219
Ind. Eng. Chem. Fundam. 1983, 22, 219-223
agement science. The program used here was implemented on a minicomputer. Linear programming is not a substitute for the algorithms (reviewed recently by Smith, 1980) that have been developed for the computation of equilibria in complex systems when precise compositions are required and there are adequate resources and staff. However, linear programming provides a rapid and efficient way of surveying wide ranges of independent variables and getting a better intuitive understanding of the principal reactions. Acknowledgment The author is indebted to Professor Thomas L. Magnanti of the Sloan School of Management at MIT for making available a linear programming routine on their Prime computer. Financial support was received from the Dreyfus Foundation. Literature Cited
(moles of various elements present) and yields "shadow prices" for changes in the constraints. The "shadow prices" are the changes in Gibbs energy when the number of moles of an element in the system is changed by one unit, everything else being held constant. Thus a great deal more information is obtained from the linear programming calculation than the optimal solution. When it is of interest, Gibbs energies and numbers of moles of various elements can be changed to values beyond these limits to see what new optimal solution is obtained. Conclusions These calculations show that linear programming provides a useful tool for surveying the effects of temperature, pressure, and reactant composition on the equilibrium of a system for which the number of species is greater than one larger than the number of linear constraints. In the case examined there are 6 species and 3 linear constraints so that three independent chemical reactions are required to represent all possible chemical changes. However, under some conditions the equilibrium composition may be represented by a single chemical reaction, and in others by two chemical reactions. These reactions are conveniently obtained from linear programming. Linear programming may be used for systems with a very large number of species requiring a very large number of independent chemical reactions. The computer program used will handle 400 species and 100 linear restrictions. Linear programming is frequently available in connection with large computer systems because of its use in man-
Baron, R. E.; Porter, J. H.; Hammond, 0. H. "Chemical Equilibria in CarbonHydrogen-Oxygen Systems"; The MIT Press: Cambridge, MA 1976. Bjornbom, P. H. Ind. Eng. Chem. Fundam. 1975, 74, 102. Bradley, S. P.; Hax, A. C.; Magnanti, T. L. "Applied Mathematical Programming"; Addlson-Wesiey Publishing Co.: Reading, MA, 1977. Dantzig, G. B. "Linear Programming and Extensions"; Princeton University Press: Princeton, NJ, 1963. Smith, W. R.; Missen, R. W. Can. J . Cbem. Eng. 198% 46, 269. Smith, W. R. I n "Theoretical Chemistry, Advances and Perspectives"; Academlc Press: New York, 1980 Voi. 5, p 185. Stuil, D. R.; Prophet, H. "JANAF Thermochemical Tables", 2nd ed.;NSRDSNSB 37, U S . Government Printing Office, Washington, DC, 1971.
Received for review March 11, 1982 Accepted January 3, 1983
Equilibrium Constant for the Methyl tert -Buty I Ether Liquid- Phase Synthesis by Use of UNIFAC F. Colombo, L. Corl, L. Dalloro, and P. Delogu" Centro Ricerche MONTEDIPE, Via San Pietro 50, Bollate, Milano, Italy
A predictive method is used for evaluating the equilibrium constant of a reaction carried out in the liquid phase. This method uses the values of the standard free energy of formation available in the literature for the chemicals involved and the UNIFAC estimates of activity coefficients to describe the liquid-phase nonideality. Some experimental values of the equilibrium concentrations were obtained in a micro-pilot plant. The predicted equilibrium constants agree quite well with our experimental values.
