Investigation of Effective Thermal Conductivity for Ordered and Randomly Packed Bed with Thermal Resistance Network Method

Abstract

In the present paper, the effective thermal conductivities of Li₄SiO₄-packed beds with both ordered and random packing structures were investigated using thermal resistance network methods based on both an Ohm’s law model and a Kirchhoff’s law model. The calculation results were also validated and compared with the numerical and experimental results. Firstly, it is proved that the thermal resistance network method based on the Kirchhoff’s law model proposed in the present study is reliable and accurate for prediction of effective thermal conductivities in a Li₄SiO₄-packed bed, while the results calculated with the Ohm’s law model underestimate both ordered and random packings. Therefore, when establishing a thermal resistance network, the thermal resistances should be connected along the main heat transfer direction and other heat transfer directions as well in the packing unit. Otherwise, both the total heat flux and effective thermal conductivity in the packing unit will be underestimated. Secondly, it is found that the effect of the packing factor is remarkable. The effective thermal conductivity of a packed bed would increase as the packing factor increases. Compared with random packing at similar packing factor, the effective thermal conductivity of packed bed would be further improved with an ordered packing method.

1. Introduction

Nuclear power would have great potential to provide sufficient energy to satisfy mounting demand and can be used sustainably with relatively small impact on the environment. In 2006, seven main countries or areas, including USA, EU, Russia, Japan, China, India and South Korea, joined the international thermonuclear experimental reactor program (ITER), and the Chinese helium-cooled solid breeder test blanket module (CN HCSB TBM) is one of the most important parts of the ITER test-object program. In CN HCSB TBM, Li₄SiO₄ ceramic particles with diameter of about 0.5–1.0 mm were adopted as tritium breeders in the test, and this tritium breeder would have many advantages for a nuclear fusion reactor, such as high lithium content, low neutron activation rate, high tritium release rate under low temperature conditions, etc. Since the effective thermal conductivity of a Li₄SiO₄-packed bed is quite important for the design of the test blanket module, it is necessary to predict the effective thermal conductivity accurately. In the test blanket module (CN HCSB TBM), the Reynolds number for the purge gas (helium) is very low and the gas velocity is usually under 0.001 m/s. Therefore, the thermal convection in the Li₄SiO₄-packed bed would be negligible. Furthermore, since the temperature difference in the Li₄SiO₄-packed bed is relatively small, the thermal radiation would also be small, and the heat conduction in the Li₄SiO₄-packed bed should be dominant.

