Effects of Volumetric Property Models on the Efficiency of a Porous Volumetric Solar Receiver

Abstract

A porous volumetric receiver is the key component in concentrated solar power systems. In this paper, we investigate the effects of volumetric parameter models on the heat collection efficiency of the volumetric receiver by numerical simulations with the combination of local thermal non-equilibrium and discrete ordinate methods. Seven volumetric convective heat transfer coefficient models and three extinction coefficient models were investigated. The efficiencies calculated using these models were compared among each other. The results show that volumetric convective heat transfer coefficient models have significant effects with a maximum difference of 27.7% in receiver efficiency for these models. Extinction coefficient models have less effects on receiver efficiency with a maximum difference of 7.3%.

1. Introduction

The increasing consumption of fossil fuels and resultant emissions of greenhouse gases have received considerable attention [1]. The development and utilization of renewable energy become imminent. As a clean and renewable energy, solar energy is abundant but has low energy flux density [2]. Concentrated solar power (CSP) technology is an effective way to utilize solar energy and can replace fossil-fuel-fired thermal power generation. Rather than directly converting solar energy into electricity, CSP collects solar heat, stores it as thermal energy, and supplies the stored energy to turbines that run generators [3]. Studies have shown that with the significant improvement of CSP technology and power transmission technology, the global installed capacity is expected to reach 1000 GW by 2050, which means that CSP will cover more than 15% of global electricity demand [4]. Therefore, it will be crucial to improve the energy efficiency of CSP technology.

As a key component of CSP, the efficiency of the solar receiver has a great impact on the efficiency of CSP. Compared with conventional surface receivers, the volumetric receiver has the advantages of high heat collection efficiency and good stability [5]. For volumetric receiver, we hope to achieve the “volumetric effect” [6], i.e., the highest solid phase temperature is inside the absorber. In this way, the temperature of the front surface of the receiver can be reduced, local high temperature damage to the material can be avoided, and radiation losses can be reduced; thus, heat collection efficiency can be enhanced. Generally, porous media is considered to be a promising material as the absorber of a volumetric receiver [7]. The large specific surface area and complex pore structure of porous media not only enable it to increase the convective heat transfer between fluid and solid, but also provide a larger extinction area for the receiver to improve the temperature distribution. Therefore, using a volumetric receiver composed of porous media can achieve a higher operating temperature and minimize the radiation loss of the receiver, which helps to realize the “volumetric effect” and improve the efficiency of the receiver.

Numerous studies have presented numerical simulations of volumetric solar receivers aiming at achieving volumetric effect [8,9,10,11,12,13,14,15,16,17,18,19,20,21]. Natividade et al. [10] investigated the effects of thermal conductivity, porosity, and fluid mass flow of porous media on the solar thermal conversion efficiency of a volumetric receiver. The efficiency is expected to be as high as 85% when the fluid outlet temperature is about 950 K, but no significant “volumetric effect” was found in the study. Wang et al. [11] established a 3D theoretical model of the coupling of fluid flow and internal heat transfer, and studied the effects of porosity, pore size, volumetric heat transfer coefficient, and air flow rate on the receiver performance. In the case of porosity $\phi =0.95$, the thermal efficiency reached a maximum of 72.48% and the higher the $\phi$, the more significant the “volumetric effect”. From the results of the above studies, it is still not clear whether a significant volumetric effect can be achieved. The optimization results are also diverse. For example, Wang et al. [11] reported that, for porous media with the same porosity ($\phi$ ∼ 0.85), the pore size decreases from 5.83 mm to 2.22 mm, and efficiency increases by nearly 3%. In contrast, Kribus et al. [16] reported that receiver efficiency increases with increasing pore size.

