Ind. Eng. Chem. Fundam. 1982, 21, 284-289
284
stimulating discussions concerning this work. Nomenclature
a = activity distribution function C = reactant concentration De = effective diffusion coefficient f =-function defined by eq 6 k , k = local and volume averaged reaction rate constant K = adsorption rate constant
M = parameter defined by eq 15 n = integer characteristic of pellet geometry; = 0 for infinite slab, = 1 for infinite cylinder, = 2 for sphere R = characteristic pellet dimension; half-thickness (n = 0), radius ( n = 1, 2) s = x/R s = dimensionless optimal location of the active catalyst, defined in eq 19 u = C/Cf V , = pellet volume x = distance from center of pellet Greek Letters A = half-thickness of the step-distribution (33) 6 = Dirac .. 6 function 7 = effectiveness factor 9, $o = Thiele modulus; I#I = I#Io/(l + a), +o = ( R 2 k / D , ) 1 / z u = KCP
qn = function defined by eq 25
Subscripts f = bulk value
max = value corresponding to the active catalyst optimum location
Literature Cited Becker, E. R.; Wei, J. J . Catal. 1977, 4 6 , 365. Corbett, W. E.,Jr.; Luss, D. Chem. Eng. Sci. 1974, 2 9 , 1473. Ernst, W. R.; Daugherty, D. J. AICM J . 1978, 2 4 , 935. Hegedus, L. L.; Summers, J. C.; Schlatter, J. C.; Baron, K. J . Catal. 1979, 56, 321. Kasaoka, S.;Sakata, Y. J . Chem. Eng. Jpn. 1988, 1 , 138. Michalko, E. U S . Patent 3 259 589, 1966. Minhas, S.; Carberry, J. J. J . Cafal. I9B9. 14, 270. Morbiielli, M.; Varma, A. Ind. Eng. Chem. Fundam. 1982, following paper in this issue. Morbidelli, M.;Servida, A.; Varma, A., manuscript in preparation. Nystrom, M. Chem. Eng. Sei. 1978, 33, 379. Schechter, R. S . “The Variational Method in Engineering”; McGraw-Hill: New York, 1967. Shadman-Yazdi, F.; Petersen, E. E. Chem. Eng. Sci. 1972, 27, 227. Smith, T. G. Ind. Eng. Chem. Process D e s . D e v . 1976, 15, 388. Smnh, T. G.; Carberry. J. J. Can. J . Chem. Eng. 1975, 53, 347. Summers, J. C.; Hegedus, L. L. J . Catal. 1978, 51, 185. Vllladsen, J. Chem. Eng. Sci. 1978, 31, 1212. Vllladsen, J.; Michelsen, M. “Solution of Dlfferentil Equation Modeis by Polynomial Approximation”: Prentice-Hall: Englewood Cliffs, NJ. 1978. Voitz, S. E.; Morgan, c . R.; Liederman, D.;Jacob, S. M. Ind. Eng. Chem. Prcd. Res. D e v . 1973. 12. 294. Wang, J. B.; Varma, A. Chem’.Eng. Sci. 1980, 35, 613.
Received for review July 13, 1981 Accepted March 25, 1982
Optimal Catalyst Activity Profiles in Pellets. 2. The Influence of External Mass Transfer Resistance Massimo Morbldelll‘ and Arvlnd Varma Department of Chemical Engineering, UniversHy of Notre Dame, Notre Dame, Indiana 46556
The problem of optimal activity distribution in an isothermal catalyst pellet with a Langmuir-Hinshelwood reaction, already studied in part 1, is now extended to cases involving finite resistance to external mass transfer. The same activity distribution, i.e., a Dirac 6 function, as in the case of negligible external mass transfer resistance, is also found optimal in this case. Analytic expressions for the optimal locations of the active catalyst as a function of the involved parameters (4, Bi, u), are given for three pellet geometries: slab (n = 0), cylinder (n = l), and sphere (n = 2). A general relationship, valid-for all geometries, between the normalized optimal location p = x/(Vp/Sx) and only one speclfic combination, ?! , of the normalized physicochemical parameters is also given.
