Byers, R . L., Calvert, S., Ind. Eng. Chem., Fundam., 8, 646 (1969). Deryagin. B. V.. Yalarnov, Yu., J. ColioidSci., 20, 555 (1965). Z. Phys., 5 4 , 537 (1929). Epstein, P. S.. Friedlander, S. K.. Johnstone, H. F., Ind. Eng. Chem., 49, 1151 (1957). Fuchs, N. A,, "The Mechanics of Aerosols," pp 204-212, Pergamon Press, Elmsford, N. Y., 1964. Jacobsen, S., Brock. J. R . , J . CoIloidSci., 20, 544 (1965). Keng, E. Y. H., Orr, C. Jr., J. Colloid lnterface Sci., 22, 107 (1966). Murata, M.. Kitani. S..J. Nucl. Sci. Techno/., 9, 622 (1972).
Postma, A. K., HW-70791, (Hanford), (1961). Rowland, P. R . , Corlyle, M., Sonbelet, G . C., "Diffusion des Produits de Fission," No. 16, 191. Saclay, 1969. Schmitt. K. H., Z. Naturtorsch., 14a, 870 (1959). Singh, E., Eyers. R . L., lnd. Eng. Chem., Fundam., 11, 127 (1972). Thomas, J. M., J . Air Pollut. Contr. Ass., 8, 32 (1958).
Received for reuieu November 5, 1973 Accepted May 10, 1974
Modeling the Water-Gas Shift Reaction Walter F. Podolski' and Young G. Kim* Department of Chemical Engineering, Northwestern University, Evanston, lllinois 6020I
The kinetics of the water-gas shift reaction over an iron oxide-based catalyst has been studied. The experiments were performed in a glass recycle reactor at approximately atmospheric pressure. The temperature dependence of the reaction and the activation energy were determined in conjunction with a study of the diffusional limitations of different size catalyst. The statistical procedure proposed by Box for experimental design and analysis of data in order to discriminate among the rival models was used in the experimental program. A number of representative models were examined, and it was found that only the Langmuir-Hinshelwood and the power-law models could adequately describe the reaction behavior over the temperature and concentration ranges investigated. In addition, some existing data from the literature were reexamined in the same manner as used here.
Introduction Numerous kinetic studies (Barkley, et al., 1952; Bohlbro, 1961, 1962, 1963, 1964, 1966; Giona et al., 1961; Hulburt and Vasan, 1961; Kodama, et al., 1953; Kul'kova and Temkin, 1949; Paratella, 1965; Shchibrya, et al., 1965) on the water-gas shift reaction have been conducted using iron oxide catalysts, and there is general agreement among most of the results reported which can be summarized as follows: the rate of reaction is approximately proportional to the carbon monoxide concentration; the rate is usually retarded by increasing carbon dioxide concentration and almost independent of hydrogen concentration and independent of steam concentration when it is in excess of the stoichiometric amount. Temkin was the first to publish a systematic attempt to combine the reaction mechanism with a rate equation. The mechanism assumes an alternate oxidation and reduction of the catalyst surface; carbon monoxide reacts with the catalyst surface creating oxygen ion vacancies which are immediately filled by oxygen from the water molecules. Both steps proceed a t the same rate on the catalyst surface which is assumed to be nonuniform. The oxidation-reduction models in Table I are the result of this treatment. Later, Boreskov, et al. (19701, measured the rates for the oxidation and reduction steps of the reaction over an iron oxide catalyst to determine if the rates for the individual stages and the overall rate of the reaction are in agreement. They concluded that within the experimental error the rates of the individual steps agreed with the overall rate, thus lending support to Temkin's model. The same type of experiments on copper chromite and chromium oxide catalyst (Yur'eva, et al., 1969) yielded different
' Argonne National Laboratory, Argonne, 111.
