lowed, a t a set of conditions considered by Bjornbom. These results were computed using a version of the RAND algorithm (Smith, 1966). For the restricted equilibrium case, the standard free energies of formation of C6H5CH3, Hz, and C6H6 were arbitrarily taken as zero, and that of CHI as -lo4 cal mol-'. Bjornbom (1975) states that in the restricted case, reaction 1 goes essentially to completion, and the present calculation verifies this. Bjornbom's values of (C6H5CH3,Ha, C6H6, CH4) = (0.12, 0.07, 0.56, 2.91) are slightly in error for the unrestricted case.
Summary It has been pointed out that multiphase equilibrium computations are easily treated by general-purpose algorithms that use stoichiometric reactions. In addition, any of the general-purpose algorithms can treat cases of restricted equilibria by a method of modifying the species formula vectors and atom-balance constraints, due to Schott (1964). The present discussion of these cases should serve to clear up some of the problems experienced by users of such algorithms, as expressed in recent articles in this journal (Samuels, 1971; Madeley and Toguri, 1937a; Bjornbom, 1975).
Literature Cited Abadie, J., in "Integer and Nonlinear Programming", J. Abadie. Ed.. p 191, North-Holland, Amsterdam, 1970. Balzhiser, R. E., Samuels, M. R., Eliassen, J. D., "Chemical Engineering Thermodynamics", Chapter 12, Prentice-Hall, Englewood Cliffs, N.J., 1972. Bigelow, J. H., TR 70-3, Operations Research House, Stanford University, 1970. Bjornbom, P. H., Ind. Eng. Chem., Fundam., 14, 102 (1975). Brinkley, S . R., Jr., J. Chem. Phys., 14, 563 (1946). Cruise, D.R., J. Phys. Chem., 66, 12 (1964). Kaskan, W. E., Schott, G. L.. Cumbust. Name, 6, 73 (1962). Madeley, W. D., Toguri, J. M.. Ind. Eng. Chem., fundam.. 12, 261 (1973a). Madeley, W. D., Toguri. J. M., Can. Mefall. Quart., 12,71 (1973b). Samuels, M. R., Ind. Eng. Chem., Fundam., 10, 643 (1971). Schott, G.L.. J. Chem. Phys., 40, 2065 (1964). Smith, W. R., M.A.Sc. Thesis, University of Toronto, 1966. Smith, W. R., unpublished manuscript, 1972. Smith, W. R., Missen, R . W.. Can. J. Chem. Eng., 46, 269 (1968). Van Zeggeren, F., Storey, S. H., "The Computation of Chemical Equilibria", Cambridge University Press, London, 1970. Villars, D. S.,J. Phys. Chem., 63, 521 (1959). White, W. B.. Johnson, S. M., Dantzig, G. B., J. Chem. Pbys., 26, 751 (1958). Wolfe, P., in "Nonlinear Programming", J. Abadie, Ed., p 121, North-Holland, Amsterdam, 1969. Zeleznik, F. J., Gordon, S., Ind. Eng. Chem., 60, 27 (1968).
Department of Mathematics Dalhousie Uniuersity Halifax, Nova Scotia, Canada
William R. Smith
Received for reuiew September 2, 1975 Accepted March 15,1976
Maximum External and Internal Temperature Differences in Catalyst Pellets
A straightforward derivation is presented of the maximum temperature differences between bulk fluid, catalyst pellet surface, and catalyst pellet interior. These differences can be stated in terms of the standard observable (Weisz) modulus, or in terms of a new observable group proposed by Carberry (1975).