In recent years, research on the effective thermal conductivity of a packed bed has been popular, including experimental measurements, numerical simulations, etc. For example, Donne et al. [1] have measured the effective conductivity of a Li₄SiO₄ pebble bed, which was performed at Karlsruhe Research Center. The pebble bed is contained between two concentric tubes of stainless steel with a height of 530 mm. The outer diameter of the inner tube and the inner diameter of the outer tube were 16 mm and 51 mm, respectively. The effective thermal conductivity of a lithium orthosilicate pebble bed with packing factor of about 65% was measured. They found that, the effective thermal conductivity was a linear function of the mean temperature from 0 °C to 900 °C in the Li₄SiO₄ pebble bed. Mandal et al. [2] have experimentally measured the effective thermal conductivity of a lithium-titanate pebble bed at variable gas flow rates and different bed wall temperatures. They found that the effective thermal conductivity of the pebble bed was a function of the particle Reynolds number and temperature. The effective thermal conductivities of Li₂TiO₃- and Li₂ZrO₃-packed beds were also experimentally measured by Hatano et al. [3] and Earnshaw et al. [4], respectively. As for numerical simulations, these provide another way to predict the effective thermal conductivity of packed bed, which mainly includes the finite element method (FEM) and finite volume method (FVM). Panchal et al. [5] have numerically studied the effective thermal conductivities in both ordered and randomly Li₂TiO₃-packed beds with the FEM method. In their study, the numerical results of randomly packed bed could agree well with the experimental results of Hatano et al. [3], where the maximal deviation between the numerical and experimental results was about 9%. Wang et al. [6] have numerically studied the effective thermal conductivity of different ordered packings with the FVM method. The effects of different parameters were investigated, including properties of particle and gas materials, bed porosity, particle size, gas flow rate and particle-to-particle contact area. In their study, the numerical results of ordered packed beds agreed well with the published results. Chen et al. [7] have simulated heat transfer in unitary pebble beds with computational fluid dynamics (CFD) and discrete element method (DEM) in turn to evaluate the effective thermal properties. Based on comparison with existing experimental data in the literature, their numerical results for the effective thermal conductivity could agree well with lithium ceramic pebble bed experiments within a deviation of 20%. Later, Chen et al. [8] also simulated heat transfer for the mono-sized beryllium pebble bed as well as the multi-sized Li₂TiO₃/Be1₂Ti pebble bed to evaluate the effective thermal conductivity. Numerical simulation has proved to be available and accurate to predict the effective thermal conductivity of a packed bed, even if it is sometimes CPU time consuming. Meanwhile, in recent years, it has also been popular to evaluate the effective thermal conductivity of a packed bed with the thermal resistance network method, which would be highly efficiency and time-saving. Zhao et al. [9] and Wang et al. [6] have proposed a theoretical method, based on thermal resistance network, to predict the effective thermal conductivity of a Li₄SiO₄-packed bed. Their theoretical results were obtained from some ordered packing structures, such as simple cubic packing (SC) or other simplified packing models. The thermal resistance network has also been used by Mandal and Gupta [10] to evaluate the effective thermal conductivity of a randomly packed bed in a cylindrical tube. They found that the effective thermal conductivity of a packed bed would depend on particle size and air film thickness. Antwerpen et al. [11] have calculated the effective thermal conductivity in randomly packed pebble beds of mono-sized spheres with a thermal resistance network. In their studies, the porous structure, solid and gas thermal conduction contact area, surface roughness and thermal radiation were fully considered. However, in these studies [6,9,10,11], the thermal resistance networks for the related packed beds would be oversimplified, where the infinitesimal thermal resistances were only connected with each other along the main heat transfer direction and strictly separated along other directions in the packed bed. This may lead to extra deviations. Moreover, some other related studies for prediction of effective thermal conductivity of packed beds can also be found in the recent literature [12,13,14].

In the present study, two kinds of thermal resistance networks were adopted to predict the effective thermal conductivity of Li₄SiO₄ packed beds. One is the thermal resistance network based on the Ohm’s law, where the infinitesimal thermal resistances were only connected with each other along the main heat transfer direction and separated along other directions. This thermal resistance network is similar to studies in the literature [6,9]. Based on this simplified thermal resistance network, another more elaborated thermal resistance network based on Kirchhoff’s law was proposed, where the infinitesimal thermal resistances were not only connected along the main heat transfer direction, but also connected along other directions in the packed bed. Both ordered and random packing structures were constructed for the present investigations. For ordered packings, including simple cubic (SC), body centre cubic (BCC) and face centre cubic (FCC) packings, the CFD simulations were also performed for the comparisons. For random packings, ten typical random packing units were selected for the investigations and the experimental results as reported in the open literatures were adopted for the comparisons. According to the authors’ best knowledge, almost no studies have been performed on the prediction of the effective thermal conductivity for Li₄SiO₄-packed beds with a thermal resistance network of both Ohm’s law model and Kirchhoff’s law model to date, and the results will be meaningful for the thermal design and analysis of a helium-cooled solid breeder blanket.

2. Physical Model and Material Thermal Conductivity

2.1. Physical Model

In the present study, both ordered and random packings were constructed for the investigations. As for ordered packings, three different kinds of ordered packing structures were constructed, including simple cubic (SC), body center cubic (BCC) and face center cubic (FCC) packings. The images for different ordered packing units are presented in Figure 1, and the typical packing parameters are listed in

Table 1. Typical packing parameters for ordered packing units.

