Airside friction and heat transfer performance of micro‐bare‐tube heat exchangers with bundle diameters varying gradually from 1.0 to 0.4 mm

In this paper, a micro‐bare‐tube heat exchanger structure with the tube bundle diameter varying gradually in the range of 1.0–0.4 mm is proposed. The airside friction and heat transfer performance correlations for this structure are given. This paper applies CFD simulation to develop and verify the correlation of the diameter gradually varied micro‐bare‐tube heat exchangers' performance. Compared with the traditional same diameter micro‐bare‐tube heat exchanger, the new structure not only reduces the pressure drop, but also improves the thermal hydraulic performance. The friction coefficient f is decreased by 8.59%, the pressure drop is reduced by 43.14%, the heat transfer coefficient j is increased by 12.07% compared to the constant diameter micro‐bare‐tube heat exchanger. At the same time, it reduces the void volume by 47.32% and the tube bundle manufacturing material by 28.57%. The proposed airside performance prediction equation can make the error of more than 70% source data within 10%, and the error of more than 90% source data within 20%. It is suggested to apply this empirical equation to 7–29 rows.


| INTRODUCTION
At present, the rapid development of industry increases the already great demand for energy.Most of the current energy sources are still through traditional fossil energy, which can greatly pollute the environment.Under the environmental protection policy, improving energy efficiency has become an international consensus.The efficiency of the heat exchangers has a direct impact on energy consumption, thus, improving the heat exchange performance and extending its life is of great significance for protecting the environment and saving resources.With the increasing tension of energy problems, seeking more efficient heat exchange equipment and improving heat transfer efficiency has become an important issue in the energy field.The micro-bare-tube heat exchanger has the characteristics of high heat transfer efficiency, compact structure, and strong adaptability, which makes it have great application prospects in both the traditional air-conditioner and heat pump industry and the emerging aviation realm.
The micro-bare-tube heat exchanger performance depends highly on its geometric design, aiming to achieve high heat transfer coefficient and small drag coefficient.Therefore, the research of micro-bare-tube emerges rapidly all over the world.For instance, Sanave et al. 1 studied the influence of tube layout, diameter, length, number, plate spacing ratio and plate cutting ratio on heat recovery and cost of micro-bare-tube heat exchanger.Zhou et al. 2 reported that the amount of propane charged can be reduced by using the micro-bare-tube heat exchanger as the condenser and evaporator for air conditioners.Kasagi et al. 3 adopted heat transfer and pressure drop datasets of tube bundles to train neural networks to optimize pipe diameter, transverse and longitudinal pipe spacing.Zhou et al. 4 tested a modified freezer with a micro-bare-tube air-cooled evaporator.The performance of "I shape" and "N shape" micro-bare-tube evaporator was studied by theoretical model and experiment.Jin et al. 5 used EVAP-COND simulation to calculate the heat transfer characteristics of the evaporator, and compared the refrigerant flow, pressure drop and copper consumption of the evaporator before improvement.Huang et al. 6 conducted an experimental study on the thermal and hydraulic performance of bare tube heat exchangers under dry and humid conditions, and discussed the influences of inlet air humidity, flow rate and water flow rate on total heat transfer capacity, sensible heat, latent heat and pressure drop.Grimison et al. 7 and Zuukauskas et al. 8 proposed a relatively comprehensive experimental study on the correlation of heat exchanger heat transfer performance, including the subsequent development of empirical correlation.The correlation proposed by Zuukauskas et al. 8 is also the most commonly used in current research.Lee et al. 9 developed a thermal sizing software platform TSCON for condensing heat exchangers to assess condensing heat transfer dependencies.Taler et al. 10 combined a nonlinear regression method to determine the correlation coefficient that defines liquid and airside heat transfer.Bacellar et al. 11 proposed a framework to study the large design spaces of staggered bundles using automated CFD simulations, and to assess dimensionless thermo-hydraulic performance on the airside using pipe diameter from 0.5 to 2.0 mm.Bergelin et al. 12 reported pressure drop and heat transfer data for vertical tube rows in his study of tubular heat exchangers, providing a preliminary correlation between friction and heat transfer.Fujii et al. 13 numerically solved the two-dimensional (2D) Navier-Stokes equations and the energy equations in the continuous subregion, and proposed a relationship between the inline tube set and the parallel plate in terms of heat transfer characteristics, where the friction coefficient and the mean Nusselt number agree well with the published experimental results.Using the integral method of boundary layer analysis, Khan et al. 14 derived closedform expressions for calculating heat transfer and friction of parallel and staggered bundles, which can be used in a wide range of parameters.Most of the previous studies [15][16][17][18][19][20] considered improving the performance of heat exchangers by changing the number of tube rows and the size of tube spacing.There were few reports on the structure types that enhance the heat transfer effect by varying the diameter combination of tube bundles.Most studies calculated the performance of heat exchangers based on the same tube diameter.There is almost no clear evidence or equation to predict the hydrodynamic performance of heat exchanger under the structure of diameter gradually varied microbare-tube heat exchanger.
The diameters of the traditional copper tube heat exchangers in the front and back of the tube bundles were the same.In actual use, the refrigerant was in a two-phase flow state in most of the pipe range, and the temperature of the fluid in the entire pipe was unchanged, so the temperature reflected on the wall of the front and back pipes was the same.As the air flew through the copper tubes, the temperature difference in the front row was greater than that in the back row, so that the heat transfer in the back row was less than that in the front row.To solve this problem, it is necessary to increase the heat exchange temperature difference in the back row, but the temperature of the pipe wall in the back row cannot be changed, so the heat exchange temperature difference can only be increased by changing the air temperature flowing through the back row.To change the air temperature flowing through the rear, it is necessary to vary the diameter of the copper tube from the front row to the back row.
According to the above analysis, a micro-bare-tube heat exchanger with diameter gradually varied tube bundle is proposed in this paper.The first row uses a large diameter, the subsequent rows diameter decreases in turn.The pipe diameter in the middle reaches the minimum, and the subsequent rows of copper pipe diameter increases in turn.In this way, when the air flows through the front half structure to exchange heat with the copper tube, as the diameter of the pipe decreases, the heat transfer area decreases, and the temperature of the air decreases relatively slowly.The temperature difference between the air and each row of copper tubes is relatively close, and the heat exchange temperature difference is always high, which increases the heat transfer efficiency.This structure reduces the diameter of the pipe, can save the overall manufacturing material, while reducing the airflow flow resistance, also can obtain greater air volumetric flow rate at the same fan speed.
This study demonstrated the heat transfer performance of the gradually varied tube diameter micro-bare-tube heat exchanger.Empirical correlations were developed to reflect variations in thermal hydraulic performance over the design range.Based on CFD simulation, the correlations of inline tube bundles with gradually varied diameter from 1.0 to 0.4 mm was determined.The design range also included rows varying from 7 to 29 tubes, outer wall spacing from 0.2 to 1.0 mm, and air velocities from 1 to 10 m/s.
From the literature review, there was little research on the structure of tube bundles with different diameter combinations in the study of heat exchanger performance.Based on the research gap in this investigation, a heat exchanger structure with diameter gradually varying tube bundle was proposed in this paper, and its thermal hydraulic performance was studied.Based on the numerical results, empirical correlations between j and f factors were established.

