## Abstract

The paper compares different metaheuristics for using heat exchangers as a benchmark to estimate the best design parameter values using optimization efficient algorithms. Many MATLAB algorithms are used in this study. Also, an engineering equation solver, which is commercial software, is used to solve the issue. The design calculates three variables, which are the length, and inner and outer pipe diameter of the heat exchanger. The results showed that the best algorithms are particle swarm optimization, and when using this algorithm, the optimal design of the double pipe heat exchanger is as follows: the pipe length is 5.6734·10^{−1} m, the pipe inner diameter is 8.0203·10^{−3} m, and the pipe outer diameter is 2.2439·10^{−2} m.

## 1 Introduction

The heat exchange between any two types of fluid is done by a device called a heat exchanger, and it is used in different applications. The improvements to the heat transfer processes can be classified into two main categories: active techniques and passive techniques. One of the passive techniques is the use of double pipes with helical coils.

The Double Pipe Heat Exchanger (DPHE) includes a couple of pipes. The two pipes are concentric. The double pipe heat exchanger is followed for low flow rate, excessive temperature, and excessive stress applications. These styles of heat exchangers have discovered their applications in heat recuperation processes, air conditioning and refrigeration structures, foods, and dairy operations. The DPHE is properly used for lots of non-stop structures having small to medium duties, and it is utilized in enterprise along with condensers for chemical techniques and cooling fluid techniques. A lot of studies have been executed on the layout and evaluation of a DPHE.

There are two main types of DPHE: counter-flow and parallel flow. The counter flow is the best design, and it has the optimum heat transfer coefficient (*h*), and it can heat or cool the systems according to the application. Figure 1 shows the location of the outlet and inlet of the two pipes. As it can be seen in this figure, the fluid flows in opposite directions to each other and achieves the maximum difference in temperature of the two fluids at the two ends of the pipe. The diagram shows the counter flow and will consider the cold fluid as fluid one and the hot fluid as fluid two. The cold temperature at the second outlet is T2out, and it can reach temperatures close to the first inlet time, T1in, and it is clear that this temperature is larger than the first outlet time, T1out. In this counter flow, the temperature of the outlet cold fluid can reach more than the temperature of the outlet hot fluid, but it cannot reach that in parallel.

When the inlets and outlets are on the same side, the heat exchanger is called the parallel flow heat exchanger, as it is shown in Fig. 2. The heat transfers and efficiency of counter flow DPHE are higher than parallel-flow DPHE. However, almost all applications that deal with high temperatures and pressures choose counter flow DPHE.

Optimization algorithms [1] have gained increasing importance in engineering design during the last decades because of their simplicity and rapidity in finding solutions. They have been used in robot design [2, 3], drilling performance modeling, and a variety of other scientific fields. Meta-heuristics do not obey any rules; they only obey the inspiration behind the algorithm. Researchers have been inspired by animal behaviors such as grey wolves and orcas to develop grey wolf optimization [4] and whale optimization algorithms [5]. The optimization of Fertilization Optimization (FO) algorithm [6] was inspired by the reproduction operation of mammalian animals. Dynamic Differential Annealed Optimization (DDAO) [7] stimulates the production of dual-phase steel. Impressively, even insects like ant lions have inspired the Ant Lion Optimization (ALO) algorithm [8]. The behavior of the flocks of birds and fishes was the inspiration engine behind the Particle Swarm Optimization (PSO) algorithm, while the bee colony was the inspiration engine behind the Artificial Bee Colony (ABC) [9]. The Flower Pollination Algorithm (FPA) was developed in response to the dominance of flowering plants [10]. All these algorithms have been used to solve different optimization problems in different fields, and one of those is heat exchanger design. The shell and tube optimal design of the heat exchanger was conducted by using particle warm optimization [11]. Also the shell and tube is designed by firefly optimization algorithms considering the economic criteria [12]. The optimal design for a polymer heat exchanger was studied considering the use of a multi-objective genetic algorithm [13]. Another usage for the genetic algorithm was introduced to design earth-to-air heat exchangers [14]. The Harmony search algorithm, which stimulates the jazz troy, has been used to solve the heat exchanger design problem [15]. There are three approaches to multi-objective optimization: I Reinforcement Learner Non-dominated Sorting Genetic Algorithm (NSGA-RL), II Non-dominated Sorting Genetic Algorithm (NSGA-II), and III Chaotic Non-dominated Sorting Genetic Algorithms (CNSGA). The basic aims are to maximize the thermal performance index and the Nusselt number and to reduce the Fanning friction factor to the ideal design of DPHE, with perforated baffles on the annulus side. Also, an analysis of the optimization strategies is done utilizing three performance measures, and the impact of maximizing the thermal performance index is shown [16]. A Tubular Exchanger Manufacturers Association (TEMA)-compliant Shell and Tube Heat Exchanger (STHE) design model with key restrictions was created. Mixed-Integer NonLinear Programing (MINLP) used 10 design variables (continuous, integer, discrete, binary, and type) [17].

