflect complete or equilibrium conversion, or (3) the reaction is irreversible (2A * products and A + B products). Further analysis of this problem yields a twoparameter basis which is independent of the chemical species selected for the derivation. These two parameters are considered to contain the essence of secondorder reaction and diffusion behavior. A clear picture has emerged with respect to the conditions under which the general secondorder reversible reaction system is free from diffusion limitations or is diffusion limited as a result of the reduction to two parameters. Also emerging later in this chapter will be some less obvious results. These include the manner in which the internal effectiveness factor is affected by the two parameters, bounds on the dimensionless midplane concentration and its estimation, a bound on the relative magnitudes of the two 34 parameters and asymptotic behavior of the internal effectiveness factor as a function of the two parameters. Some additional practical problems have been studied by others. These will be briefly summarized here. Bailey (1971) has analyzed the problem of optimum surface concentration for the irreversible reaction case A + B  products, when the diffusivities are unequal. Tarterelli et al. (1970) have studied the two irreversible kinetic cases in a macroporemicropore catalyst model. Thiele (1939) developed an analytic solution for the slab midplane concentration and internal effectiveness factor for the single species' irreversible reaction, 2A  products, using elliptic integrals instead of elliptic functions; see Chapter One. The selectivity of porous catalysts undergoing parallel reactions was examined by Roberts (1972), and includes secondorder reactions in parallel with either a zerothorder or firstorder reaction. Roberts also used elliptic integrals. Two early studies on nonisothermal behavior are noteworthy in that the authors disagree on the results obtained, Tinkler and Metzler (1961) and Petersen (1962). These earlier studies as well as this study neglect the effect of volume change during reaction, which is most often valid when the system concentrations remain near those at the surface, when the effective diffusivity is predominantly due to Knudsen diffusion, or when the fluids are dilute gases in an inert gas solvent or liquids. Governing Equations and Dimensionless Groups Consider the general secondorder reversible isothermal reaction of the type A + B # P + R, occurring in a slablike medium or along the axis of a cylinder as 35 illustrated in Chapter One, under the assumption of constant effective diffusivity, Di. The governing equations take the following form d2 A d 2p DA d r, DP r, dx2 dx2 2.1 DB rp DR rp dx2 dx2 where rp = k,cpCR lkACB and x is measured out from the slab midplane. These expressions can be written more generally as Si r 2.2 dx2 The rate of product formation rate expressions, rp, considered in this chapter are given in Table 2.1. The first seven stoichiometries listed there are discussed in this chapter. Zerothorder reverse reactions are not discussed here. The analysis that follows always assumes a symmetric problem, i.e. the slab with sealed midplane or the slab with two exposed faces with identical surface conditions. This leads to either a Dirichlet or Robin's problem at the exposed surface. The Dirichlet problem is the more basic of the two. Generally a solution can be found for the corresponding Robin's problem, if the Dirichlet problem can be solved. There are many steps between this point and the analytical solution. The solution strategy to be employed will be outlined here so the reader knows what to expect in the pages ahead. The initial step will be to take a set of n coupled nonlinear ordinary Table 2.1 Typical SecondOrder Kinetic Schemes Case I II III IV V VI VII VIII IX Stoichiometry A+B P + R A+A P + R A+A P+P A+B P A+A P A + B A + A A + B zerothorder A + A < zerothorder Rate Expression rp = k1CCR IACB rp = kcR kfcl rp = kCp ktCACB rp=kjCpk ktC rp = k ACB rp = k K rp = k, kfcAB rp = k, kt r =  37 differential equations (ODE's) and form n single species ODE's, where n is the number of species appearing in the rate expression. These uncoupled ODE's will be made dimensionless using a characteristic length and the species' surface concentrations. It will be prudent at this point to condense and label numerous common dimensionless groups contained in the ODE's to simplify notation. The fourth step is an optional step as far as obtaining the analytical solution is concerned. A properly chosen linear transformation of the dimensionless species concentrations leads to a species independent ODE containing fewer parameters which are themselves independent of species in the sense that they are identical regardless of which species is used as a starting point for the transformation. The variables are separated and integrated one time by conventional methods in the fifth step, see Lee (1985), leading to a concentration polynomial one degree higher than in the original reaction rate expression. Sixth, take a complete, or entire, Taylor series expansion of this polynomial about one of its zeros (no truncations). Next, make a very special and particular nonlinear transformation of the concentration variable. Then separate variables for the second time and recognize that the result is something with particular significance, i.e. the Weierstrass elliptic Pefunction. Finally, back substitute to whatever point desired to produce dimensionless or dimensional concentration profile equations. This chapter will consider the Dirichlet problem. Conventional boundary conditions for the Dirichlet problem are ci = ci, x = L d = 0 x = 0. 2.4 dx The following relationships between the species are readily derivable from the governing equations, see Chapter One, since (Dici/vi Djcj/vj) are solutions of a potential function, regardless of the coordinate system chosen, and thus equal to a constant. For the Dirichlet problem, this constant is known from the surface condition, i.e. DAA DBB = DAAs DBsA 2.5 DAA + D, = DA + D, e 2.6 and similarly for R as for P. These can also be written more generally as D. D. (c c) 2.7 v. vj These expressions are critical to continuing the analysis for nonlinear kinetic rate equations. All but one concentration can be eliminated from rp using eq. (2.7). Then the system can be reduced to a single ordinary differential equation. If A is selected, and concentration and length are made dimensionless with CAs and L, then the above second order reversible system reduces to d2cA (L2kA L2krDAC As 2 L L2k kes + + R ,L A Ak R a 2.8 DR Dp D DR DA , DPDR DR D where CA = C^/C, and z = x/L. In spite of the lengthy coefficients, eq. (2.8) is an improvement over eq. (2.1). The righthand side of this equation will be referred to as the diffusion modified reaction rate expression. Dimensionless species' solutions depend only on the values of the three coefficients of cA, CA and 1. The above equation can be written d 6a c,2 + 6acA + 2a 2.9 dz2 with revised boundary conditions CA =1 z= 1 2.10 dcA 0 z = 0. 