Effect Evaluation of Filling Medium Parameters on Operating and Mechanical Performances of Liquid Heavy Metal Heat Storage Tank

Highlights

• An integrated thermal and mechanical analysis of a liquid LBE tank.
• The effects of four filling medium parameters are evaluated.
• The viability of liquid LBE for solar TES tanks is evaluated.

Abstract

In order to evaluate the feasibility and performance of liquid lead-bismuth eutectic as the heat transfer fluid for thermocline heat storage tanks in solar power systems, we conducted an effect evaluation of filling medium parameters on the integrated operating and mechanical performances of a thermocline tank using liquid lead-bismuth eutectic using the computational fluid dynamics simulation method. Four parameters were evaluated: the porosity, thermal conductivity, specific heat capacity, and equivalent diameter of the filling medium. The results show that the liquid lead-bismuth eutectic tank operated stably. The total charging and total discharging durations were 5.7 h and 5.3 h, respectively, and the discharging efficiency was 91.94%. The effect evaluation results reveal that the discharging thermocline thickness of the liquid heavy metal tank can be decreased by increasing the specific heat capacity of the filling particles, or by decreasing the porosity, thermal conductivity, and equivalent diameter of the filling medium. The total discharging quantity of the tank increased from 2.19 × 10¹⁰ J to 3.34 × 10¹⁰ J when the specific heat capacity of the filling particles increased from 610.0 J/(kg∙K) to 1010.0 J/(kg∙K), while the other three filling medium parameters had no obvious effect on the total discharging quantity of the tank. The mechanical performance of the tank wall could be improved by decreasing any one of the four evaluated parameters of the filling medium. The results of this paper may serve as a reference for the design of actual liquid heavy metal heat storage tanks in solar power plants.

1. Introduction

Solar energy applications have continued to grow in recent years, and large-scale solar power mainly includes photovoltaic and CSP technologies. Because solar energy is naturally unstable and intermittent, energy storage devices are very important for SPSs. TES devices include dual-tank and single-tank structures. The former were used earlier, but they occupied a larger area and cost more. Therefore, researchers proposed the application of single heat storage tanks in SPSs using parabolic trough collectors. In contrast to the dual-tank structure, the single-tank structure could decrease the initial investment by 35%. Hence, the study of single-tank heat storage is very meaningful. Based on different solid heat storage materials, heat storage methods can be divided into the following three types: chemical, sensible, and latent heat storage technologies, where sensible heat storage is more mature and widely used. Potential HTFs for TES systems mainly include molten salts, heat transfer oils, liquid metals, and water (see

Table 1. Different kinds of HTFs used in TES systems.

HTFs	Advantages	Disadvantages
Molten salts	Strong heat storage capacity, difficult to burn, good safety, low working pressure, non-toxic	Easy to decompose, oxidation and corrosion under high temperature conditions, high melting temperature
Heat transfer oils	Strong fluidity, low freezing point, good heat transfer performance, low corrosiveness	Short service life, low applicable temperature, flammable
Liquid metals	Excellent thermal conductivity, low melting point, high boiling point, wide operating temperature range	High corrosiveness under high temperature conditions, toxicity of some metals
Water/steam	Low cost, innocuous, low corrosiveness, environmental protection	High temperature and pressure requirements, low heat storage capacity

) [1].

Researchers have conducted many studies on single-tank heat storage technologies. For instance, Yang and Garimella [2] employed a numerical simulation to systematically analyze the releasing behavior of the thermocline heat storage tank using quartzite as the filling medium and molten salt as the HTF. Their results indicate that the releasing efficiency can be increased with a smaller Reynolds number and a larger tank height. To study thermal ratcheting, Fernández-Torrijos et al. [3] developed a simplified two-phase model of a thermocline TES tank. The analysis results revealed that the model could be used to analyze the temperature changes in the filler material, molten salt, and tank wall. ELSihy et al. [4] conducted a study on the thermocline thickness variations of a molten salt heat storage tank using three different filling materials (quartzite rock, slag pebbles, and alumina ceramics). Their results indicate that slag pebbles as a filler material are more effective than quartzite rocks in TES tanks. Sun et al. [5] studied the heat storage characteristics of a tank using water as the HTF and PA/EG/CF phase change material as the encapsulation material. Their results show that increasing the Reynolds number, increasing the Stefan number, or reducing the dimensionless diameter can improve the average exergy storage rate. Zwierzchowski and Wolowicz [6] performed energy and exergy analyses of a large-scale cogeneration sensible heat storage tank in Poland and estimated the energy and exergy efficiencies.

