# Heat Transfer in Flat-Plate Solar Collectors

Useful thermal energy obtained from a solar collector is the difference between incident solar radiation transmitted through the glazing and absorbed as heat, and heat lost back to ambient conditions. The difference is the heat transferred to the working fluid. In a steady state, the governing equation, normalized to the absorber plate area, is

qx = (m)Aap – ULAap (tap – ta) = mcp (to – tt) (11-2)

where A = area of the absorber plate, m2

ap

cp = specific heat of the transport fluid, J kg-1 K-1 I = instantaneous direct normal solar irradiation, W m-2 m = transport fluid mass flow rate, kg s-1 qx = useful heat gained, W ta = ambient air temperature, °C

t. = temperature of the transport fluid flowing into the collector, °C Figure 11-10 Example efficiency graphs for single – and double-glazed, flat-plate solar collectors.

t = temperature of the transport fluid flowing out of the collector, °C t = average temperature of the absorber plate, °C

UL = collector thermal conductance, normalized to the absorber plate area,

W m-2 K-1

a = solar absorptance of the collector plate, dimensionless t = solar transmittance of the collector glazing, dimensionless

Defining efficiency, h, as useful energy output divided by the irradiating solar insolation, we have

П = та – Ul (tap – ta )/I (11-3)

The difficulty in using the above equations is the uncertain knowledge of the absorber plate temperature in a real installation. Moreover, absorber temperature is not likely to be uniform and a means must be found to average temperature over the plate.

ASHRAE proposed an adjustment factor, called the heat removal factor (FR), to permit using the inlet transport fluid temperature (which can be readily measured, or specified as the thermal storage temperature) to substitute for absorber plate temperature.

qu = Fr (Ta)IAap – FrUAp(ti – ta) (11-4)

n = Fr (та) – FrUl(ti – ta )/I (11-5)

This adjustment, quantified empirically as part of collector efficiency testing, can be viewed as the ratio between the net heat actually collected and the net heat that would be collected were the collector absorber plate entirely at temperature ti.

The general form of the efficiency curve is in Fig. 11-10. Single-glazed collectors transmit more solar radiation to the absorber plate but are insulated less well against heat loss to the environment, thus the usual efficiency curve has a higher intercept but steeper slope. The reverse is, of course, true for double-glazed collectors.

The performance model to describe the efficiency curve follows one of the following forms, where values of pn are determined by least-squares regression using measured data to determine the best fit and decide whether the second-order term is statistically significant and provides a higher coefficient of fitness. The first of the following is used if not.

л = р0+М_^ (ii-6)

П = в +вД – ta)/l +p2(tt – ta)/l2 (11-7)

The у-intercept of the efficiency line represents no temperature difference between the collector and the ambient air and is, thereby, the value of FR(ta). The slope of the efficiency line is the negative of FRUL.

When solar irradiation is weak and ambient air is cold, efficiency can become negative (operation below the x-axis intercept). Heat is lost from the collector more rapidly than it is collected, which bleeds heat from the thermal storage. This situation can occur early in the morning when the sun is low in the sky, at a time of low air temperature, and late in the afternoon as the sun wanes and thermal storage temperature is high after a day of collection. The concept of “critical insolation” applies. Critical insolation is the insolation at which heat gain balances heat loss.

Figure 11-11 shows the concept of how critical insolation varies over the day. As a physical interpretation of the critical insolation graph, the area under the insolation curve is the total energy available for collection during the day. The area above the critical insolation line during collection hours is the net amount available for collection and the area below the critical insolation line represents the amount not available (lost back to the ambient conditions).

In practice, the critical insolation line is not straight. The line depends on ambient air temperature, among other factors, which is likely to be highest in the middle of the afternoon, lowering the critical insolation line during those hours. However, critical insolation is generally a concern only at the beginning and end of the collection day.