The energy storage system can be approximated by a 2-D enclosure. A schematic diagram of the enclosure system is shown in Figure 1. Initially, the solid form of the PCM occupies the enclosure. The initial temperature of the PCM is assumed to be equal to its melting temperature. The left vertical wall of the enclosure is maintained at constant temperature (*T*_{h}) which is above the melting temperature (*T*_{m}) of the PCM. The remaining three walls are applied to convection boundary condition. The following assumptions are applied: the liquid phase of PCM is a Newtonian and incompressible fluid, all thermophysical properties of the PCM are assumed to be constant, the Boussinesq model is used in the buoyancy force term. In addition, in the energy equations, the internal heat generation and the viscous dissipation effect are neglected, and laminar fluid motion is assumed.

Figure 1 Schematic illustration of (a) the thermal storage system, (b) the physical model of the thermal storage unit

Conservation equations of mass (continuity), momentum, and energy (in the liquid and solid regions) are used to model the complete flow and thermal fields as shown below [3, 7].

$$\begin{array}{r}\frac{\mathrm{\partial}{\rho}_{l}}{\mathrm{\partial}t}+\frac{\mathrm{\partial}\left({\rho}_{l}u\right)}{\mathrm{\partial}x}+\frac{\mathrm{\partial}\left({\rho}_{l}v\right)}{\mathrm{\partial}y}=0\end{array}$$(1)$$\begin{array}{rl}& \frac{\mathrm{\partial}u}{\mathrm{\partial}t}+u\frac{\mathrm{\partial}u}{\mathrm{\partial}x}+v\frac{\mathrm{\partial}u}{\mathrm{\partial}y}\\ & =\frac{1}{{\rho}_{l}}\left[-\frac{\mathrm{\partial}p}{\mathrm{\partial}x}+\mu \left(\frac{{\mathrm{\partial}}^{2}u}{\mathrm{\partial}{x}^{2}}+\frac{{\mathrm{\partial}}^{2}u}{\mathrm{\partial}{y}^{2}}\right)-A(T)u\right]\end{array}$$(2)$$\begin{array}{rl}& \frac{\mathrm{\partial}v}{\mathrm{\partial}t}+u\frac{\mathrm{\partial}v}{\mathrm{\partial}x}+v\frac{\mathrm{\partial}v}{\mathrm{\partial}y}\\ & =\frac{1}{{\rho}_{l}}\left[-\frac{\mathrm{\partial}p}{\mathrm{\partial}y}+\mu \left(\frac{{\mathrm{\partial}}^{2}v}{\mathrm{\partial}{x}^{2}}+\frac{{\mathrm{\partial}}^{2}v}{\mathrm{\partial}{y}^{2}}\right)+g{\left(\rho \beta \right)}_{}\left(T-{T}_{m}\right)-A(T)v\right]\end{array}$$(3)$$\begin{array}{r}\frac{\mathrm{\partial}T}{\mathrm{\partial}t}+u\frac{\mathrm{\partial}T}{\mathrm{\partial}x}+v\frac{\mathrm{\partial}T}{\mathrm{\partial}y}=\frac{{k}_{l}}{{(\rho {c}_{p})}_{l}}\left[\frac{{\mathrm{\partial}}^{2}T}{\mathrm{\partial}{x}^{2}}+\frac{{\mathrm{\partial}}^{2}T}{\mathrm{\partial}{y}^{2}}\right]\end{array}$$(4)$$\begin{array}{r}\frac{\mathrm{\partial}T}{\mathrm{\partial}t}=\frac{{k}_{s}}{{(\rho {c}_{p})}_{s}}\left[\frac{{\mathrm{\partial}}^{2}T}{\mathrm{\partial}{x}^{2}}+\frac{{\mathrm{\partial}}^{2}T}{\mathrm{\partial}{y}^{2}}\right].\end{array}$$(5)In the above equations, *ρ*_{l} is the density of the liquid PCM, *t* is the time, *u* and *v* are the velocity components of the liquid PCM in the *x* and *y*-directions, *p* is the pressure, *μ* is the dynamic viscosity of PCM, *g* is the gravity, *β* is the coefficient of thermal expansion of the liquid PCM, *T* is the temperature, *T*_{m} is the melting temperature of the PCM, *k*_{l} is the thermal conductivity of the liquid PCM, *c*_{pl} is the specific heat at constant pressure of the liquid PCM, *k*_{s} is the thermal conductivity of the solid PCM, *ρ*_{s} is the density of the solid PCM, and *c*_{ps} is the specific heat at constant pressure of the solid PCM.