LBE is a kind of liquid heavy metal (LHM) made of lead and bismuth [7]. It has many good physical properties [8], including a low melting point (423~473 K), a high boiling point (about 1943 K), low chemical activity, high thermal mobility, strong heat storage capacity, etc. It is one of the candidate HTF materials for CSP plants and TES systems. When the liquid LBE is used in a TES system, the TES system can have a wide operating temperature range with a relatively low lower temperature limit and a high upper temperature limit. However, the disadvantage of corrosion caused by high-temperature liquid LBE in the TES tank should also be considered.

Studies of liquid LBE used in TES systems are meaningful for the development of solar thermal power to a certain extent. Currently, LBE is mostly used in nuclear reactors [9], and experimental works on the use of liquid LBE in TES systems are rarely reported. Only a few simulation studies on liquid LBE-based TES systems have been reported. For instance, Pacio and Wetzel [10] carried out an evaluation of liquid LBE research paths for its use as HTF in CSP systems. The assessment results revealed the that the use of LBE in CSP systems is somewhat feasible. Laube et al. [11] conducted a thermodynamic analysis of a TES tank using different liquid metals. The feasibility and influence of the use of liquid LBE for TES tanks were studied and analyzed.

According to a literature review, though many thermal performance research works on TES tanks have been launched, most of these studies have focused on TES tanks using molten salts; numerical studies on TES tanks using liquid LBE are relatively few. Filling medium parameters can influence the operational behavior as well as the mechanical performance of TES tanks using liquid LBE, and related studies will be of value as a reference for the design of liquid LBE-based TES systems. This research gap should be addressed by undertaking studies on the effects of filling medium parameters on the integrated thermal and mechanical performances of TES tanks using liquid LBE.

Compared with previous research works, the novelty of this study is that the effects of filling medium parameters on the operating and mechanical performances of LBE-based TES tanks are evaluated based on the numerical simulation method, and four parameters (i.e., the porosity, thermal conductivity, specific heat capacity, and equivalent diameter of the filling medium) are investigated. The viability of LBE use for TES tanks and methods of improving the operational and mechanical performances of the TES tank are discussed. The results of this paper may provide a reference for the design of an actual LHM heat storage tank.

2. Material and Methods

2.1. Governing Equations

Figure 1 illustrates a liquid LBE thermocline heat storage tank that consists of three sections: the packing bed and the upper and lower distributors. The packing bed is composed of filling particles with an equivalent diameter of 60 mm. The vertical heights of the distributor, packing bed, and whole tank are H_dis (1 m), H_packing (10 m), and H_tank (14 m). The inner diameters of the inlet tube and LHM tank are D_tube (1 m) and D_tank (6 m). The thermocline thickness of the liquid LBE heat storage tank is d_t. The wall of the liquid LBE tank is composed of three layers: fire brick, steel, and ceramic. Their densities are 2000 kg/m³, 8000 kg/m³, and 1000 kg/m³, their specific heat capacities are 1000 J/(kg∙K), 430 J/(kg∙K), and 1000 J/(kg∙K), and their thicknesses are 100 mm, 20 mm, and 50 mm respectively. The charging and discharging processes of the liquid LBE tank can also be seen in Figure 1.

For the liquid LBE tank model, some assumptions were adopted [12]:

(a)

The LBE flow and heat transfer were symmetrical about the axis.

(b)

The solid fillers in the packing region of the tank could be considered as a continuous, homogeneous, and isotropic porous medium.

(c)

The liquid LBE flow in the packing region was laminar and incompressible.

(d)

The properties of the filling particles in the packing region were constant.