60439
results. Over copper chromite, they concluded that the reaction proceeded through the formation of an active complex including both a CO and an HzO molecule according to the Langmuir-Hinshelwood scheme. Over chromium oxide, carbon monoxide was not adsorbed in noticeable amounts, and therefore they concluded that the shift reaction proceeded by reaction of an adsorbed water molecule with gaseous carbon monoxide. Kaneko and Oki (1965a,b) measured the stoichiometric number of the rate-determining step of the reaction using deuterium and 14C. These experiments gave values of the stoichiometric number close to unity. They concluded that the rate determining step, if it existed, must involve both carbon monoxide and water molecules. These results supported the Langmuir-Hinshelwood and Eley-Rideal models but not Temkin's model. Reexamination of this data and subsequent experimentation with 1 8 0 (Oki, et al., 1972; Oki and Mezaki, 1973a,b) has led to a value of 2 for the stoichiometric number, which is compatible with a mechanism with two rate- determining steps, namely, the adsorption of CO and the associative desorption of hydrogen. In the initial stages of the reaction only the adsorption of CO is rate determining, while near equilibrium both steps are rate determining. These latter data were published after this work was completed and were not considered. The models considered, shown in Table I, fall into one of the following general classes: (1) Langmuir-Hinshelwood model, (2) Eley-Rideal model, (3) oxidation-reduction model, (4) Hulburt-Vasan model, (5) Kodama model, or (6) empirical (power law) model. Experimental Strategy In this study, we have made use of some of the techniques developed by Box and his coworkers (Box and Hill, 1967; Box and Henson, 1969) for discrimination among Ind. Eng. Chem.,
Process Des. Develop., Vol.
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415
rival models. The procedure consists of three parts: analysis of experimental data, experimental design, and collection of experimental data. (a) Analysis of Experimental Data. A t any stage, all the experimental data obtained up to that point are analyzed to estimate the parameters in each model by means of nonlinear estimation. After n preliminary runs, Box and Henson proposed that the posterior probability associated with each model be calculated according to
probabilities by simply replacing the first two quantities in eq 1 by their counterparts in eq 2 because, strictly speaking, this would change the prior densities for the e', p ( 0 4 M t ) ,a t different stages of experimentation. (b) Experimental Design. If none of the models has attained the desired posterior probability, the next experiment will be performed a t a point in the experimental variable space which maximizes Box and Hili's discrimination function D, given by m
where P(MLlyn,u)is the posterior probability of model i given the data set y n and exact knowledge of u2, P ( M , ) is the prior probability of model i in the absence of any data, and S,,nis the residual sum of squares for model i containing p r parameters. A key step in the development of eq 1 is the selection of prior densities for the et, p ( e r J M , )where , e r is the vector of parameters in model i. As is often the case, very little a priori informatiog is known about the parameters in each model. Box and Henson presented an argument for dealing with this case, but they subsequently concluded that their reasoning was faulty. Box and Kanemasu (1972) developed an alternative argument which arrived a t the same result, however. After the posterior probabilities have been calculated as shown above, they can be recomputed sequentially as new observations become available. Box and Kanemasu suggest that with the addition of 1 more experiments the probabilities be updated by the following
where Xr,,, is a matrix whose elements are the partial derivatives of model i with respect to the parameters e l , evaluated a t a set of experimental conditions tu and a set of parameter estimates, 0,.
