To this point, we have developed an understanding of how solar energy is collected by flat-plate collectors…those where the entire area that the sun’s radiation falls, absorbs the incident energy. Concentrating collectors, on the other hand, use large reflectors to ´concentrate´ the incident solar energy onto a smaller receiver. The main goal for doing this is to increase the temperature of the heat collected from the sun. Increased temperature is a benefit for many industrial process uses, and is directly connected to the efficiency at which electricity can be produced from thermal sources.

In the following sections we will develop the analytical tools necessary to understand the basic concepts of concentration of solar energy, in parabolic trough, parabolic dish, central receivers and Fresnel lens solar collection systems.  These tools will be presented with the following outline:

• Why Concentration

  • Concentration Ratio

• Parabolic Geometry

  • The Parabola

  • Parabolic Cylinder

  • Paraboloid

• Parabolic and Spherical Optics

  • Two-dimensional Ray Trace Diagrams

  • Line-focus Troughs

  • Point-focus Dishes

• Reflection of Energy to the Receiver

  • Reflection of Parallel Rays

  • Reflection of Non-parallel Rays

  • Contribution of Errors

• Reflection to a Fixed Point (the Power Tower)

• Principles of Reflection and Refraction

  • Mirrors

  • Lenses

  • Total Internal Reflection

  • Dielectric Reflection and Antireflection Coatings

• Fresnel Lens Optics

• Conclusions

The operation of any solar thermal energy collector can be described as an energy balance between the solar energy absorbed by the collector and the thermal energy removed or lost from the collector. If no alternative mechanism is provided for removal of thermal energy, the collector receiver heat loss must equal the absorbed solar energy.

The temperature of the receiver increases until the convective and radiation heat loss from the receiver equals the absorbed solar energy. The temperature at which this occurs is termed the collector stagnation temperature.

For control of the collector temperature at some point cooler than the stagnation temperature, active removal of heat must be employed. This heat is then available for use in a solar energy system. The rate at which heat is actively removed from the collector determines the collector operating temperature. For removal of a large fraction of the absorbed solar energy as useful heat, the amount of heat lost from the receiver must be kept small.

Receiver heat loss can be reduced by operating the collector near the ambient temperature (such as with low-temperature flat-plate collectors) or by constructing the collector such that heat loss at elevated temperature is reduced. The most common way of reducing receiver heat loss at elevated temperatures is to reduce the size of the hot surface (i.e., the receiver) since heat loss is directly proportional to area of the hot surface. Concentrating collectors reduce the area of the receiver by reflecting (or refracting) the light incident on a large area (the collector aperture) onto an absorber of small area. With reduced heat loss, concentrating collectors can operate at elevated temperatures and still provide significant quantities of useful thermal energy.

A second reason for using concentration in the design of solar collectors is that, in general, reflective surfaces are usually less expensive than absorbing (receiver) surfaces. Therefore, large amounts of inexpensive reflecting surface area can placed in a field, concentrating the incident solar energy on smaller absorbing surfaces. However, concentrating collectors must track the sun´s movement across the sky, adding significant cost to the construction of a concentrating collector system.

The term "concentration ratio" is used to describe the amount of light energy concentration achieved by a given collector. Two different definitions of concentration ratio are in general use. They are defined briefly here so that the terms may be used.

Optical concentration ratio relates directly to lens or reflector quality; however, in many collectors the surface area of the receiver is larger than the concentrated solar image.

Thermal losses in such situations are larger than might be inferred from examination of the optical concentration ratio. Since geometric concentration ratio refers to receiver area, it is most commonly used because it can be related to collector heat loss [e.g., see Chapter 5] . Note that if the aperture insolation and receiver irradiance are both uniform over the entire area, the optical and geometric concentration ratios are equal.

