Analysis on the Effects of Different Receiver Structures and Porous Parameters on the Volumetric Effects and Heat Transfer Performance of Porous Volumetric Solar Receiver

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Introduction
In order to improve the solar-thermal conversion efficiency of the concentrated solar power (CSP) system, we design a high-efficiency solar receiver is imperative.Among different types of solar receiver, the volumetric solar receiver is a promising type due to the stable structure and volumetric effects [1][2][3].The volumetric solar receivers usually use the air as the heat transfer fluid.However, air has many disadvantages such as small heat capacity and low thermal conductivity, especially extremely low solar radiation absorptivity [4].It is well known that porous media have many advantages such as better heat resistance, antioxidant properties, larger heat transfer specific surface area, and higher thermal conductivity, which is suitable for dish volumetric solar receiver to improve the solar-thermal conversion efficiency [5].
Some researchers studied the porous materials.For instance, Fend et al. [6] analyzed multiple types of ceramic foam porous materials and concluded that the ceramic foam is a quite suitable material, which has a large specific surface area and reduces the pressure drop so as to improve the thermal efficiency of receivers.Moreira et al. [7] carried out experiments to study the receivers filled with carborundum aluminum oxide composite metal foam, which show the satisfactory pressure drop.Albanakis et al. [8] tested and compared the heat transfer performance of two foam metals in the receiver which are nickel and chromium-nickel-iron alloy and found that the form metal filled with nickel achieved the better heat transfer performance.Coquard et al. [9] performed radiation experiments on receivers filled with nickel-chromium-aluminum, iron-chromium-aluminum, and zirconia, respectively, finding that all the three materials have good thermal performance.Kamath et al. [10] analyzed the heat transfer performance of receivers filled with aluminum foam and copper foam, respectively.The results showed that the heat transfer performance of receivers filled with copper foam was better than aluminum foam.Banerjee et al. [11] experimentally investigated the high-temperature heat transfer performance of receivers filled with aluminum oxide foam and ceramic, which had a high efficiency balancing the pressure drop and heat transfer performance.Mey-Cloutier et al. [12] analyzed the characteristics of receivers filled with porous media of different materials such as carborundum, carborundum-silicon dioxide-aluminum oxide, and zirconium boride.It was indicated that the carborundum had the better heat transfer performance with the same porosity and porous diameter than other materials.Patil et al. [13] analyzed the receivers filled with carborundum, cerium oxide, and aluminum oxide metal foam and found that with the same porosity, the heat transfer performance of receiver filled with carborundum was better than those filled with the other two metal foams.Du et al. [14] studied a porous volumetric solar receiver with molten salt as the heat transfer fluid.The results showed that the outlet temperature of the porous volumetric solar receiver was 9.6% higher than that of the conventional air porous receiver.
On the other hand, some scholars studied the effects of porous parameters on the heat transfer performance of volumetric solar receiver.For instance, Du et al. [15] optimized the porosity distribution using the genetic algorithm method, thus enhancing the absorption of solar radiation and improving the efficiency and internal heat transfer of receivers with high porosity front surface and a low rear porosity surface.Zhang et al. [16] applied the nonorthogonal multirelaxation time Boltzmann method to analyze the heat and mass transfer of volumetric receiver and found that the porosity at the entrance had a significant effect on the internal heat transfer performance.Du et al. [17] experimentally and numerically investigated the volumetric receivers with the hierarchic radial porosity, which can improve the heat transfer performance and the thermal efficiency.Avila-Marin et al. [18] analyzed that the outlet temperature of the fluid was directly related to extinction coefficient.In addition, the smaller the specific surface area of material, the higher the temperature of the solid at the front.Barreto et al. [19] numerically investigated the volumetric receivers, and the results showed that higher porosity and larger porous diameter would effectively improve the thermal efficiency and significantly reduce pressure drop.Reddy and Nataraj [20] numerically investigated the porosity, heat conductivity coefficient, and inlet flow velocity and concluded that the high heat conductivity coefficient can reduce the radial temperature difference and uniform the temperature field.Rivas et al. [21] analyzed the inlet velocity, porosity, and thermal conductivity coefficient and concluded that the maximum temperature of solid decreased with increased inlet velocity.Du et al. [22] used 3D printing technology to fabricate complex porous medium samples and conducted experimental studies.It was found that the radial gradient porous volume solar receiver with large pore diameter could improve the thermal efficiency and reduce the flow resistance compared with the uniform porous receiver.Barreto et al. [23] optimized the porosity, porous diameter, and flow velocity to promote the efficiency of porous volumetric solar receiver.Godini and Kheradmand [24] selected the average cell diameter, the porosity, the absorber diameter, and the distance of glass window from the absorber surface as the optimized parameters and performed a parametric and optimization study to minimize the pressure drop in the receiver while maximizing the output temperature by experimental design method.Huang and Lin [25] investigated the effects of porous structure and materials on the solar thermal conversion.Chen et al. [26] analyzed the parameters such as the optical thickness and thermal conductivity coefficient of porous and proposed the critical value of the optical thickness, porosity, and porous diameter that would significantly reduce the solar radiation loss.
From the aforementioned discussion, it is clear that there are two methods for improving the heat transfer performance of porous volumetric solar receiver.One is changing the porous materials, and the other is changing the porosity and pore size of porous.However, there is little research concerning the structure design of the porous volumetric solar receiver.Therefore, this contribution firstly designs six different types of structure of porous volumetric receiver, applies the GDM model to simulate the actual nonuniform solar flux distribution on the optical aperture, adopts the local non-thermal-equilibrium model to analyze the heat transfer between porous media and air, and analyzes the effects of receiver structure, porosity, nonuniform heat flux boundary condition, and porous diameter on volumetric effects and heat transfer performance of the porous volumetric receiver.This contribution can provide guidance for the structure design to improve the heat transfer performance.

