Ind. Eng. Chem. Res. 1988,27, 338-343
338
GENERAL RESEARCH Equilibrium Constant for the Methyl tert -Butyl Ether Vapor-Phase Synthesis Javier Tejero,* Fidel Cunill, and Jose Felipe Izquierdo Chemical Engineering Department, University of Barcelona, 08028 Barcelona, Spain The equilibrium constant for the reaction of methyl tert-butyl ether (MTBE) vapor-phase synthesis was determined experimentally in a continuous-flow device a t atmospheric pressure. The temperature of the catalytic bed does not change when a mixture in equilibrium is fed into the reactor. Given t h a t the reaction of MTBE synthesis is fairly exothermic, we consider this a rough indicator that the composition of the mixture corresponds to equilibrium composition. This method has turned out t o be suitable, and the equilibrium constants obtained agree satisfactorily with the predictled values_calculated from thermal data found in the literature. In addition, the values of AH", ASo, and AGO a t 298 K, deduced from the variation of the equilibrium constant with temperature in the range 40-110 "C, agree with the values calculated from the bibliographical data, within the limits of the experimental error. Chemical equilibrium is the conversion limit for a chemical reaction and is characterized by the equilibrium constant. The main type of calculations in which chemical equilibrium is involved (Denbigh, 1981; Smith and van Ness, 1965; Hougen et al., 1962; Glastone, 1947) consists of determining the value of the equilibrium constant at a given temperature starting with the enthalpy and entropy changes of reaction a t the standard state and, vice versa, determining the equilibrium constant of the reaction at several temperatures to find the enthalpy and entropy changes of reaction a t the standard state. In general, the estimated values of the equilibrium constant of a reaction in the vapor phase computed from classical thermodynamicsmethods are sufficiently accurate and can be later utilized provided that the primary data of the estimation procedure are sufficiently accurate. However, when these data are not reliable or are not available, they must be determined in the laboratory. This work presents a method for the experimental determination of the equilibrium constant of vapor-phase reactions whose enthalpy change.of reaction is relatively high, so that variations in composition give rise to a measurable temperature change in the reactor. This method, which requires the use of a continuous fixed-bed microreactor, is applied to the reaction for obtaining methyl tert-butyl ether (MTBE) in the vapor phase. The reaction of MTBE synthesis has been considered suitable for our purposes because it is a reversible and fairly exothermic reaction (AH" (298 K) = -15.63 kcal. mol-l). The MTBE is obtained hom the reaction of addition of methanol to isobutene catalyzed by sulfonic ion-exchange resins with high selectivity, in both the liquid and vapor phases (Ancillotti et al., 1978; Setinek et al., 1977; Heck et al., 1981). In addition there is currently great interest in MTBE because of its remarkable antiknock properties (Reynolds et al., 1975; Garibaldi et al., 1977; Taniguchi and Johnson, 1979; Talbot, 1979; Harrington et al., 1979; Csikos et al., 1980; Farhat and Mazhar, 1984).
Computation of Equilibrium Constant from Thermal Data The equilibrium constant for a gas reaction in which n compounds take part is (Denbigh, 1981; Smith and van Ness, 1965; Hougen et al., 1962)
The temperature dependence of the equilibrium constant is given by In K = I K - IH/RT + ( a / R ) In T + (b/2R)T + (c/6R)T2 + (d/12R)T3 (2) where IH and IK are the constants of integration of the Kirchoff and van't Hoff equations, respectively. Parameters a, b, c, and d can be deduced from the molar heat capacities of _thecompounds that take part in the reaction, in the form Cp,i= a, + b,T + c,T2+ diT3(Smith and van Ness, 1965; Hougen et al., 1962), and I H can be calculated from the following equation, which expresses the temperature dependence of the enthalpy change of reaction: AI?
= IH
+ aT + ( b / 2 ) T 2+ ( c / 3 ) T 3+ (d/4)T4
(3)
The thermal data required for evaluation of equilibrium constants in this manner are the heats of formation and free energies of formation (or alternatively, the heats of formation and absolute entropies) at the standard state, of the compounds that take part in the reaction, as well as their molar heat capacities in the form of power functions. Standard-state conditions for the pure components, chosen for convenience, are the ideal gas state at 1 atm and 298 K. From the thermal data reported in Table I for methanol, isobutene, and MTBE, the following values for enthalpy and free-energy chaqges of reaction at the standard-state are obtained: AHO(298 K) = -15 630 calm mol-'; AC"(298 K) = -3080 cal-mol-'.