2.11 dz The numerical constants in eq. (2.9) are per Whittaker and Watson (1950) and, when following their solution method, these choices reduce the occurence of fractions in the Table 2.2 Coefficients of c2 for SecondOrder Reactions Case Species 6ai I A A+B PA'P+R S+^ 2 t2 t2 SB B+ rApB +A +R/ A+B I P #+ A+B/rAP I R ~, >2 2PAP+ II A 2(2A rAPp+R/4) II P I+R 42A/rA II R R+I 4A +PrA +R III A 2(i2A AAP 2P) III P 2(2&p 42A/rAP) IV A A+B IV B B +A IV P $A+B/rAP V A 22A V P 442A/rAP VI A A+B VI B B+A VII A 22 Table 2.3 Coefficients of c, for SecondOrder Reactions Case Species 6 I A I+A +B + P+R + R+P+2FAPp+R I B $+BB+A +"P+R +R+P+2AP B+AR +P/A+B ID 2 t +2 2 + I P R+P4 +R +A+B +B+A+24+B/rAP I R Pi+R R+P+ A+B B+A+ B+2 AR+P/AP P+R II A P 2+R + p+rAP +P II P D2 2 II P RLPIP+R+4 2A 82A/rAP II R IP+R,pR + 42A + 82 +P/ FApP+R III A 2(242p + 2rFApp) III P 2(24 +24A/AP) IV A 2 +A B+ IV B A+B B+A+ IV P (D 2 +D2 t2 IV P 4P+A+B' B+A+24 A+B/PAP V A p V P 4+4 +4 A/rAP VI A B+A A+B VI B 2A+B +A VII A 0 Table 2.4 Constant Coefficients for SecondOrder Reactions Case Species 2a, I A (4+p++ R+rAP P+R+ R+p/'AP) I B (42+R+ 2+PrA +A 2 +R/4 +B+ +B +P/rAP2+A I P (IA+B B+ArAP B+A ++B/rAP) I R I R ( B+ B)+A+ rAPt+A P+R/,R+P+A +B R+P/rAPDP+R) II A (p+R + +P+2p +p/FAP+ Ap4p+R/2) II P _A(4+PAp+4/PAP) II R 4A2(4 +rAPp +R/42 ++4 +p/rPAP +R) III A 2 4p(2+rAP+ /rAp) III P 2i2A(2+PAP+ I/FAP) IV A (1 + 1/PAP) IV B 4p(1 +4AB/FAP B+A) IV P 2 4p2 pot2 _4p2 IV P 4A+ B +AAB+A A +B/rAP V A 4p(1+/rAp) V P 2A(4+rAP+4/rAP) VI A 0 VI B 0 VII A 0 43 final results. Six dimensionless groups appear naturally in these three coefficients in eq. (2.8). If B, P or R had been selected instead of A, then some of these groups and some new, but similar, groups would have appeared. Expressions for the ag are given by reaction and species in Tables 2.22.4 for the seven cases free of zerothorder reactions using a notational standard described below. A shorthand notation system would be useful to help minimize the number of dimensionless subgroups appearing within the more general parameters and facilitate studies of parameter interrelationships. After all, there are 17 species dependent ODE's under consideration at this point. Such a notational system should also retain some physical significance or familiarity. Toward this end, a set of irreversible reaction Thiele moduli will be defined I2 = L2 ki,/ D 2.12 I i LJ where i,j are either both products or both reactants. The kl indicate the rate constant, either kf or k,, multiplying the concentration of species I in the rate expression. Such a group would appear naturally in the study of the irreversible reaction I + J products. Large values of such a group would suggest possible strong diffusional effects, while small values suggest a possibility of nearly diffusion free operation. The true state of affairs will be revealed by further analysis. If i = j, define the irreversible reaction Thiele modulus 0 = L2 ki ,/D, 2.13 which would arise in studying 21 products or I + I products, and define r Di is 2.14 where i and j are not both products or both reactants. This ratio serves as a bridge between product and reactant species. The set {4A+,, IB+A, P+R, IR+P, IAp} is sufficient, as can be seen from the tables, to write the dimensionless governing equations of all four species in the reaction A + B ; P + R. Another sufficient set {4A+B, 4,+R, "AB, rAP, IAR} is prototypical of many similar sets, as is {lk/k,, 2A+B, AB, AP, ,AR } The first set is also sufficient to express the internal effectiveness factor using the derivative of the dimensionless concentration. It can be seen in Table 2.2 that 4'+j is the coefficient of ci when the rate constant of the opposing reaction is zero, and that it has similar dependence on physical parameters as the generalized Thiele modulus for a secondorder irreversible reaction of a single species given in Bischoff (1965), Aris (1975), and Lee (1985) 2 = 3L2ke,/2D. 2.15 The following interrelationships between the parameters in the new notational system should be noted Fi = 1/r.. 2.16 2 r = j k 2.17 j"r+ = i 45 For the case of mixed firstorder and secondorder reversible kinetics, a first order irreversible reaction Thiele modulus will be defined in the traditional way as in Chapter One, ,I = L2'k/Di, where i indicates the species undergoing a firstorder reaction and its rate constant. With this adopted nomenclature, the coefficients of the governing equations for many secondorder systems can be readily condensed. Expansions for the coefficients, aj, for the first seven secondorder kinetic expressions listed in Table 2.1, have been formulated in terms of the irreversible reaction Thiele moduli and a single FIj. The results of employing the above nomenclature scheme are summarized in Tables 2.22.4, for d2ci/dz2 = 6a1,ic + 6a2ic, + 2a3,. Secondorder systems with a zerothorder reverse reaction can be handled by the integration methods that follow, but are not considered in this chapter. In dealing with such reactions, it is potentially possible to associate a real species with the zerothorder reaction, and to have the concentration of that species drop to zero at an intermediate point in the slab. This requires the introduction of an additional parameter to establish this position, and conventionally the rate constant for the zerothorder step would have to be made zero or set equal to the opposing rate beyond that point of penetration. Solution of the Dirichlet Problem The 17 dimensionless species dependent governing equations in the above seven conventional cases are of the same form, namely dc = 6 aic + 6a2,c + 2a3, 2.18 dz2 where ali, a2i and a3, are those coefficients summarized in Tables 2.22.4. Two generic ways are developed here to further unify the species dependent parameter problem of eq. (2.18) to a unified twoparameter boundary value expression for the general case where the aij are all nonzero. This step is not critical to solving the differential equation, but reducing the parameter space dimension is quite useful and well worth pursuing. For the first of these methods, let c' = 6a,,(c, 1). 2.19 The variable, c', could also be interpreted as a dimensionless extent of diffusion reaction variable, since it characterizes the species independent evolution of the diffusion reaction and has the value of zero at the slab surface. The result that c' is independent of species is anticipated in leaving it unsubscripted. Then eq. (2.18) becomes d2 = c2 + (12ali+6aZi)c' + 6ali(6ai,+6a2+2a3,) 2.20 dz2 which can be rewritten as d2c = 2 + 0 + T 2.21 dz2 with boundary conditions c' = 0 at z = 1 2.22 dc' d 0 at z = 0 2.23 dz 47 where ,t = 12a1, + 6a2i and = 6a1i(6a1i + 6a2i + 2a3). The sign of c' is no longer necessarily positive, unlike the case for c, in eq. (2.18). The scaling choice made in eq. (2.19) leads to a coefficient of unity for c'2. Other scaling choices are possible that would lead to a fixed coeffient other than unity for c'2, but they do not seem to offer any particular advantages. The procedure for the first integration of the general ordinary differential equation above is well known; see Lee (1985). Let q = dc'/dz, then dq c'2 + s2c + 2.24 d z . Multiply both sides by 2qdz, equivalent to 2dc', and integrate to obtain q2 = 2 '3 + ,c + 2' c' + a 2.25 3 