Numerical Study on Thermal Hydraulic Performance of Supercritical LNG in Zigzag-Type Channel PCHEs

In this paper, we study a promising plate-type heat exchanger, the printed circuit heat exchanger (PCHE), which has high compactness and is suitable for high-pressure conditions as a vaporizer during vaporization. The thermal hydraulic performance of supercritical produce liquefied natural gas (LNG) in the zigzag channel of PCHE is numerically investigated using the SST κ-ω turbulence model. The thermo-physical properties of supercritical LNG from 6.5 MPa to 10MPa were calculated using piecewise-polynomial approximations of the temperature. The effect of the channel bend angle, mass flux and inlet pressure on local convection heat transfer coefficient, and pressure drop are discussed. The heat transfer and pressure loss performance are evaluated using the Nusselt and Euler numbers. Nu/Eu is proposed to evaluate the comprehensive heat transfer performance of PCHE by considering the heat transfer and pressure drop characteristics to find better bend angle and operating conditions. The supercritical LNG has a better heat transfer performance when bend angle is less than 15◦ with the mass flux ranging from 207.2 kg/(m2·s) to 621.6 kg/(m2·s), which improves at bend angle of 10◦ and lower compared to 15◦ at mass flux above 414.4 kg/(m2·s). The heat transfer performance is better at larger mass flux and lower operating pressures.


Introduction
Natural gas (NG) is an advantageous energy source for various applications due to its clean nature and its environmental and economic advantages [1].NG is liquefied to produce liquefied natural gas (LNG) for long-distance transportation and storage, and is regasified before terminal utilization [2].Therefore, an efficient and reliable LNG vaporizer is a key component in a LNG vaporization system.The common LNG vaporizers, such as intermediate fluid vaporizers, open rack vaporizers (ORVs), super ORVs and submerged combustion vaporizers [3][4][5] do not satisfy the requirement of high efficiency and compactness in finite volume vaporization processes.Hence, the printed circuit heat exchanger (PCHE), which is a prospective plate-type heat exchanger with high compactness that can operate under high pressure and low temperature, has been investigated by many researchers [6,7].
In recently years, four PCHE channel morphologies have been studied, namely the straight, zigzag, S-shape, and airfoil shapes.Figley et al. [8] conducted numerical simulations to investigate the thermal hydraulic performance of the straight-channel PCHE using helium.The thermal effectiveness and overall heat transfer coefficient were defined and calculated to describe the overall heat transfer performance of PCHE.Kim et al. [9] predicted the thermal performance by developing a mathematical expression of geometric parameters, material properties, and flow conditions to express the effectiveness of cross, parallel, and counterflow PCHEs.Yoon et al. [10] developed a friction factor and Nusselt number relationship of laminar flow in various bend angles for a semi-circular zigzag channel PCHE.Ishizuk et al. [11] studied heat transfer and flow characteristics of zigzag-type PCHE experimentally using supercritical carbon dioxide and obtained the total heat transfer efficiency and pressure drop.Tsuzuki et al. [12] performed numerical analysis of the transition section of a zigzag-type channel PCHE, and encountered the presence of vortices and local circulation flow.Ngo et al. [13,14] conducted three-dimensional simulations of the pressure drop and Nusselt number of PCHEs with S-shaped fins in a supercritical carbon dioxide loop.Kim et al. [15] studied airfoil PCHEs using numerical analysis and indicated that the stream line was smooth and the vortices and countercurrents disappeared in the airfoil channel.Zhao et al. [16] investigated the heat transfer characteristics of an airfoil PCHE numerically using supercritical LNG and optimized the arrangement of airfoil fins.
In spite of the superior pressure drop performance of PCHEs with S-shaped and airfoil fins, compared to straight and zigzag-type PCHEs, discontinuous fins lack durability at high pressure operating conditions, which is attributed to the confined junction area of the fins in the diffuser bonding [17].This in turn leads to an increase in manufacturing and maintenance costs [18,19].Therefore, it makes sense to further investigate continuous channels and optimize channel shapes.
Previously, investigations were mainly concerned with the heat transfer performance and pressure drop of zigzag PCHEs, but few combined the heat transfer and pressure drop performance to study the optimization of zigzag PCHEs in terms of bend angle and operating conditions.In addition, supercritical fluids can be used to improve heat transfer performance due to their favorable properties, like high density, low viscosity, and high thermal conductivity compared to conventional fluids [20].Most studies on the flow and heat transfer characteristics of fluids in PCHE have been conducted using helium and water [21][22][23], and many researchers also used supercritical water and supercritical carbon dioxide to investigate PCHEs [24,25].However, studies on the heat transfer performance of supercritical LNG in zigzag PCHE under low temperature and high pressure conditions are rare [16].
In this study, the flow and heat transfer characteristics of supercritical LNG in zigzag-type PCHEs were numerically simulated at an operating pressure of 6.5-10 MPa.The effects of bend angle and mass flux on heat transfer performance were also investigated.The local convection heat transfer coefficient and pressure drop of supercritical LNG under different conditions are discussed.Dimensionless parameters such as the Nusselt number (Nu) and Euler number (Eu) are analyzed to assess the heat transfer and pressure loss performance, Nu/Eu of the bend angles and operating conditions are investigated by considering the performance of heat transfer and the pressure drop of supercritical LNG in PCHEs.