We believe that the inconsistence of these results is caused by the volumetric parameter models used in these studies. Porous media are often modeled as continuous translucent media, known as the continuum scale method [9]. Two important volumetric parameters are the volumetric convective heat transfer coefficient and the extinction coefficient, which have a significant impact on the internal heat transfer process and results of porous media. Many different models have been reported for these two parameters [22,23,24,25,26,27,28,29,30]. For example, for the volumetric convective heat transfer coefficient models, some include the length of the receiver, while some do not. The constants are also different in different extinction coefficient models. To the best of our knowledge, the effects of volumetric parameter models on the receiver efficiency have never been reported.

In this paper, we selected seven volumetric convective heat transfer coefficient models and three extinction coefficient models. The efficiency calculated using these models was compared with each other to investigate their effects on the receiver efficiency.

2. Models and Methods

Governing equations. A two-dimensional axisymmetric model was used as shown in Figure 1. The continuity and momentum equations are

$$\nabla {·(\mathsf{\rho }}_{\mathrm{f}} \overrightarrow{\mathrm{V}}) = 0$$

(1)

$$\frac{1}{\epsilon }\nabla \left({\rho }_f\frac{\underset{V}{\to }\underset{V}{\to }}{\phi }\right)=-\nabla \mathrm{P}+\nabla \left(\frac{{\mu }_f}{\epsilon }\nabla \overrightarrow{V}\right)+\overrightarrow{F}$$

(2)

where $\epsilon$ is the porosity. $\overrightarrow{F}$ is the momentum source term due to the porous media, and the correlation can be expressed as

$$\overrightarrow{F}=\frac{{\mu }_f}{k_1}\overrightarrow{V}+\frac{{\rho }_f}{k_2}\overrightarrow{V^2}$$

(3)

where $k_1$ is the permeability coefficient and $k_2$ is the inertial coefficient; the correlations are

$$k_1=\frac{d^2}{1039-1002\epsilon }$$

(4)

$$k_2=\frac{d}{0.5138{\epsilon }^{-5.739}}$$

(5)

Volumetric convective heat transfer coefficient models. A local thermal non-equilibrium model (LTNE) was used to accurately obtain the temperature distribution of the solid and fluid. The energy equations of the fluid phase and solid phase can be defined as

$$0=\nabla ·\left(k_{se}\nabla T_s\right)+h_v\left(T_f-T_s\right)-Sr$$

(6)

$$\nabla ·\left({\rho }_f{C}_{p.f}\overrightarrow{V}{T}_f\right)=\nabla ·\left(k_{fe}\nabla T_f\right)+h_v\left(T_s-T_f\right)$$

(7)

where $k_{fe}$ and $k_{se}$ are the effective thermal conductivities of the fluid and solid phases, respectively, which are computed as $k_{fe}=\epsilon k_f$ and $k_{se}=\frac{1}{3}\left(1-\epsilon \right)k_s$. $Sr$ is the source term and the correlation is:

$$Sr=k_a\left({\displaystyle \sum }_{m=1}^{n\left(n+2\right)}{I}_m{w}_m-4\sigma T^4\right)$$

(8)

In this work, we chose seven volumetric convective heat transfer coefficient models that are widely used.

Model 1:

$$N{u}_l=2.0696{\epsilon }^{0.38}R{e}^{0.438}$$

(9)

$$h_v=a_{sf}N{u}_l\frac{k_f}{d}$$

(10)

This Nusselt number model was proposed by Wu et al. [22] through simulation studies and mathematical derivation. Zhu et al. [30] applied the model in their work and gave the specific surface area formula:

$$a_{sf}=\frac{10.33H\sqrt{1-\epsilon }-5.8H^{1.6}\left(1-\epsilon \right)}{d}$$

(11)

where H = 1 is defined as the Heywood circularity factor.

Model 2:

$$N{u}_v=(32.504{\epsilon }^{0.38}-109.94{\epsilon }^{1.38}+166.65{\epsilon }^{2.38}-86.98{\epsilon }^{3.38})R{e}^{0.438}$$

(12)

$$h_v=N{u}_v\frac{k_f}{d^2}$$

(13)

This Nusselt number model was derived by Wu et al. [22] through mathematics.

