How to implement the radiation boundary condition for the object in vacuum?

I am trying to simulate heat transfer in the object that is mounted inside the vacuum chamber using the PDE toolbox. The thing is cooled internally with water running through the pipe inside. But I suppose that it can also loose some energy via the radiation from the boundaries into the surrounding vacuum. The question is how to program this radiative BC into vacuum correctly in Matlab using the PDE toolbox. Another question is if this radiative heat loss is even worth taking it into account, if it would be significant compared to other cooling mechanisms.

[David Goodmanson]

Hi Alexander,

for the second question you can do an approximate calculation for the size of the effect. For an object of area A, temperature T in kelvin, Tc the temperature of the walls of the vacuum chamber, and eps the emissivity (usually between 0 and 1) , from the Stefan-Boltmann law

W = 5.67e-8*eps*A*(T^4 -Tc^4)

where W is the radiated power in watts. So for example if A = 2, Tc = 296, T = 346, eps = .8, then W = 604 watts.

[Alexander]

Thank you!

[Umar]

Hi @Alexander ,

To address your query regarding, “The question is how to program this radiative BC into vacuum correctly in Matlab using the PDE toolbox.”

Define the radiative heat transfer through the boundaries of the vacuum chamber by specifying the radiative heat flux at the boundaries to account for the energy radiated from the object to the surroundings. Here is a simplified example of how you can incorporate radiative boundary conditions in your Matlab PDE Toolbox simulation.

% Define radiative heat flux at the boundary
q_rad = @(T) epsilon * sigma * (T^4 - T_surroundings^4);
% Apply radiative boundary condition
thermalBC(thermalmodel, 'Edge', 1:4, 'HeatFlux', q_rad, 'Temperature', T_surroundings);

For more information regarding thermalBC function, please refer to

https://www.mathworks.com/help/pde/ug/pde.thermalmodel.thermalbc.html

You already familiar with these constants used in physics but I will elaborate on them anyway, epsilon is the emissivity of the object's surface, sigma is the Stefan-Boltzmann constant, T is the temperature of the object., T_surroundings is the temperature of the vacuum surroundings. So, by defining the radiative heat flux function q_rad based on the Stefan-Boltzmann law and applying it as a boundary condition, you can simulate the radiative heat transfer accurately.

Now, addressing your next query, “Another question is if this radiative heat loss is even worth taking it into account, if it would be significant compared to other cooling mechanisms”, my comments: incorporating radiative heat loss is crucial for accurate results, especially in vacuum conditions where convection is minimal and @David Goodmanson already pointed out on how to model radiative heat transfer using the Stefan-Boltzmann law and he has provided a nice example to illustrate it. In addition to specify the Stefan-Boltzmann law in the PDE toolbox for radiative heat loss, here is an example to demonstrate it.

thermalBC(thermalmodel,'Edge',1,'HeatFlux',@(region,state) emissivity * sigma * (state.T - Tamb)^4);

So, the terms emissivity is the emissivity of the material, sigma is the Stefan-Boltzmann constant, Tamb is the ambient temperature, and state.T is the temperature at the boundary. You can adjust these parameters according to your specific case to evaluate the significance of radiative heat loss in your simulation or using @David Goodmanson’s example. Please let us know if you have any further questions.