They do not recommend that eq 1 be used to update 416
Ind. Eng. Chem., Process Des. Develop., VOI. 13, NO. 4, 1974
m
D is a function of the experimental conditions, m is the number of models under consideration, P(M,ly,) is the probability associated with model i prior to the planned experiment, is the predicted value of the response y given by model i for the ( n + 1)th experimental conditions, using the estimates for the parameters contained in model i based on prior information. Box and Hill demonstrate that u2 u,2 is simply Var(yn+llMl,yn,u). Equations 1-4 have all been developed on the assumption that us is known. In the case where u2 is unknown, analogous equations can be found in the previous references. If o2 is unknown, the mathematical computations are somewhat more involved. (c) Collection of Experimental Data. Preliminary experiments can be performed according to some convenient scheme such as a factorial design. There should be at least one more experiment than the largest number of parameters in any one model. Subsequent experiments can be made according to eq 4. It is not necessary to recompute the probabilities after each experiment, but rather a group of 1 experiments could be performed before the probabilities are updated according to eq 2 . This can frequently be the case when experiments are more economically performed in blocks. There has been much confusion in the literature concerning application of these procedures. Froment and Mezaki (1970) simulated the sequential procedure by using
+
the data of Hosten (1967) for the isomerization of n-pentane over Pt-Al203 in the presence of hydrogen. They wished to discriminate between two models and chose 3 of the 13 observations to be “preliminary” experiments, and then computed the posterior probabilities using eq 1. The next experimental condition was determined from eq 3 and chosen closest to this condition from the available data. Posterior probabilities were then computed by eq 2 . From the results the posterior probabilities of the models appear to depend on the particular set of “preliminary” observations chosen. Box and Kanemasu reexamined Froment and Mezaki’s results and showed that when all 13 observations were fitted to the models the residual variances were a t least 20 times as high as the estimated variance used by Mezaki and Froment. Box and Kanemasu concluded that the instabilities in the posterior probabilities arose either because of a gross underestimation of the experimental variance a* or because neither of the models was adequate. Computation of posterior probabilities assuming an unknown variance did not show any instability. Experimental Equipment and Procedure
A recycle reactor was chosen for the experimental program because it is more nearly able to approach gradientless operation than single-pass type reactor systems and was somewhat easier to construct than a stirred tank reactor. The inlet section of the apparatus, shown in Figure 1, consists of gas metering equipment and a water pump. The gases were mixed in a packed column before being mixed in turn with steam, and this mixture was then fed to the reactor section. The four gases used in these experiments were carbon monoxide, CP grade, approximately 99.3% pure; carbon dioxide, bone dry grade, 99.8% pure; helium, high-purity grade, 99.995% pure; and hydrogen, prepurified grade, 99.99% pure. All gases except carbon monoxide were used as received, since water and nitrogen were the major impurities. Carbon monoxide was purified by passing it over hot copper wire (200-300°C) to decompose iron carbonyl and remove traces of oxygen, through a bed of silica gel to remove water, through a bed of activated coconut charcoal to remove sulfur-bearing compounds, and through a bed of Ascarite to remove carbon dioxide, Each gas cylinder was fitted with a two-stage regulator to control the discharge pressure. The gas lines were protected from particle impurities by the fine pore filters, and a capillary tube flow meter was used to monitor the flow of each gas. The flow rate of each gas was controlled by means of a micro-flow valve. An additional shut-off valve was installed after each flow control valve to provide a positive shut-off of each gas. The gases were manifolded together and then passed through a packed-bed mixer. With this system, reasonably steady flow in the desired range could be maintained. Liquid water was fed to a vaporizer by a syringe pump which was constructed as follows. A constant speed motor was coupled, through several reducing gears, to a rotating nut which was held in a fixed vertical position. A threaded rod was attached to the plunger of a syringe to deliver a fluid a t fixed flow rate, the precision of which was dependent on the precision of the motor rpm. Several synchronous motors, l/*, 1, 2 rpm, were used to provide flexibility in the delivery rate. In addition, several syringes of different inner diameter were made from 2-in. 0.d. round Lucite stock. The use of O-rings on the plunger resulted in a leak-tight syringe. The outlet from the pump was sent through 1h-h. poly-
5
I
6
7
5
I
L i c +act>Figure 1. Gas metering system: 1, t a n k regulator; 2, fine pore filter; 3, manometer; 4, capillary tube; 5 , furnace with copper turnings; 6, silica gel column; 7 , ascarite column; 8, coconut charcoal column; 9, micro-flow valve; 10, shut-off valve; 11, packed column mixer.