Model Descriptions
2.1.Physical Model.In solar power dish plant, the concentrator is basically constructed as dish paraboloidal-shaped mirror that reflects and concentrates the solar radiation into the porous volumetric receiver located in the focal point of the dish paraboloid, which heats air and porous media in the receiver.Because the absorption rate of air to solar radiation is very low, the porous media is heated by the concentrated solar radiant firstly.Then, air is heated to high temperature through convective heat transfer occurring between porous media and air and then drives the steam turbine.This process can realize the conversion between solar energy and mechanical energy.The porous volumetric solar receiver consists of steel shell, porous material, and quartz glass aperture, as shown in Figure 1.International Journal of Energy Research Theoretically, a well-designed volumetric receiver can realize the ideal volumetric effect; that is, the porous temperature near the quartz glass aperture is lower than that of rear of the receiver, and the highest temperature region appears inside the receiver.In this case, the radiation loss is small, and the solar-thermal efficiency is greatly improved.However, in the traditional volumetric receiver, the large amount of concentrated radiation exists in the near quartz glass aperture domain and the depth of external radiation is shallow.Thus, a traditional volumetric receiver cannot achieve the ideal volumetric effects, which will limit the solarthermal conversion efficiency and even lead to the flow instability.Obviously, the volumetric effect is influenced by the receiver structure.Therefore, receiver structures should be carefully designed to improve the solar-thermal conversion efficiency of the porous volumetric receiver.As the cylindrical porous volumetric receivers are widely used in the dish thermal generation system, in this contribution, we choose the cylindrical structure as basic model, keeping the volumes and aperture areas of different structures consistent.We design six types of porous volumetric receiver structures (see Figure 2): (a) hexagonal prism type of receiver (RT-I), (b) octagonal type of receiver (RT-II), (c) drum type of receiver (RT-III), (d) composite circular platform type of receiver (RT-IV), (e) frustum type of receiver(RT-V), and (f) scaled cone type of receiver(RT-VI).The different geometry parameters are shown in Table 1, where l is the length of RI-I and RI-II, a is the length of the hexagonal prism, b is the octagonal length, c is the spheroid diameter, d is the length of RI-III, e is the inlet and outlet diameter of RT-III, f is the length of RT-IV, g is the middle circular surface of RI-V, h is the inlet and outlet circular surface of RI-V, i is the length of RI-V, j is the inlet diameter of RI-V, m is the outlet diameter of RI-V, n is the length of RI-VI, o is the inlet and outlet diameter of RI-VI, and p is the middle circular surface of RI-VI, respectively.

