# Solar thermal energy collectors

Figure 1: Typical construction of solar thermal collector

Solar collector analysis is similar to that for spacecraft, with the added complications that the collector surface usually has convective losses in addition to radiation; the collector is exposed to not only solar but also environmental radiation; and the collector is not isothermal (see Fig. 1).

So called flat plate collectors are used to gather thermal energy, such as those employed for swimming pool heating, domestic hot water production, residential and commercial building heating, and possibly for use with thermally-driven cooling cycles. The collector is made up of an absorber plate, which absorbs the solar radiation, and transfers it to a fluid flowing through channels in the plate. These are often of fin-tube design. For low temperatures such as for swimming pool heaters, the absorber surface is often uncovered. For intermediate to higher temperatures, convective losses to the environment become significant, and a transparent cover plate may be placed above the absorber plate to add additional resistance to losses. The back of the collector is insulated. The instantaneous rate of net energy collected by the working fluid with mass flow rate $\dot m$ and specific heat cp is given by a First Law energy balance as

$q = \dot m\left( {{H_{out}} - {H_{in}}} \right) = \dot m{c_p}\left( {{T_{out}} - {T_{in}}} \right)\qquad \qquad(1)$

Alternately, a heat transfer balance for the collector gives

$q = {\alpha _{eff}}{q''_{solar}}A\cos \theta - \bar hA(\overline {{T_s}} - {T_\infty }) - \varepsilon \sigma A(\overline {T_s^4} - T_\infty ^4)\qquad \qquad(2)$

where q"solar is the solar flux on a surface normal to the sun and θ is the angle of the sun relative to the normal of the solar collector.

This form includes the following assumptions: First, the value α eff is the effective solar absorptivity of the collector, which is also assumed to be a diffuse surface property. As eq. (2) is written, it is inherently assumed that the solar energy incident on the collector comes only from the direction of the sun, and is what is known as direct solar energy. The direct component is the major solar input on a clear day. However, when days are hazy or cloudy, a so-called diffuse component may be present, which is incident from many other directions than θ onto the collector. Additionally, a tilted collector may gather energy reflected from surrounding structures or the ground. The latter two components are omitted in the simplified treatment given here. The α eff in eq. (2) includes not only the absorptivity of the solar collector plate itself, but also any effects of cover glasses on the overall absorption of solar energy.

The final two terms in eq. (2) are the heat losses between the collector and the surroundings. The $\overline {{T_s}}$ is the average solar collector plate surface temperature, taken as the average of the fluid inlet and outlet temperatures,

$\overline {{T_s}} = ({T_{in}} + {T_{out}})/2$
.

Because most flat plate collectors have a fairly small temperature change in the working fluid between the collector inlet and outlet, this is a valid approximation to use in determining a temperature difference for use in the convective and radiative loss terms. Often, a sky radiating temperature that is lower than the environmental (air) temperature is used as the radiative sink temperature in the final term.

The final term in eq. (2) comes from approximating the radiative loss as an exchange between the collector surface and the surroundings, both of which are near enough in absolute temperature that the IR emissivity and absorptivity are close to the same value. If cover glasses are present, the radiative loss calculation becomes more complex, as most “transparent” materials in cover glasses actually become opaque at wavelengths longer than about 2 μm, although some materials have transparent “windows” in the infrared spectrum.

Figure 2: Natural convection patterns in parallel enclosures as found in solar collectors.

The rate of energy collection is time dependent. Even on a clear day, the angle of the sun relative to the collector, θ, will vary with time of day and day of the year. In addition, weather conditions can change the value of the direct solar radiation q"solar over short time periods.

The convective heat transfer coefficient $\bar h$ is an overall value, accounting for natural convection between the collector plate and any cover glasses, conduction through the cover glasses, and forced convection from the outer cover glass surface and the environment due to average wind conditions. Its determination is challenging, because it can involve free convective transfer between parallel surfaces (absorber plate and cover glass, and possibly between multiple parallel cover glasses.) If the plate is installed on a slanted rooftop, for example, then the free convective flows are dependent on the slant angle of the collector. The free convection patterns undergo a transition from multiple cells similar to Bénard cells for the horizontal collectors to a single large rotating cell found in a vertical collector (Fig. 2), although the figure is quite simplified (the flow patterns are three-dimensional because of the presence of corners in the closed rectangular geometry). The transition occurs at some tilt angle that depends on the ratio of length L to the plate-to-cover spacing height H, L / H. For L / H > 12 (typical of solar collectors,) the critical angle according to Hollands et al. (1976) is about 70°. Analysis of the effect of tilt and the transition angle on solar collector configurations has been carried out by Catton et al. (1974), Hollands et al. (1976), Randall et al. (1979) and Torrance and Catton (1980). Randall et al. report that the average Nusselt number in the laminar flow regime based on plate spacing H is adequately correlated (within ± 8%) for tilt angles φ from 45 to 90° by

