t'
= defined by Equation 27
U
= upper limit on flow rate deviation = dimensionless distance
x
T
4
GREEKLETTERS cy = u' for increase in &l,t), a for decrease P = PI' for increase in J(l,t), 01 for decrease P1 = LIU 61' = U / L = dummy variable of integration ( = dummy argument of X 7 0 = dimensionless temperature for heat exchanger; dimensionless concentration for chemical reactor X = functional of r, defined by Equation 5
= dummy time variable of integration = transformed process output variable, Equation 2
defined by
literature Cited (1) Kamman, D. T., Koppel, L. B., IND. END. CHEM.FUNDAMENTALS
5. 208 (1966).
(2) Koppelii: B.,'Ibid.,'i, 131 (1962). (3) Zbid., 4, 269 (1965). (4) Zbid., 5 , 403 (1966). ( 5 ) Watts, R., Ph.D. thesis in mechanical eneineerine. Purdue
University, Lafayette, Ind., 1964.
.,
"I
RECEIVED for review July 17, 1965 ACCEPTEDApril 11, 1966
OPTIMUM CROSSCURRENT EXTRACTION WITH MISCIBLE SOLVENTS T. C . C H E N A N D N . H . C E A G L S K E Department of Chemical Engineering, University of Minnesota, Minneapolis, Minn. The optimal solvent distribution in staged crosscurrent extraction with miscible solvents is determined by a modified maximum principle algorithm. For the case of linear equilibria an analytical solution is developed which enables the amount of the second solvent to be chosen such that the total profit is maximized. Comparison of the numerical results with an earlier publication confirms the accuracy of the data obtained by a numerical technique.
Lee described the use of a gradient technique in obtaining the optimum solvent distribution in crosscurrent extraction with miscible solvents (4). The problem was solved numerically for both linear and nonlinear equilibrium relationships by the iterative method. For the case of linear equilibria we found, however, that an analytical solution is obtainable by the use of a modification of Pontryagin's maximum principle. This, resulting in an exact solution, once again demonstrates one of the advantages in using the maximum principle algorithm in solving multistage optimization problems ( 2 ) . The analytical solution can be reduced to that for the case of immiscible solvents and linear equilibria, obtained by Aris, Rudd, and Amundson using dynamic programming (7) and by Fan and Wang using the maximum principle (3). Numerical comparison confirms that Lee's computation was very accurate. ECENTLY,
given, the problem is maximization of a specified function of the final state
..
by an optimal choice of the decisions On, n = 1, 2, ., N . Here C is a constant column vector of s components and superscript T denotes the transpose. The objective function, J,in the form of Equation 3 is general, as various definitions of objective functions can be transformed into it by suitable definition of new state variable. According to a discrete version of the maximum principle ( 2 ) , if a set of maximizing decisions does exist and lie in the interior of the admissible domain, it can be found by the condition
A Modified Discrete Version of Maximum Principle
To begin with, let us first consider in general an optimization problem pertaining to a N-stage sequential process. The operation of the stages is characterized by the following set of difference equations: xn = fn(xn-*, On),
n = 1, 2,
. . .,N
(1)
where xn is the state vector of s-component quantities at the nth stage and On is the vector of d decisions that may be made for the operation of this stage. With the initial condition XO = a0 422
l&EC FUNDAMENTALS
where zn is a vector of s-component quantities defined by the adjoint equation
(2)
along with the final condition iP=C The proof of this statement can be found elsewhere-e.g., ( 2 ) . There are, however, many problems for which difference
equations characterizing the operation of stages are obtainable in the form hn(xn-l, xn,
en) = 0 ,
n = 1, 2,
,
. .,N
(7)
rather than in that of Equation 1. If we assume that Equation l is the solution for x n from Equation 7, it can be shown by the chain rule of calculus that
The N-stage extraction system under consideration is shown in Figure 1. A mixture containing partially miscible solvents A and B and solute C is being extracted by the addition of pure solvent B at each stage. We assume that the concentration and the flow rate of the mixture entering the first stage are known and the composition is such that it lies on the saturation curve in the raffinate side. If CRn,
AR" = concentration of C and A in raffinate, respec-
tively concentration of C and A in extract stream, respectively = inlet and outlet flow rate of raffinate stream, respectively = inlet and outlet flow rate of extract stream, respectively
CEn, A E n =
qn--l, q" and that the inv'erse of the matrices bhnlbxn,n = 1, 2, . . . , N, exist. Hence, what we usually do in this case is to solve Equations 8 and 9 for the matrix coefficients bfn+l/bxn and bp/dOn,which are then substituted into the optimal condition (Equation 4) and the aijjoint equation (Equation 5). However, if we define a set of new adjoint variables, En, by the linear transformation
wn,un
we have the following relationships from the material balances on the total flow rate, solute C, and solvent A around the nth stage, n = 1, 2, . , ., N
+ wn
qn-l
(14)
+ unCEn = QnARn+ unAEn
qn-lARn-'
(11)
+ un
= qnCRn
qn-ICRn-'
then the substitution of Equations 8, 9, and 10 into Equations 4, 5, and 6 yields
= qn
(15) (16)
If equilibrium is attained a t each stage, there is a set of three additional equations between the four quantities, C R n , CEn, A R n , and A E n , which may be obtained from the saturation curve and tie lines in the form : ARn = S(CRn)
and
(17)
CE"
= A!?(CRn)
(18)
AEn
= T(CRn)
(1 9 )
n = 0 , 1 , 2 , . . . ,N Thus we have six equations in the six variables CR", C E " , AR", qn, and un and the process is uniquely described if the distribution of wn is known for n = 1, 2, . . . , N. The problem here is maximization of the profit from the operation of the process, this being defined as the value of solute C recovered in the extract stream less the cost of solvent B used. Therefore, we are to maximize the objective function AEn,
As will be seen, use of the new optimal condition (Equation 11) and the new adjoint system (Equations 12 and 13) gives a solution with fewer ma thematical difficulties and computations whenever the stage equations are given in the form of Equation 7, which may not be easily rewritten in the form of Equation 1. The new equations involve only the partial derivatives of hn with respect to its arguments. Optimum Solvent Distribution in Crosscurrent Extraction with Miscible Solvents
N
=
To illustrate the use of the modified algorithm, let us consider the following extraction problem, which is slightly different from the one considered by Lee. We show below that Lee's problem can be easily recovered from the present one.
(qn-iCRn-i
n=l
- q"CR" -
W2
W"
WN
V'
V2
V"
VN
c:
c2E
c:
CEN
.
(20)
by an optimal choice of wn, n = 1, 2, . . ., N, where X is the relative cost of solvent B to solute C. To solve this problem by the maximum principle algorithm, it is convenient to redenote CRn, qn, and wn by x l n , xZn, and en,
W'
Figure 1
Awn)
Crosscurrent extraction with miscible solvents VOL. 5
NO. 3 A U G U S T 1 9 6 6
423
respectively, and eliminate the rest of variables from the six equations. We have then X*,
hl'(xn-1,
0") =
- Xl'XZ'
x1'-lxz"--I
hzn(Xn--l, x*,
e")
-
+ xzn + e")E(xln) = o
(X2n--l
- S(x1*)xZn (xZn-l + en) = 1, 2, . . ., N
(21)
= S(X1R-')X2n-1
q x l f l )
xZfl
n
= 0
(22)
with given initial values of x? and xz0, and N
J
=
- xl"xZn -
(xln-lx2"--l
n=l
(23)
he")
To put J into the form of Equation 3, we define a third state variable x3" by h3"(xn-l,
x",
Substituting Equations 30 and 31 into adjoint Equations 27 and 28, rearranging, and simplifying, yields
e")
=
+
- x3"
~3"-'
Equations 30, 32, 33, and 34 are to be solved simultaneously and, in general, this will result in a set of recursive equations in XI' and 51'. However, if the equilibrium relationships (Equations 17, 18, and 19) are linear and are of the following form $(XI")
- xl'x2' xe') = o (24)
(xln-lxzn-l
= a
+ 6x1'
(35)
E(x1') = gx1' T(xln)
=
(36) (37)
exl'
Taking x j = 0, we have simply N
J =
- x3"-')
( ~ 3 %
n-1
(25)
= xsN
the solution of the four simultaneous equations simplifies drastically. I t can be shown that Equations 33 and 34 reduce to
and hence the problem is equivalent to that of maximizing by an optimal choice of one-component decision vector en,, = 1 , 2 , . . . , N . From the comparison of Equations 3 and 25, we immediately see that
xgN
c=
[;I [p]
The optimum condition (Equation 30) can now be written (26)
=
gZln
Computing now the matrix coefficients bhn/bx"and b h n + l / b x f l from Equations 21, 22, and 24 and substituting them into the adjoint Equations 12 and 13 yields [XZ"
+
(xz'-1
-
xZfl
[xZ*] [Xl"
~ 3 '
=
[xZ']
+ 8.) +
XZ"
[S(xln)
+
+
-
xZfl
[S(xi")xz"]Zz'
+' +
[ ~ z ' ] b3" '
+
- -x
(40)
Xl'
and hence
[XI']
Substituting Equations 38 and 39 into Equation 41 gives (27) 'xlfl]
- E ( X l * ) l El' [ S ( X l " ) - T(x1")lbz' [Xl'l Z 3 ' [XI' - E(Xl"+')] i l * + 1 + [ S ( X l " ) - T ( x l n + 1 ) ] Z 2 ' + 1 + &3'+'
=
&' =
(28)
(2%
&'+1
where = d / d x l f l . These adjoint equations are valid for n = 1 to N - 1 and, for n = N , the right-hand sides are simply replaced by 0, 0 and 1, respectively, by virtue of Equation 26. It follows that ~ 3 ' = 1 for all n, n = 1, 2, . . . , N . O n the other hand, the optimum condition (Equation 11) gives the following relationship
+ T(x1")bzn + X
=
0,
n = 1, 2,
. . ., N
x 1'X
(30)
= ag
X l '
2'
n = 1,2,
and
E(x1") El'
=
.E'(Xl')] bl"
(x2fl-1
&in+'
+ + e*) ! P ( X ~ ~zz") I f
+ e&'
[L - $1 Xl'+l
..., N -
1
(42)
But, from Equation 32 and the linear equilibrium relationships, we also have
--
XZ'-~
xzn
-
+ 6x1') g(a + g(a
bxlfl-l)
- exln - exl'-l'
n = 1, 2,
. . .,N
(43)
Substituting this into Equation 42 and separating terms containing X I " from others, we obtain a set of difference equations in xln alone
g(a
+
bxlfl+l)
- exlfl+l
Moreover, by combining Equations 21 and 22, we have
n = 1, 2,
424
I&EC FUNDAMENTALS
. . ., N
(31)
which is valid from n = 1 to N - 1. For n = N , we have, by setting the right-hand sides of Equations 27 and 28 zero and making use of Equations 30, 31, 3 5 , 36, and 37.
Solving for ~1~ and z Z N ,substituting the results along with Equation 43 into Equation 40 for n = N , we obtain, after fairly complicated rearrangements,
g(a
+
bxl"+')
]
-
XlN+1
where
(47)
is a constant defined by
X~N+I
X1"+l
x g
=
( e- bg)
1+-
that its composition lies on the saturation curve and that its flow rate, xzo, and the solute concentration, xlO, are known. In practice, a feed containing only solvent A and solute C is usually fed to the first extractor, as in the problem considered by Lee, and hence its composition does not lie on the saturation curve. However, if the solute concentration in this feed is represented by XI' (and hence the concentration of solvent A by 1 X I ' ) and its feed rate by X Z ~ there , are definite relations between the four quantities, x l F , x z F , xlO, and xqo. To obtain these relationships, we may imagine that a hypothetical mixing tank is attached at the entrance point of the first extractor. The feed is first introduced into the mixing tank, in which pure solvent B of amount OF is also added, so that its composition is brought onto the saturation curve, where the concentration of solute C is x:. Material balances on the total flow rate, solute C, and solvent A then yield
-
Comparing Equations 44 and 47 we see that the validity of Equation 44 can be extended from n = 1 to N , with the understanding that xlN+1 is a known constant. The difference Equation 44 can be solved analytically and has general solution of the form g(a
+
bxl")
- ex1"
=
n
=
0, 1 , . . ., N , N
+1
Xln
(49) where a and p are two constants to be determined by the initial condition a t n = 0 and the final condition a t n = N 1 . We found then
+
a = -g ( a
+ bx+) -
exlo
(50)
x lo
x10
+.-
+ bxPfl) g(a + - ex+
g(a
X I ~ + ~
exlN+l
bxlo)
I'
N+1
(51)
Thus, by considering now a and /3 as known constants, xln is explicitly given by
Having obtained this, explicit expressions of xzn, follow immediately. They are x p x 20 xzn = -[a
ag
f
en, and J
w] e
from which values of xlO, xzO, and OF can be obtained in terms of .xlF and xzF. We then imagine that this raffinate stream, with flow rate xz0 and solute concentration x:, is fed into the first extractor. In actuality, the feed as well as the additional solvent B used in the imaginary tank is all directly fed into the first extractor. Therefore, the amount of solvent B used in the first stage is the sum of OF and 0' given by Equation 54. This fact leads Lee to the observation that "there is a fairly large drop in the amount of solvent used between the first and second stages and then it rises slowly from the second to the Arth stage." This extra amount of solvent B used, OF, which is independent of the number of extractors in the system, also gives rise to an additional cost, AB', which must be subtracted from the profit, J, given by Equation 55. For comparison and confirmation, Lee's numerical data are used in the computation :
bg
Table 1.