Packing	dp (mm)	Unit Size	Packing Factor (%)
SC	0.5	1dp × 1dp × 1dp	50.8
BCC	0.5	1.155dp × 1.155dp × 1.155dp	66.0
FCC	0.5	1.414dp × 1.414dp × 1.414dp	71.9

. As shown in Figure 1, the SC packing unit consists of eight 1/8 particles at eight cubic corners, the BCC packing unit consists of eight 1/8 particles at eight cubic corners and one full particle at the cubic center, the FCC packing unit consists of eight 1/8 particles at eight cubic corners and four 1/2 particles at four cubic face centers. In Table 1, it should be noted that the packing factor of FCC packing is the highest and it is the lowest for SC packing. Here, the packing factor means the volume ratio of solid particles in the packing unit.

As for random packing, it was generated with discrete element method (DEM) as reported in literatures [15,16,17]. 1000 particles were fallen freely from a high plane to a square container with dimensions of 10d_p × 10d_p × 20d_p, where the container’s height is 20d_p. When the maximal speed of particles is lower than 10⁻⁶ m/s, the packed bed was considered to be in steady state. As shown in Figure 2a, the dimensions of the full size randomly packed bed are about 10d_p × 10d_p × 10d_p. For the current computational condition, direct simulation of heat transfer in the full size randomly packed bed with so many particles is quite difficult. Therefore, representative packing units with different dimensions were selected for the present study, which is shown in Figure 2b–d, where the grey areas are filled with helium gas. Ten typical random packing unites were extracted from the full size randomly packed bed, and their detailed locations, packing factors and unit sizes are listed in

Table 2. Typical packing parameters for random packing units.

Packing Unit	x (mm)	y (mm)	z (mm)	dp (mm)	Packing Factor (%)	Unit Size
Unit-1	1.0–1.5	1.0–1.5	1.0–1.5	0.5	62.67	1dp × 1dp × 1dp
Unit-2	1.5–2.0	1.5–2.0	2.0–2.5	0.5	65.32
Unit-3	2.0–2.5	3.0–3.5	3.5–4.0	0.5	68.73
Unit-4	2.5–3.0	2.0–2.5	2.5–3.0	0.5	63.11
Unit-5	3.0–3.5	3.5–4.0	1.5–2.0	0.5	65.82
Unit-6	1.0–2.0	1.5–2.5	3.0–4.0	0.5	63.99	2dp × 2dp × 2dp
Unit-7	1.5–2.5	2.0–3.0	2.5–3.5	0.5	68.88
Unit-8	2.0–3.0	3.0–4.0	1.0–2.0	0.5	65.56
Unit-9	1.5–3.0	1.5–3.0	1.5–3.0	0.5	64.17	3dp × 3dp × 3dp
Unit-10	2.0–3.5	1.0–2.5	2.5–4.0	0.5	65.05

. It should be noted that the dimensions of unit-1 to unit-5 are the same, which is 1d_p × 1d_p × 1d_p. The dimensions of unit-6 to unit-8 are the same, which is 2d_p × 2d_p × 2d_p. The dimensions of unit-9 to unit-10 are the same, which is 3d_p × 3d_p × 3d_p. In order to avoid the wall effect caused by the container, the packing unit should not be too close to the container wall, and the distance should be kept at least 2d_p away from the container wall in the present study. The average packing factor of these 10 packing units is 65.33%, which agrees well with the experimental data (about 65%) as reported by Donne et al. [1].

2.2. Material Thermal Conductivity

In Li₄SiO₄-packed bed, the effective thermal conductivity is significantly affected by the thermal conductivities of Li₄SiO₄ particle and helium gas. The thermal conductivities of Li₄SiO₄ particle (k_s) and helium gas (k_f) as reported in References [9,18] are calculated as follows:

$$\{\begin{array}{l}{k}_{\mathrm{s}}=\frac{(1.98+850/T)(1-\epsilon )}{1+\epsilon (1.95-8\times {10}^{-4}T)}\hspace{1em}\hspace{1em}{\mathrm{Li}}_4{\mathrm{SiO}}_4\mathrm{particle}\\ k_{\mathrm{f}}=2.774\times {10}^{-3}{T}^{0.701}\hspace{1em}\mathrm{Helium} \mathrm{gas}\end{array}$$

(1)

where T is the mean temperature of packed bed. ε is the porosity of pebble.