ethylene tubing to the vaporizer which was made from a small cold finger. To the bottom of the cold finger, a short length of capillary tubing of 1-mm i.d. was attached. The vaporizer was heated to around 200°C by a heating tape which was controlled by a variable power transformer. It was necessary to vaporize the water in the capillary tube to prevent bumping and to provide a relatively smooth flow of steam into the reactor. The other reacting gases entered the vaporizer through a different inlet to mix with the steam before entering the reactor. A micro-flow valve was placed at this inlet to prevent any backward flow of steam due to the action of the recirculation pump. Part of the outlet from the vaporizer was packed with glass beads to further mix the steam and other gases. The recycle reactor is shown in F i g q e 2. The recycle pump, made by Metal Bellows Corp., is a stainless steel bellows pump in which the bellows are welded to the piston and to the pump head to provide leak-proof performance. Parts of the recycle loop were made from stainless steel tubing to provide support and rigidity while the reactor section was made from Pyrex glass. Glass to metal seals were made using Swagelok or similar fittings in which the front ferrule was made of Teflon. The opening a t the top of the reactor facilitated the loading of catalyst into the reactor. This opening was sealed using a 3/8-in. Swagelok plug in which the front ferrule was replaced by a specially made Teflon ferrule. Indentations were made in the reactor to hold the glass wool plug on which the catalyst was placed, and a thermowell was provided at the exit of the catalyst bed. The reactor section was heated by an electric furnace which was controlled by an automatic temperature controller. The control thermocouple was mounted on the outside of the reactor section. The temperature in sections of the recycle loop outside the furnace was maintained just above the condensation level by heating tapes which were controlled by variable power transformers. This was necessary to preclude the catalyzing of the reaction by the steel portions of the system and to prevent any possible damage to the pump from an excessive temperature. The pressure in the reactor was measured with a stainless steel Bourdon tube gauge made by Heise Bourdon Tube Co. The range of the gauge was 0-50 psia, and pressures could be determined to within 0.01 psia. A trap was placed between the gauge and the reactor to prevent any Ind. Eng. Chern.. Process Des. Develop., Vol. 13,No. 4 , 1974
417
Diffusion Limitations
rror
IIFOIITII
Figure 2. Recycle reactor: 1, recirculation pump; 2, mixing section (packed with glass beads); 3, pressure gauge; 4, pressure control valve; 5, reactor section; 6, furnace; 7, temperature controller.
condensate from reaching the gauge. The reactor pressure was controlled by means of the same type of microflow valve used elsewhere. This valve enabled the reactor pressure to be set quite accurately. After leaving the recirculation loop, the gas mixture passed through a condenser cooled by laboratory water and then through a cold finger partially immersed in Dry Ice to remove most of the remaining water. The dry gas mixture was sampled for analysis, passed through a soap bubble flow meter, and vented outdoors. The outlet gas was analyzed for carbon monoxide and carbon dioxide by means of a gas chromatograph equipped with a thermal conductivity detector and a Porapak QS column. Several preliminary experiments were conducted which included a study of the mixing characteristics of the reactor to ensure stirred tank behavior. In addition blank runs were made to determine the extent of reaction without catalyst. As a result of these latter experiments, a glass reactor was used because the original stainless steel reactor section was found to catalyze the reaction to a significant extent a t normal operating temperatures (400°C). Catalyst Preparation a n d Characterization The catalyst containing 93% iron oxide and 7% chromium oxide was prepared by coprecipitation of the iron and chromium compounds according to fairly standard procedures. Hydrated iron(II1) and chromium(II1) oxides were precipitated with a slight excess of 3 M NH40H. The precipitate was filtered through a Buchner funnel, washed thoroughly with distilled water, and filtered again. The filter cake was broken up and dried overnight a t 130°C and then overnight again a t 400°C. The dried chunks of filter cake were then crushed and screened into various size fractions for later use. The BET surface area of the catalyst, determined by the low-temperature adsorption of nitrogen, was about 18 m2/g. The adsorption/desorption isotherms showed a nearly vertical hysteresis loop preceded by a relatively flat section. This type of isotherm has been identified with the presence of both transitional pores and micropores. The pore size distribution was calculated from the desorption isotherm by means of the Kelvin equation which indicated a mean pore diameter of approximately 65 A. The catalyst that was used varied in diameter from 0.13 to 1 mm. The amount of catalyst used varied from about 0.3 to 1 gm, which resulted in a bed depth of 318 to 3/4 in. Since small amounts of catalyst were used, no separate reduction of the catalyst was found to be necessary. The catalyst was reduced "on stream" with a CO-H20 mixture a t around 400°C. 418