Model 3:

$$N{u}_v=\left[0.0426+1.236\left(\frac{d_m}{L}\right)\right]Re$$

(14)

where $d_m$ is the “passage” diameter and the correlation can be defined as:

$$d_m=\sqrt{\frac{4\epsilon }{\pi }}$$

(15)

This Nusselt number model was proposed by Fu et al. [23] through experimental measurements on samples of different materials, and the deviation was within ±20%.

Model 4:

$$N{u}_v=0.819\left[1-7.33\left(\frac{d}{L}\right)\right]R{e}^{0.38\left[1+15.5\left(\frac{d}{L}\right)\right]}$$

(16)

This model was proposed by Younis [24] based on measurements on samples with porosity of 0.83–0.87 and the maximum deviation was 27.1%.

Model 5:

$$N{u}_v=1.5590+0.5954{\mathrm{Re}}^{0.5954}P{r}^{0.4720}$$

(17)

where $Pr$ is the Prandtl number and the correlation is:

$$Pr=\frac{C_p\mu }{k_{fe}}$$

(18)

This model was derived by Petrasch [25] using tomography to measure a real porous medium sample with a pore size of 2.54 mm and a porosity of 0.857.

Model 6:

$$N{u}_l=2.832n_{PPI}^{-1.1}R{e}^{0.70905}$$

(19)

$$h_v=a_{sf}\frac{N{u}_l·k_f}{d_h}$$

(20)

$$d_h=\sqrt{\left(1-\epsilon \right)\frac{4s^2}{3\pi }}$$

(21)

$$s=\frac{0.0254}{1.5n_{PPI}}$$

(22)

$$a_{sf}=35.7n_{PPI}^{1.1461}$$

(23)

$$n_{PPI}=\frac{2.54\times {10}^{-2}}{d_c}$$

(24)

where $n_{PPI}$ is the pore density, usually expressed by the number of pores within a length of 2.54 cm; and $s$ is the pore diameter. This model was proposed by Xu [26]. By applying the least square method, Xu fitted the coefficients with the experimental data.

Model 7:

$$N{u}_v=0.34{\epsilon }^{-2}{\mathrm{Re}}^{0.61}P{r}^{\frac{1}{3}}$$

(25)

This model was proposed by Chen [27] through a large amount of experimental data fitting and applied to his subsequent work.

Extinction coefficient models. The porous media is treated as participating media, and the air is treated as transparent with a refractive index of 1 [31]. The radiative transfer equation can be expressed as:

$$\frac{\partial I_{\lambda }\left(r,\theta \right)}{\partial r}=-\beta I_{\lambda }\left(r,\theta \right)+{\kappa }_{\alpha }{I}_l^b\left[T\left(r\right)\right]+\frac{{\sigma }_{\lambda }}{4\pi }{{\displaystyle \int }}_{4\pi }\Phi \left({\theta }^{\prime }-\theta \right)I_{\lambda }\left(r,{\theta }^{\prime }\right)d{\theta }^{\prime }$$

(26)

where $\beta$ is the extinction coefficient, ${\kappa }_{\alpha }$ is the absorption coefficient, and ${\sigma }_{\lambda }$ is the scattering coefficient, expressed as:

$$\beta ={\kappa }_{\alpha }+{\sigma }_{\lambda }$$

(27)

$${\kappa }_{\alpha }=\alpha \beta$$

(28)

$${\sigma }_{\lambda }=\left(1-\alpha \right)\beta$$

(29)

In this work, we selected the following extinction coefficient models:

$$\beta =\frac{C\left(1-\epsilon \right)}{d}$$

(30)

This model was obtained by Hsu et al. [28] based on optical geometric analysis. In an ideal case, C = 3, and by adjusting the value of C, the model can be extended to actual porous media materials. Through comparison with the actual measured porous media material data, it was found that for SiC, C = 4.8 fitted best [29]. Therefore, the ideal cases C = 3 and C = 4.8 were used in this work.

