Thermal performance of a single-layer packed metal pebble-bed exposed to high energy fluxes

Abstract

It is difficult to accurately measure the temperature of the falling particle receiver since thermocouples may directly be exposed to the solar flux. This study analyzes the thermal performance of a packed bed receiver using large metal spheres to minimize the measurement error of particle temperature with the sphere temperature reaching more than 700°C in experiments in a solar furnace and a solar simulator. The numerical models of a single sphere and multiple spheres are verified by the experiments. The multiple spheres model includes calculations of the external incidence, view factors, and heat transfer. The effects of parameters on the temperature variations of the spheres, the transient thermal efficiency, and the temperature uniformity are investigated, such as the ambient temperature, particle thermal conductivity, energy flux, sphere diameter, and sphere emissivity. When the convection is not considered, the results show that the sphere emissivity has a significant influence on the transient thermal efficiency and that the temperature uniformity is strongly affected by the energy flux, sphere diameter, and sphere emissivity. As the emissivity increases from 0.5 to 0.9, the transient thermal efficiency and the average temperature variance increase from 53.5% to 75.7% and from 14.3% to 27.1% at 3.9 min, respectively. The average temperature variance decreases from 29.7% to 9.3% at 2.2 min with the sphere diameter increasing from 28.57 mm to 50 mm. As the dimensionless energy flux increases from 0.8 to 1.2, the average temperature variance increases from 13.4% to 26.6% at 3.4 min.

Graphical abstract

Keywords

packed bed / solar thermal power plants / high heat fluxes / radiative heat transfer

1 Introduction

Solar energy is the cleanest and most abundant renewable energy in the world. Solar power plant receivers, an important part of the light-heat conversion process in concentrating solar thermal power plants, directly affect the costs of the power plants.

Solar power plant receivers are classified, based on their working media, as water/steam receivers, molten salt receivers, and air receivers or particle receivers. The allowable energy flux for water/steam receivers is lower than that for molten salt receivers [1]. To achieve higher efficiencies, the medium working temperature needs to be increased which increases the safety requirements for the receivers and the whole system. Molten salt receivers can withstand higher energy fluxes while operating at normal pressures with better safety margins than water/steam receivers [2]. However, molten salts have high solidification temperatures and low decomposition temperatures, and strongly oxidize and corrode the metal walls. Air receivers have wide working temperature ranges and the air will not plug the pipelines under natural conditions. However, the air physical properties change with temperature, and air can easily cause non-uniform flows in the receiver that can eventually result in local overheating. In addition, air receivers cannot withstand the higher energy flux [3]. Particle receivers do not have these disadvantages since the particles are easy to obtain and can withstand high temperatures, and therefore, they can be used as the receiver material [4]. However, particle receivers also have disadvantages such as particle overflow loss and particle agglomeration, and the particles are not easily heated to high temperatures in a single cycle. In addition, the opacity of the falling particle receiver decreases with the increasing drop distance within the receiver, and heat losses increase [5].

Investigations into pebble flow have mainly focused on pebble-bed high temperature gas-cooled reactors with experimental and numerical researches on the pebble motion inside high temperature gas-cooled reactors [6–8]. Wang et al. [9] proposed pebble flow receiver for solar receivers. The spheres flow through the receiver to absorb and transfer thermal energy. The biggest difference between a gas-cooled reactor and a solar receiver is that the spheres are the heat source for a gas-cooled reactor but are the energy absorber material in a solar receiver.

