A Review of Radiative Heat Transfer in Fixed-Bed Particle Solar Receivers

Abstract

A highly efficient receiver is required because re-radiation loss increases dramatically with increased working temperature. Among a large number of receivers, the fixed-bed Particle Solar Receiver (PSR) represents a new pathway to high temperature with maximum overall thermal efficiency. The incoming solar radiation can penetrate deeper into the fixed-bed PSR filled with semi-transparent quartz and ceramic particles (spheres or Raschig rings), resulting in an increased volumetric effect. Reports show that an optimized PSR can realize overall receiver efficiency of around 92% at outlet temperatures above 1000 K, and achieve the annual temperature above 1000 K over 65% annual operating hours integrated with a concentrated solar power (CSP) system. To fully understand radiative heat transfer characteristics and provide deep insight into thermal efficiency, radiation energy is classified as incident solar radiation and radiative heat exchange in two parts. The transfer mechanism, the solution method and the progress of the investigation for each section are summarized and discussed in detail. Then, challenges and future directions, including an innovative design method, an improved experimental approach and an effective simulation method are proposed to put forward this receiver to be a preferred substitute in advanced, high-temperature power cycles.

1. Introduction

The growing problems of CO₂ emissions and concerns about energy security have strengthened interest in alternative non-oil energy sources. Solar energy is regarded as one of the most promising renewable energies because of its abundance and wide applications in nature [1]. However, although solar radiation is a high-quality energy resource due to the high temperature of its source, the low flux density limits its industrial applications. Concentrated solar power (CSP) that concentrates the incoming solar radiation to a high flux density level by the mirrors is an important candidate for becoming a major clean, renewable energy resource in the future. In the framework of the CSP, the incoming solar radiation is concentrated, and simultaneously delivered to the solar receivers [2].

The solar receiver is one the most important components in high-temperature solar thermal applications, which can make use of the incoming solar radiation as high-temperature processed heat. With the help of the high operating temperature, the solar receiver can increase the solar thermochemical reaction rate and the solar thermodynamic efficiency [3]. However, high-temperature receivers also face several challenges, including the development of geometric configurations to minimize heat loss, absorber materials and HTF to own high reliability at high temperatures over repeated cycles [4]. To a certain extent, it is very important and urgent to improve energy efficiency to alleviate current energy and environmental challenges.

Among various high-temperature solar receivers, particle solar receivers (PSRs) are currently being designed and developed as a means to achieve higher operating temperatures [5]. Unlike the traditional heat transfer fluid, solid particles have several advantages, including a wide operating temperature range, direct usage as a thermal energy storage medium, stable chemical properties, safety under high temperature and so on. These characteristics enable a PSR to achieve high-temperature applications exceeding 1000 °C and can drive efficient cycles [6].

The PSR uses solid particles to absorb incoming solar radiations either directly or indirectly [7]. The direct PSR irradiates solid particles directly when they pass through the receiver aperture, while the indirect PSR irradiates the tubes or other enclosures first, and then transfers heat to the solid particles. In addition, the solid particles may also be mobile (fluidized-bed) or static (fixed-bed). For mobile solid particles, the driving power includes gravity force, centrifugal force, buoyant force, conveyer belt and so on. For the fixed bed, the solid particles maintain static during the heating process. Once heated, solid particles can be stored in the receiver or delivered to an isolated reservoir and used to heat a secondary working fluid.

Compared with the mobile particles, the fixed-bed PSRs filled with static particles have attracted increasing attention due to their several advantages: high specific surface area, chemical stability, thermal energy density and cost effectiveness [8]. The fixed-bed PSRs have been extensively used as a medium for sensible and latent thermal energy storage and as a reacting medium in thermal chemical reactors [9,10]. However, the fixed-bed PSR has not been explored as a solar receiver for its inherent drawbacks, including low porosity and poor radiation penetration [11]. Recent studies show that these issues would be addressed by using semi-transparent glass particles as the heating medium. The semi-transparent glass particles with various shapes can considerably improve the radiative and convective heat transfer performances [12]. This type of solar receiver represents one potential way to realize a high operation temperature (>1000 K) with high efficiency.

When the solid particles absorb the incoming solar radiation to reach the high temperature, the absorbed heat is subsequently transferred to the working fluid through the convective heat transfer and the other particles via the radiative heat transfer, as well as the contact heat transfer. Among the process of the whole heat transfer within the receivers, the radiative heat transfer is fundamental to the thermal performance of the fixed-bed PSRs [13]. On the one hand, the thermal process of the PSR is derived by incoming solar radiation, and the thermal performances (e.g., maximum operating temperature, thermal efficiency) are determined by the absorption performances (or penetration depth) of the irradiation inside the PSRs. On the other hand, an obvious temperature gradient would appear inside the receiver domain due to the extremely non-uniform incoming solar flux distributions. The radiative heat transfer dominates the heat flux between the solid particles due to the high temperature level and the low contact heat transfer coefficient.

Furthermore, unlike the convective heat transfer and heat conduction, the radiative heat transfer is specified by volumetric characteristics, which means that, in theory, any solid particle can exchange radiation energy between the other particles over the whole volume [14]. In view of the polymorphic fixed-bed with different particle shapes and materials, the radiative heat transfer process that occurs inside the fixed-bed PSRs would be dramatically complex. So, the governing equation and the algorithm of the radiative heat transfer inside the fixed-bed PSRs should be processed carefully to retain accuracy and stability. Moreover, the key to improving thermal performance needs to be fully indicated to promote the development of solar thermal applications.

To comprehend the radiative heat transfer in the high-temperature fixed-bed PSR, the present work is subdivided into the following parts. Section 2 reports the types of the PSR from the heat transfer point of view. Section 3 provides the radiative heat transfer characteristics. Section 4 discusses the progress of radiative heat transfer investigations inside fixed-bed PSRs. Section 5 refers to the operating principle and heat transfer characteristics of the fixed-bed PSRs. Section 6 presents the challenges and future direction of the fixed-bed PSRs. Section 7 outlines the conclusions.

A graphical diagram of the systematics for the referenced works and scientific issues in this manuscript is represented in Figure 1.

2. Types of High-Temperature Fixed-Bed PSRs

In reviewing the reports on high-temperature solar receivers, it is clear that increasing the penetration depth of the incoming solar radiation can potentially result in a higher outlet temperature and thermal efficiency [15]. This phenomenon is well known as the volumetric effect. Although the thermal radiation (e.g., incident solar radiation, re-radiation emitted from the solar particles) can penetrate into the fixed bed with a large number of pores, the penetration depth is quite small because of the low porosity and the tortuous passageway. Thermal efficiency can be increased by increasing the solar absorptance and/or decreasing the thermal emittance [16]. However, this method is very expensive when all of the particles are deposited with spectrum-selected coating.

Another promising method to produce the volumetric effect is adopting semi-transparent particle materials, such as quartz glass spheres, quartz glass Raschig rings (a short hollow cylinder with equivalent height and outer diameter) and so on [17,18]. In practice, particles make up the major portion of the fixed-bed PSR. Herein, the radiation properties (extinction coefficient, albedo and scattering phase function) of the fixed-bed PSRs are significantly dependent on the transmission property of the particle materials. According to the transmission property of the particle materials, the fixed-bed PSRs could be classified into three types, including semi-transparent PSRs, opaque PSRs and mixed PSRs.

1.

Semi-transparent PSRs

The semi-transparent PSR is full of the quartz glass spheres and/or quartz glass Raschig rings with smooth surfaces for high-temperature applications. The quartz glass is the preferred particle material due to its several advantages, including spectrum-selected properties, good thermostability and superior abrasion resistance. This is especially true for the spectrum-selected properties, which have a low absorption coefficient for the solar radiation while having a high absorption coefficient for the infrared re-radiation. It is key to enforce the volumetric effect and improve thermal efficiency [19,20].

