tN
Conclusion
I n this paper we have shown how a simple method quickly gives the optimum analytical solution for the design of a sequence of adiabatic packed bed reactors with cold-shot cooling [Vincent (1971) presents the adaptation of this method to the case of interstage cooling by exchangers]. But we have also shown that this solution was not recommendable. Nevertheless, with the help of Figure 7, the project engineer will be able to choose values of the variables near the optimum, and such errors of construction do not produce too important a fall in the economic criterion. I n another paper, we will show how, given a n existing sequence of reactors, one can optimize its working operation in spite of some perturbations. Nomenclature
F G g g1 gt’
N R, t t, t,’
objective function to be maximized total mass flow rate = conversion as defined by Lee and Aris (1963) = conversion in inlet stream to bed i = conversion in outlet stream from bed i = total number of beds = reaction rate in bed i = reduced temperature as defined by Lee and Aris (1963) = inlet stream temperature to bed i = outlet stream temperature from bed i = =
preheating temperature
=
W i = weight of catalyst in bed i = GA, 2
auxiliaryvariable = X N t N
=
GREEKLETTERS economical coefficient related to amortized cost of catalyst p = economical coefficient related to preheater cost X i = fraction of total mass flow through bed i 6
=
literature Cited
Bhandarkar, P. G., Narsimhan, G., Ind. Eng. Chem. Process Des. Develop., 8 , 142 (1969). Buzzi Ferraris, G., Ing. Chim. Ital., 4 , 5 (1968a). Buzzi Ferraris, G., ibid., 31 (196813). Calderbank, P. H., Chem. Eng. Progr., 49,585 (1953). Hellinckx, L. G., Van Rompay, P. V., Ind. Eng. Chem. Process Des. Develop., 7, 595 (1968). Hellinckx, L. G., Van Rompay, P. V., 97th event of the European Ami1 27-30, 1970, Federation of Chemical Engineering, -, Florence, Italy. Lee, K. Y . , Aris, R., Ind. Eng. Chem. Process Des. Develop., 2, 300 119631. MalengB, J. P., ibid., 8,596 (1969). MalengC, J. P., Villermaux, J., ibid., 6 , 535 (1967). Vincent, L. M., Thkse Docteur-Inghieur, Universite de Nancy, 1971. I
\
-
-
~
,
RECEIVED for review July 22, 1971 ACCEPTEDJune 15, 1972
Optimal Operating Conditions of a Sequence of Adiabatic Reactors with Cold-Shot Cooling Submitted to Perturbations Jean P. Malengel and louis M. Vincent Centre de Cinktique Physique et Chimique du C.N.R.S., 6+$-Villers-les-Nancy, France
The method previously presented for the optimization of the design of an adiabatic reactor with cold-shot cooling i s applied to the search of optimal operating conditions of this reactor.
I n a preceding paper (MalengB and Vincent, 1972), we described a simple and quick method for the optimum design of a sequence of adiabatic packed bed reactors with cold-shot cooling. The aim of this paper is to optimize its working operation. This study is necessary because of the possible variations in the numerical values of the parameters used for the optimal design-e.g., catalyst activity, composition, and the feed rate. I n the first part of this paper, modifications required for the use of our method for solving this new problem are presented; in the second part, some numerical results are given. Yotations are those of the preceding paper.
Method for Optimization of Working Operation
When the reactors are working, it is not practical to stop t h e production to change the catalyst weight, W t ,of each reactor, wherefrom the constraints
I n this expression, TVt has a fixed value which is the one obtained by t h e optimization a t design time. If one retains the same expression for the economic criterion as into the preceding paper, the application of the Lagrange method leads to the optimization of the function: N
Present address, Universite de Nice, Parc Valrose, 06-NiceJ France. To whom correspondence should be addressed. 1
468
Ind. Eng. Chern. Process Des. Develop., Vol. 1 1 , No.
4, 1972
F
=
Ggl‘
-~
Gz 6
C z=1
+
.v z= 1
ItF,
(2)
I
I Find
g;
1
whlch satlrfies
I
I
’
I
I
, I
I Compute the economic c d l r r h
1.5
I
-
I
/’ 8’
8
WITHOUT CONTROL
8
’a5 ‘OF
Figure 1. Flow diagram of the method
where I are Lagrange multipliers. The necessary conditions for this function to be optimal are obtained by putting to zero its first derivatives with respect t o t h e 2N chosen independent variables [(gi’ (i = 1 , . . N ) , X i (i = 2 . . . ,N-1), LN, ZN = X N t N ] and to the N Lagrange multipliers :
bF = 0 bg1’ ~
+ RI(S1’)
=
11
(3)
Figure 4. Variations in % of SOz
As we explained (1972) for the design study, the use of these relationships will markedly reduce the computation time for the search of optimum, as compared with the time needed by a direct search method on the set of variables. T h e flow chart given in Figure 1 shows the application in t h e case of a three-stage sequence; this flow chart can be simplified further as shown in Figure 2 to emphasize the arrangement of the searches of the different variables. Results
The results are given for a three-stage sequence. We have first computed the effect of each perturbation without T h e condition bF/btN = 0 gives a relationship similar to Equation 5 with g N = 0; so t h a t Equation 5 is valid for i = 2 to N . The condition bF/bz = 0 gives an unusable relation. T h e conditions bF/bXi = 0 give Equation 1. Flnd lg
Figure 2. Simplified flow diagram
Flnd 9;
30
I 40
I
:
50
PRODUCTION
I
I
60
70
I 1 0
(TIDAY)
Figure 5. Variations of production Ind. Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 4, 1972
469
any modification of the optimal working conditions; then, we have computed the maximal value of the economic criterion that could be reached by modification of the three remaining decision variables: preheater temperature output ( t 3 ) , bypass (A%
X3).
