Computing Chemical Equilibrium Compositions in Multiphase Systems The first-order free energy minimization technique has been adapted for use in multiphase systems. It has been found that the number of phases that m a y b e included in a computation i s independent of the phase rule.
R e c e n t l y , Samuels (1971) and Smith (1972) have discussed the maximum number of phases which may be included iii immiscible solid-gas equilibrium comput,ations using the RXSD method (13oyntoii, 1959; Kubert and Stephanou, 1960; Oliver, et al., 1962; Van Zeggeren and Storey, 1970; White, et al., 1958). If too many phases are included in a computation, the system of C P linear equations in C P uiiknowns, which must be solved for each iteration, is singular (Smith, 1972), where C is the number of components and P the number of phases. The maximum number of phases which caii be handled by the RASD is given by the phase rule and is equal to the number of components in tlie system, for fixed temperature and pressure. Sirice the R.I?;D cannot accommodate systems of solids which do not coiitain a reacting gas phase, the number of condensed phases must be C - 1. Selecting appropriate phases for a R-iKD computation may be considered as part of the problem of determining a suitable iiiit,ial guess. One solution to the initial guess problem is to use a n extended version of the first-order method (Storey arid Van Zeggereri, 1970; Van Zeggereii and Storey, 1970) to compute starting estimates for the R.\SI) method, or the first order may be used alone if accurate mole numbers for minor species are not required. Starting estimates for the first-order method can be arbitrary and the extension of the method to include condensed phases, either miscible or immiscible, is straightforward. 1lie first-order algorithm minimizes the total free energy G, subject to tlie mass balance constraints. This is accomplished by determining the direction of maximum dope on the free energy surface a t an estimate of t’he equilibrium composition, taking a small step in this direction, and computing a iiew estimate. The procedure is repeated until the change i i i mole numbers between subsequent estimates is sufficiently small. The estimates approach both the minimum free energy and the true mass balance, as determined by the itiput to the syst,em. On each iteration two sets of C linear equations in C 1111kiiowns are generated. These are of the form A0 = s and A0 = t , where A is the same in both equations. Since P , the number of phases, does not’ affect the order of A, any 1)hases suspected of being present a t equilibrium or used as input may be iiicluded in the computation. Segative mole numbers cannot be generated as the substitution
+
+
r.
/)lip
=
esp([rp)
(where mip is an est,imate of the equilibrium mole number of the itli specie iii the pth phase) is made and computatioii is is carried out using the dummy variables tip and atip. This substitutipn allows the use of a poor or arbitrary initial guess to start tlie computation. Contrary to Storey and Vau Zeggereii’s recoinrneiidatiori of a constant stel) size, it was found that a variable step size is desirable. Smith and Nissen’s (1968) colivergence forcer has
Table 1. Blast Furnace Problem (Temperature = 775OC) Species
0 2
sz HzO CHI
co
COY H2 CHO CHiO OH FesOc FeO Fe
c
CaC03 CaO
Input (moles)
Initial guess
ldeal Gas. 046 X 10’ 0 001 871 X l o z 20 0 775 1 0 255 0 001 10 0 10 0 0 001 0 001 1 0 0 001 Solid Phases 1 310 X 10’ 10 0 1 0 3 527 2 0 8 559 X 10’ 100 0 1 499 x 10-1 1 0 6 063 x lo-’ 1 0
2 1 1 2
First-order solution
x 10-4 1 871 X 102 6 873 1 362 x 10-3 7 875 X 10’ 4 871 1 584 x 10-3 8 584 X 4 594 x 10-6 6 745 X 4 508
1 9 4 4 2 7
979 X l O W 3 097 x 281 x 10’ 661 633 x 10-1 559 X lo-’