The variations of thermal conductivities for Li₄SiO₄ particles and helium gas are presented in Figure 3. It shows that, as temperature increases, the thermal conductivity of Li₄SiO₄ particle (k_s) decreases, while the conductivity of helium gas (k_f) increases.

3. Thermal Resistance Network and Computational Fluid Dynamics (CFD) Methods

3.1. Thermal Resistance Networks Based on Ohm’s Law and Kirchhoff’s Law

In the present study, two kinds of thermal resistance networks were developed to predict the effective thermal conductivity of a Li₄SiO₄ packing unit, which is shown in Figure 4. Based on the thermal-electrical analogy principle, both Ohm’s law and Kirchhoff’s law were adopted to establish the equivalent thermal resistance networks for the packing units. As shown in Figure 4, the temperatures on the top and bottom surfaces of the packing unit are fixed at T_t and T_b, respectively, and the temperature difference is assumed to be small (such as 10 K). The other four surfaces of the packing unit are kept adiabatic. Furthermore, in the present study, since the gas velocity in the Li₄SiO₄ packing unit is assumed to be very slow and the temperature difference in the packing unit is relatively small, the convection and radiation heat transfer in the packing unit are not considered. Therefore, only the conduction heat transfer is calculated in the packing unit with the equivalent thermal resistance networks.

In both the thermal resistance networks of the Ohm’s law model and Kirchhoff’s law model, the packing unit was divided into small thermal resistance elements with total nodes of I × J × K. As for the Ohm’s law model (see Figure 4a), the thermal resistances were only connected with each other along the main heat transfer direction (z direction). While for the Kirchhoff’s law model (see Figure 4b), the heat transfer normal to the main heat transfer direction was also considered, and the thermal resistances were connected with each other along the x, y and z directions. For both thermal resistance networks, the black element (R_s) represents the thermal resistance inside the Li₄SiO₄ particle, and the white element (R_f) represents the thermal resistance in the helium gas. Therefore, the black-white element (R_m) should represent the thermal resistance between Li₄SiO₄ particle and helium gas. The definitions for the thermal resistances of R_s, R_f and R_m are formulated as follows:

$$R_{\mathrm{s}}=\frac{{\delta }_{\mathrm{element}}}{k_{\mathrm{s}}{A}_{\mathrm{element}}},R_{\mathrm{f}}=\frac{{\delta }_{\mathrm{element}}}{k_{\mathrm{f}}{A}_{\mathrm{element}}},R_{\mathrm{m}}=\frac{R_{\mathrm{s}}+R_{\mathrm{f}}}{2}$$

(2)

where k_s and k_f are the thermal conductivities of Li₄SiO₄ particle and helium gas, respectively. δ_element is the thickness in the thermal resistance element along the heat transfer direction. A_element is the area in the thermal resistance element normal to the heat transfer direction.

Based on above analysis, the total thermal resistance (R_total) in the packing unit for the Ohm’s law model (see Figure 4a) is calculated as follows:

$$R_{\mathrm{total}}={({\displaystyle \sum _{i=1}^I{\displaystyle \sum _{j=1}^J\frac{1}{{\displaystyle \sum _{k=2}^K{R}_{\mathrm{z}}(i,j,k)}}}})}^{-1}$$

(3)

where R_z is the thermal resistance element along the main heat transfer direction (z direction). And the effective thermal conductivity (k_eff) of the packing unit for the Ohm’s law model would be expressed as follows:

$$k_{\mathrm{eff}}=\frac{{\delta }_{\mathrm{unit}}}{R_{\mathrm{total}}{A}_{\mathrm{unit}}}$$

(4)

where δ_unit is the total thickness of the packing unit along the main heat transfer direction. A_unit is the total cross-section area in the packing unit normal to the main heat transfer direction.

