Theory for Heat Transfer with Phase Change

The Phase Change Material subnode is used to solve the heat equation after specifying the properties of a phase change material according to the apparent heat capacity formulation.

Instead of adding a latent heat L in the energy balance equation exactly when the material reaches its phase change temperature Tpc, it is assumed that the transformation occurs in a temperature interval between Tpc − ΔT ⁄ 2 and Tpc + ΔT ⁄ 2. In this interval, the material phase is modeled by a smoothed function, θ, representing the fraction of phase before transition, which is equal to 1 before Tpc − ΔT ⁄ 2 and to 0 after Tpc + ΔT ⁄ 2. The density, ρ, and the specific enthalpy, H, are expressed by:

where the indices ph1 and ph2 indicate a material in phase 1 or in phase 2, respectively. Differentiating with respect to temperature, this equality provides the following formula for the specific heat capacity:

which becomes, after some formal transformations:

Here, θ1 and θ2 are equal to θ and 1−θ, respectively. See Figure 4-1 for details.

Figure 4-1: Phase indicators, phase change temperature, and transition interval.

The mass fraction, αm, is defined from ρph1, ρph2 and θ according to:

It is equal to −1 ⁄ 2 before transformation and 1 ⁄ 2 after transformation. The specific heat capacity is the sum of an equivalent heat capacity Ceq:

and the distribution of latent heat CL:


	In the ideal case, when 1 − θ is the Heaviside function (equal to 0 before Tpc and to 1 after Tpc), dαm ⁄ dT is the Dirac pulse. Therefore, CL is the enthalpy jump, L, at temperature Tpc that is added when you have a pure substance.

The latent heat distribution CL is approximated by

so that the total heat per unit volume released during the phase transformation coincides with the latent heat:


	The latent heat, L, can depend on the absolute pressure but should not depend on the temperature.

Finally, the apparent heat capacity, Cp, used in the heat equation, is given by:

The effective thermal conductivity reduces to:

and the effective density is:

To satisfy energy and mass conservation in phase change models, particular attention should be paid to the density in time simulations. When the fluid density is not constant over time, for example, dependent on the temperature, the transport velocity field and the density must be defined so that mass is conserved locally.


	The Moving Mesh Interface (described in the COMSOL Multiphysics Reference Manual) can be used to account for model deformation.