0888-5885/88/2627-0338$01.50/0 0 1988 American Chemical Society
Ind. Eng. Chem. Res., Vol. 27, No. 2, 1988 339 Table I. Thermal Data of Methanol, Isobutene, and Standard State: Ideal Gas at 1 atm and 25 "C" methanol isobutene A@Of,,(298K), kcal-mol-' -48.08 -4.04 +GorJ298 K), kcalsmol-' -38.84 13.88 C, = a + b T + c T 2 + dT3, cal-mol-'-K-' a 5.052 3.834 10b 0.1694 0.6698 105c 0.6179 -2.607 108d -0.6811 0.2173 @,(298 K), kcabmol-' 9.067h 4.938' S0,(298 K), cal-mol-'.K-' 56.8d 70.16d So1(298K), cal.mol-'.K-' 30.3d 53.65f
MTBE. 21
MTBE -67.751' -28.040' -0.5498' 1.302 -7.162 5.530 7.262' 86.48O 63.399
"Data source: "Property Data Bank" (Reid et al., 1977). 'Fenwick et al., 1975. cEstimated by the Rihany and Doraiswamy method (Reid et al., 1977). dHougen et al., 1962. OEstimated by the Anderson, Beyer, and Watson method (Reid et al., 1977; Hougen tt al., 1962). !Estimated by means of the expression Sol= So, ?,IT. BAndon and Martin, 1975. hYaws and Hopper, 1976. 'Yaws, 1976.
2.
2.1
-
2.!
In K
P 24
26
27
28
29
30
3.1
3.2
( i / ~ 103 ) . (K-~)
Figure 2. Plot of In K + f ( T ) against 1/T. Comparison between the values obtained experimentally (continuous line) and those predicted from thermal data (broken line).
absolute temperature in Figures 1 and 2.
Experimental Section
26
27
28
20
30
31
32
( 1 1 ~ )l .o 3
(K-~)
Figure 1. Plot of In K against 1/T. Comparison between the values obtained experimentally (continuous line) and those predicted from thermal data (broken line).
From these, the following equations for the temperature dependence of the gas-phase thermodynamic equilibrium constant and enthalpy change of reaction are obtained: In K = 73155'-' - 4.749(1n 2') + 1.169 X 4.339 X 104T2 + 2.514 X 10-'T3 + 4.58 (4) A& = -14534 - 9.436T + 2.313 X 10-2T21.724 X 10-5T3+ 1.499 X 10-sT4 cal-mol-' (5) The values of the equilibrium constant predicted by eq 4 are shown as a semilogarithmic function of reciprocal
(i) Materials. Methanol (Scharlau, Barcelona), with a minimum purity of 99% containing less than 0.1% water, MTBE (Merck, Schuchardt), with a minimum purity of 99% containing less than 0.1% each water and methanol, and isobutene (S.E.O., Barcelona), with a minimum purity of 99%-main impurities were isobutane and linear butenes-were used without further purification and fed into the reactor diluted in nitrogen (S.E.O., Barcelona) to obtain a wide variation in their partial pressures as well as to promote evaporation and transport of the methanol-MTBE mixtures. Nitrogen with a minimum purity of 99.998% containing less than 2 ppm water was used. The sulfonic ion-exchange resin Amberlyst 15 (Rohm and Hass, Co), used as the catalyst, is a macroporous copolymer of styrene-divinylbenzene containing 20% divinylbenzene with a surface area (B.E.T. method) of 43 f 1 m2.g-' and an exchange capacity determined by titration against standard base of 4.5 mequiv of HSO,.g-' of dry resin. Catalyst samples with a particle diameter of between 0.063 and 0.1 mm were obtained by crushing and sieving the manufactured resin, dried a t 110 "C for 12 h and stored in a dessicator over sulfuric acid. The catalyst was diluted in quartz to obtain a catalytic bed of sufficient length to guarantee a good contact pattern between reactants and catalyst. The catalytic bed was preheated for 2 h a t 110 "C inside the reactor with a nitrogen flow rate of about 10 cm3& before decreasing it to the working temperature.
