Consider the system shown in Figure 8.1 with the gap between the reflector and the PV arrays removed so that wE = wC3 = 10cm, wC1 = 6cm, wC2 = 9cm, and h = 0.2cm. Also, assume the same properties as used in Figure 8.2. Using the Mathematica program in Appendix F the efficiencies, temperature, and electrical power output have been calculated as a function of the thermal input power, QE + Qb. These results are shown in Figure 8.6.
100 150 200 250
Thermal input power, Q +Q Watts
Figure 8.6a) – Effect of thermal input power on cavity efficiency, q, approximate cavity efficiency, q (eq. 8.55), PV efficiency, qpv,
and TPV efficiency, ■q of a planar, square geometry TPV
system with the same conditions as in Figure 8.2 with the gap removed between the reflector and PV arrays so that w =w =10 cm.
The photovoltaic efficiency, qPV, remains nearly constant while the cavity efficiency, qc, increases with increasing input power. As a result, the TPV efficiency,
qTPV, also increases. For constant optical properties for 0 < X < X„ = o, as is the case
being considered, and if Rs = 0 and Rsh then pPV is given by equation (5.252). The
two integral terms in that equation are functions of ug = ^. For the results in Figure
8.6, ug > 5.5. Therefore, the integrals can be evaluated using the approximation, ug » 1 (problem 8.4).
Figure 8.6b) – Effect of thermal input power on the emitter and reflector temperatures, T=T, and electrical power output, P£L
of a planar, square geometry TPV system with the same conditions as in Figure 8.2 with the gap removed between the reflector and PV arrays so that w =w =10 cm.
In that case, equation (5.252) becomes the following.
The open circuit voltage, Voc ~ ln Isc and the fill factor, FF, is a slowly varying function of Voc. Thus, both Voc and FF will be slowly varying functions of Isc, which depends upon TE [equation (5.251)]. Thus, qPV is nearly independent of TE and therefore, also of the input thermal energy, as Figure 8.6 indicates. Now use equation (8.49) to
where Nj = 25 is the number of junctions in each of the four arrays that fill area AC1. Therefore,
0 9 ( 9 4 і
hPV * —(1 – 0.1) 0.9 (0.68)1 —1 = 0.31 (8.50a)
which agrees with the result in Figure 8.2.
For the results in Figure 8.6, the optical properties are constant in the regions 0 < X < Xg and Xg < X < да. As a result, a simplified result for the cavity efficiency, qc, can be derived. Since the area of the reflector, Ab, is much less than the area of the emitter, Ae, the contribution to QC of the reflector can be neglected. As a result, since
eE is a constant, the cavity efficiency is given approximately by the following expression,
where Fec = FEC1 + FEC2+FEC3. The flux leaving the emitter, qoE, is given by equation (8.10), where the terms are functions of the optical properties and view-factors. For the results in Figure 8.6, the optical properties are constants except the PV cell reflectance, where Peg = Pc1 = Pc2 = Рсз = 0.1 for 0 < X < Xg, and pot = Pc1 = Pc2 = Рсз = 0.9 for
Xg < X < да. And since TCi = TC2 = TC3 « TE the parameter, a, in equation (8.10) can be approximated as a « eE eb(X, TE). Also, B2 « 1, aB2»bE2, E1B2 »E2B1, and E1 = 1 — PePc§(Fc1eFec1 + Fc2eFec2 + Fc3eFec3) for 0 < X < Xg, and E1 = 1 – PeP«(Fc1eFec1 + Fc2eFec2 + Fc3eFec3) for Xg < X < да. When the emitter and PV arrays are very close together, as they are for the results in Figure 8.6, where h = 0.2cm, then the sum of the view-factor products in the expression for E1 is nearly 1. Therefore, E1 « 1 – pEpCg for 0 < X < Xg, and E1 « 1 for Xg < X < да, since pEpCg = 0.4(0.1) = 0.04 « 1. Using these results in equation (8.10) to calculate qoE, results in the following expression for qoE.
q0E ^Єь (X, Te ) 0 <X<Xg (8.51)
Using equations (8.51) and (8.52) in (8.50b) and eE = 1 – pE yields the following result.
Now use the approximation eug »1 where ug becomes the following (problem 8.5).
The dark saturation current increases rapidly with increasing PV array temperature [see equation (5.132)]. As a result the PV efficiency will decrease with increasing PV array temperature.
Figure 8.7 shows the effect of PV array temperature on pc, pPV, and pTPV for the same conditions as in Figure 8.2 with the gap removed between the reflector and PV arrays. At TC1 = TC2 = TC3 = 300K the dark saturation current density was Js = 8 x 10-7 A/cm2. At other temperatures, Js was calculated using equation (5.132).
The cavity efficiency experiences a very small increase as the PV array temperature increases. This results from useful above bandgap energy emitted by the PV arrays that is added to the optical cavity radiation. However, the PV efficiency shows more than a factor of four decrease in going from pPV « 0.3 at TC = 300K to pPV « 0.07 at TC = 500K. As a result the TPV efficiency also shows a similar decrease.
In a space application PV array temperature is a critical parameter. Since the waste heat must be rejected by radiation, the size of the waste heat radiator is proportional to TC4. Therefore, to minimize the size of the waste heat radiator TC should be large.
However, TC means low efficiency, which in turn results in increased mass of the thermal energy source. Thus, there will be some optimum efficiency and PV array temperature that will yield the lowest mass system (problem 8.6).
For terrestrial applications the waste heat can be rejected by thermal conduction and convection. As a result, the waste heat radiator size is not as strongly dependent on TC as for a space application.