In this work, a comparative study is presented to investigate the best optimization algorithm for the sake of solving the heat exchanger optimization problem. Seven meta-heuristics had a fair competition on the problem with the same run conditions and the host machine. The following PSO, ABC, Grey Wolf Optimization (GWO), DDAO, FPA, Whale Optimization Algorithm (WOA), and ALO have been employed in this study, and the comparison was in terms of the best solution (best), the worst solution (worst), the average solution (average) among independent runs, and the STandard Deviation (STD) for the solutions over the independent runs. Particle swarm optimization, whale optimization algorithm, and ant lion optimizer have the best performance among other meta-heuristic algorithms in terms of minimizing the objective function. However, particle swarm optimization had returned the most feasible solution that was used for further calculations.

Engineering Equation Solver (EES) is a strong tool for solving engineering issues and is comparatively easy to train. It is especially good at solving thermodynamic and heat transfer issues since it provides various built-in functions that cover thermodynamic and thermo-physical issues, so there is no need to look at tables. One main feature of EES is that it can deal with a system of synchronic equations, which is not easy to deal with in Excel Code [18].

## 2 Methodology

Using a numerical approach to investigate and optimize DPHE design by different methods including EES and MATLAB algorithms.

### 2.1 The test section

The DPHE is schematically represented in Fig. 3, the test section consists of a double pipe; the material of the pipe is copper and it has a length X1, inner diameter X2, and outer diameter X3. The working fluid is water- R134a, and the mass flow rate of the hot fluid was of varied from 0.0330 kg s^{−1}–0.260 kg s^{−1}.

### 2.2 Theory

*T*

_{ho},

*T*

_{hi}inlet and outlet temperature of the fluid.

#### 2.2.1 Parallel flow DPHE

*U*is the overall average heat transfer coefficient.

#### 2.2.2 Counter flow DPHE

From these equations it can be seen that the heat exchanger efficiency depends on many parameters includes: the temperature of the two fluids *T*_{h1},*T*_{h2}, *T*_{c1},*T*_{c2}; the surface area of heat transfers *A = π D L;* the area parameters are: *L* and *D*; fluid flow velocity and properties *V*, *ρ*, *C*_{p}; and thermal resistance between the two fluids *R* or *U*. In the following progresses some parameter and the calculation the others parameters to calculate the optimal performance of heat exchanger are fixed.

### 2.3 EES and MATLAB algorithms for DPHE design

The design procedure will be as follows:

Input parameters:

The temperature of the two fluids (

*T*_{h1}*, T*_{h2}*, T*_{c1}*, T*_{c2});Working fluids type and velocity (

*V, ρ, C*_{p});Thermal resistance or DPHE material;

Solutions algorithms:

ESS algorithm;

MATLAB algorithm;

Output parameters:

to calculate the optimal design of DPHE;

to calculate (

*L, D*and Nu).

Used these algorithms as a tool to design heat exchanger as follow:

#### 2.3.1 Cold fluid

Refrigerant R134a, by using condensation unit with capacity 1.6 kW and make the shell of the heat exchanger is the evaporator of the condensation unit to make constant wall temperature process. The temperature of evaporator is *Tc* = 5 °C was designed the capillary tube for that purpose.

*,*

*RE*is the refrigerant effect,

Parameters for calculate mass flow rate of refrigerant

I (A) | V (V) | cosϕ | η_{iso}% | η_{mech}% | T_{c} (^{o}C) | |

0.50 | 220 | 0.85 | 85 | 80 | 0.0041 | 5 |

*Q*

_{av}is used in the calculation from predict the hot and cold sides, as it was shown by Baba et al. [19]:

*A*

_{in}

*= πD*

_{in}

*L*; and Δ

*T*

_{LM}is the logarithmic mean temperature difference,

*h*

_{o}is the outside heat transfer coefficient can be calculated from the following Eq. (15),

#### 2.3.2 Hot fluid

*T*

_{hin}= 40,

*mh*= 0.033 (kg s

^{−1})–0.26 (kg s

^{−1}) the water flow rates. The rate of heat transfer for the hot fluid (water) flowing in the inner tube will be expressed as [21]:

*h*

_{in}, can be calculated from the following equation [20]:

*U*

_{in}, ho and

*h*

_{i}are the individual heat transfer convection coefficients of the fluids outside and inside the pipes respectively and

*k*is the thermal conductivity of the pipe wall. Equation (19) represents the Nusselt number, which is estimate based on thermal conductivity of hot water, the pipe side heat transfer coefficient, and the pipe diameter, [20].

