In order to use solar energy to produce steam and hence power a Rankine cycle to generate electricity, one must concentrate the solar radiation. The concentrated insolation is absorbed by a pipe that has the heat transfer fluid flowing through it. It was demonstrated in the previous chapter that a flat plate solar energy collector can heat water fairly efficiently to order 50 °C, however, much higher temperatures, approaching 400 °C, are required for electricity generation. Here the use of parabolic reflectors is considered for electricity production.
Previously we found that concentrating the sunlight can produce a large temperature rise over a short length of pipe in a solar concentrator, in Example 3.6. The maximum concentration ratio for a linear collector, a pipe positioned in a mirrored device of some sort that concentrates the insolation, is
Cr, i = (1-Dimension) (3.35)
sin (Osub )
where CR, i is the maximum, ideal concentration ratio and 0sub is the angle subtended between the Earth and Sun (0.2640). Using the value of the angle in the above equation one finds the maximum concentration ratio is 216. In practice, the concentration ratio, Cr, is of order 20-30 in power systems and these will be the values used here.
It is possible to estimate how Cr affects the maximum temperature the absorber can obtain. To do this we start with the definition of the extraterrestrial irradiance discussed in Chapter 2,
where as is Stefan’s constant (5.670 x 10-8 W/m2-K4) and Ts is the Sun’s temperature when modeled as a black body radiator (5793 K). Here CR, i2 is the ideal concentration ratio in two dimensions which will yield the greatest absorber temperature, see eqn (3.34). Basically, this concentration ratio is equivalent to taking all the power that the Sun emits and focusing it on the absorber, and is equal to CR, i2 = 1/sin(ds)2 = 46, 700, as discussed in Chapter 3.
The maximum absorber temperature Tabs, m is found by assuming the absorber loses heat through radiation only and is in equilibrium with the Sun emitting Pext irradiance; attenuation by the atmosphere is ignored. This analysis is similar to that performed in the previous chapter when the stagnation temperature was found and is a balance of heat gain from insolation to heat loss from radiative heat transfer, one can write
Сії Aap = asTas, m A [=] W
С R, i2
where the absorber has been assumed to be equal to a perfect black body radiator and the aperture area Aap and absorber area A multiply each term for generality. The ratio of these two areas is the concentration ratio, Cr = Aap/A, and the above equation can be rearranged to (9.5)
The result of this model for the maximum absorber temperature is shown in Fig. 9.4. Convective heat transfer has been ignored, which will tend to reduce the temperature, and the emissivity for the absorber was allowed to be one or is a perfect black body. If the absorber is assumed to be a selective surface with an emissivity of 0.05-0.10, the fourth root of this
number will be taken and the temperature for a concentration ratio of 1.0 will be approximately 500 0C, which is substantial, rather than 120 °C. But, as mentioned above, convective heat transfer will reduce this value, as will attenuation of the irradiance by the atmosphere.
However, as the absorber becomes hotter, at very high concentration ratios, its behavior will tend towards a black body. Thus, this calculation is most accurate near the highest concentration ratios, yet, the values for the lower concentration ratios are of the correct order and the maximum temperature rise for CR = 25 is of order 600 0C. This is the concentration ratio used in the most mature STEGE plants.
The details of the reflector used to concentrate the insolation have not been considered yet. There is a vast literature on this topic and references which compare the various types of reflectors can be found at the end of this chapter. So, rather than an exhaustive study of the various types of reflectors, a simple one is considered, the parabolic reflector. The equation for a parabola in Cartesian coordinates is y = x2/[4/], where x and y are the coordinates and / is the focal distance. Polar coordinates are more suitable for analysis and with reference to Fig. 9.5 one can write the equation for a parabola as
2/ 1 + cos(4>) where r and ф are the radial and angular coordinates, respectively. The maximum radial position and angle, rm and фт, respectively, are shown in the figure and are important in considering the amount of insolation trapped by the parabolic reflector. These details will not be considered here as the references at the end of the chapter adequately describe this aspect of parabolic reflectors. So, we consider the parabolic reflector as 100% efficient. The parabolic reflector’s linear axis, of length L, is usually located in the north-south direction and tracking is employed by tilting it around that axis to follow the Sun through the day.
It is possible to use a truncated cylinder as a ‘parabolic’ reflector in some circumstances. Reference to Fig. 9.5 shows a circle whose center has been displaced from the origin along the y-axis by the radius R. The equation for the circle is x2 + [y – R]2 = R2 which can be rearranged to y = R – R 1 – x2/R2. If x « R then a Maclaurin series expansion can be performed to arrive at y и R – R[1 – x2/2R] = x2/2R. This is the equation for a parabola with / = R/2. In some cases a truncated cylinder may be easier to manufacture and as long as W « R it is adequate as a concentrator of insolation. Most parabolic reflectors used in practice, though, have W = 5.8 m and / = 1.8 m, meaning that if a truncated cylinder were used R = 3.6 m which invalidates the approximation. So, a true parabolic profile must be followed for systems used in the field.