Another extinction coefficient model was also chosen:

$$\beta =\frac{a_{sf}}{4\epsilon }$$

(31)

This model was obtained by Zhu [30] through using the Monte Carlo method. In a subsequent work, Zhu verified the correctness of the model by comparing with analytical equations.

The scattering phase function for an open cell foam structure under the assumption of diffuse reflection can be expressed as

$$\Phi \left(\theta \right)=\frac{8}{3\pi }\left(sin\theta -\theta cos\theta \right)$$

(32)

A discrete ordinate method was used to solve the radiative transfer equation.

Boundary conditions. In this study, the concentrated solar radiation energy flow received by the front surface of the collector was set as a uniform incident energy flux of $\varphi =1.04 \mathrm{MW}/{\mathrm{m}}^2$, and the mass flow rate at the entrance was set as $0.85 \mathrm{kg}/\left(\mathrm{s}·{\mathrm{m}}^2\right)$. The front surface was assumed to be a pseudo-surface with a surface porosity same as the volume porosity. The lateral wall was adiabatic and the inlet air temperature was set at 300 K. The back wall was assumed to be a perfect reflector without any radiation loss, and the gauge pressure at the outlet was set as 0 Pa.

Material properties. Since the porous media solar receiver operates at high temperatures, the variations of the air physical properties with the temperature cannot be ignored during the operation of the receiver. The viscosity coefficient of air is calculated via the Sutherland law, and the expressions of air specific heat capacity and thermal conductivity are [32]:

$$C_p=1.93\times {10}^{-10}{T}_f^4-8\times {10}^{-7}{T}_f^3+1.14\times {10}^{-3}{T}_f^2-4.49\times {10}^{-1}{T}_f+1.06\times {10}^3$$

(33)

$$k_f=1.52\times {10}^{-11}{T}_f^3-4.86\times {10}^{-8}{T}_f^2+1.02\times {10}^{-4}{T}_f-3.93\times {10}^{-3}$$

(34)

The raw material of the solid porous medium used in this paper was SiC with its specific heat capacity $C_p=750 \mathrm{J}/\left(\mathrm{kg}·\mathrm{K}\right)$, density $\rho =3210 \mathrm{kg}/{\mathrm{m}}^3$, and thermal conductivity $k_s=80 \mathrm{W}/\left(\mathrm{m}·\mathrm{k}\right)$.

Numerical method. This work used the finite volume method, the SIMPLE algorithm, and the second-order upwind method to solve the governing equations. The second-order upwind method was applied to the momentum equation, energy equation, and radiation equation. The convergence criterion for all equations was set to ${10}^{-6}$. In order to test the grid independence of the physical model, three groups of two-dimensional structured grids were established in this study. The number of elements were 8000, 16,000, and 32,000, respectively. By comparison, it is found that when the number of grids were 16,000 and 32,000, the difference between the calculated results was within 0.1%, so a structured grid with 16,000 elements was used.

Validation. This work referred to the case and data in reference [30], and compared the two-dimensional temperature distribution in the porous media region. The obtained solid region temperature contour results and the results in reference [30] are shown in Figure 2. Through comparison, it is found that the solid temperature distribution results obtained by the two methods are very close with the temperature difference within 20 K at each position. Therefore, the model and the method in this work are reliable.

3. Results and Discussion

Two kinds of porous media with different parameters, $d=2 \mathrm{mm}, \epsilon =0.92, \alpha =0.9$ and $d=1.5 \mathrm{mm}, \epsilon =0.8, \alpha =0.95$, were used to study the effects of volumetric parameter models on the receiver efficiency. The heat collection efficiency is expressed as [33]:

$$\eta =\frac{\dot{m}{{\displaystyle \int }}_{T_{in}}^{T_{ex}}{C}_p\left(T\right)dT}{q_{in}}$$

(35)

where $q_{in}$ is the solar radiation heat flux converging at the entrance of the receiver, $\mathrm{W}/{\mathrm{m}}^2$; m is the mass flow rate per unit area, $\mathrm{kg}/\left(\mathrm{s}·{\mathrm{m}}^2\right)$; $T_{in}$ and $T_{ex}$ are the fluid temperatures at the inlet and outlet of the receiver, K, respectively.