For the liquid LBE, the governing equations were [13]:

$$\frac{\partial \left({\epsilon }_{\mathrm{filler}}{\rho }_{\mathrm{fluid}}\right)}{\partial t}+\nabla \cdot \left({\rho }_{\mathrm{fluid}}\overline{u}\right)=0$$

(1)

$$\frac{\partial \left({\rho }_{\mathrm{fluid}}\overline{u}\right)}{\partial t}+\nabla \cdot \left({\rho }_{\mathrm{fluid}}\frac{\overline{u}\overline{u}}{{\epsilon }_{\mathrm{filler}}}\right)=-{\epsilon }_{\mathrm{filler}}\nabla \cdot p+\nabla \cdot \overline{\overline{\tau }}+{\epsilon }_{\mathrm{filler}}{\rho }_{\mathrm{fluid}}\overline{g}+{\epsilon }_{\mathrm{filler}}\left(\frac{\mu }{K}+\frac{F_{\mathrm{filler}}}{\sqrt{K}}{\rho }_{\mathrm{fluid}}\left|\overline{u}\right|\right)\overline{u}$$

(2)

$$\begin{array}{l}\frac{\partial \left[{\epsilon }_{\mathrm{filler}}{\rho }_{\mathrm{fluid}}{C}_{\mathrm{P},\mathrm{fluid}}\left(T_{\mathrm{fluid}}-T_{\mathrm{c}}\right)\right]}{\partial t}+\nabla \cdot \left[{\rho }_{\mathrm{fluid}}\overline{u}{C}_{\mathrm{P},\mathrm{fluid}}\left(T_{\mathrm{fluid}}-T_{\mathrm{c}}\right)\right]\\ =\nabla \cdot \left(k_{\mathrm{eff}}\nabla T_{\mathrm{fluid}}\right)-p\nabla \cdot \overline{u}+\mathrm{tr}\left[\nabla \left(\frac{\overline{u}}{{\epsilon }_{\mathrm{filler}}}\right)\cdot \overline{\overline{\tau }}\right]+\frac{\overline{u}\cdot \overline{u}}{2{\epsilon }_{\mathrm{filler}}}\times \frac{\partial {\rho }_{\mathrm{fluid}}}{\partial t}+h_{\mathrm{inter}}\left(T_{\mathrm{filler}}-T_{\mathrm{fluid}}\right)\end{array}$$

(3)

where k_eff represents the effective heat conductivity of the combination of the liquid LBE and filling medium.

The energy equation for the solid fillers was [13]:

$$\frac{\partial }{\partial t}\left[\left(1-{\epsilon }_{\mathrm{filler}}\right){\rho }_{\mathrm{filler}}{C}_{\mathrm{P},\mathrm{filler}}\left(T_{\mathrm{filler}}-T_{\mathrm{c}}\right)\right]=-h_{\mathrm{inter}}\left(T_{\mathrm{filler}}-T_{\mathrm{fluid}}\right)$$

(4)

The heat conduction and heat balance equations for the wall of the liquid LBE tank were [13]:

$$\rho C_{\mathrm{P}}\frac{\partial T}{\partial t}=-k\nabla \cdot T$$

(5)

$${\left.\frac{\partial T_1}{\partial r}\right|}_{12}={\left.\frac{k_2}{k_1}\frac{\partial T_2}{\partial r}\right|}_{12}$$

(6)

$${\left.\frac{\partial T_2}{\partial r}\right|}_{23}={\left.\frac{k_3}{k_2}\frac{\partial T_3}{\partial r}\right|}_{23}$$

(7)

where T₁, T₂, and T₃ are the temperatures of the fire brick, steel, and ceramic layers and k₁, k₂, and k₃ are the corresponding thermal conductivities.

The mixed boundary condition for the wall surfaces of the tank was [13]:

$${\left.\frac{\partial T_3}{\partial r}\right|}_{\mathrm{w}}=-\frac{h_{\mathrm{w}}}{k_3}\left(T_{\mathrm{w}}-T_0\right)-\frac{{\epsilon }_{\mathrm{w}}{\sigma }_{\mathrm{r}}}{k_3}\left(T_{\mathrm{w}}^4-T_{\mathrm{amb}}^4\right)$$

(8)

where T_w and ε_w are the temperature and emissivity of the ceramic surface and T_amb is the environment temperature. Figure 2 presents a flow diagram of the solutions of the relevant governing equations.

