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Effects of Fin Arrangements on Thermal Hydraulic Performance of Supercritical Nitrogen in Printed Circuit Heat Exchanger

: The printed circuit heat exchanger (PCHE) with discontinuous ﬁns is a novel type of compact and highly efﬁcient plate heat exchanger, which has superior thermal hydraulic performance. The morphology and characteristics of the ﬂow channel greatly affect the performance of the PCHE. The discontinuous airfoil ﬁns are used in PCHE channel design because they can affect the ﬂow and heat transfer by increasing the heat transfer area and the disturbance in the channel. In this paper, the effects of different staggered distance (L s ) and transverse distance (L v ) of airfoil ﬁn arrangements on the heat transfer and ﬂow of supercritical nitrogen in the PCHE are numerically simulated using ANSYS Fluent. Simulation results and subsequent analysis show that the appropriate decrease in L s and reduction in L v between the two rows of ﬁns can improve the convective heat transfer of the PCHE. A fully staggered arrangement of ﬁns (L s = 1.2) and an appropriate increase in the L v can mitigate pressure drop. The comprehensive performance of different channel geometries is compared by the performance evaluation criteria (PEC) in this study. It is shown that considering ﬂow resistance and heat transfer, the comprehensive heat transfer performance can be enhanced by properly increasing the staggered distance and the vertical distance between ﬁns. When L s = 1.2 mm and L v = 1.25 mm, the PEC value of the staggered channel is the highest, which is 11.6% higher than that of the parallel channel on average.


Introduction
The printed circuit heat exchanger (PCHE) is highly valued with high heat transfer efficiency which can be used in harsh conditions, such as high temperature and pressure. The PCHE, as one of the most promising candidate heat exchangers in the terms of compact heat transfer, is composed of diffusion bonded plate arrays [1]. Narrow spaced microchannels are formed on these metal plates to serve as flow channels through chemical etching. In this paper, the supercritical fluid was used as the working fluid in the airfoil fin PCHE. Supercritical fluids are broadly applied in industrial fields including refrigeration engineering [2], power engineering [3], chemical engineering and aerospace engineering [4]. These fluids have special physical properties, having states between liquid and gas. They have attractive flow and transport characteristics, for example, diffusivity and viscosity close to gas and density nearly the same as liquids.
In recent years, different structures and cross-section shapes of channels in the PCHE have been widely researched. The heat transfer performances of PCHEs with straight channels were investigated experimentally by Wang et al. [5] and the effects of pressure and pseudo-critical points on heat transfer performance were studied. Zhao et al. [6] investigated the heat transfer and hydraulic characteristics of supercritical nitrogen in a straight PCHE. They studied the effect of inlet pressure and established a correlation between the Nusselt number and Fanning factor. Nikitin [7] investigated the heat transfer 1.
The heat transfer and flow resistance of supercritical nitrogen in the PCHE with different arrangements of airfoil fins are investigated numerically.

2.
The local characteristics along the flow direction in the PCHE according to different arrangements of airfoil fins are analyzed. 3.
The overall pressure drops and heat transfer performance of the PCHE with respect to varying vertical and staggered distances of fins are investigated and the performance evaluation criteria (PEC) are compared to find the best arrangement of fins in this study.

Physical Model and Boundary Condition
The entire airfoil flow channel PCHE is composed of tens of thousands of periodically repeating tiny flow channels. It requires a high calculation cost to simulate the internal flow and heat transfer characteristics of all flow channels in the whole core. In this section, the flow and heat transfer of the airfoil fins channel are mainly studied. When the boundary conditions of the airfoil fins channel are the same, the flow and heat transfer characteristics are the same in the direction perpendicular to the flow. Therefore, for the convenience of simulation and research, the 3-D model for numerical calculations is simplified as a flow channel with two rows of airfoil fins in the red frame in Figure 1. The structure of the flow channel shown in Figure 1 is periodic in both transverse and longitudinal direction. The NACA0050-4.5/2.6 airfoil (the airfoil relative maximum thickness as a percentage of the chord is 50%, the leading-edge radius index is 4.5 and the chordwise-position of airfoil maximum thickness is 26%) is used as the shape of the fin. The symmetrical airfoil fins are defined by the fin height (H), fin width (W) and fin length (L), which are set to 0.75 mm, 0.6 mm and 1.2 mm, respectively, as shown in Figure 1. The different fin structures and arrangements of the channel do not increase the processing cost. Figure 1 further shows the flow channel with airfoil fins and different distances between the airfoil fins (L s , L v , L h ). The L s presents staggered pitch, L v indicates the vertical pitch of the adjacent rows and L h indicates the horizontal pitch of the adjacent fins. The height of the flow channel in the model is 0.75 mm and the length is 400 mm, which is consistent with the height and length of the experimental heat exchanger. Figure 2 shows the boundary conditions of the 3-D model. The inlet temperature was 103 K, the mass flow rate at the inlet ranged from 0.0003461 kg/s to 0.0004384 kg/s, and the inlet pressure was set at 6 MPa. The outlet was the pressure outlet, the left and right sides were periodic, and the upper and lower sides were set as constant heat flux.

