Fast Calculation of Supercritical Carbon Dioxide Flow, Heat Transfer Performance, and Mass Flow Rate Matching Optimization of Printed Circuit Heat Exchangers Used as Recuperators

Abstract

Printed circuit heat exchangers (PCHEs) are widely used as recuperators in the supercritical carbon dioxide (S-CO₂) Brayton cycle design. The variation of heat sources will have a great impact on the heat transfer effect of the recuperator. It is of interest to study the fast calculation of flow and heat transfer performance of PCHEs under different operating conditions to obtain the optimal comprehensive performance and provide guidance for the operation control strategy analysis. Herein, a fast calculation method is established through a one-dimensional model of a PCHE based on Modelica. The effects of working medium mass flow rate and inlet temperature on the flow and heat transfer process are analyzed from the three aspects of heat transfer rate, flow pressure drop, and comprehensive performance, and the mass flow rate matching optimization is realized. The results show that increased mass flow rate increases heat transfer rate and flow pressure drop. The efficiency evaluation coefficient (EEC) has a maximum value at which the mass flow rate values of the cold and hot channels are best matched, and the comprehensive performance is optimal. When the mass flow rate of the heat channel is 4.8 g/s, the maximum EEC is 1.42, corresponding to the mass flow rate of the cold channel, 4.2 g/s. Compared with the design condition, the heat transfer rate increases by 62.1%, and the total pump power increases by 14.2%. When the cold channel inlet temperature increases, EEC decreases rapidly, whereas EEC increases when the hot channel inlet temperature increases. The conclusions can provide theoretical support for the design and operation of PCHEs.

1. Introduction

The supercritical carbon dioxide Brayton cycle (SCBC) has a wide application prospect in nuclear power [1,2], ship waste heat recovery [3,4], and solar energy [5,6], owing to its advantages of high-power density, a wide range of applicable heat sources, and high thermal efficiency [7]. The simple regeneration layout of the SCBC is always considered the reference layout [5], which consists of a turbine, compressor, cooler, heater, and recuperator. As the system operates beyond the critical point, the cycle pressure ratio of the SCBC is much smaller compared with the steam Rankine cycle, and the turbine outlet temperature is relatively high. Therefore, a large amount of heat must be recuperated to increase the thermal efficiency [7]. The internal recoverable heat of SCBC systems can reach 60–70% of the total heat exchange amount [8], and the recuperator is one of its core components [6], which has a significant impact on the thermal efficiency of the cycle.

A printed circuit heat exchanger (PCHE) is widely used in the SCBC, as heat exchangers benefit from its excellent performance. Printed circuit heat exchangers (PCHEs) have compact structures, can withstand high temperatures and high pressure [9], and are small, lightweight, and easily modularized [10]. The heat exchange surface density can reach up to 2500 m²/m³ [11], and the heat exchange efficiency is as high as 98% [12]. They are often used as a recuperator for the SCBC, and their performance will have an important impact on the SCBC.

A PCHE is a dividing wall-type heat exchanger. Cold and hot fluids are separated by a heat exchanger matrix. Heat transfer from hot fluid to cold fluid is realized by convection and conduction. Its flow and heat transfer characteristics are mainly affected by channel structure, fluid property, and operating parameters. Theoretical analysis [13], experimental tests [14,15], numerical simulation [16,17], and other methods are commonly used to study heat exchangers. Some scholars use three-dimensional numerical simulation methods to conduct the calculation of straight channel [18,19,20,21], Zigzag channel [22,23,24,25], S-shaped fins channel [26,27,28,29], and airfoil fins channel [30,31,32,33] structures and obtained detailed temperature and velocity fields. Straight channels have a simple structure but poor heat transfer performance. Zigzag channels have better heat transfer performance but implement turns that cause flow separation and significant pressure loss. Discrete-fin channels, especially airfoil fin channels, have better comprehensive heat transfer performance compared to continuous channels. Furthermore, they analyzed the influence of geometric structure and operating parameters on the flow and heat transfer performance of supercritical carbon dioxide (S-CO₂) in PCHEs.

However, 3D numerical simulation is a complex process with a high calculation cost, so it is impossible to carry out a wide-range calculation. The results obtained correspond to specific geometric structures and operating conditions, which are highly targeted but difficult to expand and use.

In transient calculations, operation control strategy analysis, and other application scenarios, the overall parameters, such as heat transfer rate and pressure drop of PCHE, are often focused and require high calculation speed, so using a one-dimensional model for simulated calculation is more appropriate. The one-dimensional model of S-CO₂ flow and heat transfer in PCHE can be embedded in the whole model of the S-CO₂ Brayton cycle for transient simulation, control strategy analysis, and optimization.