Governing Equations.
For a steady-state laminar flow and incompressible convective heat transfer process in the porous volumetric receiver, the corresponding governing equations can be expressed as follows: Continuity equation is as follows: where ρ and U are the density and the velocity, respectively.Momentum equation is as follows: where P, μ, and K are the pressure, the dynamic viscosity, and the permeability, respectively.The permeability K i is modeled based on the Ergun equations: where φ and d are the porosity and diameter of the porous particle, respectively.Considering the local thermal nonequilibrium model, energy conservation equations are as follows: Fluid region: 3 International Journal of Energy Research where c p , T f , T s , and λ eff ,f are the specific heat capacity, the temperature of fluid, the temperature of solid, and the effective thermal conductivity of fluid, respectively.In addition, h v is the bulk heat transfer coefficient of porous, which is expressed as follows by Wu et al. [28], and this correlation is valid for 0:66 < φ < 0:93 and 70 < Re < 800: λ f 32:504ϕ 0:38 − 109:94ϕ Solid region: where λ eff ,s , G c , and G d are the effective thermal conductivity of solid, thermal radiation of porous media, and solar radiation, respectively.The effective thermal conductivity of solid λ eff ,s is modeled based on [29] where λ s and λ f are the thermal conductivity of porous media and fluid, respectively.The P1 approximation radiation transport model is applied to describe the radiation transport process: where k is the absorption coefficient.
where σ s is the scattering coefficient.
where βis the extinction coefficient.In addition, the thermal radiation of porous media G c is modeled as follows: where qðx, yÞ is the solar radiation with GDM.

Boundary Conditions.
In order to obtain the temperature distribution of receiver, the corresponding boundary conditions are as follows: Inlet boundary condition: Outlet boundary condition: In order to simulation the actual solar radiation distribution on the quartz glass aperture, we apply GDM from reference [30]; the boundary condition of wall is as follows: C R , η, and DNI are the concentrating ratio, the heat collection efficiency, and the solar direct normal irradiance, W m -2 , respectively.In addition, x and y are the coordinates of the aperture, m.Thus, the solar radiation distribution on the aperture is much higher in the central domain than in the edge domain, which presents significantly nonuniformity.

Numerical Method.
The geometric model is created and meshed using the CFD preprocessing software ICEM.The CFD program, FLUENT 16.0, is applied to simulate the flow, radiation, and heat transfer processes in the receiver.The laminar model is applied, the momentum and energy equations are discretized using the second-order upwind scheme, and the velocity and pressure are linked using the SIMPLE algorithm.The user-defined function (UDF) in FLUENT is utilized for source of momentum and energy equations and nonuniform heat flux boundary condition.The userdefined scalar (UDS) in FLUENT is applied for energy equation in the solid domain and radiation transport equation.The convergence criterion for the residual of energy equations is restricted to be less than 10 -6 , and the residuals of other equations are restricted to be less than 10 -3 .
In an actual solar dish power system, the inlet temperature of working fluid is chosen as ambient temperature.Thus, the inlet temperature is 300.15K.Besides, in order to keep the Reynolds number ranging from 70 to 800, the inlet velocity is chosen as 0.72ms -1 .The thermal physical properties of air are as follows: ρ = 1:225kgm -3 and μ = 1:789 × 10 −5 kgm -1 s -1 , which keep constant during the convective heat transfer process.The SiC is chosen as the solid phase.The thermal physical properties of SiC are as follows: ρ = 3:2 kgm -3 , c p = 750Jkg -1 K -1 , and λ s = 80Wm -1 K -1 .The temperature of heat transfer fluid exiting the porous volumetric receiver is more than approximately 1000 °C.Because of the high working temperature of receiver, here, the specific capacity is defined as variable [31]:

Model
Validation.The numerical model of the porous volumetric receiver developed by the authors was compared with the experimental values that were obtained by Wu et al. [32] to validate its accuracy.The case represents with a porosity 0.7, inlet mass flow 3.36 g s -1 , and inlet temperature 295 K. Figure 5 shows the radial temperature distribution of solid at the cross-section x = 0 mm, T s,radial and the axial temperature distribution of solid along the centerline T s,axial in cylinder porous volumetric receiver.It can be seen that T s,radial and T s,axial predicted by this paper agreed well with the values tested by Wu et al.The maximum errors of T s,radial and T s,axial between the predicted data by this study and the experimental data of Wu et al. are both less than 10%.In this section, the inlet velocity is set to 0.72 ms -1 and the porosity and the porous diameter are 0.7 and 2 mm, respectively.Figure 6 shows the temperature distributions of different receiver structures at the x = 0 cross-section.It is clear that the trend of temperature distributions in all receiver structure seems to be similar, which is high in the central axis domain and low in the surrounding domain due to the Gaussian distribution heat flux.It also can be   7 International Journal of Energy Research found that the temperature distribution of the fluid is significantly different from that of the solid.The temperature of fluid has a rapid temperature rise near the quartz glass aperture and a gradual decrease afterwards, while the temperature of solid keeps decreasing.This trend can be seen clearly in the average temperature as shown Figure 7.
Figure 7 shows the average temperature of fluid and solid in different structures along the z-direction.It can be found that the average fluid temperature of each receiver structure is always higher than that of solid from z = 10 mm to z = 70 mm, showing the evident volumetric effect of the volumetric solar receiver.Moreover, it is clear that the average temperature of fluid in all volumetric receiver structures increases from z = 0 to z = 10 mm initially and then decreases along the z-direction.The incident radiation can only reach a few distances from the optical aperture, and the location of the peak value for the different volumetric receiver structures is approximate.It also can be seen from Figure 7 that the temperature distributions of fluid and solid of RT-III and RT-IV are more uniform and higher than that of other receiver structures.The RT-III has the highest peak average temperature of fluid, i.e., 1050 K, and the highest average outlet temperature of fluid, i.e., 851 K, which is 41 K higher than that of RI-VI.The RT-VI has the lowest peak average temperature of fluid, i.e., 1018 K, and the lowest average outlet temperature of fluid, i.e., 810 K.
Furthermore, the average temperature of solid in all volumetric receiver structures decreases along the z-direction.The RT-III also has the highest average temperature of solid near the aperture, i.e., 1183 K, and the highest average temperature in the end-section, i.e., 844 K, which is 38 K higher than that of RI-VI.
It can be concluded that the structure has a certain influence on the volumetric effect and heat transfer performances of the porous volumetric receiver.The RT-III structure displays a better volumetric effect and heat transfer performance than other structures while the RT-VI displays the worst performance.
4.2.The Choice of the Porous Diameter.The porous diameter is an important parameter for the porous volumetric solar receivers.The low porous diameter would enlarge the local coefficient of convective heat transfer and improve the heat transfer performance between the solid and fluid, but it can also result in the increased flow resistance.Thus, the porous diameter should be carefully chosen to improve the radiation transport properties and heat transfer performance.
The diameter of porous is usually chosen from 1 mm to 4 mm in literatures [19,33].But if the diameter of porous sets as 1 mm, the Reynolds number in the porous is lower than 70.Hence, the diameters 2 mm, 2.5 mm, 3 mm, 3.5 mm, and 4 mm are used in this work to see the effect of the diameter on the receiver performance.In addition, the RT-III and the cylinder receivers are analyzed.The inlet velocity is set to 0.72 m s -1 , and the porosity is 0.8, respectively.
Figure 8 presents the cross-section average temperature of fluid along the z-direction with the different porous diam-eter, respectively.It is clear that the maximum temperature of the RT-III receiver with the porous diameter of 2.0 mm, 2.5 mm, 3 mm, 3.5 mm, and 4 mm is 1084 K, 1040 K, 1002 K, 970 K, and 941 K, respectively.It can also be seen that the maximum temperature of a cylindrical receiver has the same rule that the maximum temperature increases with  Moreover, along z-direction, the cross-section average temperature with small porous diameter decreases faster than that of large porous diameter.When the fluid passes over around the 35 mm distance away from the optical aperture, the cross-section average temperature of small porous diameter is lower than that of large porous diameter, resulting in the outlet average temperature of the large porous diameter being higher than that of the small porous diameter.This is because the solar radiation becomes weak at the 35 mm cross-section as shown in Figure 9. Figure 10 gives the outlet average temperature of fluid versus the porous diameter.The average outlet fluid temperature increases with the increase of the pore diameter.This also shows that the large porous diameter is conducive to the solar radiation into the volumetric receiver and can improve the overall thermal efficiency of receiver.In addition, the RT-III with different porous diameters still has the better heat transfer performances than the cylinder receiver, which is consistent with the conclusions of the previous section.This also means that the porous diameter would influence the temperature distributions and heat transfer performances.Figure 11 displays the pressure drop of the receivers; the porous diameter has a main impact on the pressure drop of the receiver.A slight outlet average temperature rise of 16 K (1.8% increase) in the RT-III is found between the d p = 2 mm and d p = 4 mm.However, the pressure drop is significantly decreased from 33 Pa/m to 10.3 Pa/m (68.8% reduction).Besides, the pressure drop of the RT-III is also slightly lower than the cylinder receiver.This also verifies that the RT-III is a better receiver structure which has the advantage of good heat transfer performance and small pres-sure drop at different diameter.It concludes that increasing the porous diameter can suppress the pressure drop and does not lower the outlet average temperature.Thus, the solar volumetric receiver with the small porous diameter needs more careful analysis and design, and the porous diameter of 2 mm is used in the following study.be considered.Therefore, in this contribution, we further choose a small diameter to analyze what the appropriate porosity is needed to achieve better thermal efficiency.The inlet velocity is set to 0.72 m s -1 , the porous diameter is 2 mm, and the porosity is 0.7, 0.8, and 0.9, respectively.Figure 12 shows the average temperature of fluid along the z-direction, which means that the different porosity would impact the temperature distributions of fluid and heat transfer performances between fluid and solid.It is clear that the RT-III still has the better transfer performances than that of the cylinder receiver.RT-III has the maximum average temperature of fluid 1048 K, 1083 K, and 1072 K with φ = 0:7, 0.8, and 0.9, respectively, which are higher than those of cyl-inder receiver.Figure 13 gives the outlet average temperature of fluid versus the porosity.RT-III has the outlet average temperature of fluid 852 K, 886 K, and 964 K with φ = 0:7, 0.8, and 0.9, respectively, which are also higher than those of the cylinder receiver.Moreover, the fluid average outlet temperature in the receiver structures both increase with the increase of porosity, and the position along the z-direction of maximum average temperature of fluid moves from 0.01 m to 0.02 m.This is due to that the higher porosity would properly increase the depth of the radiation flux.Figure 14 illustrates that in pressure drop of fluid versus the porosity of the receivers, the porosity has a main impact on the pressure drop of the receiver.The pressure drop is significantly decreased from 107 Pa/m to 8 Pa/m (92.5% reduction) from the φ = 0:7 to φ = 0:9.Besides, the pressure  International Journal of Energy Research drop of the RT-III is also slightly lower than the cylinder receiver.This also verifies that the RT-III is a better receiver structure which has the advantage of good heat transfer performance and small pressure drop with different porosity.It concludes that increasing the porosity can suppress the pressure drop and does not lower the outlet average temperature.

