Ind. Eng. Chem. Prod. Res. Dev. 1985, 2 4 , 512-516
512
organic templates is given as Figure 14. Amorphous sodium aluminosilicate gel just after preparation is chemically active and easily dissolved to give aluminate ions and silicate ions in an alkaline aqueous solution at high temperatures. In this system, a part of these ions combines to make aluminosilicate ions. The aluminosilicate ions are used to form the ZSM-5 species even in the absence of organic templates. After ZSM-5 species formation, they diffuse on the surface of the seed crystals. In the seeded system containing a small amount of acetone, the crystallization rate of ZSM-5 was remarkably accelerated and the crystal size and shape of the product were quite uniform. One of the reasons for accelerated crystallization with the addition of acetone is that the fluidity of the reaction mixture is increased, and the diffusion rate of the ZSM-5 species becomes very fast. Therefore, the formation of the spontaneous ZSM-5 nuclei was inhibited, and the crystallization of ZSM-5 occurred only on the seed crystals. From experiments similar to that previously described, it was also found that methyl ethyl ketone and cyclohexanone exhibited the same effect as acetone and the degree of the effect was almost equal. The fact that these ketones never play the role of templates for the crystallization of ZSM-5 was proved by the result-all of the products in the unseeded system containing a small amount of ketones at 190 " C for 12 h were amorphous. The crystallization of ZSM-5 in the seeded system containing mixtures of alcohols with ammonium hydroxide had been reported (Plank et al., 1979,1980). We tried the crystallization of ZSM-5 according to these patents and compared the results with that previously described. The percent of ZSM-5 crystallization in an acetonelwater system (100%) was higher than that in an ethanol-ammonialwater system (89% 1.
These results clearly demonstrate the significance of crystal size and shape uniformity on ZSM-5 crystallization and provide a practical and economical method that employs neither organic templates nor the calcination process at high temperatures. Further clarification of the crystallization and decomposition processes of ZSM-5 in this system and the characterization of the product are currently under way. Acknowledgment
We thank Haruhito Sato of Idemitsu Kosan Co., Ltd. for his valuable comments and Nissan Chemical Ind., Ltd. for supplying colloidal silica used in this study. L i t e r a t u r e Cited Butter, S. A.; Jurewicz, A. T.; Kaeding, W. W. U S . Patent 3894 107, 1975. Chang, C. D.; Lang, W. H.; Siivestri. A. J. US. Patent 3894106, 1975. Chang, C. D.; Silvestri, A. J.; Smith, R. L. US. Patent 3928483, 1975. Chao, K. J.; Tasi, T. C.; Chen, M. S. J. Chem. Soc., Faraday Trans. 11981, 77, 547. Chen, N. Y.; Miale, J. N. Braz. Pedido PI, BR82 00547, 1982. Culfaz, A.; Sand, L. B. Adv. Chem. Ser. 1973, 121, 140. Dai, F. Y.; Saito. Y.; Takahashi, H.,presented at the 48th National Meeting of the Chemical Society of Japan, Sapporo, 1983. Erdem. A,; Sand, L. B. J . Catal. 1979, 60, 241. Grose, R. W.; Fianigen, E. M. U.S. Patent 4257885, 1981. Inui, T.; Ishihara, T.; Morinaga, N.; Takeuchi. G.; Matsuda, H.; Takegami, Y. Ind. Eng. Chem. Prod. Res. Dev. 1983, 22, 26. Koizumi, M.; Ueda, S. Shokubai 1983, 25, 21 1. Meisei, S. L.; McCuiiough, J. P.; Lechthaier, C. H.; Weisz, P. B. Chemtech 1976, 6 , 86. Milestone, N. 8.; Bibby, D. M. J . Chem. Techno/. Biotechnoi. 1983, 3 4 , 73. Narita, E.; Sato, K.; Okabe, T. Chem. Lett. 1984, 1055. Plank, C. J.; Rosinski, E. J.; Rubin, M. K. U.S. Patent 4 175 114, 1979. Plank, C. J.; Rosinski, E. J.; Rubin, M. K. US. Patent 4 199556, 1980. Takahashi. H.; Nakamoto, H. Nendo Kagaku 1982, 22, 137. Roliman, L. D. "Zeolite: Science and Technology"; Martinus Nijhoff: The Hague, 1984; "NATO AS1 Series E", p 109. Weisz, P. B. Pure Appl. Chem. 1980, 52, 2091.
