# Thermohydraulic Characteristics of Printed Circuit Recuperators in a Supercritical CO2 Brayton Cycle with Nonuniform Minichannels

Authors: Morteza Khoshvaght-Aliabadi; Parvaneh Ghodrati; Yong Tae Kang
Journal: International Journal of Energy Research
 (2023)

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## Abstract

Supercritical carbon dioxide (SCO2) technologies have been under the spotlight of developed countries because they believe that such applied sciences can be ingenious solutions to resolve energy and environmental problems. The SCO2 power cycle is one of the most promising technologies in this field due to its high efficiency and low expenditure. However, optimal designs of heat exchange devices are a vital issue in the SCO2 power cycle because they account for almost 80% of the capital costs. In this study, the comparison of diverse combinations of convergent and divergent minichannels inside the low temperature recuperator (LTR) and high temperature recuperator (HTR) is suggested to recognize the strengths and weaknesses of nonuniform minichannels at high operating pressures and temperatures. Three-dimensional conjugate simulations are carried out, and existing data in the literature are used to validate the obtained results. The comparison reveals a slight deviation (less than 10%) in temperature and pressure profiles. After the evaluation of the studied cases in terms of thermal and hydraulic characteristics, different criteria are employed to explore optimal combinations of the hot side and cold side minichannels. Totally, the minichannel convergence enhances thermal performance of recuperators (up to 26%), and their divergence improves the hydraulic behavior. Nonetheless, diverging the minichannels imposes greater positive effects than converging them. The overall analysis specifies that the most efficient LTR and HTR would be attained by diverging the hot side minichannel, while the cold side minichannel is kept unchanged. This combination of minichannels allows reaching the highest SCO2 temperature at the outlet of LTR and HTR (456 K and 656 K) with the minimum pressure drops (around 0.02% of inlet pressures) and achieving the best performance indexes (approximately 2). Finally, the comparative study discloses that at low Reynolds numbers (less than 14000 for the hot side and 10000 for the cold side), the nonuniform patterns of minichannels perform better than the complicated structures, such as zigzag, wavy, and other optimized geometries. The average performance enhancements for the hot and cold sides are 36.3% and 37.4%, respectively.


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## Body

## 1. Introduction

Compared to the conventional power generation (or conversion) cycles, the supercritical carbon dioxide (SCO2) Brayton (or Joule) cycle is a prospective power cycle. Simple design, compact layout, high thermal efficiency, acceptable cleanliness, and potential capital cost savings are the distinct advantages of this cycle [1]. Thus, it seems that the SCO2 Brayton cycle can be used as an attractive thermal-to-power system in future generations of fossil-fired plants as well as other novel plants working with waste heat recovery, concentrated solar, and nuclear power sources [2].One of the reasons for the high thermal efficiency of the SCO2 Brayton cycle is its internal heat recuperation, forming more than 60% of the total thermal energy absorbed from the heat source in the main heat exchanger [3, 4]. Recuperators are internal heat exchange devices of a SCO2 Brayton cycle in which heat is transferred from the turbine exhaust gas (low-pressure and high-temperature SCO2) to the turbine inlet gas (high-pressure and low-temperature SCO2). Consequently, the thermal efficiency and total capital costs of the cycle are strongly coupled to the performance and price of the recuperators. Typically, there are two recuperators in the SCO2 Brayton cycle called low temperature recuperator (LTR) and high temperature recuperator (HTR), as shown in Figure 1. An effective method to handle and enhance the high heat recovery in the cycle is using the compact flow structures in LTR and HTR.Figure 1
A schematic of SCO2 Brayton cycle and its operating conditions.A printed circuit heat exchanger (PCHE), first introduced by Johnston in the early 1980s, could be a viable option for recuperators due to its small size (about1/5 of a shell-and-tube heat exchanger under the same load), large compactness (surface area to volume ratio greater than 2500 m2/m3), variable temperature-pressure resistances (>1100 K and >60 MPa), and exemplary high efficiency (up to 98%) [5]. Therefore, the heat transfer enhancement of SCO2 flow inside PCHEs has been the focus of many researchers in recent years.The continuous zigzag flow path is one of the most broadly studied patterns of minichannels inside PCHEs working with SCO2. Nikitin et al. [6] experimentally investigated the flow and heat transfer characteristics of the SCO2 flow inside a PCHE with zigzag minichannels. The effects of different inlet temperatures and operating pressures were explored, and new correlations as a function of the Reynolds number were proposed. Meshram et al. [7] evaluated the performance of the zigzag minichannel compared with the straight one. They found that the size of PCHE designed based on the zigzag minichannels was smaller than that of PCHE with straight minichannels. It was also found that the larger bend angle showed a better performance than the smaller one. Zhang et al. [8] reported that the bend angles between 110° and 130° led to the best overall performance. Saeed and Kim [9] optimized the structure of zigzag minichannels and proposed a new configuration. The overall performance of the considered minichannel was found to be up to 20% better than that of the conventional case. A coupled technique was employed by Bennett and Chen [10] to analyze the SCO2 flow performance inside the zigzag minichannels. An experimental parametric study on the precooler PCHEs with zigzag minichannels was conducted by Cheng et al. [11]. Saeed et al. [12] also focused on the performance of a PCHE with zigzag minichannels under the precooler operating conditions (high Prandtl numbers of SCO2). The effects of different mass flow rates and inlet temperatures were studied, and an optimum configuration was suggested. They extended their study using the Artificial Neural Network (ANN) and found that the trained technique could estimate 99% of the data with 90% accuracy [13].The wavy structure is one of the other continuous minichannels studied by some researchers. Yuchuan and Zhenqian [14, 15] explained that the wavy minichannels could enhance the thermal performance of PCHEs with a slight increase in the flow resistance compared with the straight ones. We et al. [16] investigated the impacts of the design parameters of the wavy minichannels and found that increasing the wave amplitude and decreasing the wavelength led to higher heat transfer rates inside PCHEs. Lv et al. [17] proposed three hybrid straight-wavy minichannels to reduce the pressure drop through the PCHEs. It was found that, compared to the full-wavy case, the hybrid minichannels could improve the overall performance of the PCHEs due to noticeable reductions in the pressure drop. Recently, Khoshvaght-Aliabadi et al. [18] proposed novel configurations of wavy minichannels by changing the wave amplitude and wavelength. It was concluded that increasing the wave amplitudes and decreasing the wavelengths at upstream of the PCHEs could enhance the overall performance index up to 1.5.Pin-shaped flow paths have also been considered to further improve the hydraulic performance of PCHEs. Xu et al. [19] compared the airfoil minichannels with the zigzag minichannels and found that a sparser staggered arrangement of these fins could result in a better overall performance in the PCHEs. Other comparative studies [20, 21] confirmed this finding and noted that the efficiency of PCHEs may enhance by about 1.0% using airfoil fins. Experimental investigations were carried out by Refs. [22, 23] to examine the flow features and thermal performance of the airfoil minichannels and develop universal correlations. Cui et al. [24] introduced two modified airfoil fins to further improve the performance of PCHEs. Another structural modification was performed by Han et al. [25]. They showed that one of the proposed structures (front-sparse and rear-dense) could improve the performance of PCHEs by up to 30%.It can be concluded that the complicated structures can disturb the SCO2 flowing along the minichannels and increase the turbulence intensity inside the PCHEs. Thus, complex flow features such as longitudinal swirl flows or normal vortices will be appeared, leading to higher local heat transfer coefficient values. However, it is an undeniable fact that high flow resistances and pressure drops in heat exchange devices of the SCO2 Brayton cycles may limit the expansion ratio across the turbine, leading to a decrease in the power output. In this paper, in order to decline the flow resistance inside the PCHEs and simultaneous improvement to their thermal performance, converging and diverging the hot side and cold side minichannels are proposed and studied. Different combinations of convergent and divergent minichannels are considered, and their effects are investigated under the LTR and HTR operating conditions. Several dimensional and dimensionless parameters are employed to appraise and compare the overall thermohydraulic performance of the mentioned combinations. It should be emphasized that although several studies have been done on thermal and hydraulic characteristics of the SCO2 flow inside PCHEs with semicircular straight minichannels, it is not clear yet how converging and diverging the minichannels affect the performance of recuperators.


## 2. Designs of Minichannels in LTR and HTR Recuperators

In order to promote the performance of LTR and HTR in the SCO2 Brayton cycle, new geometries are recommended for the minichannels of PCHEs in this study. As nonuniform patterns of flow path are an indispensable part of thermal engineering applications, converging and diverging arrangements for the hot side and cold side minichannels of LTR and HTR are investigated. According to the adopted numerical approach, this analysis aims to determine the best combinations of converging and diverging minichannels in PCHEs operating under LTR and HTR conditions. Figure 2 portrays an elaborate presentation of the front and back views of the considered cases in which the minichannels have a semicircular cross-section with a total length of 200 mm. The small, medium, and large diameters of the semicircular geometries are 0.9, 1.2, and 1.5 mm, respectively. Totally, four different combinations of converging and diverging minichannels in LTR and HTR are examined as follows:
(i)
Class 1. The hot side is always convergent and the cold side is uniform, convergent, or divergent(ii)
Class 2. The hot side is always divergent and the cold side is uniform, convergent, or divergent(iii)
Class 3. The cold side is always convergent and the hot side is uniform, convergent, or divergent(iv)
Class 4. The hot side is always divergent and the hot side is uniform, convergent, or divergentFigure 2
A layout of studied cases.The cases with a uniform combination of minichannels (conventional designs) are used for comparison purposes. It should be noted that the vertical occupied space (width×height), horizontal footprint to land surface (width×length), and total active heat transfer area of all cases are identical and equal to 4.32, 360, and 616.89 mm2.


## 3. Details of Mathematical Descriptions

### 3.1. Simulated Domain and Boundary Conditions

With reference to Figure2, each combination is composed of SCO2 minichannels along with their surrounding solid domain. The substance of the solid domain is stainless steel (SS316L) with constant thermophysical properties [26]. The periodic condition is chosen for all lateral surfaces of the solid domain, referring to the conductive heat transfer rate between the elementary unit and similar units around it. Further, the adiabatic wall condition is imposed on the front and back surfaces of the solid domain. Therefore, all LTR and HTR models are simplified to a single unit consisting of a hot side minichannel, a cold side minichannel, and their surrounding solid domain, which are, respectively, marked with red, blue, and gray colors in Figure 3 [27]. Regarding Table 1, the input and output parameters of each model are the inlet mass flux/temperature and pressure of the hot stream and the cold stream of SCO2. In this study, the magnitudes of boundary conditions are specified based on the actual operating conditions of the SCO2 Brayton cycle as quantified for LTR and HTR in Figure 1 [22].Figure 3
Simulated domain and corresponding boundary conditions.Table 1
Details of applied boundary and operating conditions.
BoundaryConditionOperating conditionLTRHTRInletMass flux and temperatureHot streamCold streamHot streamCold stream700 kgm-2 s-1442 kgm-2 s-1700 kgm-2 s-1700 kgm-2 s-1468.45 K354.65 K677.85 K458.45 KOutletPressure7.78 MPa20 MPa7.86 MPa19.99 MPa


### 3.2. Governing Equations and Turbulence Model

The data presented in Table1 can be used to appraise the nature of SCO2 flowing inside the hot side and the cold side minichannels of LTR and HTR. It is found that the Reynolds numbers of both the hot and the cold streams are larger than 17290, confirming a turbulent flow regime. The SST (Shear Stress Transport)k-ω turbulence model is employed to simulate the SCO2 flow inside both the hot side and the cold side. It is the most widely used model in previous studies on the SCO2 flow inside the recuperators [10, 16, 28]. The SST k-ω turbulence model combines the precise simulation of the near-wall SCO2 flow resulting from the k-ω model with the freestream independence in the core SCO2 flow resulting from the k-ε model [21]. Details of the governing equations under the steady-state condition are as follows [29]:
(i)
Continuity equation(1)∂∂xiρui=0,where ρ is the SCO2 density and u is its flow velocity
(ii)
Momentum equation(2)∂∂xjρuiuj=−∂p∂xi+∂∂xjμ+μt∂ui∂xj+∂uj∂xi−23∂uk∂xk+ρgi,where ρ is the flow pressure, μ is the SCO2 dynamic viscosity, μt is the turbulent viscosity, and g is the gravity acceleration
(iii)
Energy equations(3)∂∂xiρuicpT=∂∂xikf∂T∂xi,where cp is the SCO2 specific heat capacity, T is the temperature, and k is the SCO2 thermal conductivity
(4)∂∂xiks∂T∂xi=0,where ks is the solid thermal conductivityMeanwhile, the kinetic energy and dissipation rate equations for the SSTk-ω model are estimated by [30]
(5)∂ρk∂t+∂ρkui∂xi=∂∂xjΓk∂k∂xj+Gk−Yk+Sk,∂ρω∂t+∂ρωui∂xi=∂∂xjΓω∂ω∂xj+Gω−Yω+Dω+Sω,where G and Y are the generation and dissipation of kinetic energy (k) and specific dissipation rate (ω), respectively


