Natural Convection Heat Transfer in Enclosures With Multiple Vertical Partitions

Abstract

of air or vacuum, N 2 = n + IK is the complex refractive index of the lamina material, and 9 2 is the (complex) angle of refraction, which is related to 9 t by Snell's law: N, sin #, = N 2 sin 9 2 . Since r 2l = -r i2 , the reflectance at both interfaces is equal to p = r n rf 2 , where * denotes the complex conjugate. The internal transmittance T is related to the (complex) phase change 6 by r = exp . After carefully examining the transmittance formulae of a lamina, this work shows that the geometric-optics formula may result in a significant error for a highly absorbing medium even in the incoherent limit (when interference effects are negligible). Introduction Consider the transmission of electromagnetic radiation through a lamina with smooth and parallel surfaces. In the incoherent limit when radiation coherence length is much smaller than the thickness of the lamina, the transmittance (or reflectance) may be obtained either by tracing the multiply reflected radiant power fluxes (ray-tracing method) or by separating the power flux at each interface into an outgoing component and an incoming component (net-radiation method), viz. ( where p is the reflectance at the interface and r is the internal transmittance. This formula is also called the geometric-optics formula since it is obtained without considering interference effects. For a plane wave, p equals the square of the absolute value of the complex Fresnel reflection coefficient (i.e., the ratio of the reflected electric field to the incident electric field at the interface). The Fresnel reflection coefficient is (Heavens, 1965) r\ 2 = { cos 9 2 -N 2 cos f?i JVi cos (2) N, cos 0, -N 2 cos 6*2 , . , for s -polarization ,7V, cos 9 t + N 2 cos 9 2 where 9 l is the angle of incidence, /V, = 1 is the refractive index where d is the lamina thickness and X is the wavelength in vacuum. In the coherent limit, the transmittance of a lamina may be obtained from thin-film optics (i.e., wave optics) either by tracing the reflected and transmitted waves (Airy's method) or by separating the electric fields into a forward-propagating component (forward wave) and a backward-propagating component (backward wave), viz. (Heavens, 1965; Analysis and Discussion The power transmittance at the interface between the air (or vacuum) and the medium (lamina) is where (1 + r !2 ) is the Fresnel transmission coefficient. The power transmittance at the second interface between the medium and the air can be obtained by exchanging the subscripts 1 and 2 in Eq. (6). At normal incidence, r 12 = (1 -n -('K)/(1 If both K and Im(r 21 ) are nonzero, T 2 \ =t= 1 -p. As discussed by Journal of Heat Transfer AUGUST 1997, Vol. 119/645 Copyright © 1997 by ASME Zhang The above equation is identical to Eq. (5). However, it is not a simple replacement of (1 -p) 2 in Eq. As an example, suppose the lamina is a LaA10 3 wafer of thickness d = 100 p,m. The optical constants are calculated from the Lorentian dielectric function determined by (1) and the transmittance for a LaA10 3 lamina at wavelengths from 9 to 14 p,m at normal incidence are shown in The difference between the wave-optics formula and the incoherent formula is shown in For a highly absorbing lamina (i.e., r < § 1), multiple reflections may be neglected. The transmittance obtained from Eq. (1), when multiple reflections are negligible, is (1 -pfr. The transmittance calculated from Eq. (8) for T < 1 is where the last expression is for normal incidence only. Eq. Concluding Remarks By inspecting the energy balance at the second interface, this work reveals an implicit assumption associated with Eq. Certain important applications require the determination of transmittance below 10~4. Examples are in the characterization of attenuation filters, bandpass filters, and materials with strong absorption bands Acknowledgments This work has been supported by the University of Florida through a start-up fund and an Interdisciplinary Research Initiative award. / Vol. 119, AUGUST 1997 Transactions of the ASME A. A., 1994, "Modelling of the Reflectance of Silicon," Infrared Physics and Technology, Vol. 35, pp. 701 -708. Frenkel, A" and Zhang, Z. M" 1994, "Broadband High Optical Density Filters in the Infrared," Optics Letters, Vol. 19, pp. 1495-1497 Gentile, T. R., Frenkel, A" Migdall, A. L., and Zhang, Z. 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Zhang, Z. M., 199