Optimization of heat transfer performance of a micro‐bare‐tube heat exchanger using a genetic algorithm

This study presents the optimization of the heat transfer coefficient of a micro bare tube heat exchanger. A physical and mathematical model of a micro bare tube heat exchanger was built in Matlab using the simplified ε‐number of transfer unit method. Using a temperature of minus 30°C and a 0.5–1.0 mm tube outer diameter, a 1.7–8.0 mm longitudinal tube pitch, a 1.7–5 mm transverse tube pitch, and a 1.0–5.0 m/s velocity at minimum free flow area, with carbon dioxide as the refrigerant, a comparative variation analysis was performed to optimize the heat transfer coefficient of a micro‐bare‐tube heat exchanger. The results demonstrate that the heat transfer coefficient increases as the inlet air velocity is increased from 1.0 to 5.0 m/s, with a final gain of 93.17%. The growth rate of the heat transfer coefficient steadily decreases with increasing inlet air velocity and gradually approaches zero. The effect of the gradual decrease of the tube outer diameter from 1.0 to 0.5 mm on the heat transfer coefficient is 22.52% greater than that of the gradual decrease of the transverse tube pitch from 2.2 to 1.7 mm. The study also carried out an optimization analysis on the distribution of four different variables in the heat exchanger. With the use of a genetic algorithm, the study found an optimal distribution to maximize the heat transfer coefficient. The following parameter values resulted in the maximum heat transfer coefficient: a maximum inlet air velocity of 5.0 m/s, a minimum tube outer diameter of 0.5 mm, a 1.7 mm longitudinal tube pitch, and a 1.7 mm transverse tube pitch.


| INTRODUCTION
A micro-bare-tube heat exchanger is a heat exchanger with a hydraulic diameter of less than 1 mm in the heat exchange channel, which realizes the energy transfer. It was first proposed by Tuckerman 1 in 1981. Due to the limited capacity of current heat exchangers, researchers have proposed different types of new heat exchangers that could provide higher heat exchange efficiencies, require less refrigerant charge, and combine the smallest possible mass with an efficient and compact volume, achieving energy savings and emission reductions.
Much research has been done searching for a way to improve the heat transfer performance of micro-baretube heat exchangers. Acosta et al. 2 found that existing correlations that hold for smooth channels of larger hydraulic diameter also hold for narrow channels. Lee et al. 3 explored the validity of classical correlations based on channels of regular size to predict thermal behavior in single-phase flow through rectangular microchannels. Numerical predictions obtained based on the classical continuum method were found to be in good agreement with experimental data (with an average deviation of 5%), indicating that traditional analytical methods can be employed to predict heat transfer behavior in microchannels of the dimensions considered. Basing his work on topological structures found in nature, Bejan 4 introduced a tree-shaped fractal flow channel structure into the microchannel heat exchanger and successfully applied this structure in a micro-bare-tube heat sink. Li et al. 5 conducted experiments on microchannel heat exchangers with nitrogen as the medium and proposed experimental correlations for calculating the flow resistance of semicircular cross-section channels. Zhou and colleagues 6,7 also used ε-number of transfer unit (NTU) method to predict the performance of "type I" and "type N" micro-bare tube evaporator and carried out experimental tests on air-cooled freezers with micro-bare-tube evaporator. The results showed that the defrosting problem of refrigerators and freezers could be effectively solved. It was also found to reduce the amount of R290 propane needed for air conditioning and heat pumps.
In this article, inlet air velocity, tube outer diameter, longitudinal tube pitch, and transverse tube pitch are used as variable parameters affecting the micro-bare-tube heat exchanger, with the goal of maximizing the heat transfer coefficient.
The ε-NTU method refers to the thermal efficiency-heat transfer unit number method， which calculates the heat exchange rate of a heat exchanger (especially a heat exchanger with countercurrent exchange) without a logarithmic mean temperature difference (LMTD). Sepehr Sanaye et al. 8 used several design parameters for the thermal model using the ε-NTU method. Bacellar et al. 9 used a computational fluid dynamics-based model to optimize the design of a micro-bare-tube heat exchanger with a tube outer diameter of less than 2 mm.
The ε in ε-NTU refers to the heat transfer efficiency, the ratio of the actual heat transfer rate to the theoretically possible maximum heat transfer rate, as shown in Equation (1′) In Equation (2), Q is the heat transfer rate, and (WCp)min refers to the heat capacity flow rate of the colder of the two fluids when energy is exchanged from the hot fluid to the cold fluid. This fluid may also be referred to as the minimum fluid in current micro-baretube heat exchangers, In Equation (3), NTU stands for the number of heat transfer units, and UA is the comprehensive heat transfer coefficient.
The relationship between ε and NTU will vary depending on the type of heat exchanger since a complex heat exchanger model is not used in this article. Only basic micro-bare-tube heat exchangers are considered, and the appropriate formula relating ε to NTU is shown in Equation (4).

