# Boundary conditions at solid-liquid interface

For a solid-liquid phase change of a PCM with a single melting temperature, the solid-liquid interface clearly delineates the liquid and solid phases. The boundary conditions at this interface must be specified in order to solve the problem. As shown for the one-dimensional melting problem illustrated in Fig. 1, the solid-liquid interface separates the liquid and solid phases. The temperatures of the liquid and solid phases near the interface must equal the temperature of the interface, which is at the melting point, Tm. Therefore, the boundary conditions at the interface can be expressed as

${T_\ell }(x,t) = {T_s}(x,t) = {T_m},\quad \quad x = s(t) \qquad \qquad(1)$

where ${T_\ell }(x,t)$ and Ts(x,t) are the temperatures of the liquid and solid phases, respectively.

(a) Melting (T0 > Tm > Ti) (b) Solidification (T0 < Tm < Ti)

Figure 1: Two-region melting and solidification.
Figure 2: Solidification of a multicomponent PCM

For the melting process, if the initial temperature of the PCM, Ti, is below the melting point of the PCM, Tm, (or above, for solidification), the temperature distribution of both the liquid and solid phases must be determined; this is called a two-region problem. Figure 1 shows the temperature distribution of one-dimensional two-region melting and solidification problems.

For a multi-component PCM, the solid-liquid phase change process occurs over a range of temperatures (Tm1, Tm2), instead of a single temperature. The PCM is liquid if its temperature is above Tm2 and solid when its temperature is below Tm1. Between the solid and liquid phases there is a mushy zone where the temperature falls within the phase change temperature range (Tm1, Tm2). Successful solution of phase change problems involving these substances requires determination of the temperature distribution in the liquid, solid, and mushy zones; therefore, these are referred to as multiregion problems. The temperature distribution of one-dimensional solidification in a multicomponent PCM is shown in Fig. 2, where it can be seen that the solution requires tracking of two interfaces.

In solid-liquid phase change problems, the location of the solid-liquid interface is unknown before the final solution is obtained and this presents a special difficulty. Since the interface also moves during melting or solidification, such problems are referred to as moving boundary problems and always have time as an independent variable.

For a solid-liquid phase change of a PCM with a single melting temperature, the solid-liquid interface clearly delineates the liquid and solid phases. The boundary conditions at this interface must be specified in order to solve the problem. As shown for the one-dimensional melting problem illustrated in Fig. 2, the solid-liquid interface separates the liquid and solid phases. The temperatures of the liquid and solid phases near the interface must equal the temperature of the interface, which is at the melting point, Tm. Therefore, the boundary conditions at the interface can be expressed as

${T_\ell }(x,t) = {T_s}(x,t) = {T_m},\quad \quad x = s(t) \qquad \qquad(1)$

where ${T_\ell }(x,t)$ and Ts(x,t) are the temperatures of the liquid and solid phases, respectively.

The density of the PCM usually differs between the liquid and solid phases; therefore, density change always accompanies the phase change process. The solid PCM is usually denser than the liquid PCM, except for a few substances such as water and gallium. For example, the volume of paraffin, a very useful PCM for energy storage systems, expands about 10% when it melts. Therefore, the density of the liquid paraffin, ${\rho _\ell },$ is less than the density of the solid paraffin, ρs. When water freezes, however, its volume increases, so the density of ice is less than that of liquid water. The density change that occurs during solid-liquid phase change will produce an extra increment of motion in the solid-liquid interface. For the melting problem in Fig. 2(a), where the liquid phase velocity at x = 0 is zero, if the density of the solid is larger than the density of the liquid (i.e., ${\rho _s} > {\rho _\ell }$) the resulting extra motion of the interface is along the positive direction of the x- axis. Assume that the velocity of the solid-liquid interface due to phase change is up = ds / dt, while the extra velocity of the solid-liquid interface due to density change is uρ. The density change must satisfy the conservation of mass at the interface, i.e.,

${\rho _s}({u_p} - {u_\rho }) = {\rho _\ell }{u_p} \qquad \qquad(2)$

The extra velocity induced by the density change can be obtained by rearranging eq. (2) as

${u_\rho } = \frac{{{\rho _s} - {\rho _\ell }}}{{{\rho _s}}}{u_p} \qquad \qquad(3)$

which is also valid for case ${\rho _s} > {\rho _\ell },$ except the extra velocity becomes negative. Another necessary boundary condition is the energy balance at the solid-liquid interface. If the enthalpy of the liquid and solid phases at the melting point are ${h_\ell }$ and hs, the energy balance at the solid-liquid interface can be expressed as:

${q''_\ell } - {q''_s} = {\rho _\ell }{u_p}{h_\ell } - {\rho _s}({u_p} - {u_\rho }){h_s}\quad \quad x = s(t) \qquad \qquad(4)$

where ${q''_\ell }$ and q''s are the heat fluxes in the x-direction in the liquid and solid phases, respectively. Substituting eq. (3) into eq. (4), one obtains