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From the pore size calculations, it seemed that Knudsen diffusion was the major diffusional resistance. Accordingly, calculations following Satterfield (1970) were made to determine the effective diffusivity. From the adsorption data, Deff = 0.0059 cm2/sec while from the desorption data, Deff = 0.031 cm2/sec. These values were then used to calculate the Weisz-Prater criterion (Weisz and Prater, 1954) to assess the effect of pore diffusion on the rate of reaction. The calculations were unsatisfactory, however, because of the uncertainty in the value of the effective diffusivity which was due to the fact that the sorption isotherms were not of the type to yield accurate BET surface areas. A more satisfactory and safer method to assess the effect of pore diffusion is to use the same amount of different-size catalyst under the same conditions. When the rate per unit mass of catalyst no longer changes as the particle size is reduced, pore diffusion limitation has been eliminated. A series of experiments with different-size catalyst was performed with the result that particle sizes less than 0.5 mm were necessary to eliminate significant pore diffusion limitation of the reaction rate a t an operating temperature of 400°C. From these rate data, it was possible to calculate a value of effective diffusivity. For these calculations a triangle method was applied to a plot of effectiveness factor us. Thiele modulus. In this method, the rates of reaction per unit mass of catalyst, r1 and r2, are measured for two different particle sizes, R1 and R2, under otherwise identical conditions. The ratio of the rates is a measure of the ratio of the effectiveness factors: r1/r2 = 71/72. The known ratio of the particle sizes is the ratio of the Thiele moduli: Rl/R2 = hl/hz. On a double logarithmic plot of 7 us. h a certain ratio q1/q2, obtained from experiment, represents a line of fixed length on the ordinate while the known ratio hl/h2 represents a certain length on the abscissa. These two lengths form two sides of a right triangle which can be fitted to the plot. This procedure was followed and Deff was found to be smaller than that calculated on the basis of the BET measurement, thus reinforcing the idea that caution must be exercised when calculating values of effective diffusivitY. The idea that diffusional limitations on the rate of reaction were very significant was again emphasized in experiments to determine the temperature dependence of the reaction. A series of experiments in the temperature range 350 to 460°C was performed on the smallest size catalyst (-0.18 mm 0.12 mm) to determine the activation energy of the reaction. A slight bend in the Arrhenius plot was observed around 400°C which showed the onset of significant pore diffusion limitation on the rate of reaction. In addition, the activation energy for the lower temperature region was found to be about 24 kcal/mol, which is in agreement with values reported in the literature.
+
Discrimination Experiments The discrimination criterion was examined a t each of three temperature levels, and then the data were reanalyzed by incorporating temperature dependence into each model. Since the data were easily and rapidly obtained, a half-replicate of a 24 factorial design was used to organize the preliminary data a t each temperature level. Thus more preliminary data than absolutely necessary were obtained, but they should have no effect on discrimination. The temperature levels a t which the data were collected were 360, 381, and 404"C, at which temperatures the equilibrium constant K had the values 18.07, 14.29, and 11.22,
respectively. The results of the three series of runs are shown in Table 11. These runs were all made with catalyst in the -0.18 mm 0.12 mm size range. Helium was used as a diluent in all the experiments and represents the balance of the partial pressure up to a total pressure slightly more than 1 atm; its partial pressure has been omitted from Table 11. The first series of experiments were run a t a temperature of 404°C. All six of the models under consideration were fitted to the data by a nonlinear least squares technique. The constants in most of the models were positive and of reasonable value. Moreover, after the eight initial experiments, the Langmuir-Hinshelwood model had a posterior probability of 1. Therefore no additional experiments were run a t this temperature. The second series of runs were made on the same catalyst as before a t a temperature of 360°C using the same procedure. After the initial eight runs, only the empirical and the Langmuir-Hinshelwood models had significant probabilities, 0.60 and 0.40, respectively. Another experiment was then planned according to the design criterion which specified very low concentrations of reactants and high concentrations of products. R-165 was as close to these conditions as could be achieved. With the inclusion of the ninth run, the posterior probabilities for the empirical and the Langmuir-Hinshelwood models changed to 0.68 and 0.32, respectively. The design criterion next specified essentially the same conditions as it had before R-165, which could not be exactly attained. Another set of experiments was then performed a t the intermediate temperature level of 381°C to observe the discrimination behavior. After the initial eight runs, again only the empirical and Langmuir-Hinshelwood models had significant probabilities, 0.97 and 0.03, respectively. At this point it seemed clear that only the empirical model and the Langmuir-Hinshelwood model could adequately fit the data. It was also found that discrimination between the models, a t least between the two that could adequately explain the data and those that could not, was very rapid; however the discrimination did not consistently favor the same model. Temperature dependence was now taken into account and all the data in Table I1 were reanalyzed simultaneously and treated as preliminary runs, to avoid Froment and Mezaki’s difficulty, again resulting in more than the minimum number of preliminary runs necessary to inititate the design sequence. For this portion of the analysis, each parameter in all of the models, except the empirical model, was restated as follows
Table 11. Kinetic Runs
+
(5) This equation was then reparameterized as follows
where T* = 1 -
F / T , A H ~ *= -
-
AHJT,
and A,Si* = A S i
-C
AHi*.