When the solar radiation strikes at the semi-transparent PSR, it may be encountered by either the pore channels or the quartz glass particles. For the former, the solar radiation can penetrate directly inside the pore channels until it touches the particles. For the latter, a part of the radiation is reflected back to the pore channels; the other part is refracted into the particle bodies. The proportion of these two parts is determined by reflectance, which is calculated by the incident angle measured between the local normal direction and the incident direction through Snell’s law. Minor solar radiation would be absorbed by the particle materials when it enters the particle bodies for the low absorption coefficient. Major solar radiation would be able to reach the opposite side and be refracted out from the particle bodies, and then enter the pore channels again until meeting other particles [21]. So, repeatedly, the incoming solar radiation can penetrate deeply into the semi-transparent PSR.

Generally, the effective extinction coefficient of the semi-transparent PSR is mainly determined by the absorption coefficient of the quartz glass materials associated with the reflectance of the radiation beam, which is about several per meter (m⁻¹) magnitude for the incoming solar radiation, roughly [22].

2.

Opaque PSRs

The opaque PSR utilizes the ceramic material (e.g., silicon carbide, zirconium oxide and aluminium oxide) or a high-temperature alloy as heating particles. These materials are opaque to any thermal radiation despite their favorable properties at high temperatures, and are widely used as porous absorbers [23,24]. When the incoming solar radiation reaches the opaque PSR, most of it is absorbed by the particles at the inlet wall, and a minimum amount can penetrate into the receiver through the pore channel. In this case, the peak temperature will appear at the inlet wall, resulting in a large amount of thermal radiation loss emitted from the receivers. In accordance with the porous ceramics, the extinction coefficient of the opaque particle fixed-bed would be up to hundreds per meter [25].

Multi-layer designs have been proven to be helpful to ameliorate this situation, and are widely applied to the porous volumetric receiver [15,26]. Inside each layer, the porous absorbers with given parameters (i.e., porosity, pore sizes, geometric configurations and spectral properties) produce particular performances, such as a volumetric heat transfer coefficient and extinction coefficient. For the entire multi-layer receiver, stepwise performances may be realized to enhance the volumetric effect [27,28]. Similarly, the opaque PSR would be divided into several sections along the radius direction and/or the asymmetry axis, and each section would be filled with different particles (configurations, materials and sizes) to improve thermal efficiency.

3.

Mixed PSRs

In this work, the mixed PSR is defined as a PSR that is full of mixed particles with various optical performances, namely, some quartz glass particles and other opaque particles. In the mixed PSR, the proportion of the opaque particles should be increased gradually along the penetration direction of the incoming solar radiation. Consequently, the axial average extinction coefficient of the mixed PSR would be increased sharply since the extinction coefficient of the opaque particle is almost two orders of magnitude larger than that of the quartz glass particle.

According to Beer’s law, the source term of the incoming solar radiation decreases exponentially with the product of the extinction coefficient and the penetration depth, approximatively [4]. If the growth of the axial average extinction coefficient of the mixed PSR is designed particularly, the source term of the incoming solar radiation would be increased smoothly along the penetrating direction. In theory, the HTF can be heated gradually with an increased source term level when it passes through the mixed PSRs. As a result, a significant volumetric effect would be demonstrated in the mixed PSRs.

The overall temperature performances of these three types of PSRs are drawn schematically in Figure 2. For the semi-transparent and opaque PSRs, the source term of the incoming solar radiation decreases with increasing penetration depth. Additionally, the decreasing speed of the source term curve in the opaque PSR is much faster than that in the semi-transparent PSR with a smaller extinction coefficient. These situations result in the front wall temperature being higher than the outlet HTF temperature. The volumetric effects produced by these two types of PSRs are rather poor. However, the source term curve within the mixed PSR may increase with increasing penetration depth, leading to a lower front wall temperature. In a nutshell, the mixed PSR with a particular design is more appropriate for the heat transfer process in pursuit of a higher working temperature and thermal efficiency.

3. Radiative Heat Transfer Characteristics

Compared to the solid particles, the radiation intensity emitted from a gas is negligible, even for carbon dioxide (CO₂) and water vapor (H₂O), as the HTF or the intermediate product of thermochemical reactions at moderate temperature performance levels (i.e., before ionization and dissociation occurs) [14]. From the standpoint of the radiative heat transfer object, the radiative heat transfer inside PSRs occurs primarily between particles (e.g., the surfaces of the opaque particles, and the bodies of the semi-transparent particles). Furthermore, the secondary part exists between the particles and the enclosure wall, as shown in Figure 3.

In terms of the radiation component, the radiation energy is composed of the incoming solar radiation flux and the radiative heat exchange flux. On the one hand, the incoming solar radiation flux would be reflected or refracted by the particles until it is absorbed inside the cavity or returned back to the external environment. On the other hand, the radiation heat exchange flux appears between the solid particles because of the temperature difference. In high-temperature PSRs, the radiant exchange heat flux would be the major portion of the total heat flux between the solid particles, where the thermal contact conductance is quite poor.

3.1. Models of Radiative Heat Transfer

For the most fixed-bed PSRs, the effective radius of the solid particles is up to the order of a millimeter or a centimeter to pass through the HTF fluently. This size scale is sufficiently greater than the wavelength of thermal radiation, and geometrical optics are adaptable to deal with the radiative heat transfer process [29,30]. In this situation, there are two models to describe the radiative heat transfer according to geometrical optics [14]. One is the radiative exchange between the surfaces (called surface radiative heat exchange), while the other is the radiative heat transfer in the participating medium (called the medium radiative heat transfer).

1.

Surface Radiative Heat Exchange

When the radiative heat exchange arises between the opaque particles with large scales, the surface radiative heat exchange model is rather adaptable to describe the radiative exchange process. Based on the assumptions of the gray and diffuse surface, the net radiative heat flux of the surface element with uniform temperature and emissivity could be calculated through the following expression [14]:

$$q_i=\frac{A_i{\epsilon }_i}{1-{\epsilon }_i}\left(\sigma T_i^4-J_i\right)$$

(1)

with

$$J_i={\epsilon }_i\sigma T_i^4+(1+{\epsilon }_i){\displaystyle \sum _{k=1}^n{J}_k{\phi }_{k,i}}$$

(2)

where the symbol $q_i$ is the net radiative heat flux of the surface element (W/m²). The symbols $A_i$ and ${\epsilon }_i$ are the area (m²) and emissivity of the surface element, respectively. The symbol $J_i$ is the radiosity which represents the total radiation leaving the surface element per unit time and per unit area (W/m²). The symbol ${\phi }_{k,i}$ is the view factor, which indicates the fraction of radiation energy leaving surface element $k$ that reaches surface element $A_i$. The symbol $n$ is the total number of the surface elements. In addition, the symbol $\sigma$ is the Stefan–Boltzmann constant, which is equal to 5.67 × 10⁻⁸ (W/m²·K⁴).

The linear equations of the radiosity in Equation (2) could be solved in a straightforward way using the conventional numerical method, such as the Gaussian-elimination method, inverse matrix method and so on. However, most coefficients of the matrix of these equations are non-zero. Computational time would increase remarkably with an increasing number of surface elements. In addition, the sum of the view factor is equal to the square of the number of the surface element. Thus, the solution and the storage of these view factors should be concerned in particular.

2.

Medium Radiative Heat Transfer

When the number of the solid particles is extremely great (e.g., the size of the solar particle is quite small or the receiver scale is rather huge), the surface radiative exchange model becomes invalid because the solution of the radiosity and the view factors are practically impossible. In this case, the solid particle cluster should be treated as an equivalently absorbing, emitting and scattering medium (called the participating medium). The transport action of thermal radiation in the participating medium is governed by the radiative transfer equation (RTE), which is given by [31,32].

$$\widehat{\mathrm{s}}\cdot \nabla I(r,\widehat{\mathrm{s}})=\kappa (r)I(r,\widehat{\mathrm{s}})-\beta (r)I(r,\widehat{\mathrm{s}})+\frac{{\sigma }_{\mathrm{s}}}{4\pi }{\displaystyle {\int }_{4\pi }I(r,\widehat{\mathrm{s}})\Phi (r,{\widehat{\mathrm{s}}}^{\prime },\widehat{\mathrm{s}})d{\mathsf{\Omega }}^{\prime }}$$

(3)

where the symbol $I(r,\widehat{\mathrm{s}})$ is the radiant intensity at the position of $r$, and the transmission direction $\widehat{\mathrm{s}}$ in units of (W/sr·m²). The symbols $\kappa$, $\beta$ and ${\sigma }_s$ are the absorbing coefficient, extinction coefficient and the scattering coefficient of the participating medium, respectively, (m⁻¹), $\beta =\kappa +{\sigma }_s$. The symbol $\Phi$ is the scattering phase function (sr⁻¹). The symbols ${\widehat{\mathrm{s}}}^{\prime }$ and ${\mathsf{\Omega }}^{\prime }$ are the transmission direction and the solid angle (sr) of the scattering radiation.