where a is an undetermined constant of integration. It is possible to substitute the definition of q, separate variables, choose integration limits, etc. at this point, but the analytical solution of that integral would not be known. Note that q(c'(0)) = 0 by the boundary conditions. This establishes a relationship between a and c'(0), namely 2 22.26 a = c'(O)3 ~tc'()2 2Yc'(0) 2.26 The value of q(l) will be needed later for determining the internal effectiveness factor which depends on the surface concentration gradient. It is given by q(1) = i = c'(0) 0 c '(0)2 2c'(0) 2.27 48 where the same sign as is generally correct physically. If a reliable estimate of c'(0) is available, and concentration profiles are not required, then it is possible to avoid performing the second integration, and to proceed directly to the calculation of the internal effectiveness factor. Assuming the contrary, it is important to recall that c'(0) is a root of q2(c'(0)), since (c'" c'(0)) = (c' c'(0)) times a (polynomial of order n1) for all relevant n in this study, CRC Standard Mathematical Tables, (1975). The fact that the remaining boundary condition is at z = 1, not at z = 0, must be temporarily ignored in the following steps. First expand q2(c') about q2(c'(0)) in a Taylor series. q2(c') = (2c'(0)2 +2~ 0f c'(0) + 2')(c' c'(0)) + 2.28 (4c'(0) + 2 ,)(c' c'(0))2 + 4(c' c'(0)) Note that this equation is complete, and not an approximation, since eq. (2.25) is a cubic polynomial. The point of this step is to eliminate, or at least disguise, the constant term with c'(0) in eq. (2.25) by moving it into the differences with c'. Then define q2(c') = 4A3(c' c'(0)) + 6A2(c' c'(0))2 + 4A1(c' c'(0))3 2.29 Substitute dc'/dz for q, separate the variables c' and z, and then take the integral from (0, c'(0)) to (z, c'). Z C S= fda 2.30 0 c0) )4A3(ac'(0)) + 6A2(ac'(0))2 + 4A(ac'(0))3 Make a change of variables to r = (a c'(0))': z= dr 2.31 t 14A33 + 6A22 + 4A, Note that if the original kinetics had been thirdorder in c', it would only have added a constant term to the quantity in brackets, which would have been called Ao; see Chapter Three. This is therefore a powerful substitution, which is not fully exploited here but will be in Chapter Three. The next substitution is conventional and simultaneously reduces the coefficient of the cubic term to four and eliminates the quadratic term. Let a = A3r + A2/2, then z = do 2.32 o 403 (3A2 4A1A) a (2AIA2A A3) Sf do. 2.33 a 44o3 g2 g3 The numbers, g2 and g3, are invariants of the original thirdorder polynomial, Whittaker and Watson (1950), and are also (and more readily) given by ( s,4 Y) 2.34 12 4c'(0)3 + 6 t c'(0)2 + 12 c'(0) + 6 ,s't 6t 2.35 3 216 50 The relationship between a and c'(0) is needed to verify the g3 equivalence. The invariant, g2, is independent of c'(0) and therefore ab initio known. Eq. (2.33) is one defining relation of the Weierstrass elliptic Pefunction. It is also written with an elaborate script capital P, p, Whittaker and Watson (1950). The variable transformations following eq. (2.30) were made with this end in mind. The cubic polynomial in the denominator will be referred to as the canonical polynomial of the problem, even though it is not quite the canonical form of a cubic polynomial given in most mathematical handbooks. These lack the factor of four and use addition rather than subtraction of the lower order terms. The defining relation for the Weierstrass elliptic Pefunction translates eq. (2.33) to S= p(z; g2,g3) 2.36 Substituting r for a and then c' for 7 gives the analytical solution for the dimensionless concentration profile in terms of the still undetermined midplane concentration 6(c'(0)2 + l2,,c(0) + ) c'(z) = c'(0) + 6( 1c'() + 2.37 12 P(z; g2,g3) 2c'(0) 4I,, where c'(0) must be evaluated from eq. (2.37) using the boundary condition of eq. (2.22). Eq. (2.37) can be inserted into eq. (2.19) to give the dimensionless species concentration profiles. An alternative approach, Chapter One, to reducing eq. (2.18) is to take c" = lc c' 2.38 6al, + 6a2, + 2a3, T or equivalently in terms of original problem parameters 2.39 C, ( i ^) L2' vi r where rp, is rp evaluated at the surface concentrations. This leads to the following revised governing equation d2c" cu+c" dz2 c' + 1St c 1 dz2 2.40 where 1,, and I are identical with those defined by eqs. (2.20) and (2.21). The boundary conditions for c" are identical with those for c'. The analytical solution for eq. (2.40) is derived exactly as eq. (2.37) and is given by c"(z) = c "(0) + 6( c"(0) + c "(0) 1) 12p(z;g2,g3) + 2Yc"(0) 2, with invariants 4lst 4 T) 82  ^  9g2 12 43c"(0)3 +6V2 tc" (0)2 12 2c"(0) + 6T2t lst 3 = 216 2.41 2.42 2.43 52 The same results could be obtained by direct substitution of eq. (2.38) into eqs. (2.34), (2.35) and (2.37). The numerical values of the two invariants are unaffected by the change of variables. The integration technique could have been applied to the original ODE with three parameters as well. Remember that the reduction to two parameters was not required to solve the ODE. Interested readers may consult Koopman and Lee, 1991, where this form is derived in full. Inclusion of this integration is redundant in this chapter, and the threeparameter form is also less useful for what follows with regard to secondorder reaction and diffusion systems than the twoparameter form. Using either eq. (2.37) or eq. (2.41), it is now possible to present general solutions for internal effectiveness factors, slab midplane concentrations, etc. as functions of 41t and \I. The Two Fundamental Parameters The two fundamental parameters, ,t and *', have significance beyond reducing the problem from one with three parameters, the aj, to one with two parameters. The first of these parameters, 4{, is never negative and is numerically independent of the species chosen in the derivation. As seen in Table 2.5, it takes the form of a sum of irreversible reaction Thiele moduli, i.e. moduli that one would derive from each species' governing equation by ignoring reverse reactions s, = (vi ri)2 2.44 where the sum is over all species, and ,o i is either 4,+j, '2i, or 4,i depending on the kinetics. This expression is reminiscent of the one for the Thiele modulus of a firstorder 53 Table 2.5 Defining Relations for the Parameters of the Governing Equations for Seven SecondOrder Reaction Systems alt ,2 it2 +iA2 +4h2 A+B+ +B+A P+R R+P 42A +P+R + +P 4.tA + 404p 2 2 SA+B + IB+A +P 4 2A + 42 42A 4,D22 Case I II III IV V VI VII kA+BB +A +P+R4R+ArAP~B+A P+R A+BR +P AP 42A +p RR+prA 2A P+R4A I'+P/rAP 44t4+4.t4 42 2 4 +442, 4A2P(rIA+ I/PAP) 2 t2 A+B B+A A+BBPAP 4A 4A2A AP A+B B+A 4 .A 54 reversible reaction, 4c = A + Cl given in Satterfield (1970) and in eq. (1.28). A Thiele modulus symbol has been chosen for ist because of the direct relationship with irreversible reaction Thiele moduli in eq. (2.44), as well as because it multiplies the first power concentration term. It also has a similar influence on calculated solutions as that of the Thiele modulus in firstorder problems when higher order effects are minor. The firstorder modulus, 4,, is also independent of the scaling choice of eqs. (2.19) or (2.38). A general