Numerical Model and Grid Independence
The numerical solutions for the PCHE with different fin arrangements were obt using ANSYS Fluent [25]. The tested shear stress transport (SST) κ − ω model [2 showed that the model was suitable for turbulent flow and heat transfer at superc pressure. Therefore, in this paper, the SST κ − ω model with enhanced wall trea and the QUICK scheme were employed for numerical simulations and the mome equation, respectively. Other equations were discretized using a second-order up scheme. In addition, the semi-implicit method pressure linked equation (SIMPLE used to couple the velocity with pressure. The governing equations for the simu have been described in detail in our previous work [24]. The mesh was optimized to ensure that the non-dimensional length (y+) at th node close to the wall was less than 0.5, so as to calculate the viscous sub-layer accur In order to ensure the accuracy of calculation, a boundary layer was applied to the w the flow passage to carry out mesh encryption. The boundary layer was equipped eight columns of meshes. The thickness of the first mesh and increasing factor wer mm and 1.1, respectively. Figure 3 shows the zoomed-in image of the mesh and bou layer.

Numerical Model and Grid Independence
The numerical solutions for the PCHE with different fin arrangements were obtained using ANSYS Fluent [25]. The tested shear stress transport (SST) κ − ω model [26][27][28][29] showed that the model was suitable for turbulent flow and heat transfer at supercritical pressure. Therefore, in this paper, the SST κ − ω model with enhanced wall treatment and the QUICK scheme were employed for numerical simulations and the momentum equation, respectively. Other equations were discretized using a second-order upwind scheme. In addition, the semi-implicit method pressure linked equation (SIMPLE) was used to couple the velocity with pressure. The governing equations for the simulation have been described in detail in our previous work [24].
The mesh was optimized to ensure that the non-dimensional length (y+) at the first node close to the wall was less than 0.5, so as to calculate the viscous sub-layer accurately. In order to ensure the accuracy of calculation, a boundary layer was applied to the wall of the flow passage to carry out mesh encryption. The boundary layer was equipped with eight columns of meshes. The thickness of the first mesh and increasing factor were 0.01 mm and 1.1, respectively. Figure 3 shows the zoomed-in image of the mesh and boundary layer.

Numerical Model and Grid Independence
The numerical solutions for the PCHE with different fin arrangements were obtained using ANSYS Fluent [25]. The tested shear stress transport (SST) κ − ω model [26][27][28][29] showed that the model was suitable for turbulent flow and heat transfer at supercritical pressure. Therefore, in this paper, the SST κ − ω model with enhanced wall treatment and the QUICK scheme were employed for numerical simulations and the momentum equation, respectively. Other equations were discretized using a second-order upwind scheme. In addition, the semi-implicit method pressure linked equation (SIMPLE) was used to couple the velocity with pressure. The governing equations for the simulation have been described in detail in our previous work [24].
The mesh was optimized to ensure that the non-dimensional length (y+) at the first node close to the wall was less than 0.5, so as to calculate the viscous sub-layer accurately. In order to ensure the accuracy of calculation, a boundary layer was applied to the wall of the flow passage to carry out mesh encryption. The boundary layer was equipped with eight columns of meshes. The thickness of the first mesh and increasing factor were 0.01 mm and 1.1, respectively. Figure 3 shows the zoomed-in image of the mesh and boundary layer.  To ensure the accuracy of numerical results and save computational resources, grid independence needs to be verified. Five sets of grid numbers: 3,523,635, 4,804,956, 5,724,613, 6,406,608 and 7,687,930, were compared to analyze the influence of mesh density on simulation results in Table 1. Considering the accuracy of outlet temperature and grid quantity, the mesh number is selected as 6,406,608.