The RTE belongs to the differential-integral equation because of the scattering behavior, which is somewhat complex as a function of six independent variables (three space coordinates, two directional coordinates and wavelength). The analytical solution to the RTE is unimplementable unless applied to a very simple problem. Therefore, the numerical method is necessary to solve the RTE in consideration of practical solar thermal applications.

3.2. Simulation Methods of Radiative Heat Transfer

In the past decades, many simulation methods have been developed to obtain the radiant intensity from the RTE. So far, the typical simulation method of the RTE can be classified into four types, including the spherical harmonics method (P_N-approximation), the discrete ordinates method (S_N-approximation), the Monte Carlo method, and the Rosseland approximation (or the diffusion approximation) [4,13,14]. These methods have been included in several important commercial CFD codes (e.g., Fluent software, Star-CCM+ software, etc.), and used to solve combined heat transfer problems together with other methods of the CFD.

The spherical harmonics method converts the RTE into a series of partially derived equations using spherical harmonics. In theory, this method provides a vehicle to obtain an approximate solution of arbitrary accuracy [4]. However, the accuracy improves very slowly with increasing order of the spherical harmonics, while the mathematical complexity increases extremely rapidly. To compromise the solution accuracy and the mathematical complexity, low-order approximations (i.e., P₁-approximation, P₃-approximation) are usually adopted, which are generally accurate in optically thick media with near-isotropic radiative intensity [33,34]. The radiative source term in the energy equation by the P₁-approximation is computed as [14]

$$\nabla \cdot {\mathit{q}}_r=-\kappa \left(4\sigma T_{\mathrm{s}}^4-G_d\right)$$

(4)

with

$$\nabla \cdot \left(\frac{1}{3\beta }\nabla G_d\right)=-\kappa \left(4\sigma T_{\mathrm{s}}^4-G_d\right)$$

(5)

where the symbol $G_d$ is the incident radiation, which is defined as the direction-integrated intensity. The symbol $\nabla \cdot {\mathit{q}}_r$ is the radiative source term.

The discrete ordinates method (S_N-method) solves the transfer equation for a set of discrete directions spanning the total solid angle range of 4π. The radiative parameters (e.g., radiative source term, radiative heat flux, etc.) are evaluated through numerical quadrature with the directional radiative intensity over the total solid angle. Like the P_N –method, the S_N –method may obtain arbitrary levels of accuracy through the refinement of spatial and directional discretization. It is worth noting that this method gives very accurate results for optically thin media, but is difficult in optically thick media because of the computational cost [35,36]. Fortunately, the S_N –method can solve problems involving surface-to-surface radiation and participating radiation with anisotropic scattering [37,38]. The radiative source term in the energy equation calculated by the discrete ordinates method can be expressed as [14]

$$\nabla \cdot {\mathit{q}}_r=\kappa \left({\displaystyle \sum _{m=1}{I}_m{w}_m}-4\sigma T_{\mathrm{s}}^4\right)$$

(6)

where the symbols $I_m$ and $w_m$ are the radiative intensity and the quadrature weights associated with the direction ${\widehat{\mathrm{s}}}_m$.

The Monte Carlo method is a statistical sampling technique, which is delightfully convenient to apply without the additional direction discretization [4]. The outcome of the radiative heat transfer can usually be evaluated in terms of the radiation distribution factor [39]. The radiation distribution factor is determined by tracing the history of a large number of rays from their points of emission to their points of absorption [40]. The advantage of the Monte Carlo method is that even the most complicated problem may be solved readily, which includes problems ranging from optically thin regions to optically thick regions [41]. Additionally, this method more accurate compared to other available models when the number of rays is sampled enough. However, it has a higher computational cost and memory requirements when the large grid numbers are encountered [42]. To reduce the computational time, some particular means (e.g., octree data representation and search algorithm) are proposed and developed by authors [43,44].

With the Monte Carlo method, the radiative heat absorbed by element i can be calculated by the following expression [39]:

$$Q_i^a={\displaystyle \sum _{j=1}4\sigma \Delta V_j{\kappa }_j{n}_j^2{D}_{ji}{T}_j^4}+{\displaystyle \sum _{l=1}\sigma T_l^4\Delta A_l{\epsilon }_l{n}_l^2{D}_{li}}$$

(7)

where the symbols $\Delta V_j$, ${\kappa }_j$, $n_j$ and $T_j$ are the volume, absorption coefficient, refractive index and the absolute temperature of volume element i, respectively. The symbol $D_{ji}$ represents the radiation distribution factor, which is defined as the fraction of the total radiation emitted from volume element j that is absorbed by volume element i. The symbols $\Delta A_l$, ${\epsilon }_l$, $n_l$ and $T_l$ are the surface, emissivity, refractive index and the absolute temperature of surface element l, respectively. The symbol $D_{li}$ is the fraction of the total radiation emitted from surface element l that is absorbed by element i. Both $D_{ji}$ and $D_{li}$ should be determined using the Monte Carlo method.

The Rosseland approximation is extremely convenient to use without the solution of the RTE. This approximation reduces the radiation problem to a simple conduction problem with a strongly temperature-dependent conductivity (called radiative conductivity). Furthermore, the radiative heat flux could be calculated by applying the radiative conductivity as Fourier’s law of heat diffusion [45]. However, the Rosseland approximation can only be used for optically thick and isotropically scattering media [46]; it is recommended for use in Fluent software when optical thickness exceeds 3 [37]. The Rosseland diffusion coefficient is given by

$$k_r=\frac{16}{3\beta }\sigma T^3$$

(8)

where the symbol $k_r$ is the Rosseland diffusion coefficient.

The characteristics of the above simulation methods are summarized and tabulated in

Table 1. The features of the simulation methods of the RTE.

Methods	Accuracy	Computational Cost	Adaptability	Improvement
P1-approximation	Moderate	Low	Moderate optical thickness (β > 5 m−1)	Improve the calculation accuracy on unstructured grids
SN-approximation	High	Moderate	Favour optically thin (β < 10 m−1)	Expand the application scope accounting for the variation of radiative properties with temperatures
Monte Carlo method	Very high	Very high	Favour uniform medium	Reduce the computational time using an octree data representation and search algorithm
Rosseland approximation	Low	Few	Optical thick media (β > 3 m−1)	/

. Generally, the higher accuracy method needs more computational cost. Otherwise, the accuracy would be lost. The applicable conditions and guidelines for improvement are also suggested and listed in Table 1.

4. Progress of Radiative Heat Transfer Investigations

4.1. Radiative Properties of the Large Solid Particles

The radiative properties (e.g., emissivity, absorptivity, refractivity) of the solid particles are important for the precision of the solution regardless of the surface radiative heat exchange or the medium radiative heat transfer. On the one hand, the radiative properties determine the penetration and the absorption of solar radiation. On the other hand, the radiative properties affect the radiative heat exchange between particles. These two roles dominate the heat balance and the temperature distribution performances, and the subsequent thermal efficiencies or other output performances, as shown in Figure 4.

For the solid particle with a large scale, the radiative properties are roughly equal to those of bulk materials [14]. However, even for the bulk materials, the radiative properties are still rather unsettled, which vary with the wavelength, temperature and directions. Up to now, the literature has presented numerous useful radiative properties of bulk materials over a wide range of temperatures and wavelengths.

1.

Semi-transparent Particles

For the semi-transparent materials (e.g., fused silica, sapphire), they are semi-transparent in the visible and the near-infrared spectrum range, where the radiative properties refer to the absorptive index and the refractive index. While in the far-infrared spectrum range, they are opaque, and the radiative properties mainly refer to the spectral emittance.

In as early as 1971, Beder et al. [47] had measured the transmissivities of three types of fused quartz and the absorption coefficients were derived between 0.22 and 3.5 μm from 300 K to 1773 K.

