Optimal Design of a Sequence of Adiabatic Reactors with Cold=Shot Cooling Jean P. Malenge’ and louis M. Vincent Centre de Cinitigue Physique et Chimique du C.N.R.S. 54-Villers-les-Nancy, France
A method which quickly solves the problem of the optimization of a sequence of adiabatic reactors with cold-shot cooling is presented. Results of a simulation which shows that the “optimum” reactor is too sensitive to perturbations a r e given.
T h i s paper and the following paper have three goals which, considering generality and importance, are, in the following order of import’ance: To be a closing remark on a topic which raised some controversy in the past, and give full det’ails on what we think is the correct met.hod To insist on the fact that the theoretical optimum operating point of a n installation may be unrealistic because of too much instability T o give a n example of the efficiency of the algebraic methods for solving optimization problems. The development of computers and the sophisticat’ion of optimization methods lead us to think that algebraic methods may be outdated for they need a lot of painful analytical calculus for each new application. K e agree that. there are many “handmade” calculus to apply algebraic methods, but we think that they are a good tool for engineers because they allow the solution of problems whose complexity would have exceeded the computing facilities, as big as they may be, that one has a t one’s disposal. Lee and h i s (1963) proposed a n algorithm based on dynamic programming to optimize a sequence of adiabatic reactors with cold-shot cooling. Maleng6 (1969) published a short critical review of the works of Bhandarkar and Sarsimh a n (1969), Buzzi Ferraris (196Sa,b), Hellinckx and Van Rompay (1968, 19$0), ilIaleng6 and Villermaux (1967) which followed that of Lee and .Iris (1963). In this paper, we present a method which quickly solves the problem, and we give the results of a simulation which shows t’liat this optimum reactor would be far t’oo sensitive to perturbations. To compare our results with those of preceding authors, we adopt t’he notations (Figure l ) , the objective function, and t’he kinetics of Lee and Aris (1963). The objective function is
where 6 and p are two economic parameters related to the prices of catalyst and heat; R,(g) is the expression of the rate of reaction proposed by Calderbank (1953).
Principle of the Method
Many workers have tackled this problem but none have proposed the very simple method of transforming this 2N variable problem into 2 5 problems of only one variable using the relationships obtained by putting to zero the first derivatives of the objective function with respect to the independent variables. The main interest in this transformation over the use of t,he direct search method is the reduction of the number of computations of t’he objective function to reach t’he optimum. This number can be evaluated as lo2.\- for a 2N variable problem-Le. nearly 11 days of computation time with a n I B l J 1800. Our method will need only 2K X 10’ computat,ions-Le., only 1 min. I n fact, we will show later that this limit could not be reached; one of the derivatives leads to a relationship depending on all the variables, so that the number of computations is ( 2 s - 1)101 X 101-i.e., some 8 min of computation time. Choice of t h e Independent Variables. T h e 2s variabIes needed to build and operate the reactor are: the lengths of each reactor (or, equivalent,ly, the weight of catalyst X); the t’emperature after preheating, tN; and = 1). TVS (1 - X i ) i = 2 , . , . S (for If those natural variables are choseii as independent (decision) variables, t’he expression of the derivat,ives is quite impossible t o use, and this is probably t’he explanation of why this method has never been used. K e choose as independent variables the conversion a t the output of each reactor (gi’, Let us write S),the Xiand 2 =, t ,X instead of the object,ivefu nctionF with respect to these decision variables ~
Necessary Conditions of Optimality. At the price of a small amount of analytical derivations and transforniations, one easily obtains the following 2 5 necessary conditions of optimality: (3)