| Theoretical model
Condensers and evaporators with micro bare tubes are modeled using the NTU method, with the following two assumptions: 1. The two-phase flow of evaporation and condensation is ignored. 2. Due to the small wall thickness of the stainless steel tube, its thermal resistance is ignored. The model is shown in Figure 1. u is the inlet air velocity on the face area, D o is the tube outer diameter, P t is the transverse tube pitch, P l is the longitudinal tube pitch, and N is the number of tubes. The computational domain is the two-dimensional crosssectional portion of the heat exchanger, assuming that any end effects are negligible. The inlet boundary has a uniform velocity and uniform temperature (−30°C), while the outlet boundary is at constant atmospheric pressure. The upper and lower boundaries are periodic, and the tube wall is maintained at a constant temperature. The ideal gas model is used for fluid properties, and the k-ε achievable model is used to evaluate turbulence. The model itself determines the air-side thermal resistance and hydraulic resistance, so no additional thermal resistance needs to be considered. With a constant wall temperature, the capacitance ratio yields C min /C max = 0, and the heat transfer coefficient can be easily calculated using the ε-NTU method. Assuming negligible local losses, the pressure drop is determined as the difference between the inlet and outlet static pressures, as shown in Equations (5)- (8).
The model is derived from an empirical formula. The calculation of the j factor and the f friction coefficient is inseparable from the empirical formula. Equations (9)-(12) must be used to obtain the j factor and the f friction coefficient: The values of c1-c13 are determined by the number of tubes, using the empirical formula of Daniel Bacellar 10 ; the values are shown in Tables 1 and 2. Air-side thermal resistance is modeled from the airflow through the bare tube. The Reynolds number is determined by Equation (13): where Re D is the Reynolds number outside the micro bare tube, ρ air is the air density, u air is the air velocity, and D out is the outer diameter of the bare tube. μ air represents the dynamic viscosity of air. The Nusselt number Nu D is determined by the Zukauskas correlation, as shown in Equation (14): With five or six rows of micro bare tubes in the flow direction, the calculated Reynolds number for outdoor F I G U R E 1 Micro-bare-tube heat exchanger and cross-section.
λ out is the heat transfer coefficient outside the bare tube and A out is the total external area of the bare tube. Ignoring the thermal resistance of the stainless steel tube, the thermal resistance inside the micro bare tube can also be calculated as follows: L tot is the total tube length, th is the stainless steel tube wall thickness, and λ in is the CO 2 heat transfer coefficient. The Reynolds number inside a miniature bare tube is about 161, so the Nusselt number for laminar hydraulic, dynamic, and thermally welldeveloped flow in a circular tube can be determined using Equation (17), which assumes a uniform wall temperature, The number of heat transfer units can be calculated using Equation (18): The heat exchanger efficiency is given from the previous formula (4), thus the maximum heat transfer rate can be determined using Equation (19): In addition, the heat transfer rate can be calculated from Equation (1) MAX In this study, the parameters deemed to have the greatest impact on the objective function were selected. Although this article only conducts numerical simulations, several key heat exchanger parameter values are chosen according to the following practical standards: The thickness of the tube wall is th=0.18 mm; the height of the face area of the heat exchanger is H = 0.15 m, the width of the face area of the heat exchanger is W = 0.14 m; the length of the heat exchanger in the airflow direction is L = 0.014 m. The pressure is the standard atmospheric pressure, the temperature is −30°C, and CO 2 is used as the refrigerant.

| The origin of genetic algorithms
Genetic algorithms are a new type of random search algorithm with strong adaptability. They attempt to imitate the phenomenon of "natural selection," whose theory as developed by Charles Darwin became a cornerstone of modern evolutionary biology. The basic notion in biology is that within a population of a species, variations in the gene pool result in some organisms having a competitive advantage over others. Those organisms with advantageous traits (more "fit") tend to survive and reproduce more than individuals without them. With a mechanism of genetic inheritance, whose theory was initially pioneered by Mendel, this can lead to a shift in the genetic makeup of the population over time. Since the environment can favor certain types of advantageous traits, it effectively "selects" which genes get passed on in greater numbers. Critically, this selection is called "natural" because no conscious thought or planning is performed to obtain the optimization. The genes of the fittest individuals, that is, those who do best in that particular environment, pass on to the ensuing generations. This optimization framework can be applied to engineering optimization problems. The biggest advantage of the genetic algorithm is not just that it can directly and stably find the best answer, but that it does so through the repeated elimination of wrong answers.