(53) n 1
J
=
ag2
[(I
-
-&)
2 (ag2
+ X(g - l ) ( e - b g ) ) -
3
Discussion and Numerical Comparison
We have thus far assumed that the mixture entering the first stage contains both solvents A and B and solute C such
0.12333 0.07829 0.04806
4 5 6
7 8
9 10
= 3 and 10 Solvent B Added
0.94161 0,86126 0.81460 N
1 2 3
Equation 54 shows that, unlike the case of immiscible solvents (3, 4), the optimum solvent distribution is not constant even though the equilibrium relationships are linear. On either increase or decrease monotonously, depending on the sign of constants in the expression.
Optimum Results for N Flow Rate of Concn. of Solute Rafinate N = 3
0.16078 0.13853 0.11855 0,10082 0.08528 0.07180 0,06020 0.05030 0.04190 0.03482 N 3 10 20
=
0.36664 0.30841 0.32104
10 1 .02081 0.97223 0.93237 0.89966 0.87282 0.85079 0.83271 0.81788 0.80570 0.79571
0.16026 0.08424 0.08752 0.09019 0.09239 0.0941 9
0.09568 0.09689 0.09789 0.09871 Pro@ J
0.11104 0.12240 0.12596 0.18525
m
VOL. 5
NO. 3 A U G U S T 1 9 6 6
425
e = 0.4 g = 1 . 6
a = 1.0
b = -1.4
A = 0.05
xi F = 0 . 2 xzF = 1 . 0
If now solvents A and B are immiscible, we have X Z " - I = XZ" for all n-Le., the flow rate of raffinate streams is constant. I t follows that (e
From these primary data, we obtain
0.2 xlo = - xZo = 1.08 1.08
a = 6.0
p
= (8.7973)m
Some final results computed from these data are listed in Table I for comparison. All computations check perfectly with Lee's results. This not only confirms the accuracy of Lee's computation but also indicates that the gradient technique is a promising optimization tool. Trivial extension of the above notion applies as well to the case when the initial feed consists of both solvents A and B and solute C but its composition does not lie on the saturation curve. By a set of material balances similar to Equations 56, 57, and 58 the values of xlo, X Z ~ and , OF can again be obtained explicitly in terms of the initial concentrations and the initial feed rate. Finally, it is interesting to observe how the solutions (Equations 52, 53, 54, and 55) reduce to that for the case of immiscible solvents and linear equilibria, obtained by Aris, Rudd, and Amundson using dynamic programming (7) and by Fan and Wang using the maximum principle ( 3 ) . When the equilibrium relationships are linear, we have from Equations 21 and 22 that a(xZn-l
- x Z n ) - (e - bg)(xZn-l - xZ n + 0") xln = 0 n = 1. 2,
= 0,
n = 1, 2, . . ,, N
(60)
But xln and 0" cannot vanish for all n and hence we must have
(e
1
eF = 0.08
- bg) x l n P
(59)
. . .,N
- bg)
= 0,
n = 1, 2,
. . ., N
(61)
Substituting Equation 61 into Equations 48, 50, and 51 yields
Equation 52 subsequently reduces to
which is the analytical solution obtained by the previously mentioned authors. Solutions for 0" and J simplify accordingly. Acknowledgment
The financial assistance of the National Science Foundation is gratefully acknowledged. literature Cited (1) Aris, R., Rudd, D. F., Amundson, N. R., Chem. Eng. Sci. 12,88 (1960).
(2j-Fan: L. T., Wang, C . S., "Discrete Maximum Principle," Wiley, New York, 1964. ( 3 ) Fan, L. T., Wang, C. S., IND.END.CHEM.FUNDAMENTALS 3, 38 (1964). (4) Lee, E.'%, Zbid., 3, 373 (1964). RECEIVED for review July 1, 1965 ACCEPTED December 20, 1965
EFFECT OF MASS TRANSFER ON T H E STABILITY OF MISCIBLE DISPLACEMENT FRONTS IN POROUS MEDIA DAVID 0 . COONEY
Department of Chemical Engineering, University of Wisconsin, Madison, Wis.
hydrodynamic instabilities which often occur during miscible displacement operations in porous media have recently become the subject of intensive research. Some operations in which such flow instabilities can seriously decrease process efficiencies are : secondary petroleum recovery operations, certain ion exclusion operations (Z), and, in general, most fixed-bed displacements in which the viscosities THE
426
l&EC FUNDAMENTALS
and/or densities of the displaced and displacing fluids are significantly different. It has recently been shown by Perrine (5,6), through use of a perturbation analysis of the equations governing fluid behavior in porous media, that longitudinal and lateral dispersion can exert a stabilizing influence on displacement fronts which are near the borderline of instability. The analysis of Perrine