## 3 Optimum design

In this section, a comparison had been performed among PSO, ABC, GWO, DDAO, FPA, WOA, ALO and EES optimization algorithms as it is shown in flow chart (Fig. 5). The run conditions were: population size 10; maximum number of iterations 100; 30 independent runs. Table 2 reveals that PSO, WOA, and ALO have equal performance on the heat exchanger design optimization problem.

Statistical results of the minimization of the heat exchanger problem

Best | Worst | Average | STD | |

PSO | 1.6055·10^{−04} | 1.6055·10^{−04} | 1.6055·10^{−04} | 0.0000·10^{+00} |

ABC | 1.6055·10^{−04} | 5.0981·10^{−03} | 8.0255·10^{−04} | 1.0464·10^{−03} |

GWO | 1.6055·10^{−04} | 2.3212·10^{−03} | 7.0471·10^{−04} | 6.0895·10^{−04} |

DDAO | 1.4821·10^{−02} | 2.4427·10^{+00} | 4.6434·10^{−01} | 4.9002·10^{−01} |

FPA | 1.6055·10^{−04} | 3.0137·10^{−03} | 4.3905·10^{−04} | 5.6742·10^{−04} |

WOA | 1.6055·10^{−04} | 1.6055·10^{−04} | 1.6055·10^{−04} | 0.0000·10^{+00} |

ALO | 1.6055·10^{−04} | 1.6055·10^{−04} | 1.6055·10^{−04} | 0.0000·10^{+00} |

## 4 Results and discussion

From the Nu-Re number relation in Figs 6 and 7 it is showed that the new algorithms (ESS, PSO) gave good performance in comparison with ideal Gnielinski's theoretical equation.

The results show that there is an excellent coincides between the results of theoretical and the results from ESS and other algorithms design, and this indicates that the design is the best for testing liquids under constant wall temperature.

### 4.1 The optimal solution results

PSO, WOA, and ALO have returned solutions with lower value of the cost function (objective function) with different collections of lengths, inner and outer diameter. The solution of the particle swarm optimization is more feasible than other solutions and it has been chosen because it gave the best results as it is shown in Table 3 and the cost function of PSO is steady performance and fast convergence as it is shown in Fig. 8. The parameters chosen for PSO solution is: *D*_{in} = 8.0203·10^{−3} (m); *D*_{out} = 2.2439·10^{−2} (m); *L*_{pipe} = 5.6734·10^{−1} (m).

Best solutions founded by the best algorithms

Algorithm | L_{pipe} (m) | D_{in} (m) | D_{out} (m) |

PSO | 5.6734·10^{−1} | 8.0203·10^{−3} | 2.2439·10^{−2} |

WOA | 6.0328·10^{−1} | 7.0691·10^{−3} | 1.9840·10^{−2} |

ALO | 4.1279·10^{−1} | 1.5000·10^{−2} | 3.2000·10^{−2} |

## 5 Conclusion

This research discussed the design of the DPHE to test fluids for heat transfer by using EES program and MATLAB algorithms. Using the property that the evaporation of liquids is under a constant temperature, this is used to obtain a constant temperature procedure using a mini-freezing system and design its parts. And there are excellent coincide between the empirical results and the results from algorithms design, and this indicates that the design is the best for testing liquids under constant wall temperature. The best MATLAB algorithms was PSO algorithm because it was perfect in solution (the value of worst, best and average solution are equal) with STD = 0, for that this algorithm advises to design DPHE. In addition to another good algorithm that maybe benefit for another types of heat exchanger like WOA, and ALO, etc.

## Acknowledgements

The research was supported by the Hungarian National Research, Development and Innovation Office under the project number K 134358, and by the NTP-SZKOLL-20-0022 identifier “Focus'21-Focus on community by developing digital competencies” project, supported by the Ministry of Human Resources and Human Resources Support Manager.

