CFD investigation of a sensible packed bed thermal energy storage system with different porous materials

2.1 Mathematical model

The packed bed storage system geometry considered in the investigation is a 2D axisymmetric cylindrical tank whose diameter is 0.60 m and height is 1.0 m. Packed bed configuration with sphere is considered for porous media. Rock, steel and cordierite are used as solid storage material and air as heat transfer fluid. This storage system operates at atmospheric pressure (P= 0.1 MPa). As shown in Fig.1, pressure has a strong effect on the air density, and a minor effect on the heat capacity. In addition, it has no effect on the thermal conductivity and the dynamic viscosity of dry air. As a result, according to working conditions of storage system for fluid thermophysical properties as a function of temperature were represented respectively with Eq. (1) from [7], Eq. (2) from [9] and Eq. (3) from [8] while solid thermophysical properties are assumed constant [2]. Because, it is hard to find appropriate correlation for solid material. The geometry and thermophysical properties of solid materials are given in Fig. 2 and Table 1, respectively.

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Figure 1. Thermophysical properties of air as a function of temperature. a) Density, b) Thermal conductivity, c) Specific heat, d) Dynamic viscosity [10]

$\begin{align} & {{c}_{p}}=1.9327\times {{10}^{-10}}{{T}^{4}}-7.9999\times {{10}^{-7}}{{T}^{3}}+1.1407\times {{10}^{-3}}{{T}^{2}} -4.4890\times {{10}^{-1}}T+1.057\times {{10}^{3}} \\\end{align}$  (1)

$\begin{align}  & \mu =2.5914\times {{10}^{-15}}{{T}^{3}}-1.4346\times {{10}^{-11}}{{T}^{2}}+5.0523\times {{10}^{-8}}T +4.1130\times {{10}^{-6}} \\\end{align}$    (2)

$k=1.5207\times {{10}^{-11}}{{T}^{4}}-4.8574\times {{10}^{-8}}{{T}^{2}}+1.0184\times {{10}^{-4}}$     (3)

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Figure 2. Physical system in charging phase.

Table 1. Thermophysical properties of solid storage materials

In this study first charging cycle of thermal energy storage system is examined. In the charging case air enters the system at 1473 K and the initial temperature of packed bed TES system is equal 1073 K. Heat loses with the external environment is neglected. 

Numerical simulations are carried out two different mass flow rate and three porosity values and three different solid materials. Results are given in terms of stored thermal energy as a function of time. The stored thermal energy is evaluated using following equations as given in [2].

${{Q}_{stored}}={{\rho }_{eff}}{{c}_{eff}}\left[ \underset{V}{\overset{{}}{\mathop \int }}\,TdV-{{T}_{i}}V \right]$    (4)

where V=0.283m³ and T_i =1073K

${{\rho }_{eff}}=\varepsilon {{\rho }_{f}}+\left( 1-\varepsilon  \right){{\rho }_{s}}$           (5)

${{c}_{eff}}=\varepsilon {{c}_{f}}+\left( 1-\varepsilon  \right){{c}_{s}}$    (6)

2.2 Governing equations

Brinkman-Forchheimer-extended Darcy model is used for momentum equation and the porous medium is assumed to be in local thermal non-equilibrium with the fluid. In addition, it is assumed that the fluid is incompressible and the flow is unsteady. Heat transfer by radiation and heat losses to the surrounding are neglected. The flow is fully developed in the channel. The governing equations for mass, momentum and energy are, [1-3]:

$\frac{1}{r}\frac{\partial }{\partial r}\left( r{{v}_{p}} \right)+\frac{\partial {{u}_{p}}}{\partial z}=0$      (7)