340 Ind. Eng. Chem. Res., Vol. 27, No. 2, 1988
(ii) Apparatus. The experiments were performed in a fixed bed tubular microreactor (downflow) preceded by a feed supply for each of the gases and the liquid and followed by a sampling and analysis system. From a Mariotte-type buret equipped with a mercury presostate, methanol-MTBE mixtures passed through a rotameter and a flow control valve to the vaporizer (precision of the device, f1.5%). Nitrogen and isobutene were fed independently into the vaporizer from pressure bottles through a flow control valve and an orifice meter (precision of the device, f0.270 1. In the electrically heated vaporizer, packed with metallic Raschig rings, the mixture of nitrogen, isobutene, and vaporized methanol and MTBE was homogenized and from there passed to the reactor. The reaction chamber was a Pyrex tube (length, 45 cm; internal diameter, 1.2 cm) with a porous plate to hold the catalyst bed. This chamber was housed concentrically in another Pyrex tube. Water containing 30-50% propylene glycol as a thermostatic fluid was pumped from a thermostatic bath to the reactor jacket to keep the catalyst bed isothermal. The inside-outside temperature gradient of the reaction chamber was measured by means of two platinum resistance thermometers (accuracy, f0.1%), the first one inserted axially in the center of the catalyst bed and the second one located in the reactor jacket close to the catalyst bed. In the temperature range explored, the temperature gradients were not higher than 0.3 "C. A sampling valve a t the reactor outlet allowed for the analysis of the gas leaving the reactor by means of an HP-5830A gas chromatograph equipped with a heat conductivity sensor. (iii) Analysis. A stainless steel column (length, 3 m; internal diameter, 1/8 in.) packed with Chromosorb 101 (80/100 mesh) was used with helium (S.E.O., Barcelona) with a minimum purity of 99.998% as carrier gas flowing a t a rate of 30 mL-min-'. Column temperature was held at 110 O C for 4 min, increased at a rate of 15 "C-min-l up to 220 "C, and held for 2 min. The elution volume was 0.5 cm3. Corrections for the methanol content of MTBE were included in the calculations. (iv) Procedure. The methodology followed makes use of the fact that the reaction of synthesis of MTBE is fairly exothermic. It consists of feeding into the reactor at a given temperature a mixture of methanol, isobutene, MTBE, and nitrogen whose composition corresponds to that of equilibrium and checking that the composition of the mixture does not become altered in the reactor. If the composition of the feed mixture is not in chemical equilibrium, i t will vary in the reactor whenever the contact time is sufficiently long. Given that this reaction is sufficiently exothermic, these changes in composition produce a measurable alteration in temperature (for instance, kinetic experiments show that conversions of approximately 1-3% give rise to an increase of 0.5-2 "C in temperature for the MTBE synthesis and a decrease of the same order for the decomposition of the ether, using an equivalent space velocity and a similar catalyst bed (Tejero, 1986)). Therefore, the fact that the temperature of the catalyst bed remains constant when a mixture passes through the reactor can be used as a rough indicator that the composition of the mixture corresponds practically to that of equilibrium. This should be verified by chromatographic analysis. Briefly, the experimental procedure consists of the following. 1. A catalyst bed (0.2-0.6 g of resin diluted in 2-3 volume of quartz; depth, 3-4 cm) was prepared in the above-mentioned manner. In addition, another quartz bed (depth, 5 cm) was placed over the first catalyst bed to
assure that the gases would arrive at the catalyst bed at exactly the temperature of the experiment. Once the catalyst bed had been dried at 110 "C, it was cooled to the working temperature under a stream of nitrogen with a flow rate equivalent to that of the mixture that would be fed into the reactor in the next step. 2. In each run, isobutene, nitrogen, and a liquid mixture of methanol and MTBE were fed to the reactor at such flow rates that the composition of the mixture arriving in the reaction chamber corresponded roughly to that of equilibrium composition. In order to do this, prior knowledge of the reaction equilibrium constant at the temperature explored was required (see eq 4). Then, the flow rates of isobutene, of the methanol-MTBE mixture, and of nitrogen were slightly modified in order to get the catalytic bed to reach the initial temperature again. 3. To ensure that the mixture was in chemical equilibrium, we proceeded to slightly vary the flow rate of the gases or the liquid feed. Thus, a decrease in the flow rate of isobutene gave rise to a decrease of reactor temperature; an increase in the flow rate of isobutene gave rise to an increase of reactor temperature, etc. Once we confirmed that the final mixture of step 2 was roughly in chemical equilibrium, we proceeded to restore the flow rate and to check that the composition of the gas leaving the reactor agreed within the limits of the experimental error with that deduced from the flow rate and composition of the feed. 4. The equilibrium constant that corresponds to the run temperature is computed by means of eq 1from the composition and total pressure of the system.