Recently, Zhang et al. [48] developed a high-temperature FTIR (Fourier Transform Infrared Spectrometer) apparatus to measure the apparent transmittances of the fused silica samples for the wavelength from 0.8 to 5 μm, as shown in Figure 5. The fused silica samples were heated in a vacuum-Argon furnace with a uniform temperature volume. The absorptive index and the refractive index have been derived from the apparent transmittances measured. The results show that spectral properties depend on both wavelength and temperature, as shown in Figure 6.

To reveal the differences among the transmittances of various slabs of different thicknesses, Zhang et al. [49] proposed a Multi-Layer-Combination (MLC) method in which multiple sheets composed of a transmittance combination is proposed to provide obviously different transmittances. The radiative properties of C-plane sapphire were realized between 0.8 and 7.0 µm from room temperature to 1800 K, as shown in Figure 7. It can be seen that the spectral refraction indices increase steadily with increasing temperature and decrease slowly with increasing wavelength, whereas the spectral absorption indexes increase gradually with temperature.

In addition, Milton et al. [50] measured the high-temperature infrared emittance of sapphire, fused silica and other materials in temperature from 600 to 2000 K and 2 μm to 20 μm in spectrum. A vacuum emissometer was designed and built for use with a FTIR, where the samples were heated by utilizing a carbon dioxide laser.

2.

Opaque particles

For the opaque materials (e.g., zirconia, silicon carbide and alumina), the spectral properties are only the emissivity in the wide spectrum range or the spectral absorptivity to the specific incident radiation. However, some ceramic materials would change from opaque to semi-transparent with increasing temperature.

In early 1987, Cabannes et al. [51] had measured the infrared absorption of ZrO₂–Y₂O₃ crystals in the temperature range of room temperature to 1930 K. A significant increase in uptake was observed between 1600 and 1930 K.

Rozenbaum et al. [52] developed a spectroscopic method to measure directional spectral emissivity for both the homogeneous and heterogeneous semi-transparent materials, covering a large spectral 0.8–1000 µm range and 600 to 3000 K temperature range. The normal spectral emissivity of the single crystal alumina and the alumina ceramic have been proposed and discussed. Results show that the strengthening of emissivity in the phonon zone is due to the surface roughness of the ceramic. While in the semi-transparent region, changes in emissivity are due to the diffusion of volume in the texture of the ceramic.

Cagran et al. [53] implemented a temperature-resolved measurement of the spectral directional emissivity of SiC in the spectral range of 2–20 µm, over a temperature range of 573 to 1173 K at normal incidence. Wang et al. [54] established a new ultra-high temperature spectral emissivity measurement system covering a 373~2673 K temperature range and a 2~25 μm spectral range. The spectral emissivity of ZrB₂-SiC has been presented with total emissivity of 0.73 at 2073 K.

Zhao et al. [55] developed an experimental method to directly measure the emissivity and absorption performances of solar particles at elevated temperatures of up to 1200 K. The emissivity and absorption function measurements of Al₂O₃ and SiC particles were performed for utilization in concentrated solar receivers, as shown in Figure 8. It was found that the emissivities are constant, with values of 0.75 ± 0.015 and 0.92 ± 0.012 for Al₂O₃ and SiC, respectively, when the temperature ranges from 300 to 1200 K.

In summary, for both the semi-transparent and opaque materials as potential solar heating materials, their radiative properties obviously change with the wavelength. However, for the semi-transparent material, the radiative properties considerably vary with the working temperature. Although, for the opaque material, the variation of its radiative properties with the temperature is insignificant, which could be treated as a constant, approximately.

4.2. Approaches of the Incident Solar Radiation

There are three models to treat the transfer process of the incident solar radiation within volumetric receivers, including the surface absorption model, the exponential decay model and the volumetric absorption model [4].

1.

Surface Absorption Model

The surface absorption model assumes that the incident solar radiation is absolutely absorbed at the front surface of the absorber. The incident solar flux density profile may contain either a uniform distribution [56] or a Gaussian distribution [57,58,59,60]. The following Gaussian distribution is widely used for the CSF at the inlet wall of the receivers, as

$$q_{\mathrm{S}}(r)=A\exp (-B{r}^2)$$

(9)

where the symbol $r$ is the radius of the CSF and the symbols $A$ and $B$ are the fitting coefficient obtained experimentally, respectively.

This model is quite simple in that the incident solar flux is treated as boundary conditions. However, the surface absorption model loses its precision and is progressively replaced by the following improved models.

2.

Exponential Decay Model

The exponential decay model gets an attenuation by absorption of the incident solar radiation according to Beer’s law [14]. This model implies that the parallel solar beam enters the medium without the scattering effect. Like the surface absorption model, the incident solar flux may be a uniform distribution or a Gaussian distribution. In the energy equation, a volumetric heat source has to be defined to compute the contribution of the incident solar radiation as follows [26]:

$${\dot{S}}_{\mathrm{S}}(z,r)=\beta q_{\mathrm{S}}(r)\exp (-\beta z)$$

(10)

where the symbol $\beta$ is the extinction coefficient of the porous absorber, while the symbol $z$ is the penetration depth of the incoming solar radiation, and the symbol $q_S$ is proposed in Equation (9).

The radiation details of the exponential decay model are weak, which does not include radiative interactions due to the multiple reflections of the rays [61]. However, the exponential decay model could be considered a possible approach because of the very low computational cost required and the acceptable accuracy offered [4,23].

3.

Volumetric Absorption Model

The volumetric absorption model contains a volumetric heat source contributed by incident solar radiation under more practical conditions. This model considers that incident solar radiation is of a non-uniform distribution, in both the directional and spatial domain, as well that the medium has reflection, absorption and scattering behavior. In other words, the volumetric absorption model is an evolution of the exponential decay model. Generally, the Monte Carlo method is particularly suitable for the volumetric absorption model, taking into account the complex transfer process of the radiation. It can provide a better approximation at the cost of moderate computational resources. With this model, the volumetric heat source within control unit i can be expressed as [62]

$${\dot{S}}_{\mathrm{S},i}=\frac{e_{\mathrm{S}}\cdot N_i}{\Delta V_i}$$

(11)

where the symbol $e_{\mathrm{S}}$ represents the radiation energy carried by each solar ray. The symbol $N_i$ is the number of solar rays absorbed by control unit i and the symbol $\Delta V_i$ is the volume of control unit i.

By applying the Monte Carlo method, the fixed bed could be modeled using either the discrete particle model or the continuous particle model. The discrete particle model needs to describe the real geometry of each particle, including the surface, the volume and the position, along with the orientation of asymmetric particles [63,64]. The continuous particle model regards the fixed-bed as an absorbing, emitting and isotropic scattering media with homogeneous radiative properties. In the continuous particle model, the grids used to count the volumetric heat source are generated based on the volume-average method [65,66,67]. Generally, the discrete particle model can provide more details of the thermo-physical parameters based on real geometry of the fixed bed, relative to the continuous particle model with the volume-average method. However, the discrete particle model requires prohibitive computational costs unless the number of the meshes is modest (e.g., less than million orders coarsely).

The comparison characteristics between these three approaches of the incident solar radiation are tabulated in

Table 2. Features of the three incident solar radiation models.

Methods	Accuracy	Computational Cost	Adaptability	Recommendation Index
Surface absorption model	Low	Low	Optical thick medium (β > 10 m−1)	★
Exponential decay model	Moderate	Moderate	Moderate optical thickness medium (5 m−1 < β < 10 m−1)	★★
Volumetric absorption model	High	High	No restriction	★★★

. Generally, the accuracy of the approaches increases with increasing computational cost. The volumetric absorption model is preferred because it is more accurate.

4.3. Approaches of the Radiative Heat Exchange

Like the volumetric absorption model of the incoming solar radiation, the approaches of the radiative heat transfer inside the PSRs can be classified into the discrete particle model and the continuous particle model. The radiative heat exchange process is basically the same between the discrete and continuous particle model. Nevertheless, the radiation simulation for these two models are somewhat different in detail, which should be discussed to better understand and adopt the appropriate numerical method.

