Pospisil, J., Ettel, V.,Chem.Prum.,7,244 (1957): Chem. Ahstr., 5 2 , 14561 (1958). Sadana. A. 3I.Ch.E. Thesis. Universitv of Delaware. Newark. 11e1.,’19ii. Sadana, A,, Katzer, J . R . , Znrl. Eng. Chem., Fundam., submitted for publication. Satterfield, C. X., ‘‘Mass Transfer in Heterogeneous Catalysis,” 1I.I.T.Press, Cambridge, Xass., 1970, pp 129-148. Schlatter, J. C., Klimish, R.L., Taylor, K. C., Scimce, 179, 789 ~
(1973).
Stein, G., Weiss, J., J. Chem.SOC.,3265 (1951). Turk, A,, Chem.Eng., 76 (No. 34), 70 (Sov 3, 1969). RECEIVED for review March 14, 1973 ACCEPTEDJune 18, 1973 The work upon which this publication is based was supported in part by funds provided by the Vnited States Ilepartment of Interior through the University of 1)elaware Water Resources Center as authorized under Water Resources Research Act of 1964, Public Law 88-370.
COMMUNICATIONS
Approximate Design of a Multi-Stage Plant
Differeniial Optimization A procedure i s described for using differential optimization to estimate the optimum number of stages in a multi-stage plant. Since the method involves first-order approximations of unknown accuracy, no generality i s claimed, but the method has worked well in a practical problem involving a single effect multi-stage flash desalination plant, which i s used as an example.
T h e discrete optimum principle (Wilde aiici Beightler, 1967) is useful in determining the optimal values of design variables if the number of stages in a system is specified. In many applications, however, the number of stages is a n important design variable. .ln example is in the design of any multi-stage chemical process, such as a desalination plant. *In infinite number of stages would be physically most efficient but’ there is a set-up cost associated with each stage. This note describes how differential optirnization may be used to develop an efficient ,algorithm to optimize t h e number of stage.: xhich should work for most, .but not all, designs. T o apply the method, arbitrarily choose the number of stages, S,and design the plant’ using the discrete optimum principle. The criterion given below can then be checked t o determine if the optimal plant has been found. If it has not been located, the criterion indicates whether S should be increa3ed or decreased. This niet’hod will locate the optimum number of stages more rapidly than the trial and error approach of designing plants of varying numbers of stages to locate the optimum. For esaniple, suppose one knew from prior experience t h a t t h e number of stages in the optimal plant is some\vhere betv-een 45 and 60. If one simply calculated the optimal cosi: of earl1 plant for various iiumbers of stages, the most efficient search procedure to use t o determine the optimal nuniber of stages would be baied on a Fibonacci search procedure, and six designs n-ould be required ( the olitinial costs for alternative numbers of stages are unimodal with respect .to the number of stages). Using the Bolzano procedure, 01: ly four designs would be required. This can be an appreciable saving if each design involves extensive computer time (including optimizing all design variables). -1 by-product of the present method is that the sensitivity of the design to a change in the number of stages is given, S O t h a t one can estimate what a plant, wit’h one more or one less number of stage:; will cost, without actually designing the plant. The optimal values of the design variables for this
alternative design can also be estimated to a first-order allproximation. .\ssunie that the plant can be described hy the following transition functions (using t,he notation of Kilde and Beightler)
i
SnLl,i= T,,(S,,d,)
=
0 , . . . . P; n
=
I,. .
, )
.Y
(I)
+
where S,and d, are P 1 and Q dimensioiial column vectors, re,qiectirely. T R ois the stagewise objective function, which \ d l be assumed to he the total cost through stage 7 1 , l.;sume :ilso the constraint5
SI,* 2 Ai
s,v+l,i= B ,
i i
=
1 , ..
=
1,..
., P ,R
.