${q''_\ell } - {q''_s} = {\rho _\ell }{h_{s\ell }}\frac{{ds}}{{dt}}\quad \quad x = s(t) \qquad \qquad(5)$

where ${h_{s\ell }} = {h_\ell } - {h_s}$ is the latent heat of melting. If convection in the liquid phase can be neglected and heat conduction is the only heat transfer mechanism in both the liquid and solid phases, the heat flux in both phases can be determined by Fourier’s law of conduction:

${q''_\ell } = \mathop {\left. { - {k_\ell }\frac{{\partial {T_\ell }(x,t)}}{{\partial x}}} \right|}{x = s(t)} \qquad \qquad(6)$

${q''_s} = \mathop {\left. { - {k_s}\frac{{\partial {T_s}(x,t)}}{{\partial x}}} \right|}{x = s(t)} \qquad \qquad(7)$

The energy balance at the solid-liquid interface for a melting problem can be obtained by substituting eqs. (6) and (7) into eq. (5), i.e.,

${k_s}\frac{{\partial {T_s}(x,t)}}{{\partial x}} - {k_\ell }\frac{{\partial {T_\ell }(x,t)}}{{\partial x}} = {\rho _\ell }{h_{s\ell }}\frac{{ds(t)}}{{dt}}\quad \quad x = s(t) \qquad \qquad(8)$

For a solidification process, the energy balance equation at the interface can be obtained by a similar procedure:

${k_s}\frac{{\partial {T_s}(x,t)}}{{\partial x}} - {k_\ell }\frac{{\partial {T_\ell }(x,t)}}{{\partial x}} = {\rho _s}{h_{s\ell }}\frac{{ds(t)}}{{dt}}\quad \quad x = s(t) \qquad \qquad(9)$

The only difference between eqs. (8) or (9) is the density on the right-hand side of the equation. If the temperature distributions in the liquid and solid phases are known, the location of the solid-liquid interface can be obtained by solving eqs. (8) or (9). It should be noted that density change causes advection in the liquid phase, which further complicates the problem. The above boundary conditions are valid for one-dimensional problems only. For multi-dimensional phase change problems, the boundary conditions at the interface can be expressed as

${T_\ell }(x,y,t) = {T_s}(x,y,t) = {T_m}\quad \quad x = s(y,t) \qquad \qquad(10)$

${k_s}\frac{{\partial {T_s}(x,y,t)}}{{\partial n}} - {k_\ell }\frac{{\partial {T_\ell }(x,y,t)}}{{\partial n}} = {\rho _\ell }{h_{s\ell }}{v_n}\quad \quad x = s(y,t) \qquad \qquad(11)$

where ${\mathbf{n}}$ is a unit vector along the normal direction of the solid-liquid interface, and vn is the solid-liquid interface velocity along the n-direction. It is apparent that eq. (11) is not convenient for numerical solution because it contains temperature derivatives along the n-direction. Suppose the shape of the solid-liquid interface can be expressed as

x = s(y,t)

Equation (11) can then become the following form (see Problem 3.55; Ozisik, 1993):

$\left[ {1 + \mathop {\left( {\frac{{\partial s}}{{\partial y}}} \right)}^2 } \right]\left[ {{k_s}\frac{{\partial {T_s}}}{{\partial x}} - {k_\ell }\frac{{\partial {T_\ell }}}{{\partial x}}} \right] = {\rho _\ell }{h_{s\ell }}\frac{{\partial s}}{{\partial t}}\quad \quad x = s(y,t) \qquad \qquad(12)$

Similarly, for a three-dimensional melting problem with an interface described by

z = s(x,y,t)

the energy balance at the interface is

$\left[ {1 + \mathop {\left( {\frac{{\partial s}}{{\partial x}}} \right)}^2 + \mathop {\left( {\frac{{\partial s}}{{\partial y}}} \right)}^2 } \right]\left[ {{k_s}\frac{{\partial {T_s}}}{{\partial x}} - {k_\ell }\frac{{\partial {T_\ell }}}{{\partial x}}} \right] = {\rho _\ell }{h_{s\ell }}\frac{{\partial s}}{{\partial t}}\quad \quad z = s(x,y,t) \qquad \qquad(13)$

For solidification problems, it is necessary to replace the liquid density ${\rho _\ell }$ in eqs. (12) and (13) with the solid-phase density ρs. The density change in solid-liquid phase change is often neglected in the literature in order to eliminate the additional interface motion discussed earlier. This section presents a number of prototypical problems of melting and solidification; its goal is to establish the physics of this form of phase change and to demonstrate the variety of tools available for its solution.

## References

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

Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.

Ozisik, M.N., 1993, Heat Conduction, 2nd ed., Wiley-Interscience, New York.