In the momentum equations Eqs. (2) and (3), Kozeny-Carman relation is used to model the flow within the interface. The parameter *A(T)* in Eqs. (2) and (3) is defined to achieve a gradual reduction of the velocities of the liquid PCM from a finite value in the liquid zone to zero in the solid zone. To implement Kozeny-Carman relation, parameter *A( T)* is defined as [8]

$$\begin{array}{r}A(T)=\frac{C{(1-B(T))}^{2}}{(B{(T)}^{3}+q)}\end{array}$$(6)where *C* and *q* are arbitrary constants of value of 10^{5} and 10^{−3}, respectively. *B( T)* can be defined as [8]

$$\begin{array}{r}B(T)=\left\{\begin{array}{l}{0,}_{}T<({T}_{m}-\mathrm{\Delta}T)\\ \frac{T-{T}_{m}+\mathrm{\Delta}T}{2\mathrm{\Delta}T},({T}_{m}-\mathrm{\Delta}T)<T<({T}_{m}+\mathrm{\Delta}T)\\ {1,}_{}T>({T}_{m}+\mathrm{\Delta}T)\end{array}\right.\end{array}$$(7)where *ΔT* is the range of temperatures over which the melting process occurs. If the PCM is a pure material, *ΔT* is zero, and the mushy zone is thin. On the other hand, if the PCM is an impure material, *ΔT* is greater than zero, and the mushy zone is wider than that for pure material.

*B(T)* is zero when the temperature is lower than *T*_{m}, while it is one when the temperature is higher than *T*_{m}. Equations (6) and (7) can be used to calculate the thermophysical properties of the PCM, as follows [8]

$$\begin{array}{r}\rho (T)={\rho}_{s}+({\rho}_{l}-{\rho}_{s})B(T)\end{array}$$(8)$$\begin{array}{r}k(T)={k}_{s}+({k}_{l}-{k}_{s})B(T)\end{array}$$(9)$$\begin{array}{r}{c}_{p}(T)={{c}_{p}}_{s}+({{c}_{p}}_{l}-{{c}_{p}}_{s})B(T)+{h}_{f}D(T)\end{array}$$(10)$$\begin{array}{r}\mu (T)={\mu}_{l}(1+A(T)).\end{array}$$(11)*s* and *l* stand for the solid and liquid phases of the PCM, respectively, and *h*_{f} is the latent heat of fusion of the PCM. *D*(*T*), which is a Gaussian function, is used to determine the latent heat over a temperature range *ΔT*. *D*(*T*) can be calculated from [8]

$$\begin{array}{r}D(T)=\frac{{e}^{\frac{-T{(T-{T}_{m})}^{2}}{\mathrm{\Delta}{T}^{2}}}}{\sqrt{\pi \mathrm{\Delta}{T}^{2}}}\end{array}$$(12)The boundary and initial conditions of the thermal storage unit can be written as:

$$\begin{array}{rl}& \text{lower}\phantom{\rule{thinmathspace}{0ex}}\text{horizontal}\phantom{\rule{thinmathspace}{0ex}}\text{wall}:\\ & -k\frac{\mathrm{\partial}T\left(x,0,t\right)}{\mathrm{\partial}y}={h}_{}\left(T\left(x,0,t\right)-{T}_{\mathrm{\infty}}\right),u=v=0,\\ & \text{upper}\phantom{\rule{thinmathspace}{0ex}}\text{horizontal}\phantom{\rule{thinmathspace}{0ex}}\text{wall},\\ & -k\frac{\mathrm{\partial}T\left(x,L,t\right)}{\mathrm{\partial}y}={h}_{}\left(T\left(x,L,t\right)-{T}_{\mathrm{\infty}}\right),u=v=0,\\ & \text{right}\phantom{\rule{thinmathspace}{0ex}}\text{wall}:\\ & -k\frac{\mathrm{\partial}T\left(x,L,t\right)}{\mathrm{\partial}y}={h}_{}\left(T\left(L,y,t\right)-{T}_{\mathrm{\infty}}\right),u=v=0,\\ & \text{left}\phantom{\rule{thinmathspace}{0ex}}\text{wall}:\\ & T(0,y,t)={T}_{h}{,}_{}u=v=0,\\ & \text{interface}\phantom{\rule{thinmathspace}{0ex}}\text{condition}:\phantom{\rule{thinmathspace}{0ex}}T(D,y,t)={T}_{m},\\ & \rho {h}_{f}\frac{\mathrm{\partial}D}{\mathrm{\partial}t}=-{k}_{}\left(\frac{\mathrm{\partial}T\left(D,y,t\right)}{\mathrm{\partial}x}-\frac{\mathrm{\partial}D}{\mathrm{\partial}y}\frac{\mathrm{\partial}T\left(D,y,t\right)}{\mathrm{\partial}y}\right),\\ & \text{initial}\phantom{\rule{thinmathspace}{0ex}}\text{condition}:\\ & T(x,y,0)={T}_{m},\phantom{\rule{thinmathspace}{0ex}}u=v=0.\end{array}$$(13)where *h* is the convection heat transfer coefficient, *T*_{∞} is the ambient temperature, *L* is the height of the unit, and *D* is the position of the melting interface starting from the left wall.

The averaged Nusselt number is calculated from [9]

$$\begin{array}{r}Nu=\frac{1}{\mathrm{\Delta}T}\underset{0}{\overset{L}{\int}}-{\left.\frac{\mathrm{\partial}T}{\mathrm{\partial}x}\right|}_{x=0}dy\end{array}$$(14)
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