I am using the finite Volume Method and I am looking for books and articles dealing with the modelling of radiation in Heat transfer. ( View Method and so on). I am glad for every contribusion

Hi there, Let me give you the diffusion approximation for the radiative tranfer of energy in the energy equation. (assume you have a continuity equation, 3 velocity - or momentum - equations and one energy equation). The energy equation can be only for the internal energy (3/2 R T, where R is the gas constant, sometimes multiplied by the density rho) but it can also include in addition the kinetic energy of the gas (e.g. + 1/2 rho*v**2). On the right hand side of the energy equation you will have the source and leak terms. The leak term is the one that will include the lost due to the radiation of energy emitted by the flow. A common approximation for this term is in the diffusion approximation, i.e. a second derivative (in space) term. Usually it is written as -div.F , where div is the divergent and F is the radiative flux vector and can be given by F= - X grad T, where X is the coefficient of radiative conductivity, and grad T is the gradient of the temperature. The coefficient X itself is given by: X=(4 a c T**3)/(3 K rho), where the constant a = 7.565x10**15 ergs cm**(-3) deg**(-4) , c is the speed of light, K the opacity and rho the density. You need to know the opacity K as a function of the density rho and the temperature T. Once the equation is written explicitly you end up with a diffusive term for the temperature (second derivative in space of T). If you solve for time dependent equations then you will solve this term implicitly, if you solve for a steady state (d/dt=0) then you will probably need to invert a matrix for the tempature. This is a common treatment in Physics and the main unknown is ususally the opacity. This treatment is valid for optically thick media, where the radiation is in equilibrium with the gas and it covers a wide range of temperatures and densities. Since I am not sure what is the application of your flow problem, I though this general approach will certainly help (though it might be called differently by engineering people, than by physicists). A very good book about this approach is the field of Astrophysics where radiation is always important in flows. J.L. Tassoul, Theory of rotating stars, Princeton Series in Astrophysics (Princeton, NJ, 1978). See pages 52-57. The equations can also be found in Appendix B of this book (sometimes this approximation is also found as the Roessland approximation). More complete treatments do exist but are much more difficult to approach (e.g. Chandrasekhar, Radiative Transfer, 1960 by Dover publication). I hope this helps as a starting point. I have myself used this approach in one-dimensional flow problems. PG.

Please visit the following webpage. We published a few papers on radiation heat transfer calculation using the finite volume method.

http://jcchai.me.tntech.edu/vitae/jchai.html

Please feel free to contact me with questions.

HI, DIFFUSION APPROXIMATION IS VALID FOR OPTICALLY THICK MEDIA LIKE GLASS MELTS. YES IT REDUCES MATEMATICAL COMPLEXITY BUT RADIATION ITSELF IS NOT A DIFFUSION PROCESS. FOR COMBUSTION SPACE WHICH SHOULD BE TREATED AS OPTICALLY THIN MEDIA, RADIATIVE TRANSPORT EQUATION (RTE) SHOULD BE SOLVED USING AN APPROXIMATE RADIATION MODEL LIKE DISCRETE TRANSFER, DISCRETE ORDINATES OR FINITE VOLUME METHOD.

LATEST PUBLICATION RELATED TO FINITE VOLUME METHOD, S.H. KIM AND K.Y.HUH NUMERICAL HEAT TRANSFER PART B: VOL.35 NO 1, 1999

DR. NURAY KAYAKOL SISECAM GLASS RESEARCH CENTER TURKIYE

Hi there. Yes, I mentioned that the approximation I suggested is for optically thick media, where the radiation is in equilibrium with the gas. There is even a similar treatment for optically thin, but I guess that it does not apply here for the specific application. Cok merci. Dr. P. Godon, STScI.

P1 method,which is a kind of diffusion equation, is used for solving the opticallly thin media, and for the optically thick media, we must use another algorithm, like as S4 method. S4 method is accurate compared to p1 method but the cpu time is very long,

so the new alogorithm adopting the advantages of those two method(p1 method and s4 method) is needed, that is accurate prediction with reduced cpu time.

for more details , see the book "RADIATIVE TRANSFER EQUATION" BY OZISIK

Both P1 and S4 are belong to flux type radiation model known as most computationally efficient technique among the others. P1 method is not suggested for optically thin media like natural gas combustion. (Mengüç and Wiskanta, 1987 "Radiative tranfer in Combustion sytems" Prog.Energy,Combustion Science, vol 13. pp.97-160)

S4 uses 24 rays which means 24 pass over all control volumes within the enclosure. In large scale furnace modeling at least 200 000 nodes are necessary. Each dependent variable requires 200 000 visit but intensity which is a directional quantity needs 24x200 000 visit. When the walls are non-black multiply that quantity at least with 10. I think large CPU times are inevitable for large systems.

Beside the accuracy, new radiation model should be compatible with the solution of the partial diffrential equations used for CFD. That is why Sn or finite volume methods are incorparated into CFD codes. Control volume approach is used for both flow and radiation.