4.______________________________
Collecting Solar Energy

In Chapter 2, we developed an understanding of how to determine the rate and amount of energy coming from the sun. We introduced in that section the concept of the cosine effect or cosine loss representing the difference between the amount of energy falling on a surface pointing at the sun, and a surface parallel to the surface of the earth.

In order to collect solar energy here on the earth, it is important to know the angle between the sun’s rays and a collector surface (aperture). When a collector is not pointing (or more exactly, when the collector aperture normal is not pointing) directly at the sun, some of the energy that could be collected is being lost.

In this chapter, we develop the equations to calculate the angle between a collector aperture normal and a central ray from the sun. This development is done first for fixed and then for tracking collectors. These equations are then used to provide insight into collector tracking and orientation design by predicting the integrated solar radiation energy that is incident on the collector aperture and could be collected. These concepts will be developed with the following outline:

 

4.1 Aperture-Sun Angles; The Angle of Incidence  

In Chapter 3, we defined the sun's position angles relative to earth-center coordinates and then to coordinates at an arbitrary location on the earth’s surface ( ) and (A) and a functional relationship between these angles, i.e., Equations (3.13), (3.14) and (3.15). In the design of solar energy systems, it is most important to be able to predict the angle between the sun’s rays and a vector normal (perpendicular) to the aperture or surface of the collector. This angle is called the angle of incidence . Knowing this angle is of critical importance to the solar designer, since the maximum amount of solar radiation energy that could reach a collector is reduced by the cosine of this angle.

The other angle of importance, discussed in this section is the tracking angle, . Most types of mid- and high-temperature collectors require a tracking drive system to align at least one axis and often two axes of a collector aperture normal to the sun’s central ray. The tracking angle is the amount of rotation required to do this.

In this section we derive expressions for the angle of incidence for both fixed-axis and single-axis tracking apertures. We also derive equations for the tracking angle for both single-axis and two-axis (full) tracking apertures. Zimmerman (1981) has reviewed a number of sun pointing programs and evaluated their accuracy

4.1.1 Fixed (non-tracking) Apertures

The cosine of the angle of incidence for an arbitrarily oriented surface or aperture that does not track may be described in terms of the orientation of the collector and the solar altitude and azimuth angles. An expression for this is developed by taking the scalar or dot product of a unit vector S pointing from the collector aperture toward the sun and a unit vector N normal to the collector aperture. The unit vector S was defined in Chapter 3 by Equation (3.9) in terms of direction cosines given in Equation (3.10) which are functions of the solar altitude and azimuth angles.

To define N using the same axes, we will define an aperture tilt angle and an aperture azimuth angle as pictured in Figure 4.1. The sign convention for is the same as for the solar azimuth (A); that is, north is zero and clockwise angles are positive. The direction cosines of N along the z, e, and n axes, respectively, are

                     (4.1)

 

Figure 4.1 A fixed aperture with its orientation defined by the tilt angle and the aperture azimuth angle . The aperture normal N and sun position vector S are also shown.

The cosine of the angle between a sunray and the aperture normal is the dot product of these two unit vectors:

                     (4.2)

Applying Equations (3.10) and (4.1), we have for the cosine of the angle of incidence for a fixed aperture:

                     (4.3)

It is sometimes useful to write Equation (4.3) in terms of latitude, declination, and hour angle rather than solar altitude and azimuth. Using Equation (3.14), (3.15) and (3.16) does this, and the result is

                     (4.4)

Special cases of Equation (4.3) are often of interest.

For horizontal apertures, the tilt angle is zero and Equation (4.3) becomes

                     (4.5)

For vertical apertures, Equation (4.3) becomes

                     (4.6)

and for a tilted aperture facing south, we have

                     (4.7)

4.1.2 Single-Axis Tracking Apertures

Certain types of concentrating collector are designed to operate with tracking rotation about only one axis. Here a tracking drive system rotates the collector about an axis of rotation until the sun central ray and the aperture normal are coplanar. Figure 4.2 shows how rotation of a collector aperture about a tracking axis r brings the central ray unit vector S into the plane formed by the aperture normal and the tracking axis. The angle of incidence is also shown. The tracking angle measures rotation about the tracking axis r, with when N is vertical.

 

Figure 4.2 A single-axis tracking aperture where tracking rotation is about the r axis. The sun ray vector S is kept in the plane formed by the r axis and the aperture normal N by this rotation.