$\begin{align} \frac{\partial \varepsilon {{\rho }_{f}}{{u}_{p}}}{\partial t}+\varepsilon {{\rho }_{f}}{{v}_{p}}\frac{\partial {{u}_{p}}}{\partial r}+\varepsilon {{\rho }_{f}}{{u}_{p}}\frac{\partial {{u}_{p}}}{\partial z}=-\frac{{{\partial }_{p}}}{\partial z}  +\varepsilon \mu \left( \frac{{{\partial }^{2}}{{u}_{p}}}{\partial {{r}^{2}}}+\frac{1}{r}\frac{\partial {{u}_{p}}}{\partial r}+\frac{{{\partial }^{2}}{{u}_{p}}}{\partial {{z}^{2}}} \right)-{{\varepsilon }^{2}}\frac{\mu }{K}{{u}_{p}}  -{{\varepsilon }^{3}}\frac{{{C}_{2}}{{\rho }_{f}}}{2}\sqrt{{{u}_{p}}^{2}+{{v}_{p}}^{2}{{u}_{p}}} \\\end{align}$      (8)

$\begin{align} \frac{\partial \varepsilon {{\rho }_{f}}{{v}_{p}}}{\partial t}+\varepsilon {{\rho }_{f}}{{v}_{p}}\frac{\partial {{v}_{p}}}{\partial r}+\varepsilon {{\rho }_{f}}{{u}_{p}}\frac{\partial {{v}_{p}}}{\partial z}=-\frac{{{\partial }_{p}}}{\partial z}  +\varepsilon \mu \left( \frac{{{\partial }^{2}}{{v}_{p}}}{\partial {{r}^{2}}}+\frac{1}{r}\frac{\partial {{v}_{p}}}{\partial r}+\frac{{{\partial }^{2}}{{v}_{p}}}{\partial {{z}^{2}}} \right)-{{\varepsilon }^{2}}\frac{\mu }{K}{{v}_{p}}  -{{\varepsilon }^{3}}\frac{{{C}_{2}}{{\rho }_{f}}}{2}\sqrt{{{u}_{p}}^{2}+{{v}_{p}}^{2}{{v}_{p}}} \\\end{align}$       (9)

$\begin{align} \varepsilon {{\left( \rho {{c}_{p}} \right)}_{f}}\frac{\partial {{T}_{f}}}{\partial t}+~\varepsilon {{\left( \rho {{c}_{p}} \right)}_{f}}\left( {{v}_{p}}\frac{\partial {{T}_{f}}}{\partial r}+{{u}_{p}}\frac{\partial {{T}_{f}}}{\partial z} \right)=  \varepsilon {{k}_{f}}\left( \frac{\partial {{T}_{f}}}{\partial {{r}^{2}}}+\frac{1}{r}\frac{\partial {{T}_{f}}}{\partial r}+\frac{\partial {{T}_{f}}}{\partial {{z}^{2}}} \right)+{{h}_{sf}}{{a}_{sf}}\left( {{T}_{s}}-{{T}_{f}} \right) \\\end{align}$      (10)

$\begin{align} \left( 1-\varepsilon  \right){{\left( \rho c \right)}_{s}}\frac{\partial {{T}_{s}}}{\partial t}=\left( 1-\varepsilon  \right){{k}_{s}}\left( \frac{\partial {{T}_{s}}}{\partial {{r}^{2}}}+\frac{1}{r}\frac{\partial {{T}_{s}}}{\partial r}+\frac{\partial {{T}_{s}}}{\partial {{z}^{2}}} \right)  -{{h}_{sf}}{{a}_{sf}}\left( {{T}_{s}}-{{T}_{f}} \right) \\\end{align}$    (11)

where C_{2 =} 2CK^-0.5

The permeability K and inertia coefficient C of porous media for packed bed configuration are based on two relations [3],

$K=\frac{{{d}_{p}}^{2}{{\varepsilon }^{3}}}{175{{\left( 1-\varepsilon  \right)}^{2}}},~C=\frac{1.75}{\sqrt{150}{{\varepsilon }^{1.5}}}$       (12)

The convective heat transfer coefficient and interface area per volume of packed bed are based on two relations, [4]

${{h}_{sf}}=\frac{{{k}_{f}}\left( 2+1.1P{{r}^{1/3}}R{{e}_{p}}^{0.6} \right)}{{{d}_{p}}};~{{a}_{sf}}=\frac{6\left( 1-\varepsilon  \right)}{{{d}_{p}}}$       (13)

$R{{e}_{p}}=\frac{\left| v \right|{{d}_{p}}}{v}$        (14)

Coefficients were calculated using the permeability values given in the reference [2].

