Ind. Eng. Chem. Process Des. Dev. 1984, 23, 620-622
620
40 “C are in better agreement with the published data than those computed from the five-term power series (Cramer, 1980). The most extreme example of this improvement is the result for water at 0 “C where the oxygen solubility measured by Truesdale et al. (1955) was 14.16 ppm, that determined from the generalized smoothing and interpolation formula (eq 5) was 14.29 ppm, and that from the five-term power series was 15.36 ppm. Except for SSGB, there was less than a 5% difference between the atmospheric pressure solubility values given by the two equations at temperatures above 40 “C. In summary, the generalized smoothing and interpolation formula (eq 5) fit the experimental oxygen solubility data well. It is based upon a widely used heat capacity equation and the application of thermodynamic principles, which facilitates further interpretation of the results. The solubility of oxygen in water, in three sodium chloride brines, and in two geothermal brines, expressed in terms of the Henry’s law constant, is reported for temperatures from 0 to 300 OC. The solubility of oxygen in these brines at atmospheric pressure is reported for temperatures from 0 “C to their normal boiling point.
Acknowledgment N. A. Gokcen, Bureau of Mines, Albany, OR, suggested the use of eq 1 to develop the smoothing and interpolation formula. Registry No. Oxygen, 7782-44-7. Literature Cited Cramer, S. D. Ind. Eng. Chem. Process D e s . Dev. 1980, 19, 300-305. Gokcen, N. A. “Thermodynamics”; Techscience: Hawthorne, CA, 1975; pp 77-86, 180-183. Gokcen, N. A.; Chang, E. T. Denkl Kagaku 1875, 43(5), 232-237. Hougen, 0. A.; Watson, K. M.; Ragatz, R. A. “Chemical Process Principles”, Part 1; Wiley: New York, 1959; pp 252-256. 345-347. Miller, R. B.: Wichern, D. W. “Intermediate Business Statistics”; Holt, Rinehart, and Winston: New York, York, 1977; pp 205-210, 308-324. Smith, J. M.; Van Ness, H. C. “Chemical Engineering Thermodynamics”: McGraw-Hill: New York, 1959; pp 118-128, 144-147, 409-412. Truesdale, G. A.; Downing, A. L.; Lowden, G. F. J . Appl. Chem. 1955, 5(2), 53-62.
Auondale Research Center Bureau of Mines U.S. Department of the Interior Auondale, Maryland 20782
Stephen D. Cramer
Received for reuiew August 16, 1982 Accepted October 6, 1983
Optimal Policy for a Consecutive Reaction in a CSTR with Concentration-Dependent Catalyst Deactivation An optimal time-temperature policy to maximize the cumulative yield of the desired product has been developed for a firstsrder consecutive reaction, subject to concentration-dependent catalyst deactivation conducted in a continuous stirred tank reactor. A graphical methodology utilizing Pontryagin’s Mlnlmum Principle has been used. In general, the optlmal policy is a nonconstant conversion policy. This technique can be extended to include various deactivation mechanisms.
Introduction In commercial catalytic reactors undergoing slow catalyst deactivation, it is the usual practice to gradually increase the reactor temperature to maintain a constant conversion at the outlet. This constant conversion policy constitutes a rigorous optimal policy only for the simple reaction A B. Commercial reactions are generally more complex and selectivity considerations are usually more important. The optimal temperature policy for complex reactions has been investigated by several authors (Dalcorso and Bankoff, 1972; Ogunye and Ray, 1971; Reiff and Kittrell, 1981). In general, numerical solutions are necessary. Recently, Reiff (1981) presented a graphical methodology for obtaining the optimal temperature policy for a series reaction undergoing concentration-independent deactivation in a CSTR. Except for a few cases, concentration-independent deactivation is not applicable and usually the deactivation is concentration dependent. This note extends his analysis to the case of concentration-dependent deactivation for the first-order series reaction A B C conducted in a CSTR.
