argument, the descending solids from A to B add gas to the leaving stream, Y . Since roughly the same amount of gas is gained at B as lost a t C, a material balance will show that flow X agrees with flow Y . But this is not the flow through the experimental section. Since Professor Happel used a large volumetric flow rate of gas compared to that of solid, only a small error is made in assuming that flow X or Y is the flow through the bed (maximum error of about 3% for his run No. 104). However, if we would have used this procedure with our system of fine particles, our error would have been ridiculously large. This is why we used a tracer technique (tagging the flowing molecules in the experimental section) to measure flow rates.
4. D r Happel's final question asks what provision was made to counter gas leakage with the solids (the equivalent of gas entrainment from A to B , or from C to D ) . We did not make any such provision, and we do expect that significant amounts of gas were so entrained. Note that equalizing pressures up and dowpstream does not guard against gas leakage. I t actually guarantees such leakage. We measured the actual gas flow within the experimental section by tracer methods.
Daizo Kunii Department of Chemical Engineering University of Tokyo Bunkyo-Ku, Tokyo, Japan
Minimization of Capital Investment for Batch Processes
SIR: We would like to add a few comments to a recent paper by Y. R . Loonkar and J. D. Robinson (1970), in which they deal with the following optimization problem: Minimize
592 Vo.65+ 582 Vo3' + 1200 v5* + 370 (V/81)022 + 250 (V/82)o.40 + 210 (V/~I~)O.~* + 250 (V/03)040+ 200 (V/83)'@ (1) subject to
50 5 V/(10
+ 81 + 8 2 + 8,)
or
500 V-'
+ 50 0iV-l + 50 8zV-l + 50 83V-'
S 1 (2)
As the authors correctly remark, this problem has a formulation suitable for geometric programming. However, they consider the degree of difficulty as being too high to allow an advantageous use of the latter technique. We would like to stress that recent developments in this field (Passy and Wilde 1969; Duffin, 1970; Duffin and Peterson, 1970; Hellinckx and Rijckaert 1970) have provided efficient algorithms for problems with larger degrees of difficulty. The degree of difficulty of the above problem is seven. In the terminology of Loonkar and Robinson, 1970, this corresponds to M + N - 1 and not to 2 M + N 2 as reported by them. We solved this problem on an IBM 360144 using the solution procedure presented by Hellinckx and Rijckaert (1970). The algorithm was started from six different arbitrary initial points. I n each case the algorithm converged to the optimal solution without any difficulty. The total computing time for the six runs was 1 min 5 secan average of 11 sec for a complete computation. In an appendix, the problem is reformulated and solved by geometric programming. Furthermore, one should notice that the objective function is expressed in the so-called posynomial form (sum of positive terms) and that also the constraint is in posynomial form. Hence, by a simple logarithmic transforma422
Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 3, 1971
tion, as used in geometric programming, one can prove that the necessary conditions for a minimum are also sufficient in this case and that this minimum must be a global minimum. Appendix
To solve the problem by geometric programming, a minor change in the formulation of the constraint (Equation 2) was made by introducing the inequality sign so that Equation 2 states that the production capacity must be 50 ft3/hr or more. It is clear that the minimal cost will occur at the lowest possible capacity (50 ft3/ hr) so that Equation 2 will be automatically tight a t the optimum. The constraints in the dual formulation are, if we use the notation of Passy and Wilde (1969) and Wilde and Beightler (1967): Normality conditionWO1
+
WO2
+ W03 + 0 0 4 + 0 0 5 + W06 + W 0 7 + WOE = 1 (3)
Orthogonality conditions-
0.65 wO1+ 0.39 woz + 0.52 W03 + 0.22 W04 + 0.40 W O +~ 0.62 W E + 0.40 W O + ~ 0.85 W O ~ W11 - W12 - W13 - W14 = 0 for 01: 0.22 wa, + w12 = 0 for 02: 0.40 W E - 0.62 W06 + W13 = 0 for 0 3 : 0.40 W O ~ 0.85 WOE + W14 = 0
for
v
Equilibrium conditions can be formulated as
m=O i = l
with
m=0
l = l
(4)
The vectors vd, d = 1,.. . , D (linearly independent solution vectors of the homogenized normality and orthogonality conditions) are: d YO 1
d = 1 2 3 4 5 6 7
1 1 -0.55 1 1 4.55 2.50
1 2 3 4 5 6 7
d YO2
d YO3
d PC.8
d YE
-1 0 0 0 0 0 0
0 -1 0 0 0
0 0 0 0 0 -4.55 0
0 0 1.55 0 0 0 -2.5
0 0
d VC€
0 0 -1 0 0 0 0
d Yo7
d YO8
d YII
d
d
Y12
Yl?
d VI4
0 0 0 -1 0 0 0
0 0 0 0 -1 0 0
0.26 0.13 -0.36 0.65 0.65 2.95 1.62
0 0 0 0 0 -1 0
0 0 0 0 0 0 -1
0 0 0 -0.40 -0.85 0 0
Solving Equations 3-6 gives the optimal dual solution: w01
w02
= 0.2966 O= ~ 0.0205
wm W
= 0.3461 = 0.0608
wgj
= 0.0242
=0.0804 woi = 0.0171 WOE = 0.1543 W I O = 0.6049
WE
WII wi2 w13 W I
= 0.4029 = 0.0045 = 0.0595 = ~ 0.1380
This means for the primal problem:
SIR: We would like to make the following remarks on the above correspondence by L. J. Hellinckx and M. J. Rijckaert: 1. They are correct and indeed the degree of difficulty for applying geometric programming to the general problem as defined in the original article is M + N - 1. 2 . The illustration was a simple example to demonstrate the formulation and solution to the problem presented. I n general, the degree of difficulty for real plants is much larger than seven (50 is not uncommon). Although
tl
=3
01 = 0.11
t? = 1 R? = 1.47
t j = 6 83 = 3.42
and I = 126.103. Substitution in Equation 1 of the optimal values given on page 627 of Loonkar and Robinson (1970) yields the same result for I . Litera t ure Cited
Duffin R. J., “Linearizing Geometric Programs,” S I A M Rev., 12 (2), 211-27 (1970). Duffin R . J., Peterson E. L., “Geometric Programming with Signomials,” Report 70-38, August 1970. Hellinckx L. J., Rijckaert M. J., “A Solution Procedure for Geometric Programming Problems with Degrees of Difficulty,” 7th Mathematical Programming Symposium, The Hague, 1970. Loonkar Y. R., Robinson J. D., “Minimization of Capital Investment for Batch Processes,” Ind. Eng. Chem. Proc. Des. Develop., 9 (4),625-9 (1970). Passy U., Wilde D. J., “Mass Action and Polynomial Optimization,” J . Eng. Math., 3 ( 4 ) , 325-35 (1969). Wilde D. J., Beightler C. S., “Foundations of Optimization,’’ Prentice Hall, Englewood Cliffs, N. J., 1967.
L . J . Hellinckx M . J . Rijckaert Chem. Eng. Dept. Kath. Univ. Leuven Belgium
algorithms for solution of such problems do exist utilizing geometric programming, the algorithm presented in the original article is simple and easy to program. I t requires but two single variable searches, one imbedded within another. This is true whether the degree of difficulty is seven or 70.
J . D. Robinson Y . R . Loonkar American Cyanamid Co. Wayne, N . J . 07470
Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 3, 1971 423