Conclusions
By designing six different receiver structures, applying the local non-thermal-equilibrium model and the Gaussian distribution model (GDM), which is to simulate the actual nonuniform heat flux boundary condition, we find that the structure has a certain influence on the volumetric effect and heat transfer performances of porous volumetric receiver, and the drum-type structure is suitable for improving the volumetric effects and heat transfer performance.Next, we investigate the porous diameter in RT-III and cylinder receiver.For a prescribed nonuniform heat flux boundary, the maximum average temperature increasing with the decrease of the porous diameter, both two receiver structures will make the same rule.Moreover, the average temperature along the z-direction of small porous diameter decreases faster than that of large porous diameter.Furthermore, the RT-III with different porous diameters still has the better heat transfer performances than the cylinder receiver, which is consistent with the conclusions of the previous section.This also means that the porous diameter would improve the temperature distributions and heat transfer performances.Increasing porous diameter can slightly improve heat transfer performance, whereas the effect on reducing pressure drop is very obvious.
Finally, we investigate the porosity in RT-III and the cylinder receiver.For a prescribed nonuniform heat flux boundary, with the increase of porosity, the average temperature of outlet fluid increases.Moreover, the RT-III with a different porosity still has the better heat transfer performances than the cylinder receiver, which is consistent with the conclusions of the previous section.This also means that higher porosity would properly increase the depth of the radiation flux, thereby improving volumetric effect and heat transfer performances.Increasing porosity can significantly improve heat transfer performance and reduce the pressure drop.
In conclusion, the structure has a certain influence on the volumetric effect and heat transfer performances of the porous volumetric receiver, whereas the porous parameters such as the porous diameter and porosity have major impacts on the on the volumetric effect and heat transfer performances of the porous volumetric receiver.
In the future work, in the future research, we will analyze the optimum length of the porous volumetric solar receiver and apply the gradual porous parameters to obtain a higher solar-thermal conversion efficiency.