Received for review November 21, 1984 Accepted June 5 , 1985
Reaction Kinetics on a Commercial Three-way Catalyst: The CO-NO-02-H20 System Bala Subramanlam+and Arvlnd Varma' Department of Chemical Engineering, University of Notre Dame, Notre Dame, Indiana 46556
Kinetic data were obtained for the CO-NO-0,-H,O reaction system on a commercial three-way catalyst (TWC) under experimental conditions of relevance in automotive exhaust catalysis. A fixed-bed with external recycle reactor operating at high recycle ratios was used. Depending upon the temperature and oxidizing potential, a parallel network of reactions, CO 0.50, COP( k , ) , CO NO C02 0.5N2 (k2),and 2.5CO NO -I-1.5H20 NH, 2.5C0, ( k 3 ) ,with k , >> k , and k,, was found to occur over the range of experimental conditions studied. The implications of such a network in kinetic parameter evaluation are discussed. Optimized kinetic parameters, which permit prediction of reaction rate given the bulk gas composition and temperature, were derived from the experimental kinetic data for each of these reactions by using a model that accounts for pore-diffusion as well as external transport limitations. The rather unique nature of the reaction network warranted the use of such a model, which was later justified by the results.
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Introduction
The CO-NO-02-H20 reaction system includes two of the major reactions (viz. CO-02 and CO-NO) associated with catalytic cleanup of automotive exhaust gas. Intrinsic
* To whom correspondence shouid be addressed. 'Present address: Department of Chemical and Petroleum Engineering, University of Kansas, Lawrence, KS 66045. 0 196-432 1/85/1224-05 12$0 1.50/0
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kinetic parameters are reported here for this reaction system on a commercial three-way catalyst (TWC). The kinetic data for this purpose were obtained by using a fixed-bed with external recycle reactor. Intrinsic kinetic expressions are necessary for the rational design of catalytic converters. Kinetic parameters have been reported for oxidation of CO and C3H, in the presence of NO and H 2 0 on a Pt/r-A120, catalyst (Voltz 0 1985 American Chemical Society
Ind. Eng. Chem. Prod. Res. Dev., Vol. 24, No. 4, 1985 513 G A S MIX
EXHAUST
7 8+ BYPASS
REACTOR PRESSURE
BUBBLE METER PREHEATER
E
g
R E ACTOR
RECYCLE R E A C T O R
I t
HEAT EXCHANGER 4 WAY VALVE
PUMP
ANAL.
I' 1 '
/
Figure 1. Experimental setup.
et al., 1973). However, the formation of NH3 was not considered. A knowledge of the extent of NH3 formation is of practical significance, since in a dual-bed cleanup scheme any NH3 formed on the TWC is oxidized back to NO on the oxidation catalyst that follows the TWC (Gandhi et al., 1978); this is clearly undesirable. Reactions of CO, NO, and 0, on a TWC have recently been studied (Bettman and Otto, 1983). However, their analysis required extensive reactor modeling to extract kinetic parameters from their tubular reactor conversion data. The reliability of the reactor model can obviously be questioned. In the present work, a recycle reactor operating at high recycle ratios (thus simulating a CSTR) was used to obtain kinetic data. Intrinsic kinetic parameters were obtained through a relatively simpler model. Experimental Section Catalyst. A commercial three-way catalyst, supplied by Ford Motor Co., was used in the kinetic experiments. It consists of a corderite honeycomb monolith with square channels of approximately 1-mm side and 0.35" wall thickness. The washcoat, approximately 25 pm thick and consisting of 10% Ce02, 2% NiO,, and balance yA1203, is applied on the monolith. Pt and Rh, in the ratio Pt/Rh = 10, are coimpregnated on the support. The total noble metal loading is approximately 0.1 w t % of the washcoat. The BET area was determined to be 10 m2/g. The catalyst was aged in burner exhaust for 16 h a t 790 "C under approximately stoichiometric conditions. This aging process reduces the activity of the catalyst to that which remains after 50 000 mi of driving. Experimental Setup. The experimental setup is the same as that used previously for the CO-0, reaction system (Jothi, 1982; Kosanovich et al., 1985). However, an auxiliary fixed-bed tubular reactor was designed and added to the existing setup in order to permit continuous measurement of NH3 (Figure l) (Subramaniam and Varma, 1983). The recycle reactor effluent is sent to this auxiliary reactor containing Pt/ y-A1203 catalyst, over which NH3 is oxidized to NO in the presence of excess oxygen. The increase in NO concentration in the main reactor effluent stream when passed through the auxiliary reactor is attributed to the NH3 formed in the main recycle reactor. For NH3 measurement, favorable auxiliary reactor operating conditions are when the NH3 entering the reactor is quantitatively oxidized by O2to NO (rather than to N,) and when the NO entering the reactor or produced by the