Thermal Physical Properties of Supercritical Nitrogen
The thermal physical properties of supercritical nitrogen are special. The physical properties of supercritical nitrogen vary significantly around the pseudo-critical point, which changes with the increase in pressure. Physical properties (thermal conductivity, specific heat, density and viscosity) of nitrogen at 6 MPa could be obtained according to the Standard Reference Database (REFPROP) [30]. Figure 4 shows that the thermal conductivity and dynamic viscosity decrease sharply and then increase slightly with increasing temperature. The specific heat capacity reaches a peak value near the pseudo-critical point, then decreases sharply and reaches a nearly constant value. It can be concluded that the thermal physical properties of nitrogen depend significantly on temperature, which varies greatly in the heat exchanger. The temperature difference between the inlet and outlet of the PCHE is relatively large, which leads to the thermal physical properties of supercritical nitrogen changing very sharply with the temperature. Therefore, we fitted the main physical property parameters as a function of temperature into a polynomial, and used the PIECEWISE-POLYNOMIAL function in FLUENT to input the fitting polynomial into the physical property of the software material according to the temperature interval. Table 2 shows the piecewise-polynomial function of physical properties of nitrogen at 6 MPa.

Thermal Physical Properties of Supercritical Nitrogen
The thermal physical properties of supercritical nitrogen are special. The physical properties of supercritical nitrogen vary significantly around the pseudo-critical point, which changes with the increase in pressure. Physical properties (thermal conductivity, specific heat, density and viscosity) of nitrogen at 6 MPa could be obtained according to the Standard Reference Database (REFPROP) [30]. Figure 4 shows that the thermal conductivity and dynamic viscosity decrease sharply and then increase slightly with increasing temperature. The specific heat capacity reaches a peak value near the pseudo-critical point, then decreases sharply and reaches a nearly constant value. It can be concluded that the thermal physical properties of nitrogen depend significantly on temperature, which varies greatly in the heat exchanger. The temperature difference between the inlet and outlet of the PCHE is relatively large, which leads to the thermal physical properties of supercritical nitrogen changing very sharply with the temperature. Therefore, we fitted the main physical property parameters as a function of temperature into a polynomial, and used the PIECEWISE-POLYNOMIAL function in FLUENT to input the fitting polynomial into the physical property of the software material according to the temperature interval. Table 2 shows the piecewise-polynomial function of physical properties of nitrogen at 6 MPa.   Table 2. Piecewise-polynomial function of physical properties of nitrogen at 6 Mpa.

Model Validation
The thermal hydraulic performance of supercritical nitrogen was investigated numerically in this study. Experimental data and numerical data were compared to verify the accuracy of the numerical model. Figure 5 shows photographs of the experimental PCHE and system. The airfoil fin PCHE in the experiment was manufactured by 316 L grade stainless steel. The chord length of the airfoil fin was 1.2 mm and the maximum thickness was 0.6 mm. The inlet temperature of nitrogen was 103 K where the mass flux ranged from 0.0003461 kg/s to 0.0004384 kg/s at 6 Mpa, corresponding to the numerical operating conditions [24]. The differences between the experimental and the numerical data in temperature and pressure drop were defined as the following form: The differences of outlet temperature and pressure drop between experimental results and numerical data for different mass flux are shown in Table 3. The uncertainty for the outlet temperature was within 0.84% and the maximum error of pressure drop was 4.51% under the operating pressure of 6 Mpa. As these errors are small, we can conclude that the numerical results were consistent with experimental results, and the numerical models and methods were reasonably accurate.