As for the thermal resistance network of the Kirchhoff’s law model (see Figure 4b), according the energy conservation principle, the energy equation at each internal node of the network can be formulated as follows:

$$\begin{array}{l}[\frac{T(i-1,j,k)-T(i,j,k)}{R_{\mathrm{x}}(i,j,k)}+\frac{T(i+1,j,k)-T(i,j,k)}{R_{\mathrm{x}}(i+1,j,k)}]+[\frac{T(i,j-1,k)-T(i,j,k)}{R_{\mathrm{y}}(i,j,k)}\\ +\frac{T(i,j+1,k)-T(i,j,k)}{R_{\mathrm{y}}(i,j+1,k)}]+[\frac{T(i,j,k-1)-T(i,j,k)}{R_{\mathrm{z}}(i,j,k)}+\frac{T(i,j,k+1)-T(i,j,k)}{R_{\mathrm{z}}(i,j,k+1)}]=0\\ (i=2, 3\dots , I-1; j=2, 3\dots , J-1; k=2, 3\dots , K-1)\end{array}$$

(5)

where R_x, R_y and R_z are the thermal resistance element along the x, y and z directions.

The energy equation at each boundary node of the network can be formulated as follows:

$$\{\begin{array}{l}\mathrm{For} \mathrm{the} \mathrm{adiabatic} \mathrm{boundary} \mathrm{at} i=1 (j=2, 3\dots , J-1; k=2, 3\dots , K-1)\\ 2[\frac{T(2,j,k)-T(1,j,k)}{R_{\mathrm{x}}(2,j,k)}]+[\frac{T(1,j-1,k)-T(1,j,k)}{R_{\mathrm{y}}(1,j,k)}+\frac{T(1,j+1,k)-T(1,j,k)}{R_{\mathrm{y}}(1,j+1,k)}]\\ +[\frac{T(1,j,k-1)-T(1,j,k)}{R_{\mathrm{z}}(1,j,k)}+\frac{T(1,j,k+1)-T(1,j,k)}{R_{\mathrm{z}}(1,j,k+1)}]=0\\ \mathrm{For} \mathrm{the} \mathrm{adiabatic} \mathrm{boundary} \mathrm{at} i=I (j=2, 3\dots , J-1; k=2, 3\dots , K-1)\\ 2[\frac{T(I-1,j,k)-T(I,j,k)}{R_{\mathrm{x}}(I,j,k)}]+[\frac{T(I,j-1,k)-T(I,j,k)}{R_{\mathrm{y}}(I,j,k)}+\frac{T(I,j+1,k)-T(I,j,k)}{R_{\mathrm{y}}(I,j+1,k)}]\\ +[\frac{T(I,j,k-1)-T(I,j,k)}{R_{\mathrm{z}}(I,j,k)}+\frac{T(I,j,k+1)-T(I,j,k)}{R_{\mathrm{z}}(I,j,k+1)}]=0\end{array}$$

(6)

$$\{\begin{array}{l}\mathrm{For} \mathrm{the} \mathrm{adiabatic} \mathrm{boundary} \mathrm{at} j=1 (i=2, 3\dots , I-1; k=2, 3\dots , K-1)\\ [\frac{T(i-1,1,k)-T(i,1,k)}{R_{\mathrm{x}}(i,1,k)}+\frac{T(i+1,1,k)-T(i,1,k)}{R_{\mathrm{x}}(i+1,1,k)}]+2[\frac{T(i,2,k)-T(i,1,k)}{R_{\mathrm{y}}(i,2,k)}]\\ +[\frac{T(i,1,k-1)-T(i,1,k)}{R_{\mathrm{z}}(i,1,k)}+\frac{T(i,1,k+1)-T(i,1,k)}{R_{\mathrm{z}}(i,1,k+1)}]=0\\ \mathrm{For} \mathrm{the} \mathrm{adiabatic} \mathrm{boundary} \mathrm{at} j=J (i=2, 3\dots , I-1; k=2, 3\dots , K-1)\\ [\frac{T(i-1,J,k)-T(i,J,k)}{R_{\mathrm{x}}(i,J,k)}+\frac{T(i+1,J,k)-T(i,J,k)}{R_{\mathrm{x}}(i+1,J,k)}]+2[\frac{T(i,J-1,k)-T(i,J,k)}{R_{\mathrm{y}}(i,J,k)}]\\ +[\frac{T(i,J,k-1)-T(i,J,k)}{R_{\mathrm{z}}(i,J,k)}+\frac{T(i,J,k+1)-T(i,J,k)}{R_{\mathrm{z}}(i,J,k+1)}]=0\end{array}$$