Results and Discussion The experiments were performed at atmospheric pressure in the temperature range 40-110 "C. A t 40 "C, however, the method does present some problems due to the low reaction rate of MTBE synthesis. As a result, the sensitivity of the method to the changes in composition is substantially decreased. Table I1 shows the experimental results of this research: equilibrium composition and equilibrium constant K , obtained for each temperature explored. We checked that the space velocity does not affect the K , values obtained. To do this, some experiments were performed at different space velocities but while maintaining a constant composition of the feed. Experiments E-18, E-19, and E-20 at 60 "C and experiments E-10 and E-11 at 80 "C show that the values obtained for the equilibrium constant present no trends, and the dispersion between them is within the limits of the experimental error. In addition we checked that the K , values were not affected by the composition of the feed in the reactor. To do this, runs in which mixtures of different composition were fed into the experimental device were performed at approximately the same space velocity. Experiments E-7, E-8, and E-9 at 80 "C, E-11, E-12, and E-13 also at 80 "C, and E-23 and E-24 at 50 "C show that the dispersion of the values of the equilibrium constant is once again within the limits of the experimental error. However, as can be observed in Figures 1 and 2, the values for K, obtained experimentally are 20% greater in the range of temperatures explored than the values for K deduced theoretically. This discrepancy could be attributed to the following causes (once we have considered the analytical error): (1)the nonideality of the mixtures used and/or the presence of side reactions (systematic errors or errors of method); (2) the lack of accuracy of the thermal data used to evaluate the thermodynamic equilibrium constant.
Ind. Eng. Chem. Res., Vol. 27, No. 2, 1988 341 Table 11. Equilibrium Experiments: Experimental Conditions and Results run P, atm T,, K V, mol.h-'.g-' YM YT 0.124 383.2 14.43 0.161 E-1 1.138 0.100 E-2 383.3 3.41 0.223 1.080 0.143 14.22 0.130 E-3 1.135 373.2 0.100 0.209 E-4 1.086 373.5 6.52 0.134 0.107 E-5 1.136 363.4 13.85 0.073 0.277 1.087 363.2 7.07 E-6 0.082 14.71 0.171 E-7 1.138 353.2 0.167 0.068 1.127 353.3 13.55 E-8 0.261 0.029 1.131 353.3 14.90 E-9 0.042 0.323 E-10 353.2 7.47 1.088 0.042 E-11 353.2 3.73 0.322 1.087 0.139 353.2 3.51 0.147 E-12 1.080 0.233 0.106 E-13 1.077 353.2 3.83 7.52 0.219 0.070 E-14 1.154 353.2 0.092 0.126 E-15 14.56 1.136 343.6 E-16 3.61 343.3 0.049 0.254 1.085 0.267 0.035 E-17 8.08 1.158 343.2 7.71 0.135 0.059 E-18 1.148 333.3 0.135 0.060 3.89 E-19 1.173 333.3 0.135 1.91 E-20 1.166 333.3 0.060 7.67 328.4 0.254 0.013 E-21 1.157 6.92 0.085 0.046 E-22 1.125 323.4 323.2 0.130 0.015 6.78 E-23 1.146 7.55 323.1 0.257 0.011 1.158 E-24 6.33 0.130 0.013 1.157 E-25 318.3 0.059 0.026 1.141 6.64 E-26 313.2 0.124 1.139 6.94 E-27 313.2 0.014 Table 111. Fit of a Linear Model to In K and 1/T Data: Analysis of Variance In K + f ( T ) In K , = bo + eauation bi/T = b,' + bl'/T -2.094 X 10' 4.653 intercept 7.837 X lo3 7.336 X lo3 slope 9.995 x lo-' 9.995 x lo-' regression coeff 1.578 x 10-3 1.566 X residuals variance 1.355 X 1.355 X error variance 2.190 2.190 tabulated F(25, 17) for (Y = 0.05 1.16 1.16 exptl F 6.22 X lo2 1.18 x 104 intercept variance 4.57 x 10' 4.00 X 10' slope variance 8.75 X lo6 4.59 x 105 exptl F for intercept 2.95 x 104 3.37 x 104 exptl F for slope 4.451 4.451 tabulated F(1,17) for (Y = 0.05 b< f 0.256 confidence interval for intercept bo f 0.256 bl f 87.9 b i 87.9 confidence interval for slope
*