Ivhere A i and Bi are coilstants. These may be production requirements, or other constrairits on the syst,em. S o t all derision variables may affect the Rtate variables, S n i ,.. . , S n p . Let d n q , (I = 1 , . , , ! Q be those decisioii variables that i = 1 , . . . , P. occur iit the traiisitioil fuiictions Tni. If the plant were to be reduced by one st:ige, the values of the design variables in each of the remaining stages m u 4 be changed so that the constraints 2 viill he met. ‘The size of the changes required can be approximated to a first degree as discussed below. The change in the cost in t,he 71th stage due t o i t change in the decision variable d,, is, to a first-order approximation, (bT,o’3d,,)Adnq. If S is large so that t,he change in each decisioii variable, Adnq, is small, or if T,o is not too iionliiiear, this will be a good aliprosimation. The total change in the cost of the illant, taking into account the change of each of the decision variables d,,, n = 1 , . . , S - 1 : q = 1, . . . ,Q; and elirninat’ingstage S will then be
.v--l Q dTno --
Ad,,
- [cost of stage S)
(3)
n = q~ = b ~dnq
Ind. Eng. Chem. Process Des. Develop., Vol. 1 2 , No. 4, 1973
481
Since cost is to be minimized, the number of stages should be decreased if (3) is negative or increased if (3) is positive. If required, a neiv value, N’, is selected, the discrete optimum principle is used to design the plant, and the criterion 3 is rechecked to determine if the optimal plant has been found. The changes Adnq, q = 1,., ., &; n = 1,. . ,, A‘ - 1 are chosen to minimize the cost increase in stages 1 , .. . , N - 1 mhile satisfying the constraint,s 2. The change in S n + ~ ,due i to a change in d,, is, using a Taylor series first-order approximation, (bTnilbdnq)Adnq.If one st,age is eliminated the remaining stages must be altered so that the constraints are still met. Assuming t h a t the S t h stage is on the average equal to all ot,lier stages mit’h respect bo the changes in the state variables across it,, then SN+l,i- S x i 2 (Ai - B i ) / S . To meet the constraints, i t is required that
From ( 3 ) , the Adna should be chosen to .V-1 Q dT min on Adnq n =1 Q =1 bdn,
TnOISn, d,) is the total cost through stage n and is a function of all the state and decision variables. S,I is the amount of product mater leaving the nth stage and Sn2 is the temperature at the tube-side stream leaving the nth stage. A41 is the production requirement. In this case, d l- B1 = Z , = ~ ~ ( a d , l )If. dnl is approximately equal for all n and if the plant consists of N - 1 stages, d l - BI = (S - l ) a d n l . Equation 9 is then
If aTno,lddnlis approximately equal for each stage, the criterion 6 for determining the optimal number of stages can be written as
subject to (4). This is a standard linear programming problem, so that the Adnp can be optimally chosen rrithout difficulty. The values of the decision variables, ( d n g Adnp), then approximate the optimal values for the decision variables in the new plant. If all stages are similar in design, as is usually t’he case, then (dTni/bd,,) Ad,, will be approximately equal for all values of n. In this case, ( 3 ) , (4),and ( 5 ) can be written as
+
decrease N if (dT,O,lbdnl)dnl
< cost of
stage N
This criterion was used to determine the optimal number of stages in the desalination plant. The result was verified by direct search. Such success should be attainable in moat engineering design situations, although the unknown accuracy of the approximations precludes any general guarantee of convergence. Acknowledgments
This research was supported by the Office of Saline Water, Grant No. 14-0-1-0001-699. Dr. Beamer was a National Science Foundation graduate fellow while working on this research.
If & = P
=
Liferature Cited
1
Beamer, J. H., Wilde, D. J., Desalination, 9, 259 (1971). Wilde, D. J., Beightler, C. S., “Foundations of Optimization,” Prentice-Hall, Englewood Cliffs,N.J., 1967, Chapter 9.
An application of this technique has been made to the design of desalination plants (Beamer and Kilde, 1971). The problem in simplified form is formulated as follows S ~ + I=, OTno(Sn, d n ) n = 1, Sn+1?1= Snl - ad,] Sn+1,2
482
= Sn1
3
+ bdnl
Ind. Eng. Chem. Process Des. Develop., Vol. 12, No. 4, 1973
JOHS H. BEAMER DOUGLASS J. WILDE* Mechanical Engineering Department Stanford Cnicersify Stanford, Calijornia 94506 RECEIVED for review March 17, 1972 ACcEPTEDJune27, 1973