Physical Model and Boundary Conditions
In this study, the thermal hydraulic performance of supercritical LNG in zigzag PCHEs is investigated.The cross flow PCHE core model with a full length of 400 mm using supercritical LNG in the cold side and R22 in the hot side is shown in Figure 1a.The supercritical LNG and R22 flow in the semicircular channel with a diameter of 1.5 mm, the solid is composed of steel with thermal conductive coefficient of 16.27 W/(m•K).In this paper, we only study the performance of supercritical LNG in the cold channel.Considering that the cold side of the PCHE contains hundreds of channels, it is unrealistic to consider all the channels, and it is therefore necessary to simplify the cold channels' model.Supercritical LNG flows in parallel in each cold channel, so some assumptions on its flow in the cold side are made.The mass flux is the same in every channel, and there is no temperature difference and heat loss between neighboring channels.The flow of supercritical LNG is steady and uniformly distributed.Based on these assumptions, the cold channel can be simplified to a single model with a geometry of 2 mm × 1.75 mm.The cross-section of the fluid channel is semicircular with a diameter of 1.5 mm (Figure 1b).The bend angles α vary from 0 • (which is a straight channel) to 45 • (Figure 1c).supercritical LNG in cold channels absorb heat from top and bottom hot channels.Three types of boundary conditions were applied in the model: fluid inlet, fluid outlet, and wall.The mass flow rate boundary condition was set at the inlet of the supercritical LNG channel whereas at the outlet the pressure-outlet boundary condition was applied (Figure 1b).The left and right walls of the single model are set to adiabatic boundary conditions, and the constant heat flux condition was used at top and bottom walls.The details of the boundary conditions are presented in Table 1.The adjacent cold channels do not exhibit temperature difference and heat transfer loss; only the supercritical LNG in cold channels absorb heat from top and bottom hot channels.Three types of boundary conditions were applied in the model: fluid inlet, fluid outlet, and wall.The mass flow rate boundary condition was set at the inlet of the supercritical LNG channel whereas at the outlet the pressure-outlet boundary condition was applied (Figure 1b).The left and right walls of the single model are set to adiabatic boundary conditions, and the constant heat flux condition was used at top and bottom walls.The details of the boundary conditions are presented in Table 1.In this paper, the operating pressure of LNG considered varies from 6.5 MPa to 10 MPa, which is supercritical pressure.Supercritical LNG has gas-like properties, such as low viscosity, and liquid-like characteristics, like high density and high thermal conductivity.The thermo-physical properties of supercritical LNG, i.e., density, specific heat, thermal conductivity and viscosity, are affected by temperature and pressure.The properties' values were obtained from the NIST Standard Reference Database (REFPROP) [26].For the numerical simulations, the temperature was changed from 121 K to 385 K.At such a large temperature difference, the properties of supercritical LNG change dramatically, using the average values will cause the inaccurate calculation results in ANSYS Fluent.Therefore, as shown in Figure 2, the thermal properties of supercritical LNG were approximated as piecewise-polynomial functions of temperature.The piecewise-polynomial functions of temperature at 10 MPa is shown in Table 2.The error percentages of various properties using the proposed approximation are shown in Figure 3.The errors were within ±2.5%, which indicates the fitted piecewise-polynomial function approximations are suitable.

Thermo-Physical Properties of Supercritical LNG
In this paper, the operating pressure of LNG considered varies from 6.5 MPa to 10 MPa, which is supercritical pressure.Supercritical LNG has gas-like properties, such as low viscosity, and liquidlike characteristics, like high density and high thermal conductivity.The thermo-physical properties of supercritical LNG, i.e., density, specific heat, thermal conductivity and viscosity, are affected by temperature and pressure.The properties' values were obtained from the NIST Standard Reference Database (REFPROP) [26].For the numerical simulations, the temperature was changed from 121 K to 385 K.At such a large temperature difference, the properties of supercritical LNG change dramatically, using the average values will cause the inaccurate calculation results in ANSYS Fluent.Therefore, as shown in Figure 2, the thermal properties of supercritical LNG were approximated as piecewise-polynomial functions of temperature.The piecewise-polynomial functions of temperature at 10 MPa is shown in Table 2.The error percentages of various properties using the proposed approximation are shown in Figure 3.The errors were within ±2.5%, which indicates the fitted piecewise-polynomial function approximations are suitable.