| Methodology
The method applied CFD simulation to evaluate dimensionless aerodynamic thermal-hydraulic performance factors ( j and f ) on the airside.The research process (Figure 1) consisted of four main parts, including problem specification, modeling, correlation development and random data verification.Each part was described and corresponding results were given below.
The first part is the problem specification, where tube bundle design features such as simulation operating conditions, tube and air capacity and flow are defined.The second part is to establish the heat exchanger model on the CFD platform according to the problem specification, and to carry out the simulation.The main objective of this work is to find the empirical correlations of the tube bundle ranging from 1.0 to 0.4 mm, because the existing correlation results cannot predict the models of bundles with gradually varied diameter.To solve this problem, the correlation was developed based on the CFD platform, which can save a lot of computational work during the design and research phase.Finally, a randomized simulation was conducted to compare and verify the evaluation results.

| Problem description and design of simulation
The diameter of the heat exchanger tube bundle proposed here changed gradually in the range of 1.0-0.4mm.This paper developed the correlation for its thermal and hydraulic performance of the airside.The correlations mentioned consider pipe rows ranged from 7 to 29 (see Figure 2).After determining the boundary conditions and structural parameters in the design scheme, the heat exchanger model correlation was established by CFD simulation.The heat exchanger in the design problem was composed of several tubes with gradually varied diameter, which were arranged in parallel, in a cross-flow structure, as shown in Figure 2. The pipeline flow path was the same as the micro-channel heat exchanger, that is, they were connected to the D-shape collector.