The reparameterization in eq 6 is used to reduce the interaction between the parameters and aid convergence by centering the data. More details on the above can be found in Blakemore and Hoerl (1963) and Mezaki and Kittrell (1967). For the empirical model, only the rate constant was
Partial pressure Run no.
Rate
CO
HzO
COB
HZ
0.048 0.140 0.032 0.144 0.044 0.157 0.052 6.158 0.220
0.048 0.028 0.142 0.156 0.044 0.053 0.176 0.157 0.209
0.076 0.208 0.091 0.194 0.085 0.188 0.049 0.173
0.076 0.083 0.221 0.189 0.085 0.044 0.172 0.157
0.144 0.144 0.169 0.263 0.234 0.145 0.234 0.124
0.145 0.144 0.290 0.160 0.240 0.144 0.118 0.236
Kinetic Runs at 360°C R-156 R-157 R-158 R-159 R-161 R-162 R-163 R-164 R-165“
0.00112 0.00072 0.00074 0.00095 0.00104 0.00114 0.00126 0.00087 0.00060
0.277 0.199 0,197 0.303 0.259 0.349 0.349 0.266 0.198
0.445 0.462 0.457 0.451 0.221 0.218 0.214 0.233 0.240
Kinetic Runs at 381°C R-166 R-167 R-168 R-169 R-170 R-171 R-172 R-173
0.00173 0.00189 0.00213 0.00139 0.00197 0.00105 0.00113 0.00129
R-146 R-147 R-148 R-149 R-150 R-152 R-153 R-154
0.00338 0.00338 0.00399 0.00377 0.00299 0.00340 0.00284 0.00294
0.234 0.330 0.331 0.238 0.299 0.167 0.161 0.250
0.197 0.189 0.178 0.211 0.404 0.446 0.439 0.430
Kinetic Runs at 404°C
a
0,268 0.263 0.383 0.377 0.277 0.255 0.229 0.225
0.125 0.123 0.098 0.105 0.136 0.341 0.363 0.352
Runs made according t o design criterion.