Marchionni et al. [34] modeled the heat transfer process in the PCHE channel in one and three dimensions, and the results show that the results obtained by the two modeling methods are very consistent. Ding et al. [35] developed the PCHE one-dimensional flow and heat transfer model in Modelica-based simulation software according to the correlation between S-CO₂ flow and heat transfer. Both steady and transient results verify the reliability of the PCHE model. Sui et al. [36] realized one-dimensional PCHE numerical simulation by using Modelica-based simulation software and simulated two typical transient working conditions, “shutdown and startup” and “load following,” which can better simulate the law of working medium temperature changing with time in the hot and cold channels. Chen et al. [37] designed a reduced-scale zigzag channel PCHE model and studied the dynamic behavior subject to inlet temperature variations and mass flow rate step changes on both the hot and cold sides. Jiang et al. [38] developed one-dimensional design and dynamic models for PCHEs utilized in the SCBC. The design model is used to determine the optimal geometry parameters by minimizing the metal mass. The dynamic model is used to predict transient behavior and can be easily implemented into system-level models. Hu et al. [39] introduced a one-dimensional dynamic model to study the thermodynamic parameter evolution process of the recuperator in an S-CO₂ Brayton cycle, which considered the S-CO₂ physical properties of its temperature and pressure, as well as the distribution of the thermodynamic parameters along the flow direction. The recuperator’s startup and generation load ratio reduction processes were simulated. Albright et al. [40] investigated control methods for mitigating oscillations in flow conditions and power demand load, especially as the load ramp rate is increased and the load setpoint is more closely tracked. The PCHEs are modeled using a one-dimensional dynamic model.

All these indicate that the one-dimensional modeling and simulation of PCHEs meet the needs of the above application scenarios in terms of calculation accuracy and solution speed.

The SCBC suits various kinds of heat sources, and its operating conditions will change greatly with the change of external conditions. The SCBC system applied to nuclear or coal-fired power needs to track the load change of user demand. Solar energy and industrial waste heat as heat sources for SCBC systems are fluctuating. The variation of the heat source will have a great impact on the heat transfer effect of the recuperator and then cause inefficient or unstable operation of the whole system. Therefore, it is necessary to quickly calculate the flow and heat transfer characteristics of PCHEs, analyze the flow and heat transfer performance of PCHEs under different operating conditions, and provide theoretical support for the design and operation of PCHEs.

As one of the key pieces of equipment of the SCBC, the heat transfer of the recuperator has an important influence on thermal efficiency, which is greatly influenced by the inlet parameters. Accurate and rapid acquisition of the flow heat transfer performance of CO₂ in recuperators under different working conditions is of great significance for the application of the SCBC. The previous analysis shows that one-dimensional modeling simulation has both fast calculations under multiple working conditions and the accuracy of overall parameters.

This paper first presents the one-dimensional flow and heat transfer mathematical model of PCHEs and verifies the established model’s accuracy by comparing it with the actual thermal-hydraulic performance parameters. Second, the influence of working medium mass flow rate and inlet temperature on the flow and heat transfer process is analyzed from the perspectives of heat exchange rate, flow pressure drop, and comprehensive performance to further achieve mass flow rate matching optimization and obtain the optimal comprehensive performance. Finally, conclusions are given on the influence of the mass flow rate and inlet temperature on PCHE performance and the optimal matching of cold and hot mass flow rates.

2. Mathematical Model and Result Verification

2.1. Mathematical Model

A PCHE is composed of a large number of cold and hot channels, which have good symmetry. One pair of cold and hot channels can be selected for modeling and simulation, as shown in Figure 1. It consists of a group of cold and hot channels separated by a metal matrix, and the fluids on both sides flow in opposite directions. The heat transfer from hot fluid to cold fluid is realized by heat convection between the fluids and the wall and conduction in the metal matrix. The size of the PCHE channel in the mainstream direction is two orders of magnitude higher than that in other directions, and the variation of thermal-hydraulic parameters in the mainstream direction is more significant. Therefore, the flow and heat transfer process of S-CO₂ in the PCHE is reduced to one-dimensional flow and heat transfer. The one-dimensional flow and heat transfer model is unable to obtain the parameter distribution over the channel cross-section.

The following assumptions have been made to simplify the solving process of the flow and heat transfer of PCHE:

(1)

The velocity only changes in the mainstream direction;

(2)

The physical parameters are evenly distributed in the cross-section;

(3)

The cross-section area is identical everywhere along the channel length;

(4)

The effects of gravity are ignored;

(5)

The heat losses to the environment are ignored.

The established 1D model for the PCHE can quickly obtain the overall parameters, such as heat transfer rate and pressure drop, and can be easily implemented into system-level models for cycle performance evaluations.