(7)

$$\{\begin{array}{l}\mathrm{For} \mathrm{the} \mathrm{fixed} \mathrm{temperature} \mathrm{boundary} \mathrm{at} k=1, K\\ T(i,j,1)=T_{\mathrm{b}},\hspace{1em}T(i,j,K)=T_{\mathrm{t}}\\ (i=2, 3\dots , I-1;\hspace{1em}j=2, 3\dots , J-1)\end{array}$$

(8)

The energy equations at internal and boundary nodes (Equations (5)–(8)) are solved with Gauss Seidel iteration method, and the residual of the iteration is less than 10⁻⁶. Finally, the effective thermal conductivity (k_eff) of the packing unit for the Kirchhoff’s law model would be expressed as follows:

$$k_{\mathrm{eff}}=\frac{\varphi {\delta }_{\mathrm{unit}}}{A_{\mathrm{unit}}(T_{\mathrm{t}}-T_{\mathrm{b}})}\hspace{1em}(\varphi ={\displaystyle \sum _{i=1}^I{\displaystyle \sum _{j=1}^J\frac{T_{\mathrm{t}}-T(i,j,K-1)}{R_{\mathrm{z}}(i,j,K)}}}={\displaystyle \sum _{i=1}^I{\displaystyle \sum _{j=1}^J\frac{T(i,j,2)-T_{\mathrm{b}}}{R_{\mathrm{z}}(i,j,2)}}})$$

(9)

where $\varphi$ is the total heat flux in the packing unit. R_z (i, j, K) and R_z (i, j, 2) is the thermal resistance element along the main heat transfer direction (z direction) near the top surface or bottom surface of the packing units, respectively. δ_unit is the thickness of the packing unit along the main heat transfer direction. A_unit is the cross-section area in the packing unit normal to the main heat transfer direction. T_t and T_b are the fixed temperatures on the top and bottom surfaces of the packing unit, respectively.

In the present study, the effective thermal conductivities of the packing unit for the Ohm’s law model and Kirchhoff’s law model were both solved with self-developed code based on mathematical software MATLAB. The total nodes of the thermal resistance network for both models are from 10 × 10 × 10 to 100 × 100 × 100. It was found that, the network with total nodes of 50 × 50 × 50 would be suitable for the calculations, where the deviation of k_eff between the case of 50 × 50 × 50 and 60 × 60 × 60 is less than 0.5% for the random packing unit-1. Therefore, the node setting of 50×50 × 50 was finally adopted for the following calculations with thermal resistance network methods.

3.2. CFD Simulations

In this part, the CFD simulation method was adopted to predict the effective thermal conductivity of ordered Li₄SiO₄ packing units (SC, BCC and FCC), and the simulation results were used as comparisons with those extracted from thermal resistance network methods.

As shown in Figure 5, the SC packing is selected as an example to present the CFD simulation process. The temperatures on the top and bottom surfaces of the packing unit are fixed at T_t and T_b, respectively, and the temperature difference is assumed to be 10 K. The other four surfaces of the packing unit are kept adiabatic. These physical settings are the same to those of thermal resistance networks as presented in Figure 4. Furthermore, in the CFD simulations, the convection and radiation heat transfer were also not considered. The governing equations for the conduction heat transfer in the packing unit are formulated as follows:

$$\{\begin{array}{l}\mathrm{Fluid} \mathrm{Region}: \hspace{1em}0=\nabla \cdot [k_{\mathrm{f}}\cdot (\nabla T)]\\ \mathrm{Solid} \mathrm{Region}: \hspace{1em}0=\nabla \cdot [k_{\mathrm{s}}\cdot (\nabla T)]\end{array}$$

(10)

where k_s and k_f are the thermal conductivities of Li₄SiO₄ particle and helium gas, respectively.