Introduction Methyl tert-butyl ether is a chemical whose production will greatly increase in the near future because of its good antiknock properties. It is industrially manufactured from the isobutene contained in olefinic C4 cuts and from methanol through the reversible reaction i-C4H8 CHSOH * t-C,HgOCH3
+
This reaction is exothermic; therefore the equilibrium is shifted to the right at low temperatures and to the left at high temperatures. In all industrial processes the reaction is carried out in the liquid phase, but no data on the equilibrium constant are available. Classical methods of thermodynamics allow the computation of the gas-phase equilibrium constant; the derivation of the liquid-phase constant, in terms of mole fractions, requires knowledge of the liquid-phase activities 0196-4313/83/ 1022-021 9$01.50/0
for all components of the reaction mixture. The reaction is usually performed in the presence of inert linear C4 olefins; therefore the system is a multicomponent one, and its behavior is not simple to predict. We have tried to evaluate the equilibrium constant only by means of predictive methods, using available literature values of AGf" for the chemicals involved and UNIFAC predictions of activity coefficients (Fredenslund et al., 1977) to describe the liquid phase nonideality. The results are compared with the experimental values measured in our laboratory. Thermodynamic Framework Tabulated values of the standard free energy of formation (AGof,298)iallow the computation of the standard free energy of reaction for the MTBE synthesis in the gas phase n
(AG
0 1983 American
298)reaction
=
Chemical Society
2 v i (AG i=l
f,298)i
(1)
Ind. Eng. Chem. Fundam., Vol. 22, No. 2, 1983
220
Table I. Thermodynamic Data for Gas Phase Equilibrium Computation
The gas-phase equilibrium constant is given by
Equilibrium constants at different temperatures can be obtained through the Van't Hoff equation d 1' KP -= dT
(Ho)reaetion
(3)
RT2
where
AH029a,
AG0293,
COPZ98'
component
cal/mol
cal/mol
cal/(mol K)
isobutene methanol MTBE
-4040' -480501 -67751
1 3 880a -38 810' -28 0 4 0 b
21.30' 10.4ga 32.49'
a API (1979). (1976).
Fenwick et al. (1975).
Benson
Table 11. Clausius-Clapeyron Equation Parameters and Liquid Molar Volumes
The free energy change for the liquid-phase reaction is related to that for the gas phase through the equation
acTP(i) = n
AGTo(g)+ c~,[O.O242u,(~)(P - P;") + RT In Pi"] ( 5 ) r=l
Pi" is the saturation pressure of the pure component i at temperature T. AGTp(l)is the free energy change of the process i-C4H8(1,T,P,xi.~4H8 = 1) + CH30H(1,T,P,xc~,o~ = 1) 2
a
component
A
isobutene methanol MTBE
10.42892a 13.51821; 11.07472
B, K 2776.60' 4564.92; 3634.82
u(l), mL/mol
93.33 44.44 118.8
Alm and Ciprian (1980).
API (1979).
considered for the K , calculation. From eq 5 and 8 we finally obtain In KnT= 1
R
- P;") - RT In P;"] (10) In KpT - - ~,[0.0242u,(~)(P R T i=l t-C4H90CH3(1,T,P,Xt-C4HSOCH8 = 1) Equations 5 and 9 have been derived for the particular reaction of MTBE synthesis, but they are in fact general carried out at constant temperature and pressure. This relations, applicable to all reactions carried out in the process can be divided into the following steps liquid phase. AG' K , Computation for MTBE Synthesis i-C4H8(1,T,P,Xi.~4~8 = 1) __+ i-CqH8(S01,T,P,Xei.c4~*)(i) Equation 3 has been integrated supposing the Cpo values SG independent of temperature and equal to the values at 298 CH@H(~,T,P,~CH,OH = 1) K. This is possible because of the narrow range of temCH,OH( Sol,T,P,XeCH30H)(ii) peratures of practical interest. Equation 2 provides the Kp298value. The basic data for (Cp0298),,(Afi0f,298)j and i-CqH8(SO1,T,P,xei.cI~*) + (AGof,298)i are reported in Table I. As the Cp" value of CH~OH(S~~,TQ,X~CH,OH) MTBE is not available in the literature, it has been computed by means of the group contribution method for t-CqHg0CH3(S01,T,P,Xe~.~4~g~~~8) (iii) thermochemical quantities proposed by Benson (1976). JGW Pure component saturation pressures have been ex~-C~H~OCHB(SO~,T,P,~~~.C~H~OCH,) pressed as a function of temperature by the Clausius~-C~H~OCH,(~,T,P,~,.C,HSOCH, = 1) (iv) Clapeyron equation in order to simplify the form of the final K , equation AGr, AG'r, and AGIV refer to dilution of reactants and product from the pure state to the equilibrium concenIn Pi" = A i- (Eli/?") (11) tration in the reaction mixture; they can be computed from A and B parameters shown in Table I1 have been obtained the liquid-phase fugacity ratio from the vapor pressure data reported in the original (fi)sol references. &I~= RT In -= RT In a, (6) For our reaction eq 10 yields "