| CFD models of correlations development
The existing correlation for tube bundles with the same diameter structure was improved and the correlation for tube bundles with gradually varied diameter was proposed.The coefficients are determined by solving the minimum of the square differences summation (Equation 1).
CFD calculation domain (Figure 3) was a 3D model structure of heat exchanger, the calculation domain included the tube wall, but the influence of inlet and outlet ports were ignored in the simulation.The speed of the water inside the tube was maintained at stable 0.5 m/s, the temperature of the water was kept at 278 K, the temperature of the air inlet was set at near 300 K, and had a uniform speed varying from 1 to 10 m/s, and the pressure of the water and airside outlet boundary was set as constant.Both water and air were set at the ideal model.The pressure base solver was adopted to improve the calculation efficiency, reducing the error in numerical calculation and improving the accuracy of simulation results.The flow model took turbulence which was evaluated using the k-epsilon realizable model and scalable wall function.Based on the advantages of strong robustness and fast convergence speed, the coupled scheme of the pressure-velocity coupling method was used for steady state calculation.When it reached the order of 10 −6 , it was regarded as computational convergence.The Navier-Stokes equation was adopted in the CFD model to govern the air and water fluid flow: F I G U R E 3 CFD computational domain and mesh.
Also, to determine the temperature and entropy flow in the computational domain, the energy equation in the CFD model is shown in Equation (3).
The continuity equation ensures the mass balance in the simulation model, which is shown in Equation (4).
The ideal gas equation is adopted in Equation (5).
The cloud diagram of heat exchanger temperature and velocity change simulated by CFD was shown in Figures 4 and 5. From the velocity contour plot, the airflow through the diameter gradually varied tube bundle created a smooth flow from the large diameter to the small diameter and then to the large diameter tube.The streamline looked like an airplane wing or a fish.Therefore, it can reduce the pressure drop across the tube bundle significantly.
The CFD model is applied to determine the thermal and hydraulic performance of the airside of the heat exchanger.The extra thermal resistance and local pressure drop loss are ignored in the simulation design, so the heat transfer coefficient can be easily calculated by using ε-NTU method, as per Equations ( 6)- (9).The accuracy of the subsequent correlation was assessed by comparing it with the calculated results.

| Grid independence analysis
The grid independence was tested in the simulation of CFD model.For the same model, different mesh cell sizes and total mesh numbers were selected to analyze the grid independence, and the results were shown in Figure 6: After the number of grid nodes used in the simulation, the temperature difference and pressure drop on the airside of the micro-bare-tube heat exchanger did not change significantly with the increase of the number of grids.Therefore, the number of grids used in the model was feasible and independent in the simulation.

| CFD simulation results
The simulation designs and source data calculations were carried out according to the sequential flow of geometric design, meshing and operations.Based on the CFD simulation results, the correlation coefficient was obtained using the least square method, and the correlation was compared with the source data and random data, respectively (see Tables 1-3).The correlation prediction proposed was compared with the source data as shown in Figure 7.Ten related subsets of airside friction and heat transfer performance factors were proposed for different number of tube rows.
The grid independence analysis.

| Random data verification
The CFD correlation was verified using 60 randomized simulation samples with different simulation designs from the source data.The result is shown in Figure 8.