The solution domain is meshed in the flow direction, and the governing equations of fluid in cold and hot channels are the same. According to the conservation of mass, momentum, and energy, the governing equations of fluid flow and heat transfer in a one-dimensional channel [36] can be established as follows.

$$V_i\frac{d{\rho }_i}{dt}=w_{i-1:i}-w_{i:i+1}$$

(1)

$$\frac{l}{A}\frac{d{w}_i}{dt}=p_{i-1:i}-p_{i:i+1}-\Delta p_{fric,i}$$

(2)

$$V_i{\rho }_i{c}_{v,i}\frac{d{T}_i}{dt}=w_{i-1:i}{h}_{i-1:i}-w_{i:i+1}{h}_{i:i+1}+Q_i$$

(3)

In the above equations, V_i (m³), l (m), A (m²), ρ_i (kg·m⁻³), w_i (kg·s⁻¹), c_v,i (J·kg⁻¹·K⁻¹), and T_i (K) represent the volume, flow direction length, cross-sectional area, average density, average mass flow rate, average specific heat at constant volume, and average temperature of fluid element i, respectively. w_i:j (kg·s⁻¹), p_i:j (Pa), and h_i:j (J·kg⁻¹) represent the average mass flow rate, the average pressure, and the average specific enthalpy at the interface between the fluid element i and j. ∆p_fric,i (Pa) indicates the frictional pressure loss in fluid element i. Q_i (W) indicates the heat transfer rate by convection in fluid unit i.

∆p_fric,i is calculated as Equation (4), where the flow pressure drop can be calculated by the Fanning friction coefficient, which can be calculated according to the empirical correlation.

$$\Delta p_{fric,i}=C_{f,i}\frac{l{\rho }_i{u}_i^2}{D_{hyd}}.$$

(4)

In the above equation, C_f,i, and u_i (m·s⁻¹) are the average Fanning friction factor and the average velocity in the mainstream direction of fluid element i, respectively, and D_hyd (m) is the equivalent hydraulic diameter of the flow channel.

Q_i is calculated in Equation (5), where the total heat transfer coefficient can be obtained by the heat resistance network method. The calculation formula is shown in Equation (6), the convection heat transfer coefficient between the fluid and the wall can be calculated by the Nu, as shown in Equation (7), and the Nu can be calculated by the relevant empirical formula.

$$Q_i=k_i{A}_{s,i}(T_i-T_{s,i}),$$

(5)

$$\frac{1}{k_i}=\frac{{\delta }_s/2}{{\lambda }_{s,i}}+\frac{1}{h_i},$$

(6)

$$N{u}_i=\frac{h{c}_i{D}_{hyd}}{{\lambda }_i}.$$

(7)

In the above equations, k_i (W·K⁻¹·m⁻²) is the overall heat transfer coefficient of element i, and A_s,i (m²), T_s,i (K), δ_s (m), and λ_s,i (W·K⁻¹·m⁻¹) are the area of contact with fluid, average temperature, average thickness, and average thermal conductivity coefficient of metal matrix element i, respectively. hc_i (W·K⁻¹·m⁻²), Nu_i, and λ_i (W·K⁻¹·m⁻¹) are the average heat convection coefficient, average Nusselt number, and average thermal conductivity coefficient of fluid element i, respectively. Fanning friction factor, heat convection coefficient, and Nusselt number are calculated using the Gnielinski correlation [39].

The energy conservation equation in the metal matrix is:

$$V_{s,i}{\rho }_{s,i}{c}_{ps,i}\frac{d{T}_{s,i}}{dt}=\sum Q_i.$$

(8)

In the equation, V_s,i (m³), ρ_s,i (kg·m⁻³), and c_ps,i (J·kg⁻¹·K⁻¹) denote the volume, average density, and average specific heat at constant pressure in metal matrix element i, respectively. ∑Q_i (W) represents the net heat transfer rate between the metal matrix and fluid.

To quantitatively analyze and compare the comprehensive performance of heat exchangers, the efficiency evaluation coefficient (EEC) [41] is referenced:

$$EEC=\frac{Q/Q_0}{N_p/N_{p0}},$$

(9)

$$N_p=\frac{w}{\rho }\Delta p.$$

(10)

In Equation (9), Q (W) and N_p (W) are the heat transfer rate and the pump power, respectively. Subscript 0 represents the design condition; the numerator is the income ratio of the heat exchanger; the denominator is the cost ratio of the heat exchanger. EEC reflects the heat exchanger’s comprehensive enhanced heat transfer performance under certain energy consumption. The larger the EEC, the greater the heat transfer rate under the same pump power. In Equation (10), w (kg·s⁻¹), ρ (kg·m⁻³), and ∆p (Pa) are mass flow rate, average density, and inlet and outlet pressure drop, respectively.

Modelica is an open, object-oriented, equation-based computer language that can facilitate the modeling of complex physical systems. The one-dimensional flow and heat transfer calculation model of the PCHE is established using Modelica [42]. The thermophysical properties of S-CO₂ were obtained from the ExternalMedia Library [43] based on Coolprop [44]. The PCHE one-dimensional model, shown in Figure 2a, includes a set of cold and hot channels, an intermediate metal matrix, and topological layers used to mark the relative flow directions of the cold and hot channels. The one-dimensional model simulation calculation loop based on Modelica is shown in Figure 2b, with a given inlet mass flow rate, inlet temperature, and outlet pressure.