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--
Problem Formulation and Solution The objective is to maximize the yield of B for the first-order series reaction A B C undergoing catalyst deactivation in an isothermal, continuous stirred tank reador. For convenience, the nomenclature of Reiff (1981) will be retained. For no products present in the feed and assuming that the pseudo-steady-state approximation is valid, we have
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0196-4305/84/1123-0620801 50f0
where, in general (3)
The aim is to find an optimal temperature policy such that the total yield of B over the catalyst life is maximized subject to the constraints on the reaction temperature. In order to find the optimal temperature policy for this problem, the run length, 0, has to be specified (Reiff, 1981). Thus, the objective is to minimize I $ F dt subject to kD, IkD(t) 5 kD* where kD rather than T is taken as the control variable because of mathematical convenience in calculating the partial derivative of the Hamiltonian with respect to the independent variable. The graphical solution has been discussed in detail by Reiff (1981). Briefly, the Hamiltonian, H , is defined as H=F+Af (4) The first necessary condition for optimally can be obtained from Pontryagin’s Minimum Principle as
or
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 3, 1984 621
''
'\
\
-1.6
\
\ I
670
I
680
I
I
690
\
'\
I
I
700
I
710
0
Temperature, K
Figure 2. Optimal path and stopping condition for consecutive reaction with concentration-dependent deactivation; 0 = 16 000 h. Other parameter values same as in Figure 1. Temperatura. K
Table I. Summary of the Optimal Policy % yield % time, temp, of B conversion h K activity
~
Figure 1. Variation of Hamiltonian with temperature and activity. Parameter values: El= 169000kJ/kg-mol; Ez= 227000 kJ/kg-mol; ED = 157000kJ -mol; AI = 8.02X 10l2l/h; A2 = 8.13 X 10lel/h; A D = 2.38 X 101z3/(kg-mol h); T = 2.15 h; n = 1; m = 0.
Differentiating eq 2 and 3 with respect to kD and substituting the resultant expression for X in eq 4, the Hamiltonian can be expressed as a function of the catalyst activity and the deactivation rate constant. The Hamiltonian can then be plotted as a function of kD with catalyst activity as a parameter and the second necessary condition for optimality and the stopping condition superimposed on this plot. Expressing kl and k2 in terms of kD, we have
where
substituting eq 6 in eq 2 yields
Similarly, substituting eq 1, 2, and 6 in eq 3, we obtain
Using eq 7 and 8 analytical expressions for aF/dkD and af/dkD can be obtained and these substituted in eq 5 to yield an analytical expression for X in terms of known constants, catalytic activity, and kD. Thus, the contours of the Hamiltonian vs. temperature can be constructed for different constant values of the activity. Results and Discussion For this study, the values of n and m were taken to be unity and zero, respectively; i.e., deactivation was assumed
~~
0 2 000 4 000 6 000 8 000 10 000 12 000 14 000 16 000
670 673 676 680 685 689 696 703 720
1.0 0.87 0.75 0.63 0.51 0.40 0.29 0.19 0.10
39.6 39.1 38.5 37.8 37.1 36.0 34.0 32.8 30.4
53.8 53.7 53.4 53.4 53.6 51.8 51.2 47.9 48.9
to be caused by the reactant only. The other parameters are given in Figure 1 and except for A D and AI were the same as those reported by Reiff (1981). C, was taken to be 1.0 kg-mol/m3. The variation of Hamiltonian with temperature and catalyst activity is shown in Figure 1, whereas the region of interest has been enlarged in Figure 2 so that the stopping condition (dF/dkD = 0) and the optimal path for 0 = 16000 h can be depicted clearly. The optimal path was constructed by guessing a start of run temperature on the contour of unit activity and moving horizontally on Figure 2 until the stopping condition was reached after a certain run length. The trial and error procedure was continued until the guessed value of the starting temperature yielded the desired run length of 16000 h. The optimal policy is given in Table I. For the parameter values used the outlet conversion decreases gradually from an initial value of 53.8% to 48.9% at the end of the run. For this particular example, the product ADCA is approximately equal to the preexponential factor for the deactivation rate constant used by Reiff (1981),and therefore the optimal policy is quite similar to the one reported by him for independent deactivation. However, in general, the optimal policy is a nonconstant conversion policy. A similar procedure can be followed for other values of n and m. Conclusion Through construction of optimal time-temperature progression, it is possible to replace the conventional constant conversion policy. The above analysis can be easily extended to include cases of product inhibition as well as deactivation orders other than unity. It is also possible to extend the graphical method to include cases of deactivation with diffusional limitations. Further work
622
Ind. Eng. Chem. Process Des. Dev. 1904, 23, 622-625
on this aspect is in progress. Nomenclature a = catalyst activity Al, A2, AD = preexponential factors C = concentration, kg-mol/m3 E,, E2,ED = activation energies, kJ/kg-mol f = rate of catalyst deactivation, l / h F = negative of yield of B H = Hamiltonian k l , k2, kD = rate constants k,o, k2o = constants defined in eq 6c n = order of catalyst decay with respect to A m = order of catalyst decay with respect to B t = time, h T = reactor temperature, K Greek Symbols P = El/ED E = E~/ED 7 = reactor space time, h
8 = run length, h X = adjoint variable in eq 4
Subscripts A, B , C , = chemical species 0 = initial conditions * = maximum (*) or minimum (,) bound
Literature Cited Dalcorso, J. P.; Bankoff. S. G. Chem. f n g . J . 1972, 3, 62. Ogunye, A. F.; Ray, W. H. AICM J . 1971, 17, 43. Reiff, E. K. Ind. Eng. Chem. PIOcessDes. Dev. 1981, 20, 558. Reiff, E. K.; Klttrell, J. R. Chem. Eng. J . 1981, 21, 71.