Research on the heat transfer in packed beds can be classified as the unit cell method and the particle scale method depending on the model scale. van Antwerpen et al. [10] reviewed the packing structure and the effective thermal conductivity of randomly packed beds consisting of mono-sized particles, and specially showed the heat transfer by the solid conduction, gas conduction, contact area, surface roughness, and thermal radiation. An effective thermal conductivity correlation based on a unit cell proposed by Zehner and Schlünder [11] was widely applied in simulations of packed beds consisting of mono-sized spheres. Kuipers et al. [12] modified the correlation and combined it with a two-fluid model to simulate the heat transfer in a gas-fluidized bed. Martinek and Ma [13] and Marti et al. [14] used the modified effective thermal conductivity to simulate a near-blackbody (NBB) enclosed particle receiver and a dense gas-particle suspension particle receiver, respectively. The comparison of the experimental results showed that the model proposed by Marti et al. [14] accurately predicted the heat transfer in dense gas-particle systems. Feng et al. [15] presented a simplified discrete thermal element method (DTEM) which used a thermal pipe-network model to link the centers of two spheres without considering the direct heat transfer between contact zones. Oschmann et al. [16] derived an implicit three-dimensional heat transfer model to represent the heat conduction within spherical and non-spherical particles in a fluid. The model considered particle-wall, particle-particle, and particle-fluid-particle conduction, particle-fluid convection, and particle-particle radiation heat transfer. The heat transfer model was verified against resolved CFD simulations. Tsory et al. [17] integrated particle-particle and particle-wall heat transfer models into a discrete element model that considered the effects of surface roughness, and the numerical results were in good agreement with the experimental data. An analytical model for the thermal conductivity of spherical particle beds in a static gas proposed by Slavin et al. [18] considered the effect of surface roughness on the thermal conductivity, and the results conformed to measurements using alumina particles in helium at 100°C–500°C up to a pressure of 100 kPa. Gómez et al. [19] implemented the effective conductivity model of Slavin et al. [18] into ANSYS Fluent and modified the DO radiation model, and obtained a good agreement with the experimental data. Cheng and Yu [20] presented a new approach using a Voronoi network model to model the bed structure and calculate the radiative heat transfer. However, this method needs to calculate the view factors between particles, and therefore, is difficult to apply when the void porosities differ greatly at different positions in the computed field. Wu et al. [21] considered the effect of particle-particle thermal radiation at the particle scale in packed pebble beds and coupled the particle thermal radiation model into a CFD-DEM model with the numerical results in good agreement with the results predicted by other codes. However, the model was not verified for large temperature differences between adjacent particles. Grena [22] developed a three dimensional transient simulation model to investigate a single particle falling and absorbing solar energy, which considered the effects of particle diameter and rotational angular velocity on the temperature gradient inside the sphere.

Because of the disadvantages of modeling particle receivers with small particles, pebble bed receivers have been proposed using larger diameter spheres as a thermal absorber material. Yet, neither study has ever been reported nor thermal efficiency has been compared with that of particle receivers. More than 80% of the absorbed incident flux is absorbed by the first layer of spheres in packed bed solar receivers with the incident flux almost completely absorbed within two diameters [23]. Therefore, the thermal performance of a single-layer packed bed is investigated in this study. Unlike small particles, large particles can be filled with phase change materials to increase the thermal storage and reduce the material weight to reduce the power consumption of the mechanical transport. In addition, according to the Ergun equation [24], the direct heat transfer between large diameter spheres and the fluid (such as supercritical carbon dioxide) can effectively decrease the pressure drop in the packed bed heat exchanger compared with that of smaller particles. This study of the heat transfer in a single-layer packed bed is very necessary. To achieve an ideal high temperature when spheres are away from the receiver, the spheres can be coated with a high-temperature selective absorbing coating to enhance heat absorption, and the sphere motion can be controlled by mechanical transmission or other means to maximize their heat absorption time.

In this paper, the experimental and numerical studies of the heat absorption in a packed metal-pebble bed was described and the temperature characteristics of larger diameter spheres exposed to high fluxes were predicted. The initial experiments were first conducted in a solar furnace and then in a solar simulator. Besides, the Monte Carlo Ray Tracing (MCRT) method was combined with a heat transfer model to form a three-dimensional simulation model for a packed metal pebble-bed receiver.

