(371d) An Alternative Superstructure and Solution Strategies for Global Optimization of Heat Exchanger Networks

The objective of this work is to present an alternative superstructure to be used in global optimization of heat exchanger networks (HEN). In addition several solution strategies are proposed.They rely on solving a single multilevel lower-bounding convex MINLP (LB-MINLP) to obtain lower bound tight enough to close the gap between the original nonconvex MINLP and its convex relaxation to a specified tolerance (ε_r = 1 %).

The stage-wise HEN superstructure proposed by Yee and Grossman (1990) offers a relatively large number of possible matches among the process streams of which only a small fraction is selected in the resulting HEN. In addition, inherently embedded in Yee's HEN superstructure is a problem of an almost exponential increase in the number of nonconvex terms as the size of the problem increases. To overcome the before mentioned limitations, a reformulation of Yee's HEN superstructure is proposed, which now serves exclusively as a matrix for determining thermodynamically feasible matches among hot streams and cold streams. Only the existing matches (heat loads, temperature differences) are then assigned to a staggered substructure comprising N possible heat exchangers, where N is a number slightly larger than the theoretically minimal number of heat exchangers. To properly and efficiently assign the heat loads and temperature differences to the staggered substructure all of the original big-M formulations of logical conditions were replaced by convex hull formulations and additional equations and logical constraints were added to the model.

The nonconvex terms are underestimated by piecewise linear underestimators and piecewise convex nonlinear underestimators (Quesada and Grossmann, 1995, Zamora and Grossmann, 1997). To tighten the convex relaxation the partitioning of the domains of all the variables involved in the nonconvex terms is performed. The tightness of the lower bound increases with the increasing number of domain partitions (sub-regions), however, since the underestimators in each of the sub-regions are (de)activated by a corresponding binary variable an increase in the number of sub-regions also increases computational effort needed to solve the convex relaxation. To keep the number of binaries as low as possible throughout the solution procedure a multilevel LB-MINLP is formulated and solved. For illustrative purposes basic steps of one of the proposed solution strategies are presented.

1. First level

Define O sub-regions and solve the convex LB-MINLP to optimality. Calculate the upper bound for a given optimal solution of the LB-MINLP by solving the original nonconvex MINLP at fixed (LB-MINLP optimal) topology of HEN. Check the convergence criterion. If satisfied, stop the procedure. The solution obtained is globally optimal. If not, continue to the next level.

2. Second level

Partition only the sub-regions which are active in the optimal solution of the LB-MINLP at the previous level to P sub-regions and solve the convex LB-MINLP to optimality. Calculate the upper bound for a given optimal solution of the LB-MINLP by solving the original nonconvex MINLP at fixed (LB-MINLP optimal) topology of HEN. Check the convergence criterion. If satisfied, stop the procedure. The obtained solution is globally optimal. If not, continue to the next level.

The mathematical model and the solution strategies were implemented in a MINLP process synthesizer MIPSYN a successor of PROSYN (Kravanja and Grossmann, 1993). Illustrative examples will be given to demonstrate the proposed approach.

REFERENCES:

Yee, T.F., Grossmann I. E. (1990). Simultaneous optimization models for heat integration II, Computers and Chemical Engineering, 14, 1165–1184.

Quesada, I., Grossmann, I. E. (1995). A global optimization algorithm for linear fractional and bilinear programs, Journal of Global Optimization, 6, 39–76.

Zamora, J. M., Grossmann, I. E. (1997). A comprehensive global optimization approach for the synthesis of heat exchanger networks with no stream splits, Computers and Chemical Engineering, 21, 65–70.

Kravanja, Z., Grossmann, I. E. (1993). Prosyn - an automated topology and parameter process synthesizer, Computers and Chemical Engineering,17, 87–94.