The determination of the maximum temperature differences between bulk fluid, catalyst pellet surface, and catalyst pellet interior in terms of directly observable quantities is a very useful tool in the study of catalytic reactions. Only if these temperature differences are significant need one be concerned with further extensive analysis of the transport phenomena. Lee and Luss (1969) provided such results in terms of the observable (Weisz) modulus and the external effective Sherwood and Nusselt numbers. Carberry (1975) recently presented an "audible" analysis showing that the fraction of the total temperature difference external to the pellet can be found in terms of a new observable quantity and the ratio of the effective Sherwood to Nusselt numbers, thus obviating the need to have precise values of both of them. However, his analysis was rather tortuous and did not completely enumerate the useful catalog of such results that can be expressed in this way. Our purpose is to provide a straightforward derivation of these several results. The steady-state mass and heat balances for an arbitrary reaction, using slab geometry, are d2C de^ = r(C,T) dz d2T A, 7 = AHr(C,T) dz The particle surface boundary conditions are
Following the procedure of Prater (1958), eq 1 and 2 can be combined (5) which when integrated once from the pellet center to surface gives, utilizing eq 3 and 4
A second integration and rearrangement gives the overall temperature difference
nH
T - T b = [ T ( L )- T b ] i- _e [C - C ( L ) ] Ae
-
k,
(-H)
[cb
- c ( L ) ]-k
(-AH)
D
e [ c ( L ) - c] (7) Xe
The right-hand side of eq 7 is the sum of the external and internal temperature differences, as pointed out by Hlavacek and Marek (1970). The maximum temperature difference is for complete reaction, when C = 0
Ind. Eng. Chem., Fundam., Vol. 15, No. 3, 1976
229
The find step is to obtain C(L)/Cb in terms of an observable rate, which is the volume-averaged rate in the pellet which is eq 16 of Carberry (1975). Equations 10-18 are then a summary of the various temperature differences in terms of two possible observable groups.
Using the observable (Weisz) modulus
Substituting eq 10 into eq 8 gives the result of Lee and Luss (1969) (their eq 11,which was in spherical geometry)
Lee and Luss also presented results for the maximum surface-to-interior temperature difference. Carberry (1975) defined a new observable group
Then, eq 11can be written in terms of Ca and only the ratio Sh’/Nu’
Nomenclature C = reactant concentration group of Carberry (1975) Ca = Lr,b,/k,Ct,-observable De = effective diffusivity in catalyst pellet h = pellet external heat transfer coefficient AH = enthalpy change of reaction k c = pellet external mass transfer coefficient L = pellet size, volume to external surface ratio Nu’ = hL/Ae = effective Nusselt number r(C,T) = reaction rate Sh’ = k,L/D, = effective Sherwood number T = temperature /3b = (-AH)D,Cb/A,Tb = Prater number evaluated at bulk fluid conditions A, = effective thermal conductivity in catalyst pellet 9 = L2robs/DeCb = observable (Weisz) modulus evaluated at bulk fluid conditions Subscripts b = bulk fluid value obs = directly observable rate
Likewise, the interior temperature difference is
Literature Cited Carberry, J. J., Ind. Eng. Chem., Fundam., 14, 129 (1975). Hlavacek, V., Marek, M.. Chem. Eng. Sci., 25, 1537 (1970). Lee, J. C. M., Luss, D., Ind. Eng. Chem., Fundam., 8, 597 (1969). Prater, C. D., Chem. Eng. Sci., 8 , 284 (1958).
and the external temperature difference is
Instituut uoor Chemie-ingenieurstechn~ekKenneth B. Bischoff’ Kutholieke Uniuersiteit te Leuuen Leuuen, Belgium = PbCa
Sh’ (Nu‘)
Received for reuiew April 7 , 1976 Accepted April 14,1976
Finally, the fractional external temperature difference is the ratio of eq 17 to eq 13
1 Permanent address: School of Chemical Engineering, Cornell University, Ithaca, N.Y. 14853.
On Size-Dependent Crystal Growth
Observed size-dependent crystal growth rates have previously been explained in terms of either the influence of bulk diffusion to the crystal surface or the Gibbs-Thomson effect. In this communication we show that these effects are unlikely to be important over the size range 1-100 pm. A more likely cause, size-dependent surface integration kinetics, is suggested and evidence to support this view is presented.
Introduction The size distribution equation for mixed-suspension, mixed-product-removal crystallizers, generally written in the form n = no exp(-L/GT)
(1)
is derived assuming that the crystal growth rate, G , is independent of crystal size. When plotted as In n against L this equation gives a straight line and many published data appear to follow such a law. There are a number of results, however, which show significant curvature when plotted in such a manner, data for potassium sulfate (White et al., 1974) and 230
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1976
potash alum (Ottens et al., 1972; Jancic and Garside, 1975) being examples. Assuming that classification effects both in the crystallizer and in the product removal line are absent, nonlinear population density plots could be obtained as a result of either size-dependent growth kinetics or of the “birth” of secondary nuclei into the small size ranges. This second mechanism has been suggested by Youngquist and Randolph (1972) and Randolph and Cise (1972)who explained experimental crystal size distributions by a combination of size-dependent growth kinetics and the production of new nuclei in the range 2-20 Pm.
