Refractive Index Effects on Radiation in an Absorbing, Emitting, and Scattering Laminated Layer

Abstract

IntroductionThe use of ceramic materials for parts or coatings is of interest for high-temperature applications. Some of these materials have reinforcing fibers or are laminated so it is necessary to consider heat transfer in composite regions. The surrounding temperatures are high, which provides substantial radiative heating. Since some of the materials are semitransparent, their temperature distributions depend on the internal radiative heat flow. The refractive index of a semitransparent material can have a considerable effect on its temperature distribution. The refractive index governs the amount of transmission into the interior of the material and affects the internal reflections that occur. Of more significance, the emission within-a material depends on the square of its refractive index; hence internal emission can be many times that for a blackbody radiating into a vacuum. Since radiation exiting through an interface into ávactium cannot èxcéëd that of a blackbody, the amount of internal reflection can be substantial. It redistributes energy within the layer and tends to make the temperature distribution more uniform than for a material with refractive index close to one.To obtain the general solution for the temperature distribution in a composite semitransparent layer requires solving the integral equations of energy transfer in each region, including heat conduction, and matching temperatures and heat flows at the internal interface. A spectral calculation can be carried out in each significant wavelength band and the total energy flow found by summing over the bands. A numerical solution by computer is required. A simple limiting solution is obtained here that is helpful and informative. The purpose of this paper is to show that in the limit with no heat conduction and for gray layers with refractive indices greater than one, the solution for a composite region can be obtained from information that is already known and results calculated very easily. The effects of surface reflections and isotropic scattering are included. The solution of integral equations is not required. The limiting result can be helpful in initiating iterative computer solutions for a more general analysis and for checking the validity of numerical solutions.The development here builds on the analysis of Siegel and Spuckler (1992) where it was shown that for radiative equilib-

Transient coupled radiative and conductive heat transfer in a composite of semitransparent media

“…[5] (Siegel and Spuckler, 1993). The parameters are: T r = 500 K, qrl = 34, qr2 = 1.54 , n 1 = 1, n 2 = 3, 6=0.5, C21=1.…”

“…This is because two approximate functions of dimensionless radiative heat flux and temperature distribution, which have a little error compared with the exact solution for a small optical thickness, were used in Ref. [5].…”

“…The results under the diffuse reflection are separately compared with the steady-state results in Ref. [5](1993) and Ref. [6](1994) and the transient results in Ref.…”

“…The isotropic scatting is included and the external boundaries are assumed to be covered by a thin opaque layer. While Siegel and Spuckler (1993) [ 5] , and Spuckler and Siegel (1994) [ 6] using a numerical solution of the exact transfer equations investigated the steadystate temperature distributions in two plane layers with diffuse boundaries. By the two-flux method in combination with the Green's function, Siegel and Spuckler (1994) [ 7], Siegel (1996Siegel ( , 1997 [ 8,9] respectively gained the steady and transient temperature distributions in the scattering single layer and the composite layers with diffuse interfaces.…”

Radiative heat transfer in a participating medium with specular–diffuse surfaces

“…If the diffuse surface S 2 is semitransparent, the determination of the reflectivity of the diffuse surface S 2 can be seen in Ref. [23].…”