# Review of Radiation Transfer Method

As stated at the beginning of the chapter, the radiation transfer method assumes isotropic radiation and uniform fluxes incident and leaving a surface. The isotropic assumption is necessary in order to remove the intensity, i, from the double integral that determines the radiation flux leaving a surface [equation (6.4)]. This results in the double integral becoming merely a geometrical factor, the view-factor. Without the isotropic assumption, the method would be very cumbersome since not only would the double integral depend upon the intensity, but the radiation transfer equation [equation (1.139)], which determines the intensity, would depend upon direction, 0, as well as distance, s. For a TPV system with a non-isotropic emitting emitter, the radiation transfer method is not appropriate.

Although the radiation transfer method assumes uniform fluxes incident and leaving a surface, non-uniform fluxes over a surface area can be included by splitting the area into several parts. Each part will have different incident and leaving fluxes. This was done in the earlier examples by splitting the filter-PV area into three parts. Each surface area that is included results in a linear algebraic equation for the leaving flux, qo, [equation (8.4)]. Thus, the number of algebraic equations to be solved will equal the number of surface areas. This set of equations will be a function of the reflectances and emittances of each of the surfaces, which are in turn functions of the wavelength. Therefore, the radiation transfer method easily accounts for any rapid changes with wavelength, such as those in the spectral emittance of a selective emitter or the reflectance of an interference filter. Whereas a method that split the spectrum into several wavelengths regions with constant optical properties in each region cannot account for rapid changes.

8.8 Summary

The objective of this chapter has been to apply the radiation transfer method to calculate the performance of a TPV system. This method assumes that the radiation incident and leaving the various surfaces in the optical cavity is isotropic and uniform over the various surfaces. The method is applied to a hypothetical planar, square geometry, optical cavity. Significant results from that example are the following.

1) Leakage of radiation from the optical cavity, even from very small openings, is a significant loss.

2) Large radiation power fluxes circulate about the optical cavity. As a result, if a front surface filter is used for spectral control, its absorptance must be negligible in order to avoid a large absorptance loss.

3) To obtain high efficiency, the method for spectral control may be less important than avoiding radiation leakage from the optical cavity.

4) In a system with constant properties for 0 < X < Xg, the PV efficiency is nearly independent of the emitter temperature and thus the input thermal power. Whereas the cavity efficiency increases with emitter temperature and thermal input power resulting in a similar increase in the TPV efficiency.

5) Since PV efficiency is a sensitive function of the PV array temperature, the PV efficiency decreases rapidly with increasing PV array temperature. The cavity efficiency is nearly independent of the PV array temperature. As a result, TPV efficiency decreases with increasing PV array temperature.

[1] Electromagnetic wave propagation (radiation) is described by Maxwell’s equations. For a homogeneous, stationary medium, the electric and magnetic fields are determined by the so-called harmonic plane wave solution to Maxwell’s equations.

[2] A conductive medium (о Ф 0) is described by an index of refraction, n, which has both real and imaginary parts. The imaginary part, which determines the dissipation, is related to the medium absorption coefficient, a.

[3] Application of the proper boundary conditions at an interface between two media with indices of refraction n1 and n2 leads to the law of reflection for a specular interface and Snell’s Law n1 sin 0; = n2 sin 0r where 0; is the angle of incidence and 0r is angle of refraction.

[4] Reflectivity, R, and transmittivity, T, at a specular interface have been calculated for so-called natural light (direction of electric field is random), which is appropriate for TPV applications.

[5] Radiation transfer theory is required to analyze a medium that is emitting, absorbing, and scattering. The radiation intensity, i, has been defined and the radiation transfer equations and the energy equation appropriate for TPV applications have been presented.

[6] Application of radiation transfer theory to a one-dimensional medium with negligible scattering produced results for the radiation flux, qo, leaving a

[7]

R™qb = |(1 – Rfo)eb (Te)d^

[8] o V ^2E