## 1. Introduction

Natural gas is one of the most profitable fuels to replace conventional fuels, such as diesel and gasoline, around the world [

1]. One of the reasons is a large number of reserves of fossil fuels around the world that are becoming considered as a suitable alternative in the industrial energy sector [

2]. All devices such as heat exchangers implicate energy losses during the process, which is equivalent to increasing the total operational costs and reducing the energy performance of the system. Therefore, improvement in the thermal performance of a gas engine was proposed by incorporating an exhaust gas heat recovery system using an organic Rankine cycle (ORC) [

3]. However, this proposed configuration requires a thermal oil circuit, in which the residual exhaust gas is used to evaporate the organic fluid with the help of a thermal oil that regulates the amount of heat transferred and decreases the ORC evaporation temperature.

The optimization of equipment used for waste heat recovery (WHR) has been studied by many researchers using different methods and formulations. Technical challenges, such as the high acquisition costs and entropy generation inside the heat exchanger, represent an improvement opportunity to increase ORC performance [

4]. In these cases, mathematical tools can be used to find the best solutions through a stochastic search according to the objective selected. Holland [

5] and De Jong [

6] introduced the concept of genetic algorithms in publications, although these were not applied to the heat transfer field of knowledge.

Several researchers have applied optimization techniques to design industrial equipment using thermodynamic and economic approaches, specifically in heat exchangers in the last years. Martin et al. [

7] used a dimensionless function proportional to the sum of annual investment costs and operating costs, where the minimum of this function made it possible to determine the optimal Reynolds number, which depends on the type of heat exchanger chosen. In other research, Niclout et al. [

8] describe an optimization problem in which objective functions, such as manufacturing cost and heat exchanger volume, as well as operating and manufacturing constraints were studied considering as decision variables the geometric parameters of the fins. To solve this problem, the author developed nonlinear programming of mixed integer numbers, like other options of the solution. Nevertheless, his work is limited by not considering the thermodynamic parameters involved in the system but only the geometric parameters, although this means a possible over-dimensioning of the equipment.

An economic point of view has been studied in heat exchanger design. Selbaş et al. [

9] used genetic algorithms to design a shell and tube heat exchanger that varied design variables, such as the outer tube diameter, the tube arrangement, and the outer shell diameter, among other geometric parameters to obtain the optimal heat transfer area for the desired configuration using the logarithmic media temperature difference (LMTD) method. Likewise, using genetic algorithms, Ozkol et al. [

10] determined the optimal geometry to obtain the minimum area and acquisition cost, considering the limits of performance and operation, in a specific thermal application. These formulations use similar principles to those used in this research, but they do not consider economic indicators and, moreover, operational limitations since they differ completely from this study.

Muralikrishna et al. [

11] proposed a methodology to determine the feasible range of a heat exchanger design based on pressure drop criteria. Although, this research does not give concrete results about the design and sizing of this equipment because it only developed a valid design range to determine important knowledge about the limitations for designing this equipment widely used in the industry.

These results indicate a global concern to improve the thermal performance of this equipment and, thus, the global efficiency of the systems. Jarzebski et al. [

12] calculated the minimum cost of the operation and maintenance of a plate heat exchanger based on constant geometry and variable flow of the working fluid, and a simple analytical correlation for the pressure drop is presented. In this case, the authors did not apply a genetic algorithm. On the other hand, Zhu et al. [

13] performed geometric optimization of a plate heat exchanger in a geothermal application through the use of iterative programming to determine the optimal configuration that satisfied the objective (minimum area). This study has concrete similarity to the current study because the decision variables are similar. However, the author did not perform multiobjective optimization, which restricts the analysis of their investigation. Ahmadi et al. [

14,

15] performed a multidimensional thermo-economic analysis using multiobjective optimization of an integrated biomass system using a genetic algorithm. The integrated system is an ORC system cooled through an absorption chiller, a hydrogen production unit, a domestic water heater, and a reverse osmosis desalination system. For this purpose, the authors used the design variables as selection criteria, while the total costs and energy efficiency indicators of the system were selected as objective functions. Additionally, a sensitivity analysis was performed to evaluate the effect of the decision variables on exergy destruction rate, CO

_{2} emissions, and energy efficiency, which is similar to the approach presented in this paper but with different objective functions.