It can be used to simulate the flow and heat transfer process of a heat exchanger under different operating parameters so as to obtain the overall parameters such as heat transfer rate, pressure drop, and temperature difference between inlet and outlet, as well as the distribution values of temperature, pressure, and physical property in the flow direction. The one-dimensional model of the PCHE can quickly obtain the distribution of thermodynamic parameters along the mainstream direction, but the distribution of parameters on the cross-section is unavailable, the change rules of flow and heat transfer cannot be analyzed from the mechanism, and the accuracy of the calculation results depends on the empirical formula. The one-dimensional model is suitable for conducting rapid calculation, optimized design, or dynamic characteristics analysis based on the known flow heat transfer characteristics.

The model can be applied to different working mediums, channel structures, and operating conditions by changing different flow and heat transfer correlations and has good expandability. The simulation errors are mainly from the simplified assumptions of the 1D model and the errors of the empirical formula for calculating the convection heat transfer coefficient and fanning friction coefficient.

2.2. Verification of Results

To verify the accuracy of the established one-dimensional flow and heat transfer model, the simulation results have been compared with the actual PCHE thermal-hydraulic performance parameters from Ref. [34]. The material properties and geometric parameters of the PCHE are shown in

Table 1. The material properties and geometric parameters of the PCHE.

Parameter	Unit	Value
Cold channel fluid	/	CO2
Hot channel fluid	/	CO2
Metal matrix material	/	Stainless steel 316 L
Channel type	/	Zigzag
Channel shape	/	Semi-circular
Diameter	mm	2
Channel length	mm	1012
Wetted perimeter	mm	5.14
Cross-sectional area	mm2	1.57
Plate thickness	mm	1.63
Density of metal	kg/m3	7990
Thermal conductivity of metal	J/(kg·K)	500

. Using the established one-dimensional flow and heat transfer model of the PCHE, the five working points shown in

Table 2. Simulation operation conditions.

Parameters	Unit	Design Condition	Deviation Condition-# 1	Deviation Condition-# 2	Deviation Condition-# 3	Deviation Condition-# 4
Total mass flow rate of each channel	kg/s	2.06	1.57	2.09	2.09	2.62
Inlet temperature of cold channel	°C	72.9	72.9	87.5	62	72.9
Inlet pressure of cold channel	bar	125	125	125	125	125
Inlet temperature of hot channel	°C	344.3	344.3	344.3	344.3	344.3
Inlet pressure of hot channel	bar	75	75	75	75	75

from Ref. [34] are simulated. The results are shown in

Table 3. Comparison between simulation results and measured results in reference [34].

Parameters		Cold Channel			Hot Channel
		Outlet Temperature (°C)	Flow Pressure Drop (kPa)	Heat Absorption (kW)	Outlet Temperature (°C)	Flow Pressure Drop (kPa)	Heat Release (kW)
Design condition	Measured value	284.90	120.00	623.67	80.50	130.00	629.77
	Calculated value	287.45	119.94	629.85	81.52	130.89	626.79
	Deviation%	0.90	−0.05	0.99	1.27	0.68	−0.47
Deviation condition 1	Measured value	287.19	74.00	479.39	78.59	79.00	485.01
	Calculated value	290.10	73.18	484.76	79.34	79.09	483.32
	Deviation%	1.01	−1.10	1.12	0.95	0.11	−0.35
Deviation condition 2	Measured value	294.49	139.00	580.71	99.69	145.00	584.80
	Calculated value	297.45	138.66	587.98	99.59	145.04	585.07
	Deviation%	1.01	−0.24	1.25	−0.10	0.03	0.05
Deviation condition 3	Measured value	269.29	106.00	674.42	66.59	122.00	682.57
	Calculated value	272.74	105.16	682.93	67.53	122.68	679.44
	Deviation%	1.28	−0.79	1.26	1.41	0.56	−0.46
Deviation condition 4	Measured value	282.29	184.00	785.55	82.07	202.00	793.46
	Calculated value	284.54	185.61	792.50	83.95	204.76	786.57
	Deviation%	0.80	0.88	0.88	2.29	1.37	−0.87

. Compared with the measured values in the reference, the maximum relative deviation is 2.29%, which shows that the established one-dimensional model can effectively simulate the flow and heat transfer process of S-CO₂ in a PCHE.

3. Results and Analysis

Based on the established one-dimensional flow and heat transfer model of PCHE, the influence of operating parameters on the flow and heat transfer process is carried out, and the performance of PCHE under different working conditions is accordingly obtained. The range of pressure and inlet temperature ensure that CO₂ is in a supercritical state.

