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Thick wall temperature BC with radiation

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Old   June 1, 2021, 07:28
Default Thick wall temperature BC with radiation
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António Galhardo
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Hi everyone,

I am working on a problem with a fire in an enclosure, which i am modelling using firefoam. I have a question regarding which temperature boundary condition I should use at the wall, because it seems to me like OpenFOAM does not offer the right BC for this kind of situation. Heat transfer to the wall is significant and therefore an adiabatic BC is not adequate. Moreover, the effects of both convection and radiation need to be taken into account. The problem is transient, so the wall surface temperature will vary with time, as the wall will heat up due to the fire. As I understand it, the boundary condition to be applied at the wall surface should be an energy balance, meaning that the heat flux from the fire products should equal the heat flux inside the wall:

q_{conv} + q_{rad} = k_{wall}\frac{\partial T}{\partial n}.

Looking through the boundary conditions in OpenFOAM, the closest one I found was wallHeatTransfer, which calculates the wall temperature as:

T_{wall} = wT_{\infty} + (1-w)T_c, \, \:\:  w = \frac{1}{ 1+\frac{ k_{eff} }{\alpha_{wall} \delta}},

where T_c is the temperature in the cell closest to the boundary and T_{\infty} is the ambient temperature. Understanding why this BC works is probably easier after rearranging it to:

\frac{ k_{eff}(T_c - T_{wall}) }{\delta} = \alpha_{wall}(T_{wall} - T_{\infty})

This equation is therefore making an energy balance: the left-hand side has the convective flux from the fluid (with an "effective thermal conductivity", that takes into account the effects of turbulence), while the right-hand side expresses the conductive heat flux in the solid wall, by using a heat transfer coefficient called \alpha_{wall}.

This BC does almost what I want, but there is something missing! It doesn't take into account the radiative heat flux. Therefore I compiled my own custom BC, simply adding the value of the radiative flux, as calculated by the radiation model:

T_{wall} = wT_{\infty} + (1-w)\left(T_c + \frac{q_r\delta}{k_{eff}}\right),

which can be rearranged as:

q_r + \frac{ k_{eff}(T_c - T_{wall}) }{\delta} = \alpha_{wall}(T_{wall} - T_{\infty})

However, when I tried to apply this BC I ran into some issues: although cells have realistic values of temperature, a few wall cells in the vicinity of the fire start showing very high and fast variations of both temperature and radiative heat flux. I plotted these variations and attached the results in figures to illustrate the issue. Therefore, I guess this isn't the ideal way to set up this BC.

I'd be grateful if anyone could offer any ideas on how to tackle this problem differently. Has anyone had similar problems where you needed to apply this sort of BC? Do you see any problem with my approach that I should correct?

Thanks in advance,
Attached Images
File Type: png wallT_1.png (12.5 KB, 18 views)
File Type: png wallT_2.png (27.8 KB, 16 views)
File Type: png qr_2.png (31.6 KB, 15 views)
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Old   June 15, 2021, 06:19
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António Galhardo
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I believe the problem has to do with the fact that the radiative heat flux depends on the wall temperature itself. Therefore, when the BC is calculated as I presented in my previous post, the value of q_{r} used is the one from the previous iteration. To attempt to improve this, I replaced q_{r} in the equation with its two components: incoming radiative flux and emission. Mathematically:

q_{r} = q_{in} - \epsilon \sigma {T^4_{wall}}

The advantage here is that q{in} is from the same iteration, although {T^4_{wall}} is from the previous iteration. This improved the stability of the BC only slightly and unfortunately the same problem as described above appeared after a larger period of time. I guess using the value of {T^4_{wall}} would improve the stability further, but then I wouldn't be able to write this as a mixed boundary condition as described in

I would like to ask again if anyone can think of a different approach for this boundary condition. As said before, the problem is a compartment fire using fireFoam. The fire gases lose heat to the wall, so an adiabatic BC won't work. The wall heats up during the fire, so a fixed temperature BC won't work either. Which BC would you use instead, keeping in mind that the fluid loses heat to the wall due to both convection and radiation?

Thanks in advance,

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boundary condition, energy balance, firefoam, radiation, temparature

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