derivation for the four species reaction, vAA + vBB # vpP + vRR, and rp = kl C~ R kCACB, gives the following expression for V,, 2 VpL2kdp, + vRLk2R, e vAL2kA vBL2kfB 2.45 DR D DB DA R D Simple rules apply for deriving the expressions in Table 2.5 from eq. (2.45) when, for example, A and B are identical species, or when the reverse reaction is firstorder (no R). Since complete expansions are given in Table 2.5, discussions of these rules are deferred until Chapter Three and the more complicated expressions for thirdorder kinetics. The second dimensionless parameter, 6a1,(6ai, + 6a2i + 2a3) or NI, has no first order analog, and in eq. (2.40) it is the coefficient of c"2. Symbolic expansions for *I are given in Table 2.5. There it is seen to be composed of irreversible reaction Thiele moduli as t,, plus the bridging parameter, lAp. The irreversible reaction Thiele moduli appear only as products with themselves or other moduli. For the general secondorder 55 problem just discussed above, can be expressed in terms of original physical constants as P = L VPVRkr VAk )(CR kfces)e 2.46 Dp R DAD, )rAe 2.46 It is seen that is composed of a difference times the reaction rate at the slab surface. Both parts of this expression are of arbitrary sign, so can be either positive or negative. In fact could be zero for nonzero surface reaction rates through the first term in eq. (2.46). This leads to precise firstorder behavior of the solution as will be discussed later in this chapter. Note that for irreversible reactions, k, or kf is zero, and * is always positive. The parameter, *, will be referred to as the secondorder modulus. It is dimensionally similar to the fourth power of a Thiele modulus, L4k2',/D2. The secondorder modulus, *, is independent of the chemical species selected as a derivational basis as was the case for V,, and unlike the case for the numerous aij. A discussion of simple rules to use eq. (2.46) to form the expressions in Table 2.5 will be deferred to Chapter Three as mentioned above. The feasible ranges for 4,, and define a region somewhat smaller than a semi infinite plane, (4it 2 0, f). Although can be positive or negative, its positive range is limited physically by the condition that all species' concentrations, ci(x), must be real if the diffusion modified reaction expression (righthand side of the diffusion equation) is set equal to zero. The dimensional, diffusion modified "equilibrium" concentrations are related to the dimensionless diffusion modified equilibrium concentration, c",(z), which constrains c"(z) by 2 Y st + Ols . 4 2.47 0 c c"(z) < c"eq(z) = 22.47 where the righthand side arises from applying the quadratic formula to the righthand side of eq. (2.40). This diffusion and reaction driven concentration shift generally does not lead to an equilibrium state if applied to the set of external surface concentrations in the absence of diffusional effects. In other words the extent of reaction of the diffusion modified rate expression is a physically different concept from that in homogeneous kinetics. Applying the requirement of real valued concentrations at equilibrium leads to the following physical parameter constraint 4 Slst 2.48 4 which ensures c",(z) is real and also implies g2 > 0. The same result, i.e. that ,I/4 is the largest value available to I, is also obtained by solving the problem: maximize *(4,+B, I,+A, Pp+R, R+P, rAP), subject to constant ,,, and the species' parameters constrained to be positive. An additional word is in order about the constraint in eq. (2.48). The equality in eq. (2.48) can be examined by expanding {,,/4 I for Case I of Table 2.1 in the physical constants of the original problem to give S4 = L'4 k ( e _( eft) + k,( Ps ) + 1st D D DR D D ADB'2.49 L'kfk (DA A + DP P) (DcBS" + DR ) > 0 DADBDpDR If species A and B are identical and k, = 0, or if cA,/D = CB,/DA and k, = 0, then 14/4  9 = 0. Similar conditions occur if the roles of (A, B, k,) are switched with (P, R, kf). Therefore the equality of eq. (2.48) always holds for the case of the irreversible reaction, 2A , products, and can occur in the case of the irreversible reaction A + B  products. The equality is not expected to ever occur for reversible reactions. This reflects the fact that even if products are not present at the surface, they will be present in the slab and will influence the achievable reaction rates in the slab. The slab midplane concentration, either c'(0) or c"(0), must be obtained by solving a nonlinear equation to complete the analytical solution of eqs. (2.37) or (2.41). The development so far is such that c"(0) has a much narrower range than c'(0), which ranges from +oo to oo in the ( *,t, I) parameter space. The sign of c'(0) changes with the sign of I, and when I I is large, c'(0)  is also large. The rescaling to c" in eq. (2.38) yielded some surprising results as seen in Figure 2.1, where only feasible combinations of 4,,, and I were used per eq. (2.48). Not only was c"(0) always positive, as expected, but it was seemingly bounded above by onehalf. Integration of eq. (2.40) assuming the righthand side is a constant equal to 1 gives onehalf as the midplane concentration. If the magnitude of the dimensionless reaction rate decreases 58 0 00 5%, LO0 ~J 1) 0 0 c;0 cn (U 9 I / '~a 00 U LOO U? ~ ~ 6 o  C c 59 r O s I I (U E I 0 U, 0 o* I 6i u cdJ 5i Ln ~*S 0O OU 60 d 0 0 0 o 04 C 0E 0 1 0 o I O no 4 61 in the slab, the midplane concentration falls below onehalf. Figures 2.2 and 2.3 provide some twodimensional perspective on the surface of Figure 2.1 for selected parameter values. The limit of 41,t 0 at constant 9 gives values of c"(0) that are within 0.5% of those shown for 4, = 0.01. This limit has more relevance in less conventional kinetic schemes. Figure 2.1 is somewhat deceptive. Since the surface concentration, c"(1), equals zero, the maximum difference between c"(l) and c"(0) occurs at parameter ranges that correspond to high internal effectiveness factors. This is counterintuitive in the sense that small concentration differences from surface to center are normally associated with diffusion free cases and large concentration differences are normally associated with diffusion limited cases. This phenomenon has to do with the inverse I scaling constant in the definition of c". Eq. (2.47) had given one upper bound for c"(0), c"q(z=0), but it is only closely approached when c"(0) goes to zero. For example, at (c1,, ') = (0.1, 105), c",(0) = 91.6, implying 0 < c"(0) < 91.6, which is a much broader range than zero to one half. Figure 2.4 plots the scaled departure of either c'(0) or c"(0) from the diffusion modified equilibrium conversion as given in eq. (2.47) and where ,C'st t 4 2.50 eq 2 for c',. Note that (c'e c'(0))/c'q = (c", c"(0))/c" The departure from equilibrium conversion is negligible for strongly diffusionlimited cases. It is not 62 formally proper to discuss the solution of eq. (2.40) at I = 0, since the equation was derived using division by *. The solution to eq. (2.21) for c', however, passes through c'(0) = 0 at = 0, changing sign in the process, i.e. c" approaches zero over zero as I  0. Figure 2.4 further substantiates one of the claims made