2 Experimental investigation using a packed bed in a solar furnace

Heat absorption and release experiments were first conducted in a solar furnace, as shown in Fig. 1, to study the heat absorption characteristics of the metal spheres, using the energy flux measurement device, as shown in Fig. 2, the Agilent data acquisition system, and the metal pebble-bed heat absorption test bed, as shown in Fig. 3. The sunlight was concentrated onto the receiver through primary and secondary reflectors, while calibrated thermocouples were placed on the back of the spheres for temperature measurements. The poor tracking accuracy of the primary mirror resulted in the fact that the spot center in the solar furnace did not coincide with the center of the receiver at different times during the day. The energy flux distribution on the receiver was difficult to be measured, which affected the accuracy of the results since the sphere temperatures are closely related to it. The simulation model was validated using the data from an indoor solar simulator.

Based on the assumption of the Gauss distribution energy flux, thermocouples were placed on spheres 2, 4, 5, 7, 10, 12, 13, and 15, as demonstrated in Fig. 4. The sphere temperatures then changed as the direct normal irradiance, DNI, changed in the solar furnace, as depicted in Fig. 5(a). Due to the tracking errors of the solar furnace and the shape of the concentrated spot changing with the change of the relative position between the sun and the ground, spheres 4 and 13 had the lowest temperatures while spheres 7 and 10 had the highest temperatures. Although the distances between the spot center and the receiver center in each experiment were different, some interesting phenomena were still observed in these experiments.

Because the geometric concentrating ratio of the spot center is 1100 and higher than other positions, spheres 7 and 10 near the spot center in Fig. 5(b) were heated more than other spheres while the irradiance was about 50 W/m² at 2.0–2.8 min. The front surfaces of the spheres directly receive the incident solar flux from the secondary mirror while the back surface does not directly receive any solar flux. When the irradiance suddenly drops to 0 W/m² at 2.8 min, a temperature gradient exists inside the sphere. Thermal energy is transferred from the high temperature zone to the low temperature zone by the conduction inside the sphere. The thermocouples are located on the back of the sphere, therefore, the measured temperatures continue to rise during this period and the temperature gradients gradually decrease as the heat losses continue.

As can be seen from Fig. 5(a), sphere 10 has the highest temperature rise at 4–7 min, which indicates that sphere 10 has the maximum difference between the transient energy input and the heat loss. At 8.3–10.3 min, the temperatures of high-temperature spheres decrease while the temperatures of low-temperature spheres increase as the receiver interior tends toward the thermal equilibrium. Based on the physical properties of steel spheres provided by Tan [25], Fig. 6 is obtained by fitting. As exhibited in Fig. 6, when the sphere temperature is less than 755°C, higher temperatures lead to a smaller thermal diffusivity. Therefore, higher temperatures lead to larger temperature gradients. At about 7.6 min, because of the low DNI and the heat loss, the temperatures on the back of all the high-temperature spheres are higher than those on the front about 0.7 min later. So far as low-temperature spheres are concerned, it takes a shorter time for the backside temperatures to become higher than the front temperatures. Although the average irradiance at 9.4–10.2 min is higher than that at 2.0–2.8 min, the temperatures of spheres 2, 7, and 10 are higher than those at 2.0–2.8 min, and the temperatures of spheres 2, 7, and 10 continue to decrease because the incident flux is less than the heat loss from the spheres. However, the temperatures of low-temperature spheres continue to increase due to the thermal radiation from high-temperature spheres and the irradiance from the secondary mirror.

The use of large diameter spheres as the heat absorbing materials will reduce the temperature differences between spheres. The intermittent sunshine or the rotation of sphere around its own axis can improve the temperature uniformity to reduce the heat losses and improve the receiver efficiency.

3 Mathematical model

The simulation model contains the calculation of energy, E_i, absorbed by the spheres from incident solar irradiance; the calculation of view factors, F_i,j; and the calculation of heat transfer. Since only a few spheres are used and the sphere positions are easily determined, E_i and F_i,j are calculated using the Monte Carlo ray tracing method [26].