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Table I. Range of Operating Conditions Used in Kinetic Study -1 atm pressure temp 268-480 "C SO2 -20 ppm N2 balance co 0.1-2 vol % 0 2 0.1-2 vol % NO 0-0.15 vol % H20 -10 vol % COZ -10 vol %
NH3-O2 reaction does not undergo any reaction on the Pt catalyst. It was found that these two criteria depend upon reactor operating conditions such as temperature, spacevelocity, and excess oxygen present. Catalyst temperature of 440 OC, space velocity of 80 kh-', and presence of excess oxygen were determined earlier (Subramaniam and Varma, 1983) to be appropriate for the measurement of NH,; these conditions were therefore employed for the auxiliary reactor. The experimental conditions employed in the main reactor for kinetic studies on the TWC are summarized in Table I. The reactor pressure was maintained close to atmospheric. The recycle flow was maintained constant throughout (-50 L/min), while typical external flow rates ranged between 300 and 400 mL/min. This gives sufficiently high recycle ratios (125-165) so that the gas phase can be considered well mixed. Kinetic data were obtained at seven different gas-phase temperatures between 268 and 480 OC. Below 268 OC the conversions are two low for reliable measurement, while above 480 "C the conversions are severely hampered by mass-transfer limitations. At each temperature, the concentrations of CO, 02,and NO were varied in the range shown in Table I in a background gas consisting of 10% H20, 10% CO,, 20 ppm SO,; and the balance N2-as found in a typical automotive exhaust gas. Experimental Data. Kinetic data were obtained as outlet concentrations of CO, 02, NO, and NH3 for a set of reactor gas-phase temperature, inlet concentrations, and feed flow rate. The following reactions were found to occur: R1: CO + 0.502 CO2 (1) R2: CO + NO C02 + 0.5N2 (2) 2.5CO + NO + 1.5HzO NH3 + 2.5CO2 (3) R3:
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The extent of water-gas-shift reaction was found to be negligible. Thus, evaluating the conversions of CO, NO, and O2 and measuring NH, formation rate verified the mass balance. Following this, the rates of CO consumption via reactions R1, RP,and R3were evaluated. Details of this procedure are presented elsewhere (Subramaniam, 1984).
Data Analysis The experimental data were analyzed for kinetic form and kinetic parameters. This was accomplished in various steps. First, selected pairs of data points were qualitatively analyzed in order to infer a possible kinetic form. The data base was then divided into two temperature regions. In the first region (268-305 "C), only reactions R1and R2 occur; also, calculations showed that intrinsic kinetics control the measured reaction rates (Carberry, 1976). The next step was to analyze these data in order to get an idea of intrinsic kinetic parameters for reactions R1and, R2. In the second region (340-480 "C), all three reactions (Rl, RS,R3)proceed a t measurable rates. However, calculations to check for diffusion limitations indicated a unique situation; while R1is diffusion-limited over part of this regime, R2 and R3are both slow compared to diffusion over the entire range. In order words, because of the relatively high reaction rate of R, (as compared to the diffusion rates of CO and Oz), CO and O2 exhibit concentration gradients within the catalyst pores. However, reactions involving NO (viz. R2 and R3)are much slower compared to diffusion, and hence the NO concentration profile is expected to be flat within the pore. Nevertheless, since reactions Rz and R3 also involve CO, they are indeed subject to CO diffusional limitation. Therefore, in order to derive intrinsic kinetic parameters for reactions R1,R2, and R3,it is imperative that concentration profiles of CO and O2be obtained. This is done by using the model described later, which incorporates both interphase mass transfer as well as porediffusion resistances. The model serves to derive kinetic parameters from transport-influenced reaction rates. Since NH3 formation occurs only at the higher temperatures of the study (340-480 "C), there are fewer data for reaction R3(as compared to reactions R1 and R2) from which its kinetic parameters are derived. Qualitative Analysis. Sets of data were chosen so that the qualitative effects of CO, NO, and O2 on the various possible reactions could be qualitatively inferred readily. Based on this analysis, the reaction rate expressions take the following form:
Hence optimization of the various parameters in eq 4 and 5 was done by using the Marquardt-Levenberg optimization technique, which is a nonlinear least-squares technique (Marquardt, 1963). Since the rates of reactions R, and R2 are known individually, parameter optimization was done separately for each of the reactions as follows: 1. A reasonable initial guess was made for each of the kinetic parameters, viz., the k , E,m, n, q , p , K , and AH values. 2. The reaction rates R1 and R2 were then evaluated. Normalized error, defined as Erri = (rekxpti - rCalcd)/rekxptl, was calculated. 3. Step 2 was repeated for all the data points. The normalized error values were then supplied to the optimization routine. 4. Kinetic parameters were modified by the optimization routine such that Erri2tended toward a minimum. Steps 2 and 3 were repeated until the above sum was indeed minimized. The kinetic parameters thus obtained are given elsewhere (Subramaniam, 1984). The values of these parameters served as starting guesses for the solution of the model described below. Analysis for the Complete Data Set. As mentioned before, analysis of the entire data was done by using a model that accounts for concentration gradients within the pores. The steady-state differential mass balances describing CO and O2 concentrations are as follows:
xzl
(7)
where De,COand De,Ozare the effective diffusivities of CO and O2 within the catalyst washcoat in the presence of inerts. The concentrations used are the local concentrations (in the washcoat) and are unknowns to be determined as functions of the position variable x, measured from the washcoat-substrate interface in a direction normal to the catalyst surface. Although the recycle reactor was operated at high recycle flow rates, external heat and mass transfer resistances may still persist, especially at the higher temperatures. Thus these were included in the model as boundary conditions at the interface between the catalyst and the bulk gas. As the ceramic substrate is catalytically inert, there is no diffusion of mass into it at steady state. The relevant rl = ~kl[ex~(-El/RT)l(C0)"1(02)n1~/[(1 + boundary conditions (BCs) for the differential eq 7 and Kl[exp(AH1/RT)1(CO) + K 2 1 e x ~ ( ~ 2 / R T ) l ( N O ) l p i l 8 are as follows: (4) at x = 0, k2[exp(-E2/RT)] (CO)m2(02)nz(N0)q2 r2 = (5) I1 + K3[exp(AH3/R~1(CO)lP~
r, = {k,[e~p(-E,/RT)](CO)"3(0~)~3(N0)4~)/[(1 + K, x [exp(AK/RT)1(CO)JP3(1+ K S [ e x p ( W / R n 1(OJlp41 (6) Note that reactions Rz and R3proceed at measurable rates only in the presence of oxygen; this is why eq 5 and 6 contain explicit positive-order dependence (n2,n3 > 0 ) on oxygen despite the fact that oxygen is not involved in the stoichiometry of these reactions. Data Analysis for the Lower Temperature Region. As noted before, in this temperature region (268-305 "C) only reactions R1 and R2 occur. Both eq 4 and 5 are nonlinear and cannot be rearranged to become linear.
at x = L,
As discussed above, the concentration profiles for NO and NH, within the pore are flat. Their concentrations
Ind. Eng. Chem. Prod. Res. Dev., Vol. 24, No. 4, 1985
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Table 11. Rate Exoressions rl =
6.28 X 1021[exp(-29000/R~l(C0)(02)
(1
+ 5.22 X 106[exp(2000/R7'l(CO) + 9.85X 106[exp(1500/RT)](NO))2
r2 = 1.32 X 1016[exp(-20 200/RT)](CO)'.4(02)0.3(NO)o.13 (1 + 10.0 x 106[exp(1300/RT)](CO))2
TRIRNGLE-COtOZ CIRCLE-CO+NO
/
RRTE1268~480°CI
R R T E I 2680-48OoC1
PLUS-CO+NO*H20
RATE1 39O0-48O0C1
Temperature L
r
O
Figure 2. Temperature and concentration profiles in the interphase and the washcoat.
within the pore are therefore given by overall mass balances over the entire pore length:
The temperature within the pore is similarly given by an overall heat balance, as dictated by the internal isothermal model (Carberry, 1976; Pereira et al., 1979): [(-AHl)rl
IU"
'
"""I V ' ' " " " \ ut '
' '
""'I u" ' " " " I u a 7"''11
E X P E R I R E N T A L RRTE.GROL
IU'
CO/lRIN-CCRTI
Figure 3. Plot of experimental vs. calculated reaction rates for each reaction.
