## 1 Introduction
The results showed that entropy generation in heat pipe decreased when nanofluids were using working fluids instead of base fluid. Figure 1: Schematic of an air-to-air heat exchanger core with cross-corrugated triangular ducts. The design of air-to-air heat exchanger involves many geometric and operating variables. These geometric and operating variables are a part of the search for an exchanger geometry that meets the heat duty requirement and a given set of design constraints. The conventional optimization techniques become very cumbersome and laborious for optimizing the design parameter of the heat exchanger. In recent decades, many researchers have applied intelligent algorithms for optimization of air-to-air heat exchangers. [PERSON] et al. (2018) used a multi-objective genetic algorithm to obtain the optimal values of the pitch and height of undulated plate and the height of the corrugated plate by using the Pareto optimal strategy. In their studies, the heat transfer capability and pumping power were the objective functions. [PERSON] and [PERSON] (2019) investigated the optimum design of the heat recovery steam generator using the genetic algorithm method under the fixed total volume condition. The total entropy generation was considered as the objective function. [PERSON] and [PERSON] (2019) demonstrated the successful application of the genetic algorithm combined with back propagation neural networks for the optimal design of plate-fin heat exchangers. Several investigators used other intelligent algorithms like the biogeography-based optimization algorithm ([PERSON] et al., 2019), adaptive simulated annealing algorithm ([PERSON] et al., 2019), bees algorithm ([PERSON] et al., 2019), cuckoo search algorithm ([PERSON] et al., 2019) and the Jaya algorithm ([PERSON] et al., 2019), to optimize the heat exchangers. The particle swarm optimization (PSO) algorithm is a relative recent heuristic search method. It is based on the idea of collaborative behavior and swarming in biological populations. PSO is a population-based search approach, and depends on information sharing among their population members to enhance their process using a combination of deterministic and probabilistic rule. Recently, some researchers have used the PSO for heat exchanger and thermal system optimization. [PERSON] and [PERSON] (2019) used the PSO to solve optimization problems for building design and control. [PERSON] and [PERSON] (2019); [PERSON] and [PERSON] (2019) applied the PSO to optimize the cross-flow plate-fin heat exchanger and shell-and-tube heat exchangers. The results demonstrated successful application of PSO techniques for the thermodynamic optimization of heat exchangers. [PERSON] et al. (2019) investigated the optimal design of a compact heat exchanger using PSO. Numerical results indicated that the PSO generated the optimum configuration with higher accuracy and a higher success rate. [PERSON] et al. (2019) employed PSO to determine the optimal micro-finned geometric which maximizes the thermal efficiency of micro-finned double pipe heat exchangers. From the references mentioned above, there are few studies on optimizing the design of air-to-air heat exchangers with cross-corrugated triangular ducts using the PSO. In the present work, the mathematical model of air-to-air heat exchangers with cross-corrugated triangular ducts is developed. The entropy generation rate \(\dot{S}_{\text{gen}}\), and total annual cost \(C_{\text{total}}\) are considered as the objection functions by single objective and multi-objective optimization with specified mass flow rate under a given space. To guide the design of air-to-air heat exchangers with cross-corrugated triangular ducts, the \(j\) factor, \(f\) factor, and \(JF\) factor are defined as objective functions.