4.3.1. Discrete Particle Model

The discrete particle model solves the RTE of the fixed bed directly based on real discrete particles. Considering the computational cost and the memory requirement, the feasible method of the RTE simulation for the discrete particle model, at present, mainly contains the radiative effective thermal conductivity method, the discrete ordinates method and the surface-to-surface radiation method.

1.

Radiative Effective Thermal Conductivity Method

In this method, the radiative effective thermal conductivity is employed to compute the contribution of the radiative heat transfer between particles. The radiative effective thermal conductivity is commonly grouped with thermal conductivity to account for the combined radiative heat transfer and the thermal conduction (or thermal contact conductance between the particles). This method is much like the Rosseland approximation in form. However, the radiative effective thermal conductivity is derived from the surface radiative exchange model between the real discrete particles. At a low temperature gradient, the radiative effective thermal conductivity for the structured fixed bed is formulated as [68]

$${\lambda }_{\mathrm{R}}=\frac{2}{3}{\epsilon }_{\mathrm{R}}A{T}^3{\rho }_n\sigma {\displaystyle \sum _{j=1}^N{X}_j{\widehat{r}}_j^2}$$

(12)

where the symbols ${\epsilon }_R$, $A$, $T$ are the surface emissivity, surface area (m²) and the particle temperature (K), respectively. The symbol ${\rho }_n$ is the number density of the bed. The symbol $X_j$ is the view factor between two particles. The symbol $N$ is the particle number with $X_j>0$ and the symbol ${\widehat{r}}_j$ is the distance between two particles (m).

The view factor involved in Equation (12) is the geometrical parameter, which can be calculated using the short-range radiation model, the long-range radiation model, or the microscopic radiation model [69]. The short-range radiation model only calculates the radiative heat exchange between adjacent particles to simplify the calculation. Mehrabian and colleagues [70] proposed a classical short-range radiation model applied in a packed bed, which considered the radiation heat transfer within the short-ranges of 1.5 times the particle diameter, as shown in Figure 9.

In the work by Cheng and Yu [71], a numerical model only estimates the radiative flux between particle O with its Voronoi neighbors contacted directly (i.e., particles A–F), while the possible heat transfer between particles O and G is ignored.

Wu and colleagues [72] developed a short-range radiation model coupled with a complete CFD-DEM (Discrete Element Method) method for packed pebble beds. Ruiz and coworkers [73] implemented an experimental and numerical analysis of the heat transfer in a packed bed, in which the short-range radiation model is proposed for considering thermal radiation in a discrete elements system due to its easy implementation and low computational cost.

The short-range radiation model underestimates the radiative heat flux at high operating working temperature (e.g., higher than 1488 K) and the high surface emissivity (e.g., higher than 0.8) [69]. In this condition, the long-range radiation model is applicable to calculate the high-temperature particle radiation. The long-range radiation model calculates all of the radiative heat exchange fully between any two particles, and from the particle to the enclosure wall over a long distance.

Wu and colleagues [69] utilized the long-range radiation model to predict the radiation heat transfer with three peripheral layers of Voronoi neighbors, as illuminated in Figure 10. Results indicated that the long-range radiation model is better than the short-range model for predicting the heat exchange in packed beds when the effective thermal conductivity of radiation is much smaller than the solid conductivity. More recently, radiative heat flux and effective thermal conductivity were derived mathematically using a combined matrix model and a multi-layer neural network to present the thermal radiation with the long-range radiation model by the authors [68].

The microscopic radiation model should definitely be used to treat the non-uniform temperature distribution of particulates. In the microscopic model, the particle surface is split into a series of meshes and each mesh is considered an isothermal surface. Asakuma and colleagues [74] analyzed the effective thermal conductivity with radiation using the microscopic model. A simple periodic composite structure of the packed bed was considered to reduce the computational process. Wu and colleagues [69] computed the thermal radiation heat exchanges by these three models, respectively, to analyze the effect of the spatial scales based on the DEM data of particle packing. By comparison, it has been found that the microscopic model is computationally unacceptable although it can provide the most accurate solutions. The long-range radiation model is limited in that it can provide precision in the state where the solid conductivity is much greater than the effective thermal conductivity of radiation. However, this particular requirement is generally not met in practice. The short-range radiation model could be one of the choices, which is sufficiently feasible when the operating temperatures are lower than 1488 K and the solid conductivity is in the same order of the effective thermal conductivity of radiation.

More empirical radiation models for the packed bed of particles were reviewed and evaluated by the authors [75,76]. With the empirical approach, the effective thermal conductivity is usually used to represent the overall heat transfer, which includes heat conduction through the solid material and the contact areas between spheres, as well as thermal radiation between particles [76]. In comparison with the results of CFD simulations, Qian et al. [75] recommended the Zehner–Bauer–Schlünder (ZBS) model for calculating the effective thermal conductivity of a packed bed under low-temperature conditions. For high temperature conditions, the Breitbach and Barthels (B–B) correlation was the optimal method for various particle diameters, emissivities and cancellations.

2.

Discrete ordinates Method

The control unit of the discrete particle model in numerical simulations consists of the surface elements and the volume elements. As mentioned in Section 3.2, the discrete ordinates method is considerably powerful, which can solve both the surface-to-surface radiation and the participating medium radiation. It also allows the solution of radiation with semi-transparent walls, and the non-gray radiation using a gray-band model as well. Therefore, the discrete ordinates method can directly resolve the radiative heat exchange between the semi-transparent particles and the opaque particles under the discrete particle model.

To implement the discrete ordinates method, theta divisions and phi divisions should firstly be defined to make discrete each octant of the angular space. A finer angular discretization can be specified to better resolve the influence of small geometric features or strong spatial variations in temperature, but larger numbers of theta divisions and phi divisions will increase the computational cost [37]. Simultaneously, theta pixels and phi pixels need to be defined to control the pixelation that accounts for any control volume overhang. For specular or semi-transparent boundaries produced by glass particles, a pixelation of 3 × 3 is needed to achieve acceptable results.

Clearly, the discrete ordinates method requires excessive computational effort unless considering a representative volume section to reduce the computational domain with modest grids. Sedighi and colleagues [8] carried out a novel 3-D pore-scale analysis of the packed-bed receiver filled with glass particles, as shown in Figure 11. The discrete ordinates method was applied to solve the thermal radiation between the particles. A division of 3 × 3 and a pixelation of 3 × 3 were selected in the study according to the sensitivity analysis, respectively.

3.

Surface-to-surface Radiation Method

For the opaque particles with a diffusion surface, the surface-to-surface radiation method could be adopted to calculate the radiative heat exchange inside the receiver. As mentioned in Section 3.1, the surface-to-surface radiation method is an easy method to apply. However, it is very expensive in terms of computational effort and memory requirements when there are a large number of radiating surfaces. One of the approaches to reduce the number of radiating surfaces is grouping faces together to form surface clusters. The surface cluster information (coordinates and connectivity of the nodes, surface cluster IDs) is used by Ansys Fluent in radiosity calculations to compute the view factors [37]. Another alternative is considering only a representative domain of the solar receivers to reduce numerical grids by an order of magnitude. Zhu and colleagues [77] carried out a pore-scale numerical simulation of heat transfer and flow in structured packed-bed receivers with three packed types, including simple cubic, face-centered cubic and body-centered cubic, as shown in Figure 12. In the work, a quarter unit cell of packed channels was used during simulation to reduce the computational cost.

Sidighi and colleagues [18] aimed to find the limits and the key limiting factors for the overall thermal performance of a semi-transparent packed-bed solar receiver by exploring a wide range of geometrical and operational parameters. The parametric study was conducted by applying a heat transfer model associated with ray tracing and a surface-to-surface radiation model.

In conclusion, the aforementioned radiation methods for the discrete particle model limit the balance between accuracy and calculation cost. The accuracy increases from the radiative effective thermal conductivity method to the surface-to-surface radiation method, while the computational cost also increases accordingly.