Perhaps the simplest solar concentrator is a concentrating flat-plate collector. An example of such a collector is shown in Figure 8.1. This type of concentrator is a flat-plate collector surrounded by mirrors that reflect sunlight incident outside the normal perimeter of the flat plate onto the absorber plate of the collector. This type of collector is usually constructed to reduce the cost of a flat-plate collector by reducing the area of the absorber and cover plates, which are frequently expensive. A concentrating flat-plate collector such as that shown in Figure 8.1 is typically limited to geometric and optical concentration ratios of 2-3.

Figure 8.1 A concentrating flat-plate collector

If higher concentration ratios are desired, curved mirrors or lenses are used. Circular and parabolic mirror optics are described in Section 8.3, and Fresnel lenses are examined in Section 8.7.

Figure 8.2 The parabola.

Figure 8.3 Segments of a parabola having a common focus F and the same aperture diameter.

this area should not be confused with the reflecting surface area of a parabolic trough or dish or their aperture areas, the equations for which are given in the following section.

Often in evaluating parabolic geometry and related optical derivations, the casual reader becomes confused with the many forms used to represent the geometry of a particular parabolic shape. The following equivalencies are given for the convenience of the reader:

The reflective surface area is found by using the arc length developed in Equation (8.10):

  (8.19)

 Figure 8.4 Paraboloid of revolution.

The surface area of the paraboloid may be found by integrating Equation (8.19) over the appropriate limits. We can define a circular differential area strip on the paraboloid as shown in Figure 8.5 as

Figure 8.5 Parameters defining a circular strip of differential area.

The concentrator aperture area, of most importance to the solar designer for

An equation for the aperture area may also be cast in terms of the focal length and the rim angle. Using Equation (8.6), which is the polar form of the equation of a parabola, we have

Both spherical and parabolic geometries are represented in solar thermal concentrators. In some concentrators, however, a spherical (or more usually cylindrical) geometry is used as an approximation to a true parabola. As will be shown in Chapter 9, spherical optics allow for fixed- aperture (i.e., non-tracking) concentrators. Since parabolic geometries strongly dominate solar concentrators, a rather thorough examination of the analytical description of parabolic geometry is presented in this chapter.

In order to facilitate a discussion of spherical and parabolic optics, it is convenient to define the term ‘plane of curvature’, as illustrated for a parabolic trough in Figure 8.6.

The plane of curvature (i.e., cross section) can be considered as a two-dimensional slice normal, in the case of a parabolic trough, to the linear axis of the concentrator. For a spherical or a parabolic dish, the plane of curvature is rotated to generate the dish geometry. The optical principles of spherical and parabolic mirrors are examined below through ray trace diagrams and by initially restricting the discussion to the plane of curvature. The impact of translation or rotation of the plane of curvature is then discussed in order to evaluate the optical characteristics of actual concentrator geometries.

Figure 8.6 Definition of plane of curvature

The purpose of a general discussion at this point, prior to examining individual concentrator concepts, is to develop an understanding of the basic limitations imposed by spherical and parabolic geometries and how these limitations impact basic concentrator design.

which reduces to the familiar form

A parabola (see Figure 8.7b) can be defined as any section of a right circular cone resulting from a plane slicing through the cone parallel to, but not including, the axis of the cone. The equation for a parabola, with focal length f, in the Cartesian coordinates is

These basic equations are useful in examination of the optical properties of spherical and parabolic mirrors.

Figure 8.8 Definition of rim angle

1. All parallel rays reflected from a circular mirror pass through a line drawn through the center of the circle and parallel to the incident rays (see Figure 8.9a).

In addition, a circular mirror is symmetrical with respect to rotations about its center. This means that if the sun’s rays (assumed to be parallel for this discussion) are not normal to the mirror’s aperture as in Figure 8.9a, the pattern of reflected rays looks the same but is rotated (see Figures 8.9b and 8.9c).

A parabolic mirror, on the other hand, is not symmetrical to rotations about its focal point. As shown in Figure 8.10, if the incident beam of parallel rays is even slightly off normal to the mirror aperture, beam dispersion occurs, resulting in spreading of the image at the focal point. For a parabolic mirror to focus sharply, therefore, it must accurately track the motion of the sun to keep the axis (or plane) of symmetry parallel to the incident rays of the sun.

