relation (Temkin et al., 1963) for ammonia synthesis in the absence of NH3 is used
rNnS = kPN,aPH21-U
(a
%)
(3)
then k can be determined if the pulse shape is known and the NH3 is adsorbed very rapidly onto the sieves. Since the partial pressure of N2 varies through the column, this must be considered in applying eq 3. 0
8
I6
24
32
T I M E (mini
Chromatogram for reactor column 2 operated with temperature programming. After final N2 pulse injection, 20 Wmin was begun. A t 470 O C , temperature was held constant for 10 min before returning to initial 400 “C. F i g u r e 4.
Acknowledgment The authors are indebted to Mr. J. H. Miller of Girdler Chemical, Inc., for generously providing us with the ammonia-synthesis catalyst, G-82. They also thank H. Dale Wilson, a doctoral student a t UCSB, for stimulating discussions. Literature Cited
and adsorbant could be pelletized to form a composite catalyst packing. Optimum pulse size, flow rates, temperatures, and pressures could also be considered. Use of the pulsed chromatographic reactor in NH3 manufacturing might prove to be economical if used as a first stage in series with a conventional NH3 synthesis reactor. The NH3 would be removed between stages. It might also be feasible to inject N2 at various locations along the chromatographic reactor to replace the depleted N2 pulses. combining this with temperature programming would very significantly increase the NH3 production. The results of the chromatographic reactor studies for NH3 synthesis could also be extended to the study of low-pressure, continuous-flow synthesis reactors in which the solid phase consisting of catalyst-adsorbant mixture is fluidized or carried through the reactor as a moving bed. Separation of ammonia-synthesis catalyst and loaded sieves adsorbant can be accomplished magnetically followed by degassing of the adsorbant. Finally, the chromatographic reactor may also be used to obtain kinetic data rapidly for certain reactions. If the Temkin
Dinwiddie, J. A,. US. Patent 2 976 132 (1961). Gore, F. E.,hd. fng. Chem., Process Des. Dev., 6, 11 (1967). Langer, S.H., Patton, P. D., Chem. lnd. (London), 1346 (1970). Langer, S.H.,Patton, P. D., “New Developments in Gas Chromatography,” pp 352-354, J. H. Purnell, Ed., Wiley, New York, N.Y., 1973. Langer, S.H., Yurchak, J. Y.. Patton, P. D., Ind. Eng. Chem., 61, 10 (1969). Magee, E. M., Ind. Eng. Chem., Fundam., 2, 32 (1963). Matsen. J. M., Harding, J. W., Magee, E. M., J. Phys. Chem., 69, 523 (1965). Roginskii, S. Z., Yanovskii, M. I., Gaziev, G. A,, AM. Akad. Nauk SSR, (fng. Ed.), 140, 771 (1961). Roginskii, S. Z.,Yanovskii. M. I., Gaziev, G. A,, Kinet. Catal. (USSR), 3, 464 (1962). Semenenko, E. I., Roglnskii. S.Z., Yanovskii, M. I. Kinet. Catal. (USSR), 5, 426 (1964). 4, 224 Temkin, M. I., Morozov, N. M., Shapatina, E. N., Kinet. Cafal. (USSR), (1963a). 4, 494 Temkin, M. I., Morozov, N. M.. Shapatina, E. N., Kinet. Cafal. (USSR), (1963b). Unger, 6. D., M.S. Thesis, University of California, Santa Barbara, Calif., 1974.
Chemical and Nuclear Engineering Department Uniuersity of California Santa Barbara, California 93106
Bernard D. U n g e r Robert G.Rinker*
Received for reuiew August 25,1975 Accepted April 8,1976
Some Remarks on the Calculation of Complex Chemical Equilibria by General Methods
The alleviation of problems arising in the computation of complex chemical equilibria concerning multiphase systems and restricted equilibria is discussed. General-purpose algorithms using the concept of stoichiometric reactions are particularly suitable for multiphase systems.
Introduction It may be desirable to calculate the equilibrium composition of a reacting chemical system for any of a number of reasons. Numerous general-purpose algorithms have appeared in the literature to solve this problem. Zeleznek and Gordon (1968) and Van Zeggeren and Storey (1970) have reviewed many of these methods. Recently, a number of articles have appeared in this journal suggesting that difficulties may be encountered for specific types of equilibrium problems when such methods are used (Samuels, 1971; Madeley and Toguri, 1973a; Bjornbom, 1975). Samuels (1971) and Balzhiser et al. (1972) suggested that the RAND “free-energy minimization method” (White et al., 1958) fails when too many solids are included in the compu-
tation. Madeley and Toguri (1973a) suggested use of an alternative algorithm to partially overcome this problem. Bjornbom (1975) suggested that many of the general algorithms would fail in the calculation of certain kinds of “restricted” equilibrium problems, in which not all reactions are allowed. It is the purpose of this communication to illustrate simple ways of avoiding the aforementioned difficulties using general-purpose calculation methods.