Calculation Methodology
As the characteristic length of the flow channel, the hydraulic diameter D h , defined as four times the cross-sectional area divided by the perimeter of the circular and non-circular tubes, is of great significance for dimensional analysis. The arrangement of the airfoil fins shown in Figure 1 is periodic, and the hydraulic diameter can be calculated as follows [24]: where S a and P a indicate the area at the top and the perimeter of the airfoil fin, respectively, as shown in Figure 6, and S and V are the side surface area and volume of the airfoil fin, respectively, while L a indicates the length of the airfoil fin, as shown in Figure 6.  The average convective heat transfer coefficient was selected to evaluate the transfer performance of the PCHE with airfoil fins. The average convective heat tr coefficient, h, is calculated from the solver data as: where indicates the area-average wall heat flux, and is the area-average temperature. The bulk mean temperature of nitrogen is represented by , which culated as [31]: where , , and → are the nitrogen temperature, density and velocity vector in t cell, respectively, and → is the flow area vector of the jth cell. The physical proper nitrogen are dependent on the bulk mean temperature. The intensity of convective heat transfer can also be characterized by the N number, which is expressed in the following form: The average convective heat transfer coefficient was selected to evaluate the heat transfer performance of the PCHE with airfoil fins. The average convective heat transfer coefficient, h, is calculated from the solver data as: where q w indicates the area-average wall heat flux, and T wall is the area-average wall temperature. The bulk mean temperature of nitrogen is represented by T b , which is calculated as [31]: is the flow area vector of the jth cell. The physical properties of nitrogen are dependent on the bulk mean temperature. The intensity of convective heat transfer can also be characterized by the Nusselt number, which is expressed in the following form: where λ is the thermal conductivity of nitrogen. The pressure drop, ∆P, of the channel in the heat exchanger is determined by acceleration and friction losses. It is given as follows: and can be measured at the inlet and outlet of the airfoil PCHE through the pressure sensor. The friction pressure drop can be calculated from the ∆P and the acceleration pressure drop. The pressure drop due to acceleration pressure drop is calculated as: The frictional pressure drop is given as: where f is the Fanning friction factor, which can be calculated using the above equations as: The Performance Evaluation Criteria (PEC) [32] are computed using Equation (12): where the subscript "0" corresponds to the parallel arrangement of the channel.

Results and Discussion
The effects of different staggered distance (L s ) and transverse distance (L v ) of airfoil fin arrangements on the thermal hydraulic performance of supercritical nitrogen in the PCHE are numerically simulated using ANSYS Fluent in the present study. The fin arrangements mainly affect the thermal hydraulic performance of the PCHE by increasing the heat transfer area and interfering with the flow. In fact, the height of the channel is very small, which results in the fin having a small side area. The lateral area of the fin counteracts the heat transfer area occupied by the fin on the inner surface of the channel. As a result, the enlargement of the heat transfer area is very limited. Therefore, the fin arrangement plays a significant part in improving the performance of the airfoil PCHE. In this study, the staggered distance (L s ) of the airfoil fins was varied as 0 mm, 1 mm and 1.2 mm and the transverse distance (L v ) was varied as 1 mm and 1.25 mm. Figure 7a shows the velocity vector of supercritical nitrogen in the channel near the outlet arranged in parallel and staggered fins with a mass flux of 227.7 kg/(m 2 s). As can be seen from the velocity vector distribution of the top three flow channels, when L v is fixed at 1.25 mm, it is observed that the velocity distribution becomes more stable as the L s increases from 0 mm to 1.2 mm. The distribution of red and blue velocity vectors is significantly different in parallel arrangement, and the velocities suddenly increase and then decrease between the two adjacent fin heads. The maximum velocity of the parallel flow channel at the minimum section is larger than that of the staggered flow channel, which can be seen from the degree and number of red vector arrows. When the fins were completely staggered (L s = 1.2 mm), the vertical distance of 1 mm and 1.25 mm between the adjacent rows of fins had no obvious difference in the velocity distribution, and L s further decreased to 0.8 mm, indicating a higher velocity and a more uniform distribution. Since the fin structure is wide head and narrow tail, it is easier to form a streamline when the fins are staggered. As the tail of the fin is narrow, there is nearly no vortex at the tail of the fin. The local average velocity of the cross section along the channel is shown in Figure 8a. It can be observed that the increasing trend of the velocity along the channel is in the consistent trend and becomes steeper under different fin arrangements. This is due to the rapid decrease in the density and viscosity of supercritical nitrogen in the flow channel. However, within the distance from 315 mm to 340 mm from the entrance, the increase in velocity disappears and becomes a decrease in velocity instead. This trend is because the density and dynamic viscosity of supercritical nitrogen change from a sharp decrease to a  Figure 7b shows the temperature distribution near the inlet with varying L s and L v , where the inlet temperature and pressure are 103 K and 6 MPa, respectively. The temperature of all the flow channels increases by 10 K at x = 28 mm. The temperature near the fin tail is higher than that in the main flow area, especially in the case of a complete staggered arrangement of L s = 1.2 mm and L v = 1.25 mm. The reason for this high temperature is the velocity of the fin tail region being lower than that of the middle region of the flow channel. The sudden increase and decrease in velocity in the cross section of the parallel channel is beneficial for distributing the temperature uniformly in the fin tail. At the inlet section, the effect of the L s = 1.2/L v = 0.8 arrangement on supercritical nitrogen is relatively severe, resulting in a relatively strong local heat transfer capacity in this region, so there is a local rise in temperature around the fin.