For the circumferential direction of the steel wall, the strain ε_L was [14]:

$${\epsilon }_{\mathrm{L}}\left(x,r\right)={\epsilon }_{\mathrm{t}}+{\epsilon }_{\mathrm{m}}$$

(9)

$${\epsilon }_{\mathrm{t}}\left(x,r\right)=\alpha \left[T_{\mathrm{w},2}\left(x,r\right)-T_{\mathrm{ref}}\right]$$

(10)

$${\epsilon }_{\mathrm{m}}\left(x,r\right)=\frac{1}{E}\left[{\sigma }_{11}-\nu \left({\sigma }_{22}+{\sigma }_{33}\right)\right]$$

(11)

The MMS of the steel wall was [14]:

$${\sigma }_{\max }\left(x,r\right)=\alpha E\left[T_{\mathrm{w},2,\max }\left(x,r\right)-T_{\mathrm{w},2,\min }\left(x,r\right)\right]$$

(12)

2.2. Numerical Model

In this study, the CFD simulation method was utilized to conduct the operation simulations for the LHM thermocline tank, and the commercial software used for the CFD simulations was Fluent. For the numerical model of the liquid LBE-based TES tank, the turbulent model was the standard k-ε model. A 2-D dual-precision solver and PISO algorithm were employed. Gravity was taken into account, as can be seen in Figure 1. There was no inner heat source in the liquid LBE tank. For the outer top and side surfaces of the TES tank, the mixed boundary condition, which considers both convective and radiation heat losses (see Equation (8)), was selected. The bottom surface temperature of the tank was set to be constant. According to the mesh sensitivity analysis results, the total mesh number of the numerical model was about 1.2 × 10⁶.

The expressions of the thermophysical properties of the liquid LBE were [15]:

$${\rho }_{\mathrm{fluid}}=11096-1.3236T_{\mathrm{fluid}}$$

(13)

$$k_{\mathrm{fluid}}=3.61+1.517\times {10}^{-2}{T}_{\mathrm{fluid}}$$

(14)

$${\mu }_{\mathrm{fluid}}=4.94\times {10}^{-4}\exp \left(757.1/T_{\mathrm{fluid}}\right)$$

(15)

$$C_{\mathrm{P},\mathrm{fluid}}=159-2.72\times {10}^{-2}{T}_{\mathrm{fluid}}+7.12\times {10}^{-6}{T}_{\mathrm{fluid}}^2$$

(16)

Based on the experimental results of a relevant study [16], the applicability of the numerical model of the liquid LBE thermocline tank was validated. According to the literature, the height and inner diameter of the experimental prototype were 5.9 m and 1.5 m. The HTF in the experimental facility was molten salt consisting of NaNO₃ and KNO₃. As presented in Figure 3, the results reveal a relatively acceptable agreement between the test and simulation results.

3. Results

3.1. Operation Performance

This section evaluates and presents the operation performance of the liquid LBE thermocline tank. For the initial parameters for the simulation, the inlet flow rate u_in was 0.000278 m/s. The initial cold temperature T_c of the liquid LBE was 566 K. The initial hot temperature T_h of liquid LBE was 723 K. The filling medium porosity ε_filler was 0.2. The equivalent diameter of the solid fillers was 0.06 m. The thermal conductivity k_filler, specific heat capacity C_P,filler, and density ρ_filler of the solid fillers were 5.8 W/(m∙K), 810 J/(kg∙K), and 2550 kg/m³, respectively. The equivalent diameter D_filler of filling particles was 60 mm. The ambient temperature T_amb was 298 K.

The charging and discharging processes of the liquid LBE heat storage tank were simulated. Figure 4 shows the liquid LBE temperature distributions of the tank during the discharging process. It can be seen that the thermocline was effectively formed and kept during the whole process. It could go up and through the tank stably, which means the liquid LBE tank had good operating stability. According to the simulation results, the total charging and total discharging durations were 5.7 h and 5.3 h, respectively, and the discharging efficiency was 91.94%, which is close to that reported in the previous literature (about 92%) [11]. The total discharging quantity of the tank was 2.75 × 10¹⁰ J (about 7.64 MWh). These results can provide reference values for the heat release capacity and operation parameter designs of actual TES tanks using liquid LBE.