To write expressions for and in terms of collector orientation and solar angles, we transform the coordinates of the central ray unit vector S from the z, e, and n coordinates used in Equation (3.10) to a new coordinate system that has the tracking axis as one of its three orthogonal axes. The other two axes are oriented such that one axis is parallel to the surface of the earth. This new coordinate system is shown in Figure 4.3, where r is the tracking axis, b is an axis that always remains parallel to the earth’s surface, and u is the third orthogonal axis. Note that the coordinates remain fixed as the aperture normal N rotates in the u-b plane.

 

 

Figure 4.3 Single axis tracking system coordinates. The collector aperture rotates about the r axis, where N is a unit vector normal to the collector aperture.

Inspection of this figure reveals that both and can be defined in terms of the direction cosines of the central ray unit vector S along the u, b, and r axes, denoted as Su,, Sb, and Sr respectively. The tracking angle is then

                     (4.8)

Remembering that S is a unit vector, the cosine of the angle of incidence is

                     (4.9)

In the sections that follow we develop equations for and , first for cases where the tracking axis is arbitrarily oriented but still parallel to the surface of the earth, and then for cases where the tracking axis is inclined relative to the surface of the earth. Both cases require a coordinate transformation.

Horizontal Tracking Axis. To describe this category of tracking schemes, we must rotate the u, b, and r coordinates by an angle from the z, e, and n coordinates that were used to describe the sun ray unit vector S in Equation (3.10). Since the tracking axis is to remain parallel to the surface of the earth, this rotation takes place about the z-axis as shown in Figure 4.4, with the u and z-axes coincident. Note that this rotation is in the negative direction based on the right-hand rule. The rotated direction cosines of S take the form

                     (4.10)

Figure 4.4 Rotation of the u, b, r coordinates from the z, e, n coordinates about the z-axis. Diagram shows view looking downward on the surface of the earth.

Solving and substituting into Equation (4.8), we have for the general case of a collector aperture tracking about a single, horizontal axis, the tracking angle

                     (4.11)

and from Equation (4.9), the angle of

                     (4.12)

A special case of this equation commonly found in practice is when the tracking axis is oriented in the north-south direction

                     (4.13)

and

                     (4.14)

When the tracking axis is oriented in the east-west direction, Equations (4.11) and (4.12) become

                     (4.15)

and

                     (4.16)

Tilted Tracking Axis. Starting with the arbitrarily oriented horizontal tracking axis r from the previous section, we now tilt this axis from the horizon by an angle as shown in Figure 4.5. Again, describing the direction cosines for the sunray unit vector in these new coordinates, we take a column matrix describing the solution to Equation (4.10) and rotate it in a positive direction through an angle . The direction cosines are found by solving

                     (4.17)

 

Figure 4.5 Rotation of the collector coordinates about the horizontal axis b.

Applying the solution of this to Equations (4.8) and (4.9), we have

                     (4.18)

and

                     (4.19)

If the tracking axis r is tilted toward the south as is often done, the equations above simplify on setting to 180 degrees.

Offset Aperture In some tracking designs, the collector aperture is offset relative to the tracking axis by an angle as shown in Figure 4.6. For this design, the tracking angle is still as computed by Equation (4.18) since the aperture normal N remains in the plane formed by the tracking axis r and the sunray vector S. However, the angle of incidence is no longer that computed by Equation (4.19). The angle of incidence for this case is simply the sum of the angle of incidence found in Equation (4.19) and the aperture offset . Appropriate signs must be used for each. The aperture offset is considered positive when offset counterclockwise (based on the right-hand rule) about the b axis. The sign for the angle found by Equation (4.19) would be the same as the sign of Sr found in Equation (4.17).

 

Figure 4.6 Tracking with the collector offset with respect to the tracking axis by an angle . The sign of the aperture offset in this sketch is positive.

Two special cases of Equation (4.18) and (4.19) that occur in practice are the case of the vertical collector and the collector whose tracking axis is tilted toward the south at the local latitude angle .

Vertical Tracking Axis For the vertical tracking axis case, the tracking axis become collinear with the zenith axis and Equations (4.18) and (4.19) simplifies to

                     (4.20)

where by convention the tracking angle is counterclockwise about the zenith.

Vertical Tracking Axis With Offset Aperture In some designs the collector aperture is offset by an angle relative to the vertical tracking axis. In this case, the tracking angle and angle of incidence become

                     (4.21)

where when the tracking axis is aligned with the plane of the aperture. Examples of this scheme are large concentrator arrays floating in a pond and arrays on circular railroad-like tracks.