Keywords

1 Introduction

North America faces great challenges during the winter season, making it critical to find different ways to store energy. In the current study, some concepts involving storing thermal energy using PCM or Bentonite or a combination of PCM and Bentonite in a cavity were analyzed. PCM can absorb, store and release large amounts of latent heat [1]. Bashar and Siddiqui [1] conducted a study to investigate the melting process and thermal behavior of Phase Change Material (PCM) due to heat transfer convection by a U-shaped tube. The results revealed the importance of using paraffin as a means of storing energy. Touatia et al. [2] focused their study in extracting heat from a solar system and storing it in a phase change material. Plotze et al. [3] conducted a study describing the thermal properties, heat conductivity, heat capacity, and the thermal diffusivity of Bentonite. The aim of this paper is to investigate the effectiveness of using PCM as heat storage media.

2 Model Description

A three-dimensional rectangular cavity filled with PCM (paraffin wax) and/or Bentonite was used in this investigation. The length of the cavity was 149.86 mm, the height was 115.22 mm, and the depth was 13 mm. A U-shaped copper tube was used inside then cavity with a thickness 0.2 mm, an outer diameter of 4.76 mm, and a vertical length of 113.22 mm. This U shape tube was used as a heat source. A clearance of 14.95 mm was maintained between the two legs of the tube. Hot water (39 °C) was flowing through the tube at a flow rate of 0.37 l/min for six hours. The inlet temperature of the heat transfer fluid (HTF) selected was 39 °C, because this temperature generally comes from the solar collector during the day time.

2.1 Governing Equations and Boundary Conditions

The governing equation used in this simulation are;

The energy equation:

$$ \rho c_{p} \frac{\partial T}{\partial t} + \rho c_{p} u\,.\,\nabla T + \nabla \,.\,q = Q $$
(10.1)

Navier-Stokes equation:

$$ \rho \frac{\partial u}{\partial t} + \rho (u\,.\,\nabla )u = \nabla \,.\,[ - PI + \mu (\nabla u + \nabla u)^{T} ] + F $$
(10.2)

The continuity equation:

$$ \rho \nabla \,.\,(u) = 0 $$
(10.3)

No slip boundary condition was used in the rectangular cavity. Open boundary condition was used in inlet, and outlet of U-shaped copper tube.

2.2 Mesh Sensitivity Analysis and Convergence Criteria

In the present study, a tetrahedral element is used to perform the numerical model. To observe the grid dependency, the calculation of maximum and minimum temperature on the surface for heat transfer fluid flow through the U-shaped copper tube was performed for different numbers of domain, as shown in Fig. 10.1. The convergence criteria depend on the temperature, the velocity, the pressure, and the number of iterations. The convergence is deemed acceptable when the change in variable between two consecutive iteration was less than 1e–5.

Fig. 10.1
figure 1

Normal element mesh

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3 Results and Discussions

In the first case the cavity was filled with paraffin and flow is circulating inside the U shape pipe. The flow duration was for six hours. It is clear from Fig. 10.2a, b and c that by increasing the circulation time, the surface temperature inside the cavity increased significantly. This is happened because, heating source U-shaped copper tube transfer heat to the PCM continuously during the simulation time and PCM absorbed the heat.

Fig. 10.2
figure 2

Temperature contours at different times

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A horizontal temperature distribution in the cavity at different times is shown in Fig. 10.3. This time difference showed the heat absorbed by the PCM toward the two sides of the cavity. One can see that the temperature of the PCM inside the heating source was approximately 31 °C, and the temperature inside the U-shaped tube was approximately 39 °C for all three-time steps, respectively. However, outside the heating source, the temperature decreased gradually. Near the two side walls, the temperature was approximately 24 °C after two hours and increased to 26 °C after six hours. This is further proof of the low conductivity of the PCM.

Fig. 10.3
figure 3

Temperature distributions at y = 0 m of the cavity along x-axis

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The other model consisted of embedding the heat source with Bentonite then paraffin outside the bentonite and vice versa. The last case involved Bentonite only. A horizontal temperature distribution of all the models is shown in Fig. 10.4. The model consisted of three hours of heat added to the cavity and three hours of heat removed from the cavity.

Fig. 10.4
figure 4

Temperature distributions at y = 0 m of the cavity along x-axis

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Figure 10.4a, b show the surface temperature distributions of all four models under heating and cooling conditions inside the rectangular cavity, respectively. The results revealed that the heat transfer rate during the heating period was higher for Bentonite than paraffin. Figure 10.4b also shows that the heat transfer rate during cooling was higher for Bentonite than PCM. This is because Bentonite has a higher thermal conductivity than the PCM. The heat transfer rate (Q) of the cavity is related to the mass flow rate (\( \dot{m} \)), heat capacity (\( {\text{c}}_{\text{p}} ) \) and temperature difference (ΔT). The heat transfer rate was calculated using the following equation:

$$ Q = \dot{m}c_{p} \Delta T $$
(10.4)

The ratio of the energy supplied (\( {\text{Q}}_{\text{S}} \)) to the energy absorbed \( ({\text{Q}}_{\text{A}} ) \) after six hours in all four models are presented in Table 10.1. In model A, the entire cavity is filled with paraffin, so it stored large amounts of heat and released large amounts of heat when the heat was extracted. In model C, the heat source was surrounded by paraffin and Bentonite was placed on the outside region. During heat absorption, all of the heat was released to the paraffin because the heat source and paraffin were too close to each other. That is why the efficiency of model C is higher than any other models. In models B and D, the efficiency is low because model B is filled with Bentonite only and in model D, the heat source was surrounded by bentonite and on the outside is the PCM.

Table 10.1 Total heat transfer analysis
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4 Conclusion

This numerical study was conducted to investigate a time-dependent heat transfer process in the presence of a phase change material and bentonite inside a rectangular cavity. The results revealed that the geometry of the heating source has a great impact on the melting behaviour of the PCM as well as the melting period. It is also clear from the results that the position of the heating source has a significant effect on the heat transfer rate in a thermally insulated cavity. It is clear from the results that bentonite is more sensitive to temperature variations in a cavity than PCM because of higher thermal conductivity.