Question

Concentrating Optics - Assume the direct sun incidence half angle is 0.27degrees. A lens with diameter-focal length, and f-0.8m at ?=500nm is used to concentrate the light. The spot size is estimated to be 2cm 2 4. a) What is the output angle of the concentrator (incident on receiver)? (deg) What is the theoretical maximum concentration corresponding to these input/output angles? What receiver area is needed to capture all of the spot size for a tracking error of+/- 1degree? what is the effective geometric concentration for this system? b) c) and

Answer 1

ince the sun is not really a point source, solar beam incident on the reflector is represented as a cone with an angular width 0.53o(so the half-angle between the cone axis and its side is 0.27o). Being reflected at a point on the parabolic surface, the beam hits the focal plane, where it produces an image of a certain dimension, centered around the focal point. The diameter of the cylindrical receiver (D), which would intercept the entire reflected image can be theoretically calculated using aperture width (a), and rim angle (? ) as follows

D = asin0.27/ sin?

reflected beam cylindrical receiver 0.267o flat receiver 0.267o focus parabolic reflector

For the linear receiver, the width of the image (W) produced on the focal plane can be determined as follows:

W=asin(0.267) / sin?cos(?+0.267)

The equations presented here can be used to estimate the size of the reflected light image on the receiver for different shapes of parabolic reflectors. The formulas include a as a chosen aperture of the reflector (width of the trough), and (? ) as a measure of parabolic curvature. Note that these are the minimal theoretical dimensions of the reflected image that would be produced by the ideal parabolic mirror that is perfectly aligned. If there are any flaws in the mirror surface or trueness of the angle, additional spreading of the image may occur.

The above-described geometrical concepts apply to the cross section of a parabolic reflector. In reality, the reflector itself is three-dimensional shape, i.e., parabolic cylinder with a finite length (l). So, the cone-shaped ray reflected at a point on the surface of a parabolic reflector will produce an ellipse-shaped image on the focal plane. We can see that as the reflection point is moved away from the vertex towards the rim, the ellipse transforms from a circular to a more and more elongated shape (because the cone would be sectioned by the focal plane at greater and greater angle.

Knowing the angular width of the cone, the dimensions of the ellipse image can be theoretically derived and presented as a function of ? (angle of deviation from the parabola axis). Below are the equations describing the length of the minor and major axes of the ellipse.

Minor axis =2rsin(0.2670)

Major axis ={ rsin(0.2670) / cos(??0.2670) } + { rsin(0.2670)cos(?+0.2670) }

where r is the distance between the focus and reflection point (local radius) on the parabolic mirror (r=f at the vertex); ? is the angle between the parabola axis and the ray; and 0.267o is the half-angle of the ray cone width.

The superposition of these individual ellispes produced by each element of the reflector form the total image, which is not uniform, but rather has a distribution of light intensity. The focal length (which is related to the rim angle of the reflector) is responsible for image size, while the aperture is responsible for the total amount of energy concentrated by a collector. So, the total image intensity (brightness) at the receiver should be a function of a/f. The image brightness essentially reflects the energy flux concentration:

Energy flux concentration ~ a/f

The larger the aperture, the more energy is concentrated within a certain image size. The smaller the focal length, the smaller the image size within which the energy is concentrated.