Numerical Method and Grid Independence
The commercial software FLUENT was used to solve the 3D numerical model.Considering the inlet parameters, the flow corresponded to turbulent flow regimes.Some turbulence models have been studied and used in the literature; these include the κ-ε standard model, the RNG κ-ε model, the shear-stress transport (SST) κ-ω model and the low Reynolds number turbulence model [27,28].In this study, the SST κ-ω model was used because of its more accurate results on heat transfer of supercritical fluids [29][30][31][32].
The governing equations for heat transfer were the continuity, momentum, and energy equations, respectively: Continuity equation: where ρ is the density, and u i is the velocity vector.
where p is the pressure, µ and µ t are the molecular and turbulent viscosities, respectively.Energy equation: where k eff is effective conductivity, k e f f = k + k t , and k t is the turbulent thermal conductivity.
The transport equations are expressed as follows: ) where ε is the turbulent kinetic energy dissipation rate, Ω is vorticity, and y is the distance from the wall.The constants and damping functions of the SST κ-ω model are shown in Table 3.The local convective heat transfer coefficient was calculated using Equation ( 10): where q is the constant heat flux from the top and bottom walls, T w is the wall temperature and T b is average temperature of the inlet and the outlet.Nu was defined as: where D h is the hydraulic diameter and λ is the local thermal conductivity of LNG, A is the cross-sectional area of the semicircular fluid channel and l is the circumference of the semicircular fluid channel section.
The local Fanning friction coefficient ( f ) was defined in terms of the pressure drop and is expressed by Equation ( 12): Energies 2019, 12, 548 7 of 19 where ∆P is the total pressure and was obtained from Fluent directly, ∆P f and ∆P a are the frictional and accelerated pressure drops, respectively, L is the channel length, ρ b and v b are the bulk density and velocity of LNG, respectively.The Reynolds number (Re) is given by Equation ( 14): The Euler number (Eu) is defined as Equation ( 15): For the solution methods, the SIMPLE algorithm was applied to establish the coupling of velocity and pressure.The momentum, turbulent kinetic energy, turbulent dissipation rate and energy were discretized using the second order upwind scheme.The calculation was considered to converge when the residuals were less than 10 −6 .
The mesh on the computational domain was generated using GAMBIT.The grid independence was verified to confirm numerical result accuracy.The mesh size of solid, fluid and boundary layer's scale in fluid affect the grid numbers.The influence of the grid numbers on the convective heat transfer coefficient is shown in Table 4. Case 4 has a larger relative error compared to the other cases.The heat transfer coefficient in Cases 1, 2, and 3 is nearly the same.The relative error of the heat transfer coefficient between Cases 1 and 7 is only 0.08%.Therefore, considering the calculation accuracy and time, the 2,988,329 grid nodes (Case 1), showing in Figure 4, was selected in the present work.
where ∆ is the total pressure and was obtained from Fluent directly,∆ and ∆ are the frictional and accelerated pressure drops, respectively, L is the channel length, and are the bulk density and velocity of LNG, respectively.The Reynolds number (Re) is given by Equation ( 14): The Euler number (Eu) is defined as Equation ( 15): For the solution methods, the SIMPLE algorithm was applied to establish the coupling of velocity and pressure.The momentum, turbulent kinetic energy, turbulent dissipation rate and energy were discretized using the second order upwind scheme.The calculation was considered to converge when the residuals were less than 10 −6 .
The mesh on the computational domain was generated using GAMBIT.The grid independence was verified to confirm numerical result accuracy.The mesh size of solid, fluid and boundary layer's scale in fluid affect the grid numbers.The influence of the grid numbers on the convective heat transfer coefficient is shown in Table 4. Case 4 has a larger relative error compared to the other cases.The heat transfer coefficient in Cases 1, 2, and 3 is nearly the same.The relative error of the heat transfer coefficient between Cases 1 and 7 is only 0.08%.Therefore, considering the calculation accuracy and time, the 2,988,329 grid nodes (Case 1), showing in Figure 4, was selected in the present work.

Model Validation
To validate the accuracy and reliability of the model, the simulation results of temperature difference and pressure drop were compared to previous experimental results [30].The experimental setup is shown in the PCHE cold channel was 520 mm, the inlet temperature was 102 K, and the operating pressure varied from 6.5 MPa to 10 MPa.In the simulation, a straight channel model with a length of 520 mm was selected, which was the same as the experimental case.The boundary conditions are shown in Figure 1b.Supercritical nitrogen was used as the working fluid to confirm the correctness the experiment.The inlet pressure changed from 6.5 MPa to 10 MPa and the inlet temperature was 102 K.The comparison of temperature difference and pressure drop between simulation and experimental results is listed in Table 5.The maximum errors of temperature difference and pressure drop are 2.1% and 10.25%, respectively.The simulation pressure drop was less than the experimental, which may be attributed to the following factors: (1) the channel was assumed to be smooth in the numerical study while the PCHE channel of the experiment was rough, (2) the header pressure drop of inlet and the outlet were neglected in the numerical study, but may have been large in the experiment, and (3) the deviation of temperature and pressure transmissions.The simulation results are in accordance with the experiment, illustrating that the simulation model and method were credible.

Model Validation
To validate the accuracy and reliability of the model, the simulation results of temperature difference and pressure drop were compared to previous experimental results [30].The experimental setup is shown in Figure 5. Since LNG is flammable and explosive, the straight-channel cross flow PCHE used supercritical nitrogen as the cold side fluid and R22 as the hot side fluid.The length of the PCHE cold channel was 520 mm, the inlet temperature was 102 K, and the operating pressure varied from 6.5 MPa to 10 MPa.In the simulation, a straight channel model with a length of 520 mm was selected, which was the same as the experimental case.The boundary conditions are shown in Figure 1b.Supercritical nitrogen was used as the working fluid to confirm the correctness the experiment.The inlet pressure changed from 6.5 MPa to 10 MPa and the inlet temperature was 102 K.The comparison of temperature difference and pressure drop between simulation and experimental results is listed in Table 5.The maximum errors of temperature difference and pressure drop are 2.1% and 10.25%, respectively.The simulation pressure drop was less than the experimental, which may be attributed to the following factors: (1) the channel was assumed to be smooth in the numerical study while the PCHE channel of the experiment was rough, (2) the header pressure drop of inlet and the outlet were neglected in the numerical study, but may have been large in the experiment, and (3) the deviation of temperature and pressure transmissions.The simulation results are in accordance with the experiment, illustrating that the simulation model and method were credible.