3.1 Monte Carlo ray tracing method

E_i is the energy absorbed from the light rays and q_i is the radiant heat transfer between the surfaces. The incident energy on each surface changes over time. The Monte-Carlo method is used to calculate E_i and the view factors, F_i,j. The calculation of E_i divides each participating radiant surface i into multiple blocks, and tracks each light ray emitted from the inside surface of the receiver to determine which surface absorbs each beam of light or whether the light ray escapes from the receiver. F_i,j is equal to the ratio of the number of rays reaching surface j to the total number of rays emitted from surface i.

The light emission points in each infinitesimal surface are determined by random numbers. The random numbers R_x, R_y, R_z, R_q, and R_ϕ specify the locations of emission points. R_x, R_y, and R_z specify the locations of the points on rectangular surfaces, and R_θ and R_ϕ specify the locations of the emission points on spherical surfaces.

For a rectangular surface, as shown in Fig. 7(a), z = f(x, y), the location is given by

(1)x=x_{\min }+R_x(x_{\max }-x_{\min }),

(2)y=y_{\min }+R_y(y_{\max }-y_{\min }),

(3)z=z_{\min }+R_z(z_{\max }-z_{\min })\mathrm{.}

For a spherical surface, as shown in Fig. 7(b), (x– O_x)² + (y– O_y)² +(z– O_z)² = R², the location is given by

(4)x=O_x+R\cdot \sin ({\theta }_{\min }+R_{\theta }({\theta }_{\max }-{\theta }_{\min }))\cdot \cos ({\phi }_{\min }+R_{\phi }({\phi }_{\max }-{\phi }_{\min })),

(5)y=O_y+R\cdot \sin ({\theta }_{\min }+R_{\theta }({\theta }_{\max }-{\theta }_{\min }))\cdot \sin ({\phi }_{\min }+R_{\phi }({\phi }_{\max }-{\phi }_{\min })),

(6)z=O_z+R\cdot \cos ({\theta }_{\min }+R_{\theta }({\theta }_{\max }-{\theta }_{\min })),

where O is the center of the sphere, R_θ and R_ϕ also describe the ray emission direction, q is the polar angle, and ϕ is the azimuthal angle. Since the radiation between surfaces is diffuse reflection, q and ϕ are described by

(7)\theta =\arcsin \sqrt{R_{\theta }},

(8)\phi =\mathrm{2}\mathrm{\pi }{R}_{\phi }.

Once the emission position and direction of a ray are determined, the intersections of the ray and the other surfaces can be determined. If the intersection is located on the receiver inlet, the ray escapes and is no longer tracked. The random number R_a is used to determine whether the ray is absorbed when it intersects a surface. If R_a≤e, the ray is absorbed; otherwise, the ray is reflected where e is the emissivity. The calculation process is displayed in Fig. 8. The optical data including the number, directions and locations of the rays are supplied by the solar simulator manufacturer to determine E_i. The structure calculated by using the Monte Carlo ray tracing method is shown in Fig. 9.

3.2 Heat transfer model

There are many heat transfer paths in the receiver as shown in Fig. 10.

The assumptions are ① the large elastic modulus (210 GPa) of the sphere leads to point contact, so mechanisms 1 and 2 in Fig. 10 are neglected; and ② the radiation is the main heat transfer mode in the receiver because the gas thermal conductivity is very small; therefore, mechanisms 4, 5, 6, and 14 in Fig. 10 are neglected.

The heat conduction through a gas layer existing between particles is expressed as [27,28]

(9){\lambda }_{\mathrm{gap}}={\displaystyle {\int }_{r_{\mathrm{1}}}^{r_{\mathrm{2}}}{\displaystyle \frac{\mathrm{2}\pi r\mathrm{d}r}{[\sqrt{R^{\mathrm{2}}-r^{\mathrm{2}}}-r(R+HH)/r_{ij}]\cdot \mathrm{2}/\lambda {}_{\mathrm{s}}+\mathrm{2}[(R+HH)-\sqrt{R^{\mathrm{2}}-r^{\mathrm{2}}}]/{\lambda }_{\mathrm{f}}}}}/{\displaystyle {\int }_{r_{\mathrm{1}}}^{r_{\mathrm{2}}}\mathrm{2}\mathrm{\pi }r\mathrm{d}r},\mathrm{\hspace{0.17em}}HH\le \mathrm{0.5}R,

(10){\lambda }_{\mathrm{gap}}=\mathrm{0},HH>\mathrm{0.5}R.