__*
-
__*
(filpure
In this way the activity of component i in the liquid phase is defined in relation to the same reference state used in vapodiquid equilibrium data correlation (Prausnitz 1969), namely: ai= 1when xi = 1. Substitution of eq 6 for each component in AGTP(l)= AG' + AG" + AG"' + AGIV (7) gives n
AGTp(l) = RT Cvi In ai= -RT In Ka
(8)
i=l
where (9) It must be recognized that the summation of eq 8 is extended only to the reacting species, but all the species present in solution, both reacting and inert, have to be
In Ka7'= -10.0982
+ 4254'05 + 0.2667 1nT T ~
-
Numerical inspection of this equation shows that the last term on the right-hand side is negligible up to 20 atm total pressure. Therefore K, is a function of temperature only. The full line of Figure 1shows this function in the temperature range of practical interest. K , Computation for the Reaction System The equilibrium constant as calculated from eq 12 is the true equilibrium constant, independent of the reaction mixture composition. In order to use it to compute the equilibrium conversions of the alcohol and the olefin, we have t o know K , in eq 9. It is defined as
Ind. Eng. Chem. Fundam., Vol. 22, No. 2, 1983 221 0 I MEOH
o
I
iC4
t
0
IO
05 XIC.
I J
290
L
300
310
320
330
340
350
Figure 2. Comparison between 7’spredicted by UNIFAC and experimental values from Churkin et al. (1977) for isobutene-methanol system at 323 K.
370
360 T(K1
Figure 1. K , vs. T as calculated by eq 12 and comparison with experimental values.
where the activity coefficients depend on temperature and composition. Methanol is much more polar than the other components and it forms very nonideal solutions with hydrocarbons. Then K, values will be very different from unity and they will depend on alcohol concentration. In the past, only extensive experimentation afforded the y vs. composition correlations, mainly for multicomponent systems. In our case the system is formed by the C4olefins, the ether and the alcohol, and by other minor components. Some experimental information has been published in the recent literature on the possible binary subsystems (Alm and Ciprian, 1980; Churkin et al., 1977), but they are not complete. Instead of performing experiments to obtain the missing information, we thought it more interesting to try the UNIFAC predictive method recently developed (Fredenslund et al., 1977). This is a method for the prediction of activity coefficients in nonelectrolyte liquid mixtures based on the group solution concept. Instead of considering a liquid mixture as a solution of molecules, the mixture is considered as a solution of groups. The activity coefficients are then determined by the properties of the groups rather than by those of the molecules. The parameters needed for the use of UNIFAC are group volumes, Rk,group surface areas, Qk,and group interaction parameters, umnand urn. The last ones have been taken from the table published by Skjold-Jerrgensen et al. (1979). Some difficulties arise in the use of those parameters for our particular problem, namely (i) the interaction parameters were estimated from vapor-liquid equilibria, a phenomenon very different from a chemical reaction equilibrium; in order to verify that UNIFAC can be used in the computation of chemical equilibria a comparison between experimental and calculated equilibrium mole fractions must be carried out; (ii) the vapor-liquid equilibrium data which were used for parameter estimation in the table of Skjold-Jargensen et al. (1979) included only components with normal boiling point greater than 303 K. Thus the validity of UNIFAC predictions for systems containing C4 hydrocarbons has been checked by comparing the calculated with the available experimental values for isobutene-methanol at 323 K (Churkin et al., 1977). Figure 2 shows a quite good agreement. Also, the agreement between calculated and experimental values for the system methanol-MTBE (Alm and Ciprian, 1980) is equally good, as can be expected because the boiling points
iooa
100
1
L
1005
IO
15
I-IIUTENEIYETHANOL
20
INITIAL RATIO
Figure 3. Calculated mole fraction equilibrium ratio vs. the initial ratio of the reactants. The 1-butene/isobutene ratio is set at 1.22 mol/mol.