Results and Discussion
Compared with traditional vaporizers, the heat transfer performance of PCHE is better, but the pressure drop is larger due to the small size of the channel, resulting in an increase of operating costs.A number of studies have shown that a supercritical fluid can enhance heat transfer and reduce pressure drop.The supercritical LNG vaporized by the PCHE is suitable for long distance transport and utilization.In this paper, the flow and heat transfer characteristics of supercritical LNG were Energies 2019, 12, 548 9 of 19 studied at 6.5-10MPa, the performance of supercritical LNG at 10 MPa was discussed in detail in the following.

Effect of Bend Angles of the Zigzag Channel
The heat transfer performance of supercritical LNG is influenced significantly by the bend angle in the zigzag channel of a PCHE.The effect of the bend angles on the local convection heat transfer coefficient and bulk temperature of the LNG along the streamwise direction at a mass flux of 207.2 kg/(m 2 •s) and operating pressure of 10 MPa are shown in Figure 6.As the bend angle increases, the local convection heat transfer coefficient and bulk temperature increase, which is due to the fact that with the increase of the bend angle, disturbance and turbulence will increase.Moreover, the local heat transfer coefficient increases and then decreases along the streamwise direction, and reaches a peak when the bulk temperature is near the pseudo-critical temperature at bend of 0 • -15 • .This is because that the thermo-physical properties of supercritical LNG vary drastically at different temperatures, and the specific heat in particular reaches an extremum near the pseudo-critical temperature (Figure 2).However, at bend angles of 25 • -45 • , the convection heat transfer coefficient is greater at Np = 2-4.When the LNG flows into the channel, its velocity is not large in the Np = 2-4.The flow separation is not dramatic and the velocity of vortices is not much different that of the fluid.As the flow develops, the velocity increases.The velocity difference of vortices and fluid becomes large and the vortices and flow separation increase (as shown in Figure 7b), leading to a decrease in the convection heat transfer coefficient.In addition, the heat transfer coefficient is larger at the inlet due to the entrance effect.Figure 8 shows the pressure drop and Fanning friction coefficient ( ) at different bend angles along the streamwise direction.The pressure drop increases with the bend angles and increases along the streamwise direction.The density and viscosity of the supercritical LNG decrease as the temperature increases (as shown in Figure 2a,d), resulting in a velocity increase and thus in pressure drop.Figure 7 shows the velocity vectors of different cross-sections along the channel and velocity vectors of Np = 10-12 at different bend angles.The velocity increases with the increase of bend angles, which is attributed to the increase of turbulence (Figure 7a).Flow separation and reverse flow appear in larger bend angles (Figure 7b), which increases flow resistance, resulting in an increased pressure drop.The velocity increases along the flow direction, which also increase the pressure drop along the streamwise.The Fanning friction coefficient increases with the bend angle, which is the same trend observed in pressure drop.However, decreases along the flow direction; this difference between pressure drop and f is due to the increase of velocity.Figure 8 shows the pressure drop and Fanning friction coefficient ( f ) at different bend angles along the streamwise direction.The pressure drop increases with the bend angles and increases along the streamwise direction.The density and viscosity of the supercritical LNG decrease as the temperature increases (as shown in Figure 2a,d), resulting in a velocity increase and thus in pressure drop.Figure 7 shows the velocity vectors of different cross-sections along the channel and velocity vectors of Np = 10-12 at different bend angles.The velocity increases with the increase of bend angles, which is attributed to the increase of turbulence (Figure 7a).Flow separation and reverse flow appear in larger bend angles (Figure 7b), which increases flow resistance, resulting in an increased pressure drop.The velocity increases along the flow direction, which also increase the pressure drop along the streamwise.The Fanning friction coefficient f increases with the bend angle, which is the same trend observed in pressure drop.However, f decreases along the flow direction; this difference between pressure drop and f is due to the increase of velocity.
which is attributed to the increase of turbulence (Figure 7a).Flow separation and reverse flow appear in larger bend angles (Figure 7b), which increases flow resistance, resulting in an increased pressure drop.The velocity increases along the flow direction, which also increase the pressure drop along the streamwise.The Fanning friction coefficient increases with the bend angle, which is the same trend observed in pressure drop.However, decreases along the flow direction; this difference between pressure drop and f is due to the increase of velocity.Figure 9 shows the Nusselt (Nu) and Euler (Eu) numbers along the streamwise direction at different bend angles.It can be seen that Nu increases with the bend angle, and reaches its maximum value near the pseudo-critical temperature, then decreases along the streamwise direction.This is because the thermal conductivity decreases intensely as the temperature rises before pseudo-critical temperature and rises slightly when the temperature surpasses the pseudo-critical value, reaching a minimum near the pseudo-critical temperature, which leads to a maximum for Nu near the pseudo-critical temperature.The Euler number increases as the bend angle rises, which is consistent with the change in pressure drop.Figure 9 shows the Nusselt (Nu) and Euler (Eu) numbers along the streamwise direction at different bend angles.It can be seen that Nu increases with the bend angle, and reaches its maximum value near the pseudo-critical temperature, then decreases along the streamwise direction.This is because the thermal conductivity decreases intensely as the temperature rises before pseudo-critical temperature and rises slightly when the temperature surpasses the pseudo-critical value, reaching a minimum near the pseudo-critical temperature, which leads to a maximum for Nu near the pseudocritical temperature.The Euler number increases as the bend angle rises, which is consistent with the change in pressure drop.The objective of improving the heat exchanger's performance can be realized by increasing heat transfer performance and reducing pressure drop.It is therefore essential to comprehensively consider the heat transfer and pressure loss characteristics of supercritical LNG in PCHE.In this study, the ratio of Nusselt to Euler numbers (Nu/Eu) is proposed to evaluate the performance of supercritical LNG in the channel, where a larger ratio indicates better heat transfer performance.Figure 10 shows Nu/Eu at different bend angles.Nu/Eu reaches its peak value at a bend angle of 10°.When the bend angle exceeds 10°, the growth rate of Nu is much less than that of Eu.Nu and Eu at 0° and 5° are nearly the same, while from 5° to 10°, the increase of Nu is much greater than that of Eu.The objective of improving the heat exchanger's performance can be realized by increasing heat transfer performance and reducing pressure drop.It is therefore essential to comprehensively consider the heat transfer and pressure loss characteristics of supercritical LNG in PCHE.In this study, the ratio of Nusselt to Euler numbers (Nu/Eu) is proposed to evaluate the performance of supercritical LNG in the channel, where a larger ratio indicates better heat transfer performance.Figure 10 shows Nu/Eu at different bend angles.Nu/Eu reaches its peak value at a bend angle of 10 • .When the bend angle exceeds 10 • , the growth rate of Nu is much less than that of Eu.Nu and Eu at 0 • and 5 • are nearly the same, while from 5 • to 10 • , the increase of Nu is much greater than that of Eu.
study, the ratio of Nusselt to Euler numbers (Nu/Eu) is proposed to evaluate the performance of supercritical LNG in the channel, where a larger ratio indicates better heat transfer performance.Figure 10 shows Nu/Eu at different bend angles.Nu/Eu reaches its peak value at a bend angle of 10°.When the bend angle exceeds 10°, the growth rate of Nu is much less than that of Eu.Nu and Eu at 0° and 5° are nearly the same, while from 5° to 10°, the increase of Nu is much greater than that of Eu.