For sphere-fluid-sphere, the heat conduction coefficient is about 10^-2 W/(m²∙K), which is far less than the radiative heat transfer coefficient. ① The air is treated as a transparent medium in which diffusion and absorption are neglected. Thus, mechanisms 9, 10, and 11 are neglected. ② The receiver outer surface is well insulated. The temperature gradients in the radial and longitudinal directions in the wall are much less than those across the wall. Therefore, a one-dimensional model can be used to simulate the conduction through the wall while the conduction in the radial and longitudinal directions in the wall is neglected. ③ The emissivities of the spheres and walls do not vary with temperature and the spheres and walls can be treated as gray bodies. ④ Cooling air is blown into the receiver, and the forced convection is considered. However, the convective heat transfer inside the receiver is neglected and only the convective heat transfer to the front surface of the spheres is considered. The convective heat transfer coefficient in the front surface of spheres is calculated by Eq. (15).

The heat transfer model is listed in Table 1. All the participating radiant surfaces are divided into multiple elements. The incident rays absorbed by each surface element are calculated by the Monte Carlo ray tracing method which is combined with the energy of each ray to determine E_i. q_i is the radiant heat exchange between surface meshes. These parameters are then applied as the boundary conditions for solving the energy equations for the spheres and walls.

4 Model verification

Figure 11 shows the energy flux distribution used for the model verification in the solar simulator where the cross represents the center of the target. Figure 12 displays the solar simulator and the experimental bench. Using the metal balls with a diameter of 5 cm as endothermic materials, the properties of metal balls (S45C) are expressed by Eqs. (25–28) [25].

(25){\lambda }_{\mathrm{s}}=\mathrm{1.811}\times {\mathrm{10}}^{-\mathrm{10}}{T}^{\mathrm{4}}-\mathrm{5.477}\times {\mathrm{10}}^{-\mathrm{7}}{T}^{\mathrm{3}}+\mathrm{5.911}\times {\mathrm{10}}^{-\mathrm{4}}{T}^{\mathrm{2}}-\mathrm{0.292}T+\mathrm{95.043},\mathit{T}\le \mathrm{1}\mathrm{0}\mathrm{73\; K},

(26)

(27)c_{\mathrm{ps}}=\mathrm{2.823}\times {\mathrm{10}}^{-\mathrm{11}}{T}^{\mathrm{5}}-\mathrm{8.364}\times {\mathrm{10}}^{-\mathrm{8}}{T}^{\mathrm{4}}+\mathrm{9.641}\times {\mathrm{10}}^{-\mathrm{5}}{T}^{\mathrm{3}}-\mathrm{0.053}{T}^{\mathrm{2}}+\mathrm{14.15}T-\mathrm{975.67},\mathit{T}\le \mathrm{1}\mathrm{0}\mathrm{28\; K},

(28)

Experiments were first conducted with a single sphere to measure the absorbed thermal energy as shown in Fig. 13. There are several energy input paths to the sphere surface: the incident solar flux from the solar simulator; the radiant heat transfer from the walls; and the radiant and convective heat transfer between the sphere and the environment. The boundary condition, Eq. (29), on the front surface was implemented in Fluent [30] using a user-defined function with the back surface assumed to be adiabatic. Figure 13 shows that the predicted temperature variations with time are in good agreement with the measured temperatures from the thermocouples located on the back of the sphere. The maximum error is 23°C at 8.87 min with an error of 3.29%. Before the experiment started, the sphere had a uniform temperature distribution. Energy was then conducted from the front to the measuring points on the back. At 0–3 s, the energy has not yet reached the measuring point whose temperature is constant. As the experiment continues, the energy is conducted to the back, and the temperature of the measuring point gradually increases. However, as the temperature increases, the convection and radiation heat losses of the sphere increase and the energy absorbed by the sphere per unit time decreases, and the slope gradually decreases.

