Ind. Eng. Chem. Res. 1992,31, 2804-2805
2804
Gray, D. F.; Watson, I. D. Heat of mixing of carbon tetrachloride with diethylether, dimethylsulfide, and pyridine. Aust. J . Chem. 1968, 21, 739. Hanson, D. 0.; Van Winkle, M. Relation of binary heats of mixing and distribution of ketone between phases in ketone-water solvent ternaries. J . Chem. Eng. Data 1960, 5, 30. Kojima, K.; Tochigi, K. Prediction of Vapor-Liquid Equilibrium by the ASOG Method; Elsevier: Amsterdam, 1979; p 11. Krug, J. (1985) In Gmehling, J.; Holderbaum, T. DECHEMA Chemistry Data Series; DECHEMA Frankfurt/Main, 1989; Vol. 111, Part 3, p 1630. Larkin, J. A. Thermodynamic properties of aqueous nonelectrolyte mixtures. 1. Excess enthalpy for water + ethanol at 298.15 to 383.15K. J . Chem. Thermodyn. 1975, 7, 137. Letcher, T. M.; Baxter, R. C. Excess enthalpies and excess volumes of (benzene or cyclohexane or n-hexane + an alkene or an alkyne) at 298.15K. J. Chem. Thermodyn. 1987, 19, 321. Marquardt, D. W. An algorithm for least-squares estimation of nonlinear parameters. J . SOC. Ind. Appl. Math. 1963, 11, 431. Mato, F.; Berrueta, J. Heat of mixing in exothermic systems. An. Quim. 1978, 74, 1290. McLure, I. A.; Rodringuez, A. T. Excess Functions for (n-alkanenitrile +n-alkane) liquid mixtures. 2. Excess enthalpies at 298.15K for propanenitrile and n-butanenitrile with some C5 to C14 n-alkanes. J. Chem. Therrnodyn. 1982, 14, 439. Moelwyn-Hughes, E. A.; Missen, R. W. Thermodynamic properties of MeOH-MeCl solutions. J . Phys. Chem. 1957, 61, 518.
Nagata, I.; Nagashima, M.; Kazuma, K.; Nakagawa, M. Heat of mixing for binary systems and data reduction based on a triplet model of Guggenheim. J . Chem. Eng. Jpn. 1975,8, 261. Nakanishi, K.; Touhara, H.; Watanabe, N. Associated solutions. I1 Heat of mixing of methanol with aliphatic amines. Bull. Chem. SOC. Jpn. 1970, 43, 2671. Nicolaides, G. L.; Eckert, C. A. Experimental heat of mixing of some miscible and partially miscible nonelectrolyte systems. J . Chem. Eng. Data 1978, 23, 152. Ramalho, R. S.; Ruel, M. Heat of mixing for ternary systems: nalkanes + two n-alcohols. Can. J . Chem. Eng. 1968, 46, 467. Roach, M.; Van Ness, H. C. Excess thermodynamic function for ternary systems. 10. HE and SE for ethanol/chloroform/1,4-Dioxane at 50 “C. J . Chem. Eng. Data 1984,29, 181. Schnaible, H. W.; Van Ness, H. C.; Smith, J. M. Heat of mixing of liquids. AIChE J . 1957, 3, 147. Siddigi, M. A.; Lucas, K. Excess enthalpy of the system chloroform + carbontetrachloride and a thermodynamic evaluation of its state dependence. Fluid Phase Equilib. 1984, 16, 87. Valero, J.; Gracia, M.; Gutierrez Losa, C. Excess enthalpies of some chloroalkane + n-alkane mixtures. J . Chem. Thermodyn. 1980, 12, 621.
Received for review May 21, 1992 Accepted September 15, 1992
CORRESPONDENCE Comments on “An MINLP Process Synthesizer for a Sequential Modular Simulator” Sir: In a recent paper, Diwekar et al. (1992) present an interesting MINLP process synthesizer used in conjunction with the ASPEN sequential process simulator. The authors apply the MINLP solver OA/ER/AP (Duran and Grossmann, 1986; Kocis and Grossmann, 1987, 1988; Viswanathan and Grossmann, 1990) coupled with the partitioning variable strategy of Floudas et al. (1989),viz., the GBD/OA/ER/AP algorithm, to two illustrative examples. The first example can be written as min y1 + 1 . 5 ~ 0~ . 5 + ~ x12 ~ xZ2
+
subject to
xq
+
+ ( x , - 2)2 I 0 2y, - X l I0
1 - y1 - x , IO
3y3 - x 1 Y1
x2
50
+ Y2 + Y3
2 1
= (0,1,0,1.0,0.0;2.5000). Since the optimum integer variables have the same values with both algorithms, the NLP subproblem was analytically solved, confiiing the answer obtained with the MSGA algorithm. Also, the necessary Kuhn-Tucker conditions for local constrained minimum (Vanderplats, 1984) are here verified, contrary to the reported solution. Thus, the reported solution is neither the global optimum nor a local optimum. It is possible that there is some typographical error in the constraints reported for this example, due to the robustness verified by the OA/ER/AP in solving MINLP problems for the global optimum. However, if such is not the case, then the reason for failure should be investigated. The application of the MSGA algorithm to the second example gave the same answer as the GBD/OA/ER/AP algorithm. Nomenclature MINLP = mixed-integer nonlinear programming NLP = nonlinear programming MSGA = MINLP Salcedo-GonCalves-Azevedo algorithm OA/ER/AP = outer approximation with equality relaxation and augmented penalty function algorithm GBD/OA/ER/AP = generalized Benders decomposition/ outer approximation with equality relaxation and augmented penalty function algorithm