Nomenclatures c p :
Specific heat capacity, Jkg -1 K -1 d p : Porous diameter, mm G c : Thermal radiation from porous G d : Solar radiation h v : Volumetric heat transfer coefficient, wm -3 K -1 k: Absorption coefficient K i : Permeability L: Length of receiver, m P: Pressure, Pa Q: Heat flux, W m -2 Re: Reynolds number S r : Source term of energy equation T: Temperature, K u, v, w: Velocity component in x-, y-, and z -directions, m s -1 U: Velocity vector, m s -

Figure 2 :
Figure 2: Six different structures of porous volumetric receiver.

Figure 3 :
Figure 3: The average outlet temperature of difference structures with different grids.

Figure 4 :
Figure 4: Computation meshes of the x-y plane and y-z plane.

Figure 5 :
Figure 5: Comparison of the temperature distribution alone the receiver between the results of the validation case and results obtained from Ref. [32].

Figure 6 :
Figure 6: The temperature distributions of different receiver structures at the cross-section of x = 0.

Figure 7 :
Figure 7: The average temperature of fluid and solid in different receiver structures along the z-direction.

Figure 8 :
Figure 8: The average temperature of fluid with the different porous diameter along the z-direction.

4. 3 .Figure 9 :
Figure 9: The distribution of solar radiation in the RT-III and cylinder receivers.

Figure 10 :
Figure 10: The outlet average temperature of fluid versus the porous diameter.

Table 1 :
Dimensions of six receiver structures.
3.1.Code Checking.Since the unified grid solution is used for all volumetric solar receivers with different structures, it is just needed to investigate the gird independence test for a steady-state incompressible laminar convective heat transfer process in the different receivers.Five different grid systems are tested: (1) cylinder receiver: 6146 ðcross-sectionÞ × 90 ðz − directionÞ, 6587 × 95, 8356 × 100, 9568 × 110, and 9237