A comparison between the presented study and reference [2] is given for different porosities and two time of computational simulation for a packed bed with steel spheres and same geometry with [2] in Table 2. As we can see from Table 2, the obtained results best match with the results of reference [2].

Table 2. A comparison about stored thermal energy values for different porosities and two time of computational simulation for a packed bed steel spheres

Numerical solutions are carried out for 0.2 and 0.4 constant mass flow rate with 0.2, 0.4 and 0.6 porosity values. The charging phase is considered completed when the initial temperature of the system becomes equal to inlet fluid temperature which is 1473 K. Results are presented in terms of stored energy profiles for steel, cordierite and rock spheres.

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Figure 3. Stored energy in charge cycle for G=0.2 kg/m²s   and different porosity values in packed bed with steel spheres a) Steel, b) Cordierite, c) Rock

Fig. 3 and 4 shows the energy stored values as a function of time, for porosity values 0.2, 0.4 and 0.6 and mass flow rate values equal to 0.2 and 0.4 kg/m²s, respectively. At each porosity, the stored thermal energy reaches steady state conditions, which indicates the thermal saturation of the tank. Also, the trends show that the stored energy is higher for the lower porosity values because of thermal capacity increase as seen in Fig. 3 and 4. Three different materials which are steel, cordierite and rock are compared as storage material. As seen in the Table 1, they are different in respect to heat capacity, thermal conductivity and solid density. Among the materials, steel has the highest volumetric heat capacity while rock and cordierite have lower value. The most important property of the storage material is the volumetric heat capacity whereas the thermal conductivity of the solid only has a small effect on the stored energy. So, the maximum heat stored is achieved in the packed bed with steel spheres, the stored energy of cordierite and rock spheres are in close values as seen Fig.3 and 4. For assigned porosity values in Fig.3 and 4, when the mass flow rate increases the stored energy values are not change.

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Figure 4. Stored energy in charge cycle for G=0.4 kg/m²s and different porosity values in packed bed with spheres a) Steel, b) Cordierite, c) Rock

The effect of the mass flow rate is showed also in Fig. 5 according to the average solid temperature fields in 3600 s charge cycle for 0.6 porosity value and the packed bed with steel spheres. When the mass flow increases the stored energy are not change, storage time decreases due to more efficient convective heat transfer between the fluid and solid matrix for same charge cycle as seen in Fig.5. Figure 6 shows the effect of porosity values on the average solid temperature fields in 3600 s charge cycle for the packed bed with cordierite spheres and for 0.4 kg/m²s mass flow rate. When the porosity value increases the stored energy decreases in the packed bed at the same charge cycle.

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Figure 5. Average solid temperature fields in 3600 s charge cycle for 0.6 porosity values packed bed with steel spheres a) 0.2 kg/m²s, b) 0.4 kg/m²s

The stored energy values are given for the first six hours of simulation time in Table 3 and 4. For all storage material, after 3 hours, thermal energy stored reaches almost maximum value In Table 3, increasing mass flow rate from 0.2 kg/m²s to 0.4 kg/m²s is no effect on the amount of stored energy while storage time decreases due to more efficient convective heat transfer between the fluid and solid matrix. Table 4 helpful to compare the stored energy for three studied porosity values, 0.2, 0.4 and 0.6 for 0.2 kg/m²s.

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Figure 6. Average solid temperature fields in 3600 s charge cycle for 0.4 kg/m²s mass flow rate for packed bed with cordierite spheres a) ε=0.2, b) ε=0.4, c) ε=0.6

Table 3. Stored thermal energy values for porosity 0.6 and different time of computational simulation with different solid storage materials. a) G=0.2 kg/m²s, b) G=0.4 kg/m²s

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Table 4. Stored thermal energy values for mass flow rate 0.4 kg/m²s and different time of computational simulation with different solid storage materials. a) ε=0.2, b) ε=0.6

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