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, and nonparallel solar rays provides the information needed to examine the principles used in selecting the size of the receiver.

An examination of the resulting concentration ratios yields information about the approximate rim angle for a parabolic collector. Heat-loss considerations discussed in Section 9.2 will further constrain the design of the receiver.

where

Figure 8.17 Reflection of a light ray from a parabolic mirror.

Application of Figure 8.17b yields

Thus

and

where the following substitution was used (see Figure 8.17).

Figure 8.18 Reflection of nonparallel rays from a parabolic mirror.

This is the minimum beam spread that will occur. Any errors in, for example, the trueness of the mirror slope will cause additional spreading of the image size.

In addition to the finite angular size of the sun, independent errors in the conformity of the mirror to a true parabolic shape (i.e., slope errors), non-specular reflection of the incident light beam, tracking errors, and receiver alignment errors all contribute to further apparent spreading of the reflected light beam. Typically, these errors are assumed to be random and are reported in standard deviation units. Their combined effect is statistically determined. Although the sun’s intensity is not normally distributed across its disc, it may be approximated as a standard distribution so that it can be treated similarly to the concentrator errors.

Note also that Table 8.1 is divided into two main parts: one-dimensional errors, and two-dimensional errors. One-dimensional errors are those errors that contribute to beam spreading in the plane of curvature. They are combined as

Note that the slope error is multiplied by 2. This is a result of Snell’s law (Equation 8.54) and the fact that with a slope error, the mirror surface direction is changed whereas the receiver position is not (see Figure 8.19).

Figure 8.19 Impact of slope error on beam deflection.

Figure 8.19a shows the reflection of the beam with no slope error. If, as illustrated in Figure 8.19b, there is a slope error of 5 degrees, the reflected beam is misdirected by an amount of 10 degrees from the original reflected path. In the case of a tracking error, for example, the receiver moves together with the mirror surface so that the factor of 2 does not appear for this type of error in Equation (8.41). The factor of 2 appears only when an error is the result of a movement on the mirror surface and not the receiver.

With central receiver systems, the sun 's rays that hit a field of movable mirrors (heliostats) are reflected to a single fixed point in space (the receiver). The tracking angles for the reflector surface, along with the angle of incidence to this surface, may be derived using the vector techniques developed in Chapter 4.

The appropriate geometry can be best described by using zenith, east, and north (z, e, n) coordinates with their origin O at the base of the aim point A as shown in Figure 8.20. The location of a reflecting surface B can be defined as z₁, e₁ and n₁ and the aim point located at a distance z₀ above the origin.

The third vector of interest is a unit vector normal to the reflector surface. This is defined as

 Table 8.3 summarizes the sign conventions used to develop the angles related to heliostat design.

Table 8.3  Sign Convention for Important Heliostat Angles

Using Equations (3.9) and (8.46), we have

from which the angle of incidence or reflection can be obtained, if we know the position of the sun and the position of the aim point relative to the reflection surface.

and

The third expression is redundant but is written here for completeness:

where the symbols are defined as in Figure 8.21.

Non-specular reflectance is measured by use of a device that integrates the amount of light reflected by the surface in all directions.

An example of a specular reflector is a highly polished silver mirror that gives a clear undistorted reflected image. An example of a diffuse non-specular reflector is snow that yields no reflected image. The properties of curved mirrors to concentrate solar energy through specular reflection are examined in the following paragraphs.

It is important to recognize that, for a solar concentrator, specular reflection of the entire terrestrial solar spectrum is important. Not all materials exhibiting high specular reflectance reflect equally well at all wavelengths. The terrestrial solar spectrum is reported in Figure 2.10. Figure 8.23 shows the specular reflectance of several materials as a function of wavelength and illustrates why silver is widely used as a reflector material for solar collectors. Gold, for example, has a very high reflectance in the IR spectrum, which leads to its use as a radiation barrier in thermal insulation applications. However, it absorbs in the blue part of the visible spectrum (hence its color), decreasing its effectiveness as a solar reflector. The specular reflectance of several materials integrated over the entire solar spectrum are given in the Appendix.

Mirrors used in solar concentrators are frequently back-surface or second surface mirrors (i.e., the reflector surface is behind a protective sheet of glass or plastic). When light passes through a material, part of the light energy is absorbed. The absorption of light by transparent materials is an important factor in the design of the mirrors used in solar concentrators. Figure 8.24 is an example of a back-surface reflector. Since the light beam must traverse the thickness of the protective layer twice (once on incidence and again on reflection), any material that absorbs light can significantly reduce the solar energy reaching the receiver. The ability of a material to transmit light is characterized by its transmittance.

Figure 8.24 A back-surface reflector

As in the case of reflectance, transmittance can be either specular or diffuse. Light transmission through a homogeneous material such as glass or clear plastic is typically specular in that the transmitted ray follows Snell's law (Equation (8.54). A translucent material such as fiberglass may have high transmittance but localized discontinuities (e.g. glass fibers) that scatter the light, giving rise to a diffuse non-specular transmittance. Since solar concentrators typically employ only materials having high specular transmittance, the term "transmittance" in this book refers only to specular transmittance unless specified otherwise.

The ability of a material to transmit light is frequently a function of the wavelength of the light. Figure 8.25 shows the reflectance of back-surface glass mirrors with varying amounts of iron present as Fe²⁺. Since Fe²⁺ absorbs light in the visible part of the spectrum, the transmittance of glass (reflectance of a mirror) for the solar spectrum decreases with increasing concentrations of Fe²⁺. Thus, low-iron (i.e., 0.06 percent Fe²⁺ or less) glass and glass manufactured in a manner to convert Fe²⁺ to Fe³⁺, which does not absorb strongly in the solar spectrum, is of growing interest. In addition, thin mirrors are of interest since reduction of the path length that the light must traverse through the glass will reduce losses that are due to light absorption by the glass.

