Radiation Transfer Equations for TPV Systems with Close Coupled Emitter-Window and Filter-PV Cells

In a practical TPV system it is likely that the emitter and window will be close together in order to minimize the conductive and convective heat loss from the emitter. Also, it is likely that the filter will be physically connected to the PV cells in order that the same cooling system can be used for the filter and PV cells. Therefore, consider a TPV system with the emitter and window, and the filter and PV cells very close together. In that case the following view-factor approximations can be made.

Few = FWE’ = FfC = FCf = 1


F — F — F — F

rE’f rW’f _ rW’C _ rE’C


F — F — F — F

rfE’ rfW’ “ rCW’ _ rCE’


Ff = Ff = Ff = F = F

fC A Cf A CC A ff A 22


FIW _ pfW _ pfW _ pfW _ pfW _ vW _ vW _ fw _ f _ f _ о (7 49e)

-‘-cc if...

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Cavity Efficiency for Cylindrical Filter and Selective Emitter TPV Systems without a Window

For a cylindrical geometry, equations (6.72) and (6.73) must be solved before the cavity efficiency, qc, can be determined. Similar to the planar geometry system (section 7.2), neglect the window loss (xW = 1, pW =pW= qW =qW= 0) the filter and PV cell emissive powers (qf = qj =qC = 0) and assume the filter and PV cells are together so that FEf = Fe, c , FfE, = Fce, , and FfC = FCf, = 1. In addition, the following view-factor conditions apply if the filter and cells are together; Fff + Ff = FfC + FfW = F, Cf + FCW = FCc + FCW = F22, Fcc = 0 . Where, F22 is given in Figure 6.3d with R1 = RE (emitter radius), R2 = RC (PV cell radius), and f is the length of the cylinder (emitter, filter, and PV cells all the same length). Using the conditions just described, equations (6.72) and (6...

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Cavity Efficiency for Planar Filter and Selective Emitter TPV Systems without a Window

Consider the cavity efficiency, qc, for a planar TPV system. To calculate QC and Qe, the solution to the radiation flux equations for qoE and qoC [equations (6.53) and (6.54)] must first be obtained. To simplify the problem, several approximations can be made. First, since the window, filter, and PV cells are at much lower temperatures than the emitter, it is reasonable to neglect the emissive powers of the window (qW and qW), filter (qf and q[), and PV cells (qC). Also, assume the window reflectances and absorption are small (pW = pW, = aW « 0) so that xW = 1 and there is no window loss (Qlw = 0). Neglecting the emissive power of the window, filter, and PV cells and assuming xW = 1 (pW = pW, = aW = 0) yields the following result for equations (6.53) and (6.54).

C1 _Pe PfFE’fFfE’)qoE _Pe ...

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Radiation Losses in Optical Cavity

At this point it is appropriate to consider the factors that determine the efficient utilization of the radiation that leaves the emitter. In other words, how much of the emitted radiation that reaches PV cells can be converted to electrical energy? Radiation leaving the emitter is lost in two ways. First, radiation can escape from the system by “leaking” out between the components. This loss is controlled by the view-factors, which depend upon the system geometry (planar or cylindrical). The second radiation loss process is absorption. There are absorption losses for the window, filter, and any other component lying between the emitter and PV cells. Radiation leakage is the main topic of this chapter.

It is the cavity efficiency, pc, defined by equation (6...

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Efficiency of TPV Systems

6.5.1 Overall Efficiency

At the beginning of Section 6.3 [equation (6.35)], the TPV system efficiency, qT, is defined. To evaluate the effect upon qT of the various parts of the system, split the efficiency into three parts as follows,

hT = dth hc PpV (6.83)

where qth is the thermal efficiency, qc is the cavity efficiency, and qPV is the photovoltaic efficiency.

6.5.2 Thermal Efficiency

The thermal efficiency, qth, accounts for conductive, convective, and radiative heat loss from the emitter, plus any heat loss that occurs between the thermal source, Qth, and the emitter

The net power leaving the emitter is QE and is obtained from equation (6.34c) by including only the positive terms (primed quantities). Any negative going radiation (unprimed quantities) is included in the loss term, QLE ...

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Radiation Energy Transfer in Cylindrical TPV System

Figure 6.8 shows a general cylindrical TPV system consisting of the same four elements as the planar system in Figure 6.7. The cylindrical geometry results in additional view-factors not present in a planar system. As already mentioned, the concave inner surface has a “self” view-factor. In addition, transmission through adjoining elements allows radiation from the concave side of an element, qom, to be incident on an adjoining element’s concave side. Therefore, the simple results for qim and q|m for a planar system given by equations (6.39) and (6.40) are no longer applicable. However, the same relations [equations (6.30) and (6.31)] between qom and qim and qj, m and q’m that apply for the planar geometry also apply for the cylindrical geometry.

First, consider the window in Figure ...

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Energy Balance on a Component of a TPV System

Remember that the radiative energy reaching Aj from A;, given by equation (6.5), assumes that ioi is uniform across A; and is also independent of 0;. As already stated, a surface that emits and reflects independent of angle 0, is called a diffuse surface. If ioi is a function of x; and 0;, then it must be included in the integration of equation (6.4). In that case, the integration yields a complicated function of Ai and Aj geometry, as well as temperature, T;, and wavelength, X. However, if ioi is uniform across A; and independent of 0I, then the rather simple expression given by equation (6.5) is obtained for Qij. Also, note that Q;j is the radiation that reaches Aj from A;. It is not the radiation energy absorbed by Aj.

Now consider a single element, m, of the system as shown in Fi...

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Optical Properties of Components

To obtain the equations that govern the TPV system performance, it is necessary to perform an energy balance on each of the elements in the system. Each of the elements is characterized by the optical properties: emittance, є; absorptance, a; reflectance, p; and transmittance,!. In general, these properties are functions of direction and wavelength.

Interference effects are neglected in the following analysis so that the results do not apply for interference filters, which are treated in Chapter 4. However, it is assumed that the components behave in a diffuse manner so that the optical properties are independent of direction and that the radiation field is also independent of direction. The diffuse approximation is also used in deriving equation (6.5)...

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