## References

- [2]↑
H. N. Ghafil and K. Jármai, “Kinematic-based structural optimization of robots,”

., vol. 14, no. 3, pp. 213–222, 2019.*Pollack Period* - [3]↑
S. Alsamia, D. S. Ibrahim, and H. N. Ghafil, “Optimization of drilling performance using various metaheuristics,”

., vol. 16, no. 2, pp. 80–85, 2021.*Pollack Period* - [4]↑
S. Mirjalili, S. M. Mirjalili, and A. Lewis, “Grey wolf optimizer,”

, vol. 69, pp. 46–61, 2014.*Adv. Eng. Softw.* - [5]↑
S. Mirjalili and A. Lewis, “The whale optimization algorithm,”

, vol. 95, pp. 51–67, 2016.*Adv. Eng. Softw.* - [6]↑
H. N. Ghafil, S. Alsamia, and K. Jármai, “Fertilization optimization algorithm on CEC2015 and large scale problems,”

., vol. 17, no. 1, pp. 24–29, 2022.*Pollack Period* - [7]↑
H. N. Ghafil and K. Jármai, “Dynamic differential annealed optimization: New metaheuristic optimization algorithm for engineering applications,”

, vol. 93, 2020, Paper no. 106392.*Appl. Soft Comput.* - [9]↑
H. N. Ghafil and K. Jármai, “Comparative study of particle swarm optimization and artificial bee colony algorithms,” MicroCAD International Multidisciplinary Scientific Conference, Miskolc-Egyetemváros, Hungary, September 5–6, 2018, pp. 1–6.

- [10]↑
X. S. Yang, “Flower pollination algorithm for global optimization,” in

, vol. 7445, J. Durand-Lose and N. Jonoska, Eds., Lecture Notes in Computer Science 2012, pp. 240–249.*Unconventional computation and natural computation* - [11]↑
V. K. Patel and R. V Rao, “Design optimization of shell-and-tube heat exchanger using particle swarm optimization technique,”

, vol. 30, nos 11–12, pp. 1417–1425, 2010.*Appl. Therm. Eng.* - [12]↑
D. K. Mohanty, “Application of firefly algorithm for design optimization of a shell and tube heat exchanger from economic point of view,”

, vol. 102, pp. 228–238, 2016.*Int. J. Therm. Sci.* - [13]↑
U. Han, H. Kang, H. Lim, J. Han, and H. Lee, “Development and design optimization of novel polymer heat exchanger using the multi-objective genetic algorithm,”

, vol. 144, 2019, Paper no. 118589.*Int. J. Heat Mass Transf.* - [14]↑
R. Kumar, A. R. Sinha, B. K. Singh, and U. Modhukalya, “A design optimization tool of earth-to-air heat exchanger using a genetic algorithm,”

, vol. 33, no. 10, pp. 2282–2288, 2008.*Renew. Energy* - [15]↑
M. Fesanghary, E. Damangir, and I. Soleimani, “Design optimization of shell and tube heat exchangers using global sensitivity analysis and harmony search algorithm,”

, vol. 29, nos 5–6, pp. 1026–1031, 2009.*Appl. Therm. Eng.* - [16]↑
A. B. Colaço, V. C. Mariani, M. R. Salem, and L. dos Santos Coelho, “Maximizing the thermal performance index applying evolutionary multi-objective optimization approaches for double pipe heat exchanger,”

, vol. 211, 2022, Paper no. 118504.*Appl. Therm. Eng.* - [17]↑
Ö. Aras and M. Bayramoğlu, “A MINLP study on shell and tube heat exchanger: Hybrid branch and bound/meta-heuristics approaches,”

, vol. 51, no. 43, pp. 14158–14170, 2012.*Ind. Eng. Chem. Res.* - [18]↑
L. J. Habeeb, A. A. Mohmmed, A. H. Askar, and H. M. Hussain,

. Publisher: Independently published, 2019.*How to Use Engineering Equation Solver (EES): Refrigeration and Heat Transfer Applications* - [19]↑
M. S. Baba, M. B. Rao, and A. V. S. R. Raju, “Experimental study of convective heat transfer in a finned tube counter flow heat exchanger with Fe3O4-water nanofluid,”

, vol. 8, no. 11, pp. 500–509, 2017.*Int. J. Mech. Eng. Technol.* - [21]↑
Y. Cengel, J. Cimbala, and R. Turner,

. McGraw Hill, 2012.*Fundamentals of Thermal-Fluid Sciences (SI Units)* - [22]↑
V. Gnielinski, “New equations for heat and mass transfer in turbulent pipe and channel flow,”

, vol. 16, no. 2, pp. 359–368, 1976.*Int. Chem. Eng.*