Non-gray radiative transfer in plane-parallel media with reflecting boundaries

THE EVALUATION of the energy transfer by radiation has been of interest to scientists working with high:temperatpre ,gases over the years. ,Early investigations of radiative transfer were mostly in astrophysical applications, and the analyses were based on approximate methodq oj, ;@$ion. HOPF”’ appears to be the first investigator to present exact results for several radi@ive invariance principles to solve exactly various problems of astrophysics. With the introductionof the norjnalmode-expansion technique by CASE,(~) a new era was opened in the field of radiative transfer. SIEWERT and ZWEIFEL,‘~’ utilizing this powerful technique, developed the formalism for the e%act solution of the equation of radiative transfer in the two-group picket-fence model. Their exact method of solution was applied by SIMMONS and FERZIGER,“’ SIEWERT and ~ZISIK,‘~’ BOND and SIEWERT’~ and REITH et ~1.‘~’ to radiative transfer problems. Later SIEWERT and SHE&@’ SIEWERT and ISHIGURO,“~’ REITH and SIEWERT’“’ utilized this method and obtained the elementary solutions of the equation of radiative transfer in the two-group picket-fence model with a scattering term included (isotropic and linearly anisotropic scattering) and established the related half-range expansion and orthogonality theorems. YENER et a1.(“’ have applied the method to analyze a generalized form of the equation of radiative transfer in the two-group picket-fence model for a half space. In the present analysis, radiative heat transfer in an absorbing, emitting and isotropically scattering stagnant gas confined between two emitting and diffusely reflecting parallel boundaries and with uniform internal heat sources is considered in the two-group picket-fence model. The solution of the problem is expressed by the superposition of simpler problems which are in turn solved by the normal-mode-expansion technique. 2. GENERAL FORMULATION The generalized equation of radiative transfer for one-dimensional plane-parallel, emitting, absorbing, isotropically scattering non-gray media is (CHANDRASEKHAI?) ~~~"(x,~)+(K.i(T,)I,(X,p)=(Kv+e~")g~[T(X)l+f~~(~-~,) I 1