# One-region melting in porous media

Convection-controlled melting in a porous medium.

Melting in an enclosure filled with a porous medium and solid PCM, studied by Jany and Bejan (1987) will be presented here (see figure on the right). It is assumed that all of the walls of the two-dimensional rectangular cavity are impermeable, and that the initial temperature of the porous medium and PCM, Ti, is at the melting point of the PCM, Tm. At time t = 0, the temperature of the left side of the cavity is increased to Tw, while the right side, top, and bottom of the cavity remain insulated. The following assumptions are made:

1. Liquid flow is very slow, so Darcy’s law is applicable.

2. The PCM and porous medium are in thermal equilibrium.

3. The porous medium is isotropic.

4. Constant thermal properties for both PCM and porous medium.

Since the temperature in the solid region is equal to the melting point, this is a one-region melting problem. The governing equations for the liquid region are

$\frac{\partial \left\langle {{u}_{\ell }} \right\rangle }{\partial x}+\frac{\partial \left\langle {{v}_{\ell }} \right\rangle }{\partial y}=0 \qquad\qquad(1)$

where $\left\langle {{u}_{\ell }} \right\rangle \text{ and }\left\langle {{v}_{\ell }} \right\rangle$ are the components of extrinsic phase-averaged velocity in the x- and y-directions and they can be obtained by Darcy’s law:

$\left\langle {{u}_{\ell }} \right\rangle =-\frac{K}{{{\mu }_{\ell }}}\frac{\partial {{\left\langle {{p}_{\ell }} \right\rangle }^{\ell }}}{\partial x}\qquad\qquad(2)$
$\left\langle {{v}_{\ell }} \right\rangle =-\frac{K}{{{\mu }_{\ell }}}\left\{ \frac{\partial {{\left\langle {{p}_{\ell }} \right\rangle }^{\ell }}}{\partial y}+{{\rho }_{\ell }}g\left[ 1-\beta \left( \left\langle T \right\rangle -{{T}_{m}} \right) \right] \right\}\qquad\qquad(3)$

The energy equation is

$\Upsilon \frac{\partial \left\langle T \right\rangle }{\partial t}+\left\langle {{u}_{\ell }} \right\rangle \frac{\partial \left\langle T \right\rangle }{\partial x}+\left\langle {{v}_{\ell }} \right\rangle \frac{\partial \left\langle T \right\rangle }{\partial y}={{\alpha }_{m}}\left( \frac{{{\partial }^{2}}\left\langle T \right\rangle }{\partial {{x}^{2}}}+\frac{{{\partial }^{2}}\left\langle T \right\rangle }{\partial {{y}^{2}}} \right)\qquad\qquad(4)$

where

$\Upsilon =\varepsilon +(1-\varepsilon )\frac{{{(\rho {{c}_{p}})}_{sm}}}{{{(\rho {{c}_{p}})}_{\ell }}}\qquad\qquad(5)$

is the ratio of the heat capacity of the porous medium saturated with PCM over the heat capacity of the PCM. ε is the volume fraction of PCM, ${{(\rho {{c}_{p}})}_{\ell }}\text{ and }{{(\rho {{c}_{p}})}_{sm}}$ are heat capacities of the liquid PCM and porous matrix. The thermal diffusivity of the porous medium saturated with the PCM is ${{\alpha }_{m}}={{k}_{eff}}/{{(\rho {{c}_{p}})}_{\ell }}$.

Equations (1) – (4) are subjected to the following boundary conditions:

$\left\langle {{u}_{\ell }} \right\rangle =0\begin{matrix} , & \left\langle T \right\rangle ={{T}_{w}}, & x=0 \\ \end{matrix}\qquad\qquad(6)$
$\left\langle {{u}_{\ell }} \right\rangle =0\begin{matrix} , & \left\langle T \right\rangle ={{T}_{m}}, & x=s \\ \end{matrix}\qquad\qquad(7)$
$\left\langle {{v}_{\ell }} \right\rangle =0\begin{matrix} , & \frac{\partial \left\langle T \right\rangle }{\partial y}=0, & y=0,H \\ \end{matrix}\qquad\qquad(8)$

The energy balance at the solid-liquid interface is (Jany and Bejan, 1987)

