the value of the summation is approximately 0.35; however, as n decreases below 10, the value sharply increases. Furthermore, when n is sufficiently large, [l - l/n2] in the second term in Equation 12 goes to 1. For sufficiently large numbers of side ports, Equation 12 reduces to
Lapple (1951) has derived the pressure drop through a perforated pipe distributor assuming ideal momentum recovery-i.e., k = 1-and uniform flow of fluid along the entire length of the pipe distributor-Le., the ports are very close to one another, simulating a continuous slot distributor. Lapple’s pressure drop equation for a horizontal distributor is given as:
This is equivalent to Equation 13 for k = 1 and n very large. For small values of n-Le., less than 10-Equation 12 would be more appropriate to use than Equation 14. The value of k to be used in the momentum recovery term in Equation 12 still needs additional investigation. Acrivos et a/. (1959) present some values of k for air systems, ranging from approximately 0.6 to 0.9 and apparently not correlating with fluid maldistribution. O n the same basis, data based on the work of Soucek and Zelnick (1945) in long 6-inch square channels with square side ports were compared with the air system. The value of k for water varied from approximately 0.7 to 0.4 with increasing maldistribution. I t has been suggested that k may be better correlated with fluid velocity or pressure drop through the side ports ; however, additional data are needed to substantiate this.
Nomenclature
distributor cross-sectional area, sq. ft. port cross-sectional area, sq. ft. D distributor diameter, ft. f Fanning friction factor gravitational constant, 32.2 (ft. lb.) (lb. force sec.’) g, i = section number k = momentum recovery correction factor L = distributor length, ft. n = number of side ports or orifices PI = pressure a t distributor inlet, p.s.i. = pressure a t closed end of distributor, p.s.i. P, AP = total pressure drop over distributor, p.s.i. V = velocity, ft./sec. = velocity through side ports, ft./sec. V, = inlet velocity to distributor, ft./sec. VI X,Y = direction vectors = liquid density, lb./cu. ft. pL
A1 A,
= = = = =
literature Cited
Acrivos, A., Babcock, B. D., Pigford, R. L., Chem. Eng. Sci. 10, 112-24 (1959).
Lapple, C. E., “Fluid and Particle Mechanics,” 1st ed., pp. 14-15, University of Delaware, Newark, Del., 1951. Soucek, E., Zelnick, E. W., Trans. A m . SOC.Civil Eng. 110, 1357-401 (1 945).
E. J. G R E S K O V I C H J. T. O’BARA Esso Research and Engineering Co. Florham Park, N . J . 07932 RECEIVED for review February 1, 1968 ACCEPTED May 20, 1968
CORRESPONDENCE O P T I M A L ADIABATIC BED REACTOR W I T H COLD SHOT COOLING SIR: The optimal design of adiabatic bed reactors with cold shot cooling has been treated in detail by Lee and Aris (1963). I n a recent communication MalengC and Villermaux (1967) have shown that the optimizing algorithm proposed by Lee and Aris does not lead to the optimal design conditions; in fact, by a direct search method on the set of six decision variables appearing in the expression for the profit of a three-bed reactor they could substantially improve the profit as obtained by Lee and Aris. However, here it is shown that even the solution of Malengt and Villermaux does not yield the true optimum and that neither of the previous solutions, although giving a profit close to the maximum profit, leads to the optimal design conditions. Although the optimizing algorithm used by Lee and Aris fails, their mathematical formulation of the problem is correct and it suits perfectly a discrete maximum principle approach. I n this note we use the notation of Lee and Aris, although this notation is more appropriate to a dynamic programming formulation than to the maximum principle formulation used by us. A stage consists of the catalyst bed and the preceding bypass mixing chamber or the preceding heater (for the Nth stage).
The state of the process stream a t each stage can be described by the set of state variables: entrance conversion g, exit conversion g’, exit temperature t’, cumulative relative mass flow rate A/XN, cumulative profit per unit of mass flow through that stage P. The decisions to be made a t each stage are the entrance temperature, t , and the holding time, e. The following set of equations results, corresponding to Equations 15, 14,16, and 18 of Lee and Ark, respectively :
Equations 1 to 5 are implicit forms of the “performance equations” of Fan and Wang (1964). Since the total mass VOL. 7
NO. 4
OCTOBER 1 9 6 8
595
Table I.