earlier on the behavior of the slab midplane concentration. The surface shown characterizes the approach to either diffusion modified equilibrium conversion (for reversible reactions), or complete conversion (for irreversible reactions), at the slab midplane. This approach is quite strong as 9 oo and/or 4k,, * oo. When , t,/4 (g2 = 0) as 4,, oo; however, the approach to high midplane conversions appears to be retarded, i.e. along the diagonal boundary of the parameter space. The accuracy of eq. (2.54), below, for ci(0) is also qualitatively indicated in Figure 2.4. If one defines 1 1 / cosh(Os,) c" = lim(c"(0)) = 2.51 0 Ol then the solution of eq. (2.41) for c"(O) should match that of the firstorder problem, see Chapter One, for d 2 ,stc 1 2.52 dz2 given by c"' above. The group, (co' c"(0))/cc', represents a scaled measure of the departure of the dimensionless midplane concentration for secondorder reactions from 01 00 C.o c o ci 0 *a I C O U o S ,4 0 C O C o U, / \ \ \ \ \ \ \ \ \ 64 that of the firstorder problem. This group is plotted in Figure 2.5 against 1,, and *. The plot includes contours showing the demarcation between values within 1% of first order behavior and beyond. Figures 2.4 and 2.5 are somewhat complimentary in their regions of small departure. Eq. (2.52) gives c"(0) as a function of i,, at = + 10 to within 0.1%. The double limit II * 0, I,., * 0 gives c"(0) = 0.50 exactly. Numerically, c"(0) < 0.5 at all 1092 values of 4,, and examined. Combining eqs. (2.19) and (2.51) gives an approximate relationship between the dimensionless species' concentration, c,(0) and ai,, 1,,, and * F( 11/cosh(i )) 4,(0) c,, 1 . ) 2.53 6i 6li, ist approximately valid for I < 0.1 by derivation, but applicable at some larger \II\ when 4),t is also large per the limits in Figure 2.5. Alternatively, when (c"q  c"(0))/c"q in Figure 2.4 is small, c"(0) is well approximated by eq. (2.51). This leads to a second approximate expression for ci(0) 12 ao. ( 2 / ei(0) 1 eis I + 0 12 aJ 2.54 which is acceptable for it,, > 5 or < 1000. Eqs. (2.53) and (2.54) together cover much of the parameter space. Therefore, if one does not wish to work with the analytical solution, then eqs. (2.53) and (2.54) offer one alternative method for estimating the midplane concentration and ultimately the internal effectiveness factor. U2 I 0, 0 U 0) 0i 0) .0 0) tC. 0 o U, U, 0) S e, 0 0 .0a vs0 0)0o bO4 0 (0 CM uo b 66 Finally, one should not generally think in terms of ,,st going to zero at constant '. This can be seen by noting in Table 2.5 that when < 0,  < M s,,t, where M is a constant, such as rAP/2 for Case I (when rAp > 1). In other words, the magnitude of is generally constrained by the magnitude of t'I and can not be chosen independently in most actual physical systems; however, there are possibilities for exceptions. The point is fairly subtle. How can 9 go to zero much more slowly than ?',? First note that 4,s goes to zero if (1) L 0, (2) kf * 0 and k, 0, (3) Di oo for all species, or (4) CA,/DB 0, CB,/DA 0, p,/DR 0, and CRs/Dp " 0. Then note that I goes to zero under the first three limits above, but the fourth must be modified to (4') CAsCBs/DADB 0, CpRs/DDR 0 0, CAsBs/DpDR 0, and CPCRs/DADB 0. Since only the first half of condition (4') is forced by condition (4) for 1,t, it is still possible for I I to remain large, but the rest of condition (4') often follows from condition (4) in practice. Still the key terms are CA,CB,/DpDR and CP,CRs/DADB. Consequently, only under unusual circumstances will I be large when 41,, is small. The Weierstrass PeFunction. oi The Weierstrass elliptic Pefunction, p, is discussed in Whittaker and Watson (1950) and other texts. One of its original uses was in the study of dynamics of nonlinear pendulum motion, with friction dependent on displacement squared and/or cubed, and the sin(x) expansion increased to include the cubic as well as linear term. In this situation the Weierstrass elliptic function provided an analytical solution to an approximate physical model. Here, if the assumptions on diffusivity, etc. are valid, the 67 Weierstrass elliptic function is an analytical solution to a true physical system, since the reaction rate forcing function is not approximated. The Pefunction, p, is a doubly periodic function in the complex plane, i.e. p(z + 2mc + 2nwo) = p(z) for any integers m,n. The ratio of w /ow2, the two halfperiods, is not purely real, and the only singularities in the finite part of the plane are double poles at z 2mw, 2nw2 = 0. The two periods, 2 0 and 2c2, define a single function. Many equivalent period pairs can define the same function; however, all such pairs are linked by common invariants, g2 and g3. Abramowitz and Stegun (1964) give a procedure for determining a base pair of frequencies from the two invariants, but this is not germane to the present problem. They also give an expression for p in terms of the two invariants, g2 and g3, 1 2t2 2.55 P(z; g2, g3) = 2 + 2.55 Z k=2 where there is no constant term, and C2_ 2 C g 2.56 20 28 k2 3 E Cm Cm 2.57 Ck= m=2 k 4 S(2k + 1)(k 3) Eq. (2.55) is the definition of choice for evaluating the Pefunction in this work. Abramowitz and Stegun (1964) give the coefficients up to C,9 in terms of C2 and C3 only. If g2 = 20 and g3 = 28, then C,9 = 6.04*106. (Evaluation of any midplane Values of the Weierstrass PeFunction Invariants g2 g3 0.0 1.0 0.0 3.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.1 0.1 0.1 0.1 3.0 3.0 5.0 2.0 5.0 2.0 20.0 28.0 20.0 28.0 40.0 20.0 P(1;g2,g3) 1.035813 1.108031 1.087146 1.014733 0.986156 0.915691 1.008586 0.991443 1.046923 1.3487 1.1955 3.856 1.180 3.423 First Five Terms of p 1.035714 1.107143 1.087035 1.014632 0.986061 0.915606 1.098585 0.991442 1.045974 1.3471 1.1945 3.606 1.061 3.229 Error (%) 0.01 0.08 0.01 0.01 0.01 0.01 0.0001 0.0001 0.091 0.116 0.081 6.48 10.12 5.65 Table 2.6 Typical 69 concentration from the final boundary condition requires summing the series at z = 1.) Some representative values of p(l;g2,g3) are given in Table 2.6. This summation will be time consuming, except for a digital computer, if g, and/or g, are large in magnitude. Note, however, that if (g2g3) < 0 and Ig2/g3  = 0(1), then p(1;g2,g3) and p(1;g2,g3) are closer to one than p(1;g2,g3) and p(1;g2,g3). In many regions of engineering interest g2 > 0 and g3 < 0. Convergence of the series for 0 < z 0.8 can be quite rapid, regardless of the invariant magnitudes, because powers of z tend to dominate over products of the invariants. The two invariants, g2 and g3, are approximately of the same order of magnitude as the largest irreversible reaction Thiele modulus, e.g. 4+A,, raised to the power of two or three respectively. Consequently, when the individual irreversible reaction Thiele moduli get near four, g2 can easily be greater than 20. (An exception is Case VII kinetics, where g2 is identically 0). The other invariant, g3, is frequently smaller in magnitude than g2 for weakly to moderately diffusion limited reversible reactions, and tends to be negative for the conventional secondorder reactions considered here. When one or more irreversible reaction Thiele moduli for a problem is near three, either g2 