The first porous media.Figure 3 and Figure 4 are graphs of the temperature distributions along the receiver axis obtained by using different volumetric convective heat transfer coefficient models when the extinction coefficient model is Equation (30) and C = 3 and 4.8, respectively. The corresponding heat collection efficiencies are shown in

Table 1. Material 1: heat collection efficiency for different volumetric convective heat transfer coefficient models when C = 3.

Model	Wu(a)	Wu(b)	Fu	Younis	Petrasch	Xu	Chen
Efficiency	0.821	0.817	0.663	0.675	0.673	0.824	0.606

and

Table 2. Material 1: heat collection efficiency under different volumetric convective heat transfer coefficient models when C = 4.8.

Model	Wu(a)	Wu(b)	Fu	Younis	Petrasch	Xu	Chen
Efficiency	0.819	0.815	0.638	0.671	0.651	0.821	0.577

.

As can be seen from Figure 3, when the extinction model is used as Equation (30) with C = 3, the simulation results obtained by different volumetric convective heat transfer coefficient models are quite different. When models 1, 2, and 6 are used, heat transfer is more efficient and the so-called volumetric effect is achieved; thus, the heat collection efficiency corresponding to the three volumetric convective heat transfer coefficient models is higher than that for other models. According to Table 2, it can be found that different volumetric convective heat transfer coefficient models lead to a large difference in the heat collection efficiency of the receiver. The efficiency of model 6 is the highest and the efficiency corresponding to model 7 is the lowest; the difference between the two is 21.8%.

It can be seen from Figure 4 that when the extinction model is used as Equation (30) with C = 4.8, the simulation results obtained by different volumetric convective heat transfer coefficient models are also very different. When models 1, 2, and 6 are selected, heat transfer inside the receiver is also more efficient and the volumetric effect can be achieved. The heat collection efficiency obtained by using different volumetric convective heat transfer coefficient models is shown in Table 2. The heat collection efficiency is quite different under the influence of different models. The heat collection efficiency corresponding to models 1, 2, and 6 is higher compared to other volumetric convective heat transfer coefficient model. The efficiency corresponding to model 6 is the highest and the efficiency obtained by using model 7 is the lowest; the difference between the two is 24.4%.

By comparing Table 1 and Table 2, it is found that the selection of different C, that is, the selection of different extinction coefficient models, has less effect on the heat collection efficiency of the receiver. However, when C = 3, the heat collection efficiency obtained by using different volumetric convective heat transfer coefficient models is slightly higher than that when C = 4.8. This is because a smaller C means a smaller extinction coefficient and thereby, the extinction area of the porous medium becomes larger. Thus, the solar radiation absorbed by the front end of the receiver decreases and as a result, the front-end temperature decreases. Consequently, the radiation loss is reduced and the heat collection efficiency is increased.

Figure 5 shows the temperature distribution of the receiver axis obtained by simulating seven different volumetric convective heat transfer coefficient models when the porous medium material 1 is selected and the extinction coefficient model is Equation (31). The corresponding heat collection efficiency is shown in

Table 3. Material 1: heat collection efficiency under different volumetric convective heat transfer coefficient models when using Equation (31).

Model	Wu(a)	Wu(b)	Fu	Younis	Petrasch	Xu	Chen
Efficiency	0.805	0.799	0.596	0.634	0.609	0.810	0.533

; when model 6 is selected, the efficiency is the highest and model 7 is the lowest; the difference between the two is 27.7%. The heat collection efficiency for the extinction coefficient model in Equation (31) is lower than that for the model in Equation (30).