of pure components are included in the validity range of UNIFAC parameters. Besides methanol, isobutene, and MTBE, the actual components of the reaction system are 1-butene, trans- and cis-2-butene, and minor components, as saturated C4and the dimer of isobutene. Despite the complexity of this mixture, the predicted activity coefficients of all the hydrocarbon-methanol and hydrocarbon-MTBE systems are very similar and the binary systems between the hydrocarbons are nearly ideal. Therefore, only four components have been considered: isobutene, methanol, MTBE, and 1-butene in place of the other hydrocarbons. K, is a function of the equilibrium compositions, which are related to the initial (or feed) compositions by the stoichiometric constraints. If we suppose that the initial MTBE concentration is zero, K , is a unique function of the isobutene to alcohol and 1-butene to isobutene initial ratio. Figure 3 shows the computed K, values as a function of the isobutene/methanol ratio, temperature being a parameter. The initial ratio isobutene/ 1-butene is set equal to 1.22. Experimental K, Values Experimental values of equilibrium concentrations have been obtained using a micro-pilot plant for the synthesis
222
Ind. Eng. Chem. Fundam., Vol. 22, No. 2, 1983 c
XMEO"
Figure 4. Simplified flow diagram of the experimental apparatus for K, determinations. Table 111. mpical Effluent Composition from the MTBE Micro-Pilot Plant component
mol %
isobutane n-butane trans-2-butene isobutene 1-butene + cis-2-butene dimethyl ether isobutene dimer methyl tert-butyl ether tert-butyl alcohol methanol others
0.87 1.04 11.07 5.80 34.55 0.09 2.70
40.66 0.07 2.60 0.65
of MTBE (Figure 4). It consists of a tubular reactor containing Amberlyst-15 acidic resin to catalyze the reaction. The reactor can be continuously fed with fresh reactants. All types of regimens, from the perfect plug-flow to the perfect backmixing, can be obtained by means of a recycle pump. The chosen reaction temperature was kept by means of a preheater placed before the inlet of the reactor. The equilibrium conditions for each run were reached in the following way: CH30H from D1 and i-C4H8(as a component of a commercial butadiene-free C4 cut) from D2 were fed to the reactor in a chosen ratio. When the reactor, operated in the continuous mode, had reached a steady-state condition, the feed of the reactants was interrupted, the recycle pump was kept on, and the reaction proceeded batchwise. The fluid composition was analyzed by taking samples of liquid from the valve "a" at successive times until steady conditions were reached. Samples were collected as a liquid in a chilled test tube and analyzed by gas chromatography in a two-column system. Table I11 shows the product distribution of a typical run. The minor components are mainly normal and isobutane and the dimer of isobutene. The experimental K , values were computed from the actual mole fractions of MTBE, methanol, and isobutene through the relation
Kx =
XMTBE XMeOHXi-C,H8
In our micro-pilot plant the initial reactant and product concentrations for a run depended on the holdup of the recycle loop at the switching time from continuous to batch regimen. The hold-up composition was not uniform, so
%
Figure 5. Plots of K, equilibrium ratio vs. methanol equilibrium mole fraction.
that the initial conditions were unknown and did not allow the experimental check of the trends shown in Figure 3. Experimental K , values have been plotted vs. the equilibrium methanol mole fraction in Figure 5, where the data obtained at nearly equal temperature are connected by "drawn-by-eye lines" to obtain an approximate trend. Discussion of Correlation Accuracy The experimental equilibrium concentrations can be used to calculate the related values of K , by means of eq 9, 13, and 14. These values are not true experimental values because they contain the form and the parameters of the correlation. Anyway, the agreement between those pseudo-experimental quantities and the corresponding ones calculated through eq 12 is a check of the full model. Figure 1 shows the fit obtained. The check looks quite satisfactory because: (i) the correlation correctly predicts the dependence of K , on temperature, and (ii) it interpolates the "experimental" values within the experimental accuracy. The experimental data points and the calculated curve shows a low systematic shift that may be due to very small inaccuracies of the thermochemical basic data. Only a few experimental data do not agree with this correlation. All of them have been obtained at methanol mole fractions lower than 0.01. The experimental error in this range of concentrations is very high; the standard deviations obtained from repeated measurements performed for each condition are shown in Figure 1. Nevertheless, the shift between calculated and "experimental" values is greater than the experimental error and it must be assigned to a correlation inaccuracy. It is possible to formulate some hypotheses on the reasons of this inaccuracy. As the activity coefficients for MTBE and isobutene are nearly equal and very close to unity, K7 is nearly proportional to 1/yMe0H. The relation between K , and K , is approximately K,
K X
N
YMeOH
The required values of Y M ~ Hfor translating all points at x M ~ H< 0.01 on the theoretical curve range from 10 to 30, depending on concentration, while the values predicted from UNIFAC range from 8 to 10. The predicted dependence of 7MeOH values on concentration looks too low, as Figure 6 shows. The data points shown in Figure 6 are the same as those used to compute the standard deviations of the low MeOH concentration points in Figure 1. It appears that the scattering of the experimental values is associated with a
Ind. Eng. Chem. Fundam., Vol. 22, No. 2, 1983
f = fugacity, atm
0 311.7 K A 319.5 K
0 3 2 1 . 2 ti -UNIFAC t
b
223
i
Figure 6. Comparison between ~~d~values required for a better agreement between calculated and experimental values of K , and
UNIFAC predictions.