Optimal Operating Conditions
Lee-Aris A2 A3
g1
9’2 g‘s
w1 w2 w3 f 3
0,652 0.363 4.145 2.460 2.3597 1.768 41,515 7,671 212 2.036
MalengtVillermaux 0.7237 0.4245 4.120 2.442 2.337 1,765 11,835 5,104 257 2,107
Present Calculation 0,7572 0.4870 3.455 2.450 2,372 2.040 11,796 4,917 2,059 2.122 I /
flow passes through stage 1, it is clear that the maximum of PI is really the desired result. The optimization problem then reduces to the choice of t, and e,,, n= N, N- 1, . . . , I , which maximizes PI: a direct application of the discrete maximum principle algorithm. As given by Equation 5, the profit at any stage is the profit which is realized in that stage plus the cumulative profit of the preceding stages times the appropriate weighting function. The reason for the failure of the dynamic programming approach is therefore apparent: One of the decisions to be made a t stage n-namely, the choice of the entrance temperature, determining fraction A, of the total flow going through stage n-influences directly the profit realized as the result of the preceding stages. The correctness of the proposed discrete maximum principle algorithm has been verified by solving the example treated in detail by Lee and Aris and checked by Malengt and Villermaux (6 = 0.03125; p = 0.15). The results of our solution are given in Table I. Thus the optimal profit of Malengt and Villermaux, substantially higher than that obtained by Lee and Ark, can be improved still further. But a more interesting result is that, although direct search allows improvement of the profit, the operation conditions that are obtained are still far removed from the optimal conditions, especially for the Nth stage (W3,t 3 ) . This is related to the shallow maxima that are encountered, due to the high relative value placed on the reaction conversion. Under optimal conditions the relative catalyst bed volumes (calculated by the maximum principle algorithm) are in the ratio 5.7:2.4:1.0, rather than 196:36:1 (Lee and Aris) or 46: 20: 1 (Malengt and Villermaux) (Table I). This condition is preferable from the designer’s viewpoint. In Figure 1 are represented the optimum operating conditions found by Lee and Aris, by Malengt and Villermaux, and in the present calculation. As seen in this figure, the major improvement in operating conditions is obtained in the lower inlet temperature in the primary stage, 370’ rather than437’ C.
596
l&EC
PROCESS DESIGN A N D DEVELOPMENT
0
I
2
3
4
I
I
I
5
6
7
I
Figure 1 .
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Optimum operation conditions - - Lee-Ark
-.-.-
Maleng&Villermaux Present calculation
Computational Procedure
The computation was done on an IBM 360, initiating the procedure by a guess a t the decision variables (Katz, 1962). For arbitrarily chosen values of the decision variables, not necessarily close to the optimum, the method of successive approximations converges rapidly to the solution (Demessemaekers and De Wilde, 1967; Grootjans and Vranckx, 1968) : Three iterations led to the given result. Calculation time depends merely on the accuracy in the two decision variables a t each stage; the values of t and g’ (determining e in turn) have been determined so that the error was less than 0.03 and 0.001, respectively. I n that case, calculation time was 22 minutes per iteration, 70 minutes for the entire three-stage problem. literature Cited
Demessemaekers, E., De Wilde, J., “Optimisering van een Adiabatische Multibed Reaktor,” Eindwerk, Universiteit Leuven, 1967. Fan, L. T., Wang, C. S., “Discrete Maximum Principle,” Wiley, New York, 1964. Grootjans, J., Vranckx, O., “Adiabatische Reaktor. Optimale Werking,” Eindwerk, Universiteit Leuven, 1968. Katz, S., Ind. Eng. Chem. Fundamentals 1,226 (1962). Lee, K. Y . , Aris, R., IND.ENG.CHEM.PROCESS DESIGN DEVELOP. 2, 300 (1963). Malengt, J.-P., Villermaux, J., IND.ENC. CHEM.PROCESS DESIGN DEVELOP. 6, 535 (1967). Leon G. Hellinckx Paul V. Van Rompay Instituut uoor Chemie-Ingenieurstechniek Universiteit Leuven Heverlee, Belgium