Effect of Mass Flux
The effect of mass flux on flow and heat transfer performance of supercritical LNG was investigated at the bend angle of 10° and an operating pressure of 10 MPa.As shown in Figure 11, the local convective heat transfer coefficient and pressure drop increase significantly as the mass flux increases because of the enhancement of turbulent flow.When the mass flux is increased by 2 times, the local heat transfer coefficient increased 1.4 times, and at the same time, the pressure drop

Effect of Mass Flux
The effect of mass flux on flow and heat transfer performance of supercritical LNG was investigated at the bend angle of 10 • and an operating pressure of 10 MPa.As shown in Figure 11, the local convective heat transfer coefficient and pressure drop increase significantly as the mass flux increases because of the enhancement of turbulent flow.When the mass flux is increased by 2 times, the local heat transfer coefficient increased 1.4 times, and at the same time, the pressure drop increases 3.3 times.The Nu and Eu are shown in Figure 12.The Nu increases as the mass flux is raised.However, at the last third of the channel, Nu peaks at a mass flux of 301.8 kg/(m 2 •s), and then decreases at further mass flux increase.This is because as mass flux is increased, the temperature of LNG decreases, its viscosity increases and its velocity decreases.When the heat flux is kept constant, the heat absorbed by the LNG per unit volume is reduced.Figure 14 shows the impact of mass flux on Nu/Eu at different channel bend angles.The Nu/Eu is significantly reduced when the bend angle exceeds 15°, indicating that the increase of pressure drop is much higher than that of heat transfer performance.Consequently, the comprehensive heat transfer performance is not good when the bend angles exceeds 15°.When the mass flux varies from 207.2 kg/(m 2 •s) to 621.6 kg/(m 2 •s), Nu/Eu are higher when the bend angle is less than 15°.However, Nu/Eu at bend angles of 10° and lower are increased compared to 15° at mass fluxes above 414.4kg/(m 2 •s).It can be concluded that the supercritical LNG in the PCHE has better heat transfer performance when the bend angle is less than 15° with the mass flux ranging from 207.2 kg/(m 2 •s) to 621.6 kg/(m 2 •s), and that it improves at bend angles of 10° and lower compared to 15° at mass fluxes Figure 14 shows the impact of mass flux on Nu/Eu at different channel bend angles.The Nu/Eu is significantly reduced when the bend angle exceeds 15 • , indicating that the increase of pressure drop is much higher than that of heat transfer performance.Consequently, the comprehensive heat transfer performance is not good when the bend angles exceeds 15 • .When the mass flux varies  Figure 14 shows the impact of mass flux on Nu/Eu at different channel bend angles.The Nu/Eu is significantly reduced when the bend angle exceeds 15°, indicating that the increase of pressure drop is much higher than that of heat transfer performance.Consequently, the comprehensive heat transfer performance is not good when the bend angles exceeds 15°.When the mass flux varies from 207.2 kg/(m 2 •s) to 621.6 kg/(m 2 •s), Nu/Eu are higher when the bend angle is less than 15°.However, Nu/Eu at bend angles of 10° and lower are increased compared to 15° at mass fluxes above 414.4kg/(m 2 •s).It can be concluded that the supercritical LNG in the PCHE has better heat transfer performance when the bend angle is less than 15° with the mass flux ranging from 207.2 kg/(m 2 •s) to 621.6 kg/(m 2 •s), and that it improves at bend angles of 10° and lower compared to 15° at mass fluxes above 414.4kg/(m 2 •s).

