Temperature dependent thermal diffusivity using PDE Toolbox

Hello,

This is my first implementation of a problem using the PDE Toolbox. My 3D time-dependent thermal model is the following : a sphere (radius equals to 1) with radiative flux on the outside surface, with cell at inital temperature . I would like to use a temperature dependent thermal diffusivity where . I have then an analytical expression for it given by : , which I know from documentation for my material.

The parameters used in my simulations are the following :

lambda = @(location,state) 0.46+0.95*exp(-2.3e-3*state.u);  % (W/m/K) Thermal conductivity
rho=1000;              % (kg/m**3) Density
cp=1000;               % (J/kg/K) Specific heat
T0=2000;               % (K) Initial temperature
T_out=300;             % (K) outer space temperature
eps=1;                 % Emissivity 
dt=20;                 % (s) time-step 
day=3600*24;
tmax=2*day;
tlist = [0:dt:tmax];   % time list

I then implement my model using Matlab PDE Toolbox :

thermalModel = createpde('thermal','transient');
gm = multisphere(1);
thermalModel.Geometry=gm;
generateMesh(thermalModel,'Hmax',0.2,"GeometricOrder","quadratic");
thermalModel.StefanBoltzmannConstant = 5.670373E-8;
thermalProperties(thermalModel,'ThermalConductivity',lambda,'MassDensity',rho,'SpecificHeat',cp);

% Initial Temperature
thermalIC(thermalModel,T0);

% Radiative Flux : 
thermalBC(thermalModel,"Face",1,"Emissivity",@(region,state) eps,"AmbientTemperature",T_out, "Vectorized","on");

% Solver
thermalResults = solve(thermalModel,tlist);

Does anyone know whether the implementation is correct ?

Many thanks for any help you can give me !

Regards,

Tom

[Torsten]

Since your problem is 1-dimensional at the moment (solution only depends on r, I guess), I'd compare with the "pdepe" solution.

The unit factor of 1e-6 is no longer necessary in the formula for thermal conductivity ?

[Tom]

As I had a 1e-6 factor in the diffusivity expression, I declared and , to avoid multiplying by 1e-6, as the expression of diffusivity is .

The suggested problem is indeed a 1-dimensional case, however this was to use a sphere with voids afterwards.

[Torsten]

What does it mean: sphere with voids ? A sphere with holes in it that make the temperature distribution asymmetric ?

As said: at this stage, I'd use "pdepe" for validation. Since it is a one-dimensional solver, the mesh can be chosen extremely fine such that it will ressemble comparing your PDE toolbox results with an analytical solution.

[Umar]

Based on the information provided, it appears that you have successfully implemented a 3D time-dependent thermal model in Matlab using the PDE Toolbox. Your model consists of a sphere with radiative flux on the outside surface, and you have incorporated a temperature-dependent thermal diffusivity function λ(u) = 0.46 + 0.95 * exp(-2.3e-3 * u) as well as other parameters such as density, specific heat, initial temperature, outer space temperature, emissivity, time step, and simulation duration. Your implementation involves defining the geometry, mesh generation, setting Stefan-Boltzmann constant, specifying thermal properties, initializing initial temperature conditions, defining radiative flux boundary conditions, and solving the model over a specified time interval. To verify if your implementation is correct, you can consider the following aspects: 1. Check if the physical properties and boundary conditions are correctly defined based on your problem statement. 2. Verify that the mesh generation parameters are appropriate for capturing the spatial resolution required for accurate simulations. 3. Ensure that the thermal properties function λ(u) captures the desired temperature dependency accurately. 4. Confirm that the initial and boundary conditions are consistent with your model requirements. 5. Validate the solver settings and time-stepping scheme to ensure numerical stability and accuracy. You can further assess the correctness of your implementation by comparing simulation results with analytical solutions (if available), conducting sensitivity analyses on key parameters, and performing convergence studies to evaluate mesh independence. Overall, your approach seems comprehensive and well-structured. By rigorously validating your model against known benchmarks and conducting thorough testing, you can gain confidence in the accuracy and reliability of your thermal simulation results. If you encounter any discrepancies or unexpected behavior during validation, consider refining your model assumptions or numerical settings to address potential issues. I hope this guidance helps in evaluating the correctness of your implementation. Feel free to reach out for further assistance or clarification on specific aspects of your thermal model. Good luck with your simulations!