(29){\lambda }_{\mathrm{s},i}{\displaystyle \frac{\partial T_{\mathrm{s},i}}{\partial r}}{|}_{r=R}={\displaystyle \frac{E_i}{A_i}}-h_{\mathrm{s}}(T_{\mathrm{s},i}{|}_{r=R}-T_{\mathrm{a}})-\epsilon \sigma ({(T_{\mathrm{s},i}{|}_{r=R})}^{\mathrm{4}}-T_{\mathrm{a}}^{\mathrm{4}}),

(30)T_{\mathrm{s},i\_\mathrm{av}}={\displaystyle \frac{\mathrm{1}}{V}}{\displaystyle \sum (T_{\mathrm{s},i}\mathrm{\Delta }V)}.

The predicted and measured temperatures for multiple spheres are compared in Fig. 14. The average temperatures are calculated by Eq. (30). Sphere 6, 8, and 11 are used to validate the model due to their higher temperatures. The surface emissivity increases when the surface oxidation occurs [31]. The simulation assumed that the emissivity of the sphere surfaces was constant. Therefore, the spheres absorbed more thermal energy in the predictions than in the experiments. At the end of the experiment, the maximum error occurs in the temperature of sphere 11 with an error of 4.36%. Therefore, the model can be used to investigate the temperature characteristics of a packed metal pebble-bed receiver.

5 Results and discussion

The effects of the convection inside the receiver and the metal structure that holds the spheres were not considered. The transient thermal efficiency, the temperature variance, and the maximum temperature difference inside the highest-temperature sphere were used to analyze the effects of various design parameters. The equations are

(31)\eta ={\displaystyle \frac{{\displaystyle \sum _{i=\mathrm{1}}^{i=n}{m}_i(c_{\mathrm{p},i}(T_{\mathrm{s},i}^k)\cdot T_{\mathrm{s},i}^k-c_{\mathrm{p},i}(T_{\mathrm{s},i}^{k-\mathrm{1}})\cdot T_{\mathrm{s},i}^{k-\mathrm{1}}})}{\mathrm{d}\tau \cdot {\displaystyle \underset{-W/\mathrm{2}}{\overset{W/\mathrm{2}}{\int }}{\displaystyle \underset{-H/\mathrm{2}}{\overset{H/\mathrm{2}}{\int }}{q}_{\mathrm{solar}}\mathrm{d}x\mathrm{d}y}}}},

(32)D(T_{\mathrm{s\_av}}/\mathrm{273})=\left({\displaystyle \sum _{i=\mathrm{1}}^{i=n}{(T_{\mathrm{s},i\_\mathrm{av}}/\mathrm{273})}^{\mathrm{2}}}\right)/n-{\left(\left({\displaystyle \sum _{i=\mathrm{1}}^{i=n}(T_{\mathrm{s},i\_\mathrm{av}}/\mathrm{273})}\right)/n\right)}^{\mathrm{2}}.

5.1 Effect of ambient temperature

The ambient temperature affects the heat loss. For a given incident energy flux and initial sphere temperature, changes in the ambient temperature have little effect on the maximum temperature difference inside the highest temperature sphere, the transient thermal efficiency, and the temperature uniformity as shown in Fig. 15. As the ambient temperature rises, the radiation heat loss from the receiver to the ambient decreases. Therefore, the transient thermal efficiency slightly increases. As the ambient temperature increases from 263 K to 303 K, the transient thermal efficiency increases from 69.7% to 69.9% at 4.45 min. The initial temperatures in the spheres and the wall were higher than the ambient temperature. Therefore, in the early stage, the heating rate at the back of the spheres was relatively slow and then increased, while that at the front of the spheres was initially fast and then slowed.

