View-Factors for TPV Systems

Two basic geometries are most likely for a TPV system. The first such geometry is a planar one as shown in Figure 6.2a, where each of the components are circular or rectangular disks aligned parallel to each other. The second geometry is a cylindrical one, where each component is a concentric cylinder as shown in Figure 6.2b. Most likely, the inner cylinder is the emitter and the outer cylinder contains the PV cells. However, the reverse case, where the emitter is the outermost cylinder and the PV cells are on the inner most cylinder, is also possible.

View-factors for circular and rectangular disks and concentric cylinders are shown in Figure 6.3. Note that in the case of the cylindrical geometry, the concave inner surface of the outer cylinder has a “self’ view-factor, F22...

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Radiation Transfer Theory

6.2.1 Radiation Transfer for Uniform Intensity

Consider the transfer of energy by radiation from area A; to area Aj as shown in Figure 6.1. The intensity of radiation leaving the infinitesimal area dA1 is ioi, which in general is a function of location, x;, direction, 9;, temperature, Т;, and wavelength, X. If an absorbing medium that does not alter the direction of the radiation (as a lens would, for example) exists between A; and Aj, then the radiation energy reaching area dAj from the area dAi is the following.

Where Tij is the transmittance of the medium between A; and Aj, and dffldA is the solid angle subtended by dAj at dAi.


. , cos 0; cos 0 J

dqdA, ^dAj = — (0i>xi> ^,Ti)—- s——– dAjdAj

Assuming Tij is a constant, the total radiative energy at wavelength, X, reaching Aj ...

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Governing Equations for Radiation Fluxes in Optical Cavity

In the previous chapters, the primary components (emitter, filter, PV cells) of a TPV system are considered. It is the optical properties of these components that determine the TPV system performance. Knowing the optical properties of the components allows the calculation of the efficiency and power output of the TPV portion of the energy conversion system. However, it should be remembered that the overall performance of a TPV energy conversion system will also depend upon the efficiencies of the thermal energy source, the waste heat removal system, and the power conditioning.

Chapter 6 develops the system of equations that must be solved in order to obtain the radiation fluxes incident on the various components of the system...

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PV Cell Efficiency and Power Output

Efficiency and power output are the two PV cell parameters required to calculate the performance of a TPV system. The electrical power output is the product of the voltage, VL, and current, IL, that is supplied to the load,

Pel = VlIl (5.219)

where IL is a function of VL. To determine the condition for maximum PEL, the standard maximization procedure is used. Namely, differentiate equation (5.219) with respect to VL and set the result equal to zero.

dPEL = dPEL dV = 0 dVL dV dVL


It can be shown (problem 5.11) that > 0 for all VL. As a result, to satisfy equation (5.220)

for a maximum PEL. Therefore, using equation (5.209) for VL in equation (5.219) for identical junctions yields the following result.

Using equation (5...

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Quantum Efficiency and Spectral Response

Two important and related quantities used to describe the performance of a PV cell are the quantum efficiency, ^q, and the spectral response, Sr. These wavelength dependent quantities can be used to calculate the PV array power output and efficiency, as shown in Section 5.12.

In section 5.9.1, the quantity, JF [equation (5.161)] is introduced. It is the maximum possible current density that can be produced by the incident photon flux.

where R; = Rn if photons are incident on the n side of the junction, and R; = Rp is photons are incident on the p side of the junction. Also, bo = V has been used in equation (5.161). Equation (5.201) is called the internal quantum efficiency since it is based upon the photon flux that enters the semiconductor. Obviously, qq(X) < 1...

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Current Generation in Depletion Region

To obtain the total current, the current density generated in the depletion region must be included along with the minority current densities at the depletion region boundaries. In the depletion region, the electric field will accelerate electrons and holes out of the region. Since the depletion region is narrow, it is reasonable to assume that the electrons and holes leave the depletion region before they can recombine. As a result, the steady state continuity equations for the electrons and holes are the following.

= – G (X, x) (5.187a)

e dx

Xp ^ x ^ Xn

1 dJp, 4

—– ^ = G (X, x) (5.187b)

e dx

Similar to equation (5.168), the generation rate in the depletion region is the following,

G(M) = 2Fp (*■)ad (*■)jE2 [apxp + ad (x – xp)] – umE2

where ad(X) is the absorption coefficient for the depl...

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Electron Current Density in p Region

The transport equation for the minority electrons in the neutral p region now includes the generation term given by equation (5.66a). The generation rate is the number of electrons and holes produced per unit wavelength. Therefore, the minority electron density appearing in the transport equation will be the electron density per unit


wavelength, which is denoted as np. The total density is therefore np = JnpdX. And


the transport equation is the following,

where Rp(X) is the reflectivity at the surface of the p region, a^X) is the absorption

(n Y

coefficient in the p region, and uM = 1 – — where no = 1 is the index of refraction

V I np)

of vacuum and np is the index of refraction of the p region.

Rewrite equation (5.140a) as follows,

where Np = np – npo, Ln, and Fp(X) is the photon f...

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Current-Voltage Relation for an Ideal p-n Junction Under Illumination

Now consider a p-n junction with incident radiation, q;(X), on the p side of the junction. In this case, electron-hole pairs are generated throughout the semiconductor hc

for photon energies, E = —- > Eg, where Eg is the bandgap energy. For the ideal p-n


junction in the dark, it is assumed there is no generation or recombination in the depletion region. Therefore, the electron and hole currents are constant through the depletion region. As a result, the total current density, J, is the sum of the minority electron current density, Jn, at the boundary between the p region and the depletion region and the minority hole current density, Jp, at the boundary between the depletion region and the n region.

For the illuminated p-n junction, electron-hole pair production occurs in the depletion...

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