$\frac{\partial s}{\partial t}=-\frac{{{\alpha }_{m}}{{c}_{p\ell }}}{{{h}_{s\ell }}}\left( \frac{\partial \left\langle T \right\rangle }{\partial x}-\frac{\partial s}{\partial y}\frac{\partial \left\langle T \right\rangle }{\partial y} \right)\begin{matrix} , & x=s \\ \end{matrix}\qquad\qquad(9)$

where the second term in the parentheses at the right-hand side represents the effect of the inclination of the interface. To change $\left\langle {{u}_{\ell }} \right\rangle \text{ and }\left\langle {{v}_{\ell }} \right\rangle$ into one dependent function, a stream function, ψ, is defined as

$\left\langle {{u}_{\ell }} \right\rangle =\frac{\partial \psi }{\partial y}\begin{matrix} , & \left\langle {{v}_{\ell }} \right\rangle =-\frac{\partial \psi }{\partial x} \\ \end{matrix}\qquad\qquad(10)$

Introducing the following dimensionless variables

\begin{align} & X=\frac{x}{H},\text{ }Y=\frac{y}{H},\text{ }Fo=\frac{{{\alpha }_{m}}t}{{{H}^{2}}},\text{ }S=\frac{s}{H},\text{ }\left\langle \text{ }{{U}_{\ell }} \right\rangle =\frac{\left\langle {{u}_{\ell }} \right\rangle H}{{{\alpha }_{m}}},\text{ }\left\langle \text{ }{{V}_{\ell }} \right\rangle =\frac{\left\langle {{v}_{\ell }} \right\rangle H}{{{\alpha }_{m}}} \\ & \left\langle \theta \right\rangle =\frac{\left\langle T \right\rangle -{{T}_{m}}}{{{T}_{w}}-{{T}_{m}}},\text{ }\Psi =\frac{\psi }{{{\alpha }_{m}}},\text{ }Ra=\frac{g\beta KH({{T}_{w}}-{{T}_{m}})}{{{\nu }_{\ell }}{{\alpha }_{m}}},\text{ }Ste=\frac{{{c}_{p\ell }}({{T}_{w}}-{{T}_{m}})}{{{h}_{s\ell }}} \\ \end{align}\qquad\qquad(11)

where Ra and Ste are the Rayleigh and Stefan numbers, respectively, the governing equations and boundary conditions become

$\frac{{{\partial }^{2}}\Psi }{\partial {{x}^{2}}}+\frac{{{\partial }^{2}}\Psi }{\partial {{y}^{2}}}=-Ra\frac{\partial \theta }{\partial X}\qquad\qquad(12)$
$\Upsilon \frac{\partial \theta }{\partial Fo}+U\frac{\partial \theta }{\partial X}+V\frac{\partial \theta }{\partial Y}=\frac{{{\partial }^{2}}\theta }{\partial {{x}^{2}}}+\frac{{{\partial }^{2}}\theta }{\partial {{y}^{2}}}\qquad\qquad(13)$
$U=\frac{\partial \Psi }{\partial Y}=0\begin{matrix} , & \theta =1, & X=0 \\ \end{matrix}\qquad\qquad(14)$
$U=\frac{\partial \Psi }{\partial Y}=0\begin{matrix} , & \theta =0, & X=S \\ \end{matrix}\qquad\qquad(15)$
$V=-\frac{\partial \Psi }{\partial X}=0\begin{matrix} , & \frac{\partial \theta }{\partial Y}=0, & Y=0,1 \\ \end{matrix}\qquad\qquad(16)$
$\frac{\partial S}{\partial Fo}=-Ste\left( \frac{\partial \theta }{\partial X}-\frac{\partial S}{\partial Y}\frac{\partial \theta }{\partial Y} \right)\begin{matrix} , & x=S \\ \end{matrix}\qquad\qquad(17)$

where $\left\langle {} \right\rangle$ for phase-averaging has been dropped in the dimensionless variables for ease of notation.

The dimensionless governing equations were solved numerically using a finite difference method by Jany and Bejan (1987).

## References

Faghri, A., and Zhang, Y., 2006, Transport Phenomena in Multiphase Systems, Elsevier, Burlington, MA

Jany, P., and Bejan, 1987, Melting in the Presence of Natural Convection in a Rectangular Cavity Filled with Porous Medium, Report DU-AB-6, Department of Mechanical Engineering and Materials Science, Duke University.