3.1. Effect of Mass Flow Rate on the Flow and Heat Transfer Performance of PCHE

To analyze the influence of mass flow rate on the flow and heat transfer performance of the PCHE, for the cold channel, the outlet pressure is 75 bar, and the inlet temperature is 400 K; for the hot channel, the outlet pressure is 125 bar, and the inlet temperature is 600 K. Simulation calculations are conducted for different mass flow rates of cold and hot channels.

3.1.1. Effect of Mass Flow Rate on Heat Transfer Rate

The change in mass flow rate will affect its convection heat transfer intensity with the wall and, thus, influence the overall heat transfer coefficient in the heat transfer process. Meanwhile, it will also affect the temperature variation in the fluid flow process and then impact the heat transfer temperature difference in the heat transfer process.

The variation of mass flow rate changes the velocity field of the fluid in the channel, leading to the change of the thickness of the boundary layer, and then affects the convective heat transfer coefficient hc. From Equation (6), it eventually causes an impact on the heat transfer rate. Figure 3 shows the change in heat transfer rate with different mass flow rates in the cold and hot channels. With the given hot channel mass flow rate, when the cold channel mass flow rate rises, the heat transfer rate first increases linearly and slows down to the maximum. When the heat channel mass flow rate is 2.6 g/s after the cold channel mass flow rate exceeds 4 g/s, the heat transfer rate tends to the maximum, and when the cold channel mass flow rate increases to 10.6 g/s, the heat change only increases by 2.5%. After the comparison of different curves, overlapping parts can be found among the curves in the front part; the heat transfer rate at this section is only affected by the change of cold channel mass flow rate, which is irrelevant to the hot channel mass flow rate. When the cold channel mass flow rate is 1.6 g/s, and the hot channel mass flow rate increases from 2.6 g/s to 10.6 g/s, the heat transfer rate increases by 1%. In the rear part, the heat transfer rate will not augment the growth of the cold channel mass flow rate, and the heat transfer rate is only influenced by the change of the hot channel mass flow rate, uncorrelated to the cold channel mass flow rate. When the cold channel mass flow rate is 10.6 g/s and the hot channel mass flow rate increases from 2.6 g/s to 10.6 g/s, the heat transfer rate increases by 217%. In the middle part, the heat transfer is affected by both the cold channel mass flow rate and the hot channel mass flow rate.

Figure 4 demonstrates the variation of the overall heat transfer coefficient with the change of mass flow rate in the cold and hot channels. The overall heat transfer coefficient increases with the rise of mass flow rate in the cold and hot channels, and the growth trend slows down gradually. When the mass flow rate of the cold channel is 10.6 g/s, the overall heat transfer coefficient increases by 38.9% after the mass flow rate of the hot channel increases from 2.6 g/s to 5.3 g/s and increases by 9.3% after the mass flow rate of the hot channel increases from 7.9 g/s to 10.6 g/s; when the mass flow rate of the hot channel is 5.3 g/s, the overall heat transfer coefficient increases by 160.5% after the mass flow rate of the cold channel increases from 0.53 g/s to 5.3 g/s, and increases by 6.9% after the mass flow rate of the cold channel increases from 5.8 g/s to 10.6 g/s. With constant inlet cross-sectional area, inlet temperature, and pressure, the increase of mass flow rate can conduce the rise of the inlet Reynolds number and improve the convection heat transfer between the working medium and the channel wall, thus enhancing the heat transfer. When the mass flow rate increases to a large value, the thermal resistance for convection is relatively small in the heat transfer process, and the thermal resistance for conduction in the metal matrix gradually dominates, with less effect of the thermal resistance for convection on the overall heat transfer coefficient. At this time, the effect of increasing the mass flow rate to enhance heat transfer gradually weakens.