and/or g3 could be larger than twenty in magnitude and convergence of p will take more than just a few terms. It is best to determine a few of the coefficients in eq. (2.55) before assessing the unwieldiness of the exact solution. Larger invariant magnitudes were found to be indicative of steeper concentration gradients and stronger diffusional limitations. The practical point being made is that for small invariant magnitudes, p is 70 easily handled with a pocket calculator. A computer evaluation of a lengthy truncated series is only required for larger invariant magnitudes. The dependence of p on c"(0) can be very weak for secondorder reversible reactions, since g3 is proportional to powers of the two fundamental parameters added to similar terms multiplied times c"(0), which is always bounded by onehalf for the reactions being discussed in this chapter. Furthermore, I g3 tends to be a maximum at the equilibrium conversion, i.e. a root of Ic"2 + 4 sc" 1 or c'2 + t2c' + '. For fixed p(1;g2,g3), where g3 is an estimate of the true g3, eq. (2.41) becomes a simple quadratic equation for c"(0)2 4Wc"(0)2 (50st +12p(1; g2 g3))c"(0) + 6 = 0. 2.58 The situation above is simpler for the two irreversible secondorder reactions, Cases VI and VII, but not better. These reactions should have only two or one modulus respectively for even cruder analyses. For Table 2.1, Case VI, (A + B ), the two fundamental parameters depend on only two species' parameters, A,+B and 4+A, as do the invariants 2 2 B+A A+B 2.59 12 2 2 3 32 02. 3 ( .+B BA)3 +B 2 A+BCA (0) 2 2 2 2.60 g =  +( B'A4B)cA (0) 216 6 3 71 since CA(0) is also a function of +B, and CI+A. Barring a major difference in diffusivities of A and B, or a large departure from a stoichiometric surface concentration, g2 and g3 will be manageable numbers up to quite large P2. For Table 2.1, Case VII, (2A  products), there should be only a single parameter, and 4~ and are directly related as expected. The first invariant, g2 is identically zero and 6 3 S40CA (0) 2.61 g3 27 When g2 = 0, only every third term in the expansion of p is nonzero. Nevertheless, in both Cases VI and VII there is a limiting computational aspect at large values of the irreversible reaction Thiele moduli. This situation is closely related to one degeneracy or simplification of the Pe function that occurs when the discriminant of the canonical polynomial, 4o3 g2a g3, is zero, i.e. g3 27g2 = 0. This is the traditional criterion for a cubic polynomial to have a repeated root adapted to the canonical form used with the Weierstrass elliptic function. The discriminant of the canonical polynomial characterizes its roots. When the discriminant is positive there are three real roots, and when it is negative there is only one. It is easiest to use the c' equations for further analysis of zero valued discriminants 3 82 2.62 x 27 72 which can be expanded to 4c'(0)3 + 6 ,,c'(0)2+ 12'Yc'(0) +6~ ,Y 0fst =( (, 4TY)I 2.63 This expression can be treated as two cubic equations in c'(0). All six roots have been found and are given by o +,, J st 4 Y 2.64 c'(o) = 2 Olst2 Qj 4T 2.65 2 c'(0) = 2 4 2.65 where the two roots given by eq. (2.64) are both double roots of eq. (2.63) in addition to also being the roots of the diffusion modified reaction equilibrium expression, eq. (2.50). Thus the discriminant equality is expected to be met asymptotically in the limit as the midplane concentration approaches the diffusion modified reaction equilibrium concentration. The negative root in eq. (2.66) gives a concentration further from the surface concentration than the negative root in eq. (2.65), and is physically unachievable. It is potentially possible that the surface given by the positive root of eq. (2.66) intersects the c'(0) of (4+,t, 'I) diffusion reaction solution surface, eq. (2.37). This case was left unexplored, but methods were developed to deal with zero or near zero discriminants in general in the computer codes used to generate general solution surfaces. For Table 2.1, Case VII, cA(0) goes to 0 as VA goes to infinity, and g3 goes to zero, so both g2 and g3 are zero, and again the discriminant approaches zero. When 73 there is a repeated root, eqs. (2.18), (2.21) and (2.40) can be integrated using conventional methods to give concentration profiles as functions of either sin2(z) or sinh2(z) depending on the sign of the repeated root; see below. One special case in the analysis of the discriminaant occurs when c'(0) = c' = c' = 0, i.e. a gradient free case, and therefore the second derivative is zero. This leads to a line in the (4,,, ') parameter space, = 0. Crossing this line in parameter space corresponds to changing the number of real roots of the canonical polynomial. On this line, the numerator of eq. (2.37) for c'(z) is zero, and no solutions are obtained for which c'(0) 0. If the discriminant of the canonical cubic polynomial is zero, and a double root, ri, exists, then the third root, r3, is equal to 2i,; see Abramowitz and Stegun (1964). For a positive double real root, ri, p is given by 3 , p(z; g2, g3) = fl + > 0 2.66 sinh2(Jiz) Since sinh2(x) = (cosh(2x) 1)/2, this gives approximately a firstorder concentration profile expression when sutstituted into either eq. (2.37) or (2.41). The result is that the dimensionless concentration depends on the inverse of an inverse of a hyperbolic cosine function plus supporting constants. Two other cases arise p(z; g2, 3) = P1 < 0 2.67 sin( z) (z; 2, g)=1 = 0. 2.68 z The last equation is for a triple root, which only occurs when all three roots are equal to zero. These equations give a little more insight into the nature of Weierstrass Pe function. The physical significance of the double root cases to the nature of the Pe function is that one of the two characteristic periods of p becomes infinite. The Pe function then becomes simply periodic. The periodicity of the double poles of the Weierstrass Pefunction then goes as the inverse of the square of the simple zeroes of sin(2cx) or sinh(2oiy), for z = x + iy. If g2 and g3 are both real and if the discriminant is positive, then there are three distinct real roots, r, > r2 > i3, and p is related to Jacobi elliptic functions, e.g. sn(z), by relations such as P1 fs 2.69 P(z; g2, g3) = 3 + 2.69 sn2( r3 z) Other relations with other Jacobi elliptic functions apply for other root cases. Since no roots are known in advance due to their dependence on the unknown midplane concentration, these expressions are both awkward in applications and difficult to generalize much as was the case for Thiele and his elliptic integrals (1937). Eq. (2.37) with p is preferred as being more general, since it holds throughout the parameter space regardless of the number of real roots. 