## X

0.32 X 1350

0.901

The useful energy is estimated from Eq. (3.127) using the concept of absorbed radiation:

= 0.901[500 X 68.2 - 3.14 X 13.95(220 - 25)] = 23,031W

Finally, the fluid exit temperature can be estimated from

23,031

0.32 X 1350

Another analysis usually performed for parabolic trough collectors applies a piecewise two-dimensional model of the receiver by considering the circumferential variation of solar flux shown in Figures 3.40 and 3.41. Such an analysis can be performed by dividing the receiver into longitudinal and isothermal nodal sections, as shown in Figure 3.42, and applying the principle of energy balance to the glazing and receiver nodes (Karimi et al., 1986). Longitudinal division of Cross-section of the receiver showing the receiver into sections the isothermal nodal sections

FIGURE 3.42 Piecewise two-dimensional model of the receiver assembly (Karimi et al., 1986).

Longitudinal division of Cross-section of the receiver showing the receiver into sections the isothermal nodal sections

FIGURE 3.42 Piecewise two-dimensional model of the receiver assembly (Karimi et al., 1986).

The generalized glazing and absorber nodes, showing the various modes of heat transfer considered, are shown in Figure 3.43. It is assumed that the length of each section is very small so that the working fluid in that section stays in the inlet temperature. The temperature is adjusted in a stepwise fashion at the end of the longitudinal section. By applying the principle of energy balance to the glazing and absorber nodes, we get the following equations.

For the glazing node,

?G1 + Qoi + Qg3 + QG4 + la 5 + QG6 + Qoi + QG8 = 0 (3.129)  FiGURE 3.43 Generalized glazing and absorber nodes, showing the various modes of heat transfer (Karimi et al., 1986).

For the absorber node,

Qa\ + Qa2 + ?A3 + Qa4 + ?A5 + Qa6 + Qa7 + ?A8 = 0 (3.130)

where qG1 = solar radiation absorbed by glazing node i. qG2 = net radiation exchange between glazing node i to the surroundings. qG3 = natural and forced convection heat transfer from glazing node i to the surroundings.

qG4 = convection heat transfer to the glazing node from the absorber (across the gap).

qG5 = radiation emitted by the inside surface of the glazing node i. qG6 = conduction along the circumference of glazing from node i to i + 1. qG7 = conduction along the circumference of the glazing from node i to i - 1. qG8 = fraction of the total radiation incident upon the inside glazing surface that is absorbed. qA1 = solar radiation absorbed by absorber node i. qA2 = thermal radiation emitted by outside surface of absorber node i. qA3 = convection heat transfer from absorber node to glazing (across the gap). qA4 = convection heat transfer to absorber node i from the working fluid. qA5 = radiation exchange between the inside surface of absorber and absorber node i.

qA6 = conduction along the circumference of absorber from node i to i + 1. qA7 = conduction along the circumference of the absorber from node i to i - 1. qA8 = fraction of the total radiation incident upon the inside absorber node that is absorbed.

For all these parameters, standard heat transfer relations can be used. The set of nonlinear equations is solved sequentially to obtain the temperature distribution of the receiver, and the solution is obtained by an iterative procedure. In Eqs. (3.129) and (3.130), factors qG1 and qA1 are calculated by the optical model, whereas factor qA5 is assumed to be negligible.

This analysis can give the temperature distribution along the circumference and length of the receiver, so any points of high temperature, which might reach a temperature above the degradation temperature of the receiver selective coating, can be determined.