| The application of genetic algorithms
The origin of genetic algorithms can be traced back to the 1990s. Genetic algorithms show strong robustness in complex system optimization applications and are increasingly used in the study of micro-bare-tube heat exchangers. With the continuous development of computer science and technology, the application of genetic algorithms in the field of heat transfer design has become more and more extensive. Sanaye and Modarrespoor 11 took the tube diameter, tube length, number of tubes in the vertical and horizontal directions as multiple variables, set up two optimization objectives, and used a genetic algorithm to implement a multiobjective optimization on the heat exchanger. Long Huang 12 and others proposed a generalized model based on finite volume, which can further improve the heat transfer performance and material utilization of heat exchangers.

| PERFORMANCE OPTIMIZATION OF THE HEAT EXCHANGER
To improve the heat transfer performance of the heat exchanger, the heat exchanger is taken as the test subject, and CO 2 is used as the refrigerant. The number of heat transfer units method (NTU method) is used, with the goal of maximizing the heat transfer coefficient, and the genetic code is written in Matlab. The algorithm optimizes the parameters affecting the heat transfer performance of the micro-bare-tube heat exchanger.

| Selection of decision variables and constraints
The range of u is 1.0-5.0 m/s, the value of D o is 0.7 mm, the value of P t is 4.0 mm, and the value of P l is 2.0 mm. The thickness of the tube wall is th=0.18 mm; the height of the face area of the heat exchanger is H = 0.15 m, the width of the face area of the heat exchanger is W = 0.14 m; the length of the heat exchanger in the airflow direction is L = 0.014 m. The pressure is the standard atmospheric pressure, the temperature is −30°C, and the air density, air mass flow rate, airspecific heat capacity, air viscosity, and air-to-air heat transfer coefficient will all take values at the corresponding temperature and air pressure. CO 2 is taken as the refrigerant.

| Analysis of results on inlet air velocity
The optimization of a micro-bare-tube heat exchanger was carried out. The heat transfer coefficient is used as the objective function for the optimization calculation, as shown in Figure 2. It can be seen from the figure that the maximum heat transfer coefficient is obtained at the greatest velocity. It can also be observed that when other conditions remain unchanged, the greater the flow rate of the inlet air, the greater the heat transfer coefficient. When the velocity of the inlet air is increased from 1.0 to 5.0 m/s, the heat transfer coefficient increases from 182.5 to 352.5 W/(m 2 ·K), and the two are positively correlated. 40.32%. It can be seen that the heat transfer coefficient does not grow linearly, and the slope decreases steadily after a certain inlet air velocity. It is reasonable to assume that if the inlet air velocity continued to increase, at some point, the slope of the curve would become zero. There would be a critical inlet air velocity at which the heat transfer coefficient takes a maximum value and the heat transfer coefficient stabilizes. However, that critical inlet air velocity would be too impractical, so it will not be considered.
While keeping the inlet air velocity unchanged, the values of D o , P t , and P l were varied, while continuing to observe the effect of u on the heat transfer system. The result is shown in Figure 4.
In case 1, the range of u is 1.0-5.0 m/s, the value of D o is 0.7 mm, the value of P t is 4.0 mm, and the value of P l is 2.0 mm.
In case 2, the range of u is 1.0-5.0 m/s, the value of D o is 0.9 mm, the value of P t is 4.0 mm, and the value of P l is 2.0 mm.
In case 3, the range of u is 1.0-5.0 m/s, the value of D o is 0.5 mm, the value of P t is 4.0 mm, and the value of P l is 2.0 mm.
In case 4, the range of u is 1.0-5.0 m/s, the value of D o is 0.7 mm, the value of P t is 3.0 mm, and the value of P l is 2.0 mm.
In case 5, the range of u is 1.0-5.0 m/s, the value of D o is 0.7 mm, the value of P t is 4.0 mm, and the value of P l is 3.0 mm.
As can be seen from the figure, even if the values of D o , P t , and P l are changed, the basic trend of the effect of air velocity on the heat transfer coefficient remains the same. The greater the inlet air velocity, the greater the heat transfer coefficient, and the two are positively correlated.

| Selection of decision variables and constraints
The optimization scheme of the heat transfer coefficient with respect to the inlet air velocity is discussed above, but the inlet air velocity is only an external variable of the heat exchanger, and it is only a single variable. The effect of more complex variables on the heat transfer coefficient are presented next. Still with the goal of maximizing the heat transfer coefficient, a genetic algorithm is used in Matlab to determine the optimal values of the tube outer diameter, longitudinal tube pitch, and transverse tube pitch of the micro-bare-tube heat exchanger.
The value of u is 2.5 m/s, the range of D o is 0.5-1.0 mm, the range of P t is 1.7-8.0 mm, and the range of P l is 1.7-5.0 mm. The rest of the parameters remain the same.