Multiphase Systems For solid-gas systems of the type discussed by Samuels (1971), which are important in metallurgical applications, the inclusion of too many condensed solids in the computation w ill Ind. Eng. Chem., Fundam., Vol. 15, No. 3, 1976 227
Table I. Madeley-Toguri (1973b)Blast Furnace Problema
Species
Free energy of formation, Equilibrium amounts, kcal mol-’ mol Equilibrium amounts, mol
co Fe(4 H2
co2
CaO(s) H20 CHI CHzO CHO OH
-48.91
81.62
0 0
42.83
-94.64 -126.48 -45.38 5.94 -20.58 -14.95 5.40 0
0 2
FeaO4(s) Us) CaCOs(s) FeO(s)
-182.28 0
-225.25 -46.35
6.43 6.66 0.756 0.444 0.655 X 0.236 X 0.105 X 0.793 x 10-lo 0.173 X 0 0 0 0
a T = 1050 K, P = 1 atm, 187.1 mol of inerts, (bo, bH, b,, bFe, bCJ = (96.151,13.766,88.294,42.827,0.7562).
cause the RAND method to fail (Balzhiser et al., 1972). This is because the RAND algorithm iteratively solves a system of linear equations which become singular (Smith, 1972) when the number of solids is greater than C - 1, where C is the number of components in the system (Brinkley, 1946). Balzhiser et al. (1972) essentially state that one must know which solids are present at equilibrium in order to use the RAND algorithm in such cases. Bigelow (1970) has described one means of avoiding this. He suggests including fictitious inert species in each solid phase, and performing a sequence of computations in which these fictitious species are driven to zero amounts. Apart from possible numerical problems arising from such a calculation sequence, it is undesirable to have to perform a series of calculations for a single problem. Madeley and Toguri (1973a) suggested that combined use of a first-order free-energy minimization algorithm and the RAND algorithm (Madeley and Toguri 1973b) would overcome this problem. Although this method works reasonably well it is somewhat cumbersome, since it essentially combines two existing equilibrium algorithms. Also, the first-order algorithm used to provide an estimate for the second-order RAND method is quite slow computationally. In addition, as Madeley and Toguri have noted (1973a), the technique may encounter difficulties if one is looking a t the conditions under which a phase will just disappear. General-purpose algorithms based on the use of a set of stoichiometric reactions for the system (Villars, 1958; Cruise, 1964; Smith and Missen, 1968) completely avoid these difficulties with condensed phases. Any number of solids may be inchded in the calculation. Once a solid species becomes zero, the partial derivative with respect to its amount is examined on each iteration. If this derivative is negative, the solid is re-introduced into the calculation. Even problems consisting entirely of solids can be handled by this method. The programming is trivial and is discussed by Cruise (1964). It is perhaps important to remark a t this point that such methods can also be formulated as “free-energy minimization methods” (Smith and Missen, 1968).They are mathematically related to the nonlinear programming algorithms of Wolfe (1969) and others (Abadie, 1970). As an example, the algorithm of Smith and Missen (1968) was applied to the “Blast Furnace Problem” of Madeley and Toguri (1973a). The same solution was obtained in only nine iterations using the automatic solution estimating procedure of Smith and Missen (1968). Using an arbitrary estimate of 228
Table 11. Equilibrium in the System CsH5CH3, Hz, C6&, CHI for Initial Feed of 4 mol of H2,1 mol of C6H5CH3, T = 980 K,P = 43 atm
Ind. Eng. Chem., Fundam., Vol. 15, No. 3, 1976
Species
Restricted”
Unrestrictedb
0.0020 3.0020 0.9980 0.9980
0.1183 0.0787 0.5439 2.9081
CsH5CH3 Hz CsH6 CH4
“ “Restricted” denotes only eq 1 of the text allowed. “Unrestricted” denotes all equilibria allowed. ( 0 2 , Hz, Fe, C, CaO) = (47.6904,6.878,42.823,88.289,0.7552) and 0.001 mol for the remaining species, the solution was reached in 17 iterations. The computing times in each case were 0.42 and 0.71 CP s, respectively, on a CDC 6400 machine. This compares with Madeley and Toguri’s 95 iterations for the first part of their technique plus an unstated number of iterations of the RAND algorithm. The results are given in Table I along wtth the standard free energy data used. Our results differ slightly from those of Madeley and Toguri, due possibly to the use of different standard free energy data.
Restricted Equilibria In general, in a system comprising N chemical species and C components (Brinkley, 1946), there are a t most R = N - C linearly independent stoichiometric reactions. In some cases, it is desirable to perform equilibrium computations for systems in which fewer than R reactions are allowed. Bjornbom (1975) has suggested that free energy minimization methods cannot be used in such cases. This is false. Early in the development of general purpose equilibrium algorithms Kaskan and Schott (1962) and Schott (1964) discussed such problems. Schott (1964) suggested a way of modifying the formula vectors of the species in the system (vectors specifying the number of atoms per molecule of the species) and the atom balance constraints so that any of the general purpose algorithms can perform the required calculations. Application of this technique to Bjornbom’s hydrodealkylation of toluene example, in which the only stoichiometric reaction allowed is
shows that a possible set of modified formula vectors ai are a(CgH&H3) = (l,O,O)T a ( H d = (O,l,O)T a(CsHs) = (O,O,l)T a(CH4) = (l,l,-l)T
(2)
where superscript T denotes a transpose. Using the above formula vectors, the modified atom-balance constraints for the system are
+ n(CH4) = no(C6H5CH3) + nO(CH4) b2 = n(H2) + n(CH4) = nO(H2) + nO(CH4)
bi = n(C&jCH3)
b3
= n ( C & 3 ) - n(CH4) = nO(C&)
- nO(CH4)
(3)
where superscript 0 denotes amounts initially present in the system. In Table I1 are shown the equilibrium compositions for this problem and for the situation in which all equilibria are al-
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
