Radiative transfer between two plates

Hi,

I'm new to FeaTool, and I'm trying to simulate the classic example of heat transfer between two plates separated by an air gap. The main mode of transfer is radiative transfer.

To solve this problem, I set up three different physics modes (ht, ht2, ht3), so that I get three domains with dependent temperatures (T, T2, T3). I then set the boundary conditions on the plate-air interfaces to "Heat flux", and set Tamb = T3 in the radiation term for domain 1 (plate with dependent variable T), and Tamb = T in domain 3 (plate with dependent temperature T3). Moreover, initial temperatures are T_0 = 1000 K and T3_0 = 300 K.

My thinking is that as the temperature T in domain 1 decreases, Tamb in domain 3 will decrease, since it is, again, set to T. Conversely, the temperature T3 in domain 3 should increase, as its Tamb decreases.

However, it seems that Tamb always stays at its initial condition, for both plates. So it seems to me that the heat transfer equations in the two plates are solved independently. My question is therefore how I can make Tamb for both plates update at each time step, to simulate the radiative exhange?  

simulatore wrote

To solve this problem, I set up three different physics modes (ht, ht2, ht3), so that I get three domains with dependent temperatures (T, T2, T3). I then set the boundary conditions on the plate-air interfaces to "Heat flux", and set Tamb = T3 in the radiation term for domain 1 (plate with dependent variable T), and Tamb = T in domain 3 (plate with dependent temperature T3).

Hmm, don't think that will quite work, as I believe for example in subdomain 1 (corresponding to T), T2 and T3 are not defined (null or zero), to your approach wouldn't work without some quite advanced custom function to map/average temperature from the other domains. The proper way would be to pointwise compute viewfactors (how much each boundary segment sees the others), however that approach isn't implemented yet as it does not nicely fit in with standard PDE formulations, and is also quite computationally expensive (especially in 3D).

simulatore wrote

So it seems to me that the heat transfer equations in the two plates are solved independently.

No, with the built-in solver all equations are solved simultaneously (unless deactivated).

Thanks for the quick reply!

Precise Simulation wrote

Hmm, don't think that will quite work, as I believe for example in subdomain
1 (corresponding to T), T2 and T3 are not defined (null or zero), to your
approach wouldn't work without some quite advanced custom function to
map/average temperature from the other domains. The proper way would be to pointwise compute viewfactors (how much each boundary segment sees the others), however that approach isn't implemented yet as it does not nicely fit in with standard PDE formulations, and is also quite computationally
expensive (especially in 3D).

Oh, ok. I thought this problem would have an easily attainable albeit rough solution.

Precise Simulation wrote

No, with the built-in solver all equations are solved simultaneously (unless
deactivated).

I mean that domain 1 seems to be "unaware" of what's happening in domain 3 and vice versa. But maybe that's just because, as you say, T3 (T) is currently not defined for domain 1 (3)?

Expressions are evaluated pointwise, so the expressions "T^4 - T3^4" implicitly mean "T(x,y)^4 - T3(x,y)^4", as the software cannot know you want to evaluate T3 from somewhere else. However, for this specific case you could likely use a custom/user-defined function to do this, something like:

    function [T3] = T3_offset( x, y )
    y_offset = 123;   % Your offset to get the coordinate of the boundary where T3 should be evaluated.
    p_eval = [ x(:), y(:) - y_offset ]';
    fea = evalin( 'caller', 'prob' );  % Get local fea problem data.
    T3 = evalexpr( 'T3', p_eval, fea );
    T3 = reshape( T3, size(x) );

https://www.featool.com/doc/evalexpr_8m

However, for more complicated cases it would become exponentially more complicated, and some proper viewfactor algorithm would be required.

Precise Simulation wrote

Expressions are evaluated pointwise, so the expressions "T^4 - T3^4" implicitly mean "T(x,y)^4 - T3(x,y)^4", as the software cannot know you want to evaluate T3 from somewhere else. However, for this specific case you could likely use a custom/user-defined function to do this, something like:

    function [T3] = T3_offset( x, y )
    y_offset = 123;   % Your offset to get the coordinate of the boundary where T3 should be evaluated.
    p_eval = [ x(:), y(:) - y_offset ]';
    fea = evalin( 'caller', 'prob' );  % Get local fea problem data.
    T3 = evalexpr( 'T3', p_eval, fea );
    T3 = reshape( T3, size(x) );

https://www.featool.com/doc/evalexpr_8m

However, for more complicated cases it would become exponentially more complicated, and some proper viewfactor algorithm would be required.

Thanks for the explanation. Since I intend to proceed with more complex geometries after this simple one, I think it looks like I need some other software, unfortunately. Possibly OpenFoam, since I can see that it has several radiation models available. A bit unfortunate, I think, because I've found FeaTool to be quite intuitive to work with.

Sure, although FEATool does also support using OpenFOAM as a solver, support for radiation solvers has not been implemented. Possibly the toolbox could be useful for OpenFOAM pre/post-processing, and modifying case files (to support radiation).