Figure 8.25 Effect of iron concentration on glass mirror reflectance (Clark, 1981).

Lenses operate on the principle of Snell's Law. In particular, materials such as glass and clear plastics are used to refract the incident light, transmit the light specularly through the lens medium, and then refract the light again as it leaves the lens. Properly oriented, a curved lens is capable of refracting the parallel rays of the sun and concentrating the image on a receiver as shown in Figure 8.26. The optics of Fresnel lenses are discussed further in Section 8.7.

An interesting consequence of Snell's law is the occurrence of total internal reflection. Total internal reflection can occur when light passes from a medium of high index of refraction to a medium of low index of refraction. Consider, for example, the passage of light from glass to air as shown in Figure 8.27. Total internal reflection occurs, as shown, when θ_rfr equals 90 degrees. Therefore, the incidence angle θ_i at which total internal reflection will occur can be calculated using Equation (8.54).

Figure 8.27 Total internal reflection: (a) refraction; (b) total internal reflection.

Example: Calculate the angle at which total reflection of light passing from glass to air occurs.

Thus, for incidence angles equal to or greater than 42 degrees, light passing from glass, with an index of refraction of 1.5, to air, with an index of refraction of 1.0, will undergo total internal reflection. This is termed the critical angle of the glass-air interface. Note that there is no critical angle for light passing from a medium of low index of refraction to a medium of high index of refraction.

In the discussion of reflection given above, there was no discussion of reflection from, for example, an air-glass interface. Reflection does occur, of course, as we all have observed when looking at a glass window. This is called the dielectric reflection. Referring to Figure 8.28, the fraction of the incident beam that is reflected at the interface is given by (Smith , 1966)

Figure 8.28 Dielectric reflection - definition of angles.

Figure 8.29 Effect of incidence angle on dielectric reflection from air-glass interface (Smith, 1966).

Optical coatings, so-called antireflection coatings, have been developed to reduce reflection from the air-glass interface in order to improve optical transmission. Such optical coatings have been used on the glass envelopes surrounding the receiver tubes of solar thermal concentrators.

One method of preparing antireflection glass surfaces is by etching the surface (Elmer and Martin, 1979). Antireflection surfaces on alkali-borosilicate glasses can be produced by exposing the glass surface to dilute acid solutions containing low concentrations of etchants, such as ammonium fluoride or ammonium bifluoride.

These solutions selectively extract soluble oxides within the surface layer, leaving a hydrated porous layer, largely silica, at the surface of the glass. The resulting surface layers are unique in that they maintain low reflectance losses in both the visible and the near-infrared. The reduction in dielectric reflectivity is attributable to a variable refractive index gradient caused by the interconnected pores of the silica-rich surface layer. The process is applicable to glasses that exhibit selective solubility in hot acid solutions. Etching has been used to produce an antireflection surface on the glass envelope surrounding the receiver in parabolic troughs (Anonymous, 1981). Figure 8.30 shows the transmittance of borosilicate glass treated in this manner.

Figure 8.30 Transmittance of glass with antireflection coating (Elmer and Martin, 1979).

Figure 8.31 Idealized Fresnel lens.

Several factors limit the effectiveness of Fresnel lenses. The most serious is, perhaps, the sharpness of the facets that can be produced. Any manufacturing process will produce slightly rounded facets as shown in Figure 8.32. Any ray striking the back side of a facet (e.g., ray 1) or the tip or valley of a facet (e.g., rays 2 and 3, respectively), is not properly directed to the receiver. The total fraction of the aperture area represented by these features is ineffective and reduces the ability of a Fresnel lens collector to perform efficiently.

Figure 8.32 Problems associated with Fresnel lenses

Another feature that limits the performance of a Fresnel lens (or any other lens, for that matter) is dielectric and internal reflection (see Section 8.6). As the angle of incidence of light on a facet increases, the quantity of light reflected due to dielectric reflection increases. Figure 8.29 shows the reflection of light from an air-glass interface. Note that as the angle of incidence approaches 42 degrees, total internal reflection occurs for a glass lens. Thus the edge facets of a Fresnel lens are less effective than facets closer to the center of the lens. The effect is similar for plastic lenses since they have indices of refraction similar to glass (see Appendix D).

A more serious factor limiting the use of refractive optics is the need to track the lens about two axes. This is necessary to keep the concentrated image focused on a receiver that is fixed with respect to the lens. This is true even in the case of a linear lens. Figure 8.33a shows that as the incident rays are inclined (e.g., Ray l), in the plane of curvature, the refracted rays move. Thus, as would be expected, a minimum of single-axis tracking is required with a linear Fresnel collector.

However, a somewhat more subtle effect also due to refraction, occurs if the incident ray is inclined along the linear axis of the Fresnel lens. This is illustrated in Figure 8.33b and 8.33c. The net result, over the entire lens, is the creation of a new focal line for the lens that is above the normal focal line. To maintain the refracted image focused on a receiver that is fixed with respect to the lens, therefore, one must track a Fresnel lens collector (or any lens system) to maintain the incident light normal to the lens aperture. This requires two-axis tracking.

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Torkelson, L., and D. L. Larsen (1981), "1980 Annual Report of the Coolidge Solar Irrigation Project," SAND80-2378, Sandia National Laboratories, Albuquerque.