$\overline {N{u_H}} = 0.118{\left[ {G{r_H}\Pr {{\cos }^2}\left( {\phi - 45} \right)} \right]^{0.29}}\qquad \qquad(3a)$

for $4 \times {10^3} \le G{r_H} \le 3.1 \times {10^5}$ and aspect ratios $9 \le L/H \le {\rm{36}}$. Note that within this range of L / H, the predicted Nusselt number is independent of aspect ratio. For smaller aspect ratios, other investigators have observed a dependence on aspect ratio. For angles less than the critical angle, Hollands et al. (1976) suggest the correlation

$\overline {N{u_H}} = 1 + 1.44\left[ {1 - \frac{{1708}}{{R{a_H}\cos \phi }}} \right]\left[ {1 - \frac{{1708{{\left( {\sin 1.8\phi } \right)}^{1.6}}}}{{R{a_H}\cos \phi }}} \right] + \left[ {{{\left( {\frac{{R{a_H}\cos \phi }}{{5830}}} \right)}^{1/3}} - 1} \right]\qquad \qquad(3b)$

which applies for $L/H \ge 12$ , $\phi \le {70^o}$. If any term is negative, it is set equal to zero. The efficiency of a solar collector is defined as the amount of energy transferred to the working fluid passing through the collector over the energy incident from the sun, or

$\begin{array}{l} {\eta _{coll}} = \frac{q}{{{{q''}_{solar}}A\cos \theta }} = {\alpha _{eff}} - \frac{{\bar h{T_\infty }}}{{{{q''}_{solar}}\cos \theta }}\left( {\overline {{\Theta _s}} - 1} \right) - \frac{{\varepsilon \sigma T_\infty ^4}}{{{{q''}_{solar}}\cos \theta }}\left( {\overline {\Theta _s^4} - 1} \right) \\ {\rm{ }} = {\alpha _{eff}} - \bar H\left( {\overline {{\Theta _s}} - 1} \right) - \bar S\left( {\overline {\Theta _s^4} - 1} \right) \\ \end{array}\qquad \qquad(4)$

where $\overline {{\Theta _s}} {\rm{ }} = {T_s}/{T_\infty }$ and the two dimensionless groups describing the relative importance of convection and radiation losses to incident solar flux are

$\bar H = \frac{{\bar h{T_\infty }}}{{{{q''}_{solar}}\cos \theta }}{\rm{ and }}\bar S = \frac{{\varepsilon \sigma T_\infty ^4}}{{{{q''}_{solar}}\cos \theta }}$
.

If θ s is not too far from unity, the final term in eq. (4) can be linearized by the transformation

$\begin{array}{l} \left( {\overline {\Theta _s^4} - 1} \right) = \left( {\overline {\Theta _s^2} + 1} \right)\left( {\overline {\Theta _s^2} - 1} \right) = \left( {\overline {\Theta _s^2} + 1} \right)\left( {\overline {\Theta _s^{}} + 1} \right)\left( {\overline {\Theta _s^{}} - 1} \right) \\ {\rm{ = }}\left( {\overline {\Theta _s^2} + 1} \right)\left( {\overline {\Theta _s^{}} + 1} \right)\left( {\overline {\Theta _s^{}} - 1} \right) \approx 4\left( {\overline {\Theta _s^{}} - 1} \right) \\ \end{array}\qquad \qquad(5)$

For example, if the collector average surface temperature is $\overline {{T_s}}$ = 67°C = 340K and T = 27°C = 300 K, then Θs = 340/300 = 1.13. The error in the approximation in eq. (5) is then

$\frac{{\left( {\overline {\Theta _s^2} + 1} \right)\left( {\overline {\Theta _s^{}} + 1} \right) - 4}}{{\left( {\overline {\Theta _s^2} + 1} \right)\left( {\overline {\Theta _s^{}} + 1} \right)}} = \frac{{\left( {{{1.13}^2} + 1} \right)\left( {1.13 + 1} \right) - 4}}{{\left( {{{1.13}^2} + 1} \right)\left( {1.13 + 1} \right)}} = 0.18$

or 18%. If this level of approximation is acceptable (which depends on the magnitude of $4\overline S$ relative to $\bar H$), then eq. (3) can be rewritten using eq. (5) as

$\begin{array}{l} {\eta _{coll}} = {\alpha _{eff}} - \bar H\left( {\overline {{\Theta _s}} - 1} \right) - \bar S\left( {\overline {\Theta _s^4} - 1} \right) = {\alpha _{eff}} - \bar H\left( {\overline {{\Theta _s}} - 1} \right) - 4\bar S\left( {\overline {{\Theta _s}} - 1} \right) \\ {\rm{ }} = {\alpha _{eff}} - \left( {\bar H + 4\bar S} \right)\left( {\overline {{\Theta _s}} - 1} \right) = {\alpha _{eff}} - \left( {\bar H*} \right)\left( {\overline {{\Theta _s}} - 1} \right) \\ \end{array}\qquad \qquad(6)$