The difference between the incident energy flux and the radiation heat loss determines the change in the transient thermal efficiency. Little energy falls on the walls, so the wall temperatures in the early stage are very low. The heat gains of the spheres from the walls are less than their heat losses; therefore, the transient thermal efficiency decreases. Since the wall thermal conductivity is very low and the thermal inertia in the walls is far less than that in the spheres, the wall temperatures increase faster than the sphere temperatures. When the heat transfer to the spheres from the walls and the adjacent spheres is larger than the radiation heat loss, the transient thermal efficiency increases. However, after 0.7 min, the wall temperature slightly increases and the radiation heat loss increases due to the higher sphere temperatures. Therefore, the transient thermal efficiency decreases.

The decrease in the ambient temperature increases the radiation heat loss, convective loss, and conduction loss. Thus, lower ambient temperatures lead to lower thermal efficiencies and the receiver thermal efficiency can be increased by the better insulation.

5.2 Effect of sphere thermal conductivity

The thermal diffusion coefficient affects the heat transfer inside the sphere, and the thermal conductivity is proportional to the thermal diffusion coefficient. As the temperature rises from 0°C to 755°C, the thermal conductivity of the metal sphere drops from 50.0 to 25.0 W/(m∙K). As shown in Fig. 16, when the specific heat and the energy flux are constant, a larger thermal conductivity reduces the temperature gradient in the sphere and the transient thermal efficiency decreases. The change of the thermal conductivity does not affect the sphere temperature uniformity. As the dimensionless thermal conductivities increases from 0.6 to 1.0, the transient thermal efficiency decreases from 72.3% to 71.5% at 3.58 min, and the difference between the maximum temperature and the minimum temperature in the highest temperature sphere decreases from 271 K to 165 K. At the same time, since the maximum temperature increases and the minimum temperature decreases, a smaller thermal conductivity leads to higher radiation heat losses. The heat transfer between the spheres and the walls increases with decreasing thermal conductivity of the spheres. Since a lower thermal conductivity reduces the total heat loss, the lower thermal conductivity increases the transient thermal efficiency.

5.3 Effect of energy flux

The solar irradiance and the reflectivity of the mirrors affect the energy flux that eventually arrives at the receiver inlet in the solar furnace. However, the electrical current controls the energy flux in the solar simulator. This section model changes in the energy flux by being multiplied by a dimensionless coefficient. As shown in Fig. 17, increasing the incident energy flux leads to a less uniform temperature distribution in the receiver. At 0–0.65 min, as the energy flux increases, the transient thermal efficiency increases. After 0.8 min, the radiation losses from the spheres increase and the transient thermal efficiency decreases with increasing energy flux. As the dimensionless energy flux increases from 0.8 to 1.2, the transient thermal efficiency decreases from 71.9% to 71.7% at 3.4 min, the average temperature variance increases from 13.4% to 26.6%, and the difference between the maximum temperature and the minimum temperature in the highest temperature sphere increases from 126 K to 205 K. Although the transient thermal efficiency for high energy fluxes is smaller than that for low energy fluxes, there are still more total heat gains at higher energy fluxes. With increasing energy flux, the radiation heat loss per unit time increases, but the time required to achieve the same heat gain decreases, and so does the total heat loss.

5.4 Effect of sphere diameter

The sphere arrangement in the receiver is shown in Fig. 18. As the diameters increase, the locations and total mass of the spheres change and the receiver becomes thicker since the thickness is equal to the sphere diameter. As shown in Fig. 19, smaller diameters of spheres lead to faster heating rates. Since the total surface area of all the spheres and the ratio of the external incidence absorbed by the spheres are constant as the diameter of spheres changes, the heat losses increase, the transient thermal efficiency decreases, and the average temperature distribution becomes less uniform. As the sphere diameter increases from 28.57 mm to 50 mm, the transient thermal efficiencies increases from 69.0% to 73.4% at 2.2 min, and the average temperature variance decreases from 29.7% to 9.3%, but the difference between the maximum temperature and the minimum temperature in the highest temperature sphere increases from 122 K to 195 K. If the receiver size is constant, smaller diameter spheres should be used in a multilayer packed metal pebble-bed reactor, and the ratio of the incident radiation absorbed by the spheres to the total solar irradiance and the thermal performance will change.

