COM M U N I CATION
APPLICATION OF T H E DISCRETE MAXIMUM PRINCIPLE TO CROSSCURRENT EXTRACTION W I T H A SIMPLE RECYCLE STREAM The algorithm of Fan and Wang which permits the application of the discrete maximum principle to multistage feedback processes i s applied to the problem of optimally allocating an extracting solvent to the various stages of a crosscurrent extraction unit with a simple recycle stream and a linear equilibrium relationship. The method gives the same solution as that obtained b y differential calculus and this solution agrees in the limit as the recycle rate goes to zero with solutions previously obtained for the no-recycle case, providing an interesting insight into the effectiveness of the method. Dynamic programming techniques are also applied to the problem, resulting in a similar answer.
4 previous note the discrete version of Pontryagin‘s maximum principle was applied to the problem of optimally allocating an extracting solvent to the various stages of a crosscurrent extraction unit ( 3 ) . Recently Fan and Wang have extended this principle to multistage feedback processes ( 7 , 2). In particular, they solved the problem of optimally allocating an extracting solvent to the various stages of a three-stage crosscurrent extraction unit with a recycle stream (Figure 1 ) with a markedly nonlinear equilibrium relationship. Since this equilibrium relationship is the one used in the previous application of the maximum principle to crosscurrent extraction Ivithout a recycle stream, the comparison of the two algorithms is particularly informative. This comparison can be further extended if the problem of optimally allocating solvent to a unit with recycle is solved for a linear equilibrium relationship. This note presents such a solution and comments on the Fan and LVang method.
I
x
Letting V , = M7,;’(Q equation becomes
+ R ) , a dimensionless flow term: this - x,
X,ll
ill
Vnyn
=
T h e material balance dn the recycle stream may be expressed as follows : Rxi
+
QQ
=
(Q
+ R)~.v+i
where R = flow rate of recycle stream Q = flow rate of feed stream a = concentration of solute in feed stream Letting R ! ( Q becomes
+ R ) = r and Q/’(Q + R ) rx1
+
qa =
=
q. this equation
(2)
X.v-1
Finally the equilibrium relationship is taken to be
(3)
x = cuy
Crosscurrent Extraction
T h e process of crosscurrent extraction with recycle can be described mathematically by three equations: two material balances and an equilibrium relationship. T h e material balance around the nth stage is:
The nth stage performance criterion, R,. is given by the following equation :
R,
V,(yn
=
-
xj
(4)
where X can be looked upon as the relative cost of the solvent to the solute. T h e object of the optimization is to select V,. ’\:
where ( Q
4R )
flow rate of immiscible solvent (constant) x = concentration of solute in raffinate y = concentration of solute in extract W’n = flow of solvent to nth stage =
n
=
I , 2,
,
,
,, .%’ in order
R,. the total return
to maximize n=l
or profit for the process. According to the method of Fan and Wane;. the optimum allocations at the nth stage, V,, must maximize or locate a stationary point of a Hamiltonian-type expression, H,, which can be expressed as follows :
Hn
= P n ~ n V i nf v n ( Y n
- A) = V n [ Y n P n t
( ~ n
X)I
where p, is the adjoint variable, defined by the following equation : Figure 1 .
384
Crosscurrent extraction unit with simple recycle stream
I&EC FUNDAMENTALS
T h e term a,, br,. by Equation 3, is equal to equation can be \vritten as follows:
a-l,
so the above
the material balance equation is repeated S times, it fol!ows that Xgt 1
S o w according to the algorithm of Fan and Wang, the boundary condition which serves to define p, is given as
p,\. - p"
bXi
p s - p. r
a r.y- ,
Ti,,*
=
if
bnpn
if
[ynpn
v,=v*
V, is given
+ 01,- A)] < 0 + - All > 0
LVith y n specified as a function of p n by Equation 6 and as a function of r n by Equation 3. V, is obtained from Equation 1: provided .'in+ I is known. T h a t is to say, Equation 6 provides the condition I', must satisfy if it is to lie %ithin its tolerable limits. ' r o illustrate how these equations allow optimal V,'s to be calculated. note that by Equation 1
-tV.,.,a) = ysa(1
+ v,,.:
(11)
(Y)S
This same equation can be obtained directly by means of differential calculus if so desired. T h e function to be maximized, P,may be expressed as follows :
P
+
= .u.\.(l
1
which is the fourth equation, allowhg the four variables to be obtained. If this solution is obtained for V,v>the following equation in V.b- results:
(yn
Lvhere V,z* and V,* are the lower and upper limits of V,, respectively. If. however. the expression tyicp7r ( y n - A)] = 0. I T n is not defined by this scheme. However, this means that the following equation is true:
Y,\.\.cl
XI(
= 0
Since H,, is linear in V n . the optimal allocation of by the following scheme :
Fn
=
+ V,' , a )
=
q(. -
XI)
- .VXV,.
using the fact that the V,L'sare all equal. T h e variable x 1 can be replaced by means of Equations 2 and 11. yielding as the function to be maximized the following :
Differentiation of this expression with respect to V,. gives precisely Equation 12. there is no recycle stream-Equation 12 If R = 0-i.e., reduces to the following: (1
+ Vs,'a) = ($.v+l
which is precisely the limit obtained in the optimization of the system without recycle ( 3 ) . Discussion
Then, since y., = X (p.v -f1). we have the equation I
xg,
Aa(1
, :=
+ V.v/(Y)
(p.,.