The energy equations in the packing unit were solved by with the commercial software ANSYS Fluent 16.0, and the residual of the calculation is less than 10⁻¹⁰. Then, the effective thermal conductivity (k_eff) of the packing unit was calculated similar to Equation (9). As shown in Figure 5b, the unstructured meshes with tetrahedral elements were adopted for the simulations, and the meshes were intensified on the particle surfaces. Three different computational meshes with total element numbers of 1,808,692, 2,171,376 and 2,605,716 were tested. It was found that the test mesh with a total element number of 2,171,376 would be suitable for the simulations, where the deviation of k_eff between the case of 2,171,376 and 2,605,716 is less than 0.5% for the SC packing unit. Therefore, a similar mesh setting to the test case with total element number of 2,171,376 was finally adopted for the following CFD simulations.

4. Results and Discussion

4.1. Effective Thermal Conductivity of Ordered Packings

In this part, the effective thermal conductivities (k_eff) of ordered piackings, including SC, BCC and FCC packings, were investigated with the thermal resistance network method and CFD simulations. The effective thermal conductivities of SC packing calculated with the thermal resistance network of Ohm’s law model are shown in Figure 6, where the results calculated with the same method by Zhao et al. [9] are also presented. It is found that, the present calculation results can fit well with those calculated by Zhao et al. [9], where the maximum deviation is less than 1.0% and the average deviation is less than 0.3%. This would indicate that, the present calculation method based on the thermal resistance network of Ohm’s law model should be reliable.

The effective thermal conductivities (k_eff) of different ordered packings calculated with thermal resistance networks of Ohm’s law model and Kirchhoff’s law model are shown in Figure 7, where the CFD simulation results are also presented. It is found that, the results calculated with the thermal resistance network of Kirchhoff’s law model can fit well with the simulation results for different ordered packings, and this would indicate that, the present calculation method based on the thermal resistance network of Kirchhoff’s law model should be reliable. Meanwhile, it is noted that the results calculated with the thermal resistance network of the Ohm’s law model are much lower than the simulation results. As for the Ohm’s law model (see Figure 4a), the thermal resistances were only connected with each other along the main heat transfer direction and separated along other directions, which would lead to underestimations of total heat flux and effective thermal conductivity in the packing units. For the Kirchhoff’s law model (see Figure 4b), the heat transfer normal to the main heat transfer direction was also considered, and the thermal resistances were connected with each other along the x, y and z directions. Therefore, the calculation results based on the Kirchhoff’s law model would agree well with those of CFD simulations.

4.2. Effective Thermal Conductivity of Random Packings

In the present study, the effective thermal conductivity (k_eff) of Li₄SiO₄ randomly packed bed is obtained from 10 random packing units (see Figure 2). Typical packing parameters and corresponding values of k_eff for different packing units are listed in Table 2 and

Table 3. Effective thermal conductivities for random packing units (T = 1123.15 K).

Packing Unit	Packing Factor (%)	keff (W/(m·K))	Deviation (keff) (%)	Unit Size
Unit-1	62.67	1.085	8.42	1dp × 1dp × 1dp
Unit-2	65.32	1.139	13.77
Unit-3	68.73	1.055	5.39
Unit-4	63.11	0.964	−3.72
Unit-5	65.82	0.981	−2.02
Unit-6	63.99	0.897	−10.39	2dp × 2dp × 2dp
Unit-7	68.88	0.873	−12.77
Unit-8	65.56	1.142	14.04
Unit-9	64.17	1.026	2.54	3dp × 3dp × 3dp
Unit-10	65.05	0.848	−15.26
Average value	65.33	1.001	8.83	/