Results are presented by means of curves; one can find in Vincent’s thesis (1971) numerical values, along with results for other perturbations. Catalyst Activity Decay. T h e decay of the catalyst activity leads t o a modification of t h e mathematical expression for the reaction rate R(g,t). TT’hatever this modification, the work of optimization remains the same, our method not being based on t h e form of R. K e have chosen to simulate this decay by t h e simpler manner which consists of multiplying R by a coefficient a varying from 1 t o 0.5. Figure 3 shows that the economic criterion of the optimally designed sequence is very sensitive to catalyst decay; however, one ran maintain this criterion a t an acceptable level by modifying t3, and AS. Composition at Input. Reaction rate along the reactors depends on t h e composition of t h e feed mixture. Figure 4 shows t h e effect of a variation in t h e percentage of SOz. One can see that the adjustment of t 3 , Az, and A 3 is most
appreciable when the percentage of SO2 is decreasing (it has been supposed equal to 7.8 in this project). Total Flow Rate. T h e sequence of reactors has been optimized to produce 50 tons/day of sulfuric acid. Figure 5 shows the results obtained when the production varies from 25 to 75 tons ’day, with and without adjustment of t3, Al, and x3.
Conclusion
I n a preceding paper we showed that the optimal sequence of reactors was very sensitive to a small variation of preheating temperature (t3). In this paper, we have shown t h a t this sequence is also very sensitive to other perturbations. Severtheless, our method, slightly modified, allows the engineer t o control the reactors to obtain the maximum of the economic criterion, in spite of these perturbations. The method can be implemented on a small process computer. Literature Cited
llaleng6, J. P., Vincent, L. X, Ind. Eng. Chem. Process Des. Decelop., 11(4), 465 (1972). Vincent, L. M,, these Docteur-Inghieur, Universith de Nancy, 1971. RECEIVED for review July 22, 1971 ACCEPTED June 15, 1972
Separation of Binary Mixtures of CO and H, by Permeation Through Polymeric Films F. P. McCandless Department oj” Chemical Engineering, Montana State Tniversity, Bozeman, Jlont. 69715
The pure gas permeability coefficients of various polymeric materials to carbon monoxide and hydrogen were determined, and the most promising of these were tested as separation membranes using binary rnixtures of the gases. The effect of temperature, pressure, and feed composition on permeation rate and permeate compositions was determined. Polyimide, Dacron, Parylene C, and caprolactam films proved to be the most effective. Depending on permeator conditions, actual separation factors varying from 14 to about 70 and fluxes varying from 0.03 to 1.3 scfd/ft?were observed. The actual separation and flux were somewhat lower than would b e predicted on the basis of the pure gas studies, assuming Fick’s law. All materials tested were selective for hydrogen. Mathematical models of the two limiting cases of ideal permeation stages-i.e., no-mix and perfect-mix stages-were formulated. These models used the experimental flux and permeate compositions which were determined as a function of feed composition for various temperatures and pressures. These models were used to compare the surface area requirements and separation which would result for various stage cuts. In addition, a brief comparison of polyimide and Dacron was made for the two-stage separation of CO and Hz. This study showed that the membrane area requirements and the size of the interstage streams vary widely, depending on membrane material, operating temperature, and on the type of flow in the stage. A comparison of the models with a theoretical model assuming Fick’s law was also made.
T h e r e has been a growing interest the past few years in membrane processes as alternatives to conventional separation techniques. This interest continues to -grow because membrane process equipment and methods of operation are inherently much simpler than traditional separation methods. This is especially true of the separation of gaseous mixtures such as CO and HP, which require cryogenic temperatures 470 Ind.
Eng. Chem. Process Des. Develop., Vol. 1 1 ,
No. 4, 1972
when separated by condensation, distillatiou, or adsorption. Selective permeation has shown great promise for the separation of gases, but in the past it has not been economic because of the very large membrane surface areas required to carry out industrial-scale separatiorl (Feller and Steiner, 1950). HoJyever, recent advances in membrane technology
