Temperature dependent emissivity test case

Numerical methods and mathematical models of Elmer
arved
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Re: Temperature dependent emissivity test case

Post by arved »

Hi Peter,

thanks for the new sif file! I just ran a couple simulations for comparison:

- with constant emissivity everything works well, there is a good agreement of the temperatures at all boundaries with the analytical solution
- with heater eps=0.7, abs=0.8, insulation eps=0.5, abs=0.4 again good agreement
- with heater eps=0.8, abs=0.7, insulation eps=0.4, abs=0.5 again good agreement
- with heater eps=0.6, abs=0.2, insulation eps=0.2, abs=0.6 there is again a good agreement with the analytical solution, but the result is not physical as the heater outside temperature is lower than insulation inside (I attached the sif)

I think it would help to treat the radiation as a function of wavelength and set emissivity(lambda)=absorptivity(lambda) in agreement with Kirchhoffs law of thermal radiation. Do you plan to implement that?

I slightly modified the analytical solution, it can be found here: https://github.com/nemocrys/elmer-verif ... here.ipynb

Best regards
Arved
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raback
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Re: Temperature dependent emissivity test case

Post by raback »

Hi Arved,

How can there be agreement with analytical solution but results are not physical!? Perhaps you can try to increase nonlinear relaxation. Things become more difficult with the change of parameters.

Assuming that all surfaces radiate with black body spectra I think it should be fine to have just e(T). Then a=e(T_incoming) which is considered in the spectral model (not in the standard radiosity model).

There are also cases when the spectra is not that of black body. It could be some gas burner, for example. Then we can treat those objects as pointwise radiators and use the effective absorptivity related to their spectra and known burner temperature. This is already available as

Radiator Absorptivity i = Real

For each radiating source i.

I know this leaves the combination of unknown temperature heat sources radiating at non-gray spectrum being received by another non-gray surface. If the outgoing spectrum has a distribution I guess we could compute the efficient absorptivity at meterial "i" weighting it with radiation coming from material "j" e.g.

a_ij = \int k f_j(k,T) e_i(k)*e_j(k,T) dk / int k f_j(k) e_j(k) dk

Where k=1/lambda and f_j(k,T) being the distribution related to temperature at material "j". If you are able to compute this then a_ij(T_incoming,material). So technically this could be implemented as "Absorptivity j = Real" type of keywords where "j" would refer to the material seen by the surface. All dependencies related to wave length vanish in the integration.

This is of course choice of strategy. Personally I prefer the temperature basis as it is economical. Each surface has just one temperature. Instead, if you want to integrate over the black body spectrum accurately you need some nifty integration strategies. In this strategy you would do it only once when creating a_ij(). There is no reason why it could not be integrated in the code but it could be preprocessing step.

-Peter

Edit: fixed some typos
arved
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Joined: 06 Aug 2020, 15:05
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Re: Temperature dependent emissivity test case

Post by arved »

Hi Peter,

I set up the analytical solution based on to the book H.D. Baehr, K. Stephan, Wärme- und Stoffübertragung, 7th ed., Springer-Verlag, Berlin Heidelberg New York, 2010. https://doi.org/10.1007/978-3-642-10194-6, Chapter 5.5.3 where a constant emission is assumed. To get an agreement with the Elmer model eps(T) I slightly modified the equations - and by allowing eps != abs I violated Kirchhoffs law, leading to the non-physical temperature distribution. As there is a good agreement between Elmer and my equation I don't assume that we have any problems with convergence or the implementation.

I think the problem is using constant but unequal emissivity / absorptivity (e.g. eps(500°C)=const. != abs(500°C)=const.) - with eps(T) = abs(T) everything should be ok. For that case, however, it wouldn't be easy to compute an analytical solution.

Thanks for all the explanation regarding the temperature dependency, I think it's really a matter of application if this model is useful or not. For crystal growth, an additional "true" temperature dependency as implemented in the old model would be very interesting - but this may require discretization with a spectral dLambda instead of spectral dT.

Best regards
Arved
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