Local Characteristics of Different Fin Arrangements
The local average velocity of the cross section along the channel is shown in Figure 8a. It can be observed that the increasing trend of the velocity along the channel is in the consistent trend and becomes steeper under different fin arrangements. This is due to the rapid decrease in the density and viscosity of supercritical nitrogen in the flow channel. However, within the distance from 315 mm to 340 mm from the entrance, the increase in velocity disappears and becomes a decrease in velocity instead. This trend is because the density and dynamic viscosity of supercritical nitrogen change from a sharp decrease to a gradual increase in viscosity and a gentle decrease in density, leading to a decrease in velocity.  Figure 9a compares the average convective heat transfer coefficients of the PCHE for different fin arrangements at different mass flux values. The figure indicates that the average convective heat transfer coefficient increases with increasing mass flux. With a reduction in the staggered distance, the convective heat transfer performance becomes stronger. When the Lv remains constant, the convective heat transfer performance is enhanced by decreasing the value of Ls. In the case of parallel arrangement, a sudden increase and decrease in velocity will generate strong disturbance, which can enhance convective heat transfer. In addition, when Ls is at a constant value of 1.2 mm, the vertical distance of Lv = 0.8 mm has a stronger convective heat transfer performance than that at Lv = 1 mm and Lv = 1.25 mm. Figure 9b depicts the overall Nusselt number (Nu) versus Re. The Nu also increases as the Re increases; however, when Re is greater than 1.4×10 5 , the increase degree of Nu decreases. Therefore, appropriate reductions in the staggered distance (Ls) and vertical distance between the two rows of fins (Lv) can improve the convective heat transfer of the PCHE.

