How to solve Phase Change Stefan problem with the PDEtool?

I am trying to solve the Heat transfer Stefan problem with the pdetool in Matlab (governing equations are reported in "PDE governing equations".png, without convection), with the initial and boundary conditions attached.

I don't understand how to specify the required conditions for the 2 different geometries (represented in the "1D melting process".png) and how to build them in the PDE Modeler. The attached file .m only refers to the first geometry (liquid) and the input data are the following:

Thanks in advance for the help.

rho = 1370; % density [kg/m^3] 
k_s = 0.830; % thermal conductivity solid phase [W/(m K)]
k_l = 0.660; % thermal conductivity liquid phase[W/(m K)]
cp_s = 1.69; % specific heat capacity solid phase [kJ/(kg K)]
cp_l = 1.96; % specific heat capacity liquid phase [kJ/(kg K)]
L = 227; % latent heat of the phase change [kJ/kg]
T_melt = 115+273; % melting temperature of the PCM [K]
alpha_s = k_s/(rho*cp_s); % thermal diffusivity solid phase [m^2/s]
alpha_l = k_l/(rho*cp_l); % thermal diffusivity liquid phase [m^2/s]
T_0 = -10+273; % initial temperature of the PCM [K]

[Torsten]

I don't see an equation for dX_f/dt in your attached files and a second transmission condition at X=X_f besides that the temperatures at the two sides are equal.

[Elisa Revello]

In the text of the problem is just reported the Stefan condition attached, in X=X_f.

X_f(t) is defined as the position of the liquid-solid interface, given by: X_f(t) = 2*xi*sqrt(alpha_l*t), where xi is implicitly obtained by solving a trascendental equation.

Many thanks

Answer 1

[Torsten]

There is no chance to solve the problem with a standard tool to solve partial differential equations (like the PDE Toolbox).

I suggest to non-dimensionalize the spatial coordinate as described in

https://www.sciencedirect.com/science/article/pii/S0377042714000077

and use ODE15S to solve the resulting system of ordinary differential equations for the temperature.

Look up "method-of-lines" for more details.