1 Present address, Universitk de Nice, Pare Valrose, 06-Nice, France. To whom correspondence should be addressed.
Ind. Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 4, 1972
465
-
7
I
I
I
I Oj
0.0
Figure 1 . Sequence of reactors
I
I
I
I
0.2
0.3
MU
Figure 3. Variations of Fvs. p and 6
w
t
1
Find g,l
which
satisfier (4)
5
3.9
l-
3.5
0.1
0.0
0.2
03
MU
Figure 4. Variations of ti vs. p and 6
Use of These Conditions. Here we will show how t o use these relationships to find successively the 2 s variables by solving 2N problems of one variable. (a) Let us suppose that tN is known, we can find gN’ which satisfies Equation 5 with i = N : in this relationship gN‘ is the only unknown (gN = 0). Remembering that g and t are such that in any reactor we have
Flnd g; which
satfsfier ( 3 )
g
(
Print results
1
- g < = t - t(
(8)
It is therefore very easy to compute t”
= gN’
- gN
+
t N = gN’
+
tN
(b) From the knowledge of gN’]tN’ it is easy to find gN-l, which satisfy Equation 4 with i = h7 (c) We can now iterate to (a) and (b) with i = 4 - 1, N - 2.. 2 (d) Finally we can find 91‘ by using Equation 3. The only unused relation is Equation 7 which should be used to compute tN supposed known in (a). I n fact this relation is too complicated, and we have found tN using the Golden Search method. Figure 2 gives the flowchart based on this method. tN-l
Figure 2. Flowchart
(5)
Results
One can make some remarks about these relationships: Equation 3 has a n economic explanation: when, in the last reactor a rate of reaction] related to the price of the catalyst, is reached, it is not worth continuing reaction; Equation 4 means that between two consecutive reactors] the rates of reaction must be equal before and after mixing; Equation 6 is a particular case of Equation 5 for gN = 0, so that one can apply Equation 5 for i = 2 , . . AT instead of using Equation 6. 466 Ind. Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 4, 1972
Here we give some results obtained with the numerical values of the different parameters given by Lee and Aris (1963). These results are only a part of those which are presented by Vincent (1971). Optimum Reactor. Table I gives our results in comparison with those obtained by different authors in t h e c a s e N = 3. The numerical differences between our results and those of Hellincks and Van Rompay are not significant] but our method is nearly 400 times faster and this is very important for it allows us to use our method in post-optimal studies which would have been too time-consuming and which have never been made. The big difference between our results and those of the other authors is due to a wrong use of the dynamic program-
Table 1.
Optimal Operating Conditions for a Three-Stage Reactor Lee and Ark
M a l e n g i and Villerrnaux
Dynamic programming
Method
F
Hellincks and Van Rompay
Direct search
2.036
2.107
0.652 0.363 4.150 2.460 2.360 1.768 41 ,515 7,671 212 49,398
0.723 0,424 4.120 2,442 2.337 1.765 11,835 5,104 257 17,196
Discrete maximum principle 2.122 70 min IBRI 360
Computation time XZ A3
t3 Q1I
Qz’ 93’
w1 wz w3 WT
ming principle by Lee and Aris, and use of too slow a method on a small computer by MalengB and Villermaux whose purpose was only to show that the results of Lee and hris were false. Effect of Variations of 6 and p. Figure 3 gives the variations of F vs. p for two values of 6 for a three-bed reactor, and Figure 4 gives the corresponding variations of f3* and shows that these variations do not present any discontinuity [as suggested by Lee and hris (1963)l. These results have been obtained by applying our method with different values of p and 6, but they verify the theoretical relationship:
dF -
del
Derivation 2.122 10 see CII 10070 or 8 min I B M 1800
0.757 0.487 3.455 2.450 2.372 2.040 11,796 4,917 2,059 18,872
0.759 0.488 3.456 2.450 2.374 2.041 11 , 639 5,029 2,085 18,753
a small decrease of t 3 would be disastrous. If f3 = t3* - 3”C, we get F = 0.4 F*. Physically this means that reactor 3 would be extinguished. But as it would be impossible industrially to control t 3 within l°C, the result is that the theoretical optimal reactor is a practical nonsense.
= -Ah&
Variation of K. I n industrial practice, S = 2, 3, 4, 5 , b u t our method is so fast that we could not resist the temptation of optimizing a sequence of Ai = 2, 8 reactors (Figure 5), verifying a t the same time that the computation time is only proportional to N (Figure 6). Post Optimal Analysis. I n Table I, lve gave t h e values of the optimal natural variables A*, X3, t 3 , Wl, W?,and W3. B u t for different reasons these variables have not in actual plant exactly their optimal value, and this has led us to compute the effect on the decrease in F in the case of small errors (a few percent) of any one of these variables, the others being supposed exact. The results are shown in Figure 7, and it can be seen that
+
+
+
+
+
I
t I
I
I
1
2
3
4
200
Our results
I
I
I
5
6
7
I 8
1
2
3
4
5
6
7
8
N
Figure 6. Variations of computational time vs.
N
2.
w
2 I-
G m
1.
0
0.
1
-8%
I
I
-5%
-3%
I
I
l
l
4% 0 +1%
I +3x
I 6%
4%
N
Figure 5. Variations of Fvs. N
Figure 7. Variation of F vs. small change of the natural variables for a three-bed reactor Ind. Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 4, 1972
467
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)