Under the same heat transfer rate, the variation of enthalpy of fluid is the same, and different mass flow rates have different specific enthalpy. The specific enthalpy of fluid is directly related to the temperature, so the variation of mass flow rate leads to the change in fluid temperature, which will affect the heat transfer temperature difference. Figure 5 displays the variation of mean temperature difference with the change of mass flow rate in the cold and hot channels. As the driving force of heat transfer, the temperature difference directly impacts the heat transfer process. The mean temperature difference is the mean value of fluid temperature difference in the cold and hot channels in each element. With the increase of the cold channel mass flow rate, the mean temperature difference increases first and then becomes stable. When the mass flow rate of the cold channel is small, the temperature of the cold channel changes fast and can approach the temperature of the hot channel quickly, so the mean temperature difference is small. So, the mean temperature difference increases with the increased mass flow rate of the cold channel. At the same time, the convection heat transfer in the cold channel is enhanced with the increased mass flow rate of the cold channel, and the thermal resistance between the cold channel and hot channel decreases, which is beneficial to reduce the temperature difference. Therefore, the mean temperature difference increases with the increased mass flow rate of the cold channel, and the growth trend slows down. In the front part, the cold channel mass flow rate is low, and the mean temperature difference is inversely proportional to the hot channel mass flow rate. When the mass flow rate of the cold channel is 2.6 g/s, the mean temperature difference decreases by 5.7 K after the mass flow rate of the hot channel increases from 2.6 g/s to 10.6 g/s. In this part, when the hot channel mass flow rate decreases, the convection heat transfer between the fluid and wall weakens, with insufficient heat transfer between the cold and hot fluid and a small temperature change of the fluid in the flow process. Then, the temperature difference between cold and hot fluid is large. In the rear part, the cold channel mass flow rate is high, and the mean temperature difference is proportional to the hot channel mass flow rate. When the mass flow rate of the cold channel is 10.6 g/s, the mean temperature difference increases by 27.7 K after the mass flow rate of the hot channel increases from 2.6 g/s to 10.6 g/s. The fluid flows and transfers heat in the channel, and the heat transfer amount is equal to the difference between the total enthalpies of the inlet and outlet. In this part, when the hot channel mass flow rate increases, though the heat transfer between cold and hot fluids is enhanced, the total enthalpy of the fluid also increases rapidly, and the heat transfer has little influence on the enthalpy change of the fluid. The temperature change of the fluid decreases in the flow process, and the temperature difference increases.

3.1.2. Effect of Mass Flow Rate on Flow Pressure Drop

The influence of mass flow rate on pressure drop includes two aspects: first, the pressure drop is directly proportional to the square of flow velocity. With a certain inlet cross-sectional area, temperature, and pressure, increasing the mass flow rate means increasing the flow velocity, which causes the pressure drop of this channel to rise parabolically. Second, the increase in mass flow rate will affect the heat transfer process. It will affect the temperature of the contralateral channel and then influence the thermophysical properties, such as the density and viscosity coefficient of S-CO₂, thus finally impacting the flow pressure drop of the contralateral channel.

Figure 6 shows the change of flow pressure drop in cold and hot channels with different mass flow rates in the cold and hot channels. According to Figure 6a, the pressure drop in the cold channel increases parabolically with the rise of the cold channel mass flow rate. When the mass flow rate of the hot channel is 5.3 g/s, the pressure drop of the cold channel increases by 136.4 times after the mass flow rate of the cold channel increases from 0.53 g/s to 10.6 g/s. As the hot channel mass flow rate goes up, the pressure drop also increases. When the mass flow rate of the cold channel is 10.6 g/s, the pressure drop of the cold channel increases by 39.8% after the mass flow rate of the hot channel increases from 2.6 g/s to 10.6 g/s. When the cold channel mass flow rate is constant and small, the heat transfer is limited by the cold channel. The increase in the hot channel mass flow rate affects the temperature change of the cold channel and its thermophysical properties less, so the pressure drop curves of the cold channel tend to coincide when the cold channel mass flow rate is small. When the cold channel mass flow rate is constant and large, the increase of the hot channel mass flow rate can result in increased heat transfer and raise the temperature of the working medium in the cold channel. It can be seen from Figure 7 that its viscosity coefficient increases and density decreases, which finally results in the growth of the flow pressure drop.

As can be seen from Figure 6b, the pressure drop in the hot channel first decreases and then tends to be stable with the increase of the cold channel mass flow rate and also increases with the rise of the hot channel mass flow rate. When the mass flow rate of the hot channel is 5.3 g/s, the pressure drop of the hot channel decreases by 34.1% after the mass flow rate of the cold channel increases from 0.53 g/s to 5.3 g/s and decreases by 5.3% after the mass flow rate of the cold channel increases from 5.8 g/s to 10.6 g/s. When the hot channel mass flow rate is constant, and the cold channel mass flow rate is small, increasing the cold channel mass flow rate can enhance the heat transfer, and the temperature of the hot channel decreases.

The thermophysical properties of CO₂ can be determined by temperature and pressure. During the flow and heat transfer process in the channel, the variation of pressure is negligible compared to the inlet pressure, so the thermophysical properties are mainly affected by temperature. The mass flow rate can influence the temperature and then affect the thermophysical properties. It can also be seen from Figure 7 that its viscosity coefficient decreases and density increases, which finally leads to a decrease in the flow pressure drop. When the hot channel mass flow rate is constant, and the cold channel mass flow rate is large, the increase of the cold channel mass flow rate has less effect on the heat transfer, and the fluid temperature in the hot channel changes little, so the pressure drop tends to be stable with the increase of cold channel mass flow rate.