| Analysis of results on the D o , P t , and P l
The tube outer diameter, the transverse tube pitch, and the longitudinal tube pitch are variables, which are jointly programmed into the genetic algorithm, and the heat transfer coefficient is used as the objective function for optimization calculation. The results are shown in Figure 5.
It can be seen from the figure that after 600 iterations the curve begins to level out, and the final heat transfer coefficient settles to 624.0 W/(m 2 ·K). At that point, the tube outer diameter takes its minimum value, and both the longitudinal tube pitch and transverse tube pitch take their minimum values. However, only the value of the optimal heat transfer F I G U R E 3 The ideal relationship between air velocity and heat transfer coefficient.
F I G U R E 4 Relationship between air velocity and heat transfer coefficient under different conditions. efficiency is determined, and it is not clear how the three parameters affect the trend of the heat transfer coefficient. In another simulation, a threedimensional plot of the longitudinal tube pitch and transverse tube pitch of the tubes and the heat transfer coefficient is obtained, as shown in Figure 6. It can be seen from the three-dimensional plot that the maximum value of 603.7 W/(m2·K) is obtained when the values of the vertical spacing and the horizontal spacing are simultaneously 1.7 mm, which is the minimum value of both. Only the longitudinal tube pitch and transverse tube pitch are used as independent variables. As the longitudinal tube pitch and transverse tube pitch decrease, the increase in the number of tubes increases the heat exchange surface area. Even though the tube outer diameter remains unchanged, the face area of the heat exchanger remains unchanged, the increase of the number still increases the overall cross-sectional face area, which results in a larger heat transfer coefficient.
Next, we consider the influence of the tube outer diameter and the d transverse tube pitch on the heat transfer coefficient, as shown in Figure 7.
It can be seen from Figure 7 that when the value of the tube's outer diameter is 0.5 mm and the value of the transverse tube pitch is 1.7 mm, both of which are the minimum values of each parameter within the limits we are considering, the maximum value found is 456.5 W/(m 2 ·K). Even if only the tube outer diameter and transverse tube pitch were to be used as independent variables, as the tube outer diameter and transverse tube pitch decreases, the air velocity in the tubes would correspondingly increase, and the increase in the number of tubes would increase the heat exchange surface area. Although the face area of the tubes becomes smaller, the increase in the number of tubes still increases the overall cross-sectional face area. At the same time, the increase in the air velocity increases the heat exchange efficiency, causing the heat transfer coefficient to increase even more. With a tube outer diameter of 0.5 mm and transverse tube pitch of 2.2 mm, the heat transfer coefficient is 392.2 W/(m2·K), the tube outer diameter is 1.0 mm, the horizontal distance of the tube is 1.7 mm, and the heat transfer coefficient is 328.6 W/(m2·K), and the univariate effect of tube outer diameter on heat transfer coefficient is 22.52% higher than that of tube outer diameter. Thus, the tube outer diameter has a much greater influence on the heat transfer coefficient.
The same method can also be used to analyze the effect of varying the tube outer diameter and the longitudinal tube pitch, shown in Figure 8.
The result is almost the same as when with the tube outer diameter and longitudinal tube pitch. A reduction in either the tube's outer diameter or the longitudinal tube pitch results in an increase in the heat transfer coefficient. In both situations, however, the tube's outer diameter has a greater influence on the F I G U R E 5 Genetic algorithm optimization results for D o , P t , and P l .
F I G U R E 6 Three-dimensional (3D) Plot of P t and P l versus heat transfer coefficient.
F I G U R E 7 Three-dimensional (3D) Plot of D o and P t versus heat transfer coefficient. heat transfer coefficient than either the longitudinal or transverse tube pitch.
The final optimization results are given in Figure 9. The optimal heat transfer coefficient is obtained with the maximum inlet air velocity and the minimum tube outer diameter, longitudinal or transverse tube pitch.

| CONCLUSION
In this article, a simplified ε-NTU method is used to build a mathematical model of a micro-bare-tube heat exchanger in Matlab. An optimization analysis utilizing a genetic algorithm found that increasing the inlet air velocity results in an increase in the heat transfer coefficient, but the gain in the heat transfer coefficient diminishes as the inlet air velocity continues to rise; the reduction of the tube outer diameter, the longitudinal tube pitch and transverse tube pitch within the range of the standard range lead to an increase in the heat transfer coefficient, with the tube outer diameter having the greatest influence; within the range of the standard range, the heat transfer coefficient is maximized at the maximum inlet air flow rate and the minimum outer tube diameter, longitudinal spacing and transverse spacing of the tubes. Colburn factor (-) N number of tube banks (-) P pressure (Pa) P l longitudinal tube pitch (mm) P t transverse tube pitch (mm)