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October 27, 1999, 09:39 
Analytical solution for radiation

#1 
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Hi,
I working on radiation modeling and I was looking for analytical solutions to the RTE. I found one to a 2d problem in 'Discrete Ordinates Solutions of the RTE for Rectangular Enclosures', Fiveland, J. Ht. Transfer, 6, 1984. However, the solution contains the modified Bessel function of order zero (k0) and the twice repeated integral of k0. What is the twice repeated integral of k0?? How do I calculate it?? Thank you. Chrys 

October 27, 1999, 11:02 
Re: Analytical solution for radiation

#2 
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There aren't many problems I have found in my career that are more evil to solve than radiation heat transfer. The analytical solutions are ugly for simple geometries and worse for any practical problems.
I would refer you to the book I used in my Rad HT grad class, since it has a great deal of information on how to solve simple geometries (and text books often provide more explanation of what to do than journal papers): Thermal Radiation Heat Transfer, 3rd Edition, by Robert Siegel and John R. Howell, ISBN 0891162712. YOu should be able to get it through any college library. The book is about 1000 pages and takes you through everything from calculating view factors to solving the resultant (nasty) integral equations. 

October 27, 1999, 19:08 
Re: Analytical solution for radiation

#3 
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Hi Chris,
Are you studying, only radiation or does your problem involve convection and radiation in a rectangular/square geometry. I have got several references to papers with convection and radation in such geometries and will pass it on to you if you are interested. By the way are you from University department of chemical technology, Bombay? Cheers 

October 28, 1999, 02:42 
Re: Analytical solution for radiation

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Hi,
My final goal is to study radiation coupled with convection, reactions, etc. I'm presently only trying to evaluate the radiation part. So I'm trying to compare the discrete ordinates method with analytical solutions in simple geometries. However, the analytical solution mentioned in Fiveland's paper has the twice repeated integral of the modified Bessel's function of order zero, and I have no clue how to calculate that. Chrys BTW: I did study in UDCT! 

October 28, 1999, 05:08 
Re: Analytical solution for radiation

#5 
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Hi,
I suppose it might be a good idea to go through the book by Seigel and Howell as suggested by Alton. However if your final aim is to carry out numerical modelling of radiation, convection combined with reaction it would be better to start doing the modelling immediately after you get some confidence with the basics of radiation. Believe me it takes a lot of time to do the modelling!!! Cheers 

November 12, 1999, 11:02 
Re: Analytical solution for radiation

#6 
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I used the "VDIWärmeatlas: Strahlung technischer Oberfläche, Kapitel ka 8, 1984" to make a numerical study about the influence of grid, DOMDiskretization on the precision of the results. On this book, unfortunaly in german, they have calculate radiation in simple geometries like cylinders, etc. May be it will help you. Just take a look! I have also some results of this study.
LCA 

November 25, 1999, 10:59 
Re: Analytical solution for radiation

#7 
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Firsy you can check your code on 1D paralel plate problem. available Modest book on radiative heat Transfer. http://michelle.me.psu.edu/~mfm/me514_99/general.html You don't have to calculate reference(exact or analitical) solution. On radiative heat transfer literature many test cases
(N. Selçuk and N. Kayakol, Evaluation of Discrete Ordinates Method for Radiative Transfer in Rectangular Furnaces, Int. J. Heat Mass Transfer, 40, 213222 (1997)) are available. Nuray 

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