been used t o iiidicate when the step size should be decreased. We have found that the step size should be kept constant for the first 10 iterations, then the forcer can be alqilied to subsequent iterat’ions. Use of the forcer prevents mole iiuriibers from oscillating or diverging from the true mole balance (as determilied b y the iiiput) . difficulty encountered n i t h the firsborder algoritlim is that an inordinate number of iterations are required to allproach accurate values for species present ill quantities less than, say, 10-j mole. This may not p r o w serious uiilcss one is looking for the conditioris under IThich xome iiliase will just disappear. Table I shows the result of the application of this method to a blast furnace problem at i i 5 O C having five compoiients plus nit,rogeii, which was considered to be inert a t this temperature. Six solid phases plus a gaseous phase are included iii the computatiori as shown. The input mole numbers were taken from the literature (llcGarinon, 1964) and ail arbitrary initial guess was used. Thermodynamic data w r e taken from the ,JAX.\F tables (Stull, 1965) and from Kubascliewski, et a/.(1967). A s one would expect, the first-order solution indicates that the number of moles of Fe304,FeO, arid CaC03 is decreasing and may be eliminated from the computation. The mole balance is satisfied to 11 significaiit digits iii 95 iterations. A \
literature Cited
Boynton, F. P., J . Chcm Phys. 29, 1880 (1959). Kubaschewki, 0..Evan>,E. L L , Alcock, C. B., “_\Ietnlluiqical Thermochemistry,” 4th ed, Pergamon Press, London, 1967. Kubert, B. I:,, Stephanou,8. E., ”Kinetics, Equilibria and Perforinance of High Teniperature Systems,” G. S. Bahn and E. E. Zukoski, Ed., Butterworths, London, 1960. Ind. Eng. Chem. Fundam., Vol. 12, No. 2, 1 9 7 3
261
McGannon, H. E., Ed., (‘The Making, Shaping and Treating of Steel,” 8th ed, United States Steel Corp., 1964. Oliver, R. C., Stephanou, S. E., Baier, R. W., Chem. Eng. 69, 121 (1962). Samuels, M . R., I N D . ENG.CHEM.,FUNDAM. 10, 643 (1971). Smith, W. R., private communication, Aug 1972. Smith, W. R., hlissen, R. W., Can. J. Chem. Eng. 46, 269 (1968). Storey, S. H., Van Zeggeren, F., Can. J . Chem. Eng. 48, 591 (1970). Stull, D. R., “JANAF Thermochemical Tables, PB 168 370 (1965) plus supplements I (1966), I1 (1967) and I11 (1968). U. S. Department of Commerce, National Bureau of Standards, 1965. Van Zeggeren, F., Storey, S. H., “The Computation of Chemical Equilibria,” Cambridge University Press, London, 1970.
White, W. B., Johnson, S. M., Dantzig, G. B., J. Chem. phys. 28, 751 (1958).
W. DAVID MADELEY JAMES M. TOGURI* Department Of and Materia1s Science University of Toronto, Toronto, Ont., Canada Received for review September 11, 1972 Accepted January 15 1973 Financial assistance was received from the National Research Council of Canada. W. D. M. is grateful to Falconbridge Nickel Mines Ltd. for a fellowship.
CORRESPONDENCE ~
~~~
The Compensation Effect in Chemical Kinetics SIR:In a recent article Johnson, et al. (1972), reported the results of a n experimental kinetic study of the vapor phase isomerization of cyclopropane. A linear relationship between the activat.ion energy ( E ) and the logarithm of the frequency factor (ln A) was observed. This t’ype of relationship, known as a “compensation effect,” is quite common in chemical kinetics and theoretical explanations have been discussed by Cremer (1955). It is the purpose of the present letter to suggest that in the above study the observed compensation effect may have arisen simply from the data correlation rather than from any real physical effect. Experimental rate constants measured a t temperatures within the range 454-538°C were fitted to an Arrhenius equation (111 k = In -4 - E / R T ) by a linear least-squares method. Any errors in the individual rate combants will give rise to correspoiiding errors in the values of d and E and since b 111 A/dE = 1/RT, it follows that the calculated values of In A aiid E mill show a linear relationship of slope 1/RT, where T is the mean temperature of the experimental measurements. I n the above study the mean temperature was 769’K so that l / R T = 6.6 X mole ca1-l which is close to the observed slope of the compensation plot (6.76 X 10-4) in Figure 5.
SIR:We agree with Professor Ruthven that the apparent linearity between the logarithm of the frequency factor (A) and the activation energy ( E ) may have no real physical significance and, in fact, may even be a result of our method of data correlation. We believe that this point was sufficiently emphasized in our original paper: “It wodd be premature to advance any theoretical conjectures for this empiricallg observed behavior.” On the other hand, we do not follow the argument advanced by Professor Ruthven. I n the first place, his expression, b In A / b E = 1/RT, presumably descends from differentiating the empirical Xrrlienius equation while holding the reaction rate coefficient, k , and temperature, T , const’ant. The data from which Figure 6 was derived, yielding ln ( A ) = 6.76 X Eo - 8.8828, covered a range of temperatures from 454 to 538”C, a range of pressures from 0.4 to 137 atm, and a range of reaction rate coefficieiit,s from 0.46 X to 113 X sec-I. Figure 6 demonstrates that, t,he foregoing equation, relating In ( A ) and EO,constitut’esa reasonable fit of the data obtained by five previous investigators covering a range of temperatures of 420 to 62OoC, a range of 1)Gessures of 0.013 to 137 atm, and a range of reactioii rate coefficients from 0.2 X t o 0.2 sec-l. It i s interesting t,o note that other reactions which shorn a “compensation” effect have In A us. Eo slopes between 0.60 X and 0.70 X 10-4 mole “Kjcal. The fact that Professor Ruthven was able to predict the slope of the ln 262
Ind. Eng. Chem. Fundam., Vol. 12, No. 2, 1973
If the effect of pressure is disregarded, the standard deviation of the activation energies listed in Table I is =t9.8Y0 (Eo = 66,100 f 500 cal/mole) and this is of the same order as the probable error estimated from the quoted uncertainty in the individual rate constants ( =k4Y0). It therefore seems likely that the linear relationship between the calculated values of In A and E may have no real physical significance and conclusions concerning the variation of activation energy with pressure should be treated with caution. A somewhat similar example of an artificial compeiisation effect has been discussed previously (Ruthven, 1968). literature Cited
Cremer. E. Advan. Catal. 7 . 7 5 (1955). Johnson, D. W., Pipkin, ’0. A., Sliepcevich, C. >I., IIZD. ENG. CHEM.,FUNDLM. 11, 244 (1972). Ruthven, D. M., Trans. Inst. Chem. Eng. 46 T185 (1968). Douglas M . Ruthven Department of Chemical Engineering Cniversity of Queensland St. Lucia 4067, Australia
( A )us. E plot ill Figure 5 by simply using a mean temperature of 769’K in the equation b 1n A/bE = 1!RT is somewhat fortuitous since the range of slopes obtainable from the temperature extremes of 727 to 811°K is 6.2 to 6.9. The most that can be concluded is that the Arrhenius relationship has the characteristic of desensitizing the effect of temperature and pressure on the activation energy and frequency factor without masking the wide variations in the reaction rate coefficient. For this reason we stated in our original paper that the linearity between 111 (il) aiid E “may prove to lie a useful tool in correlating data on reaction kinetics.” Whether our original paper has demonstrated the so-called “compensation effect” or not remains to be seen, particularly in view of the teiiuous proof offered by Professor Ruthven. Although we had no intention to broach the subject of compensation effect, which admittedly we do not compreheiid, we are reasonably coiifident bhat the activation energy and frequency factor are inextricably linked. Whether the apparent linearity between 111 A and E is real or a coilsequence of our method of correlation does not affect the validity of the observed effect of pressure 011 activat’ion energy. D . TI7. Johnson C. X.Sliepcevich* Cniversity o j Oklahoma IVorman, Okla. 73069