With respect to the nonideality of the mixtures, the fugacity coefficients of the components in each of the experiments performed were computed by using the virial equation as a function of the pressure (Reid et al., 1977; Perry and Chilton, 1973). The computed fugacity coefficients, like K,, are very close to unity (Tejero, 1986). Therefore, the reaction system behaves ideally, in agreement with the rule established by Hougen et d. (1962) that the fugacity coefficients are approximately unity in the neighborhood of atmospheric pressure. Consequently, K z K for the reaction system. T i e presence of side reactions causing the observed discrepancy in the values of K in the esterification reaction of acetic acid with ethanol in vapor phase (Hawes and Kabel, 1968) does not seem plausible in this case, as no byproducts in significant amounts were observed. Moreover, preliminary kinetic experiments show that the only byproduct detected in significant amounts was diisobutene a t 90-110 "C using a molar isobutene/methanol ratio higher than 2 (Tejero, 1986). Therefore, the isobutene/ methanol ratio employed a t 90-110 OC was lower than 1.3, as can be seen in Table 11. Finally, we must taken into account the lack of accuracy of the thermal data used in estimating K , reported by Kabei and Johanson (1961) as a possible cause of the
Y!? 0.014 0.015 0.023 0.023 0.031 0.041 0.053 0.044 0.030 0.050 0.051 0.080 0.098 0.063 0.080 0.083 0.072 0.124 0.127 0.125 0.074 0.122 0.058 0.090 0.069 0.109 0.117
YN
0.702 0.662 0.704 0.667 0.729 0.609 0.694 0.721 0.680 0.586 0.585 0.634 0.563 0.650 0.702 0.614 0.625 0.681 0.678 0.681 0.659 0.747 0.798 0.643 0.789 0.806 0.744
Kv 0.699 0.660 1.216 1.116 2.16 2.01 3.77 3.90 4.01 3.66 3.73 3.94 3.98 4.11 6.85 6.65 7.02 15.5 15.6 15.5 22.6 31.2 29.0 32.0 42.6 71.5 67.9
K,, atm-' 0.615 0.611 1.071 1.027 1.90 1.85 3.32 3.46 3.54 3.37 3.43 3.65 3.70 3.56 6.03 6.13 6.06 13.5 13.3 13.3 19.5 27.7 25.3 27.6 36.8 62.7 59.6
Table IV. Standard Free-Energy,Enthalpy, and Entropy Changes of the Reaction for Obtaining MTBE at 298 K in Vapor Phase A&'(298 K), &(298 K), A&(298 K), cal-mol-' calamol-' cal-mol-'-K" determined whenwe -3160 f 150 -15570 f 175 -41.61 f 0.5 assume that AHo is const determined when-we -3170 f 150 -15670 & 175 -41.55 f 0.5 assume that AHo varies with temp estd from -3080 -15 630 -42.09 bibliographical data
discrepancies observed in the reaction of dehydration of ethanol tojiethyl ether. As can be seen in Table IV, the value of AH"(298 K) deduced from the bibliographical data agrees fairly weil with that obtained experimentally, while the value of AG"(298 K) is slightly higher, by 3%, than that deduced from bibiliographical data. As the relationship between K and AGO is exponential, an error of only 3% in AG"(298 K) causes a 20% error in the integration constant IK of eq 2, which affects the values of K estimated in the temperature range explored. This magnification of the error is characteristic of calculations of equilibrium constants from thermal data (Park and Himmelblau, 1980; Hawes and Kabel, 1968) and is due to the standard enthalpy and entropy (or free energy) changes of reaction that are small in relation to many of the thermal data employed to obtain them and to the fact that the equilibrium constant is equated to these changes exponentially. Starting with the measurements of the equilibrium constant a t various temperatures, we can determine the values of AH" and AS" a t 298 K. The following two methods are widely used (Glastone, 1947). 1. Assuming that AH" is practically constant in a wide range of temperatures, In K is plotted against 1/T. Given that In K = -A@"/RT = A*/R - &/RT (6) A& is deduced from the slope and AS" from the intercept of the resultant straight line. As it is supposed that AHo
342 Ind. Eng. Chem. Res., Vol. 27, No. 2, 1988