Effect of Inlet Pressure
At a bend angle of 10 • and a mass flux of 207.2 kg/(m 2 •s), the inlet pressure was varied from 6.5 MPa to 10 MP.The corresponding local heat transfer coefficient and pressure drop values are shown in Figure 15.Before the pseudo-critical temperature, the specific heat and thermal conductivity are slightly affected by the inlet pressure (as shown in Figure 2b,c), so the convection heat transfer coefficient changes slightly with the inlet pressure.After the pseudo-critical temperature, the inlet pressure effect on the local convective heat transfer coefficient is greater because of the specific heat is influenced by inlet pressure more.In addition, the reduction rate of the specific heat rises rapidly as the inlet pressure is reduced.Therefore, the local convection heat transfer coefficient decreases with the decrease of inlet pressure along the streamwise direction.This shows that the specific heat depends on inlet pressure and has a great influence on local convective heat transfer coefficient when temperature exceeds the pseudo-critical temperature.
effected by the inlet pressure.After Np = 10, the supercritical LNG has gas-like properties, so the influence of inlet pressure on density and dynamic viscosity is greater, leading to the large effect on velocity and causing the pressure drop to change more obviously with the inlet pressure.At the last portion of the channel, the pressure drop reduced, because the last pitch of the channel is straight, which reduces turbulence.The effect of inlet pressure on Nu and Eu are shown in Figure 16, where both Nu and Eu decrease with as inlet pressure increased.Nu is inversely proportional to thermal conductivity, so as the inlet pressure increases, the thermal conductivity increased, which decreases Nu.Eu is increased as the inlet pressure is reduced, which the same behavior is observed in pressure drop, indicating that the pressure drop performance is larger when the inlet pressure is lower.With an increase of inlet pressure, the pressure drop decreases.As inlet pressure rises, the density and dynamic viscosity are larger (Figure 2a,d), which lowers the velocity of supercritical LNG.Hence, the pressure drop is reduced as the inlet pressure is increased.However, up until Np = 10, the pressure drop is uninfluenced by the changing inlet pressure.This is because the supercritical LNG has liquid-like properties, so the influence of inlet pressure on density and dynamic viscosity as well as velocity is small (as shown in Figure 2), which leads to pressure drop being only slightly effected by the inlet pressure.After Np = 10, the supercritical LNG has gas-like properties, so the influence of inlet pressure on density and dynamic viscosity is greater, leading to the large effect on velocity and causing the pressure drop to change more obviously with the inlet pressure.At the last portion of the channel, the pressure drop reduced, because the last pitch of the channel is straight, which reduces turbulence.
The effect of inlet pressure on Nu and Eu are shown in Figure 16, where both Nu and Eu decrease with as inlet pressure increased.Nu is inversely proportional to thermal conductivity, so as the inlet pressure increases, the thermal conductivity increased, which decreases Nu.Eu is increased as the inlet pressure is reduced, which the same behavior is observed in pressure drop, indicating that the pressure drop performance is larger when the inlet pressure is lower.Although both Nu and Eu increase with a decrease in inlet pressure, Nu/Eu increases, as shown in Figure 17.This is because with the decrease of pressure, the increase rate of Nu is larger than that of Eu.Nu/Eu reaches its maximum at 6.5 MPa, indicating that supercritical LNG will have better heat transfer performance in PCHE at a lower inlet pressure.Although both Nu and Eu increase with a decrease in inlet pressure, Nu/Eu increases, as shown in Figure 17.This is because with the decrease of pressure, the increase rate of Nu is larger than that of Eu.Nu/Eu reaches its maximum at 6.5 MPa, indicating that supercritical LNG will have better heat transfer performance in PCHE at a lower inlet pressure.Although both Nu and Eu increase with a decrease in inlet pressure, Nu/Eu increases, as shown in Figure 17.This is because with the decrease of pressure, the increase rate of Nu is larger than that of Eu.Nu/Eu reaches its maximum at 6.5 MPa, indicating that supercritical LNG will have better heat transfer performance in PCHE at a lower inlet pressure.