Overall Performance of Different Fin Arrangements
In addition, in order to summarize the heat transfer characteristics of supercritical nitrogen in the airfoil PCHE, a new correlation formula of average convective heat transfer is proposed. By fitting the average convective heat transfer coefficient of the simulation results, the new correlation formula for ℎ is obtained: .
. .    Figure 8b shows the pressure distribution of the flow channel with different fin arrangements along the channel. The pressure in the flow channel decreases gradually along the flow direction, and the decreasing extent is more and more large, owing to a rapid increase in velocity during the heat transfer process. In addition, when the L s increases from 0 to 1.2 mm, the change in pressure becomes lower under the condition of constant L v . At the same staggered distance (L s = 1.2 mm), reducing the vertical distance between two rows of fins can lead to an increased pressure reduction. This increase is caused by a decrease in fin vertical distance that increases the flow velocity. Figure 9a compares the average convective heat transfer coefficients of the PCHE for different fin arrangements at different mass flux values. The figure indicates that the average convective heat transfer coefficient increases with increasing mass flux. With a reduction in the staggered distance, the convective heat transfer performance becomes stronger. When the L v remains constant, the convective heat transfer performance is enhanced by decreasing the value of L s . In the case of parallel arrangement, a sudden increase and decrease in velocity will generate strong disturbance, which can enhance convective heat transfer. In addition, when L s is at a constant value of 1.2 mm, the vertical distance of L v = 0.8 mm has a stronger convective heat transfer performance than that at L v = 1 mm and L v = 1.25 mm. Figure 9b depicts the overall Nusselt number (Nu) versus Re. The Nu also increases as the Re increases; however, when Re is greater than 1.4×10 5 , the increase degree of Nu decreases. Therefore, appropriate reductions in the staggered distance (L s ) and vertical distance between the two rows of fins (L v ) can improve the convective heat transfer of the PCHE. The pressure drop of the PCHE for different fin arrangements under varying values of mass flux is shown in Figure 10a. The pressure drop slope decreases with the increase in mass flux. At the same vertical distance (Lv = 1.25 mm), the pressure drop decreases with the increase in Ls. Therefore, when Ls = 1.2 mm, the average pressure drop corresponding to each mass flow is the minimum of 9.0 × 10 Pa, only 56.6% of the parallel arrangement. The pressure loss of the completely staggered arrangement (Ls = 1.2 mm) reduces by 35.5% on average compared with that of the parallel arrangement. In addition, for the same Ls of 1.2 mm, the pressure loss decreases by 34.9% on average as the vertical distance increases from 0.8 mm to 1.25 mm. Figure 10b shows the Fanning friction factor versus the variation of Re in the flow channel with different Lv and Ls. With the increase in Re, the Fanning friction factor decreases. The Fanning friction factor of the completely staggered arrangement (Ls = 1.2 mm) reduces by 37% on average compared with that of the parallel arrangement, and the Fanning friction factor for a vertical distance of Lv = 0.8 mm increases by 35% on average compared with that for Lv = 1.25 mm. These effects are a result of the velocity of the staggered channel being more uniform than that of the parallel flow channel and the wide head and narrow tail being more likely to form a streamlined structure, which eliminates the reverse flows and eddies. Therefore, appropriate increases in the staggered distance (Ls) and vertical distance between the two rows of fins (Lv) are more suitable for reducing pressure loss. In addition, in order to summarize the heat transfer characteristics of supercritical nitrogen in the airfoil PCHE, a new correlation formula of average convective heat transfer is proposed. By fitting the average convective heat transfer coefficient of the simulation results, the new correlation formula for h is obtained: (13) where C p , ρ, ν, and µ are the specific heat capacity, density, flow rate and viscosity of nitrogen, respectively. The pressure drop of the PCHE for different fin arrangements under varying values of mass flux is shown in Figure 10a. The pressure drop slope decreases with the increase in mass flux. At the same vertical distance (L v = 1.25 mm), the pressure drop decreases with the increase in L s . Therefore, when L s = 1.2 mm, the average pressure drop corresponding to each mass flow is the minimum of 9.0×10 3 Pa, only 56.6% of the parallel arrangement. The pressure loss of the completely staggered arrangement (L s = 1.2 mm) reduces by 35.5% on average compared with that of the parallel arrangement. In addition, for the same L s of 1.2 mm, the pressure loss decreases by 34.9% on average as the vertical distance increases from 0.8 mm to 1.25 mm. Figure 10b shows the Fanning friction factor versus the variation of Re in the flow channel with different L v and L s . With the increase in Re, the Fanning friction factor decreases. The Fanning friction factor of the completely staggered arrangement (L s = 1.2 mm) reduces by 37% on average compared with that of the parallel arrangement, and the Fanning friction factor for a vertical distance of L v = 0.8 mm increases by 35% on average compared with that for L v = 1.25 mm. These effects are a result of the velocity of the staggered channel being more uniform than that of the parallel flow channel and the wide head and narrow tail being more likely to form a streamlined structure, which eliminates the reverse flows and eddies. Therefore, appropriate increases in the staggered distance (L s ) and vertical distance between the two rows of fins (L v ) are more suitable for reducing pressure loss. The PEC values for different fin arrangements are shown in Figure 11. The value of the PEC for the staggered channel geometry is higher compared with that of the base geometry of the parallel channel. The PEC values for the channel geometries of Ls = 1.2 mm, Lv = 1.25 mm and Ls = 1.2 mm, Lv = 0.8 mm decrease with increasing Re, but the channel geometries of Ls = 0.6 mm, Lv = 1.25 mm and Ls = 1.2 mm, Lv = 1 mm show the opposite trend. Under the same Re condition, when the Lv increases from 0.8 mm to 1.25 mm, the PEC value increases with the Lv increasing. These results show that among all fin arrangements, the PEC average value of the staggered channel with Ls = 1.2 mm and Lv = 1.25 mm is the highest, which is 11.6% higher than that of the parallel channel. It could be deduced that the comprehensive heat transfer performance can be improved by properly increasing the staggered distance (Ls) and vertical distance between the two rows of fins (Lv).   The PEC values for different fin arrangements are shown in Figure 11. The value of the PEC for the staggered channel geometry is higher compared with that of the base geometry of the parallel channel. The PEC values for the channel geometries of L s = 1.2 mm, L v = 1.25 mm and L s = 1.2 mm, L v = 0.8 mm decrease with increasing Re, but the channel geometries of L s = 0.6 mm, L v = 1.25 mm and L s = 1.2 mm, L v = 1 mm show the opposite trend. Under the same Re condition, when the L v increases from 0.8 mm to 1.25 mm, the PEC value increases with the L v increasing. These results show that among all fin arrangements, the PEC average value of the staggered channel with L s = 1.2 mm and L v = 1.25 mm is the highest, which is 11.6% higher than that of the parallel channel. It could be deduced that the comprehensive heat transfer performance can be improved by properly increasing the staggered distance (L s ) and vertical distance between the two rows of fins (L v ). The PEC values for different fin arrangements are shown in Figure 11. The value of the PEC for the staggered channel geometry is higher compared with that of the base geometry of the parallel channel. The PEC values for the channel geometries of Ls = 1.2 mm, Lv = 1.25 mm and Ls = 1.2 mm, Lv = 0.8 mm decrease with increasing Re, but the channel geometries of Ls = 0.6 mm, Lv = 1.25 mm and Ls = 1.2 mm, Lv = 1 mm show the opposite trend. Under the same Re condition, when the Lv increases from 0.8 mm to 1.25 mm, the PEC value increases with the Lv increasing. These results show that among all fin arrangements, the PEC average value of the staggered channel with Ls = 1.2 mm and Lv = 1.25 mm is the highest, which is 11.6% higher than that of the parallel channel. It could be deduced that the comprehensive heat transfer performance can be improved by properly increasing the staggered distance (Ls) and vertical distance between the two rows of fins (Lv).   Figure 11. Performance evaluation criteria for different fin arrangements.