4.3.2. Continuous Particle Model

The continuous particle model treats the fixed bed as pseudo-porous medium with homogenized radiative properties, including the extinction coefficient, the scattering albedo and the scattering phase function, as indicated in Equation (3). These radiative properties can be either experimentally determined or have been derived from a numerical detailed simulation method in theory [4]. The experiment approach can directly measure the hemispherical reflectance losses of the porous volumetric receiver based on UV-VIS-NIR spectrophotometer together with the integrating sphere [78,79]. However, for the reticulated porous materials with thick optical depth, the homogenized radiative properties are rather difficult to recover from spectral hemispherical reflectance and transmittance measurements using the inverse analytical techniques based on the RTE [80].

On the one hand, it is not always possible to obtain reliable results from reverse analytical techniques since the RTE is strongly non-linear. Additionally, a considerable simplification of the porous architecture should be supposed to simplify the computational complexity, such as neglecting the pore size distribution. On the other hand, the homogenized radiative properties depend on a number of geometrical parameters of the foam, such as the strut diameter or cell size. Unfortunately, these parameters are difficult to determine univocally because of the intrinsically random quality of the real foam structures [81].

More recently, a great interest has been spread on reproducing the foam structures through different approaches, including mathematical morphology operations applied on existing tomography data [63,82], simulation of the bubbling process [83] and regular [84,85] and irregular [25,86] Voronoi partitions, as shown in Figure 13. When the reconstructions of the real geometry of the porous foam are achieved, an improved Monte Carlo Ray Tracing method could be used to obtain the homogeneous radiative properties [87,88]. Similarly, these approaches and the improved Monte Carlo Ray Tracing method can also be used fairly to predict the radiative properties of the packed bed regarded as an equivalent homogeneous medium for both the semi-transparent particles and the opaque particles [82,89].

Once the radiative properties are obtained, the radiative heat exchange inside either the porous volumetric receiver or the pecked-bed PSRs can be computed by applying the typical simulation method of the RTE, such as the Rosseland approximation [15,23,90,91,92,93,94], the P₁ method [27,28,57,60,95,96,97] and the discrete ordinates method [16,66,98,99]. Since the extinction coefficient of the porous materials is up to several hundred per meter (m⁻¹) orders, there is no difference in the results of the radiative heat exchange solved by these methods.

To a certain extent, the Rosseland approximation would be more favorable than the other method for its computational efficiency, especially for the combined heat transfer and flow applications requiring tremendous computational costs. The features of the radiative heat transfer models inside the PSRs are summarized and listed in

Table 3. Features of the radiative heat transfer models inside the PSRs.

Models	Approaches	Accuracy	Computational Cost	Application Scenes
Discrete particle model	Radiative effective thermal conductivity method	Low	Low	Multiphase flow and heat transfer, or large number of grids
	Discrete ordinates method	Moderate	Moderate	Heat transfer with moderate quantity of grids (<106)
	Surface-to-surface radiation method	High	Very high	Heat transfer with a representative volume section (<104)
Continuous particle model	Rosseland approximation	Low	Low	Multiphase flow and heat transfer, or large number of grids
	P1 method	Moderate	Moderate	Combined heat transfer with moderate quantity of grids
	Discrete ordinates method	High	Quite high	Combined heat transfer with moderate quantity of grids

.

4.4. Overall Thermal Performances of the Fixed-Bed PSRs

Flamant and colleagues [100] first introduced the concept of the fixed-bed PSRs in the late 1980s, suggesting a packed bed containing glass and silica silicon carbide particles. However, there seems to have been no further advances in this type of design over the past 30 years [18]. Until recently, the overall thermal performance of the fixed-bed PSRs was numerically simulated by Zhu et al. [77] and further developed by Sedighi et al. in a series of literatures [8,12,18,101].

Zhu and colleagues [77] investigated the overall thermal performances of the structured fixed-bed solar receivers with silica silicon carbide particles, as shown in Figure 14. The Monte Carlo Ray Tracing (MCRT) method was employed to analyze the radiation propagation in the volumetric receiver, and the heat transfer and flow in pore scale models were analyzed. Numerical results showed that the incident flux would be completely absorbed in three times the particle diameter, regardless of the packed type and the incident angle, as shown in Figure 15. It was also demonstrated that the sphere surface absorptivity had a great influence on the absorption performances, where a smaller incident angle and higher porosity could slightly increase propagation depth.

In the work by Sidighi and colleagues [12], the overall thermal performance of a semi-transparent packed-bed solar receiver was conducted by coupling a comprehensive heat transfer model, as shown in Figure 16. A proposed dimensionless parameter was used to maximize the effective efficiency. For a 1000 K outlet temperature, an effective maximum receiver efficiency over 70% can be achieved using 18 rows of transparent spheres. The simulation results indicate that it is possible to use a semi-transparent packed-bed solar receiver in advanced power cycles.

Sedighi and colleagues [18] proposed and compared a series of packed bed designs (i.e., semi-transparent and high-transparency quartz spheres against an opaque bed of ceramic spheres). The optical and the thermal performances were investigated through a detailed ray-tracing analysis and a comprehensive thermal circuit model. The results revealed that the best proposed design with semi-transparent quartz glass particles can have an overall receiver efficiency of about 80% with outlet temperatures above 927 K, resulting in a high value for the elusive volumetric effect. Based upon these results, the packed bed designs have been proven as reliable and high-efficiency solar receivers.

Lately, Sidighi and colleagues [8] presented the results of a novel 3-D pore-scale analysis of a gas-phase, packed-bed receiver to characterize the optical-thermo-fluid behavior. To ensure the validity of this approach, experimental tests were conducted for a complex packed bed of spheres, as shown in Figure 17. The 3-D analysis at the pore level allows the development of innovative mechanisms to reduce radiosity losses. The final results demonstrated that the pore-scale modifications of the design can yield an optimized packed-bed receiver with a maximum efficiency of ~92% (with an outlet temperature above 927 K).

To understand the annual performance, Sidighi and colleagues [101] scaled up the semi-transparent packed-bed absorber and integrated with a concentrated solar power (CSP) system, as shown in Figure 18. A transient heat transfer analysis was conducted using real-time solar irradiance data. The calculation results showed that the gas-phase receiver could feed a gas turbine with an annual temperature above 927 K over 65% annual operating hours. It was found that this design is cost-effective for small, modular receivers.

To further investigate the optical performances of a semi-transparent packed bed, Sidighi and colleagues [102] proposed eight different ‘constant transmittance’ and ‘variable-transmittance’ combinations of transparent/semi-transparent/opaque spheres packed into a cylindrical cavity and exposed to a solar simulator for a layer-by-layer transmission test. A 3-D ray-tracing model was applied to elucidate the relative optical absorption of the cavity walls and spheres. An optimal variable-transmittance design enabled ~40% of the rays to penetrate to >70% of the absorber’s depth, and ~56% of the total rays were absorbed within the inner volume. Therefore, the optimal design yields uniform absorption, potentially leading to a uniform temperature distribution and a minimization of thermal emission losses.

5. Operating Principle and Heat Transfer Characteristics

5.1. Charging and Discharging of the Fixed-Bed PSRs

Thermal energy storage (TES) is one of the most promising ways to address the gap between the fluctuation of the solar energy and the thermal energy or electric power demands [103,104,105]. Integration of energy storage system into a concentrated solar power can avoid the frequent start-up and shut-down of electrical devices, and increase the load factor by dispatching production according to demand and current electricity price [106,107].

In recent years, fluidized-bed receivers have been widely used as a system capable of capturing concentrated solar energy for thermal energy storage [108,109,110,111,112,113,114,115,116]. Depending on the flow direction of the HTM in the receiver, the receivers could be divided into three categories: downflow receiver, upflow receiver and horizontal flow receiver [117]. Lee et al. [108] performed a numerical simulation of particulate flow in interconnected porous media for a central particle-heating receiver, where the porous structure reduces the speed of the falling particulate material and a large temperature rise to be achieved in a single pass. Diago et al. [109] measured the optical properties of desert sand to evaluate its performance as a direct solar absorber. Thermogravimetric analyses showed that the samples appear to be thermally stable between approximately 923 K to 1273 K. Nie et al. [110] measured the radiative properties of solid particles as heat transfer fluid in a gravity driven moving bed solar receiver. Three particles were found to have the best characteristics for the SPSR. Díaz-Heras et al. [111,112] tested a bubbling fluidized bed and observed that SiC presents the highest thermal efficiency during the charging process, although sand obtained lower pumping costs. Behar et al. [113] presented a practical technique to design a modular combined cycle solar power plant using the fluidized particle solar receiver technology. The nominal efficiency of the components including the heliostat field, the solar receiver, the gas turbine and the steam turbine were calculated and discussed, respectively.