1. Introduction
It is well known that improved catalyst performance can result by using a nonuniform catalyst activity distribution within a pellet. An important aspect of catalyst design is the determination, both theoretically and experimentally, of active catalyst distributions in the pellet support which improve either the conversion, or the selectivity, or if some poisoning occurs, the life of the catalyst itself. In part 1of this work, Morbidelli et al. (1982, preceding article in this issue) have determined the optimum activity distribution for an isothermal catalyst pellet in which a very common special case of bimolecular Langmuir-Hinshelwood kinetics, where one of the reactants is strongly adsorbed on the catalyst while the other one is in excess, takes place. In that analysis, external mass transfer re-
’On leave from Politecnico di Milano, Milano, Italy. 0 196-431 318211021-0284$01.25IO
sistance was assumed to be negligible. In this work, we examine the influence of external mass transfer limitations on the optimum activity distribution function, for three pellet geometries: infinite slab, infinite cylinder, and sphere. 2. Mathematical Formulation of the Problem For a symmetric porous support with a nonuniform distribution of active catalyst, the steady-state mass balance has the form (Morbidelli et al., 1982)
where n = 0, 1, 2 indicates the slab, cylindrical, and spherical geometry, respectively, and u ( x ) is the activity distribution function, defined as the ratio between the local rate constant, k ( x ) and its volume average value k. It must then satisfy the condition @ 1982 American Chemical Society
Ind. Eng. Chem. Fundam., Vol. 21, No. 3, 1982
285
and also from eq 8, 9, and 11, it is readily seen that 1
q
Assuming finite external mass transfer resistance, the boundary conditions (BCs) for eq 1 are given by dC/dx = 0
(at x = 0)
D,(dC/dx) = iz,(Cf - C)
(34
(at x = R )
(3b)
I (n + 1 ) x Ma(s)snds= M
Thus if we can find a function a(s) such that q = M , then this a(s) will indeed be the desired optimal activity distribution function. Such an expression is the same as in part 1; i.e.
Introducing the following dimensionless quantities
a(s) =
x / R ; 42 = 402/(1+a)2 = hR2/[D,(1 + CJ = KCf; Bi = kJl/D,;
u = C/Cf;
s =
f(u) = (1
+ u)2u/(l + uuy
(4)
(at s = 0)
du/ds = Bi(1 - u )
(at s = 1)
(5) (64 (6b)
where (7) As in part 1, since 1 - dVp = (n VP
We now have to determine the operating conditions under which such a position actually exists and precisely where it is located. For this purpose, we need to examine separately the two cases of u greater or small than 1. 3.1. Case of u > 1. Let us assume the existence of the position S in the pellet defined by eq 14. The integration field of eq 5 , in which the expression of a(s) given by eq 13 has been introduced, can then be divided in two zones. (i) 0 < s < S. In it eq 5 reduces to
~1
+ l)J1 a(s)sn ds = 1
(8)
= (n + l)J1f(u)a(s)sn ds
(15)
=A
(16)
where A is the remaining integration constant. (ii) S < s < 1. While eq 5 reduces also in this zone to the same eq 15, integration must now be performed with the other BC, i.e., eq 6b, obtaining
u ~ ( s=) ~ ( 1+) Bi[l - u(~)]+,(s)
(17)
where +Js) is a function depending on the pellet geometry (same as in part 1) defined as +,(s) = s - 1 (for n = 0)
or alternatively, substituting in eq 5 and 6, through the equation
4' The purpose of this work is now to determine the activity distribution function a(s) which maximizes the effectiveness factor q, that is, to define the function a(s) which maximizes the functional q(a(s), u(s)) under the constraints given by eq 5, 6, and 8. 3. Effectiveness Factor Optimization In the following the same optimization technique, previously employed in part 1for the case of negligible mass transfer resistance, will be used. I t is based on the observation that the reaction rate function f ( u ) has a maximum at u, = 1 / u (see Figure 1 of part 1). Since 0 Iu I1, the value of u, is in the interval of interest only for u 1 1. If u < 1,the function f(u) assumes its maximum value in the range 0 5 u I1,at u = 1. Thus as in part 1 max f ( u ) = f(u,) = M (11) O 0 and the optimal location starts moving toward the center of the pellet. This qualitative behavior is physically reasonable, and it can also be justified quantitatively. Let us consider the transition point between the two situations above mentioned, i.e., $ = 0. In this case, if one concentrates the active catalyst on the pellet surface, Le., s = 1, from eq 33 the dimensionless concentration value at that point is u(s) = u(1) = 1 / u
(48)
Note from Figure 1of part 1that at this value u, = l/u, where the rate of reaction reaches its maximum value. Therefore, if now the external mass transfer decreases, then u(1) < 1 / u and since the rate of reaction increases monotonically with concentration for u < 116, the optimal location is indeed the external surface where the reactant concentration is maximum. On the other hand, if the external mass transfer increases; then u(1) > l / a , and so there exists a point within the pellet where the reaction rate, and therefore also the effectiveness factor, is maximized. In this latter case the value of p > 0, which determines the optimal location for the active catalyst s, is affected also by internal mass transfer resistance, present in the first factor in the right-hand side of eq 47. In particular, for the limiting case of negligible external mass transfer resistance, i.e., E a,it is readily seen from eq 47 that the value of fl depends only on the ratio between the internal mass transfer rate and the reaction rate.