Conclusions
The effects of different arrangements of airfoil fins characterized by staggered distance (L s ) and transverse distance (L v ) on the heat transfer and hydraulic resistance of supercritical nitrogen in the PCHE were studied in the present study. The local characteristics and the overall performance of the heat exchanger for different fin arrangements were analyzed. The major conclusions of this study can be summarized as follows: 1.
When L v is constant, the velocity distribution tends to be stable gradually as the L s increases from 0 mm to 1.2 mm. Since the fin structure is wide head and narrow tail, it is easier to form a streamline when the fins are staggered. As the tail of the fin is narrow, there is nearly no vortex at the tail of the fin. In the case of completely staggered airfoil fins (L s = 1.2 mm), the velocity improvement and uniform distribution can be achieved when L s is the minimum value, i.e., 0.8 mm. The increasing trend of the velocity along the channel is similar for different fin arrangements and this increasing trend becomes steeper along the channel. In addition, when the L s increases from 0 mm to 1.2 mm, the change in pressure becomes lower under the condition of constant L v . At the same staggered distance (L s = 1.2 mm), reducing the vertical distance between two rows of fins can lead to an increased pressure reduction.

2.
When the L v remains constant, the convective heat transfer performance is enhanced by decreasing the value of L s . In addition, when L s is at a constant value of 1.2 mm, the vertical distance of L v = 0.8 mm has a stronger convective heat transfer performance than that at L v = 1 mm and L v = 1.25 mm. Hence, appropriate reductions in L s and L v are more suitable for improving the convective heat transfer of the PCHE. 3.
At the same vertical distance (L v = 1.25 mm), the pressure drop decreases with the increase in L s . Therefore, when L s = 1.2 mm, the average pressure drop is the minimum, only 56.6% of the parallel arrangement. For the same L s of 1.2 mm, the pressure loss decreases by 34.9% on average as the vertical distance increases from 0.8 mm to 1.25 mm. Therefore, appropriate increases in L s and L v can reduce the pressure loss.

4.
Among all fin arrangements, the PEC average value of the staggered channel with L s = 1.2 mm and L v = 1.25 mm is the highest, which is 11.6% higher than that of the parallel channel, especially for a smaller value of Re. The comprehensive heat transfer performance can be improved by increasing the L s and L v.
Author Contributions: Conceptualization, S.Y. and Z.Z.; visualization, formal analysis and writing -original draft preparation, S.Y., Y.Z. and Z.Z.; methodology, validation and data curation, S.Y., Y.Z., Z.C. and M.Y. All authors have read and agreed to the published version of the manuscript.
Funding: This research was funded by Jiangsu marine and fishery science and technology innovation and extension project, grant number HY2017-8, and Zhenjiang funds for the key research and development project, grant number GY2016002-1.
Institutional Review Board Statement: Not applicable.

Conflicts of Interest:
The authors declare no conflict of interest.