In addition, Gomez-Hernandez et al. [114] proposed a linear system with Fresnel reflectors, in which the fluidized particles allow the horizontal movement and the linear absorption of solar energy. Jiang et al. [115] proposed an experimental investigation on the hydrodynamic performances of fluidized-bed particle solar receivers with a gas–solid countercurrent flow pattern to enhance the heat transfer performances. It was proven that the CCFB can operate stably and adjust the solid flux rapidly. Mukherjee et al. [116] proposed a quantitative investigation and comparison between sensible energy storage and hydrogen energy storage apropos to a concentrated solar thermal power plant. Results show that sensible energy storage delivers superior energy and power density (68 kWh/m³ and 4.5 kW/m³) as opposed to 26 kWh/m³ and 1.8 kWh/m³ for hydrogen energy storage, respectively. More details on the materials, configurations and methodologies of the gas–solid fluidized-bed particle solar receivers have been reviewed and summarized by Jiang et al. [118] and Tawfik [119].

Apart from the fluidized-bed PSRs, the fixed-bed PSRs offer an alternative pathway to absorb and store the incident solar radiation [12]. Figure 19 shows the fixed-bed PSR associated with a TES tank proposed in this paper. Within the fixed-bed PSR, there are transparent solid particles, semi-transparent solid particles and opaque solid particles arranged along the penetration direction of the incident solar flux. The transparent solid particles and semi-transparent solid particles that remain motionless are used to lower the front surface temperature and capture the re-radiation loss emitted from the opaque solid particles at high temperature. Meanwhile, the opaque solid particles are used to absorb solar radiation and store heat energy.

During the charging process, the concentrated solar radiation impinges on the quartz glass window of the fixed-bed PSR, and then penetrates into the solar receiver to heat the solid particles. On the one hand, the opaque solid particles crawl toward the window and absorb the solar radiation with gradually increasing levels. When the front particles achieve the required working temperature under a sufficient heating time, these high temperature particles proceed to the storage tank. On the other hand, the HTF passes through the storage tank and is heated by the high temperature particles by convective heat transfer.

Additionally, on cloudy days, after sundown, or in the early morning, the system falls into a discharging process. The high temperature solid particles are stored in the tank and heat HTF to a required temperature demand. Therefore, the fixed-bed PSR associated with TES tank is capable of producing reliable and sustained thermal or electrical output.

5.2. Heat Transfer Characteristics of the Fixed-Bed PSRs

Apart from the radiative heat transfer, the internal heat transfer behaviors inside the fixed-bed PSRs or in the TES tank contain the contact conduction, convective heat transfer between the HTF and the particle surfaces, as well as the heat between the HTF and the tank wall, as illustrated in Figure 19. However, since there are only point contacts between the spheres, and the particles are also thermally poor conductive materials, conduction may be ignored, relative to radiation and forced convection [12,120].

For sphere-to-air convective heat transfer, the most commonly used ones are listed in

Table 4. Available Nusselt correlations for spheres-to-air heat transfer in packed beds.

Authors	Correlations	Validity Limit
Dunkle [121]	$N{u}_p=2.4R{e}_p^{0.7}Pr$	$10\le R{e}_p\le {10}^5$
Wu [122]	$N{u}_p=0.32R{e}_p^{0.59}$  $N{u}_p=8+0.004R{e}_p$  $N{u}_p=0.0032R{e}_p$  $N{u}_p=0.0022R{e}_p$	$200\le R{e}_p\le 7\times {10}^3$ and $\varphi =0.39$ $\varphi =0.48$ $\varphi =0.73$ $\varphi =0.97$
Gunn [123]	$\begin{array}{c}N{u}_p=\left(7-10\varphi +5{\varphi }^2\right)\left(1+0.7R{e}_p^{0.2}P{r}^{1/3}\right)+\\ \left(1.33-2.4\varphi +1.2{\varphi }^2\right)R{e}_p^{0.7}P{r}^{1/3}\end{array}$	$1\le R{e}_p\le {10}^5$ and  $\varphi \ge 0.35$
Wakao [124]	$N{u}_p=2+1.1R{e}_p^{0.6}P{r}^{1/3}$	$100\le R{e}_p\le {10}^5$
Lee et al. [125]	$N{u}_p=2+1.1R{e}_p^{0.6}P{r}^{1/3}\exp \left(-0.3\frac{H}{d}\right)$	$R{e}_p\le 1000$ and  $3.3\le H/d\le 43.3$
Reichelt [126]	$Sh=2+0.991{\left(R{e}_{p}Sc\right)}^{1/3}+\left[\frac{0.037ScR{e}_p^{0.8}}{1+2.44R{e}_p^{-0.1}\left(S{c}^{2/3}-1\right)}\right]$	$R{e}_p\le 1000$ and  $3.3\le \frac{H}{d}\le 43.3$
Deen [127]	$\begin{array}{c}N{u}_p=\left(7-10\varphi +5{\varphi }^2\right)\left(1+0.7R{e}_p^{0.2}P{r}^{1/3}\right)+\\ \left(1.33-2.31\varphi +1.16{\varphi }^2\right)R{e}_p^{0.7}P{r}^{1/3}\end{array}$	$1\le R{e}_p\le 100$ and  $\varphi \ge 0.4$
Sun [128]	$\begin{array}{c}N{u}_p=\left(-0.46+1.77\varphi +0.69{\varphi }^2\right)/{\varphi }^2+\\ \left(1.37-2.4\varphi +1.2{\varphi }^2\right)R{e}_p^{0.7}P{r}^{1/3}\end{array}$	$1\le R{e}_p\le 100$ and  $\varphi \ge 0.5$
Singhal [129]	$N{u}_p=2.67+0.53R{e}_p^{0.77}P{r}^{0.53}$	$9\le R{e}_p\le 180$, and  $\varphi =0.359$

. In their results, Dunkle [121], and Wu and Hwang [122] presented correlations based on experimental data over irregular packed beds using hot air as the HTF. However, Gunn and De Souza [123], and Wakao et al. [124] presented statistical analyses of the experimental data available in the literature. Some researchers [125,126] have conducted experimental mass transfer tests, such as the sublimation of naphthalene in air, because of its advantages over the single-heated sphere method. In this method, a Nusselt correlation of a packed bed is developed according to the analogy of heat transfer and mass transfer. Deen et al. [127], Sun et al. [128] and Singhal [129] developed the gas–solid heat transfer models by applying particle-resolved direct numerical simulations (PR-DNS). These Nusselt correlations based on the simulation data are derived by refitting the Gunn correlation within the narrow range of the Reynolds number.

For convective wall-to-air heat transfer in a packaged bed, most developed Nusselt correlations are valid only for a relatively small Reynolds number. The temperature difference can be measured between the cavity wall temperature and either the extrapolated wall fluid temperature or fluid bulk temperature [130]. A number of the available commonly used wall-to-fluid Nusselt correlations were reviewed and commented by Esence et al. [105] and Calderón-Vásquez et al. [9].

5.3. Advantages and Disadvantages of the Fixed-Bed PSRs

Generally, the fixed-bed PSRs and the fluidized-bed PSRs are the two types of solid particle receivers. Compared with the traditional HTM, solid particles have several benefits, including those of a wide operating temperature range, direct usage as the TES medium, stable chemical properties under high temperature and so on. These characteristics enable a PSR to achieve high-temperature TES exceeding 1300 K and can drive more efficient cycles [131,132,133].