Figure 8 shows the influence of mass flow rate on the total pump power in the cold and hot channels. It can be found that with a constant cold channel mass flow rate, the total pump power increases with the increase of mass flow rate in the hot channel. When the mass flow rate of the cold channel is 5.3 g/s, the total pump power increases by 18.9 times after the mass flow rate of the hot channel increases from 2.6 g/s to 10.6 g/s. The increase of the hot channel mass flow rate will increase the pressure loss of its own channel, and the temperature of the cold channel will rise, leading to flow pressure drop increases in the cold channel. Therefore, increasing the hot channel mass flow rate will increase the total pump power. When the hot channel mass flow rate is constant, the increase in cold channel mass flow rate causes an increase in flow resistance, while the enhanced heat transfer leads to the decrease of hot channel temperature and its pressure drop. Thus, the total pump power decreases first and then increases with the cold channel mass flow rate.

3.1.3. Analysis of the Influence of Mass Flow Rate on the Flow and Heat Transfer Comprehensive Performance

To quantitatively analyze the comprehensive effect of mass flow rate change on flow and heat transfer, the efficiency evaluation coefficient EEC is used.

Figure 9 shows the variation of EEC with the change of mass flow rate in the cold and hot channels. There is an EEC maximum in each curve. When the mass flow rate of the heat channel is 4.8 g/s, the maximum EEC is 1.42, corresponding to the mass flow rate of the cold channel, 4.2 g/s. Compared with the design condition, the heat transfer rate increases by 62.1%, and the total pump power increases by 14.2%. Based on the previous analysis, the increasing mass flow rate can enhance the heat transfer, but the growth trend gradually slows down. Meanwhile, it will also lead to an increase in flow resistance, and the growth rate gradually becomes faster. The EEC maximum is produced by the combination of the two aspects above. At this time, the comprehensive performance of the flow and heat transfer process is the best, and the configuration of the working medium mass flow rate in the cold and hot channels is optimal. Moreover, the mass flow rate in the hot and cold channels corresponding to the optimum is proportional, and the maximum of EEC decreases with the increase of the hot channel mass flow rate. When EEC is maximum, the comprehensive performance index of the heat exchanger is optimal; the heat transfer rate is maximum under the same pump power, which can provide theoretical guidance for the actual operation and control of the SCBC.

3.2. Effect of Inlet Temperature on Flow and Heat Transfer Performance of PCHE

To analyze the influence of inlet temperature on the flow and heat transfer performance of the PCHE, the mass flow rate of the cold and hot channels remains at 2.6 g/s, the outlet pressure of the cold channel is 75 bar, and the outlet pressure of the hot channel is 125 bar. Simulation calculations are conducted on the different inlet temperatures of cold and hot channels.

3.2.1. Effect of Inlet Temperature on Heat Transfer Rate

The change in inlet temperature can lead to a change in heat transfer temperature difference between the cold and hot channels in the flow process; it can also affect thermophysical properties, such as viscosity coefficient, density, and specific heat capacity. These two aspects will affect the heat transfer rate between the cold and hot channels. Figure 10 shows the change in heat transfer rate with the inlet temperature of the cold and hot channels. With the increase in the inlet temperature of the cold channel and the decrease in the inlet temperature of the hot channel, the heat transfer rate decreases approximately linearly.

For further analysis, Figure 11 shows the variation of mean temperature difference with the change in inlet temperature of the cold and hot channels, and Figure 12 shows the variation of overall heat transfer coefficient with the change in inlet temperature of the cold and hot channels. With the increase in the inlet temperature of the cold channel and the decrease in the inlet temperature of the hot channel, the mean temperature difference decreases approximately linearly, while the change range of the overall heat transfer coefficient is very small. When the inlet temperature of a hot channel is 580 K, the variation of the overall heat transfer coefficient is within 4.3%. So, the change of heat transfer is consistent with the change of the mean temperature difference and approximately decreases linearly.

With the increase in the inlet temperature of the cold channel, the overall heat transfer coefficient first decreases and then increases; with the increase in the inlet temperature of the hot channel, the overall heat transfer coefficient increases, which is mainly due to the influence of the temperature on the thermophysical properties of S-CO₂.

3.2.2. Effect of Inlet Temperature on Flow Pressure Drop

With the change in inlet temperature, the temperature change of the working medium at each position in the flow process affects thermophysical properties, such as density and viscosity coefficient, and then affects the flow pressure drop. Figure 13 shows the variation of channel pressure drop with the inlet temperature.

The pressure drop of the cold channel increases with the increase in the inlet temperature of the cold channel, and it initially increases fast, and then the rate slows down. When the inlet temperature of the hot channel is 580 K, the pressure drop of the cold channel increases by 15.9% after the inlet temperature of the cold channel increases from 320 K to 350 K and increases by 10.1% after the inlet temperature of the cold channel increases from 350 K to 490 K. With the increase in temperature, the density will decrease, and the flow velocity will increase under the condition of constant mass flow rate, which will lead to the increase of flow velocity and the final flow pressure drop. In addition, the change in density is shown in Figure 7. In the temperature range of the cold channel, the density decreases with temperature, and it initially changes fast, and then the rate slows down. This characteristic is also reflected in the change of pressure drop in the cold channel. With the increase in the inlet temperature of the hot channel, the temperature of the fluid in the cold channel will also increase through heat transfer, which will lead to the increase of the flow pressure drop. The changing trend of pressure drop in the hot channel is similar to that in the cold channel, which increases with the increase in temperature.