75 Thus an analytical solution has been derived for the Dirichlet problem for one dimensional Cartesian coordinates in terms of the Weierstrass elliptic Pefunction for all seven cases of Table 2.1, subject only to some practical computational limits as described above. This solution becomes the focal point for further analysis. Concentration Profiles From the Analytical Solution It was mentioned earlier that, if the two invariants both exceed 20, then the computational effort for determining p becomes significant. Simplifications should occur in the other extreme of small invariant magnitudes. Consider the problem of determining c'(0) using eq. (2.37). From eq. (2.56) the expansion of P has the following form 2 4 2_6 _ 1 82 g 9 gzz6 3g+g3Z 2.70 (z; g, g) = + ++ + 3g3z + 2.70 P(z; 20 28 1200 6160 In the seven discussed cases, g2 is ab initio known. Therefore the first, second, and fourth terms in p are known functions of z, while the third and fifth terms are linear in the unknown invariant, g3. Recall that 4c'(0)3 + 6 t c'(0)2 + 12Yc'(0) + 6~ 'Y 2.71 g3 216 To find the maximum feasible range of g3, set dg3/dc'(0) equal to zero, giving (after multiplying by 18) 2.72 c'(0)2 + 2,Sc'(0) + T = 0 76 which is nothing more than the dimensionless diffusion modified rate expression evaluated at the midplane concentration. So a maximum or minimum in g3 occurs at the equilibrium concentration. Thus, c'(0), bounded by the surface concentration and equilibrium concentration, sets the possible range of g3 for given parameters. Table 2.6 indicates that a five term expansion has 99.9% accuracy at z = 1 when 1g21 < 3 and Ig31 < 3. The five term expansion for p is linear in g3, so eq. (2.37) is approximately a quartic equation for c'(0). Using just the first two terms of the expansion for p is within about 10% for this condition (and gets progressively better for smaller invariants), but leads at once to an easily solvable quadratic equation for c'(0) 5 0s,, 12 0.6g2 c'(0) = 8 where the positive sign must generally be chosen. This equation estimates c'(0) better than the two term expansion estimates p, and can itself be sufficiently accurate for small Sg21 and Ig3 . A twostep method to estimate c'(0) is suggested for I g2, I g3 < 10. Use the above quadratic equation to estimate c'(0), use the c'(0) obtained to estimate g3, eq. (2.72), use the five term expansion to estimate P and use eq. (2.37), which becomes quadratic with an estimate of p, to obtain a new solution for c'(0). The following example illustrates the procedure. Given 2 1 lst =15 T = 27 2.74 calculate g2 from eq. (2.34). g2 = 9.75 2.75 Find the equilibrium concentration from eq. (2.73) cq = 2.09167 2.76 For the first step, make a crude estimate of p using Sz 1 + = 1.49 2.77 20 Find a first estimate of c'(0) from eq. (2.74) above. c'(0) 1.21 2.78 Use c'(0) to estimate g3. g3 = 5.61165 2.79 For the second step estimate p with the five term expansion, eq. (2.71). (1; g2, 3) = 1.3397 2.80 Recalculate c'(0) with eq. (2.59) at z= 1 and the above p giving c'(0) 1.945 2.81 Repeating the second step gives c'(0) = 1.948, while the exact solution is 1.946, so the above estimate is within one per cent. The magnitude of g2 is approximately ten, so this is a borderline case of the method. The center concentration is at about 93% potential conversion relative to 100% at diffusion modified reaction equilibrium, so this example is not in the diffusion free region. 78 The twostep method is not limited to c'(0) but could be applied to c"(0) or cA(0). As the magnitudes of the two invariants get larger, the simple hand calculation above is less accurate. The twostep method for quick hand calculation is thus limited to the region of small to moderate diffusional effects, even though that may cover 90% of the possible range of c'(0) values, as in the above example. In the region of strong diffusional effects, a longer expansion of the Weierstrass Pe function is required. FirstOrder Subcases of the General Problem For the cases described earlier, there are numerous subcases that exhibit pseudo firstorder or even true firstorder kinetics. Maymo and Cunningham (1966) observed that for Table 2.1, Case VI, irreversible kinetics, as DB goes to infinity, c goes to cB,, and the rate becomes approximately equal to the pseudofirstorder form (k ,B A. Similar arguments, applied to one or more diffusivities, could be extended to any of Cases IV. Cases IIII in Table 2.1 all have another property in that at quite ordinary parameter values, all the coefficients of the c"2 term, i.e. 'k, vanish as do the a1i in all of the species' governing ODE's. This arises from the lefthand difference in the equation for 'i, eq. (2.46). This degenerate firstorder behavior occurs not as a limiting behavior as in Case VI above, but in the middle of the parameter space. This phenomenon is independent of the initial coordinate geometry (slab, sphere, cylinder), and is not a statement of reaction equilibrium. This is a degenerate zero in the discriminant of the canonical cubic polynomial not clearly shown in the development 79 using c'(0). To explore this situation requires the initial dimensionless species concentration equations in terms of the aj,. Developing the discriminant in the ai, for a species, I, gives g227g3 = a [(36a2a3 64ala3) + (108ala2a 54a)a4 (27af)a4 2.82 where a4 = 4a,1c,(0) 6a2c(0) 4a3c,(0) 2.83 As seen from eq. (2.82), ai, = 0 also corresponds to a case where the discriminant of the canonical polynomial equals zero independent of the six c'(0) zeroes derived earlier. A degeneracy in the definition of c' or c" occurs when a,, = 0. Thus all solutions in the neighborhood of the plane ai = 0, or I = 0, are strongly firstorder in character. This plane does not apparently correspond to any further unusual behavior of the solution other than that numerically (a2) = (a2)j, (i,j =A,B,P,R) and 6a2i = D' ,. Something of the physical significance of 6a2, is thus revealed. In the general secondorder reversible kinetics for Cases IIII, 6a2, can be rewritten as 6a2i = s 12al. 2.84 So for ai, i 0, 6a,, carries the species dependent deviation from firstorder behavior for the coefficients of both the firstorder and secondorder terms, ci and c?, in eq. (2.18). The above relationship, eq. (2.84), holds for all species in all seven cases, although its existence cannot be motivated by the above line of reasoning (search for firstorder behavior), especially for the two irreversible cases where some balance between reactant and product Thiele moduli is impossible. The initial recognition of eq. (2.84) occurred 80 prior to the discovery of the twoparameter basis and was the first clue that more fundamental parameters might exist. When ali, or I, is zero, it makes little sense to use c', defined with muliplication by ali, or c", defined with division by 'I. For Cases I III, when a,, = 0 d2ci 2.85 2 = 6 a2 c, + 2a3, = Q c + 2a3s 2.85 dz2 which has the solution 2 12a c,(z) = s + 2as cosh(Osz) 2 2.86 2 Csth(4 ) o s t 2 1st ( i st( 1st with the center concentration given by ci(0) = 0st + [2a3 ( ist s os 2.87 0,,tcosh(O,,d) This form of the expression permits ready comparisons with the similar expressions for cylindrical and spherical coordinates, Table 2.7 and 2.8. (The functions 4 2cosh(4,t), sinh(c(P,), and tlo(4L) in Table 2.7 all have series expansions in odd powers of 4, only, beginning with the same firstorder term.) In the limit of small V,, a3i also goes to zero, thus c,(O) goes to one in this limit as expected. For some neighborhood of a1i about zero, this expression can be used to estimate ci(0) for secondorder cases. Tables 2.7 and 2.8 also