The obvious difference between them is the size of the solid particles. The fixed-bed PSRs use solid sphere (larger than 5.0 mm usually) as solar absorber materials and thermal energy storage materials [8,12,18]. While the fluidized-bed PSRs using solid powder (usually smaller than 1.0 mm) as the working medium [134,135]. Relative to the fluidized-bed PSRs, the fixed-bed PSRs mentioned in this paper have the following specific advantages:

• Realize the gradient absorption of the incident solar radiation to enhance the volumetric effect, and produce higher thermal efficiency and outlet temperature [12,18]. However, the fluidized-bed PSR cannot absorb the CSF with an increasing absorption coefficient due to its flow character, despite the fluidized solid particles directly absorbing the CSF inside the quartz tube [136,137,138].
• The heat transfer from the fixed-bed particles to the HTF is feasible via direct contact. Thereby the fixed-bed PSRs are capable of eliminating the additional heat transfer resistance from particle/fluid heat exchangers. Nevertheless, the particle/fluid heat exchangers are essential in the fluidized-bed PSR system [139,140].
• Reduce the demand of the feeding device because of the stable and flowable characteristics of the sphere particles with rather a large size [141,142,143,144].

Nevertheless, the following disadvantages of fixed-bed PSRs also should be further addressed:

• The sphere particles may be ablated and fractured due to the obvious temperature difference between the interior and the surfaces subjected to the highly concentrated solar flux [18].
• The sphere particles would be abrased and deformed during longtime absorption of the CSF, convective heat transfer with HTF and the conveying operation [8].

6. Challenges and Future Directions

To date, the fixed-bed PSRs proposed are mainly utilizing uniform spheres (glass spheres or ceramic spheres) as heating particles. This design presents several drawbacks for its low specific surface area, including a considerable pressure drop, a strong Wall Effect (the porosity near the wall is higher than the other zone) and high reflection losses. Another challenge is that the radiative properties of the semi-transparent materials that change with temperature are not yet fully understand. Particularly, its degradation law with temperature and/or the times of the heating cycles. Moreover, the efficient numerical approach is indeed essential to characterize the optical-thermo-fluid behavior and enable the development of innovative mechanisms.

Accordingly, the future directions for fixed-bed PSRs, based on the above challenges, could be summarized, outlining an innovative design method, an improved experimental approach and an effective simulation method.

1.

Innovative Design Method

It has been proven that the thermo-physical properties of fixed-bed PSRs could be enhanced by exploring different configurations and shapes of solid particles. For example, Raschig ring particles have better thermo-fluids and optical performances for their higher specific surface area than the sphere particles [145,146]. The packing form with a proper sub-channel can reduce the pressure drop and improve the heat transfer performance compared to randomly packing the particles [147,148,149]. Additionally, size distributions have a remarkable effect on the overall performance of the packed bed, and smaller particles fixed close to the tube wall are beneficial for restraining the Wall Effect and improving heat transfer [150,151].

In addition, reflection losses of incident solar radiation would be decreased obviously when the front surface of the receiver arranges the glass Raschig ring in place of the glass balls, because the reflection area of the glass Raschig ring is much less than that of the glass balls. To a certain extent, a multi-layer design with various functions may potentially enhance the overall thermal performances of fixed-bed PSRs [101]. For instance, the first layer fixed with the glass Raschig ring enables most incident solar radiation to penetrate deep into the receiver. While the second layer utilizes the glass spheres to homogenize the extremely non-uniform incident solar flux to diminish the local hot spot. Finally, the third layer enhances the convection heat transfer between the HTF and the solid phase using high specific surface area materials (e.g., small balls, reticulated porous materials).

In a word, an innovative design method with optimal particle parameters can potentially increase the thermal efficiency and the reliability of fixed-bed PSRs. The following gaps should be addressed by the innovative design through the optimization of shapes, materials and particle sizes associated with structural layouts:

• Minimizing reflection losses of the incident solar radiation.
• Enhanced heat transfer between the gaseous fluid and solid phase.
• Inhibiting re-radiation losses emitted from the high-temperature receiver.

• Improved Experimental Approach

On the one hand, the surface radiative properties (reflectivity, emissivity) of both the glass particles and the ceramic particles are generally insensitive to surface contamination, roughness, and so on. In other words, the assumption that these properties of the particles are equal to those of the bulk materials is not always precise in practice [14]. In addition, glass particles are crucial components of the packed-bed solar receiver that allow the incoming radiation to penetrate deeper within the receiver. However, the absorption coefficient of the glass particles would be degraded with cyclic temperatures through complex relationships. As a result, instrumentation and approaches must be completely improved to measure the actual spectral radiative properties of particles with different temperature orders.

On the other hand, the whole measurement of the fixed-bed PSR prototype is somewhat intricate, which includes a solar simulator, the solar receiver prototype, a working fluid supply system, a temperature measuring device for both the working fluid and the particles, a flowmeter and other auxiliary devices. The measurement results tend to influence each other. Particularly, for high-temperature measurements with a thermocouple (usually a K-type thermocouple), a considerable amount of error between the real temperature and the indicated temperature will exist because of radiation losses. Some unobtrusive deviation, such as the focal plane position deviation, mass flow rate fluctuating and inlet velocity distributions will produce obvious variations in thermo-physical parameters. Therefore, an improved experimental approach is substantially required to provide precise measuring results and insight into the thermo-physical performance as well.

In short, the following issues are necessary to be addressed by the improved experimental approach:

• The radiative properties of solid particles should be accumulated continually and the law of degradation of radiative properties in relation to temperature and cycle time should be established.
• A comprehensive experimental scheme should be proposed to measure the overall thermal performances of fixed-bed PSRs under high temperature.
• The key performances (e.g., thermal efficiency and the outlet temperature) of the experimental approach, as well as their preconditions need to be defined more clearly.

• Effective Simulation Method

As mentioned above, to address the trade-off between computational cost and accuracy, a possible approach is to select only a small representative section to perform the pore-scale simulation, while the other section adopts the homogeneous equivalent simulation. However, the process of selecting the representative section requires further investigation (e.g., pick the temperature section or the maximum temperature gradient section as the representative section). In addition, the way of obtaining the homogenized properties to implement the homogeneous equivalent simulation should be investigated thoroughly.

Moreover, there are several approaches to solve the radiative heat exchange inside the fixed-bed. However, these approaches face the limitations of accuracy and computation effort. Choosing the right approach that provides the best balance between accuracy and computational cost must be discussed in detail. Furthermore, the radiative properties are commonly treated as constant (gray medium) or changing with wavelength (non-gray medium), at most. However, as revealed in the literature [48,49,50], the radiative properties of the semi-transparent materials would vary with temperature over a wide temperature range. An additional calculation cost is required when the variation of the radiative properties with temperature is considered. Furthermore, the existing RTE simulation methodology should also be upgraded to take this variation into account.

In brief, the following challenges are deeply necessary to overcome through an effective simulation method:

• An effective pore-scale simulation method should be developed to produce the best balance between accuracy and computational cost.
• A proper criterion for extracting the representative section used for the pore-scale simulation should be further investigated to simplify the computational domain.
• RTE simulation methods need to be improved to save computation labor and to meet the variation of radiative properties with temperature.

7. Conclusions

To reveal the pivotal heat transfer performance and overcome the trade-off between the working temperature and thermal efficiency, this work presents a review of the radiative heat transfer, as well as the convective heat transfer in high temperature fixed-bed PSRs. Among the various heat transfer processes, the key to improving thermal efficiency of the receiver is to increase the penetration depth of the CSF via semi-transparent solid particles together with sufficient porosity, as well as the convective heat transfer performances between the solid particles and the HTF. The fixed-bed PSRs with a mixed model can further improve the penetration characteristics of the CSF to mitigate the trade-off between the desired outlet temperature and heat losses.

To balance the gaps between the accuracy and the computational cost, the volumetric absorption model was usually adopted for the incident solar radiation and the discrete ordinate method was widely used for radiative heat exchange between the solid particles during the numerical simulation. Simulation results show that the fixed-bed PSRs are capable of offering an effective maximum receiver efficiency of over 70% under a 1000 K outlet temperature. A modular fixed-bed PSR integrated with a 9 MWe gas turbine outperformed, by a 38% decrease, in the levelized cost of electricity relative to the conventional system. Results indicate that the fixed-bed PSRs represent a promising near-term design for cost-effective, high-temperature solar receivers, which can be used in advanced power cycles. Furthermore, the challenges and future directions, including an innovative design method, an improved experimental approach and an effective simulation method are discussed and refined to promote the development of fixed-bed PSRs. In summary, a high temperature fixed-bed PSR is a promising alternative approach for solar energy harvesting, and is useful in mitigating current energy and environmental challenges.