# Two-region melting and solidification in porous media

Figure 1: Control volume for volume averaging.

A more generalized model for solid-liquid phase change in porous media was proposed by Beckermann and Viskanta (1988). Instead of using Darcy’s law, which is valid only at very low velocity, the effect of inertia terms was included in the model. This treatment allows for nonslip conditions at the wall of the cavity, e.g., v = 0 at x = 0 (under Darcy’s law, slip occurs at the wall). The generalized governing equations that are applicable to the entire computational domain, including both liquid and solid phases, are obtained by using an enthalpy model that can be applied to a two-region problem (melting of a subcooled solid or solidification of a superheated liquid).

Beckermann and Viskanta (1988) developed their model based on volume-averaged governing equations. Figure 1 shows the illustration of a phase change process in a porous medium. The volume fraction of pore space in the porous medium is defined as

${{\varepsilon }_{p}}=\Delta {{V}_{p}}/\Delta V\qquad\qquad(1)$

where ΔVp is the volume of porous space and ΔV is the total volume. The porous space can be occupied by either liquid ($\ell$) or solid (s) phase of the PCM. The liquid volume fraction in the porous space is defined as

$\gamma =\Delta {{V}_{\ell }}/\Delta {{V}_{p}}\qquad\qquad(2)$

The volume fraction of the liquid in the porous media is therefore

${{\varepsilon }_{\ell }}=\Delta {{V}_{\ell }}/\Delta V={{\varepsilon }_{p}}\gamma \qquad\qquad(3)$

Although the PCM under consideration is single-component and its melting point is well defined, the melting front can have a finite thickness because the phase change can be simultaneously inhabited by solid and liquid in the pore. A two-phase region, similar to a mushy zone for a binary system, exists between the solid and liquid phases. The liquid fraction, γ, in this two phase region varies between 0 to 1, while the average temperature in this two phase region is equal to the melting point, Tm. It was assumed that the liquid fraction and the temperature in the two-phase region were related by

$\gamma =\frac{T-{{T}_{m}}+\Delta T}{2\Delta T}\begin{matrix} , & {{T}_{m}}-\Delta T\le T\le {{T}_{m}}+\Delta T \\ \end{matrix}\qquad\qquad(4)$

Viskanta (1988) assumed that: (1) the flow and heat transfer is two-dimensional and laminar, (2) the properties of both porous matrix and PCM are homogeneous and isotropic, (3) the porous matrix and the PCM are at thermal equilibrium, (4) the velocities of the porous matrix and solid phase are zero, (5) the flow is incompressible and the Boussinesq approximation (density change is considered only in the buoyancy term) applies, (6) the thermal physical properties are constants, (7) the dispersion flux due to velocity fluctuation is negligible, and (8) the densities of the liquid and solid phases are identical. The continuity, momentum, and energy equations are

$\nabla \cdot \mathbf{V}=0\qquad\qquad(5)$
\begin{align} & \frac{{{\rho }_{\ell }}}{{{\varepsilon }_{\ell }}}\left[ \frac{\partial \mathbf{V}}{\partial t}+\frac{1}{{{\varepsilon }_{\ell }}}(\mathbf{V}\cdot \nabla )\mathbf{V} \right]=-\nabla p+\frac{{{\mu }_{\ell }}}{{{\varepsilon }_{\ell }}}{{\nabla }^{2}}\mathbf{V} \\ & -\left( \frac{{{\mu }_{\ell }}}{K}+\frac{{{\rho }_{\ell }}{{C}_{f}}}{{{K}^{{1}/{2}\;}}}\left| \mathbf{V} \right| \right)\mathbf{V}-{{\rho }_{\ell }}\mathbf{g}\beta (T-{{T}_{ref}}) \\ \end{align}\qquad\qquad(6)
${{(\rho {{c}_{p}})}_{eff}}\frac{\partial T}{\partial t}+{{\rho }_{\ell }}{{c}_{p\ell }}(\mathbf{V}\cdot \nabla T)=-\nabla \cdot ({{k}_{eff}}\nabla T)-{{\varepsilon }_{p}}{{\rho }_{\ell }}{{h}_{s\ell }}\frac{\partial \gamma }{\partial t}\qquad\qquad(7)$

where $\mathbf{V}={{\left\langle {{\mathbf{V}}_{\ell }} \right\rangle }^{\ell }}$ is intrinsic phase-averaged velocity. For the porous medium consisting of spherical beads with diameter dp, the permeability can be obtained using the following Kozeny-Carman relation:

Figure 2: Solid-liquid phase change in a rectangular cavity filled with porous media
$K=\frac{d_{p}^{2}\varepsilon _{\ell }^{3}}{175{{(1-{{\varepsilon }_{\ell }})}^{2}}}\qquad\qquad(8)$

which indicates that the permeability is zero in the solid region. The Forchheimer inertia coefficient Cf is taken as 0.55 (Ward, 1964). The effective heat capacity ${{(\rho {{c}_{p}})}_{eff}}$ in eq. (8) is

${{(\rho {{c}_{p}})}_{eff}}={{\varepsilon }_{p}}\rho \left[ \gamma {{c}_{p\ell }}+(1-\gamma ){{c}_{ps}} \right]+(1-{{\varepsilon }_{p}}){{(\rho {{c}_{p}})}_{sm}}\qquad\qquad(9)$

where (pcp)sm is heat capacity of the porous matrix. The effective thermal conductivity of the porous medium saturated with PCM is estimated by (Veinberg, 1967)

${{k}_{eff}}+\varepsilon k_{eff}^{1/3}\frac{{{k}_{sm}}-{{k}_{\ell s}}}{k_{\ell s}^{1/3}}-{{k}_{sm}}=0\qquad\qquad(10)$

where ${{k}_{\ell s}}$ is the average thermal conductivity of the PCM:

${{k}_{\ell s}}=\gamma {{k}_{\ell }}+(1-\gamma ){{k}_{s}}\qquad\qquad(11)$

Beckermann and Viskanta (1988) applied the above model to solve solid-liquid phase change problems as shown in Fig. 2. A rectangular cavity is filled with a rigid porous matrix that is saturated with PCM. The left wall temperature is held at Th, which is above the melting point of the PCM, Tm, while the right wall is kept at a temperature Tc, below the melting point. The top and bottom of the cavity are adiabatic. The initial temperature of the porous medium and PCM, Ti, is equal to Tc for a melting problem or Th for a solidification problem.

The governing equations were nondimensionalized and numerical simulations for both melting and solidification were performed by Beckermann and Viskanta (1988). They also performed experiments on melting and solidification in a system containing spherical glass beads (dp = 6 mm, εp = 0.385) and gallium as a PCM (Tm = 29.78 °C) in a 3.81×4.76cm2 (L × H) rectangular cavity.

## References

Beckermann, C., and Viskanta, R., 1988, “Natural Convection Solid/Liquid Phase Change in Porous Media,” International Journal of Heat and Mass Transfer, Vol. 31, pp. 35-46.

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

Nield, D.A., Bejan, A., 1999, Convection in Porous Media, 2nd ed., Springer-Verlag, New York.

Oosthuizen P.H., 1988, “The Effects of Free Convection on Steady State Freezing in a Porous Medium Filled Cavity,” ASME HTD-Vol. 96, Vol. 1, pp. 321–327.

Veinberg, A.K., 1967, “Permeability, Electrical Conductivity, Dielectric Constant and Thermal Conductivity of a Medium with Spherical and Ellipsoidal Inclusions,” Soviet Phys. Dokl., Vol. 11, pp. 593-595.

Viskanta, R., 1988, “Heat Transfer During Melting and Solidification of Metals,” ASME Journal of Heat Transfer, Vol. 110, 1205-1219.

Ward, J. C., 1964, “Turbulent Flow in Porous Media,” ASCE J. Hyd. Div., Vol. 90, HY5, pp. 1-12.