and A@ do not vary with temperature, &^and AsoAat 298 K coincide with the values obtained for AH" and AS", respectively. 2. On the contrary, if hEi0 varies appreciably with temperature, we can rearrange eq 2 in the following manner: In K - (a/R) In T - (b/2R)T (c/6R)T2 - (d/12R)T3 = ZK - ZH/RT Plotting the first member of this equation against 1/T, ZH is obtained from the slope and ZK from the intercept of tbe resultant straight line. From the values of ZH and ZK,AH" and AS" at 298 K can be calculated applying eq 2, 3, and 6. Thus, the values of K obtained experimentally were plotted semilogarithmically against the reciprocal absolute temperature by means of the twogrocedures described, in the first place assuming that AH" was practically constant in the temperature range explored (Figure 1) and, then, in a more rigorous manner admitting that AH" varies with temperature (Figure 2). In both cases, the data were fitted by ordinary least squares, and the results obtained are presented in Table 111. The analysis of variance was done by taking the dispersion of the replicated values of K obtained a t various temperatures as a measure of the experimental error. Table I11 shows that the two straight lines to which the data have been equated as a function of reciprocal absolute temperature are significant. The expressions obtained for the temperature dependence of K are, respectively, In K = 7840T-I - 20.94 In K = 7340T-' - 4.749(1n T ) + 1.169 X 10-2T4.339 X 10+T2 + 2.514 X 10-'T3 4.65 In the second case, the following expressions were obtained for AH", AGO, and AS" of the reaction as a function of the temperature & = -14577 - 9.436T + 2.313 X 10-2T21.724 X 10-5T3+ 1.499 X 10-8T4 calamol-1 (7)
+
Ad"
+ 9.436T In T 10-2T2+ 8.621 X lOW6T3 4.995 X 10-'T4 cal.mol-' (8)
= -14577 - 9.246T
2.313
X
+
A& = 0.19 - 9.436(1n T ) 4.625 X 10-2T2.587 X 10-5T2+ 1.998 X 10-8T3 cal.molbl.K-' (9)
Table IV shows the values of At?", &, and A $ O at 298 K obtained through eq 6 when we suppose-AHO to be constant, and eq 7-9 when we suppose AHo to be a function of temperature and those computed from bibliographical data. We can see that the data obtained for standard free-energy, enthalpy, and entropy changes of reaction at 298 K using the two procedures mentioned before agree fairly well, within the limits of the experimental error, with those computed from therm4 data found in the literature. Moreover, the values of AHo and AS" at 298 K obtained experimentally by both methods practically coincide with those estimated from bibliographical data. In contrast to this, the values obtained for A@"(298K) by both of these methods practically coincide. However, they are somewhat higher than the value estimated from the bibliographical data (by approximately 3%). The effect of this small difference on computing the thermodynamic equilibrium constant for the reaction has been previously analyzed. Finally, starting with @,(298 K) and @",(298 K), we estimated the values for AHo1(298K)and ASo1(298K) and
then compared them with the values found in the literature for both thermodynamic quantities. At any given temperature, it is shown that
Therefore, from the standard heats of vaporization of methanol, isobutene, and MTBE (see Table I)-and from AHog,at 298 K, the thermodynamic quantity AHo1(298K) = -8930 f 175 cal-mol-' is obtained. This value agrees with the one found experimentallyby Gicquel and Torck (1983) of -9510 f 480 cal.mol-', using a kinetic procedure, as well as the values reported by Obenaus and Droste (1980) of -8840 cal-mol-I and Gupta and Prakash (1980) of -8800 cal-mol-'. In addition, at any given temperature, it is shown that
AS", = Agog - (1/T)?vi(&J, i
So, from the standard heats of vaporization of products and reactants and of AS",, at 298 K, ASo1(298K) = -19.0 f 0.5 cal.mol-'-K-' is obtained. 4 s no values in the bibliography have been found for ASo1(298K), we have estimated this thermodynamic quantity from the standard absolute entropy of isobutene, methanol, and MTBE (see Table I), obtaining a value of ASo1(298K) = -20.5 cal. mol-l-K-l, which agrees satisfactorily with the value found from experimental data.