3.2. Effect Evaluation Results of Filling Medium Parameters

3.2.1. Effects of Porosity of the Filling Medium

The effect evaluation results of the porosity ε_filler of the filling medium on the discharging and mechanical performances of the liquid LBE heat storage tank are provided in Figure 5, Figure 6 and Figure 7. The circles in the figures indicate the ordinates corresponding to the curves surrounded by them. As is shown in Figure 5 and Figure 6, the porosity of the filling medium had no obvious influence on the total discharging duration t_discharge or the maximum discharging power P_max of the tank, but did affect the thermocline thickness of the tank. The thermocline thickness of the tank during the discharging process increased as the porosity of the filling medium increased. This agrees with the results reported in the previous literature [13].

The effect of the porosity of the filling medium on the total discharging quantity Q_release of the tank was also relatively small. This is because when the porosity of the filling medium increased, the total mass of the liquid LBE in the packing region also increased. The heat carried by the added liquid LBE compensated to a large extent for the heat lost due to the reduction in solid filling materials.

Figure 7 presents the axial MMS distribution variations of the liquid LBE tank at different filling medium porosities. Figure 7 shows that as the porosity of the filling medium increased from 0.12 to 0.28, the peak MMS increased from 129.7 MPa to 142.5 MPa. This is partly because when the filling medium porosity increases, the mass of liquid LBE in the packing region increased, thus leading to a bigger temperature difference between the maximum and minimum temperatures of the steel wall. Thus, it can be concluded that by reducing the porosity of the filling medium, the thermocline thickness during the discharging process and the peak MMS can both be decreased.

3.2.2. Effects of the Thermal Conductivity of the Filling Medium

This section presents the effect evaluation results of the thermal conductivity k_filler of the filling particles on the discharging and mechanical behaviors of the liquid LBE thermocline tank. The relevant results are shown in Figure 8, Figure 9 and Figure 10.

According to Figure 8, the thermal conductivity of the filling particles essentially had no effects on the total discharging duration or the outlet LBE temperature, though it had an obvious impact on the thermocline thickness of the liquid LBE tank during discharging processes. As the thermal conductivity of the filling particles increased, the maximum thermocline thickness of the liquid LBE tank increased. This was due to an improvement in the heat transfer between the liquid LBE and the filling medium caused by an increase in the thermal conductivity of the filling particles. This conclusion agrees with that reported in the previous literature [12]. When the thermal conductivity of the filling particles was 9.8 W/(m∙K), the maximum thermocline thickness of the tank was 3.3 m. The results of Figure 9 reveal that the thermal conductivity of filling particles had little effect on either the maximum discharging power or the total discharging quantity of the liquid LBE tank.

The axial MMS distribution variations of the liquid LBE tank at different thermal conductivities of the filling particles are shown in Figure 10. It can be seen that the peak MMS increased as the thermal conductivity of the filling particles increased. This is because a lower particle thermal conductivity will weaken the heat transfer between the liquid LBE and the filling particles, as well as the heat transfer between the tank wall and the packing bed. This reduces the maximum temperature of the steel wall during the operating cycle, resulting in a smaller MMS. This was also reported in the relevant study [13]. When the thermal conductivity of the filling particles was 9.8 W/(m∙K), the peak MMS of the tank wall was 139.8 MPa. Therefore, it was demonstrated that by decreasing the thermal conductivity of the filling particles, the thermocline thickness during the discharging process and the peak MMS can both be decreased.

3.2.3. Effects of the Specific Heat Capacity of the Filling Medium

The effect evaluation results of the specific heat capacity C_P,filler of the filling particles on the discharging behavior of the liquid LBE tank are shown in Figure 11 and Figure 12. According to the two figures, the specific heat capacity of the filling particles had an obvious effect on the discharging process. As the specific heat capacity of the filling particles increased from 610.0 J/(kg∙K) to 1010.0 J/(kg∙K), the total discharging duration increased from 4.5 h to 6.5 h, the maximum thermocline thickness decreased from 3.4 m to 2.8 m, the total discharging quantity of the tank increased from 2.19 × 10¹⁰ J to 3.34 × 10¹⁰ J, while the maximum discharging power of the tank was basically unchanged. The maximum thermocline thickness decreased with the increase in the specific heat capacity of the filling particles because the temperature increase caused by the same heat quantity reduced when the specific heat capacity of the filling particles increased. The total discharging quantity increased with the increase in the specific heat capacity of the filling particles because for the same temperature variation the heat release of the filling medium with the higher specific heat capacity was larger.