, respectively. These show that when the mean temperature of packed bed is fixed at 1123.15 K, the averaged value of k_eff in the packing units is 1.001 W/(m·K), the maximal deviation of k_eff is −15.26% and the standard deviation is 8.83%. The variations of effective thermal conductivities for random packing calculated with thermal resistance network of Ohm’s law model and Kirchhoff’s law model are shown in Figure 8, where the experimental results as reported by Donne et al. [1] and Reimann et al. [19] are also presented. It shows that, the results calculated with thermal resistance network of Kirchhoff’s law model can fit well with the experimental results for random packing. As the mean temperature of Li₄SiO₄ packed bed changes from 323.15 K to 1123.15 K, the maximal deviation of k_eff between present calculate calculations (Kirchhoff’s law model) and experiments [19] is less than 12%. Meanwhile, it is also noted that, the results calculated with the thermal resistance network of the Ohm’s law model are much lower than the experimental results. This is similar to the results obtained from ordered packings. Therefore, it is proved that, the thermal resistance network method based on the Kirchhoff’s law mode and selected packing units prosed in the present study would be reliable and accurate for prediction of effective thermal conductivities in a Li₄SiO₄ randomly packed bed.

4.3. Effect of Packing Factor

The variations of effective thermal conductivity and packing factor for ordered and random packings are present Figure 9. It shows that, the effective thermal conductivity (k_eff) increases as packing factor increases. Furthermore, it also shows that, the deviations of k_eff between the Ohm’s law model and the Kirchhoff’s law model is large when the packing factor is high (such as in BCC, FCC and random packing units). Since the thermal conductivity of a Li₄SiO₄ particle is much higher than that of helium gas (see Figure 3), as the packing factor increases the volume ratio of Li₄SiO₄ particles in the packing unit increases, and the effective thermal conductivity (k_eff) increases too. Due to the same reason, the conduction heat flux should be higher in the packing units with higher packing factors, which would also lead to larger deviations of k_eff between the Ohm’s law model and the Kirchhoff’s law model. In addition, it is noted that, although the packing factors of BCC packing (66.00%) and random packing (65.33%) are quite close to each other, the effective thermal conductivity of random packing (0.991 W/(m·K)) is much lower than that of BCC packing (1.148 W/(m·K)), and the deviation is more than 15%. In BCC packing, each particle is in contact with eight neighbor particles and the coordinate number is relatively high, while for the random packing, the particles are contacting randomly and coordinate number might be lower under similar packing factor, which may lead to lower effective thermal conductivity for random packing. This may indicate that, compared with random packing under a similar packing factor, the effective thermal conductivity might be further improved with ordered packing methods due to its higher coordinate number inside.

5. Conclusions

In the present paper, the effective thermal conductivities of Li₄SiO₄-packed beds with both ordered and random packing structures were investigated using both thermal resistance network methods: Ohm’s law and the Kirchhoff’s law. For ordered packings, including SC, BCC and FCC packings, the CFD simulations were also performed for comparisons. For random packing, 10 typical random packing units were selected for the investigations and the experimental results as reported in the open literature were adopted for comparisons. The main findings are as follows:

(1)

For ordered packings, the effective thermal conductivities calculated with thermal resistance network of Kirchhoff’s law model fit well with CFD simulation results. While the results calculated with the Ohm’s law model are significantly underestimated. Therefore, when establishing a thermal resistance network, the thermal resistances should be connected along the main heat transfer direction and other heat transfer directions as well in the packing unit.

(2)

For random packings, it is proved that, the thermal resistance network method based on the Kirchhoff’s law mode and selected random packing units proposed in the present study would be reliable and accurate for the prediction of effective thermal conductivities in a Li₄SiO₄-packed bed, while the results calculated with the Ohm’s law model are also significantly underestimated for random packings.

(3)

The effect of packing factor is remarkable. As packing factor increases, the effective thermal conductivity (k_eff) of a packed bed increases, and the deviation of k_eff between the Ohm’s law model and Kirchhoff’s law model also increases. Furthermore, compared with random packing at a similar packing factor, the effective thermal conductivity of a packed bed might be further improved with an ordered packing method due to its higher coordinate number inside.