| Simulation validation
The heat exchanger composed of tube bundles with diameters varying gradually from 1.0 to 0.4 mm was simulated and verified.The number of tube rows of the heat exchanger was 7, the distance between the outer walls of the longitudinal tube was 0.2 mm, and the distance between the outer walls of the transverse tube was 1.5 mm.In the design, the cold end fluid was liquid water and the hot end fluid was air.Seven different wind speeds and three different water flow rates were applied to carry out multiple verification.
The simulation results (see Figures 9 and 10) showed that the heat transfer coefficient and frictional property decreased with the increase of Reynolds number, which was proportional to the air velocity.The deviation of heat transfer coefficient was large when the air velocity was small, the error was about 20%, which might be related to the limited heat transfer capacity of the tube bundle under the low wind speed.With the increase of the air velocity, the prediction of correlation was gradually close to the simulation results.The deviation between the prediction of friction performance and the simulation was kept in the range of 10%-20%.By correcting the correlation of prediction with a coefficient of 0.85, the prediction deviation of friction can be reduced to less than 5%.
The random data and simulation results showed that the performance correlation equations of heat exchangers with gradually varied diameter bundles had good prediction ability, and the prediction deviation was within the acceptable range.This indicated that the combination of tube bundle diameter had a direct effect on the thermal hydraulic performance in addition to the tube diameter and flow velocity.
F I G U R E 8 Correlations verification using random data.
As mentioned above, the bundle with gradually varied diameter had a significant effect on the heat exchanger performance, the amount of refrigerant charged in the pipe and the thermal hydraulic performance on the airside.This section discussed the performance advantages of using gradually varied diameter structures over the same diameter structures.
It can be seen from Figure 11 how the airside performance of the heat exchanger with gradually varied diameter tube bundle was affected by tube spacing and airflow velocity.As shown in Figure 11, the Colburn factor j and the friction factor f decreased gradually as the air velocity increased from 1 to 10 m/s.Also, under the same boundary condition described in the CFD model section, the Colburn factor j decreased from 0.02 to 0.015 as the longitudinal pitch increased from 0.2 to 1.0 mm at the air inlet velocity of 1 m/s.The friction factor f dropped slowly from 0.14 to 0.05 as the air velocity increased from 1 to 10 m/s when the longitudinal pitch was 1 mm.The model prediction results indicated that the optimum diameter gradually varied micro-bare-tube structure must be determined because the large longitudinal pitch provide higher Colburn factor but also with great friction coefficient.Therefore, to design a good diameter gradually varied micro-bare-tube heat exchanger with high heat transfer coefficient and low friction factor, its geometric structure must be well organized by using the proposed empirical correlations in this study.11.The Colburn factor j of this study was in the middle of that of the constant tube diameter tube bundles.This result illustrated the excellent thermal performance of the proposed structure.Because large diameter tube increased the refrigerant charge and the copper or stainless-steel material cost, however, tiny diameter tube bundles increase the refrigerant flow resistance and thus increased the compressor power cost.Therefore, the proposed diameter gradually varied micro-bare-tube heat exchanger had a great potential to decrease the refrigerant charge for the air conditioners and heat pumps.Besides, the friction factor calculated by the proposed geometry was much smaller than that of the traditional constant diameter tube bundles.The results showed that the tube bundle with gradually varied diameter can reduce more metal materials and provided better heat transfer performance.The novel tube bundle also had the lowest pressure drop under the same air velocity as shown in Figure 12.
Table 4 compared the volume of metal material and refrigerant charge required for diameter gradually varied tube bundle and the constant diameter tube bundle.The diameter of the heat exchanger was usually constant for the traditional heat exchanger due to the commercial feasibility; however, the micro-bare-tube heat exchanger had no fins so that it is easy to manage the geometric structure with diameter gradually varied tube bundle which can potentially decreased the pressure drop and increased the heat transfer coefficient.Furthermore, owing to the diameter gradually varied geometry, the newly proposed heat exchanger could save 47.32% refrigerant charge and 28.57% raw metal material cost compared to the constant 1 mm outer diameter tube bundle as shown in Table 4.The research studied the gradually varied diameter micro-bare-tube heat exchanger to both reduce the pressure drop and increase its thermal hydraulic performance.Compared with the traditional microbare-tube heat exchanger of the same diameter, the friction coefficient f is reduced by 8.59%, the pressure drop is reduced by 43.14%, and the heat transfer coefficient j is increased by 12.07%.At the same time, it reduced the void volume by 47.32% and the tube bundle manufacturing material by 28.57%, thus reducing refrigerant filling and copper use, making it a more environmentally friendly solution.
The empirical correlation for evaluating the thermal and hydraulic performance of the heat exchanger with gradually reducing diameter was presented, which had great prediction ability.The correlation prediction error of 83.4% of the data is within 15%, and that of 90.86% of the data was within 20%.The flow and heat transfer performance of the micro-bare-tube heat exchanger structure with the tube bundle diameter varying gradually was numerically simulated, and the performance advantages of the gradually varied tube bundle diameter structure were illustrated, which can provide a reference for the future study of the influence of tube diameter structure on the performance of the micro-bare-tube heat exchanger and the subsequent optimization.
The research on the micro-bare-tube heat exchanger was not comprehensive enough, and many parameters that affecting the heat exchange performance of the micro-bare-tube heat exchanger had not been considered.Although the micro-bare-tube heat exchanger with gradually varied tube bundle diameter had better heat transfer performance compared to the constant diameter one, its optimization was needed to be further studied.Although some factors affecting the flow and heat transfer performance of heat exchangers were analyzed, the degree of control of these factors remains to be further explored.

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I G U R E 2 (A) Tube bundles stereogram.(B) Tube bundles cross-section.

F I G U R E 4
The cloud diagram of temperature distribution.F I G U R E 5 The cloud diagram of velocity distribution.

F 4607 Figure 12
Figure12compared the variation of thermohydraulic performance of tube bundles with gradually varied diameter and those with the same diameter within the air velocity shifted from 1 to 10 m/s.The structural parameters except pipe diameter remained unchanged.The dark solid line indicated the Colburn factor of this work in Figure11.The Colburn factor j of this study was in the middle of that of the constant tube diameter tube bundles.This result illustrated the excellent thermal performance of the proposed structure.Because large diameter tube increased the refrigerant charge and the copper or stainless-steel material cost, however, tiny diameter tube bundles increase the refrigerant flow resistance and thus increased the compressor power cost.Therefore, the proposed diameter gradually varied micro-bare-tube heat exchanger had a great potential to decrease the refrigerant charge for the air conditioners and heat pumps.Besides, the friction factor calculated by the proposed geometry was much smaller than that of the traditional constant diameter tube bundles.The

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I G U R E 12 Parameter analysis of pipe diameter combination.T A B L E 4 Comparison of two different tube bundle types.