Department of Chemical Engineering Indian Institute of Technology Kanpur, India
Mritunjoy Pramanik Deepak Kunzru*
Received for review September 27, 1983 Accepted October 17, 1983
Catalytic Cracking of Heavy Ofls in Combhation wlth Pyrolysis for the Production of Light Oils A two-stage method for catalytic cracking has been developed to process heavy oils. A heavy 011 was thermally cracked at about 440 OC in the fkst stage, and the cracked oll thus produced was carried to the silica-alumina catalyst bed (the second stage). On the basis of the pyrolyzed oil carried to the catalyst bed, yields of CS-Cl1 gasoline, C13+ oil and coke from an atmaspherlc or a vacwm residue correspomlsd to those from a direct catatytic cracking of a vacuum gas oil. The asphattene fraction can be left in the first stage as a pyrolysis r e s b , so that the formation of coke may be reduced. A tar-sand Mtwnen, which is disadvantageous for the production of lower olefins by high-temperature pyrolysis, is considered to provide larger amounts of gasoline and kerosene fractions than paraffinic feed stock such as Taching vacuum residue.
Introduction Increasing demands for lighter petroleum fractions, motor gasoline in the US. and kerosene and gas oil in Japan, have produced occasional market shortages. To increase the gasoline yield, the use of the heavy end of crude as fluid catalytic cracking (FCC) feed stock has been intensively studied. For the residual oils used as feedstocks for the FCC unit, the most troublesome feature is metallic contaminants and high-molecular-weight condensed aromatic hydrocarbons contained in them, as well as the high sulfur content. To overcome these problem, numerous efforts have been made: catalyst improvement (Ritter et al., 1981), development of a riser cracking (Finneran et al., 1974), of low-conversion operation (Hemler and Vermillion, 1973; Finneran et al., 1974), combination of the hydrodesulfurization and FCC process (Yanik et al., 1977; Murphy and Treese, 1979; Masoslogitea and Beckberger, 1973),removal of metals from the catalyst (Edison et al., 1976), and so on. To treat heavy residues of crudes, coker processes are in operation and coker gas oils will be upgraded to lighter fractions. In the studies on the thermal cracking of residual oils, we have demonstrated that two-stage pyrolysis of heavy oils is an efficient process for the production of lower olefins (Suzuki et al., 1982b; Itoh et al., 1983). In this method, residual oil was first thermally cracked at 4-40 "C to give pyrolyzed oil and pitch. The pyrolyzed oil was carried to the second, high-temperature stage (750-800 "C) and was further pyrolyzed to lower olefins. A similar process was developed by Kureha Chemical Industry Co. and Chiyoda Engineering and Construction Co. as the Eureka process (Aiba et al., 1981). 0196-4305/84/1123-0622$01.50/0
In our experiments, it was elucidated that a yield of ethylene was affected significantly by the characteristics of feedstocks. From the structural investigation of heavy oils (Takegami et al., 1980; Suzuki et al., 1981a, Suzuki et al., 1982a),a straight-chain paraffin index (SPI) for heavy oils was proposed (Itoh et al., 1983). The SPI value was well correlated with ethylene yield on the two-stage pyrolysis. In this communication preliminary experimenta on the two-stage process for the catalytic cracking of heavy oils will be carried out, and yields of gasoline and gas oil fractions from various heavy oils are compared. Experimental Section Materials. Two tar-sand bitumens, Cold Lake and Orinoco, three fractions from Arabian Light crude, an atmospheric residue, a vacuum residue, and a vacuum gas oil, and Taching vacuum residue have been used as heavy oil feeds. Properties of these oils are shown in Table I. Detailed structural analyses of them have been described previously (Takegami et al., 1980; Suzuki et al., 1981a; Suzuki et al., 198213). A commercial silica-alumina amorphous catalyst used for the FCC process (Catalyst and Chemicals Ind. Co., No. 2244) was used. Apparatus. The experimental apparatus used in the present study is represented schematically in Figure 1. A quartz reactor tube (3), 150 mm long and 18 mm i.d. at the first section and 150 mm long and 8 mm i.d. at the second section, was used. The temperature of the tow section was controlled independently by the electrical resistance heaters (1). The reactor tube connecting the two sections was also heated electrically. An inner tube (5), 10 mm o.d., on which platinum wire (0.3 mm diameter) 0 1984 Amrlcan Chemical Snciatv