Figure 14 shows that the total pump power changes with the inlet temperature, and the total pump power increases with the increase in the inlet temperature. When the inlet temperature of the hot channel is 580 K, the total pump power increased by 263.3% after the inlet temperature of the cold channel increases from 320 K to 490 K. When the inlet temperature of the cold channel is 400 K, the total pump power increased by 35.9% after the inlet temperature of the hot channel increases from 540 K to 660 K. With the increase in the inlet temperature, the temperature of the working medium in both the cold and hot channels will increase. At this time, the density decreases, and the viscosity coefficient increases, which leads to an increase in the flow pressure drop, thus increasing the total pump power.

3.2.3. Effect of Inlet Temperature on Comprehensive Performance of Flow and Heat Transfer

Figure 15 shows the change of EEC with the inlet temperature of the cold and hot channels. Because the mass flow rate is greater than the mass flow rate of the design condition, the total pump power is greater than the total pump power under the design condition, resulting in an EEC of less than 1. With the increase in the cold channel inlet temperature, EEC decreases rapidly. When the inlet temperature of the hot channel is 580 K, the EEC decreases by 91.8% after the inlet temperature of the cold channel increases from 320 K to 490 K. EEC increases with the increase in the inlet temperature of the hot channel. When the inlet temperature of the cold channel is 400 K, the EEC increases by 36.3% after the inlet temperature of the hot channel increases from 540 K to 660 K. The inlet temperature of the cold channel increases, which reduces the heat transfer temperature difference and reduces the heat transfer rate between the cold and hot channels; simultaneously, the temperature of the working medium in both cold and hot channels increases, which leads to the increase of flow pressure drop, which is not conducive to heat transfer and flow, so EEC decreases with the increase in inlet temperature of the cold channel. The inlet temperature of the heat channel increases, which increases the heat transfer temperature difference and the heat transfer rate; at the same time, the increase in working medium temperature in the cold and hot channels leads to the increase of flow pressure drop, which is beneficial to heat transfer but unfavorable to flow. Overall, EEC increases with the inlet temperature of the hot channel.

4. Conclusions

The flow and heat transfer process of S-CO₂ in the PCHE was modeled and simulated using Modelica. The effects of the mass flow rate and inlet temperature on the flow and heat transfer process were analyzed from three aspects: heat transfer rate, flow pressure drop, and comprehensive performance. The optimal matching of cold and hot mass flow rates was obtained, and the comprehensive performance was optimized. The main conclusions are as follows.

(1)

The mass flow rate influences the overall heat transfer coefficient and heat transfer temperature difference, such that the greater the mass flow rate, the greater the heat transfer rate. The heat transfer rate growth rate first increases and then slows down. With the increase of mass flow rate, the flow pressure drop increases rapidly. Taking EEC as the comprehensive performance index of flow and heat transfer, EEC has a maximum value with the change of mass flow rate. At this time, the mass flow rates of the cold and hot channels match, and the comprehensive performance of flow and heat transfer processes is the best;

(2)

The inlet temperature of the channel directly affects the heat transfer temperature difference. Increasing the inlet temperature of the hot channel can enhance the heat transfer while increasing the inlet temperature of the cold channel weakens the heat transfer. Increasing the inlet temperature of both cold and hot channels will increase the flow pressure drop of the channels. With increasing cold channel inlet temperature, EEC decreases rapidly, while EEC increases with increasing hot channel inlet temperature;

(3)

The one-dimensional modeling and simulation method can quickly calculate the flow and heat transfer process of a PCHE, considering the change of thermophysical properties to obtain the distribution values of important parameters such as temperature and pressure in the flow direction. The calculation accuracy and speed can meet the needs of transient calculation, operation control strategy analysis, and other application scenarios.

Based on the influence of mass flow rate and inlet temperature on the flow and heat transfer performance of the PCHE, the operation strategy research of the SCBC can be better conducted. The optimal matching of cold and hot mass flow rates can provide guidance for the setting of operation parameters. Furthermore, the optimal design and dynamic characteristic analysis of the PCHE can be conducted. The model can be applied to different working mediums, channel structures, and operating conditions by changing different flow and heat transfer correlations. To improve the one-dimensional model of the PCHE, it is necessary to obtain empirical correlations with wider application and higher accuracy by experiments or three-dimensional numerical calculations. The one-dimensional model of the PCHE can be implemented into system-level models to conduct cycle performance analysis.