Wang et al. [

16] performed multiobjective optimization of the ORC condenser using a genetic algorithm, where the pressure drop and the heat transfer area were minimized under constant heat transfer conditions, resulting in a series of optimal solutions presented in a Pareto frontier. However, the author did not base their optimization on a thermo-economic analysis, which indicates that the effect of equipment costs on the results obtained was not considered. Consequently, minimization of equipment costs and the entropy generation number, considering geometrical parameters of the parallel plate heat exchanger (ITC2) without its effect on the heat transfer capacity of the equipment and reducing system performance, had not been developed considering some energy indicators, such as global energy conversion efficiency and energy efficiency, which decrease due to the irreversibility of the process and limits its application and commercial feasibility [

17].

From these results and their limitations, the main contribution of this research is the multiobjective optimization of a plate heat exchanger, which was used to evaporate the organic fluid in an ORC as a WHR system from a natural gas engine using the NSGA-II genetic algorithm, and a detailed thermodynamic model of the heat exchanger. The most sensitive variables of the system are determined, and the optimal configuration is selected to obtain the minimum acquisition cost and entropy generation number. In addition, the influence of the design parameters of the thermodynamic cycle, such as the evaporator and condenser pinch temperatures, the turbine and pump efficiency, and different working fluids were studied.

## 5. Conclusions

This work presents a model to optimize a parallel plate heat exchanger (ITC2), which corresponds to the evaporator of a secondary thermal circuit, through the NSGA-II approach. The acquisition cost of the equipment and the entropy generated were selected as objective functions, whereas the geometric parameters of the exchanger were considered as decision variables. As a single parameter value cannot satisfy both functions to be optimized, a series of optimal points are presented in the form of the Pareto frontier, which represents the equilibrium curve between both functions. The working fluid has a determinant role in the performance of the simple ORC cycle, so it is necessary to select the organic fluid that provides more significant benefits to the system. A study on performance parameters of the system is carried out before varying the efficiency of both the turbine and the pump as well as the temperature pinches of both the evaporator and the condenser of the system. With this, we attempted to identify which organic fluid had the highest net power values, the highest absolute increase in the thermal efficiency, and highest overall exergetic efficiency. Effects of the decision criteria on the objective functions are also studied by means of a sensitivity analysis, which showed that the length between plates is the most promising criteria, since its increase causes an elevation in the costs of the equipment up to a maximum value of USD 18,000 and a decrease in the entropy generation number.

The results found that, when applying the methodology proposed for this evaporator design through multiobjective optimization and selecting the best configuration of the five possible solutions through the TOPSIS method, point D was the best solution according to the established criteria. It was possible to minimize the entropy generation number (NGE = 0.058) and the acquisition cost of the equipment (USD 10,385.55), with an inclination angle (20.44°), plate height (2070.32 mm), plate width (205.16 mm), length between plates (800.49 mm), and heat transfer area of 22.04 m^{2}. This guarantees that heat transferred from the thermal oil (Therminol 75) to the toluene is 693.87 kW, overall thermal motor efficiency of the ORC cycle is 41.6%, and pressure drop is 980.32 mbar, which is within the permissible backpressure range of the engine.

The results of optimization may vary with the change of some configurations in the genetic algorithm, for instance: the fraction of the population, the population number, and the number of allowed iterations. In this case, optimization was performed with a total of 200 particles in the iterative space fractionated, achieving a value of 0.16 for a total of 32 optimal points, as shown in the Pareto frontier. In conclusion, the optimal geometry will be different for each case because there are infinite combinations in the iteration space, and adjustments were made in the configuration of the optimization model.

This proposed methodology can be applied to the thermodynamic and economic optimization of plate heat exchangers in any type of heat recovery system with indirect evaporation of the organic fluid. This methodology is always more relevant for cases where there are limitations in heat source backpressure, such as industrial engines with medium and high exhaust gases temperatures, and is applicable in cases where ORC technology has not been widely applied commercially.