Conclusions The dispersion of the values obtained from the equilibrium constant in the series of experiments performed varying the composition of the feed or the space velocity is of a magnitude of 5%. The experimental method proposed is satisfactory for determinations in a continuous manner for the equilibrium constant of reaction systems in the vapor phase. The use of temperature changes in the reactor as an auxiliary indicator is applicable only when the reaction studied is sufficiently exothermic or endothermic. The goodness of the thermodynamic constants determined for the reaction for obtaining MTBE is clear, as proved by the agreement of the values of standard freeenergy, enthalpy, and entropy changes of reaction in the vapor phase and of standard enthalpy and entropy changes of reaction in the liquid phase, at 298 K, obtained from the temperature dependence of K , with those computed from bibliographical data. Finally, the supposition that standard enthalpy change of reaction does not vary with temperature for the reaction of vapor-phase synthesis of MTBE is acceptable in the temperature range explored. Nomenclature a , b, c, $ = changes of molar heat capacity coefficients with chemical reaction a,, bi, ci,di = coefficients in equation for molar heat capacity of component i, C = ai + biT + ciT2+ b i T 3 bo, bl = intercept an8 slope in equalion for temperature dependence of K assuming that AH" is constant with temperature bo', bl' = intercept and slope in equation for temperature dependence of K assuming that AH" is a function of temperature 6 ,i = molar heat capacity of component i, cal.mol-'.K-' F"= variance ratio distribution fi = fugacity of component i Go = standard free energy, calsmol-I Ho = standard enthalpy, cal.mol-'
Ind. Eng. Chem. Res. 1988,27, 343-350
ZH = integration constant in Kirchoff's equation, cabmol-l ZK = integration constant in van't Hoff s equation, adimen-
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sional K = thermodynamic equilibrium constant K, = ratio of fugacity coefficients K = equilibrium constant based on partial pressures K" = equilibrium constant based on molar fractions d T B E = methyl tert-butyl ether P = total pressure, atm pi = partial pressure of component i, atm e = gas constant, cal-mol-l.K-l So = standard entropy, cal.mol-'.K-' T = absolute temperature, K T , = reactor temperature, OC V = space velocity (total flow rate/catalyst weight), mo1.h-l.g-l yi = molar fraction of component i Greek Symbols a ,= significance level AGO = standard free-energy change of reaction, cal-mol-'
AGOf,+= standard free energy of formation of component i,
calqmol-' = standard enthalpy change of reaction, calsmol-' qfi = standard heat of formation of component i, calamol-' A H 0 , i = standard heat of vaporization of component i, cal-mol-l ASo = standard entropy change of reaction, cal.mol-'-K-' vi = stoichiometric coefficient of component i pi = fugacity coefficient of component i A@O
Subscripts E = MTBE e = equilibrium f = formation g = vapor phase I = isobutene i = component 1 = liquid phase M = methanol N = nitrogen v = vaporization Registry No. MTBE, 1634-04-4; CH,OH, 67-56-1; isobutene, 115-11-7; Amberlyst 15, 9037-24-5.
Literature Cited Ancillotti, F.; Massi Mauri, M.; Pescarollo, E.; Romagnoni, L. J.Mol. Catal. 1978, 4(1), 37.
Received for review June 24, 1986 Accepted July 15, 1987
The Use of Biased Least-Squares Estimators for Parameters in Discrete-Time Pulse-Response Models N. L. Ricker Department of Chemical Engineering, BF-IO, University of Washington, Seattle, Washington 98195
T h e use of biased least-squares methods for the estimation of pulse-response coefficients in discrete-time process models is proposed. The two specific approaches considered here are partial least squares (PLS) and a method based on the singular value decomposition (SVD). Example applications include the estimation of the open-loop impulse and step responses of a pilot-scale anaerobic wastewater treatment process. The SVD and PLS methods give very similar results for the problems studied. T h e PLS method typically requires much less computer time, however. In order to use modern analytical techniques and algorithms for process control, one must provide a model of the dynamics of the subject process. Such models are usually linear and time-invariant and may be in either a state-space, transfer function, or "nonparametric" form. The third category includes pulse-response, step-response, and frequency-response formulations. Models can be ei0888-5885/88/2627-0343$01.50/0
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ther continuous or discrete-time; the latter are convenient for computer control applications. The various discretetime models are reviewed in many texts (e.g., Astrom and Wittenmark (1984)). Ideally, one would like to develop process models from first principles. Most unit operations in chemical processes are too complex for this, however. In the case of an existing
0 1988 American Chemical Society