Figure 13 presents the axial MMS distributions of the liquid LBE tank at different specific heat capacities of the filling particles. The results reveal that the peak MMS increased as the specific heat capacity of the filling particles increased. When the specific heat capacity of the filling particles was 1010.0 J/(kg∙K), the peak MMS was 145.6 MPa. Hence, by increasing the specific heat capacity of the filling particles, the thermocline thickness during the discharging process can be reduced and the total discharging duration and total discharging quantity of the tank can both be increased, but the peak MMS will also be increased.

3.2.4. Effects of the Equivalent Diameter of the Filling Medium

This section presents the effect evaluation results of the equivalent diameter D_filler of the filling particles on the performances of the liquid LBE thermocline heat storage tank. Figure 14 shows the variations of thermocline thickness and the outlet LBE temperature of the tank at different equivalent diameters of the filling particles. The results reveal that the equivalent diameter of filling particles had a very small effect on the total discharging duration, and that the thermocline thickness of the tank during the discharging process increased when the equivalent diameter of the filling particles increased. This is because increasing the equivalent diameter of the filling particles can lead to the temperature difference between the liquid LBE and the solid particle surfaces increasing, as well as that within the solid particles, which contributes to a faster thermocline expansion and thus results in a thicker thermocline layer. This conclusion is similar to that reported in [17]. When the equivalent diameter of the filling particles increased to 0.10 m, the maximum thermocline thickness of the tank was 3.1 m. According to Figure 15, the effects of the equivalent diameter of the filling particles on the maximum discharging power and total discharging quantity of the tank were both very small. These results are similar to those of the effect evaluations of the porosity and thermal conductivity of the filling particles on the discharging performance.

Figure 16 shows the axial MMS distributions of the liquid LBE thermocline tank at different equivalent diameters of the filling particles. The results show that as the equivalent diameter of the filling particles increased from 0.02 m to 0.10 m, the peak MMS increased from 129.4 MPa to 148.7 MPa. The reason for this influence law is similar to that of the filling medium porosity. Thus, it can be concluded that by decreasing the equivalent diameter of the filling particles, the thermocline thickness during the discharging process and the peak MMS can be both reduced.

4. Conclusions

Liquid lead-bismuth eutectic is a potential heat transfer fluid material for thermal energy storage tanks. This study used the computational fluid dynamics simulation method to evaluate the effects of filling medium parameters on the integrated operation and mechanical performances of a thermocline tank using lead-bismuth eutectic and a filling medium. Four filling medium parameters were evaluated: porosity, thermal conductivity, specific heat capacity, and equivalent diameter of the filling medium. The simulation results showed that the liquid heavy metal tank could achieve stable operation, indicating the technical feasibility of the use of lead-bismuth eutectic in thermal energy storage systems. The total charging and total discharging durations were 5.7 h and 5.3 h, respectively, the discharging efficiency was 91.94%, and the total discharging quantity was 2.75 × 10¹⁰ J. The effect evaluation results demonstrate that the thermocline thickness during the discharging process can be reduced by increasing the specific heat capacity of the filling particles, or by decreasing the porosity, thermal conductivity, and equivalent diameter of the filling medium. The total discharging quantity of the tank can be increased by increasing the specific heat capacity of the filling particles. As the specific heat capacity of the filling particles increased from 610.0 J/(kg∙K) to 1010.0 J/(kg∙K), the total discharging quantity of the tank increases from 2.19 × 10¹⁰ J to 3.34 × 10¹⁰ J. The maximum mechanical stress of the tank can be reduced by decreasing any one of the four evaluated parameters of the filling medium. The analysis results of this study can enrich the research of thermal energy storage tanks using liquid metals and provide a reference for the selection of filling media and the structural safety design of steel walls for thermal energy storage tanks.

There are still some limitations in this study. For instance, the packing region of the heat storage tank was simplified so as to be a continuous porous medium. A more detailed filling medium structure should be considered for the tank in the future research. In addition, this paper lacks practical experimental study results. An experimental study based on a small liquid lead-bismuth eutectic tank prototype will be a meaningful next step, and may provide test data for the validation of the numerical model of the tank as well as for a comparison with simulation results.