COMMUNICATION
APPLICATION OF THE DISCRETE MAXIMUM PRINCIPLE TO CROSS-CURRENT EXTRACTION The discrete version of Pontryagin’s maximum principle is applied to the problem of optimally allocating and extracting solvent to the various stages of a cross-current extraction unit. This problem has been solved by other investigators using dynamic programming, and this note illustrates the differences in application between the two methods.
As OF THIS WRITING, Pontryagin’s maximum principle has had several applications to chemical engineering problems. In general, these applications have made use of the continuous version of this celebrated principle, which differs very little in practice from continuous dynamic programming, although conceptually they are quite distinct. Recently, however, the interest in the discrete version of the maximum principle has generated new enthusiasm for the optimization of stagewise processes. This enthusiasm has already led to refinements and extensions of the mathematical theory. The purpose or this note is to demonstrate the applicability of the method to engineering systems and to illustrate how this application differs from dynamic programming. The application of the maximum principle to the optimization of discrete automatic control systems was first announced by Rozonoer in 1959 (6). Later, Chang developed the basic algorithm of the discrete version of the principle with heuristic arguments based on the principle of optimality ( 2 ) . Recently, Katz formalized these arguments and demonstrated that the principle was a t least a necessary condition for the best control action of quite arbitrary nonlinear systems ( 5 ) . He also applied the method to the optimization of a cascade of stirred tank reactors ( 4 ) . Cross-Current Extraction
I n this note, the discrete version of the maximum principle is applied to the problem of allocating a n extracting solvent to the various stages of a cross-current extraction unit. This problem has been solved by Aris? Rudd, and Amundson ( 7 ) by dynamic programming methods for both a linear equilibrium relationship and for one given by Figure 1. Converse (3) has also examined this same problem with a perturbationsearch routine and obtained results similar to those obtained by dynamic programming. The process under consideration has been described in detail in both reports. A schematic diagram of the process is given in Figure 2. This process can be described mathematically by two equations: a material balance and an equilibrium relationship. The material balance around the nth stage is: 4(Xn+1
-
X,)
The equilibrium expression relating p to x may be expressed by stating y = I(.). .4 linear relationship, x = ay. where a is a n appropriate constant, and one given by Figure 1 will be considered. The nth stage performance criterion, R,,is given by:
R,
=
V&n - A )
(2)
where X can be looked upon as the relative cost of the solvent to the solute. Case 1-linear
Equilibrium
Equations 1 and 2 are taken as the system expressions, together with the equilibrium expression x = ap. According to the maximum principle, the optimum allocation at the nth stage, V,, must minimize a Hamiltonian-type expression H,,: Hn =
p n ~ n v n- V n ( y n
- A)
= V n bnZ‘n
-
(yn
- All
where P, is the adjoint vector defined by the following difference equation : (3)
Two points are made here. The first concerns the boundary conditions on the system of difference equations, 1 and 3, which are : x,%+I = a
Po
(.V
= total number of stages)
= 0
0.2
Y1 0 . 1
= W,yn
where q = flow rate of immiscible solvent (constant) x = concentration of solute in raffinate
y = concentration of solute in extract W , = flow of solvent to nth stage Letting V, = W7,/q ( V , = a dimensionless flow term), this equation becomes : Xn+l
238
I&EC
-
xn =
FUNDAMENTALS
Vnyn
(1)
0
0. 1 X-
Figure 1.
Equilibrium diagram
C
n t
Feed qi
I
N
’N+ 1
)...
.
Stage
q, XN
q t
Stage X”+
n
yN
V , = V,
if
[.YJ’~ - ( y r 2- XI1 > 0
V,
if
[y,P,
-
(y,, - A ) ]