### 3.3. Numerical Method and Grid Independence Study

In order to predict the thermophysical properties of the SCO2 involved in the governing equations, the NIST (National Institute of Standards and Technology) REFPROP (Reference Fluid Thermodynamic and Transport Properties) database [30] is coupled with the ANSYS Fluent 2021R1 CFD (Computational Fluid Dynamics) commercial solver. Meantime, the FVM (Finite Volume Method) is applied to discretize the computational domain using the adopted grid pattern, the SIMPLEC (Semi-Implicit Method for Pressure Linked Equations-Consistent) algorithm is used to couple pressure and velocity fields, and the second-order upwind scheme is employed to discretize the convection terms. All iterations are continued until the variations of dependent variables of both the hot and the cold streams lie within ±0.5%.A grid study is also carried out to eliminate the dependence of simulation outputs on the grid size. For the case with uniform minichannels, diverse structured grid patterns with magnitudes adopted in the range of 0.9–2.3 million for the SCO2 (fluid) domain and 0.36–0.6 million for the SS316L (solid) domain are generated and examined. Proper refinements with a ratio of 1.2 are made towards the fluid-solid boundaries to fulfill the requirement of the SST k-ω turbulence model on the dimensionless parameter of Y-plus. After a detailed analysis, it is found that 1.52 million grids for the SCO2 domain and 0.48 million grids for the SS316L domain are sufficient for the independence of the numerical results from the grids number. As demonstrated in Table 2, for larger values, both the thermal and the hydraulic characteristics of the representative case remain independent of grid number, and the deviations are less than 0.5%.Table 2
Results of grid independence study.
SCO2SS316LTotalY-plusSideThermal (Nu)DeviationHydraulic (f)Deviation90000036000012600000.93Hot71.09—0.00600—Cold40.56—0.00970—152000048000020000000.36Hot73.112.75%0.006060.67%Cold42.564.57%0.009690.64%230000060000029000000.15Hot73.240.18%0.006090.48%Cold42.740.43%0.009670.19%


### 3.4. Reduction of Data

As demonstrated in Figure3, x, y, and z axes refer to the span-wise, normal, and stream-wise directions, respectively. In this study, a segmental-averaged discretization technique is employed for the data reduction, in which the variations of thermophysical properties and thermohydraulic characteristics in a specific axial length are considered by employing several points in both the span-wise and the normal directions. Then, by averaging the values obtained from the lines passing through these points, the local values in the stream-wise direction will be obtained. The relevant equations to the main local parameters are listed as follows:
(6)hz=qzTSS316L,z−TSCO2,z,Nuz=hzDzkSCO2,jz=NuzRezPrz1/3,Euz=10.5uzρSCO2dpzdz,fz=2ρSCO2DzGz2dpzdz,Rez=GzDzμSCO2,where qz, TSS316L,z, and TSCO2,z are, respectively, the heat flux, SS316L wall temperature, and SCO2 temperature at a fixed axial length, kSCO2, ρSCO2, and μSCO2 are, respectively, the local thermal conductivity, density, and dynamic viscosity of the SCO2. GSCO2 is the mass flux of the hot side and cold side streams, dpz/dz is the pressure gradient along the stream-wise direction, and Dz is the local hydraulic diameter estimated as follows:
(i)
For the converging pattern(7)Dz=ππ+2asmall−alargeLz+alarge(ii)
For the diverging pattern(8)Dz=ππ+2alarge−asmallLz+asmall,where a and L are the diameter and length of semicircular minichannels, respectivelyFinally, the average value of each thermohydraulic parameter can be calculated using the following equation,(9)Parameterh,Nu,f,Re,⋯Average=∫0LParameterh,Nu,f,Re,⋯LocaldzL.The average heat transfer rate between the hot stream and the cold stream of LTR and HTR is estimated by Eq. (8),
(10)Q=Qhotside+Qcoldside2.Likewise, the overall heat transfer coefficient of LTR and HTR is appraised by Eq. (9),
(11)Uove.=QAΔTm,where ∆Tm is the logarithmic mean temperature difference calculated as follows,
(12)ΔTm=Thot,out−Tcold,in−Thot,in−Tcold,outlnThot,out−Tcold,in/Thot,in−Tcold,out,where Tin and Tout represent the inlet and outlet temperatures of the SCO2 streams, respectively.


## 3.1. Simulated Domain and Boundary Conditions

With reference to Figure2, each combination is composed of SCO2 minichannels along with their surrounding solid domain. The substance of the solid domain is stainless steel (SS316L) with constant thermophysical properties [26]. The periodic condition is chosen for all lateral surfaces of the solid domain, referring to the conductive heat transfer rate between the elementary unit and similar units around it. Further, the adiabatic wall condition is imposed on the front and back surfaces of the solid domain. Therefore, all LTR and HTR models are simplified to a single unit consisting of a hot side minichannel, a cold side minichannel, and their surrounding solid domain, which are, respectively, marked with red, blue, and gray colors in Figure 3 [27]. Regarding Table 1, the input and output parameters of each model are the inlet mass flux/temperature and pressure of the hot stream and the cold stream of SCO2. In this study, the magnitudes of boundary conditions are specified based on the actual operating conditions of the SCO2 Brayton cycle as quantified for LTR and HTR in Figure 1 [22].Figure 3
Simulated domain and corresponding boundary conditions.Table 1
Details of applied boundary and operating conditions.
BoundaryConditionOperating conditionLTRHTRInletMass flux and temperatureHot streamCold streamHot streamCold stream700 kgm-2 s-1442 kgm-2 s-1700 kgm-2 s-1700 kgm-2 s-1468.45 K354.65 K677.85 K458.45 KOutletPressure7.78 MPa20 MPa7.86 MPa19.99 MPa


## 3.2. Governing Equations and Turbulence Model

The data presented in Table1 can be used to appraise the nature of SCO2 flowing inside the hot side and the cold side minichannels of LTR and HTR. It is found that the Reynolds numbers of both the hot and the cold streams are larger than 17290, confirming a turbulent flow regime. The SST (Shear Stress Transport)k-ω turbulence model is employed to simulate the SCO2 flow inside both the hot side and the cold side. It is the most widely used model in previous studies on the SCO2 flow inside the recuperators [10, 16, 28]. The SST k-ω turbulence model combines the precise simulation of the near-wall SCO2 flow resulting from the k-ω model with the freestream independence in the core SCO2 flow resulting from the k-ε model [21]. Details of the governing equations under the steady-state condition are as follows [29]:
(i)
Continuity equation(1)∂∂xiρui=0,where ρ is the SCO2 density and u is its flow velocity
(ii)
Momentum equation(2)∂∂xjρuiuj=−∂p∂xi+∂∂xjμ+μt∂ui∂xj+∂uj∂xi−23∂uk∂xk+ρgi,where ρ is the flow pressure, μ is the SCO2 dynamic viscosity, μt is the turbulent viscosity, and g is the gravity acceleration
(iii)
Energy equations(3)∂∂xiρuicpT=∂∂xikf∂T∂xi,where cp is the SCO2 specific heat capacity, T is the temperature, and k is the SCO2 thermal conductivity
(4)∂∂xiks∂T∂xi=0,where ks is the solid thermal conductivityMeanwhile, the kinetic energy and dissipation rate equations for the SSTk-ω model are estimated by [30]
(5)∂ρk∂t+∂ρkui∂xi=∂∂xjΓk∂k∂xj+Gk−Yk+Sk,∂ρω∂t+∂ρωui∂xi=∂∂xjΓω∂ω∂xj+Gω−Yω+Dω+Sω,where G and Y are the generation and dissipation of kinetic energy (k) and specific dissipation rate (ω), respectively


## 3.3. Numerical Method and Grid Independence Study

In order to predict the thermophysical properties of the SCO2 involved in the governing equations, the NIST (National Institute of Standards and Technology) REFPROP (Reference Fluid Thermodynamic and Transport Properties) database [30] is coupled with the ANSYS Fluent 2021R1 CFD (Computational Fluid Dynamics) commercial solver. Meantime, the FVM (Finite Volume Method) is applied to discretize the computational domain using the adopted grid pattern, the SIMPLEC (Semi-Implicit Method for Pressure Linked Equations-Consistent) algorithm is used to couple pressure and velocity fields, and the second-order upwind scheme is employed to discretize the convection terms. All iterations are continued until the variations of dependent variables of both the hot and the cold streams lie within ±0.5%.A grid study is also carried out to eliminate the dependence of simulation outputs on the grid size. For the case with uniform minichannels, diverse structured grid patterns with magnitudes adopted in the range of 0.9–2.3 million for the SCO2 (fluid) domain and 0.36–0.6 million for the SS316L (solid) domain are generated and examined. Proper refinements with a ratio of 1.2 are made towards the fluid-solid boundaries to fulfill the requirement of the SST k-ω turbulence model on the dimensionless parameter of Y-plus. After a detailed analysis, it is found that 1.52 million grids for the SCO2 domain and 0.48 million grids for the SS316L domain are sufficient for the independence of the numerical results from the grids number. As demonstrated in Table 2, for larger values, both the thermal and the hydraulic characteristics of the representative case remain independent of grid number, and the deviations are less than 0.5%.Table 2
Results of grid independence study.
SCO2SS316LTotalY-plusSideThermal (Nu)DeviationHydraulic (f)Deviation90000036000012600000.93Hot71.09—0.00600—Cold40.56—0.00970—152000048000020000000.36Hot73.112.75%0.006060.67%Cold42.564.57%0.009690.64%230000060000029000000.15Hot73.240.18%0.006090.48%Cold42.740.43%0.009670.19%


## 3.4. Reduction of Data

As demonstrated in Figure3, x, y, and z axes refer to the span-wise, normal, and stream-wise directions, respectively. In this study, a segmental-averaged discretization technique is employed for the data reduction, in which the variations of thermophysical properties and thermohydraulic characteristics in a specific axial length are considered by employing several points in both the span-wise and the normal directions. Then, by averaging the values obtained from the lines passing through these points, the local values in the stream-wise direction will be obtained. The relevant equations to the main local parameters are listed as follows:
(6)hz=qzTSS316L,z−TSCO2,z,Nuz=hzDzkSCO2,jz=NuzRezPrz1/3,Euz=10.5uzρSCO2dpzdz,fz=2ρSCO2DzGz2dpzdz,Rez=GzDzμSCO2,where qz, TSS316L,z, and TSCO2,z are, respectively, the heat flux, SS316L wall temperature, and SCO2 temperature at a fixed axial length, kSCO2, ρSCO2, and μSCO2 are, respectively, the local thermal conductivity, density, and dynamic viscosity of the SCO2. GSCO2 is the mass flux of the hot side and cold side streams, dpz/dz is the pressure gradient along the stream-wise direction, and Dz is the local hydraulic diameter estimated as follows:
(i)
For the converging pattern(7)Dz=ππ+2asmall−alargeLz+alarge(ii)
For the diverging pattern(8)Dz=ππ+2alarge−asmallLz+asmall,where a and L are the diameter and length of semicircular minichannels, respectivelyFinally, the average value of each thermohydraulic parameter can be calculated using the following equation,(9)Parameterh,Nu,f,Re,⋯Average=∫0LParameterh,Nu,f,Re,⋯LocaldzL.The average heat transfer rate between the hot stream and the cold stream of LTR and HTR is estimated by Eq. (8),
(10)Q=Qhotside+Qcoldside2.Likewise, the overall heat transfer coefficient of LTR and HTR is appraised by Eq. (9),
(11)Uove.=QAΔTm,where ∆Tm is the logarithmic mean temperature difference calculated as follows,
(12)ΔTm=Thot,out−Tcold,in−Thot,in−Tcold,outlnThot,out−Tcold,in/Thot,in−Tcold,out,where Tin and Tout represent the inlet and outlet temperatures of the SCO2 streams, respectively.


## 4. Results and Discussions

### 4.1. Validation

Aiming at validating the present numerical method, the variations of temperature and pressure drop along the axial length of the hot side and cold side minichannels of a recuperator are considered. A PCHE with uniform minichannels is selected for comparison purposes, and the corresponding results are compared with the data reported by Marchionni et al. [31] and Li et al. [32], as shown in Figure 4. A comparison between the present results and the previous data shows the deviations being less than 10%. It confirms the validity of the numerical method employed in this study. However, the existing differences in the parameter predictions are inevitable and can stem from different considered turbulence models. As mentioned, the SST k-ω turbulence model is employed in the present study, while the turbulence model of the standard k-ε approach was applied by the previous studies.Figure 4
Validation of present results with previous data: (top) temperature and (bottom) pressure drop variations along hot side and cold side minichannels of PCHE.


### 4.2. Thermal and Hydraulic Characteristics

In this section, the thermal and hydraulic characteristics of diverse combinations of uniform (hereinafter referred to as the “U”), converging (hereinafter referred to as the “C”), and diverging (hereinafter referred to as the “D”) minichannels are compared with those of the conventional minichannel (hereinafter referred to as the “Ref.”) under the same operating conditions of LTR and HTR. The data analysis results indicate that converging and diverging the minichannels can significantly affect the thermal and hydraulic characteristics of both recuperators (LTR and HTR), which will be discussed in the following subsections.


#### 4.2.1. Converging Hot Side Minichannel (Class 1)

Figure5 exhibits the thermal characteristics of diverse combinations of minichannels in Class 1 where the hot side minichannel is always C, while the cold side is either U, C, or D. As it can be seen, the minichannel convergence and divergence consequences in both the recuperators (LTR and HTR) are similar. However, in the same combination of minichannels, the bulk temperature difference ∆Tb=Tout−Tin and the heat transfer rate (Q) of HTR are always larger than those of LTR, which is related to the inlet conditions considered for the hot side and cold side minichannels (Table 1). For instance, both ∆Tb and Q of HTR with the conventional minichannels (Ref. case) are almost twice those of LTR. It is found that when both the hot side and the cold side minichannels have a convergent pattern (C-C), ∆Tb is basically the same as that of the Ref. case, and the visible changes are obtained when the structure of minichannels is not the same (C-U and C-D).Figure 5
Thermal characteristics of SCO2 flow inside LTR and HTR (Class 1: Converging hot side minichannel).The hot side SCO2 enters the recuperators at z=0 and transfers thermal energy to the cold side SCO2 through heat conduction in the solid domain; thus, its temperature gradually declines. By analyzing the data provided in Figure 6, it can be revealed that in the C-C pattern, the difference between the hot side and the cold side SCO2 temperatures in the axial direction remains unchanged, which is almost similar to that of the Ref. case. Also, the wall and SCO2 temperatures do not change much compared to the Ref. case when the cold side minichannel is also C, but they change noticeably when the cold side minichannel is U or D. Likewise, diverging the cold side minichannel results in the lowest temperature variation along the hot side minichannel, leading to the highest percentage difference between the hot side and the cold side SCO2 temperatures, which is 59.8% for LTR and 57.6% for HTR. Figure 6 also clarifies that when the cold side has a D pattern, the SCO2 temperature difference between the inlet of the hot side minichannel and the outlet of the cold side minichannel is small, while the SCO2 temperature difference between the outlet of the hot side minichannel and the inlet of the cold side minichannel is large. This is the main reason for the lower heat transfer rates in this case (C-D), as shown in the top-right of Figure 5. However, it is found that the simultaneous convergence of the cold side minichannel with the hot side minichannel (C-C) causes the highest heat transfer rates in the studied recuperators. When the cold side minichannel is U, converging the hot side minichannel increases Q by about 14.6% on the hot side and 8.3% on the cold side of LTR. Moreover, when the cold side minichannel is C, Q increases by 37.8% and 31.8%, respectively, compared to that of the Ref. case. The results explain that these increments in HTR are higher than those obtained for LTR.Figure 6
Variations of SCO2 temperature along axial length of minichannels in Class 1 (left) LTR (right) HTR.By analyzing the results of the Nusselt number (Nu) ratio and overall heat transfer coefficient presented in the bottom-left of Figure5, it can be found that converging the hot side minichannel results in better thermal performance on the hot side. Always developing fluid flow in converging structures can be the main reason for increasing heat transfer [33]. Compared to the Ref. case, the Nu enhancements on the hot side of the recuperators with C-U, C-C, and C-D patterns are 56.7%, 65.3%, and 75.6% for LTR and 55.5%, 60.5%, and 70.4% for HTR, respectively. Although the C-D pattern shows the best thermal performance on the hot side, its poor thermal performance on the cold side causes the lowest values of overall heat transfer coefficient. It is found that by replacing the C-D pattern with the C-C pattern inside LTR and HTR, the overall heat transfer coefficient increases 43.2% and 54.8%, respectively. As illustrated in Figure 6, when the cold side has a C pattern, the difference between the cold side SCO2 temperature and the wall temperature remains almost constant, which is very similar to the minichannel heated by a uniform heat flux, being beneficial for the heat transfer. However, when the cold side has a D pattern, the difference between the cold side SCO2 temperature and the wall temperature gradually decreases along the flow direction, which deteriorates the convective heat transfer.The thermal performance improvement of the recuperators with the C-C pattern can also be explained by the variations of SCO2 thermophysical properties and their corresponding effects on increasing the Prandtl number (Pr) and Reynolds number (Re), as presented in the bottom-right of Figure 5. It is well-known that Pr variations are analogous to the distributions of specific heat capacity (cp) and thermal conductivity (k), while Re variations follow the distributions of density (ρ) and dynamic viscosity (μ). Changing trend of SCO2 thermophysical properties and its corresponding effects on Pr and Re have already been reported in full detail by many researchers [12, 13, 34]. In models categorized in Class 1, despite the noticeable Pr variations occurred on the cold side of the recuperators, Pr values of SCO2 on the hot side are almost the same. For example, converging the hot side minichannel (refers to as the C-U pattern) in LTR reduces the cold side Pr by 9.4%, while Pr reduction is about 2.7% on its own side. Re augmentations in the C-C pattern are about 50% compared to the Ref. case. Therefore, the higher values of Pr and Re obtained by the C-C pattern result in larger Nu and better overall thermal performance, because according to the classical correlations, such as Dittus-Boelter or Gnielinski equations, Nu is a function of Pr and Re, so its improvement is directly proportional to the augmentations of these dimensionless parameters.The recuperators’ performance depends not only on the thermal characteristics but also on the hydraulic characteristics; therefore, the hydraulic performance of different combinations of minichannels should be taken into consideration in the design of LTR and HTR. The effects of the considered cases in Class 1 on the pressure drop (∆p=pin−pout) and friction factor (f) of the SCO2 flowing inside the hot side and cold side minichannels are reported in Figure 7. As similar to the thermal characteristics, minichannel convergence and divergence have significant impacts on the hydraulic specifications of the recuperators. According to the analysis carried out in the previous paragraphs, converging the hot side minichannel implies higher Nu values compared to the Ref. case as well as larger ∆p values on the hot side of LTR and HTR. At the same mass fluxes, the computed ∆p of the hot side is between 55.85 and 63.68 kPa for LTR and between 1.02 and 8.51 kPa for HTR. The maximum percentage augmentation of ∆p compared to the Ref. case is found to be by 336.2% and 332.8%, corresponding to the C-D pattern of minichannels inside LTR and HTR, respectively. Such high flow resistances in LTR and HTR may limit the maximum expansion ratio inside the turbine, affecting the net power output of the SCO2 cycle. In this class, the highest difference between ∆p of the hot side and ∆p of the cold side is also found for the C-D pattern, where ∆p occurred on the hot side is 62.4 and 25.5 times higher than that occurred on the cold side of LTR and HTR, respectively. These noticeable differences are mainly due to the changes in the cross-sectional area of the minichannels, because converging the hot side minichannel increases ∆p, while diverging the cold side minichannel causes ∆p to decrease. Likewise, a rise of more than 5-fold in the value of f is observed by converging the hot side minichannel. In particular, simultaneous diverging the cold side minichannel (refers to as the C-D pattern) causes f to augment more than 6 times as large as the Ref. case, while the corresponding decrements of f in the cold side are only 81.5% and 82.5% for LTR and HTR, respectively.Figure 7
Hydraulic characteristics of SCO2 flow inside LTR and HTR (Class 1: converging hot side minichannel).


#### 4.2.2. Diverging Hot Side Minichannel (Class 2)

Figure8 exhibits the effects of the hot side minichannel divergence on the thermal characteristics of LTR and HTR. At certain operating conditions, ∆Tb of D minichannel pattern on the hot side is higher than that of the other patterns (U and C). In comparison to the Ref. case, when the cold side minichannel is U, C, and D, diverging the hot side minichannel increases ∆Tb by 23.8%, 31.9% and 5.6% in LTR and 22.4%, 29.4%, and 5.8% in HTR, respectively. As it can be perceived from the previous minichannel combinations (Class 1), the maximum difference between ∆Tb of the hot side and that of the cold side occurs when the hot side and cold side minichannels come in the opposite arrangement. However, it is notable that this difference in the D-C pattern is about 4 times as large as that of the C-D pattern. Diverging the hot side minichannel also has a distinct effect on Q inside the recuperators. The results presented in the top-right of Figure 8 reveal that the thermal performance of both LTR and HTR is reduced by diverging the hot side minichannel. This is because the SCO2 flow in D pattern of minichannel decelerates along the flow direction [35]. Combined with the bottom-left of Figure 8, the overall heat transfer coefficient of all nonuniform combinations of minichannels is lower than that of the uniform combination. In comparison to the Ref. case, when the cold side minichannel is U, C, and D, diverging the hot side minichannel decreases the overall heat transfer coefficient by 25.1%, 26.8%, and 33.0% in LTR and 26.1%, 25.1%, and 39.6% in HTR, respectively. As discussed, Pr and Re are significant parameters affecting the thermal characteristics of the studied cases. It is found that diverging the hot side minichannel has comparable impact on Re and Pr of the SCO2 flowing inside both sides of the recuperators. For instance, by diverging the hot side minichannel with the cold side minichannel kept unchanged (refers to as the D-U pattern), about 41.1% and 39.3% falls will be witnessed in Re of the hot side of LTR and HTR, respectively.Figure 8
Thermal characteristics of SCO2 flow inside LTR and HTR (Class 2: diverging hot side minichannel).In Figure9, the effects of different combinations of minichannels with the D pattern of hot side (Class 2) on the hydraulic characteristics of the recuperators are demonstrated. The comparison of Figures 7 and 9 clarifies that ∆p in the C pattern of hot side minichannel is drastically larger as compared with the D pattern. For instance, compared to the U pattern of the hot side minichannel of LTR, the ∆p is observed to be larger by 393.5% for the C pattern, whereas it is smaller by 289.4% for the D pattern. Regarding Figure 6, the temperature of SCO2 flowing inside a divergent minichannel increases along the flow direction. This temperature increment inside the hot side minichannel not only decreases the viscosity of the SCO2 flowing in it but also affects the viscosity of the SCO2 flowing on the cold side minichannel. Therefore, ∆p of the cold side minichannel is also detected to decrease as a result of diverging the hot side minichannel. These decrements in LTR and HTR are, respectively, by 11.3% and 35.4% compared with that of the Ref. case. A higher ∆p decrement observed in HTR is due to its larger bulk temperature difference ∆Tb, as presented in the top-left of Figure 8.Figure 9
Hydraulic characteristics of SCO2 flow inside LTR and HTR (Class 2: diverging hot side minichannel).Converging the cold side minichannel results in increasing the∆p. The corresponding increments, compared to the Ref. case, are 187.6% and 135.2% for LTR and HTR, respectively. Nevertheless, these ∆p increments in the D-C pattern are lower than those of the C-C pattern, where the hot side is also converging and ∆p increments of the cold side minichannel are 259.7% and 188.2% for LTR and HTR, respectively. This is because ∆p of the cold side minichannel is influenced by converging the hot side minichannel to a great extent rather than diverging it. As previously discussed, through the C mode, the colder SCO2 passes through the hot side minichannel compared to the D mode, leading to relatively lower viscosity reduction of SCO2 flowing in the cold side minichannel. Hence, larger ∆p increments in the C-C pattern are observed as compared to the D-C pattern. The hydraulic difference between the C pattern of the cold side minichannel and other patterns (U and D) is also evident from the friction factor results. The f ratio recorded for this case is 5.56 for LTR and 4.30 for HTR. However, f values of the hot side minichannel in all combinations of Class 2 are almost comparable.


#### 4.2.3. Converging Cold Side Minichannel (Class 3)

The results presented in the top-left of Figure10 display that ∆Tb on the cold side of the recuperators decreases as a result of converging the cold side minichannel. As discussed, the maximum difference between ∆Tb of the cold side and ∆Tb of the hot side is related to the D-C combination where the minichannels have different patterns. However, the highest heat transfer rates are found for the combination with the same pattern of minichannels (C-C), as presented in the top-right of Figure 10. In comparison with the Ref. case, Q of the hot side and cold side of LTR increases by 18.1% and 25.6%, respectively. These increments are mainly due to the increase in Re along the convergent path of minichannels. Wherein Nu values of the hot side and cold side minichannels enhance by 65.3% and 74.9% in LTR and 60.2% and 73.9% in HTR, respectively. Thus, simultaneous convergence of both the minichannels can strengthen the thermal performance of the recuperators, leading to 21.8% and 26.6% enhancements in the overall heat transfer coefficient of LTR and HTR, respectively. The overall heat transfer coefficient of the D-C pattern in LTR and HTR is 26.8% and 25.1% lower than that of the Ref. case, respectively. It suggests that diverging the hot side minichannel weakens the overall thermal performance in the recuperators when the cold side minichannel has a C pattern.Figure 10
Thermal characteristics of SCO2 flow inside LTR and HTR (Class 3: converging cold side minichannel).However, the percentage deviations in∆p with respect to the Ref. case are less in the D-C pattern than the C-C pattern, as shown in Figure 11. For instance, converging the cold side minichannel increases ∆p of the SCO2 flow in this channel by 187.6% in the D-C pattern of LTR, whereas the increase of ∆p is 259.7% due to the convergence of the cold side minichannel in the C-C pattern. Similar trend is also detected for the friction factor. Figure 11 shows that f of the cold side increases by converging the minichannel. In comparison with the Ref. case, converging the cold side minichannel will increase the f ratio from 5.56 to 5.99 in LTR and from 4.30 to 4.68 in HTR.Figure 11
Hydraulic characteristics of SCO2 flow inside LTR and HTR (Class 3: converging cold side minichannel).


#### 4.2.4. Diverging Cold Side Minichannel (Class 4)

Figure12 discloses that diverging the cold side minichannel deteriorates the thermal performance of the recuperators. For instance, by diverging the cold side minichannel (refers to as the U-D pattern), 20.4 K and 38.8 K drops will be witnessed in ∆Tb of the SCO2 flowing on the hot side of LTR and HTR, respectively. The results presented in the top-right of Figure 12 reveal that Q of the cold side is almost invariant with making change in the hot side minichannel configuration, but Q of the hot side deviates somewhat from the U-D pattern. For instance, the Q values for the C-D and D-D patterns in LTR deviate by only 8.5% and 17.1% from those of the U-D pattern. As depicted in the figure, the calculated values are higher for HTR (10.5% and 21.9%). The maximum percentage deterioration in the overall heat transfer coefficient with the divergent cold side minichannel is 33.1% for LTR and 39.6% for HTR, which is found for the D-D pattern. Likewise, the percentage difference in the overall heat transfer coefficient is larger between the Ref. case and the U-D pattern than between the Ref. case and the C-D pattern.Figure 12
Thermal characteristics of SCO2 flow inside LTR and HTR (Class 4: diverging cold side minichannel).However, the pressure drop decreases with diverging the cold side minichannel, which is the results of the cross-sectional area expansion along the flow direction and also the viscosity value reduction of SCO2 (Figure 13). It is found that diverging the cold side minichannel can reduce ∆p by 57.2% and 70.8% compared to the Ref. case of LTR and HTR, respectively, while converging the cold side minichannel will augment ∆p by 234.4% and 163.9%, as shown in Figure 11. It is interesting to note that the percentage variations in ∆p of the cold side minichannel are a function of the hot side minichannel pattern. For example, in the case with the convergent cold side minichannel in LTR, the percentage increase in ∆p is 259.7% when the hot side minichannel has a C pattern, whereas it is 187.6% when the hot side minichannel has a D pattern. The corresponding variations in HTR are 188.2% and 135.2%, respectively. As mentioned, the decrease of ∆p is consistent with reduction in the viscosity value of SCO2. For example, by diverging the cold side minichannel of LTR, the viscosity of SCO2 declines from 3.53×10−5 to 3.15×10−5. The viscosity percentage reduction is larger in HTR than LTR due to higher operating temperatures. It should be noted that the friction factor and its corresponding ratio almost remain constant for all nonuniform patterns of the cold side, whereas they vary substantially for the hot side. For example, f ratio of the hot side minichannel in the C-D pattern of the recuperators even exceeds 6.Figure 13
Hydraulic characteristics of SCO2 flow inside LTR and HTR (Class 4: diverging cold side minichannel).


### 4.3. Overall Thermohydraulic Performance

In this section, the thermal and hydraulic parameters of SCO2 are converted to some performance criteria for the purpose of better comparison across diverse combinations of hot side and cold side minichannels. The applied parameters used to define these criteria are the Nusselt number (Nu), Euler number (Eu), Colburn factor (j), and friction factor (f). From these parameters, two performance indexes are created as follows [21, 36],
(13)η1=Nui/Nuref.Eui/Euref.,(14)η2=ji/jref.fi/fref.1/3.Figure14 shows the results obtained from the calculations of the above criteria, in which the histogram chart quantities show the outputs of Eq. (13) and the line type values show the outputs of Eq. (14). The comparisons clarify that among all the studied cases, the divergent configuration of both the hot side and the cold side minichannels results in the superior overall performance of the recuperators. Whenever both the hot side and the cold side minichannels diverge, η1 values of them become 1.62 and 1.26 for LTR and 1.71 and 2.32 for HTR, respectively. However, the highest value of η1 among the hot side minichannels (2.10 for LTR and 2.20 for HTR) occurs when the cold side minichannel is convergent. Likewise, the highest value of η2 for the D pattern of the hot side minichannels is 1.82 for LTR and 1.92 for HTR. At the same time, the corresponding values for the cold side minichannel are 1.57 and 2.11, respectively. As it can be seen in the previous figures, although convergent minichannels provide better thermal performance than uniform and divergent minichannels, their poor hydraulic performance leads to lower values of η1 and η2.Figure 14
Nusselt number to Euler number ratio and Colburn factor to friction factor ratio of SCO2 flow inside LTR and HTR.A low temperature of SCO2 at the exit of the hot side and a high temperature of it at the cold side exit are the main targets for designing the recuperators because the lower temperature at the hot side outlet and higher temperature at the cold side outlet effectively contribute to alleviating heat transfer loads in the cooler and heater of SCO2 Brayton cycle, respectively. The results of the outlet temperature (Tout) presented in Figure 14 reveal that diverging the minichannels is also effective in order to achieve these goals. By diverging the hot side and cold side minichannels, the minimum temperatures of 364.4 K and 472.6 K and the maximum temperature of 456.5 K and 656.9 K can be obtained at the outlet of LTR and HTR with D-C and C-D patterns, respectively. Simultaneous divergence of both the minichannels (refers to as D-D pattern) leads to the outlet temperatures of 385.1 K and 432.3 K for LTR and 609.7 K and 510.1 K for HTR.The aforementioned quantitative indexes separately analyze the effect of minichannel configuration on the overall performance of the hot side and cold side of the recuperators. To comprehensively appraise the overall thermohydraulic performance of the recuperators, the ratio of the average heat transfer rate to the total required pumping power is considered as follows [16],
(15)η3=Qm/ρΔphotside+m/ρΔpcoldside.It can be seen from Figure15 that converging the hot side minichannel deteriorates the overall performance of the recuperators. As the Ref. case is replaced with the C-U, C-C, and C-D patterns, the average heat transfer rate per the total pumping power decreases by 82.9%, 77.8%, and 89.8% for LTR and 80.2%, 75.5%, and 88.7% for HTR, respectively. However, converging the cold side minichannel when the hot side minichannel is divergent (refers to as the D-C pattern) can improve the overall performance of LTR by about 168.0%. This improvement can be attributed to noticeable decrement of the SCO2 pressure drop on the hot side, leading to lower pumping power comparing with the Ref. case, as depicted in Figure 11. Therefore, diverging both the hot side and the cold side minichannels leads to striking reductions in total pumping power. Meanwhile, the heat transferred from the hot side to the cold side is much smaller than that of the Ref. case. The trade-off between the heat transfer rate and the pumping power suggests that the best combination of the minichannels is the U-D pattern. The performance improvement value obtained by this pattern is 257.2% for LTR and 171.8% for HTR. Therefore, on account of the outstanding hydraulic behavior of the cold side, the U-D pattern can be recommended for the practical application of LTR and HTR because high pressure drops of SCO2 flowing in the cold side of the recuperators limit the maximum expansion ratio across the turbine. After, the U-D pattern, the D-D pattern can be the best choice. Compared with the Ref. case, the performance improvement value obtained by this pattern is 177.5% for LTR and 151.8% for HTR.Figure 15
Average heat transfer rate per total pumping power of SCO2 flow inside LTR and HTR.


### 4.4. Comparison with Other Patterns of Minichannel

In this section, the performance of nonuniform minichannels is compared with three more integral minichannels nominated as zigzag, wavy, and C-shaped optimized by Refs. [9, 37] previously. The comprehensive evaluation, which is conducted for a wide range of the Reynolds number, i.e., 10380<Re<35560 for the hot side and 5400<Re<29170 for the cold side, is provided in Table 3 and Table 4 in terms of the Nusselt number and friction factor and in Figures 16(a)–16(b) in terms of the performance index proposed by Webb [38] as follows,
(16)η4=NuZigzag,Wavy,C‐shaped/NuNonuniformfZigzag,Wavy,C‐shaped/fNonuniform1/3.Table 3
Comparison of Nusselt number and friction factor under operating conditions of LTR.
ReNufHot sideConvergent34919111.440.03470Zigzag194.720.02891Wavy164.410.02154C-shaped143.340.02154Divergent1355346.450.00131Zigzag86.030.03295Wavy82.400.02154C-shaped72.670.01972Uniform2355373.360.00590Zigzag145.560.03150Wavy126.500.02209C-shaped112.900.02027Uniform2241070.110.00622Zigzag138.480.03172Wavy122.420.02190C-shaped109.130.02045Convergent34313124.870.03709Zigzag191.230.02893Wavy162.880.02136C-shaped141.720.02145Divergent1378747.780.00127Zigzag89.670.03186Wavy84.770.02154C-shaped73.740.02000Convergent35558117.540.03512Zigzag197.380.02780Wavy167.680.02136C-shaped147.070.02163Divergent1313744.040.00132Zigzag85.360.03224Wavy79.840.02145C-shaped71.060.01990Cold sideUniform978142.990.01035Zigzag65.030.08566Wavy65.820.04124C-shaped69.800.03493Uniform837743.040.00976Zigzag53.290.08796Wavy58.680.04321C-shaped60.580.03520Convergent1292076.190.05842Zigzag82.540.08278Wavy84.510.04019C-shaped82.740.03388Divergent577620.760.00192Zigzag31.070.09127Wavy42.900.04473C-shaped49.240.03648Divergent594019.830.00180Zigzag34.540.09015Wavy43.510.04521C-shaped50.140.03624Convergent1204983.160.05431Zigzag80.450.08330Wavy79.800.04116C-shaped80.820.03437Convergent1374470.960.05807Zigzag94.370.08178Wavy87.840.04113C-shaped88.060.03409Divergent540322.560.00224Zigzag30.340.09066Wavy38.560.04523C-shaped46.690.03698Table 4
Comparison of Nusselt number and friction factor under operating conditions of HTR.
ReNufHot sideConvergent2686785.020.03630Zigzag165.320.03101Wavy140.950.02209C-shaped126.830.02081Divergent1093238.170.00142Zigzag71.990.03395Wavy68.970.02118C-shaped58.820.01963Uniform1767153.420.00622Zigzag113.430.03244Wavy102.100.02172C-shaped91.390.01990Uniform1726254.340.00665Zigzag109.800.03191Wavy99.410.02172C-shaped89.260.01990Convergent2614993.200.03854Zigzag160.440.03213Wavy137.500.02200C-shaped122.050.02054Divergent1126854.340.00135Zigzag74.310.03244Wavy71.780.02109C-shaped60.950.01945Convergent2760487.610.03605Zigzag167.620.03098Wavy143.890.02209C-shaped127.960.02090Divergent1038237.310.00147Zigzag68.560.03433Wavy66.090.02063C-shaped55.630.01954Cold sideUniform1703652.750.00804Zigzag103.200.08001Wavy97.650.04006C-shaped92.090.03473Uniform1766260.660.00731Zigzag105.940.07981Wavy107.470.04030C-shaped94.810.03496Convergent29170109.130.04717Zigzag166.650.07741Wavy156.620.04485C-shaped142.240.03588Divergent939929.770.00147Zigzag60.250.08615Wavy61.590.04197C-shaped55.190.03445Divergent922127.680.00152Zigzag62.320.08679Wavy61.010.04124C-shaped54.070.03445Convergent27751119.910.04453Zigzag154.290.07829Wavy143.960.04465C-shaped134.680.03494Convergent2734694.150.04851Zigzag152.680.07877Wavy142.850.04442C-shaped130.720.03374Divergent968133.000.00146Zigzag62.130.08630Wavy59.980.03517C-shaped54.550.04148Figure 16
Comparison of overall performance index under operating condition of (a) LTR and (b) HTR.
(a)(b)Regarding Table3 and Table 4, the minichannel structure has distinct effects on both thermal and hydraulic characteristics of the recuperators. The comparison confirms that all indirect minichannels (zigzag, wavy, and C-shaped) display better thermal performance than the nonuniform ones, and the corresponding Nusselt number enhancements are a function of Reynolds number. For instance, in the case of the optimized C-shaped minichannel, the percentage increase in Nusselt number of LTR hot side varies from 30.4% to 47.5% (average 40.2%) compared to that of the convergent minichannels and from 77.3% to 81.2% (average 78.6%) compared to that of the divergent minichannel. The corresponding enhancements are 47.5–65.7% (average 59.1%) and 77.1–85.2% (average 79.8%) in Nusselt number of HTR hot side. The comparative study conducted by Xie et al. [39] also disclosed that the utilization of the zigzag minichannels led to a 65% improvement in the effectiveness of PCHEs with a pressure drop augmentation of 39% compared to the straight minichannel.Although the indirect and convergent minichannels display better thermal performance than the divergent minichannels, their poor hydraulic performance causes lower values of the overall performance index at the low Reynolds numbers, as shown in Figure16. The results show that replacing the convergent minichannels with the indirect minichannels on the hot side of the recuperators can enhance η4 from 1.36 to 2.05 (average 1.74). At the same time, using the divergent minichannels instead of the indirect minichannels improves the overall performance of the recuperators from 26.4% to 53.8% (average 36.3%) due to the quite low flow resistance of the SCO2. Moreover, when the divergent minichannels are used on the cold side of the recuperators, the overall performance is improved from 25.1% to 51.6% (average 37.4%). It is worth mentioning that at low Reynolds numbers (less than 14000 for the hot side and 10000 for the cold side), all nonuniform patterns of minichannels have better overall performance than the integral complicated minichannels.The above-mentioned results can be taken into consideration in the design of recuperators for the SCO2 Brayton cycles. However, it should be noted that these points are recommended for the operating conditions of LTR and HTR with the similar sizing features; however, they may lose their accuracy once the mass fluxes, pressures, temperatures, and dimensions of minichannels vary significantly, particularly near the critical point of CO2.


## 4.1. Validation

Aiming at validating the present numerical method, the variations of temperature and pressure drop along the axial length of the hot side and cold side minichannels of a recuperator are considered. A PCHE with uniform minichannels is selected for comparison purposes, and the corresponding results are compared with the data reported by Marchionni et al. [31] and Li et al. [32], as shown in Figure 4. A comparison between the present results and the previous data shows the deviations being less than 10%. It confirms the validity of the numerical method employed in this study. However, the existing differences in the parameter predictions are inevitable and can stem from different considered turbulence models. As mentioned, the SST k-ω turbulence model is employed in the present study, while the turbulence model of the standard k-ε approach was applied by the previous studies.Figure 4
Validation of present results with previous data: (top) temperature and (bottom) pressure drop variations along hot side and cold side minichannels of PCHE.


## 4.2. Thermal and Hydraulic Characteristics

In this section, the thermal and hydraulic characteristics of diverse combinations of uniform (hereinafter referred to as the “U”), converging (hereinafter referred to as the “C”), and diverging (hereinafter referred to as the “D”) minichannels are compared with those of the conventional minichannel (hereinafter referred to as the “Ref.”) under the same operating conditions of LTR and HTR. The data analysis results indicate that converging and diverging the minichannels can significantly affect the thermal and hydraulic characteristics of both recuperators (LTR and HTR), which will be discussed in the following subsections.


### 4.2.1. Converging Hot Side Minichannel (Class 1)

Figure5 exhibits the thermal characteristics of diverse combinations of minichannels in Class 1 where the hot side minichannel is always C, while the cold side is either U, C, or D. As it can be seen, the minichannel convergence and divergence consequences in both the recuperators (LTR and HTR) are similar. However, in the same combination of minichannels, the bulk temperature difference ∆Tb=Tout−Tin and the heat transfer rate (Q) of HTR are always larger than those of LTR, which is related to the inlet conditions considered for the hot side and cold side minichannels (Table 1). For instance, both ∆Tb and Q of HTR with the conventional minichannels (Ref. case) are almost twice those of LTR. It is found that when both the hot side and the cold side minichannels have a convergent pattern (C-C), ∆Tb is basically the same as that of the Ref. case, and the visible changes are obtained when the structure of minichannels is not the same (C-U and C-D).Figure 5
Thermal characteristics of SCO2 flow inside LTR and HTR (Class 1: Converging hot side minichannel).The hot side SCO2 enters the recuperators at z=0 and transfers thermal energy to the cold side SCO2 through heat conduction in the solid domain; thus, its temperature gradually declines. By analyzing the data provided in Figure 6, it can be revealed that in the C-C pattern, the difference between the hot side and the cold side SCO2 temperatures in the axial direction remains unchanged, which is almost similar to that of the Ref. case. Also, the wall and SCO2 temperatures do not change much compared to the Ref. case when the cold side minichannel is also C, but they change noticeably when the cold side minichannel is U or D. Likewise, diverging the cold side minichannel results in the lowest temperature variation along the hot side minichannel, leading to the highest percentage difference between the hot side and the cold side SCO2 temperatures, which is 59.8% for LTR and 57.6% for HTR. Figure 6 also clarifies that when the cold side has a D pattern, the SCO2 temperature difference between the inlet of the hot side minichannel and the outlet of the cold side minichannel is small, while the SCO2 temperature difference between the outlet of the hot side minichannel and the inlet of the cold side minichannel is large. This is the main reason for the lower heat transfer rates in this case (C-D), as shown in the top-right of Figure 5. However, it is found that the simultaneous convergence of the cold side minichannel with the hot side minichannel (C-C) causes the highest heat transfer rates in the studied recuperators. When the cold side minichannel is U, converging the hot side minichannel increases Q by about 14.6% on the hot side and 8.3% on the cold side of LTR. Moreover, when the cold side minichannel is C, Q increases by 37.8% and 31.8%, respectively, compared to that of the Ref. case. The results explain that these increments in HTR are higher than those obtained for LTR.Figure 6
Variations of SCO2 temperature along axial length of minichannels in Class 1 (left) LTR (right) HTR.By analyzing the results of the Nusselt number (Nu) ratio and overall heat transfer coefficient presented in the bottom-left of Figure5, it can be found that converging the hot side minichannel results in better thermal performance on the hot side. Always developing fluid flow in converging structures can be the main reason for increasing heat transfer [33]. Compared to the Ref. case, the Nu enhancements on the hot side of the recuperators with C-U, C-C, and C-D patterns are 56.7%, 65.3%, and 75.6% for LTR and 55.5%, 60.5%, and 70.4% for HTR, respectively. Although the C-D pattern shows the best thermal performance on the hot side, its poor thermal performance on the cold side causes the lowest values of overall heat transfer coefficient. It is found that by replacing the C-D pattern with the C-C pattern inside LTR and HTR, the overall heat transfer coefficient increases 43.2% and 54.8%, respectively. As illustrated in Figure 6, when the cold side has a C pattern, the difference between the cold side SCO2 temperature and the wall temperature remains almost constant, which is very similar to the minichannel heated by a uniform heat flux, being beneficial for the heat transfer. However, when the cold side has a D pattern, the difference between the cold side SCO2 temperature and the wall temperature gradually decreases along the flow direction, which deteriorates the convective heat transfer.The thermal performance improvement of the recuperators with the C-C pattern can also be explained by the variations of SCO2 thermophysical properties and their corresponding effects on increasing the Prandtl number (Pr) and Reynolds number (Re), as presented in the bottom-right of Figure 5. It is well-known that Pr variations are analogous to the distributions of specific heat capacity (cp) and thermal conductivity (k), while Re variations follow the distributions of density (ρ) and dynamic viscosity (μ). Changing trend of SCO2 thermophysical properties and its corresponding effects on Pr and Re have already been reported in full detail by many researchers [12, 13, 34]. In models categorized in Class 1, despite the noticeable Pr variations occurred on the cold side of the recuperators, Pr values of SCO2 on the hot side are almost the same. For example, converging the hot side minichannel (refers to as the C-U pattern) in LTR reduces the cold side Pr by 9.4%, while Pr reduction is about 2.7% on its own side. Re augmentations in the C-C pattern are about 50% compared to the Ref. case. Therefore, the higher values of Pr and Re obtained by the C-C pattern result in larger Nu and better overall thermal performance, because according to the classical correlations, such as Dittus-Boelter or Gnielinski equations, Nu is a function of Pr and Re, so its improvement is directly proportional to the augmentations of these dimensionless parameters.The recuperators’ performance depends not only on the thermal characteristics but also on the hydraulic characteristics; therefore, the hydraulic performance of different combinations of minichannels should be taken into consideration in the design of LTR and HTR. The effects of the considered cases in Class 1 on the pressure drop (∆p=pin−pout) and friction factor (f) of the SCO2 flowing inside the hot side and cold side minichannels are reported in Figure 7. As similar to the thermal characteristics, minichannel convergence and divergence have significant impacts on the hydraulic specifications of the recuperators. According to the analysis carried out in the previous paragraphs, converging the hot side minichannel implies higher Nu values compared to the Ref. case as well as larger ∆p values on the hot side of LTR and HTR. At the same mass fluxes, the computed ∆p of the hot side is between 55.85 and 63.68 kPa for LTR and between 1.02 and 8.51 kPa for HTR. The maximum percentage augmentation of ∆p compared to the Ref. case is found to be by 336.2% and 332.8%, corresponding to the C-D pattern of minichannels inside LTR and HTR, respectively. Such high flow resistances in LTR and HTR may limit the maximum expansion ratio inside the turbine, affecting the net power output of the SCO2 cycle. In this class, the highest difference between ∆p of the hot side and ∆p of the cold side is also found for the C-D pattern, where ∆p occurred on the hot side is 62.4 and 25.5 times higher than that occurred on the cold side of LTR and HTR, respectively. These noticeable differences are mainly due to the changes in the cross-sectional area of the minichannels, because converging the hot side minichannel increases ∆p, while diverging the cold side minichannel causes ∆p to decrease. Likewise, a rise of more than 5-fold in the value of f is observed by converging the hot side minichannel. In particular, simultaneous diverging the cold side minichannel (refers to as the C-D pattern) causes f to augment more than 6 times as large as the Ref. case, while the corresponding decrements of f in the cold side are only 81.5% and 82.5% for LTR and HTR, respectively.Figure 7
Hydraulic characteristics of SCO2 flow inside LTR and HTR (Class 1: converging hot side minichannel).


### 4.2.2. Diverging Hot Side Minichannel (Class 2)

Figure8 exhibits the effects of the hot side minichannel divergence on the thermal characteristics of LTR and HTR. At certain operating conditions, ∆Tb of D minichannel pattern on the hot side is higher than that of the other patterns (U and C). In comparison to the Ref. case, when the cold side minichannel is U, C, and D, diverging the hot side minichannel increases ∆Tb by 23.8%, 31.9% and 5.6% in LTR and 22.4%, 29.4%, and 5.8% in HTR, respectively. As it can be perceived from the previous minichannel combinations (Class 1), the maximum difference between ∆Tb of the hot side and that of the cold side occurs when the hot side and cold side minichannels come in the opposite arrangement. However, it is notable that this difference in the D-C pattern is about 4 times as large as that of the C-D pattern. Diverging the hot side minichannel also has a distinct effect on Q inside the recuperators. The results presented in the top-right of Figure 8 reveal that the thermal performance of both LTR and HTR is reduced by diverging the hot side minichannel. This is because the SCO2 flow in D pattern of minichannel decelerates along the flow direction [35]. Combined with the bottom-left of Figure 8, the overall heat transfer coefficient of all nonuniform combinations of minichannels is lower than that of the uniform combination. In comparison to the Ref. case, when the cold side minichannel is U, C, and D, diverging the hot side minichannel decreases the overall heat transfer coefficient by 25.1%, 26.8%, and 33.0% in LTR and 26.1%, 25.1%, and 39.6% in HTR, respectively. As discussed, Pr and Re are significant parameters affecting the thermal characteristics of the studied cases. It is found that diverging the hot side minichannel has comparable impact on Re and Pr of the SCO2 flowing inside both sides of the recuperators. For instance, by diverging the hot side minichannel with the cold side minichannel kept unchanged (refers to as the D-U pattern), about 41.1% and 39.3% falls will be witnessed in Re of the hot side of LTR and HTR, respectively.Figure 8
Thermal characteristics of SCO2 flow inside LTR and HTR (Class 2: diverging hot side minichannel).In Figure9, the effects of different combinations of minichannels with the D pattern of hot side (Class 2) on the hydraulic characteristics of the recuperators are demonstrated. The comparison of Figures 7 and 9 clarifies that ∆p in the C pattern of hot side minichannel is drastically larger as compared with the D pattern. For instance, compared to the U pattern of the hot side minichannel of LTR, the ∆p is observed to be larger by 393.5% for the C pattern, whereas it is smaller by 289.4% for the D pattern. Regarding Figure 6, the temperature of SCO2 flowing inside a divergent minichannel increases along the flow direction. This temperature increment inside the hot side minichannel not only decreases the viscosity of the SCO2 flowing in it but also affects the viscosity of the SCO2 flowing on the cold side minichannel. Therefore, ∆p of the cold side minichannel is also detected to decrease as a result of diverging the hot side minichannel. These decrements in LTR and HTR are, respectively, by 11.3% and 35.4% compared with that of the Ref. case. A higher ∆p decrement observed in HTR is due to its larger bulk temperature difference ∆Tb, as presented in the top-left of Figure 8.Figure 9
Hydraulic characteristics of SCO2 flow inside LTR and HTR (Class 2: diverging hot side minichannel).Converging the cold side minichannel results in increasing the∆p. The corresponding increments, compared to the Ref. case, are 187.6% and 135.2% for LTR and HTR, respectively. Nevertheless, these ∆p increments in the D-C pattern are lower than those of the C-C pattern, where the hot side is also converging and ∆p increments of the cold side minichannel are 259.7% and 188.2% for LTR and HTR, respectively. This is because ∆p of the cold side minichannel is influenced by converging the hot side minichannel to a great extent rather than diverging it. As previously discussed, through the C mode, the colder SCO2 passes through the hot side minichannel compared to the D mode, leading to relatively lower viscosity reduction of SCO2 flowing in the cold side minichannel. Hence, larger ∆p increments in the C-C pattern are observed as compared to the D-C pattern. The hydraulic difference between the C pattern of the cold side minichannel and other patterns (U and D) is also evident from the friction factor results. The f ratio recorded for this case is 5.56 for LTR and 4.30 for HTR. However, f values of the hot side minichannel in all combinations of Class 2 are almost comparable.


### 4.2.3. Converging Cold Side Minichannel (Class 3)

The results presented in the top-left of Figure10 display that ∆Tb on the cold side of the recuperators decreases as a result of converging the cold side minichannel. As discussed, the maximum difference between ∆Tb of the cold side and ∆Tb of the hot side is related to the D-C combination where the minichannels have different patterns. However, the highest heat transfer rates are found for the combination with the same pattern of minichannels (C-C), as presented in the top-right of Figure 10. In comparison with the Ref. case, Q of the hot side and cold side of LTR increases by 18.1% and 25.6%, respectively. These increments are mainly due to the increase in Re along the convergent path of minichannels. Wherein Nu values of the hot side and cold side minichannels enhance by 65.3% and 74.9% in LTR and 60.2% and 73.9% in HTR, respectively. Thus, simultaneous convergence of both the minichannels can strengthen the thermal performance of the recuperators, leading to 21.8% and 26.6% enhancements in the overall heat transfer coefficient of LTR and HTR, respectively. The overall heat transfer coefficient of the D-C pattern in LTR and HTR is 26.8% and 25.1% lower than that of the Ref. case, respectively. It suggests that diverging the hot side minichannel weakens the overall thermal performance in the recuperators when the cold side minichannel has a C pattern.Figure 10
Thermal characteristics of SCO2 flow inside LTR and HTR (Class 3: converging cold side minichannel).However, the percentage deviations in∆p with respect to the Ref. case are less in the D-C pattern than the C-C pattern, as shown in Figure 11. For instance, converging the cold side minichannel increases ∆p of the SCO2 flow in this channel by 187.6% in the D-C pattern of LTR, whereas the increase of ∆p is 259.7% due to the convergence of the cold side minichannel in the C-C pattern. Similar trend is also detected for the friction factor. Figure 11 shows that f of the cold side increases by converging the minichannel. In comparison with the Ref. case, converging the cold side minichannel will increase the f ratio from 5.56 to 5.99 in LTR and from 4.30 to 4.68 in HTR.Figure 11
Hydraulic characteristics of SCO2 flow inside LTR and HTR (Class 3: converging cold side minichannel).


### 4.2.4. Diverging Cold Side Minichannel (Class 4)

Figure12 discloses that diverging the cold side minichannel deteriorates the thermal performance of the recuperators. For instance, by diverging the cold side minichannel (refers to as the U-D pattern), 20.4 K and 38.8 K drops will be witnessed in ∆Tb of the SCO2 flowing on the hot side of LTR and HTR, respectively. The results presented in the top-right of Figure 12 reveal that Q of the cold side is almost invariant with making change in the hot side minichannel configuration, but Q of the hot side deviates somewhat from the U-D pattern. For instance, the Q values for the C-D and D-D patterns in LTR deviate by only 8.5% and 17.1% from those of the U-D pattern. As depicted in the figure, the calculated values are higher for HTR (10.5% and 21.9%). The maximum percentage deterioration in the overall heat transfer coefficient with the divergent cold side minichannel is 33.1% for LTR and 39.6% for HTR, which is found for the D-D pattern. Likewise, the percentage difference in the overall heat transfer coefficient is larger between the Ref. case and the U-D pattern than between the Ref. case and the C-D pattern.Figure 12
Thermal characteristics of SCO2 flow inside LTR and HTR (Class 4: diverging cold side minichannel).However, the pressure drop decreases with diverging the cold side minichannel, which is the results of the cross-sectional area expansion along the flow direction and also the viscosity value reduction of SCO2 (Figure 13). It is found that diverging the cold side minichannel can reduce ∆p by 57.2% and 70.8% compared to the Ref. case of LTR and HTR, respectively, while converging the cold side minichannel will augment ∆p by 234.4% and 163.9%, as shown in Figure 11. It is interesting to note that the percentage variations in ∆p of the cold side minichannel are a function of the hot side minichannel pattern. For example, in the case with the convergent cold side minichannel in LTR, the percentage increase in ∆p is 259.7% when the hot side minichannel has a C pattern, whereas it is 187.6% when the hot side minichannel has a D pattern. The corresponding variations in HTR are 188.2% and 135.2%, respectively. As mentioned, the decrease of ∆p is consistent with reduction in the viscosity value of SCO2. For example, by diverging the cold side minichannel of LTR, the viscosity of SCO2 declines from 3.53×10−5 to 3.15×10−5. The viscosity percentage reduction is larger in HTR than LTR due to higher operating temperatures. It should be noted that the friction factor and its corresponding ratio almost remain constant for all nonuniform patterns of the cold side, whereas they vary substantially for the hot side. For example, f ratio of the hot side minichannel in the C-D pattern of the recuperators even exceeds 6.Figure 13
Hydraulic characteristics of SCO2 flow inside LTR and HTR (Class 4: diverging cold side minichannel).


## 4.2.1. Converging Hot Side Minichannel (Class 1)

Figure5 exhibits the thermal characteristics of diverse combinations of minichannels in Class 1 where the hot side minichannel is always C, while the cold side is either U, C, or D. As it can be seen, the minichannel convergence and divergence consequences in both the recuperators (LTR and HTR) are similar. However, in the same combination of minichannels, the bulk temperature difference ∆Tb=Tout−Tin and the heat transfer rate (Q) of HTR are always larger than those of LTR, which is related to the inlet conditions considered for the hot side and cold side minichannels (Table 1). For instance, both ∆Tb and Q of HTR with the conventional minichannels (Ref. case) are almost twice those of LTR. It is found that when both the hot side and the cold side minichannels have a convergent pattern (C-C), ∆Tb is basically the same as that of the Ref. case, and the visible changes are obtained when the structure of minichannels is not the same (C-U and C-D).Figure 5
Thermal characteristics of SCO2 flow inside LTR and HTR (Class 1: Converging hot side minichannel).The hot side SCO2 enters the recuperators at z=0 and transfers thermal energy to the cold side SCO2 through heat conduction in the solid domain; thus, its temperature gradually declines. By analyzing the data provided in Figure 6, it can be revealed that in the C-C pattern, the difference between the hot side and the cold side SCO2 temperatures in the axial direction remains unchanged, which is almost similar to that of the Ref. case. Also, the wall and SCO2 temperatures do not change much compared to the Ref. case when the cold side minichannel is also C, but they change noticeably when the cold side minichannel is U or D. Likewise, diverging the cold side minichannel results in the lowest temperature variation along the hot side minichannel, leading to the highest percentage difference between the hot side and the cold side SCO2 temperatures, which is 59.8% for LTR and 57.6% for HTR. Figure 6 also clarifies that when the cold side has a D pattern, the SCO2 temperature difference between the inlet of the hot side minichannel and the outlet of the cold side minichannel is small, while the SCO2 temperature difference between the outlet of the hot side minichannel and the inlet of the cold side minichannel is large. This is the main reason for the lower heat transfer rates in this case (C-D), as shown in the top-right of Figure 5. However, it is found that the simultaneous convergence of the cold side minichannel with the hot side minichannel (C-C) causes the highest heat transfer rates in the studied recuperators. When the cold side minichannel is U, converging the hot side minichannel increases Q by about 14.6% on the hot side and 8.3% on the cold side of LTR. Moreover, when the cold side minichannel is C, Q increases by 37.8% and 31.8%, respectively, compared to that of the Ref. case. The results explain that these increments in HTR are higher than those obtained for LTR.Figure 6
Variations of SCO2 temperature along axial length of minichannels in Class 1 (left) LTR (right) HTR.By analyzing the results of the Nusselt number (Nu) ratio and overall heat transfer coefficient presented in the bottom-left of Figure5, it can be found that converging the hot side minichannel results in better thermal performance on the hot side. Always developing fluid flow in converging structures can be the main reason for increasing heat transfer [33]. Compared to the Ref. case, the Nu enhancements on the hot side of the recuperators with C-U, C-C, and C-D patterns are 56.7%, 65.3%, and 75.6% for LTR and 55.5%, 60.5%, and 70.4% for HTR, respectively. Although the C-D pattern shows the best thermal performance on the hot side, its poor thermal performance on the cold side causes the lowest values of overall heat transfer coefficient. It is found that by replacing the C-D pattern with the C-C pattern inside LTR and HTR, the overall heat transfer coefficient increases 43.2% and 54.8%, respectively. As illustrated in Figure 6, when the cold side has a C pattern, the difference between the cold side SCO2 temperature and the wall temperature remains almost constant, which is very similar to the minichannel heated by a uniform heat flux, being beneficial for the heat transfer. However, when the cold side has a D pattern, the difference between the cold side SCO2 temperature and the wall temperature gradually decreases along the flow direction, which deteriorates the convective heat transfer.The thermal performance improvement of the recuperators with the C-C pattern can also be explained by the variations of SCO2 thermophysical properties and their corresponding effects on increasing the Prandtl number (Pr) and Reynolds number (Re), as presented in the bottom-right of Figure 5. It is well-known that Pr variations are analogous to the distributions of specific heat capacity (cp) and thermal conductivity (k), while Re variations follow the distributions of density (ρ) and dynamic viscosity (μ). Changing trend of SCO2 thermophysical properties and its corresponding effects on Pr and Re have already been reported in full detail by many researchers [12, 13, 34]. In models categorized in Class 1, despite the noticeable Pr variations occurred on the cold side of the recuperators, Pr values of SCO2 on the hot side are almost the same. For example, converging the hot side minichannel (refers to as the C-U pattern) in LTR reduces the cold side Pr by 9.4%, while Pr reduction is about 2.7% on its own side. Re augmentations in the C-C pattern are about 50% compared to the Ref. case. Therefore, the higher values of Pr and Re obtained by the C-C pattern result in larger Nu and better overall thermal performance, because according to the classical correlations, such as Dittus-Boelter or Gnielinski equations, Nu is a function of Pr and Re, so its improvement is directly proportional to the augmentations of these dimensionless parameters.The recuperators’ performance depends not only on the thermal characteristics but also on the hydraulic characteristics; therefore, the hydraulic performance of different combinations of minichannels should be taken into consideration in the design of LTR and HTR. The effects of the considered cases in Class 1 on the pressure drop (∆p=pin−pout) and friction factor (f) of the SCO2 flowing inside the hot side and cold side minichannels are reported in Figure 7. As similar to the thermal characteristics, minichannel convergence and divergence have significant impacts on the hydraulic specifications of the recuperators. According to the analysis carried out in the previous paragraphs, converging the hot side minichannel implies higher Nu values compared to the Ref. case as well as larger ∆p values on the hot side of LTR and HTR. At the same mass fluxes, the computed ∆p of the hot side is between 55.85 and 63.68 kPa for LTR and between 1.02 and 8.51 kPa for HTR. The maximum percentage augmentation of ∆p compared to the Ref. case is found to be by 336.2% and 332.8%, corresponding to the C-D pattern of minichannels inside LTR and HTR, respectively. Such high flow resistances in LTR and HTR may limit the maximum expansion ratio inside the turbine, affecting the net power output of the SCO2 cycle. In this class, the highest difference between ∆p of the hot side and ∆p of the cold side is also found for the C-D pattern, where ∆p occurred on the hot side is 62.4 and 25.5 times higher than that occurred on the cold side of LTR and HTR, respectively. These noticeable differences are mainly due to the changes in the cross-sectional area of the minichannels, because converging the hot side minichannel increases ∆p, while diverging the cold side minichannel causes ∆p to decrease. Likewise, a rise of more than 5-fold in the value of f is observed by converging the hot side minichannel. In particular, simultaneous diverging the cold side minichannel (refers to as the C-D pattern) causes f to augment more than 6 times as large as the Ref. case, while the corresponding decrements of f in the cold side are only 81.5% and 82.5% for LTR and HTR, respectively.Figure 7
Hydraulic characteristics of SCO2 flow inside LTR and HTR (Class 1: converging hot side minichannel).


## 4.2.2. Diverging Hot Side Minichannel (Class 2)

Figure8 exhibits the effects of the hot side minichannel divergence on the thermal characteristics of LTR and HTR. At certain operating conditions, ∆Tb of D minichannel pattern on the hot side is higher than that of the other patterns (U and C). In comparison to the Ref. case, when the cold side minichannel is U, C, and D, diverging the hot side minichannel increases ∆Tb by 23.8%, 31.9% and 5.6% in LTR and 22.4%, 29.4%, and 5.8% in HTR, respectively. As it can be perceived from the previous minichannel combinations (Class 1), the maximum difference between ∆Tb of the hot side and that of the cold side occurs when the hot side and cold side minichannels come in the opposite arrangement. However, it is notable that this difference in the D-C pattern is about 4 times as large as that of the C-D pattern. Diverging the hot side minichannel also has a distinct effect on Q inside the recuperators. The results presented in the top-right of Figure 8 reveal that the thermal performance of both LTR and HTR is reduced by diverging the hot side minichannel. This is because the SCO2 flow in D pattern of minichannel decelerates along the flow direction [35]. Combined with the bottom-left of Figure 8, the overall heat transfer coefficient of all nonuniform combinations of minichannels is lower than that of the uniform combination. In comparison to the Ref. case, when the cold side minichannel is U, C, and D, diverging the hot side minichannel decreases the overall heat transfer coefficient by 25.1%, 26.8%, and 33.0% in LTR and 26.1%, 25.1%, and 39.6% in HTR, respectively. As discussed, Pr and Re are significant parameters affecting the thermal characteristics of the studied cases. It is found that diverging the hot side minichannel has comparable impact on Re and Pr of the SCO2 flowing inside both sides of the recuperators. For instance, by diverging the hot side minichannel with the cold side minichannel kept unchanged (refers to as the D-U pattern), about 41.1% and 39.3% falls will be witnessed in Re of the hot side of LTR and HTR, respectively.Figure 8
Thermal characteristics of SCO2 flow inside LTR and HTR (Class 2: diverging hot side minichannel).In Figure9, the effects of different combinations of minichannels with the D pattern of hot side (Class 2) on the hydraulic characteristics of the recuperators are demonstrated. The comparison of Figures 7 and 9 clarifies that ∆p in the C pattern of hot side minichannel is drastically larger as compared with the D pattern. For instance, compared to the U pattern of the hot side minichannel of LTR, the ∆p is observed to be larger by 393.5% for the C pattern, whereas it is smaller by 289.4% for the D pattern. Regarding Figure 6, the temperature of SCO2 flowing inside a divergent minichannel increases along the flow direction. This temperature increment inside the hot side minichannel not only decreases the viscosity of the SCO2 flowing in it but also affects the viscosity of the SCO2 flowing on the cold side minichannel. Therefore, ∆p of the cold side minichannel is also detected to decrease as a result of diverging the hot side minichannel. These decrements in LTR and HTR are, respectively, by 11.3% and 35.4% compared with that of the Ref. case. A higher ∆p decrement observed in HTR is due to its larger bulk temperature difference ∆Tb, as presented in the top-left of Figure 8.Figure 9
Hydraulic characteristics of SCO2 flow inside LTR and HTR (Class 2: diverging hot side minichannel).Converging the cold side minichannel results in increasing the∆p. The corresponding increments, compared to the Ref. case, are 187.6% and 135.2% for LTR and HTR, respectively. Nevertheless, these ∆p increments in the D-C pattern are lower than those of the C-C pattern, where the hot side is also converging and ∆p increments of the cold side minichannel are 259.7% and 188.2% for LTR and HTR, respectively. This is because ∆p of the cold side minichannel is influenced by converging the hot side minichannel to a great extent rather than diverging it. As previously discussed, through the C mode, the colder SCO2 passes through the hot side minichannel compared to the D mode, leading to relatively lower viscosity reduction of SCO2 flowing in the cold side minichannel. Hence, larger ∆p increments in the C-C pattern are observed as compared to the D-C pattern. The hydraulic difference between the C pattern of the cold side minichannel and other patterns (U and D) is also evident from the friction factor results. The f ratio recorded for this case is 5.56 for LTR and 4.30 for HTR. However, f values of the hot side minichannel in all combinations of Class 2 are almost comparable.


## 4.2.3. Converging Cold Side Minichannel (Class 3)

The results presented in the top-left of Figure10 display that ∆Tb on the cold side of the recuperators decreases as a result of converging the cold side minichannel. As discussed, the maximum difference between ∆Tb of the cold side and ∆Tb of the hot side is related to the D-C combination where the minichannels have different patterns. However, the highest heat transfer rates are found for the combination with the same pattern of minichannels (C-C), as presented in the top-right of Figure 10. In comparison with the Ref. case, Q of the hot side and cold side of LTR increases by 18.1% and 25.6%, respectively. These increments are mainly due to the increase in Re along the convergent path of minichannels. Wherein Nu values of the hot side and cold side minichannels enhance by 65.3% and 74.9% in LTR and 60.2% and 73.9% in HTR, respectively. Thus, simultaneous convergence of both the minichannels can strengthen the thermal performance of the recuperators, leading to 21.8% and 26.6% enhancements in the overall heat transfer coefficient of LTR and HTR, respectively. The overall heat transfer coefficient of the D-C pattern in LTR and HTR is 26.8% and 25.1% lower than that of the Ref. case, respectively. It suggests that diverging the hot side minichannel weakens the overall thermal performance in the recuperators when the cold side minichannel has a C pattern.Figure 10
Thermal characteristics of SCO2 flow inside LTR and HTR (Class 3: converging cold side minichannel).However, the percentage deviations in∆p with respect to the Ref. case are less in the D-C pattern than the C-C pattern, as shown in Figure 11. For instance, converging the cold side minichannel increases ∆p of the SCO2 flow in this channel by 187.6% in the D-C pattern of LTR, whereas the increase of ∆p is 259.7% due to the convergence of the cold side minichannel in the C-C pattern. Similar trend is also detected for the friction factor. Figure 11 shows that f of the cold side increases by converging the minichannel. In comparison with the Ref. case, converging the cold side minichannel will increase the f ratio from 5.56 to 5.99 in LTR and from 4.30 to 4.68 in HTR.Figure 11
Hydraulic characteristics of SCO2 flow inside LTR and HTR (Class 3: converging cold side minichannel).


## 4.2.4. Diverging Cold Side Minichannel (Class 4)

Figure12 discloses that diverging the cold side minichannel deteriorates the thermal performance of the recuperators. For instance, by diverging the cold side minichannel (refers to as the U-D pattern), 20.4 K and 38.8 K drops will be witnessed in ∆Tb of the SCO2 flowing on the hot side of LTR and HTR, respectively. The results presented in the top-right of Figure 12 reveal that Q of the cold side is almost invariant with making change in the hot side minichannel configuration, but Q of the hot side deviates somewhat from the U-D pattern. For instance, the Q values for the C-D and D-D patterns in LTR deviate by only 8.5% and 17.1% from those of the U-D pattern. As depicted in the figure, the calculated values are higher for HTR (10.5% and 21.9%). The maximum percentage deterioration in the overall heat transfer coefficient with the divergent cold side minichannel is 33.1% for LTR and 39.6% for HTR, which is found for the D-D pattern. Likewise, the percentage difference in the overall heat transfer coefficient is larger between the Ref. case and the U-D pattern than between the Ref. case and the C-D pattern.Figure 12
Thermal characteristics of SCO2 flow inside LTR and HTR (Class 4: diverging cold side minichannel).However, the pressure drop decreases with diverging the cold side minichannel, which is the results of the cross-sectional area expansion along the flow direction and also the viscosity value reduction of SCO2 (Figure 13). It is found that diverging the cold side minichannel can reduce ∆p by 57.2% and 70.8% compared to the Ref. case of LTR and HTR, respectively, while converging the cold side minichannel will augment ∆p by 234.4% and 163.9%, as shown in Figure 11. It is interesting to note that the percentage variations in ∆p of the cold side minichannel are a function of the hot side minichannel pattern. For example, in the case with the convergent cold side minichannel in LTR, the percentage increase in ∆p is 259.7% when the hot side minichannel has a C pattern, whereas it is 187.6% when the hot side minichannel has a D pattern. The corresponding variations in HTR are 188.2% and 135.2%, respectively. As mentioned, the decrease of ∆p is consistent with reduction in the viscosity value of SCO2. For example, by diverging the cold side minichannel of LTR, the viscosity of SCO2 declines from 3.53×10−5 to 3.15×10−5. The viscosity percentage reduction is larger in HTR than LTR due to higher operating temperatures. It should be noted that the friction factor and its corresponding ratio almost remain constant for all nonuniform patterns of the cold side, whereas they vary substantially for the hot side. For example, f ratio of the hot side minichannel in the C-D pattern of the recuperators even exceeds 6.Figure 13
Hydraulic characteristics of SCO2 flow inside LTR and HTR (Class 4: diverging cold side minichannel).


## 4.3. Overall Thermohydraulic Performance

In this section, the thermal and hydraulic parameters of SCO2 are converted to some performance criteria for the purpose of better comparison across diverse combinations of hot side and cold side minichannels. The applied parameters used to define these criteria are the Nusselt number (Nu), Euler number (Eu), Colburn factor (j), and friction factor (f). From these parameters, two performance indexes are created as follows [21, 36],
(13)η1=Nui/Nuref.Eui/Euref.,(14)η2=ji/jref.fi/fref.1/3.Figure14 shows the results obtained from the calculations of the above criteria, in which the histogram chart quantities show the outputs of Eq. (13) and the line type values show the outputs of Eq. (14). The comparisons clarify that among all the studied cases, the divergent configuration of both the hot side and the cold side minichannels results in the superior overall performance of the recuperators. Whenever both the hot side and the cold side minichannels diverge, η1 values of them become 1.62 and 1.26 for LTR and 1.71 and 2.32 for HTR, respectively. However, the highest value of η1 among the hot side minichannels (2.10 for LTR and 2.20 for HTR) occurs when the cold side minichannel is convergent. Likewise, the highest value of η2 for the D pattern of the hot side minichannels is 1.82 for LTR and 1.92 for HTR. At the same time, the corresponding values for the cold side minichannel are 1.57 and 2.11, respectively. As it can be seen in the previous figures, although convergent minichannels provide better thermal performance than uniform and divergent minichannels, their poor hydraulic performance leads to lower values of η1 and η2.Figure 14
Nusselt number to Euler number ratio and Colburn factor to friction factor ratio of SCO2 flow inside LTR and HTR.A low temperature of SCO2 at the exit of the hot side and a high temperature of it at the cold side exit are the main targets for designing the recuperators because the lower temperature at the hot side outlet and higher temperature at the cold side outlet effectively contribute to alleviating heat transfer loads in the cooler and heater of SCO2 Brayton cycle, respectively. The results of the outlet temperature (Tout) presented in Figure 14 reveal that diverging the minichannels is also effective in order to achieve these goals. By diverging the hot side and cold side minichannels, the minimum temperatures of 364.4 K and 472.6 K and the maximum temperature of 456.5 K and 656.9 K can be obtained at the outlet of LTR and HTR with D-C and C-D patterns, respectively. Simultaneous divergence of both the minichannels (refers to as D-D pattern) leads to the outlet temperatures of 385.1 K and 432.3 K for LTR and 609.7 K and 510.1 K for HTR.The aforementioned quantitative indexes separately analyze the effect of minichannel configuration on the overall performance of the hot side and cold side of the recuperators. To comprehensively appraise the overall thermohydraulic performance of the recuperators, the ratio of the average heat transfer rate to the total required pumping power is considered as follows [16],
(15)η3=Qm/ρΔphotside+m/ρΔpcoldside.It can be seen from Figure15 that converging the hot side minichannel deteriorates the overall performance of the recuperators. As the Ref. case is replaced with the C-U, C-C, and C-D patterns, the average heat transfer rate per the total pumping power decreases by 82.9%, 77.8%, and 89.8% for LTR and 80.2%, 75.5%, and 88.7% for HTR, respectively. However, converging the cold side minichannel when the hot side minichannel is divergent (refers to as the D-C pattern) can improve the overall performance of LTR by about 168.0%. This improvement can be attributed to noticeable decrement of the SCO2 pressure drop on the hot side, leading to lower pumping power comparing with the Ref. case, as depicted in Figure 11. Therefore, diverging both the hot side and the cold side minichannels leads to striking reductions in total pumping power. Meanwhile, the heat transferred from the hot side to the cold side is much smaller than that of the Ref. case. The trade-off between the heat transfer rate and the pumping power suggests that the best combination of the minichannels is the U-D pattern. The performance improvement value obtained by this pattern is 257.2% for LTR and 171.8% for HTR. Therefore, on account of the outstanding hydraulic behavior of the cold side, the U-D pattern can be recommended for the practical application of LTR and HTR because high pressure drops of SCO2 flowing in the cold side of the recuperators limit the maximum expansion ratio across the turbine. After, the U-D pattern, the D-D pattern can be the best choice. Compared with the Ref. case, the performance improvement value obtained by this pattern is 177.5% for LTR and 151.8% for HTR.Figure 15
Average heat transfer rate per total pumping power of SCO2 flow inside LTR and HTR.


## 4.4. Comparison with Other Patterns of Minichannel

In this section, the performance of nonuniform minichannels is compared with three more integral minichannels nominated as zigzag, wavy, and C-shaped optimized by Refs. [9, 37] previously. The comprehensive evaluation, which is conducted for a wide range of the Reynolds number, i.e., 10380<Re<35560 for the hot side and 5400<Re<29170 for the cold side, is provided in Table 3 and Table 4 in terms of the Nusselt number and friction factor and in Figures 16(a)–16(b) in terms of the performance index proposed by Webb [38] as follows,
(16)η4=NuZigzag,Wavy,C‐shaped/NuNonuniformfZigzag,Wavy,C‐shaped/fNonuniform1/3.Table 3
Comparison of Nusselt number and friction factor under operating conditions of LTR.
ReNufHot sideConvergent34919111.440.03470Zigzag194.720.02891Wavy164.410.02154C-shaped143.340.02154Divergent1355346.450.00131Zigzag86.030.03295Wavy82.400.02154C-shaped72.670.01972Uniform2355373.360.00590Zigzag145.560.03150Wavy126.500.02209C-shaped112.900.02027Uniform2241070.110.00622Zigzag138.480.03172Wavy122.420.02190C-shaped109.130.02045Convergent34313124.870.03709Zigzag191.230.02893Wavy162.880.02136C-shaped141.720.02145Divergent1378747.780.00127Zigzag89.670.03186Wavy84.770.02154C-shaped73.740.02000Convergent35558117.540.03512Zigzag197.380.02780Wavy167.680.02136C-shaped147.070.02163Divergent1313744.040.00132Zigzag85.360.03224Wavy79.840.02145C-shaped71.060.01990Cold sideUniform978142.990.01035Zigzag65.030.08566Wavy65.820.04124C-shaped69.800.03493Uniform837743.040.00976Zigzag53.290.08796Wavy58.680.04321C-shaped60.580.03520Convergent1292076.190.05842Zigzag82.540.08278Wavy84.510.04019C-shaped82.740.03388Divergent577620.760.00192Zigzag31.070.09127Wavy42.900.04473C-shaped49.240.03648Divergent594019.830.00180Zigzag34.540.09015Wavy43.510.04521C-shaped50.140.03624Convergent1204983.160.05431Zigzag80.450.08330Wavy79.800.04116C-shaped80.820.03437Convergent1374470.960.05807Zigzag94.370.08178Wavy87.840.04113C-shaped88.060.03409Divergent540322.560.00224Zigzag30.340.09066Wavy38.560.04523C-shaped46.690.03698Table 4
Comparison of Nusselt number and friction factor under operating conditions of HTR.
ReNufHot sideConvergent2686785.020.03630Zigzag165.320.03101Wavy140.950.02209C-shaped126.830.02081Divergent1093238.170.00142Zigzag71.990.03395Wavy68.970.02118C-shaped58.820.01963Uniform1767153.420.00622Zigzag113.430.03244Wavy102.100.02172C-shaped91.390.01990Uniform1726254.340.00665Zigzag109.800.03191Wavy99.410.02172C-shaped89.260.01990Convergent2614993.200.03854Zigzag160.440.03213Wavy137.500.02200C-shaped122.050.02054Divergent1126854.340.00135Zigzag74.310.03244Wavy71.780.02109C-shaped60.950.01945Convergent2760487.610.03605Zigzag167.620.03098Wavy143.890.02209C-shaped127.960.02090Divergent1038237.310.00147Zigzag68.560.03433Wavy66.090.02063C-shaped55.630.01954Cold sideUniform1703652.750.00804Zigzag103.200.08001Wavy97.650.04006C-shaped92.090.03473Uniform1766260.660.00731Zigzag105.940.07981Wavy107.470.04030C-shaped94.810.03496Convergent29170109.130.04717Zigzag166.650.07741Wavy156.620.04485C-shaped142.240.03588Divergent939929.770.00147Zigzag60.250.08615Wavy61.590.04197C-shaped55.190.03445Divergent922127.680.00152Zigzag62.320.08679Wavy61.010.04124C-shaped54.070.03445Convergent27751119.910.04453Zigzag154.290.07829Wavy143.960.04465C-shaped134.680.03494Convergent2734694.150.04851Zigzag152.680.07877Wavy142.850.04442C-shaped130.720.03374Divergent968133.000.00146Zigzag62.130.08630Wavy59.980.03517C-shaped54.550.04148Figure 16
Comparison of overall performance index under operating condition of (a) LTR and (b) HTR.
(a)(b)Regarding Table3 and Table 4, the minichannel structure has distinct effects on both thermal and hydraulic characteristics of the recuperators. The comparison confirms that all indirect minichannels (zigzag, wavy, and C-shaped) display better thermal performance than the nonuniform ones, and the corresponding Nusselt number enhancements are a function of Reynolds number. For instance, in the case of the optimized C-shaped minichannel, the percentage increase in Nusselt number of LTR hot side varies from 30.4% to 47.5% (average 40.2%) compared to that of the convergent minichannels and from 77.3% to 81.2% (average 78.6%) compared to that of the divergent minichannel. The corresponding enhancements are 47.5–65.7% (average 59.1%) and 77.1–85.2% (average 79.8%) in Nusselt number of HTR hot side. The comparative study conducted by Xie et al. [39] also disclosed that the utilization of the zigzag minichannels led to a 65% improvement in the effectiveness of PCHEs with a pressure drop augmentation of 39% compared to the straight minichannel.Although the indirect and convergent minichannels display better thermal performance than the divergent minichannels, their poor hydraulic performance causes lower values of the overall performance index at the low Reynolds numbers, as shown in Figure16. The results show that replacing the convergent minichannels with the indirect minichannels on the hot side of the recuperators can enhance η4 from 1.36 to 2.05 (average 1.74). At the same time, using the divergent minichannels instead of the indirect minichannels improves the overall performance of the recuperators from 26.4% to 53.8% (average 36.3%) due to the quite low flow resistance of the SCO2. Moreover, when the divergent minichannels are used on the cold side of the recuperators, the overall performance is improved from 25.1% to 51.6% (average 37.4%). It is worth mentioning that at low Reynolds numbers (less than 14000 for the hot side and 10000 for the cold side), all nonuniform patterns of minichannels have better overall performance than the integral complicated minichannels.The above-mentioned results can be taken into consideration in the design of recuperators for the SCO2 Brayton cycles. However, it should be noted that these points are recommended for the operating conditions of LTR and HTR with the similar sizing features; however, they may lose their accuracy once the mass fluxes, pressures, temperatures, and dimensions of minichannels vary significantly, particularly near the critical point of CO2.


## 5. Conclusions

Nonuniform patterns of semicircular straight minichannels are investigated to improve the overall performance of SCO2 Brayton cycle recuperators. In this context, different combinations of convergent and divergent minichannels are studied. The drawn conclusions will be summarized as follows:
(i)
Nonuniform patterns of minichannels offer great advantages for improving characteristics of the recuperators. It is found that converging the hot side minichannel or the cold side minichannel individually improves the overall heat transfer coefficient by less than 5%, while the simultaneous convergence of both the minichannels results in an overall thermal performance improvement of more than 20%(ii)
Meanwhile, the divergence patterns of minichannels display better hydraulic performances. For instance, the pressure drop of the SCO2 flowing inside the divergent hot side minichannel of LTR is only about 1/3 the pressure drop in the uniform minichannel and about 1/11 of this parameter in the convergent minichannel(iii)
According to the overall performance evaluation, it is found that diverging the minichannels imposes greater positive effects than converging them. The advantages of the simultaneous divergence of both the hot side and the cold side minichannels are superior than other combinations due to the quite low flow resistances. The maximum performance indexes that appeared by this pattern are 1.62 and 1.71 for the hot side and 1.26 and 2.32 for the cold side of LTR and HTR, respectively. Nevertheless, based on the ratio of average heat transfer rate to total pumping power, only the hot side minichannel convergence can provide the best overall performance in the recuperators. This ratio for the D-U pattern of minichannels is approximately 257.2% and 171.8% larger than that of the conventional pattern of minichannels inside the LTR and HTR, respectively(iv)
The comparative study reveals that the divergent minichannels, due to their brilliant hydraulic performance, present better overall performance indexes with relatively smaller heat transfer capacity values compared to the convergent and other integral complex minichannels



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*Source: 3765515-2023-02-06.xml*