+ 1)
(7)
+ 1) it also holds that (p.,.- + 1) (p'v + 1)(1 + V v / a ) Since again by Equation 1 . r v = x.v- 1(1 + V,V- ~ / a ) (1 + V.,--l,o(). andys-1 = X/(p.v-~ + l),itfollowsthat
Since p.\--
1
=
+ V.,;
~~
pr
(Y
(p.\-
:=
1
=
Y.V-
(8) la
X
But since y,\. = X (p.,, t 1). Equation 9 shows V."-] = Vs. 'The argument can be extended to show that the V allocations a t every stage are equal---which is the same result derived for the case of no recycle ( 3 ) . iVith each V , equal to V.v, Equation 9 becomes, for the .\. - Ith stage: (ps-.i
+ 1)
=
(p,
~
+ 1)(1 + V.v!a)J
In particular. w h e n j = h', po
+1
= (p.3.
+ 1)(1 + V.v/a)s
Invoking now the boundary condition on the p variable allobvs p , to be eliminated from the above equation, resulting in the following equation : (7p.v
+ 1)
=
(ps
+ 1)(1 + V s / a ) - Y
T h e establishment by Fan and Wang of a computational algorithm for determining optimal solutions to feedback optimization problems is a notable advancement of particular interest to chemical engineers and those active in process optimization and control. In this note the method has been applied in some detail to a crosscurrent extraction problem of sufficient simplicity to allolv all of the salient points of the method to be clearly illustrated. T h e result is identical to the one obtained by the more familiar rules of differential calculus and checks, in the limit of zero recycle rate: with results obtained by the application of the discrete maximum principle. T h e same result could be obtained by dynamic programming if the problem were handled in the proper way. T h e application of dynamic programming to such recycle systems has begun to receive the attention of many workers in the field and it is interesting to note that the problem discussed here can be solved by the dynamic programming technique. This technique considers the problem to be a n .V-decision process, consisting of *V - 1 optimal decisions, Vz, V3, . . . V.,->and an optimum value assigned to XI: X I * . T h a t is? the dynamic programming algorithm is applied to the system with x 1 fixed at some value. and the optimal allocations of extracting solvent determined for stages 2 through AV for this value of xl. This strategy yields the return for the process as a function of x l . T h e procedure can be repeated for various values of Yl: until the optimum x1* is located. IVith the equilibrium relationship taken as linear. this procedure can be worked o u t analytically. as shown in the following section. If a variable S,, is defined as :
(10)
Equation 10, together with Equations 2 and 7, forms a set of three equations in four unknowns, X I : x,val? p , V , and V.,. If VOL.
3
NO.
4
NOVEMBER
1964
385
discussed in the maximum principle solution. T h e function can therefore be expressed in terms of X I , xJV+1, and V.%,:
a n d a further variable fn is defined as /,(XI, xn+ 1)
Max [&(XI, V Z , , . , , V,, x,+ ..., v n
=
f.V
111
v2,
n = 2, 3, . .
,,
h:
with/, = 0, then by the principal of optimality jn+l(,-cl,x n + d
=
Max
+
Ifn
v n +I
T h e maximum cyclic return, that SN* = Max
&il(~,+l,
Since V.V(X~/CY) = x z - X I , etc., this equation can be written as
vn+J1.
(13)
is obtained by selecting .TI so
&"y*,
(14)
[ f N ( x l , xn-+JI
Using now Equations 2, 11. and 15 to eliminate xrLl and Vv fr'om this equation givesf.\- as a function of y1 alone:
xi
For the example discussed here, the selection of x1 defines
V l by Equation 1 as
vi
=
.(5f
- 1)
Therefore, fl(x1, xp) is given by the following equation :
+
By Equation 1, x 2 = Xa/(l V,/CY),so that V Zis selected to satisfy the following condition !
X3
[.(1+;)
-
j)
T h a t is, holding X I and x3 constant, Vz is selected to maximize the term in curved brackets. If this term is differentiated with respect to Vz and set equal to zero, assuming this will give a n allowable value for Vg, the following equation results:
(
- ax3
ax1
1
(
vzyk-A)+[
+;
CY
Applying condition 14 to f.\- as expressed by Equation 16Le., differentiatingf.v \vith respect to Y I and setting the resultant expression equal to zero-gives precisely Equation 12 in terms of V.Y and demonstrates the utility of dynamic programming as well as the maximum principle in solving stagelvise optimization problems. In more complicated situations than the one considered here, either dynamic programming or the maximum principle can be applied. T h e question as to which provides a more effective method of solution is a n extremely interesting one, ivhich will no doubt receive considerable attention in the near future. On the basis of the computations made in this example, both provide workable algorithms to the problem. Dynamic programming adjusts its computational algorithm by means of the celebrated and somewhat intuitive principle of optimality. The maximum principle method, on the other hand: is somewhat less intuitive and enters the cyclic nature of the process into its algorithm by means of boundary conditions on the adjoint variable. Fan and Wang have given general rules governing these boundary conditions, and: in this case, their method is quite workable.
x3 .;,-A]-
I+literature Cited
which when simplified yields
+
But xz/xl = 1 VJa. Hence Equation 15 shows V Z = VI. Similar arguments show all the V's to be equal, a fact already
386
I&EC FUNDAMENTALS
(1) Fan, L. T., Wang, C. S., J . Electron. Control (April 1964). (2) Fan, L. T., Wang, C. S., J . Soc. Znd. Appl. .Math. 12, No. 1 (March 1964). (3) Zahradnik, R. L., Archer, D. H., IND.ENG. CHEM.FUNDAM E N T A L S 2, 238 (1963). R. L. ZAHRADKIK D . H. A R C H E R Westinghouse Electric Gorp. Pittsburgh, Pa. RECEIVED for review June 26, 1964 ACCEPTED September 10, 1964