h(T, - Tb)= L
b
+ (-AH2)r2 + (-AHJr3/2.51
dx (13)
where (-AHi) is the heat generated by the conversion of 1 mol of CO by reaction Ri. The expected temperature and concentration profiles are shown in Figure 2. The pair of coupled second-order nonlinear ordinary differential equations with accompanying BCs (eq 7-10) were solved for each data point along with the integral balance equations for NO, NH,, and temperature to predict the various reaction rates. The differential equations were numerically solved by using the orthogonal collocation method (Villadsen and Michelsen, 1978; Finlayson, 1980). Kinetic parameters need to be assumed initially in order to solve for the concentration profiles and thus obtain the calculated reaction rates of the various reactions. These parameters were altered after each iteration so that the error between predicted and experimental reaction rates was minimized. Parameter optimization was done using the Simplex technique (Ficken, 1961). A modified computer version of this technique (Chandler, 1965) supplied by Ford Motor Co. Research Staff was used. This method was previously shown to be rather stable for this particular type of problem (Jothi, 1982). The parameter optimization scheme consisted of the following steps. 1. The mass and heat transfer coefficients were calculated by using correlations (Hegedus, 1973). Since it is difficult to measure experimentally,the effective diffusivity of gases in the washcoat was obtained from the kinetic data by treating it as a parameter to be optimized. The effect of temperature and molecular weight on effective diffusivity in the washcoat was taken to be the same as for Knudsen diffusion. 2. An initial guess and search domain was supplied for all parameters. The range between the upper and lower limits was sufficiently wide so as to give convergence.
3. Surface concentrations and temperature were determined by using the experimental reaction rates, bulk gas concentrations, and temperature and the transport coefficients. 4. Using surface concentrations as initial guesses over the entire pore length, the differential equations along with BCs (eq 7-10) were solved by orthogonal collocation technique to give the CO and O2concentration profiies and catalyst temperature. 5. The reaction rates rl, r2, and r3 were evaluated by integrated the reaction rates at the collocation points over the thickness of the washcoat. 6. Steps 4 and 5 were repeated until the integral balances (eq 11-13) were satisfied to prescribed accuracy. was then Normalized error, defined as (rexptl- rcdc&/rexptl, calculated for each reaction rate. 7. Steps 3-6 were repeated for each data point. Root mean square (RMS) error, defined as
was calculated for each reaction rate and supplied to the Simplex optimization routine. 8. Parameters were modified by the Simplex subroutine such, that sum of Err for the three reactions moved to a minimum. Steps 3-7 were repeated until this sum was in fact minimized. These steps were executed by a computer program. The program was found to be efficient and stable so as to converge in most cases, even when the initial guesses for parameters caused the RMS error to be several orders of magnitude higher than its minimum value. Results a n d Discussion The intrinsic kinetic parameters obtained as above for the three reactions are shown in Table 11. At convergence,
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Ind. Eng. Chem. Prod. Res. Dev., Vol. 24, No. 4, 1985
these parameters were considered optimized only if they predicted the experimentally observed rates within acceptable error and if they were physically reasonable. The reported parameters were thus obtained after quite a number of trial runs. A plot of the reaction rates predicted by using these parameters vs. the experimental reaction rates is shown for each reaction in Figure 3. The close scatter of points about the 45" line indicates a good quality of fit for each reaction. The parameters obtained for the CO-O2 reaction also show good agreement with the limiting case when NO is absent (Kosanovich et al., 1985). The addition of NO inhibits the CO-02 reaction. However, this inhibition is relatively small compared to that of CO. It should be noted that the activation energy terms and the various exponents reported were rounded off to the values shown. When nearing convergence, the program straddled about these points with virtually no change occurring in the value of the objective function being minimized. The rounding-off procedure was adopted for the other two reactions as well and merely hastened convergence. As far as our knowledge goes, there have been no kinetic studies of the CO-NO-0,-H20 system in gradientless reactors, on either noble metal or base metal catalysts, with which our parameters for CO-NO reaction or NH, formation reaction can be compared. Recently, kinetic expressions derived from modeling of tubular reactor conversion data have been reported for this reaction system over a three-way catalyst (Bettman and Otto, 1983). The object of what paper was to point out the case of negative apparent activation energy for the CO-NO reaction when it occurs in parallel with the relatively much faster C0-0, reaction on the oxidizing side of stoichiometry (S = [202 + NO]/CO > 1). This result is due to diffusion falsification of the kinetic data. Obviously, it is improper to compare those kinetic parameters with the ones obtained from the present work. It is interesting to note, however, that our model, which accounts for pore-diffusion as well external transport limitations, yields a positive value for the activation energy in case of both the CO-NO and CO-NO-H20 reactions. The values of effective diffusivities for CO and O2range between 0.054 and 0.068 cm2/s. There are in the general range of values reported for y-Al,O, pellets. It can be clearly seen from Figure 3 that there is a difference of more than an order of magnitude between the high end of the CO-02 reaction rate and those of the CO-NO and CO-NO-H20 reaction rates. This fact was also manifest in the values of the Thiele moduli calculated from the integrated rates of the various reactions occurring in the washcoat. Further, the dimensionless surface concentrations of the various species and the dimensionless surface temperature were considerably different from unity at higher temperatures, indicating that external transport limitations ought not be ignored. Effectiveness factors ranged between 0.06 and 4.2;these values imply that under certain conditions there were considerable transport resistances. Values of the effectiveness factor greater than unity arise, of course, under diffusion-influenced conditions because of strong adsorption of CO on the catalyst surface.