Conclusions
In this study, the flow and heat transfer characteristics of supercritical LNG in zigzag channel of PCHE are numerically investigated at different operating conditions.Some conclusions can be drawn as follows: (1) The local convection heat transfer coefficient rises and then falls along the streamwise direction, with the peak value appearing at the pseudo-critical temperature.The pressure drop also increases along the streamwise direction.(2) As the channel bend angle is increased, the local convection heat transfer coefficient and pressure drop rise, and so do the Nu and Euler numbers.The enhancement of heat transfer capability of supercritical LNG is mainly owed to increased turbulence.The increase of pressure

Conclusions
In this study, the flow and heat transfer characteristics of supercritical LNG in zigzag channel of PCHE are numerically investigated at different operating conditions.Some conclusions can be drawn as follows: (1) The local convection heat transfer coefficient rises and then falls along the streamwise direction, with the peak value appearing at the pseudo-critical temperature.The pressure drop also increases along the streamwise direction.(2) As the channel bend angle is increased, the local convection heat transfer coefficient and pressure drop rise, and so do the Nu and Euler numbers.The enhancement of heat transfer capability of supercritical LNG is mainly owed to increased turbulence.The increase of pressure drop is mainly due to the rising of velocity and the increase of flow resistance caused by the existence of vortices.

Figure 1 .
Figure 1.Schematic diagram of the physical model: (a) Schematic diagram of cross flow printed circuit heat exchanger (PCHE) model; (b) The computational single channel and boundary conditions; (c) Top view of zigzag channel.

Figure 1 .
Figure 1.Schematic diagram of the physical model: (a) Schematic diagram of cross flow printed circuit heat exchanger (PCHE) model; (b) The computational single channel and boundary conditions; (c) Top view of zigzag channel.

Figure 4 .
Figure 4. Cross section of computational grids.Figure 4. Cross section of computational grids.

Figure 4 .
Figure 4. Cross section of computational grids.Figure 4. Cross section of computational grids.

Figure 5 .
Since LNG is flammable and explosive, the straight-channel cross flow PCHE used supercritical nitrogen as the cold side fluid and R22 as the hot side fluid.The length of Energies 2019, 12, 548 8 of 19

Figure 5 .
Figure 5. Schematic diagram of experimental setup.Figure 5. Schematic diagram of experimental setup.

Figure 5 .
Figure 5. Schematic diagram of experimental setup.Figure 5. Schematic diagram of experimental setup.

Energies 2019 , 19 Figure 6 .
Figure 6.Local convection heat transfer coefficient and bulk temperature at different bend angles along the streamwise direction.

Figure 6 .
Figure 6.Local convection heat transfer coefficient and bulk temperature at different bend angles along the streamwise direction.

Figure 7 .
Figure 7. (a) Velocity vectors of different cross-sections along the channel with bend angles of 0° to 45°; (b) Velocity vectors of Np = 10-12 at bend angles of 0° to 45°.

Figure 8 .
Figure 8. Pressure drop and fanning friction coefficient at different bend angles along the streamwise direction.

Figure 7 .
Figure 7. (a) Velocity vectors of different cross-sections along the channel with bend angles of 0 • to 45 • ; (b) Velocity vectors of Np = 10-12 at bend angles of 0 • to 45 • .

Figure 7 .
Figure 7. (a) Velocity vectors of different cross-sections along the channel with bend angles of 0° to 45°; (b) Velocity vectors of Np = 10-12 at bend angles of 0° to 45°.

Figure 8 .
Figure 8. Pressure drop and fanning friction coefficient at different bend angles along the streamwise direction.

Figure 8 . 19 Figure 9 .
Figure 8. Pressure drop and fanning friction coefficient at different bend angles along the streamwise direction.Energies 2019, 12, x FOR PEER REVIEW 12 of 19

Figure 9 .
Figure 9. Local Nusselt number (Nu) and Euler (EU) numbers at different bend angles along the streamwise direction.

Figure 10 .
Figure 10.Nu/Eu at different bend angles along the streamwise direction.

Figure 10 .
Figure 10.Nu/Eu at different bend angles along the streamwise direction.
Energies 2019, 12, x FOR PEER REVIEW 13 of 19 increases 3.3 times.The Nu and Eu are shown in Figure 12.The Nu increases as the mass flux is raised.However, at the last third of the channel, Nu peaks at a mass flux of 301.8 kg/(m 2 •s), and then decreases at further mass flux increase.This is because as mass flux is increased, the temperature of LNG decreases, its viscosity increases and its velocity decreases.When the heat flux is kept constant, the heat absorbed by the LNG per unit volume is reduced.Eu increases as the mass flux decreases.The pressure drop and density increase as mass flux rises, but 2 decreased.The increase rate of 2 is more than that of ∆ , which results in an increase of Eu as mass flux is reduced.

Figure 11 .
Figure 11.Effect of mass flux on local convection heat transfer coefficient and pressure drop along the streamwise direction.

Figure 11 .
Figure 11.Effect of mass flux on local convection heat transfer coefficient and pressure drop along the streamwise direction.

Figure 11 .
Figure 11.Effect of mass flux on local convection heat transfer coefficient and pressure drop along the streamwise direction.

Figure 12 .
Figure 12.Effect of mass flux on Nu and Eu along the streamwise direction.

