1884
Ind. Eng. Chem. Res. 1989,28, 1884-1887
GENERAL RESEARCH Applications of Equilibrium Sensitivity Analysis to Aromatization Processes Graeme W. Norva1,t M. Jane Phillips,* and Ronald W. Missen Department of Chemical Engineering and Applied Chemistry, University of Toronto, Toronto, Ontario, Canada M5S l A 4
William R. Smith Department of Mathematics and Statistics, University of Guelph, Guelph, Ontario, Canada N1G 2W1
A general method utilizing sensitivity analysis is presented that enables estimation of the effect on a calculated equilibrium of changes in the species list to include other isomeric forms, as well as of changes in the problem parameters or thermodynamic data. The method is illustrated with the benzene-methylbenzene system. Equilibrium distributions for this system have been calculated for the temperature range 300-1000 K and methyl group to ring ratios of 1 to 5, using the thermodynamic data of Stull et al. The discovery of the shape-selective zeolite catalyst
ZSM-5 has led to the development of a number of hydrocarbon conversion processes yielding a distribution of aromatics, primarily benzene and methylbenzenes (Chen and Garwood, 1986). Treatment of this zeolite with Mn, B, or P results in a nonequilibrium para-selective product because of the hindered diffusion of the ortho and meta isomers in the pores (Young et al., 1982). Our research into aromatization reactions over medium-pore-size molecular sieves (Norval et al., 1989) has led us to a general approach, described in this paper, that demonstrates the utility of equilibrium analysis for such processes, particularly the use of sensitivity analysis to adjust an equilibrium composition to account for changes in the thermodynamic data and/or definition of the system. We use the benzenemethylbenzene system for illustration, since comparisons are continually being made between experimental and calculated equilibrium results for this system. The application of equilibrium analysis for such systems must be approached cautiously with respect to the species considered. Alberty (1985, 1986) has used the Benson group method based on the data of Stull et al. (1969) for obtaining data for the alkylbenzene isomers up to CmH3, (there are 192 alkylbenzene species up to C12H18)and has calculated equilibrium compositions for the disproportionation of the aromatics using all these species. However, shape-selectivecatalysts are used experimentally to hinder the formation of certain isomers, and equilibrium calculations must take this into account. Most authors compare experimental product distributions with calculated equilibria using 13 species (benzene and the 12 methylbenzenes). Taking the latter as a “base” system, we show how sensitivity analysis may be used to estimate the effects of including additional isomeric species. Similarly, the application of equilibrium analysis must be approached cautiously with respect to the precision of
* Author to whom correspondence should be addressed. ‘Present address: C-I-L Research Centre, 2101 Hadwen Rd, Mississauga, Ontario, Canada L5K 2L3. 0888-588518912628- 1884$01.50/ 0
the underlying thermodynamic data. Equilibrium calculations for the benzene-methylbenzene system have been reported by Egan (1960) and by Hastings and Nicholson (1961). Previously, the latter had provided thermodynamic data for eight of the methylbenzenes (Hastings and Nicholson, 19571, and they had used these together with the data of Pitzer and Scott (1943) in their calculations. Some of their data were subsequently stated to be in error, both by Egan and by Green et al. (1971). Egan gave different values for some of the data and used these together with the remaining data of Hastings and Nicholson (1957) and those of Taylor et al. (1946) in his calculations. Egan presented his results in an unusual graphical form for two temperatures (298 and lo00 K), but the full set of graphs for these and intermediate temperatures is no longer available (Zinn, 1988). More recently, additional thermodynamic data have been reported by Draeger (1985). As noted, Alberty (1985, 1986) used the data of Stull et al. (1969). Significant differences exist among the data of Stull et al., of Draeger, and of Hastings and Nicholson (1957). The purpose of this paper is to demonstrate the use of sensitivity analysis to account for (1) the effects on a calculated equilibrium of changes in the species list involving isomeric species and (2) changes or uncertainties in the problem parameters, such as temperature, pressure, feed composition, and thermodynamic data. In the course of illustrating both of these, we present a “base set” of equilibrium compositions for the benzene-methylbenzene system, using the data of Stull et al. (19691, which we consider to be the best data currently available.
Computational Methods For each species in an ideal solution, the (dimensionless) chemical potential is p i = pi*
+ In xi
(1)
where pi* is the standard chemical potential (dimensionless) of species i at temperature T and pressure P, and x i is the mole fraction of species i. The complete set of 0 1989 American Chemical Society
Ind. Eng. Chem. Res., Vol. 28, No. 12, 1989 1885 p j (Smith and Missen, 1982, pp 196-198):
Table I. api*/apjfor the Various Parameters (Ideal Solution) Pj
ari*lapj
T
-hi*/ RT Ui*/RT
P
Pi
u
api*/ap, 0
bk
2
pjo
nz
equilibrium mole fractions {xi) may be summarized in terms of #k as (Smith and Missen, 1982, p 47) M
x i = exp(Caki#k- p i * ) , i = 1, 2, k=l
...,N'
(2)
where #k is the Lagrange multiplier of species k, ahi is the subscript of the kth element in the molecular formula of the ith species, "is the number of species excluding inert or diluent species, and M is the number of elements (we assume that the rank of the formula matrix (ski) is M). Equation 2 permits presentation of equilibrium compositions in a table of values of $k only. Sensitivity analysis can be used to estimate the effects of changes in a problem parameter, pi, on the calculated equilibrium composition. This essentially involves determination of the sensitivity coefficients, d x i / d p j . The relative magnitudes of a x i / d p j indicate the relative sensitivities of x i to changes in pj. We have used the sensitivity analysis techniques described by Smith and Missen (1982, pp 192-198), based on earlier work of Smith (1969), which are summarized briefly below. One use of sensitivity analysis is to estimate, to firstorder, the changes in x i that result from changes in pi. Since pi is linear in In xi, we express the correction to some base composition, Xi,base, by 6 In x i
In
- In xi,bae = (a In x i / d p j ) ( 6 p j ) ( 3 )
where B In xi/apj is evaluated at xi,baeeand 6pj is the change in parameter pi: sPj = Pj,new - Pj,base
(4)
The sensitivity parameters to consider are T , P, {pi*), the number of moles of an element {bk),and the number of moles of inert or diluent species, nz. The sensitivity coefficient, d In x i / d p j ,is obtained by differentiation of eq 2 as M
a In x i / a p j = kCaki a $ k / a p j - api*/apj, =l
i = 1, 2,
...,N'
(5) where the values of eJ'k/dpiare obtained from the solution of the following M + 1 linear equations in the M + 1 unknowns d q k / d p jand d In n,fdpj:
"
1 = 1, 2,
..., M (6)
i=l
and
where ni is the number of moles of species i, n, is the total number of moles, and api*/apjfor the problem parameters is listed in Table I. The sensitivity coefficients can also be used to estimate the precision of x i that results from a given precision of
~ = ~(a In ~ xi/dpj)2u2pj, , ~ i
= 1, 2,
..., N'
(8)
where r?j,, x i is the variance of the uncertainty in In xi (i.e., the precision of In x i ) , and is the variance of the underlying uncertainty in p j . This assumes that uncertainties in p . are statistically independent. The equilibria of benzene and the 12 methylbenzenes were calculated on an IBM PC (using the data obtained from Stull et al. (1969)) by means of the algorithm described by Smith and Missen (1988), as implemented in the software package EQS, reviewed by Lipowicz (1986). Ideal-gas behavior was assumed. For an ideal gas, pi* = pio In P where pio is the standard chemical potential of species i at T and is equivalent to AGfio/RT(where AGfio is the standard free energy of formation). Calculations were carried out over the temperature range 300-1000 K. We considered five overall compositions corresponding to methyl group to ring ratios of 1 to 5, i.e., to the disproportionations of toluene, xylene, trimethylbenzene, tetramethylbenzene, and pentamethylbenzene, respectively. We note that the equilibrium compositions of this system are independent of pressure. Consequently, eq 2 can be expressed as
+
x i = exp(aci#co + aHi#H0 - p i o ) (9) where the values of the pressure-independant Lagrange multipliers, $co and #Ho, refer to the standard-state pressure of 1 atm. Similarly eq 5 can be expressed as a In x i / a p j = aci a # c o / a p j + aHia # H o / a p j- a p i o / a p j (10)
Results and Discussion The equilibrium mole fractions for the benzene-methylbenzene system, calculated from the data of Stull et al. (1969), are summarized in the form of #co and #Ho in Table 11; these values are for use in eq 9 together with the values of pi0 for which Stull et al. should be consulted (the composition for toluene disproportionation at 700 K is listed in Table IV as an example). The values calculated at 298 and 1000 K agree well with those of Egan (1960), generally within 2%, and we have verified that the differences arise from differences in the thermodynamic data and not from the method of calculation. There are some significant differences, however, between our results (and those of Egan), on the one hand, and those of Hastings and Nicholson (1961), on the other, resulting from differences in the thermodynamic data. For example, in the extreme, the use by Hastings and Nicholson of a free energy of formation for hexamethylbenzene about 1 2 kJ mol-l greater than that given by Stull et al. results in mole fractions of penta- and hexamethylbenzene of 0.575 and 0.219, respectively, for pentamethylbenzene disproportionation at 1000 K, compared with 0.396 and 0.319 calculated using the data of Stull et al. (the values reported by Egan are 0.392 and 0.3228, respectively); the data of Draeger (1985) are intermediate to those of Hastings and Nicholson and of Stull et al. Since some of the thermodynamic data of Hastings and Nicholson (1957) are known to be in error (vide supra), we believe that the results of Table 11 should replace those of Hastings and Nicholson (1961). Sensitivity coefficients, d In x i / d p j ,can be summarized in terms of d#co/dpj and d#H0 f dpj for use in eq 10. For brevity, the full set of (1360) values of w k o / a p jis not given. For illustration, the values for toluene disproportionation at 700 K are given in Table I11 (the "base set" equilibrium composition for this case is given in Table IV).
1886 Ind. Eng. Chem. Res., Vol. 28, No. 12, 1989 Table 11. Values of
if+"
and
for Use in Equation 9 feed
300 400 500 600
19.461 -10.987 13.450 -6.315 9.865 -3.485 7.489 -1.587 5.792 -0.216 4.524 0.815 3.539 1.621 2.755 2.266
*CO *HO *CO
*HO *C" *H"
*Cl *H
700
k"
900
*HO *CO *HO *CO
1000
*HO *CO
800
*HO
17.514 -9.361 11.575 -4.750 8.055 -1.974 5.722 -0.112 4.062 1.229 2.823 2.236 1.861 3.023 1.094 3.654
Table 111. Values of W C o / d p and j a$,"/apj for Use in Eauation 10 ( b , / b , = 7/8. T = 700 K.n, = 0) Wc"laPj 1.109 x 10-2 2.552 -2.333 -3.333 x 10-1 6.730 X lo-' 1.488 X 10-1 -8.193 X lo-' -1.755 X lo-' -7.903 X lo-' -1.109 x 10-2 -8.795 X 10" -3.332 X lo-' -3.274 X -9.625 X -6.253 X -4.090 X -4.536 X lo*
WHOIaP, -9.085 X -2.233 1.954 1.667 X lo-' -5.514 X lo-' -7.438 X lo-' 7.816 X 1.675 X lo-' 7.540 X 1.010 x 10-2 8.010 X 3.035 X 2.941 X 8.644 X loT3 5.652 X 3.648 X lo4 4.030 X lo4
The data in Table I11 can be used to ascertain the relative effects of the problem parameters on the mole fractions. The effect of temperature on the benzenemethylbenzene equilibria is slight, since the magnitudes of Wk0/dT(and d p i o / d T ) are small. The effects of changes in bc and bH are relatively large, as may be expected. The effect of uncertainty in p.O on x i (i f j ) increases as xi increases (see also Table I$). Finally, d In x i / d n , = -1 for all species in this system. The benzene-methylbenzene equilibrium compositions can be perturbed by adding or deleting isomeric forms of existing species to represent many industrial processes.
15.177 -7.556 9.442 -3.102 6.062 -0.435 3.837 1.345 2.270 2.614 1.108 3.562 0.210 4.299 -0.502 4.887
12.008 -5.241 6.518 -0.966 3.322 1.567 1.249 3.234 -0.183 4.404 -1.235 5.272 -2.036 5.939 -2.665 6.467
8.766 -2.971 3.226 1.339 0.041 3.863 -2.003 5.511 -3.399 6.655 -4.421 7.501 -5.184 8.142 -5.783 8.648
For example, ethylbenzene is a common product of the methanol-to-gasoline and toluene alkylation processes (Chen and Garwood, 1986). Since ethylbenzene and oxylene are isomers, they can be treated as a single "lumped pseudospecies" (Smith and Missen, 1974). The standard chemical potential (dimensionless) of the pseudospecies is 46.81 (at 700 K), resulting in 6 p j o = -0.32 (based on a standard chemical potential of 47.13 for o-xylene). In Table IV, the base set composition (toluene disproportionation at 700 K and 1atm) is compared with the correct equilibrium composition of the system with ethylbenzene added (calculated using EQS) and with the equilibrium approximated through use of eq 3. The agreement between the approximate and the exact results is good. Hence, a simple hand calculation, using the sensitivity coefficients of Table 111, produces an accurate estimate of the perturbed equilibrium in this case. Table IV also compares the correct equilibrium for the alkylation of toluene with methanol at 700 K and 1 atm (C,H8:CH30H = 1:0.2) with the approximate equilibrium calculated by using three perturbations: abc = 0.2, 6bH = 0.4, and an, = 0.2 to account for the product, water. Again, very good agreement is evident, further demonstrating that a simple hand calculation suffices. Summary (1)We have provided an accurate approximate method of correcting a calculated equilibrium composition for changes in problem parameters. This method enables estimation of the equilibrium of related systems based on perturbing a base equilibrium composition. The method
Table IV. Comparison of Equilibria Calculated by Using Perturbations (Cases B and C) with Exact Equilibrium (Case A) Distributions (T= 700 K,n, = 0)
svecies C6H6
C?H8 o-CaH10 m-C6H10
p-C6H10 1,2,3-CgHiz 1,2,4-CgH12 1.3,5-CgHiz 1,2,3,4-C&14 1,2,3,5-CioHi4 1,2,4$-CioH14 C11H16
ClZH18
base set 2.99 X lo-' 4.46 X lo-' 5.18 X 1.11 x 10-1 4.50 x 3.17 x 10-3 2.52 X 9.53 x 10-3 6.05 x 10-4 1.78 x 10-3 1.16 x 10-3 5.58 x 10-5 4.91 X lo'?
"Ratio of ethylbenzene to o-xylene = 0.379:l.
addition of ethylbenzenea A B 3.02 X 10-1 3.02 X lo-' 4.39 x 10-1 4.38 X lo-' 6.82 X 6.85 X lo-* 1.07 x lo-' 1.06 X lo-' 4.77 x 10-2 4.80 X 2.95 x 10-3 2.98 x 10-3 2.34 X 2.36 x 8.94 x 10-3 8.85 x 10-3 5.54 x 10-4 5.47 x 10-4 1.61 x 10-3 1.63 x 10-3 1.04 x 10-3 1.06 x 10-3 4.99 x 10-5 4.91 x 10-5 4.28 x 10-7 4.20 X lo-? Water-free basis.
addition of 0.2 mol of CH,OHb A c 2.22 x 10-1 2.18 X 10-1 4.34 x 10-1 4.35 x 10-1 6.62 X loT2 6.71 X 1.44 X 10-' 1.42 X lo-' 6.47 X 6.38 X 5.31 x 10-3 5.48 x 10-3 4.21 X 4.34 x 10-2 1.60 X 1.65 X 1.33 x 10-3 1.39 x 10-3 3.90 x 10-3 4.09 x 10-3 2.54 x 10-3 2.66 x 10-3 1.61 x 10-4 1.71 X lo4 1.85 X lo* 2.00 x 104
Ind. Eng. Chem. Res., Vol. 28, No. 12, 1989 1887
has been demonstrated with the benzene-methylbenzene system. (2) The relative magnitudes of the sensitivity coefficients indicate the relative importance of changes in the respective parameters. (3) We have presented calculated equilibrium results for the benzene-methylbenzene s y s t e m ( t h e "base set" from the data of Stull et al., 1969) to replace those in previous reports, which are either partly erroneous or are no longer completely available. (4) The ease of calculation of sensitivity coefficients and the ability to use them in hand calculations makes sensitivity analysis an integral part of equilibrium analysis.
Acknowledgment Financial support has been received from the Natural Sciences and Engineering Research Council of Canada. G.W.N. thanks the Evald Torokvei Foundation for supp o r t i n g this research with a g r a d u a t e scholarship. Nomenclature aki = subscript to the kth element in the molecular formula of species
i
bk = moles of element k in basis amount (1 mol) of system AGfio= standard free energy (Gibbs function) of formation
of species i , kJ mol-' M = number of elements ni = number of moles of species i n, = total number of moles nz = number of moles of inert or diluent species N' = number of species excluding inert or diluent species p , = jth problem (sensitivity) parameter P = pressure, atm R = gas constant, 8.314 J mol-' K-' T = temperature, K xi = mole fraction of species i
Greek Letters 6 = change in (a quantity) 6ij = Kronecker delta ( = 1 for i = j ; =O for i # j ) pi = chemical potential of species i, dimensionless pio = standard chemical potential of species i (function of T only), dimensionless, equivalent to AGfio/RT pi* = standard chemical potential of species i (function of T and P ), dimensionless u2 = variance = Lagrange multiplier for the kth element, dimensionless $c0, qH0= Lagrange multipliers for carbon and hydrogen, respectively; referring to the standard-state pressure of 1 a t m , dimensionless $k
Subscripts C = carbon
H = hydrogen i , j , k, 1 = d u m m y indexes for elements and species
Registry No. C,&, 71-43-2; C7H8,108-88-3;o-C8H10,95-47-6; m-CsHlo, 108-38-3; p-C8Hlo, 106-42-3; 1,2,3-CgH12,526-73-8; 1,2,4-c&12, 9563-6; I13,5-C&12, 108-67-8; 1,2,3,4-C1cH14,48823-3; 1,2,3,5-C&14,527-53-7; 1,2,4,5-CloH14,95-93-2;CiiH16,700-12-9; C12Hl8, 87-85-4; CH,OH, 67-56-1; ethylbenzene, 100-41-4.
Literature Cited Alberty, R. A. Standard Chemical Thermodynamic Properties of Alkylbenzene Isomer Groups. J. Phys. Chem. Ref. Data 1985,14, 177-192. Alberty, R. A. Equilibrium Disproportionation and Isomerization of Alkylbenzenes. Ind. Eng. Chem. Fundam. 1986, 25, 211-216. Chen, N. Y.; Garwood, W. E. Industrial Applications of Shape-Selective Catalysis. Catal. Reo. Sci. Eng. 1986, 28, 185-264. Draeger, J. A. The Methylbenzenes I1 Fundamental Vibrational Shifts, Statistical Thermodynamic Functions, and Properties of Formation. J. Chem. Thermodyn. 1985, 17, 263-275. Egan, C. J. Calculated Equilibrium of the Methylbenzenes and Benzene. J. Chem. Eng. Data 1960,5, 298-299. Green, J. H. S.; Harrison, D. J.; Kynaston, W. Vibrational Spectra of Benzene Derivatives XI. 1,3,5 and 1,2,3 Trisubstituted Compounds. Spectrosc. Acta 1971,27A, 793-806. Hastings, S. H.; Nicholson, D. E. Thermodynamic Properties of Selected Methylbenzenes from 0 to 1000'K. J . Phys. Chem. 1957, 61, 730-735. Hastings, S. H.; Nicholson, D. E. Thermodynamic Equilibrium Among Benzene and the Methylbenzenes from Spectroscopic Data. J . Chem. Eng. Data 1961, 6, 1-4. Lipowicz, M. Chem. Eng. 1986, 93(7), 98. Norval, G. W.; Phillips, M. J.; Virk, K. S.; Simons, R. V. Olefin Conversion over Zeolite H-ZSM-5. Can. J. Chem. Eng. 1989,67, 521-523. Pitzer, K. S.; Scott, D. W. Thermodynamicsand Molecular Structure of Benzene and its Methyl Derivatives. J. Am. Chem. SOC.1943, 65, 803-829. Smith, W. R. The Effects of Changes in Problem Parameters on Chemical Equilibrium Calculations. Can. J.Chem. Eng. 1969,47, 95-97. Smith, W. R.; Missen, R. W. The Effect of Isomerization on Chemical Equilibrium. Can. J . Chem. Eng. 1974, 52, 280-282. Smith, W. R.; Missen, R. W. Chemical Reaction Equilibrium Analysis: Theory and Algorithms; Wiley-Interscience: New York, 1982. Smith, W. R.; Missen, R. W. Strategies for Solving the Chemical Equilibrium Problem and an Efficient Microcomputer-Based Algorithm. Can. J . Chem. Eng. 1988,66, 591-598. Stull, D. R.; Westrum, E. F.; Sinke, G. C. The Chemical Thermodynamics of Organic Compounds; Wiley: New York, 1969. Taylor, W. J.; Wagman, D. D.; Williams, M. G.; Pitzer, K. S.; Rossini, F. D. Heats, Equilibrium Constants, and Free Energies of Formation of the Alkylbenzenes. J. Res. Natl. Bur. Stand. 1946,37, 95-122. Young, L. B.; Butter, S.A.; Kaeding, W. W. Shape-Selective Reactions with Zeolite Catalysts 111Selectivity in Xylene Isomerization, Toluene-Methanol Alkylation, and Toluene Disproportionation over ZSM-5 Zeolite Catalysts. J. Catal. 1982, 76, 418-432. Zinn, K. E. Chevron Research Company, Richmond, CA, personal communication, 1988.
Received for reuiew April 3, 1989 Accepted August 29, 1989