Based on the above three simulation results, when the porous medium material is selected as $d=2 \mathrm{mm},\epsilon =0.92,\alpha =0.9$, the selection of different extinction coefficient models has less effect on the heat collection efficiency, while the selection of different volumetric convective heat transfer coefficient models has a greater impact on the simulation results. When the volumetric convective heat transfer coefficient models 1, 2, and 6 are selected, heat transfer is more efficient and the heat collection efficiency of the receiver is higher than that for other models. When the volumetric convective heat transfer coefficient model 7 is selected, the efficiency difference can reach a maximum of 7.3% between the results for the ideal extinction coefficient model and the extinction coefficient model in Equation (31). When the extinction coefficient model in Equation (31) is selected, the maximum difference is 27.7% between the efficiency calculated using volumetric convective heat transfer coefficient models 6 and 7.

The second porous media.Figure 6 and Figure 7 show the temperature distributions along the receiver axis obtained by simulating different volumetric convective heat transfer coefficient models when the extinction coefficient model is selected as Equation (30) and C = 3 and 4.8.

Table 4. Material 2: heat collection efficiency under different volumetric convective heat transfer coefficient models when C = 3.

Model	Wu(a)	Wu(b)	Fu	Younis	Petrasch	Xu	Chen
Efficiency	0.823	0.823	0.655	0.711	0.720	0.791	0.693

and

Table 5. Material 2: heat collection efficiency under different volumetric convective heat transfer coefficient models when C = 4.8.

Model	Wu(a)	Wu(b)	Fu	Younis	Petrasch	Xu	Chen
Efficiency	0.806	0.805	0.629	0.685	0.694	0.768	0.667

are the corresponding receiver heat collection efficiency. From Figure 6 and Figure 7, we find that, for material 2 using the extinction coefficient model Equation (30), the simulation results obtained by different volumetric convective heat transfer coefficient models are quite different, regardless of whether C is 3 or 4.8. When models 1 and 2 are selected, the highest temperature appears inside the porous medium to achieve the volumetric effect, and the corresponding heat collection efficiency is also higher compared to other models. With the change of the basic parameters of the porous medium, the highest temperature of the solid skeleton appears at the front end of the receiver when using model 6 and the volumetric effect no longer appears. This is because the reduction in the porosity and pore size leads to the enhancement of the extinction ability of the porous medium material; thus, the front end of the receiver receives more solar radiation resulting in increased temperature of the front surface and more re-radiation losses. Consequently, heat collection efficiency decreases.

Figure 8 shows the temperature distributions along the receiver axis obtained by the simulations of different volumetric convective heat transfer coefficient models when the porous medium material 2 is selected and the extinction coefficient model is Equation (31). The corresponding heat collection efficiency is shown in

Table 6. Material 2: heat collection efficiency under different volumetric convective heat transfer coefficient models when using Equation (31).

Model	Wu(a)	Wu(b)	Fu	Younis	Petrasch	Xu	Chen
Efficiency	0.801	0.800	0.623	0.678	0.688	0.762	0.661

. Compared with the above results obtained with different extinction coefficient models, it is found that the volumetric convective heat transfer coefficient models 1 and 2 can still achieve a more significant volumetric effect, so the receiver has a higher heat collection efficiency.

4. Conclusions

This paper investigated the effects of volumetric parameter models on the performance of porous volumetric solar receivers. From the view of convection and radiation heat transfer, which are difficult to grasp and analyze in porous media, numerical simulations were carried out for various volumetric convective heat transfer coefficient models and extinction coefficient models to obtain the temperature distributions inside the receiver and the corresponding heat collection efficiency. The results calculated using these models were compared with each other. Conclusions and outlooks are as follows.

(1)

Volumetric convective heat transfer coefficient models have a greater impact on the numerical simulation results, while the extinction coefficient model has less effect on the overall simulation results. Using different extinction coefficient models results in an efficiency difference of up to 7.3%, while using different volumetric convective heat transfer coefficient models results in a much larger difference of up to 27.7%.

(2)

These significant performance differences from various volumetric parameter models show that further studies are highly needed to determine or to obtain accurate models for porous volumetric solar receivers before we can trust any optimization results. Currently, it is hard to tell which model is more accurate.