treated this way whereas the exponents in this model were assumed to be temperature independent. Furthermore, all six of the original models were fitted to the data in case any one might behave “better” when considering all the data simultaneously. Both the Eley-Ride1 and the Kodama models diverged to unbounded parameter estimates and were not considered further. As before, only the empirical and Langmuir-Hinshelwood models had significant probabilities, 0.01 and 0.99, respectively, after the data in Table I1 were reanalyzed. No more experiments were conducted. Even though the calculation of the posterior probabilities indicated that discrimination had been achieved, the two models were examined more closely to uncover any inadequacies despite the overall acceptable fit. The mean square residuals for the empirical and the Langmuir-Hinshelwood models when fitted to all the data simultaneously were 7.67 x and 8.02 x 10-9, respectively. These values compare reasonably well with the value assumed for the error variance, about 3 x 10-9, which was obtained from equipment calibration calculations and from several runs repeated as closely as possible to determine a constant catalyst activity. From the standpoint of the overall fit of the two models to the data, the residual mean square for the empirical model is only about 5% greater than that for the Langmuir-Hinshelwood model, even though the posterior probabilities are heavily in favor of the latter model. A more detailed look a t the parameters in each model is shown in Table 111. The standard error column refers to Ind. Eng. Chem., Process Des. Develop., Vol. 13, No. 4,1974
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Table 111. Parameter Estimates Coded parameters
B1 B2 B3 B4 B5 B6 B7 B8
Decoded parameters
Std e r r o r
Langmuir-Hinshelwood Model -49.67 29,364 15.95 AH@) -3.31 40.32 0.49 AS (k) 4.68 -3,064 62.65 AH(K,~) -6.74 -2.06 1.93 AS (Kco) -9.49 9.83 A H ( K , , ~ ~ ) 6,216 3.28 12.77 0.30 AS ( K H ~ o ) 19.16 15.89 -12,542 0.70 -18.45 0.46 AS (Kco2) Empirical Model
B1 B2 B3 B4
B5 B6
-42.75 -11.36 0.81 - 0.024 -0.16 - 0.044
1.26 0.23 0.061 0.024 0.026 0.023
AH@) AS (k) m n
p q
28,428 31.39 0.81 - 0.024 -0.16 - 0.044
the coded parameter estimates. The parameters in each model are relatively poorly estimated from the data, and in the case of the Langmuir-Hinshelwood model, the enthalpy and entropy of adsorption for water are opposite in sign from what is normally expected. The term corresponding to the adsorption of hydrogen in the LangmuirHinshelwood model approached zero and was dropped from that model. In addition to the examination of the parameter estimates in the two models, the residuals for each model were plotted against the predicted rate for each model; no unusual trends were noted. From these additional considerations, the discrimination in favor of the Langmuir-Hinshelwood model becomes less tenable since the empirical model has almost the same overall fit to the data. It is obvious that the discrimination criterion cannot be used as the sole evidence for discriminating in favor of one model without further examination and testing. An attempt was made to improve the parameter estimates in the Langmuir-Hinshelwood model by means of the sequential design proposed by Box and Lucus (1959) which seeks to minimize the joint confidence region for the parameters in the model. This attempt was not successful; there was no significant change in the size of the joint confidence region after additional experimentation. The large number of parameters contained in the models for the nonisothermal case probably decreased the sensitivity of the design criterion over the accessible experimental region. Comparison with Existing D a t a The fact that the design criterion specified conditions outside the range that could be obtained in the recycle reactor used here led to a reexamination of other published data. In spite of the large body of experimental data extant, the data collected by Temkin (Shchibrya, et al., 1965) were the only set readily amenable to this type of analysis. The second oxidation-reduction equation in Table I is the model they proposed. The data were originally analyzed both through graphical means and through linear least-squares techniques. Consequently, data taken at 390°C were reanalyzed in a similar manner to the analysis shown here. All of the models considered here were fitted to the data, and only three had reasonable mean square residuals. The Lang420
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muir-Hinshelwood model gave a mean square residual of 5.17 x the oxidation-reduction model proposed by Temkin, 5.18 x 10-5, and the empirical model, 8.23 X 10-5. The authors did not indicate an estimate of the error variance so it was not possible to calculate the posterior probabilities associated with each model. If it is assumed that the rate of reaction was measured to within slightly better than IO%, then the Langmuir-Hinshelwood model is favored slightly over Temkin’s oxidation-reduction model. The probabilities for the two models are 58 and 4270, respectively. The discrimination criterion then indicated that the next experiment should be run at very low conversion, a condition that cannot readily be achieved in a recycle reactor. On the basis of this analysis it seems safe to state that there are other models that can adequately account for Temkin’s data and that by no means do Temkin’s data indicate that his model, given in Table I, is the only one or the best one. Conclusions The results of this study show the Langmuir-Hinshelwood model