systematic trend with methanol concentration. K, is very sensitive to small errors in x M ~ H .Conversely, the calculated xMeOH are insensitive to the errors in K,. Thus, the use of this model also does not introduce serious mistakes at low methanol concentrations; on the other hand, the extreme methanol mole fraction range is not of practical interest. Conclusions This work shows that it is possible to predict the equilibrium constant of a reaction in the liquid phase involving nonelectrolyte components. The standard free energies of formation of reactants and products can be obtained from the literature or predicted with Benson’s group contribution method for thermochemical quantities. The activity coefficients of the components in the reaction mixture can be computed with the UNIFAC group contribution method even if the published parameters have been derived from vapor-liquid equilibrium data, a phenomenon very different from a chemical equilibrium. On the basis of these results we think that correlation of other chemical and physical properties with the liquid-phase composition, such as reaction rates and diffusion coefficients, should be attempted using the UNIFAC predictions of the activity coefficients. Nomenclature a = activity A , B = Clausius-Clapeyron constants Cp = molar heat capacity at constant pressure, cal/(mol K)
G = Gibbs free energy, cal/mol AGfo = standard free energy of formation, cal/mol H = enthalpy, cal/mol AHfo = standard enthalpy of formation, cal/mol K , = liquid-phase equilibrium constant in terms of activity KPT = equilibrium constant for the gas-phase reaction at temperature T K , = mole fraction equilibrium ratio K7 = ratio of activity coefficients at equilibrium n = number of reacting components P = pressure, atm P,” = saturation pressure, atm R = gas constant, cal/(mol K) T = temperature, K u(l) = molar liquid volume, mL/mol x = liquid mole fraction y = activity coefficient A = any difference between two states I.L = chemical potential u = stoichiometric coefficient Subscripts g = gas phase i = component index 1 = liquid phase pure = pure component sol = solution T = at temperature T Superscripts O = standard state e = equilibrium P = at pressure P Registry No. t-ClH90CH3, 1634-04-4. Literature Cited Alm, K.; Clprlan, M. J . Chem. Eng. Date 1980, 25, 100. API Project 44, Thermodynamic Research Center, Texas A8M University, College Station, TX, 1979. Benson, S. W. “Thermochemlcal Klnetlcs”, 2nd ed.;Wiley: New York, 1976; Appendix. Churkln, V. N.; Oorshkov, V. A.; Pavlov, S. Yu.; Levicheva, E. N.;Karpache va, L. L. VINITI (No. 2494-77 Dep. from June 23, 1977). Fenwick. J. 0.; Harrop, D.; Head, A. J . Chem. Thermodyn. 1975, 7 . 943. Fredenslund, A.; Gmehllng, J.; Rasmussen, P. ”Vapor-Liquid Equilibria Using UNIFAC”; Eisevler; Amsterdam, 1977. Prausnltz, J. M. ”Molecular Thermodynamics of Fluid-Phase Equlllbria”; Prentice-Hall; Engiewwd Cilffs, NJ, 1969. SkjoldJorgensen, S.; Kolbe, B.; Gmehilng, J.; Rasmussen, P. Ind. Eng. Chem. Process Des. D e v . W7g, 18, 714.
Received for reuiew December 23, 1981 Revised manuscript received December 15, 1982 Accepted January 19,1983