5.5 Effect of sphere surface emissivity

The reflection from the spheres is assumed to be diffuse reflection with the spheres treated as gray bodies. As shown in Fig. 20, a larger emissivity leads to less energy reflected. The transient thermal efficiency increases as the emissivity increases, but the temperature tends to be less uniform. As the sphere emissivity increases from 0.5 to 0.9, the transient thermal efficiencies increases from 53.5% to 75.7% at 3.9 min, the difference between the maximum temperature and the minimum temperature in the highest temperature sphere increases from 102 K to 190 K, and the average temperature variance increases from 14.3% to 27.1%. When the emissivity is greater than 0.75, the fraction of the incident energy flux absorbed by the spheres is lower than the emissivity. The ratio of the fraction of the incident energy absorbed by the spheres to the emissivity decreases as the emissivity increases. When the emissivity is less than 0.63, the maximum transient thermal efficiency is higher than the emissivity. Thus, increasing the absorptivity is the most important path to improving the thermal efficiency of the receiver.

6 Conclusions

The thermal performances of a single-layer packed bed and a single sphere were experimentally and numerically studied. The model coupled the Monte Carlo ray tracing method and a heat transfer model to investigate the effects of the parameters on the temperature gradient in the highest temperature sphere, the transient thermal efficiency, and the temperature uniformity in the single-layer packed metal pebble-bed receiver.

The experiments show that the direct normal irradiance strongly affects the temperature rise of the spheres in the receiver. There is a temperature gradient inside each sphere, and the temperature gradient changes with changes of the irradiance. Higher fluxes lead to faster heating of the spheres.

When convection is not considered, the numerical results show that the ambient temperature only weakly influences the thermal performance. As the dimensionless thermal conductivity increases from 0.6 to 1.0, the temperature gradient (the difference between the maximum temperature and the minimum temperature in the highest temperature sphere at 3.58 min: 0.6 271 K, 1.0 165 K) and the transient thermal efficiency (at 3.58 min: 0.6 72.3%, 1.0 71.5%) decrease, and the temperature uniformity is unchanged. With the dimensionless energy flux increasing from 0.8 to 1.2, the temperature gradient (at 3.4 min: 0.8 126 K, 1.2 205 K) increases and the overall temperature distribution (the average temperature variance at 3.4 min: 0.8 13.4%, 1.2 26.6%) becomes less uniform. A higher energy flux gives a higher transient thermal efficiency (at 1.0 min: 0.8 74.1%, 1.2 74.8%) during the initial heating period and then a lower transient thermal efficiency (at 3.4 min: 0.8 71.9%, 1.2 71.7%) as the system approaches steady-state. A larger sphere diameter reduces the transient thermal efficiency (at 2.2 min: 28.57 mm 69.0%, 50 mm 73.4%) and the temperature gradient (at 2.2 min: 28.57 mm 122 K, 50 mm 195 K) but improves the temperature uniformity (at 2.2 min: 28.57 mm 29.7%, 50 mm 9.3%). A higher emissivity increases the temperature gradient (at 3.9 min: 0.5 102 K, 0.9 190 K) and the transient thermal efficiency (at 3.9 min: 0.5 53.5%, 0.9 75.7%) and makes the temperature distribution (at 3.4 min: 0.5 14.3%, 0.9 27.1%) less uniform.

According to these analyses, the spheres with a larger thermal conductivity and a larger absorptivity should be used as the thermal absorber material. The spheres with a smaller diameter can effectively reduce the thermal absorption time to reach the same high temperature. The velocities of the spheres far away from the spot center in the horizontal direction should be slowed down to increase the outlet sphere temperatures. In addition, the spin of the spheres and the mixing of the spheres in the horizontal direction inside the receiver should be reinforced to maximize the thermal efficiency. The mixing of the spheres represents that the spheres with lower temperatures can flow into the higher energy flux zone to obtain the thermal energy, hence the outlet sphere temperatures of the receiver tend to be more uniform.

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