# (500b) Application of Trust Region Methods for Optimal Heat Exchanger Network Synthesis – a Hybrid Optimization Strategy

- Conference: AIChE Annual Meeting
- Year: 2019
- Proceeding: 2019 AIChE Annual Meeting
- Group: Computing and Systems Technology Division
- Session:
- Time: Wednesday, November 13, 2019 - 12:49pm-1:08pm

Most recent approaches to HENS attempt to convexify the nonlinear terms in the MINLP or look to develop new algorithms to find global solutions to the problem. These approaches fail to recognize that in doing so, the network designs may no longer be physically realizable as the shortcut models used in the network synthesis become cruder approximations of the full model. Realizing the drawbacks of using only this MINLP formulation, with fixed heat transfer coefficients and the use of the LMTD approximation to calculate heat exchanger areas to find the solution to the HENS problem, Short et al. (2016) included a sub-optimization step where the heat balances from the MINLP are used to design exchangers using Bell-Delaware design equations to ensure feasible exchanger designs. The resulting areas, number of shells, pumping costs, and individual stream heat transfer coefficients, are then used to update the MINLP model using simple correction factors.

In NLP literature, trust region methods have been used to solve similar problems in the past through the use of trust region techniques, whereby flowsheets are optimized using combinations of reduced-order models, where derivative information can be obtained in combination with detailed, black-box models.

In this work we propose a new algorithmic approach for the design of optimal HENs through the use of a combined approach using trust region methods with an MINLP optimization to generate topologies.

*1. MINLP optimization*

In this stage, the network problem is formulated as MINLP model of Yee and Grossmann. The individual heat exchangers are modeled using shortcut models and we aim to minimize an objective function that includes fixed capital costs, area-related capital costs, as well as utility costs. In addition, an LMTD approximation is used to determine the area of each exchanger in combination with the total heat transfer of each match.

We ensure that in the MINLP formulation, parameters are chosen such that the objective function is underestimated in comparison to the solutions that we expect to find in the detailed, higher-order models. Once we have a solution to the MINLP, we then send this solution to an NLP sub-optimization. The MINLP solution serves as a current, best-estimate of the lower-bound to the objective function. This would be a guaranteed lower bound to the problem should the solution be a global one. However since we cannot guarantee global optimality, this will only act as a current lower bound in the algorithm.

*2. NLP sub-optimization*

The NLP sub-optimization, which is usually used only to optimize the flowrates and relax the non-isothermal mixing assumptions used in the MINLP, can now be set up with initializations from the MINLP with binary variables fixed. This model will act as the basis for a trust region optimization problem. We can also increase the complexity of the network NLP, in comparison to the MINLP, by including additional design variables such as numbers of tubes,shell diameter, pressure drops, etc., using reduced order representations. We are then able to use this reduced-order or â€œglass-boxâ€ NLP representation of the network in a trust region formulation with new, higher-order, black-box models for the individual heat exchanger models.

The high-fidelity individual heat exchangers are modeled based on a first-principles discrete models which captures the dynamics of heat transfer within the exchanger. We model each exchanger as a shell and tube heat exchanger and divide it into small elements. We reformulate the heat equation by discretizing the differential equation inside each element and relate the element size to the exchanger design variables like number of tubes, shell diameter etc.

We determine the number of baffles, in which fluid is placed in the tube-side, and the number of shells in advance based on a set of heuristics and if the model is infeasible we change these settings in a structured way. With the solutions to the reduced order modelâ€™s inputs and outputs to these heat balances using the first-principles model. We then use the trust region approach to find the solution to the full flowsheet problem.

By updating the reduced order model with derivative information and the solution to the true model, we are able to converge upon a solution that we will use as the current upper bound to the optimization problem. This solution cannot be proven to be globally optimal, but the solution to this problem will however be physically realizable and will contain detailed aspects of the network design.

*3. Generating new networks and termination criteria*

After obtaining the optimal solution for the network topology, we then remove that specific combination of matches from the MINLP optimization. We do this because we assume that the NLP solution is the global solution for that specific topology. We re-run the MINLP optimization to generate a new network topology and then use the NLP trust region approach again in order to optimize the selected flowsheet with fixed topology.

If we obtain a better MINLP solution for the lower bound, due to a local solution at a previous iteration, then we keep this as a new lower bound to the full problem. If we obtain a lower objective function in the NLP sub-optimization (the TR model) then we keep this new solution as an improvement and make this the new upper bound. We therefore propose a stopping criteria whereby if the solution to the MINLP solution of a particular iteration is worse than the current upper bound, we terminate, with the current upper bound as the optimal solution.

We compare this work to previous work by Short, et al. (2016) and amend their work as well with the addition of the detailed exchanger NLP formulation presented here, as opposed to the heuristic designs they employed. This will not only allow for a fair comparison of the previously-proposed approach, but also extend their work with an extension that allows for that algorithm to be automated.

**References**

1. Short, M., Isafiade, A.J., Fraser, D.M., Kravanja, Z. (2016). Synthesis of heat exchanger networks using mathematical programming and heuristics in a two-step optimization procedure with detailed exchanger design. *Chem. Eng. Sci., 144*, 372-385.

1. Yee T., Grossmann I. (1990). Simultaneous optimization models for heat integration. II. Heat exchanger network synthesis. *Comp and Chem. Eng., 14,* 1165-1184.

3. Eason, J., Biegler, L.T., (2016). A trust region filter method for glass box/black box optimization. *AIChE J*., 62 (9), 3124-3136.