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Ind. Eng. Chem. Fundam. 1982, 21, 289-298
Consequently,the same is also true for the optimal location B, as previously observed in part 1. 5. Concluding Remarks The effect of external mass transfer resistance on the catalyst activity distribution function which maximizes the effectiveness factor in an isothermal catalyst pellet in which a bimolecular Langmuir-Hinshelwood reaction occurs has been examined. The optimal distribution is a Dirac 6 function, similar to the one found in part 1for the case of negligible external mass transfer resistance. Analytic expressions for the optimal location S of the active catalyst, as a function of all the involved physicochemical parameters, are derived for the slab, cylindrical, and spherical pellets. The values of S match for Bi m with those obtained in part 1. As expected, the external mass transfer resistance has the same effect on the optimal location S as the internal one. In particular, for increasing values of the resistance the optimal location moves from the interior to the surface of the pellet, for all the examined geometries. The value of S is found to depend on only one parameter, p, which contains all the physicochemical parameters involved; Le., Bi, 42, Q, and the geometry parameter n. On the other hand, the optimum value of 7,at least when the optimum location is 0 < s < 1,depends only on the adsorption constant parameter Q. This result is also consistent with that found in the case of negligible external mass transfer resistance. A shape normalization has been developed which provides a general relationship between the normalized optimal location 9 = x/(V,/SJ - and the normalized parameter 6 = [4(a - 1)/aO2 - l/Bi], valid for all geometries. Finally, it is worth noting that although in part 1 we considered step-distributions of the active catalyst and compared catalyst pellet performance for such distributions with the optimal one, we have chosen not to do so in the present case. The influence of such distributions in the case of finite external mass transfer resistance should
-
be identical with the case of negligible external mass transfer resistance. Acknowledgment
M.M. thanks the Italian Consiglio Nazionale delle Richerche (Progetto Finalizzato Chimica Fine Secondaria) for financial support during the course of this study. A.V. is grateful to the Department of Energy for partial support of this work, as part of a joint research program between Ford Motor Company and the University of Notre Dame. Nomenclature
For symbols not listed below, refer to part 1 of this work. Bi = Biot number, @ / D e E = normalized Biot number, k,V,/D,S, Bi* = $02/[4(n + l ) ( u - l)] f = reaction rate function, defined by eq 4 F = function defined by eq 33 G = function defined by eq 33 k , = mass transfer coefficient M = parameter defined by eq 11 s = s/R S(y) = surface of the pellet at a specific position y S, = external surface of the pellet f = optimal location of the active catalyst y = normalized pellet coordinate, x / ( V , / S , ) 9 = optimal value of y Greek Symbols @ = (l/Bi*) - (l/Bi)
p = normalized parameter, [4(u - 1)/+02]- 1/E $, = function defined by eq 18 a0 = normalized Thiele modulus, ( Vp/S,)(k/D,)1/2 L i t e r a t u r e Cited Aris, R. “The Mathematical Theory of Diffusion and Reaction In Permeable Catalysts”; Vol. I; CIerendon Press: Oxford, 1975. Morbdelli, M.; Servlda, A.; Varma, A. Ind. Eng. Chem. Fundam. 1982, preceding paper in this issue.
Received for review August 3, 1981 Accepted March 25, 1982
Estimating Bounds for Process Optimization Problems. An Analogy to Rules of Thumb and Structural Modifications Michael F. Doherty, Michael F. Malone,’ Fernando E. Marquez, David L. Davldson, and James M. Douglas Chemical Engineering Department, University of Massachusetts, Amherst, Massachusetts 0 1003
The cost surfaces for many process design problems are often very flat in the neighborhood of the optimum. Thus, it might be simpler to bound the minimum cost from above and below, instead of determining the exact optimal value. Two procedures for bounding the minimum cost of a train of wastsheat boilers are discussed in this paper. The first approach merely represents an approximate solution of the classical calculus problem. The second approach is a modification of the two-level method of optimization, which always enables us to find a tight lower bound on the minimum cost and is useful for evaluating some process alternatives. The predicted bounds depend on the design and cost parameters, so that the bounds are simple to update as energy prices change. These techniques provide better estimates of the optimum approach temperatures in waste-heat boilers than previously published rules of thumb.
Introduction
It is a common practice in preliminary process design to use rules of thumb to obtain first guesses of the unknown design variables. The plant sizes and costs are 0196-4313/82/1021-0289$01.25/0
based on these values, and the costs are used to estimate the process profitability. If the profitability appears to be sufficiently attractive so that additional design effort can be justified, process alternatives are examined and 0 1982 American Chemical Society