to be favored over the other proposed models. The power-law model is the next best model, and upon additional examination of the parameters and residuals, it becomes a reasonable alternative to the Langmuir-Hinshelwood model. The data collected by Temkin was also reanalyzed in a manner similar to that shown here with the result that the model proposed by Temkin, the oxidation-reduction model, was not necessarily the best one. In this case the Langmuir-Hinshelwood model was slightly favored over Temkin’s model. It is possible, in light of the specification of experimental conditions on the limits of the operability region, that no single reactor type will be sufficient for model discrimination, In our case, the recycle reactor could not reach the specified conditions. It is also an interesting question to ask what would be the behavior of the design criterion if the operating limits are extended indefinitely. The final point to be made is that the use of the discrimination criterion alone is not sufficient to ensure certainty in model discrimination because discrimination can be made between several very poor models. The overall fit of a model to the data should be checked against the estimated error variance to ensure an acceptable fit. Also individual models should be examined more closely to check for any obvious inadequacies even if the overall fit is reasonable. The various methods of examining the residuals can be a great aid in spotting any obvious model inadequacies. Nomenclature (CO) = concentration of carbon monoxide, atm (COz) = concentration of carbon dioxide, atm Derr = effective diffusivity, cm2/sec h = Thiele modulus (H2) = concentration of hydrogen, atm (H20) = concentration of water, atm A H = enthalpy factor used in simultaneous analysis, cal/ mol k = rateconstant K = equilibrium constant for homogeneous reaction K , = adsorption equilibrium constant of component i, atm-l P ( M , ) = probability of model i in the absence of data P ( M ,Iy) = probability of model i given the data set y r = rate of reaction, mol/(g of cat)(min) R = radius of catalyst particle, mm S, = residual sum of squares for model i AS = entropy factor used in simultaneous analysis, cal/ (mol)(”K)
T = temperature T = average temperature
X
= matrix containing partial derivatives
Y n = vector containing n rate observations gnl
= predicted value of model i at Fn
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Received for review November 16, 1973 Accepted June 14,1974
Synthesis of Cascade Refrigeration and Liquefaction Systems Francisco J. Barnes' and C. Judson King* Department of Chemical Engineering. University of California. Berkeley, California 94720
Systematic synthesis procedures are employed to seek minimum-cost process configurations for cascade refrigeration and gas liquefaction systems. Through a preliminary process analysis based upon graph decomposition principles it is shown that the optimal or near-optimal configuration is defined by the answers to a limited number of identifiable choices, involving both configuration and design variables. The minimum-cost configuration is located through the application of dynamic programming to the network formed by these alternatives. The procedure is repeated iteratively as optimal values are found for design variables and the number of refrigerant levels. Certain heuristics are used or implied in this overall iterative procedure, but they do not appear to limit the method significantly. Several practical examples of synthesis of cascade refrigeration and gas liquefaction systems are considered. The results are interpreted to infer factors that appear to be dominant for refrigeration system design.
Introduction
Process design may be described as the conception or synthesis of a given process configuration, followed by the evaluation or analysis of its capabilities, equipment sizes, and cost requirements. Most often, the design process is evolutionary, following a succession of alternative steps of synthesis and analysis, wherein the results of the last analysis provide additional information to improve the next synthesis (King, e t al., 1972). Whereas process analysis has received a great deal of attention and has achieved a high level of sophistication, only recently have substantial efforts been initiated to structure the logic of synthesis of chemical processes (Hendry, et al., 1973). The bulk of this research has fol-
' Department of Chemical Engineering. National Autonomous Universityof Mexico, Mexico 20, D.F.
lowed the availability of large computers making extensive use of their large memory capacity and high speed. Morphological analysis and other more qualitative approaches have also been utilized for structuring process understanding and synthesis for more broadly defined problems (King, 1973). Systematic process synthesis for processes involving sequentially connected elements may be defined as the orderly generation and evaluation of all possible alternative process configurations, followed by the application of efficient methods to discard as soon as possible the configurations that are less attractive than those remaining for further evaluation. These methods can be based upon either rigorous tests or mathematical algorithms that necessarily lead to the optimal solution, or upon heuristic methods. Heuristics are rules-of-thumb which cannot be shown to lead necessarily to the optimal solution, and the efficiency which they give to the search method must be balanced Ind. Eng. Chem.,
Process Des. Develop., Vol. 13, No. 4 ,
1974
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