Figure 8.10 Characteristics of parabolic optics, (a) for rays parallel to the axis of symmetry, (b) for rays 1^o off of the axis of symmetry.

To form either a cylindrical or parabolic trough, the two-dimensional mirrors shown in Figures 8.9 and 8.10 must be translated normal to the plane of curvature as illustrated in Figure 8.11. Tracking requirements of the linear troughs are similar to that of the two-dimensional mirrors discussed above.

Figure 8.11 Formation of linear troughs: (a) parabolic trough; (b) cylindrical trough.

A parabolic trough as shown in Figure 8.11a has a line focus and must be tracked about its linear axis in order to maintain focus. The proper tracking angle is defined by the orientation of the trough relative to the sun’s position. Analytical expressions for the proper tracking angle of parabolic troughs are developed in Chapter 4. Basically, a parabolic trough must track about its linear axis so that when the sun’s rays are projected onto the plane of curvature, they are normal to the trough aperture.

Figure 8.12 shows a photograph of a receiver tube illuminated by direct insolation entering the trough aperture with a non-normal incidence angle. In this case, the trough has been tracked about its linear axis as indicated by the highly focused beam image on the receiver tube. As a result of the non-normal incidence angle, however, the reflected beam is translated down the receiver tube; note that the right end of the receiver tube is dark. At the far end of the parabolic trough some of the incident insolation is reflected past the end of the receiver tube. This is illustrated in Figure 8.13 for another collector. Note that some concentrated light is failing on the flexible hose and not on the receiver. This energy is lost to the collector and is called the collector endloss.

Figure 8.13 Parabolic trough endloss. Courtesy of Sandia National Laboratories.

Since linear translation does not introduce defocusing of the concentrated radiation, the aperture of a cylindrical trough need not track at all to maintain focus. However, as indicated in Figure 8.11b, a high-rim angle cylindrical trough would have a focal plane not a focal line. To avoid a dispersed focus, cylindrical troughs would have to be designed with low rim angles in order to provide an approximate line focus. The advantage of a cylindrical mirror geometry is that it need not track the sun in any direction as long as some means is provided to intercept the moving focus.

For practical applications, if the rim angle of a cylindrical trough is kept low (e.g., <20-30 degrees ), spherical aberration is small and a virtual line focus trough is achieved. Figure 8.15 shows the focusing of circular mirrors with various rim angles.

 Figure 8.15 Focusing of parallel rays of light using circular mirrors with different rim angles.

If the two-dimensional curved mirrors shown in Figures 8.9 and 8.10 are rotated rather than translated, the resulting geometric figures are spherical and parabolic dishes, respectively. A parabolic dish must be tracked in two dimensions in order to maintain the incident beam insolation normal to the dish aperture at all times to allow focusing. As with the cylindrical trough, however, the aperture of a spherical dish need not be tracked because of the symmetry of a sphere (circle) as discussed earlier. However, a linear receiver that tracks the moving focal line (see Figure 8.9) must be provided. A prototype parabolic dish concentrator is described in Chapter 9 along with a prototype non-tracking aperture spherical dish.

Although the design details of any one parabolic collector may differ from those of another parabolic collector, optical constraints define the basic configuration of all parabolic concentrators. This section reviews these constraints and examines the process of supplying concentrated optical energy to the receiver. Heat-loss considerations are addressed in Section 9.2. The basic logic flow followed in this review is summarized in Figure 8.16. The reflection of parallel rays of light, normal to the collector aperture, is reviewed. The goal is to develop an analytical equation that shows the contribution of light reflected to the focus as a function of rim angle.

Figure 8.16 Optical analysis of parabolic concentrators.

This analysis is then modified to account for the fact that the sun’s rays are not truly parallel. Angular errors (e.g., slope errors) in the conformity of the reflector surface to a true parabolic shape are incorporated at this stage. The spread of the reflected beam due to errors in mirror surface slope, tracking, a