The $\bar H*$ = ($\bar H$ + $4\overline S$) is a modified dimensionless heat transfer coefficient that now accounts for both convective and (linearized) radiative losses to the surroundings. Equation (6) shows that after this linearization, the collector efficiency is directly dependent on the temperature difference (Θs -1). Examining eq. (6), it is clear that collector efficiency is affected by the radiative surface properties. Choosing a collector plate coating with large α eff is a benefit; however, a spectrally selective coating that has large absorptivity for solar energy and a small emissivity at large wavelength will not only have a large value of α eff, but also a small value of hemispherical emissivity at the plate temperature, which will make the value of $\overline S$ small and thus help to reduce radiative loss to the environment. At some high collector plate average temperature, the absorbed energy is balanced by the heat transfer to the environment, so no useful energy is collected and the efficiency equals to zero. This can occur if flow is lost to the collector. At this condition, the collector reaches its maximum or stagnation temperature given by

${\Theta _{stag}} = 1 + \frac{{{\alpha _{eff}}}}{{\left( {\bar H*} \right)}}{\rm{ or }}{T_{stag}} = {T_\infty } + \frac{{{\alpha _{eff}}{T_\infty }}}{{\bar H + 4\bar S}} = {T_\infty } + \frac{{{\alpha _{eff}}{{q''}_{solar}}\cos \theta }}{{\bar h + 4\varepsilon \sigma T_\infty ^3}}\qquad \qquad(7)$

Because the sun traverses the sky during the day, and the path traversed varies with time of year, and the radiation reaching the collector depends on local weather conditions, it is very difficult to accurately predict the annual integrated energy that can be produced by a given collector. Some methods that have been developed to account for these factors (Duffie and Beckman, 2006). Year-to-year variations can also be significant. It is possible to use the collector to reject energy from the working fluid. During the hours of night, the collector energy balance, eq. (2), can be applied with q"solar = 0 to give

$q = - \bar hA\left( {\overline {{T_s}} - {T_\infty }} \right) - \varepsilon \sigma A\left( {\overline {T_s^4} - T_{sky}^4} \right)\qquad \qquad(8)$

The effective night sky temperature Tsky has been substituted for T in the radiative loss term, since the sky temperature, especially on a clear night in a dry climate, may be much lower than the air temperature that governs convective loss. Both loss terms are negative as long as the plate temperature is above the environment and sky temperatures, so energy will be lost from the collector, and the temperature of the fluid passing therough the collector will be lowered as given by eq. (1). There is no particular advantage to linearizing eq. (8) because the sink temperatures of the two heat transfer modes are different, so the linearized form can't be combined with the convection term.

To increase the rate of energy collection per unit area by a solar collector, the incident solar flux can be multiplied by using concentrators. In this case, eq. (1) is modified to give

$q = C{\alpha _{eff}}{q''_{solar}}A - \bar hA\left( {\overline {{T_s}} - {T_\infty }} \right) - \varepsilon \sigma A\left( {\overline {T_s^4} - T_\infty ^4} \right)\qquad \qquad(9)$

In this form, C is the concentration ratio, or the number of effective sun fluxes incident on the collector due to multiple concentrators. Some concepts for solar collection have concentration ratios of from C = 2-4 (flat plates with mirrored grooves), 30-60 (parabolic mirrored troughs), to over C = 600 for solar tower collectors with many heliostats to direct sunlight onto the collector surface. Solar towers can operate at very high temperatures, and linearization of the radiative loss term is inappropriate.

## References

Catton, I., Ayyaswamy, P.S., and Clever, R.M., 1974, ‘‘Natural Convection Flow in a Finite, Rectangular Slot Arbitrarily Oriented with Restpect to the Gravity Vector,” Int. J. Heat Mass Transfer, Vol. 17, pp. 173-184.

Duffie, John A. and Beckman, William A., 2006, Solar Engineering Of Thermal Processes, 3rd ed., John Wiley & Sons, Inc., New York, NY.

Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.

Hollands, K.G.T., Unny, S.E., Raithby, G.D., and Konicek, L., 1976, ‘‘Free Convection Heat Transfer. Across Inclined Air Layers,” J. Heat Transfer, Vol. 98, 189-193.

Randall, K. R., Mitchell, J. W., and El-Wakil, M. M., 1979, ‘‘Natural Convection Heat Transfer Characteristics of Flat Plate Enclosures,” J. Heat Transfer, Vol. 101, pp. 120-125.

Torrance, K.E. and Catton, I., 1980, ‘‘Natural Convection in Enclosures,” Proc Nineteenth National Heat Transfer Conf., HTD-Vol. 8, Orlando, FL, July 27-30, 1980