give the dimensionless concentration profiles and center concentrations 81 for the Robin's problem in all three coordinate systems. The general Robin's problem is discussed in Chapter Four, but solutions for the firstorder subcase are included here for convenience. The internal effectiveness factor is also defined more formally than in Chapter One by ldc (p + 1)D.d., 2.88 lint = L2( is, js'" ) where p is zero for a slab, one for a cylinder and two for a sphere; see Aris (1975) or Lee (1985) for general details. For the slablike medium, the solution is tanh( Q ) l int anh( a, =0, T' =0 2.89 01st which is identical in form to those for simple firstorder irreversible or reversible kinetics in Cartesian coordinates, Chapter One. When ali = 0, an analytical solution to the reduced firstorder kinetics problem can be derived in spherical or cylindrical coordinate systems as well. Internal effectiveness factors for the three common geometries are given in Table 2.9 for the Dirichlet and Robin's problem. It is worth noting that, given fixed surface concentrations, temperature could, in theory, be manipulated for Cases IIII, such that a1i = 0, through temperature's effect Table 2.7 Degenerate FirstOrder Concentration Profiles Dirichlet Problem ci. Slab ~ S + 2a3. 2a coh(i ) 3J cosh(40 1z) 2 0cs(cosh(l i 05 Cylinder 0 + 2a3, 2a3, 2 0(01z) 2 Io lst ) 2 " Sphere 'Lst + 2a3, sinh($tz) 2a3 0 sinh(4D) 0 z 2 Robin's Problem Bim(l + 2a3i/ O.,,) Slab 1i,,sinh( O,) + Bicosh(i,,,) cosh(01,,z) 2a3,/ st Bim(1 +2a3t/ Ct) Cylinder 1st I (01st) + Bimlo( 0 ,) *Io(1$,tz) 2a3i,/L 0 Bim,(1 + 2asI 2 (Bim 1) cosh(Qlst) sitih(N/ st ) Sphere cosh( + st ih sinh(4i1,z) 2a3, 1stZ 1st 83 Table 2.8 Degenerate FirstOrder Midplane Concentrations Dirichlet Problem i Slab lt + 2a3i[O s, cosh( )l1l / 02st Slab 1sst Is 0s,,cosh(Old) Cylinder lr2 isto (air s 1stIo ( Ist,) Sphere 0t + 2a3,[ s ch(s)] / ist sinh(Ols,) Robin's Problem 1 +2a3il/ st 2a3, Bim Slab (+s2 1 2a 2 1 + 2a3i/01lst 2a3i Cylinder Io(1) I ( t) 'Lit 2 2a3, Shr1 + 2asDis, sinh(ez,,) cosh(ol,,) sinh(ei,) 4D Sphere 01st Bim 4 IstBim 84 on the forward and reverse rate constants, forcing firstorder behavior to occur. No obvious benefit results. Internal Effectiveness Factors of SecondOrder Reactions The internal effectiveness factor measures the ratio of actual consumption of a species inside a medium relative to the consumption possible in the absence of concentration gradients, i.e. at surface compositions. For the single reaction systems considered in this article, this expression is given in numerous texts including Aris (1975) and takes the following form for secondorder systems in terms of c' or c" (dc' (dc"\ dz )l dz= 2.90 'lint m 1 where (dc' _2c,(0)3 2 2 2.91 = 3 2c'(O ,c'(O) 2c'(0) 2.91 Sdz )i s 3 where the sign is chosen to match the sign of I, and where ( dc" 2'Pc"(0)' 3 c( 2 c0) 2.92 dc 2 0 1c "(0)2 + 2c(0) 2.92 dz i \ 3 Table 2.9 Degenerate FirstOrder Effectiveness Factors Dirichlet Problem Slab Cylinder Sphere Robin's Problem Slab Cylinder Sphere tanh(oqb,,) lstlo(Ou ) 3 3  a2 stIst 1 1Qi13 13) 2 Bim 3coth(d,,,) 3/1st + 0Stcoth(lt) 1 Bim Alternatively, the dimensionless species' concentration eq. (2.88) can be used dci =, /4a1i(1 ci(0)) + 6a2,(l c(0)) + 4a3,(l c,(O)) 2.93 The remaining portion of eq. (2.88) for the internal effectiveness factor can also be expressed in terms of the aj. The external dimensionless rate occurs at ci = 1, the surface concentration, in the righthand side of eq. (2.18). It can be shown that the internal effectiveness factor is given by (dc, ) +1)dz 2.94 = ____ \dz 2 Tlint 6as, + 6a2i + 2a3, Table 2.10 gives the expression in the denominator for each species in all seven cases for the slablike geometry (p = 0). It should be observed that although ali, a2i and a3, vary from species to species within a case when using these expressions, I,, and do not, and r can be determined from these alone, rather than the a,j! No species dependence is suggested for it. Methods exist for estimating the internal effectiveness factor based on the generalized Thiele modulus approach, Lee (1985), Aris (1975). The asymptotic form of the internal effectiveness factor based on a generalized Thiele modulus is often sufficiently accurate for engineering requirements. This expression matches the asymptotic behavior at both the diffusion free and diffusion dominated limits, but is Table 2.10 Denominators for the Internal Effectiveness Factor Expression Case Species (dci/d~z=.int I A B +A R+P/rAP C A p2 C IB B IA+B RP/PAPAIB+A I P + IAP +A IR PR RAPI B+AP+R/R+P II A 242A 24p+/rAP II P p rAP2A II R P+R A2A +R +P III A 2.t 24p /rAP III P 2p rAP A IV A +A ~P rAP IV B IA+B 'IA+BP'/ rApB+A IV P PAP B+A V A 2IA 24/rAP V P p rA 2A VI A $+A VI B A+B VII A 2 2A 88 uncertain, and not guaranteed to be even approximately correct, in intermediate ranges. Conventional approaches to estimating midplane concentrations are observed to fail badly, although limiting asymptotes of the internal effectiveness factor are usually approximately correct. This is unfortunate, since the internal effectiveness factor for the slab geometry can be written as an algebraic function of the ab initio known parameters plus the midplane concentration. The slab midplane concentration is immediately available from the new analytical solution presented here, plus the analysis suggests several new ways to estimate it under various limiting situations. Separate plots of the internal effectiveness factor versus the relevant irreversible reaction Thiele moduli can be prepared for both Case VI and Case VII kinetics. Because these two irreversible reaction systems have been studied by others numerically, these plots will be presented here using the analytical solution. The results for these two cases are also part of the general plots of the internal effectiveness factor versus l,, and *I presented later. Internal effectiveness factors have been obtained numerically by Bischoff (1965) for the problem Dd k'cn 2.95 dx2 where n is the reaction order and k' is a rate constant. This plot is given in many standard heterogeneous kinetics texts. The comparable problem in this paper, Case VII of Table 2.1, is Dd2 = 2k,f2 2.96 A dx2 CkA with the two fundamental moduli given by 4L 4L 4 bk2 A2 s 4Lf As 2.97 t D 4 2o There is really only a single parameter, since 41,t and P are related. It is almost certainly the best known simultaneous secondorder reaction and diffusion problem result. Plots of qit are made in Bischoff versus a generalized Thiele modulus, $o, given by 2 L2(n+1)k' , L23(2kf)A 2.98 2G = D C 2DA45A 2.981A 20 2,A modified here to match the current nomenclature and to coincide with secondorder reaction systems. A feature of the generalized Thiele modulus development is that 1 int I o 2.99 G Noting that k' = 2kf, substitution of eqs. (2.97) and (2.98) into eq. (2.99) yields 2 Sint Ol r 00 2.100 V3 st as the equivalent expression. Fig. 2.6 compares the exact solution for the irreversible power law kinetics of Case VII with an asymptotically correct expression based on the generalized Thiele modulus approach. The asymptotic curve is based on 9 4 c 6 0 o I 0 r. 0 0 *M 4. *a o S 4> U, o r4 I Lo '4 ^ 