Figure 12 .
Figure 12.Effect of mass flux on Nu and Eu along the streamwise direction.Eu increases as the mass flux decreases.The pressure drop and density increase as mass flux rises, but v 2 b decreased.The increase rate of ρ b v 2 b is more than that of ∆P, which results in an increase of Eu as mass flux is reduced.The influence of mass flux on Nu/Eu is shown in Figure 13.The Nu/Eu is increased as the mass flux is raised.However, at the last third of the channel, Nu/Eu peaks at a mass flux of 301.8 kg/(m 2 •s).The figure suggests that the heat transfer performance of the whole channel is improved as the mass flux increases, but with the development of the fluid flow, the local heat transfer performance is reduced at the last third of the channel, owing to the reduction of heat absorbed capacity by the unit volume fluid.Energies 2019, 12, x FOR PEER REVIEW 14 of 19

Figure 13 .
Figure 13.Effect of mass flux on Nu/Eu along the streamwise direction.

Figure 13 .
Figure 13.Effect of mass flux on Nu/Eu along the streamwise direction.
2 kg/(m 2 •s) to 621.6 kg/(m 2 •s), Nu/Eu are higher when the bend angle is less than 15 • .However, Nu/Eu at bend angles of 10 • and lower are increased compared to 15 • at mass fluxes above 414.4kg/(m 2 •s).It can be concluded that the supercritical LNG in the PCHE has better heat transfer performance when the bend angle is less than 15 • with the mass flux ranging from 207.2 kg/(m 2 •s) to 621.6 kg/(m 2 •s), and that it improves at bend angles of 10 • and lower compared to 15 • at mass fluxes above 414.4kg/(m 2 •s).

Figure 13 .
Figure 13.Effect of mass flux on Nu/Eu along the streamwise direction.

Figure 14 .
Figure 14.Effect of mass flux on Nu/Eu at different bend angles.Figure 14.Effect of mass flux on Nu/Eu at different bend angles.

Figure 14 .
Figure 14.Effect of mass flux on Nu/Eu at different bend angles.Figure 14.Effect of mass flux on Nu/Eu at different bend angles.

Figure 15 .
Figure 15.Effect of inlet pressure on local convection heat transfer coefficient and pressure drop along the streamwise direction.

Figure 15 .
Figure 15.Effect of inlet pressure on local convection heat transfer coefficient and pressure drop along the streamwise direction.

Energies 2019 , 19 Figure 16 .
Figure 16.Effect of inlet pressure on Nu and Eu along the streamwise direction.

Figure 16 .
Figure 16.Effect of inlet pressure on Nu and Eu along the streamwise direction.

Figure 16 .
Figure 16.Effect of inlet pressure on Nu and Eu along the streamwise direction.

Figure 17 .
Figure 17.Effect of inlet pressure on Nu/Eu along the streamwise direction.

Figure 17 .
Figure 17.Effect of inlet pressure on Nu/Eu along the streamwise direction.

( 3 )
The local convective heat transfer coefficient and pressure drop increase significantly as the mass flux is increased due to the enhancement of turbulent flow.When the mass flux is increased by 2-fold, the local heat transfer coefficient rises by 1.4 times, and the pressure drop increases 3.3 times.The Nu increases as mass flux is increased.However, at the last third of the channel, Nu decreases as the mass flux is raised because of the decreased heat per unit volume absorbed by the LNG.This suggests that when the mass flux is raised, the heat transfer performance of the whole channel is better, but with the development of the fluid's flow, the local heat transfer performance is reduced at the last third of the channel owing to the reduction of heat-absorbed capacity by the unit volume fluid.(4) The improvement of heat transfer performance with bend angle depends on the mass flux.The supercritical LNG has better heat transfer performance when the bend angle is less than 15 • when the mass flux ranges from 207.2 kg/(m 2 •s) to 621.6 kg/(m 2 •s), and improves at bend angles of 10 • and lower compared to 15 • at mass fluxes above 414.4kg/(m 2 •s).(5) Before the pseudo-critical temperature, the local convective heat transfer coefficient changes little with the inlet pressure, while it increases when the temperature surpasses pseudo-critical

Table 1 .
Boundary conditions in detail.

Table 1 .
Boundary conditions in detail.

Table 3 .
Constants and functions used in the shear-stress transport (SST) model.

Table 3 .
Constants and functions used in the shear-stress transport (SST) model. *

Table 5 .
Relative error of simulation and experiment results.

Pressure (MPa) Temperature Difference (K) Relative Error (%) Pressure Difference (Pa) Relative Error (%) Experiment Simulation Experiment Simulation
point.The pressure drop is reduced as the inlet pressure increases.Nu and EU decrease with increasing inlet pressure, while Nu/Eu reaches a maximum at 6.5 MPa.The results show that supercritical LNG has a better heat transfer performance in zigzag channel of PCHE at lower operating pressures.Conceptualization and supervision, Z.Z.; formal analysis and data curation, Y.Z.; validation, methodology and software, X.M., X.C., S.L., S.Y. and Z.Z.; writing-original draft preparation, Y.Z. and Z.Z.The authors gratefully acknowledge that this work was supported by Jiangsu marine and fishery science and technology innovation and extension project (HY2017-8), Zhenjiang funds for the key research and development project (GY2016002-1) and National Key R&D Program of China (2018YFC0310400).The authors declare no conflict of interest. Funding: