Analysis of heat transfer performance of the absorber tube with convergent-divergent structure for p

Abstract

Keywords: Solar energy; Parabolic trough collector; Monte Carlo ray tracing; Finite Volume method; Convergent-divergent structure; Heat transfer enhancement

Introduction

Eliminating the negative environmental impact of using fossil fuels, meeting the growing energy demand brought about by development, and seeking renewable and environmental friendly alternatives have become topics of great concern in recent years [1,2]. Although renewable energy, especially solar energy, accounts for only a small part of

today’s energy supply, it has various direct or indirect advantages that make it a key to sustainable development program [3]. The great potential of solar energy to satisfy the energy needs in many fields is increasingly recognized by relevant researchers [4–6]. In the past decade, the application of direct (photovoltaic) and indirect (concentrated solar) solar power

*Corresponding author. *E-mail address: wangfuqiang@hitwh.edu.cn This paper was recommended for publication in revised form by Regional Editor Jaap Hoffman Hoffman Published by Yıldız Technical University Press, İstanbul, Turkey Copyright 2021, Yıldız Technical University. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

generation has been steadily increasing [7,8]. The use of mixing, and the drop of fluid resistance increases signifisolar energy for electricity production promises to be one cantly. Pandey et al. [33] numerically calculated the effect of the most viable options to replace fossil fuel power plants of inserting an arc plug into the PTR absorption tube on the [9–11]. The concentrated solar power (CSP) system that heat transfer performance of the system. In order to obtain focuses solar radiation energy onto the receiver can reduce the optimal size R of the arc plug, a series of calculations are the requirement of collector materials and provide lower carried out. It is observed that for arc plugs with factors R=1 heat losses due to the reduced target area [12–14]. The par- and R=0.879, PTC has the highest thermal efficiency. abolic trough collector (PTC), as a CSP technology that has Another approach to improve the thermal performance been extremely developed, has already had relatively abun- is related to improving the thermophysical properties of dant operational experience, and currently has great pros- the materials [34–37]. Due to good properties of radiation pects [15,16]. Due to the characteristics of wide distribution absorption and high thermal conductivity, nanofluid are and low energy flow density of solar energy, centralized often used as the working medium in the heat exchange solar power plants generally occupy a large area. Parabolic equipment [38]. Hussein et al. [39–41] gave comprehensive trough mirrors, receivers (PTR) and related power genera- overviews of theories, simulations and experiments related tion equipment are usually the three main components of to the application of nanotechnology in different types of their facilities [17,18]. solar collectors. The reviews pointed out the importance Parabolic trough receiver (PTR) is the major part of the of accurate selection of nanoparticle volume fraction and traditional PTC. The receiver is a concentric sleeve struc- diameter for the performance of the collector, and further ture, with its outer layer is a thin transparent glass envelope, guided that future research must be devoted to inventing and the inner layer is a metal absorber tube [19]. And a efficient energy transport methods of nanofluid in solar vacuum environment is set between the two layers, which collectors. can effectively reduce the heat loss of convection. The input The temperature distributed around the circumference solar radiation is reflected and converges at the focal line of the absorber tube is inhomogeneous. This is because of the parabolic mirror. The concentrated solar radiation is the upper half of absorber tube directly receives incident received by the receiver located here and stored as thermal solar radiation, while the lower half receives the light that is energy into heat transfer fluid (HFT) [20,21]. concentrated by the reflector, resulting in not-uniform disThe demand for more efficient energy conversion tribution of circumferential flux [42]. The not-uniform temequipment has promoted the development of solar energy perature distribution will cause deformation and bending utilization technology [22,23]. The method of optimizing of the absorber tube, which may further damage the PTR. the structure to improve the thermal performance of the The optical performance analysis on PTC by Jebasingh et al. system is widely used in scientific research and production [43] showed that the optical efficiency should be optimized [24–27]. The research on structure modification of PTR to withstand against environmental conditions, many optiabsorber tube is earlier and the technology is more mature cal factors have been studied to improve the heating condi[28,29].In order to improve the heat transfer performance tions of the absorber tube [44]. To reduce the temperature and reliability of the PTR system, Wang et al. [30] numer- gradient of the absorber tube, scholars also have conducted ically calculated the thermal performance of the system many studies on certain specific heat transfer processes in using symmetrical outward corrugated tube as the metal PTR technology [45–47]. A new type of perforated plate tube of the PTC. The effect of various detailed parameters of inserts was proposed for use in parabolic trough collecthe bellows on the heat transfer performance of the system tor, and the thermodynamic performance of the system is studied, including corrugation height (H), corrugation was simulated [48]. The geometric parameters of different spacing (p), corrugation wave crest radius (R) and corru- perforated plates are analyzed. Studies have shown that the gation groove radius (r). The results show that when Re = thermal conditions of the receiver have improved signifi81728 and p/D=4.3, the effective heat transfer coefficient cantly. Wang et al. [49] applied the elliptical-circular secusing symmetrical outward corrugated tube can increase by tion glass cover in the solar energy absorber system. The 8.4%. Zhang et al. [31] conducted experiments on natural influence of the refractivity and cross-sectional parameconvection boiling heat transfer in smooth tube, ordinary ters of the glass cover on the heat flux distribution is also convergent-divergent tube and improved convergent-­ studied by Monte Carlo Ray Tracing (MCRT) method. The divergent tube. The improved convergence-divergence tube numerical simulation results show that using glass cover has better heat transfer performance than ordinary conver- with an elliptical-circular cross-section with a high refracgence-divergence tube, and its enhancement ratio is about tivity for the receiver can minimize the heat flux distri1.18 times. Jin et al. [32] simulated the heat transfer and flow bution gradient and effectively reduce thermal stress. The resistance performance in particular convergent-divergent work of Manikandan et al. [50] includes numerous techtube with different structural parameters. It is pointed out niques for enhancing the optical and thermal efficiency of that fluid separation near the solid wall effectively strength- parabolic trough collectors. A series of factors that affect ens heat transfer, changes in fluid flow line enhance fluid the optical efficiency of the parabolic trough collector are

summarized, including coating of selective surface on receiver tube, reflectivity of the mirror, intercept coefficient of the absorber tube, Incorporating secondary reflector, etc. In addition, they recommended the black body cavity receiver to reduce heat loss and improve optical and thermal efficiency. To enhance the collection and utilization performance of PTR technology for solar energy and improve the availability of the system, the authors proposed to apply convergence-divergence structure to PTR technology to obtain a novel parabolic trough receiver (CD-PTR). The Monte Carlo ray tracing (MCRT) method is used to determine heat flux distribution on collector tubes in the study [51]. The numerical simulation is carried out using the concept of Finite Volume Method (FVM) to investigate the advantages of CD-PTR compared to ordinary PTR system [52] [53]. The influence of the geometric parameters variation on the system performance was also studied, which also provides theoretical instructions for further research and practical application.

The parabolic trough receiver (PTR) is an important component of the PTC system, and Figure 2 is the schematic cross-sectional view of PTR and describes the complete heat transfer process within the system [51]. The solar collector is concentric sleeve structure, which the outer layer is a thin transparent glass envelope, and the inner layer is a metal absorber tube. A vacuum environment is provided between the sleeves, and the heat transfer fluid flows into internal metal absorber tube from the inlet. The absorber tube is coated with selective coating for higher solar spectral absorption. And the selective coating on glass envelope can increase the solar radiation transmitted into the PTR system, thus reducing the heat loss outside the receiver. The vacuum environment was set to reduce heat loss caused by convection during heat transfer process. Table 1 lists some detailed parameters of the RTC system. They are obtained by the existing PTR system, which are the same as the reference [54]. Some researchers have studied various tubes as PTR absorber tubes to obtain higher solar energy utilization

Physical Model

The simplified schematic diagram of general PTC system [51] is presented in Figure 1. As we can see, solar radiation is incident along the y axis in the negative direction, and then it is reflected by the trough condensing system and collected on the receiver. Highly concentrated solar radiation accumulates at the bottom periphery of the receiver, while other part is directly irradiated by sunlight with low heat flux density.

Figure 2. Schematic diagram of the heat transfer process of PTR [51]. Table 1. Detailed parameters of the PTC studied by the ­authors [54]

Figure 1. Schematic diagram of parabolic trough solar collector (PTC) system [51].

Length of PTR Outer diameter of glass envelope Outer diameter of metal tube Thickness of metal tube Aperture of PTC Rim angle Non-parallelism angle Reflectivity of PTC Absorptivity of metal tube

Momentum Conservation Equation 2  ∂v ∂v v ∂v v ∂v  ρ  r + vr r + ϕ r − ϕ + v x r  = ρ gr  ∂t ∂r r ∂ϕ r ∂x   (2)  ∂ 2vr 1 ∂vr 1 ∂ 2vr ∂ 2vr 2 ∂vϕ vr  ∂P +µ  2 + + + − − − r ∂r r 2 ∂ϕ 2 ∂x 2 r 2 ∂ϕ r 2  ∂r  ∂r ∂v v ∂v ∂v vv   ∂vϕ + vr ϕ + ϕ ϕ + v x ϕ + r ϕ  = ρ gϕ ∂r r ∂ϕ ∂x r   ∂t

 ∂ 2v 1 ∂vϕ 1 ∂ 2vϕ ∂ 2vϕ 2 ∂vr vϕ  1 ∂P + µ  2ϕ + + + + − −  ∂r r ∂r r 2 ∂ϕ 2 ∂x 2 r 2 ∂ϕ r 2  r ∂ϕ  (3)

 ∂v x ∂v v ∂v ∂v  + vr x + ϕ x + v x x  = t r r ϕ ∂ ∂ ∂ ∂x  

 ∂ 2v 1 ∂v x 1 ∂ 2v x ∂ 2v x  ∂P + + ρ g x + µ  2x + − r ∂r r 2 ∂ϕ 2 ∂x 2  ∂x  ∂r

Figure 3. 3D view and axial section schematic of CD-PTR studied in this paper. efficiency and convergent-divergent structure have been applied in various engineering practices. But no researchers had proposed to apply convergent-divergent tube as the PTR absorber tube in solar thermal power utilizations. In this study, an innovative convergent-divergent tube is proposed to enhance the heat transfer and homogenize the temperature distribution in PTR system. The structural schematic of convergent-divergent tube used for simulation in this study is presented in Figure 3, and the relevant dimensions are added to the diagram. The convergent-divergent tube simulated in this study is the structure of contraction-expansion.

Where v is the fluid velocity, ρ is the fluid density, x represents the length direction of the tube, r and φ represent the radial and circumferential directions of tube cross section, respectively. By comparing the numerical results with the experimental test conducted by Roldán et al. [55] in DISS, Wang et al. [30] proved that the standard k-ε model can not only agree well with the experimental test, but also reduce the average deviation between the simulation results and the experimental test temperature. Therefore, the standard k-ε model was chosen for numerical analyses. Taking turbulent kinetic energy (k) and its dissipation rate (ε) as important parameters, the standard k-ε model is a classic turbulence model in practical engineering calculations. The above two parameters are obtained by following two separate transport equations [56]: ∂ (ρk) ∂t

Continuity Equation vr ∂vr 1 ∂vϕ ∂v x 0 + + + = ∂r r ∂ϕ ∂x r

µ  ∂k  ∂   µ + i   + G + Gb − ρε − YM + Sk σ k  ∂x j  k ∂x j 

Mathematical Model

In computational fluid dynamics (CFD) analysis in this paper, governing equations that reflect the dynamic characteristics of PTR solar thermal utilization process include continuity equation, momentum and energy conservation equation and radiation transfer equation. The Finite Volume Method(FVM) is used to discretize the computational domain and governing equations.

The calculation method of the turbulent kinetic energy (TKE) generated by the fluid average velocity gradient is analyzed with reference to the Modeling Turbulence Generation of k-ε model. This part of TKE is represented by the symbol Gk [56]. With reference to the effect of Buoyancy

on turbulence in k-ε model, the TKE generated by buoyancy is analyzed and represented by the symbol Gb. YM characterizes the effect of fluctuating dilatation on overall dissipation rate in compressible turbulence, and it was analyzed by referring to the effect of compressibility on turbulence in k-ε model. C1ε, C2ε, and C3ε are definite constants determined by physical model and flow characteristics. σk is defined as turbulent Prandtl numbers of k, and σs is defined in the same way. Sk and Sε are source terms.

Energy Conservation Equation The solid and fluid phases follow different governing equations. For the solid phase

average incident solar irradiance is 1000 W/m2, and is projected through the glass envelope with the transmissivity of 0.96. The absorption rate of absorber tube coated with selective coating is 0.95. The lower half of the circumference of the metal tube is irradiated by concentrated solar radiation reflected from collector: In this paper, the Finite Volume Method(FVM) is used to discretize the governing equations, and the steady-state implicit scheme is used to solve them. The SIMPLE scheme is used to solve the pressure-velocity coupling equation, the discrete format of the convection term is QUICK, and the momentum equation adopts the second-order upwind style. The convergence criterion of the residual of the energy equation is set to be 1×10–8, and the criterion of other equations are set to be 1×10–6.

∂Ts k  ∂ 2Ts 1 ∂Ts 1 ∂ 2Ts ∂ 2Ts  = + + +   ∂t ρ c p  ∂r 2 r ∂r r 2 ∂ϕ 2 ∂x 2 

 ∂t  ∂ 2T f 1 ∂T f 1 ∂ 2T f ∂ 2T f  k 2 + + +   ∂r r ∂r r 2 ∂ϕ 2 ∂x 2    ∂P ∂P vϕ ∂P ∂P  +α vT  + vr + + vx +Φ ∂r r ∂ϕ ∂x   ∂t

where Φ is the dissipation function, and αv is the expan1  ∂ρ  sion coefficient which is defined as α v = −  . T is ρ  ∂T  P the fluid temperature, P is the absolute pressure, cp and k represent the heat capacity and thermal conductivity of the heat transfer fluid, respectively. This paper studies incompressible fluids and dissipation effect is ignored, which allows us to simplify the above equation as:  ∂T f

∂T f  v ∂T f + ϕ + vx = r ∂ϕ ∂r ∂x   ∂t (9)  ∂ 2T f 1 ∂T f 1 ∂ 2T f ∂ 2T f  k 2 + + +   ∂r r ∂r r 2 ∂ϕ 2 ∂x 2  

Boundary Conditions For Analysis

The boundary conditions of the numerical calculation are listed in Table 2 [57]. As described in Table 2, the numerical calculation in this paper defines the tube inlet as velocity inlet, the outlet side as fully developed conditions, and the inner wall of metal tube is set to be non-slip boundary. The upper half of the circumference of the metal tube is directly irradiated with uniform solar irradiation. The

Vx = Vin, Vr = Vφ = 0m/s, Tf =Tin = 400K Fully developed conditions

R = Ro, 0 ≤ L ≤ 4.06 0° ≤ φ ≤ 180°, qu = 1000 × 0.96 × 0.95 = 912W/m2 180° ≤ φ ≤ 360°, ql = qcal

Model of Concentrated Ray Solar radiation is distributed outside the receiver to provide thermal energy to heat transfer fluid(HTF). And the heat flux field is extremely inhomogeneous, where the top is exposed to direct sunlight and the bottom is exposed to sunlight focused by optical elements. The distribution of heat flux collected by optical elements was predicted by MCRT method. When the MCRT method is applied to calculate the radiation field, the radiation input of the system is realized by the ray input, which is set to carry the same energy. The change in the direction of the ray projection is determined by the optical characteristics of the elements and appropriate probability density functions [49]. The method of combining the fitting curve and the user-defined functions introduces the obtained concentrated heat flux distribution to CFD models to further analyze heat transfer and flow performance, which causes tiny interpolating error. Thermophysical Properties of D12 In this paper, the D12 thermal oil is applied as PTR heat transfer fluid(HTF), whose physical properties are sensitive to temperature changes within its operating temperature range. To obtain accurate simulation results, the thermophysical properties of D12 are fitted to make its physical parameters become polynomial functions of temperature [29,51] and imported into CFD analysis through UDF method. The thermophysical properties of the HTF used in the numerical calculations in this paper are shown in Table 3.

Table 3. Thermophysical properties of D12 thermal oil [29],[51] Properties

–6.96982 × 10-1 × T – 1.31384 × 10-4 × T2 – 2.09079 × 10–6 × T3 + 776.257 3.86884 × 10–3 × T + 2.05029 × 10–6 × T2 – 1.12621 × 10–8 × T3 + 3.8628 × 10–11 × T4 + 2.01422 –1.4781 × 10–4 × T – 1.6142 × 10–7 × T2 + 1.1299 × 10–1 exp[530.944/(146.4 + T) – 2.68168)] × 10–6

Meshing And Verification Of GRID Independence

Meshing During the grid generation process, a structured O-grid was generated for PTR and CD-PTR, the meshes of zoom area of convergent-divergent tube were generated by using the method of separate block association and were refined. Figure 4 and Figure 5 are schematic diagrams of the cross section and axial section (partial) of the mesh. Verification of Grid Independence The average heat flux of the absorber tube inner surface is adopted as an evaluation parameter to verify the grid independence due to its importance in characterizing the heat transfer conditions of the coupled-wall. Furthermore, it is suitable to use Nu as another evaluation parameter for grid independence verification because the average Nu at the tube outlet is a key parameter that investigates heat transfer performance. Considering the denser grid in the zoom area of convergent-divergent tube, in addition to the grid independence verification of the smooth tube, this study also verified the grid independence of convergent-divergent tubes. For the smooth tube, seven cases with different numbers of grids were set for the grid independent verification tests, respectively: 580,000, 760,000, 950,000, 114,000, 155,000, 183,000, 202,000. For the convergent-divergent tube, seven cases with different numbers of grids were set for grid independent verification tests, respectively: 850,000, 1,020,000, 1,260,000, 1,550,000, 1,830,000, 2,020,000, 2,210,000. Figure 6 presents the coupled-wall heat flux and Nu variation as the number of grids changes in PTR and CD-PTR, which are marked as symbols qw-f and Nu. It can be seen that when the grid numbers of the smooth tube and the convergent-divergent tube reach 950,000 and 1,830,000, respectively, the values of evaluation parameters remain almost constant. Considering comprehensively computational time and accuracy, the PTR model with 950,000 grids and the CD-PTR model with 1,830,000 grids were selected for further CFD simulation.

Model Validation

An experimental study on temperature distribution of PTR absorber tube with superheated steam as HTF has

Figure 5. Axial section diagram of CD-PTR mesh. been carried out by Roldán et al. [55]. In order to verify the model used in the numerical calculations in this study, Table 4 lists the detailed test information of several groups of PTR in the Spanish DISS used for model validation. The maximum and minimum temperature on the PTR outlet outer surface were taken as research indicators using the initial conditions of the experiments for numerical simulation.

Figure 6. The heat flux (qw) and Nu variation with the increase of number of grids. Table 4. Specific experimental conditions for PTC thermal performance test conducted by Roldán et al. [55]

Figure 7 depicts the comparison between the experimental [55] and the CFD simulation results when the maximum and minimum temperature at the outlet are respectively used as parameters. It can be clearly seen that the maximum and minimum temperatures of the PTR outlet simulated by the author are almost consistent with the experimental results. Table 5 lists the specific experimental results in Spanish DISS [55] and the simulation results obtained in this study under the above initial conditions. Numerical simulation always cannot accurately describe the real flow. For example, although the turbulence model has been verified with experimental results, there are still calculation errors [30]. Furthermore, the experimental temperature of Roldán et al. was measured by thermocouples, the experimental results may also contain measurement errors. These factors have accumulated errors between the simulation results and the experimental results. Table 5 also listed the relative errors between numerical calculations and experimental results,

Max-Experimental[55] Min-Experimental[55] Max-Numerical Min-Numerical

Figure 7. Comparison of temperature between experimental [55] and numerical results. where the relative error is defined as .The maximum relative error in the cases is 3.823%, and the average relative error is just 1.103%, which proves the reliability of the numerical calculation model in this study.

Results And Discussion

Analysis of PTR Heat Transfer Performance To achieve the goal of improving system thermal conditions and enhancing the heat transfer performance, it is necessary to first study heat flux distribution outside PTR metal absorber tube. Figure 8 represents the distribution of heat flux outside PTR absorber tube for Re= 65000. Obviously, the heat flux

Table 5. Comparison between the experimental results of Roldán et al. and the results simulated in this paper Case TMax.Exp (K) TMax.Num(K) TMin.Exp (K)

Figure 8. Heat flux distribution outside PTR absorber tube for Re = 65000.

Figure 9. Temperature distribution outside PTR absorber tube for Re=65000.

shows a highly uneven distribution. Specifically, the bottom of absorber tube is distributed with highly concentrated solar heat flux, the maximum value of which is close to 40,000 W, while the other parts only exposed to non-­ concentrated rays with an average heat flux is only about 100W. According to the operating experience of solar thermal power stations, the high non-uniformity of the heat flux may seriously compromise the operation safety of the PTR. Figure 9 displays the distribution of temperature outside PTR absorber tube for Re=65000. Determined by the distribution characteristics of heat flux, the part of absorber (a) Re = 18900 tube that receives concentrated solar radiation has a much higher temperature than other parts. Heat transfer fluid (HTF) is continuously heated as it flows, causing the temperature of the fluid to continue to rise, and finally reach the maximum value at the outlet. Figure 10 presents the temperature distribution at PTR outlet section in the cases with Reynolds numbers of 18900 and 65000 respectively. By comparing the two figures, we can see that as Reynolds number increases, the temperature distribution becomes more uniform. This shows that increasing the Reynolds number leads to increased flow mixing, which effectively reduces the temperature difference and provides better thermal conditions for absorber tube. Figure 11 presents the change trend of maximum tem(b) Re = 65000 perature difference of PTR absorber tube as Re increases. The maximum temperature difference decreases grad- Figure 10. Temperature distribution of PTR outlet with difually as Re increases, which also means better thermal ferent Re.

Figure 11. The trend of maximum temperature difference of PTR with the increase of Re.

Figure 13. Temperature distribution on the inner wall of absorber tube.

Cd-Ptr

conditions. And the decreasing in maximum temperature difference slows down gradually. Turbulent Kinetic Energy (TKE) is an important indicator to measure the development or decline of turbulence. In general, TKE can simultaneously reflect the changes in fluid heat transfer performance and resistance performance. The variation of mass-weighted average TKE of heat transfer fluid in PTR absorber tube with the increase of Reynolds numbers is demonstrated in Figure 12. It can be seen that the TKE of heat transfer fluid increases with the increase of Re. And as Re increases, the relationship between the TKE and the Reynolds number is close to a linear relationship. Analysis of CD-PTR Heat Transfer Performance For higher system heat transfer efficiency, a tube adopted convergent-divergent structure is introduced for

PTR absorber tube. Figure 3 has previously illustrated the schematic diagram of the convergent-divergent PTR (CD-PTR) described in this paper. The temperature distribution on the inner surface of the absorber tube of both PTR and CD-PTR(N=5) with Re=65000 are presented in Figure 13. It can be found that the average temperature is significantly reduced when convergent-divergent structure is used for PTR absorber tube, and the temperature around the zoom segment is lower than that of the straight segment for CD-PTR. Figure 14 presents the turbulent kinetic energy (TKE) field of PTR and CD-PTR axial section for Re = 65000. As we can see, flow mixing is enhanced, and the turbulent intensity of the boundary layer is significantly increased because of the variable cross section channels in the convergent-divergent tube, thus accelerating heat transfer from tube surface to HTF. Effects of N on CD-PTR Heat Transfer Performance The number of zoom sections (N) of CD-PTR absorber tube has an effect on heat transfer process coupled with flow resistance characteristics, this part is a specific study. Five groups of different models of convergent-divergent tube are investigated: N=0, N=5, N=10, N=20, N=25. The case N=0 indicates the PTR model.

Figure 15. The trend of Nu of CD-PTR with different number of zoom sections(N).

Cd-Ptr

Figure 14. TKE distribution of axial section of PTR and CD-PTR.

Where h represents the convective heat transfer coeffiqw − f cient of the coupled wall which is defined as h = . Tt −a − T f −a Figure 15 presents the effect of increased Re on Nu in models with different number of zoom sections (N). It can be seen that when the value of N is determined, Nu gradually increases with the increase of Re. This is because the increase of Re means the increase of the fluid velocity of the HTF in the heat collecting tube for a certain model. The higher the velocity, the thinner the thermal boundary layer thickness of the fluid was, which further reduced the smaller the wall thermal resistance and intensified the convection heat transfer. In addition, the increase of the number of zoom sections leads to intensified disturbance to the flow, and Nu, which characterizes the thermal performance, increases. When Re=86400 and N=25, the average Nu increased by 66%. Figure 16 presents the trend of the heat flux of coupled wall as Re increases, and it is obvious that heat flux increases continuously because of the eddy current in the zoom sections, which strengthens the heat transfer intensity. And as the value of N increases, the average heat flux on coupled wall increases gradually, meaning that the heat transfer is enhanced.

In order to study the heat transfer performance in the collector tube, the average Nusselt number of the fluid is used as a parameter for description and analysis.

Figure 16. The trend of heat flux(q) of CD-PTR with different number of zoom sections(N). Figure 17 exhibits the effect of increased Reynolds number on pressure drop () of the HTF between tube inlet and outlet of the models for both the PTR and CD-PTR with different numbers of zoom sections (N). Obviously, the ∆P between the inlet and outlet of CD-PTR absorber tube is always higher than that of PTR absorber tube. The change in the cross section of the flow channel caused by the convergent-­divergent structure will affect the velocity and greatly reduce the fluid pressure in the CD-PTR absorber tube. The introduction of convergent-divergent structure inevitably brings an increase in pressure drop, which will lead to a significant increase in operating power consumption, thereby affecting the improvement of system efficiency. The evaluation of heat exchanger efficiency should not only focus on its heat transfer performance, but also consider its resistance performance which are characterized by Fanning friction factor(f).

Figure 18 shows the trend of friction factor (f) in several set of models with different numbers of zoom sections (N) as Re increases, and the case where the N value is zero indicates the PTR model. In both PTR and CD-PTR, the Fanning friction factor(f) decreases with the increase of Re. And the f in CD-PTR absorber tube are always much higher than that in PTR. As the number of zoom sections (N) increases, the friction factor in CD-PTR absorber tube increases rapidly, which is caused by the obvious rise of the pressure drop caused by the introduction of the convergent-divergent structure. Based on the analysis, when the N is increased, Nu is higher but pressure drop down is also enhanced. In order to comprehensively consider the influence of Nu and friction, the authors proposed the Thermal Efficiency Factor(TEF) to further analyze the evaluation of the thermal performance of the PTR system. The TEF is defined as follows: Figure 19 presents the variation of the system thermal efficiency factor(TEF) with the increase of Re for CD-PTR with different number of zoom sections(N). It can be seen from the figure that the thermal efficiency factor decreases with the increase of Re, since rather than the increase of Nu, the increase of the friction factor has a more significant effect on the efficiency. It can also be seen that as the value of N increases, the thermal efficiency factor gradually decreases, which means that as the number of zoom sections increases, the impact of the sharply increased friction factor on the system efficiency is much greater than the increase of Nu. Figure 19 also shows that the thermal efficiency of CD-PTR is not always higher than that of PTR. For models with different N, within a certain Re range, the TEF of CD-PTR is higher than 1.0, which means that the overall thermal efficiency of CD-PTR is higher than that of PTR. If Re exceeds this range, the introduction of the convergent-divergent structure cannot improve the system thermal efficiency. Take the thermal efficiency factor of CD-PTR with N=10 as an example, when the Re is less than

Figure 18. The trend of friction factor (f) of CD-PTR with different number of zoom sections(N). 1.8 1.6

Figure 17.The trend of pressure drop (∆P) of CD-PTR with different number of zoom sections(N).

Figure 19. The variation of TEF of CD-PTR with different number of zoom sections (N). 26000, the introduction of convergent-divergent structure can enhance the overall thermal efficiency and the thermal efficiency factor reaches 1.51 when Re=7000.

Conclusions

This study proposed to apply convergent-divergent tube as PTR absorber tube (CD-PTR) for improving heat transfer performance of PTC system. The FVM method coupled with MCRT method was adopted to investigate the heat transfer performance and flow characteristics of parabolic solar collector system. The simulation results draw the conclusions. 1. With the increase of Re, the temperature distribution of cross-section become more uniform, thereby

2. The TKE of HTF increases as the Re increases. At the

same Re, the variable cross-section of the CD-PTR significantly increases local TKE, thereby accelerating the heat transfer of the coupled surface.

3. The average Nu of CD-PTR increases as the number

of zoom sections increases, and is always higher than that of PTR. When Re=86400 and N=25, the average Nu increased by 66%.

4. The friction factor (f) gradually decreases as Re increases.

And as the number of zoom sections increases, f in the CD-PTR absorber tube increases rapidly. 5. The improved CD-PTR that applied the convergent-divergent structure in PTC technology proposed in this paper has higher heat transfer efficiency than PTR within a certain range of Re.

Nomenclature

Diameter of receiver, m Heat capacity, J/(kg∙K) Density, kg/m3 Contribution of the fluctuating dilatation in compressible turbulence to the overall dissipation rate Generation of turbulent kinetic energy due to buoyancy Generation of turbulent kinetic energy due to the mean velocity gradients User-defined source term Solar irradiance, W/m2 Pressure, Pa Temperature, K Velocity, m/s Reynolds number Nusselt number Heat flux, W/m2 Heat transfer coefficient, W/(m2∙K) Fanning friction factor Prandtl number

Greek symbols v Kinematic viscosity, m2/s α Absorptivity of receiver αv Coefficient of expansion Ф Dissipation function σk Turbulent Prandtl numbers for k σε Turbulent Prandtl numbers for ε Subscripts Max Maximum temperature Min Minimum temperature Exp Experimental test Num Numerical simulation

Refers to smooth tube Refers to fluid Refers to environment Refers to glass envelope

Data Availability Statement

The authors confirm that the data that supports the findings of this study are available within the article. Raw data that support the finding of this study are available from the corresponding author, upon reasonable request.

Conflict Of Interest

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Ethics

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References

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