Y E ~4113 where the original variables xl17x12,and ~ 1 have 3 been expressed in terms of the decision variables xl and x q . These authors present the global optimum as (y1,y2,y3,x1,x2;F) = (0,1,0,1.0,1.0;3.5000~, obtained by the Literature Cited application of the above-mentioned algorithms. The application of a different MINLP solver (the MSGA Diwekar, U. M.; Grossmann, I. E.; Rubin, E. E. An MINLP Process algorithm of Salcedo (1992) and Salcedo et al. (1990)) to Synthesizer for a Sequential Modular Simulator. Ind. Eng. Chem. this example gives the global optimum as ( ~ ~ ~ y ~ , y ~ , x ~ , x ~Res. r F J1992, 31 (l), 313-322. 0888-5885/92/2631-2804$03.00/0
0 1992 American Chemical Society
Ind. Eng. Chem. Res. 1992,31, 2805-2806 Duran, M. A.; Grossmann, I. E. A Mixed integer nonlinear programming approach for process systems synthesis. AZChE J . 1986,32 (4),592-606. Fluodas, C. A.; Aggarwal, A.; Ciric, A. R. Global optimum search for nonconvex NLP and MINLP problems. Comput. Chem. Eng. 1989,13 (lo),1117-1132. Kocis, G. R.; Grossmann, I. E. Relaxation strategy for the structural optimization of process flow sheets. Znd. Eng. Chem. Res. 1987, 26 (9),1869-1880. Kocis, G. R.; Grossman, I. E. Global optimization of nonconvex mixed-integer nonlinear programming (MINLP) problems in process synthesis. Znd. Eng. Chem. Res. 1988, 27, 1407-1421. Salcedo, R. Solving Nonconvex Nonlinear Programming and Mixed-Integer Nonlinear Programming Problems with Adaptive Random Search. Znd. Eng. Chem. Res. 1992, 31 (l),262-273.
2806
Salcedo, R.; Goncalves, M. J.; Fey0 de Azevedo, S. An improved random-search algorithm for nonlinear optimization. Comput. Chem. Eng. 1990,14 (lo),1111-1126. Vanderplaats, G. N. Numerical Optimization Techniques for Engineering Design-with Applications; McGraw-Hill: New York, 1984;pp 17-19. Viswanathan, J.; Grossmann, I. E. A Combined Penalty Function and Outer-Approximation Method for MINLP Optimization. Comput. Chem. Eng. 1990,14 (9),769.
R. L. Salcedo Centro de Engenharia QuEmica Instituto Nacional de Inuestigaqdo Cientifica Rua dos Bragas, 4099 Porto Codex, Portugal
Response to Comments on “An MINLP Process Synthesizer for a Sequential Modular Simulator” Sir: Professor Salcedo is correct in pointing out that there is an inconsistency in the illustrative example given on p 315 of our recent paper (Diwekar, U. M.; Grossmann, I. E.; Rubin, E. S. Ind. Eng. Chem. Res. 1992,31,313-322). The correct MINLP formulation for that example is as follows: minimize y l + 1.5(y2) 0.5(y3) x l l x12
+
+
+
subject to x l l - x12 = 0 x12 - x22 = 0 ~ 1 -3x2
+ ( x l - 2)’
0
~l - x2 yl
+ 4y2 I4
+ y2 + y3 I1
Y l , Y2, Y3 = 0, 1
The typographical errors were an incorrect sign in the third constraint and the exclusion of the eighth and ninth inequalities. It is for this reason that Professor Salcedo found a different solution. The optimum solution of the problem EBgiven above is indeed the one reported in Table I of the original paper: y l = 0, y2 = 1, y 3 = 0, xl = 1.0, x2 = 1.0, x l l = 1.0, x12 = 1.0, x13 = 0.0, F = 3.5 We regret the typographical errors and would like to thank Professor R. L. Salcedo for bringing them to our attention.
x13 I0 2(yl) - x l = 0
Urmila M. Diwekar, Ignacio E. Grossmann* Edward S. Rubin
1-yl -xl I O
Department of Engineering and Public Policy and Department of Chemical Engineering Carnegie Mellon University Pittsburgh, Pennsylvania 15213
3(y3) - x l - x2 I0 x2 - y 2 1 0
Comments on “Direct Oxidative Methane Conversion at Elevated Pressure and Moderate Temperatures” Sir: Walsh et al. (1992) reported that with direct oxidation of methane, product selectivity depended on residence time, temperature, and the catalyst. The main reactions are as follows:
- + - + - +
CHI + ‘/202 CH30H CH4 + O2
CHI + Y2O2 CH,
+ 202
(1)
CH20
H20
(2)
CO
2H20
(3)
C02
2Hz0
(1)
The percent conversion of methane and product distribution depend strongly on which of the four reactions is dominating. The percent conversion of oxygen in almost all the experiments reported (with the exception of run 6) oa8a-58a5192J 2631-2ao5$03.00JO
was loo%, which means that oxygen was the limiting reactant. The theoretical conversion of methane therefore lay between 2.0 and 38.7%. Since the conversion of oxygen was 100% at low and high residence times, the interpretation of data on product distribution would pose a problem. Maximum conversion of methane was achieved at a residence time of 0.2 s (runs 5 and 7). A longer residence time will only promote the coupling of radicals or reactions between products. CH3 + CH3 C2HG (5) CHSOH + CO
-
COZ + CH,
(6)
For the same reason, the effect of reaction temperature on product selectivity could not have been fully evaluated, as the conversion of oxygen was 100% at all the reaction temperatures considered. 1992 American Chemical Society