The foregoing remarks support the use of the diffusion-reaction model with its accompanying BCs and integral equations. This model represents the general procedure to derive intrinsic kinetic parameters in the case of a parallel network of heterogeneous gas-solid catalytic reactions (viz., A B (kJ,A C ( k 2 ) ) where , k, >> k,, occurring on supported catalysts. Nomenclature (CO) = CO concentration, mol/cm3 De = effective diffusivity, cm2/s AHi = inhibition term activation energy, cal/mol Ei = frequency factor activation energy, cal/mol h = heat transfer coefficient, cal/ (cm2.0C.s) K i= inhibition constants, cm3/mol k l = frequency factor, mol/[min.g~at.(mol/cm~)~~+~l] k, = frequency factor, mol/[min-gcat.(mol/~m~)~2+~2+42] k3 = frequency factor, mol/[min-g~at.(mol/cm~)"~+~3+q3] k, = mass transfer coefficient, cm/s L = washcoat thickness, cm mi, ni,p i , q i = indices, dimensionless (NO) = NO concentration, mol/cm3 (NH,) = NH3 concentration, mol/cm3 (0,) = oxygen concentration, mol/cm3 R = gas constant, cal/mol.K ri = rate of reaction Ri, mol/(min.gcat) T = temperature, K x = position variable, cm Subscripts b = bulk calcd = calculated via the model exptl = experimental s = surface Acknowledgment This work was supported by the Department of Energy, as part of a joint research program between the Ford Motor Co. and the University of Notre Dame. Registry No. CO, 630-08-0; NO, 10102-43-9; Ce02,1306-38-3;
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Ni02, 12035-36-8; Rh, 7440-16-6; Pt, 7440-06-4.
Literature Cited Bettman, M.; Otto, N. C. Chem. Eng. Sci. 1983, 38, 491. Carberry, J. J. "Chemical and Catalytic Reaction Engineering"; McGraw-Hill: New York, 1976; Chapter 5. Chandler, J. P. Quantum Chemistry Program Exchange (OCPE), Program 307, Chemistry Department, Indiana University, 1965. Ficken, F. A. "The Simplex Method of Linear Programming"; Holt, Reinhart. and Winston: New York, 1961. Finlayson, B. A. "Nonlinear Analysis in Chemical Engineering"; McGraw-Hill: New York, 1980. Grandhi, H. S.; Yao, H. C.; Stepien, H. K.; Shelef, M. SAE Technical Paper 780606, 1978. Hegedus, L. L. Prepr.-Am. Chem. SOC.,Div. Pef. Chem. 1973, 18, 487. Jothi, N. Ph.D. Thesis, University of Notre Dame, Notre Dame, IN, 1982. Kosanovich, M.; Jothi, N.; Vallejo, R. M.; Varma, A. manuscript in preparation. Marquardt, D. W. J. SOC. Ind. Appl. Math., Ser. A 1983, 2 , 431. Pereira, C. J.; Wang, J. B.; Varma, A. AIChE J. 1979, 25, 1036. Subramaniam, B. Ph.D. Thesls, University of Notre Dame, Notre Dame, IN, 1984. Subramaniam, 0.; Varma, A. Chem. Eng. Commun. 1983, 2 0 , 81. Villadsen, J. V.; Michelsen, M. L. "Solution of Differential Equation Models by Polynomial Approximations"; Prentice-Hall: Engiewood Cliffs, NJ, 1978. Voltz, S. E.; Morgan, C. R.; Liederman, D.; Jacob, S. M. Ind. f n g . Chem. Prod. Res. Dev. 1973, 12, 294.
Received for review July 5, 1984 Accepted June 16, 1985
