The size of the focal spot of an ideal parabolic dish solar concentrator

Manuel J. Blanco, Ph.D.

Director, Solar Thermal Energy Department. National Renewable Energy Centre (CENER).
E-mail: mblanco@cener.com

### Introduction

Consider a simplified optical model of a parabolic dish solar concentrator. This model consist of a perfect parabolic dish mirror and a small circular flat disk receiver positioned normal to the parabolic dish optical axis, and located at a distance to the parabolic dish vertex equal to the focal length of the parabolic dish.

Consider, also, a simplified model of the direct sunlight reaching the Earth surface. For a given instant of time  and a given location, this simplified model of the direct sunlight assumes that:

• The solar energy power per unit area that reaches a surface perpendicular to the main direction of the solar radiation is constant.

• The sunshape, or angular distribution of the radiance, i.e., the power per unit solid angle and unit surface perpendicular to the main direction of the propagation of direct sunlight is a pill-box of cut-off angle  .

Now, move the parabolic dish and the flat receiver as a rigid body, i.e., without changing their relative positions with respect to each other, until the axis of the parabolic dish points towards the Sun. In this position, the parabolic dish will be fully illuminated by direct sunlight, with the exception of a small region around its vertex, which will be shadowed by the small circular flat receiver. Both sides of the flat disk receiver will be illuminated. The side oriented towards the parabolic dish will exhibit a very small and very bright spot of highly concentrated sunlight, while the other side will be uniformly illuminated by direct sunlight.

In this post we will derive formulae to determine the size of the bright spot of concentrated sunlight that the parabolic dish produces in the inner side of the small circular flat receiver, and to compute the geometric concentration ratio that a given parabolic dish may achieve. We will also derive the formula to compute the maximum geometric concentration that can be achieved with a parabolic dish concentrator and determine under what circumstances this maximum is obtained. Finally, we will compare the analytical results obtained with the results obtained using Tonatiuh.

### Size of the spot of concentrated sunlight

From Figure 1, it is clear that the following equalities hold:

(1)

(2)

 Figure 1. Geometric construction.

By properly combining these two equations to eliminate h, after some trigonometric manipulations, the following expression for the radius of the spot of concentrated sunlight is obtained:

(3)

### Geometric concentration ratio

Defining the three-dimensional geometric concentration ratio, CDish, as the ratio of the system’s input aperture area to the system’s output aperture area, the following expression holds:

(4)

The derivative of the previous expression with respect to ϕ is:

(5)

Within the interval [0,2π] there are two values of ϕ that make this derivative equal to zero. One of them is associated to the maximum value of the three dimensional geometric concentration ratio; the other with the minimum. The value of ϕ associated with the maximum three-dimensional geometric concentration ratio that can be achieved in a parabolic dish is:

(6)

Using this value of the rim angle in the expression of the geometrical concentration ratio we obtain the expression of the maximum concentration achievable using a parabolic dish.

This expression is:

(7)

In the above expression, for small values of θs the value of the first term in parenthesis is very close to 0.25. The second term of the expression is equal to the maximum theoretical concentration ratio derived from the second law of Thermodynamics. Thus, the maximum concentration achievable with a parabolic dish solar concentrator is of the order of one fourth of the one allowed by the second law of Thermodynamics.

### Rim angle and f-Number

Considering that h can be expressed in terms of the focal length of the parabolic dish, f, and is radius, Rdish, by:

(8)

Since the f-number, N, of an optical system is the diameter of the entrance pupil expressed in terns of the focal length, the following relation holds:

(9)

From Equations 1, 8, and 9 the expressions of the rim angle of the parabolic dish in terms of the f-number, and of the f-number in terms of the rim angle are obtained:

(10)

(11)

Figure 2 shows the relationship that exist between the rim angle and the f-Number. Notice that when the f-Number is lower than 0.25 the associated rim angle is negative. However, according to the geometrical construct in which the rim angle definition is based (see Figure 1) negative rim angle do not have physical meaning. Thus, there no physically realizable parabolic dishes with f-Number lower than 0.25.

 Figure 2. Rim angle of a parabolic dish as a function of f-number.

### Geometric concentration ratio as a function of f-Number

Figure 3 shows the variation of the concentration ratio with the f-Number for different values of θs expressed in terms of the average half-angle subtended by the Sun as seen from Earth, which is 16 minus of arc, approximately.

 Figure 3. Concentration as a function of f-Number for values of the half-cone angle of the incident solar radiation expressed in terms of the average half-angle subtended by the Sun disk as seen from Earth.

### Comparison of theoretical results with Tonatiuh-derived results

To see if Tonatiuh can accurately predict the functional relationship between the geometric concentration ratio of an ideal parabolic dish system and the f-Number, we can model in Tonatiuh a series of parabolic dishes with different f-Number and compute, using brute-force ray tracing, the minimum area of a circular target placed at the focal plane of the dish, perpendicular to its optical axis, that will contain all rays reflected by the parabolic dish mirror. Based on this information and on the geometric characteristics of the parabolic dished simulated in Tonatiuh the corresponding geometric concentration ratios can be computed and compare to its theoretical values.

Five parabolic dishes were modeled in Tonatiuh. All the parabolic dishes had a radius of 5 meters and all the flat circular targets had a radius of 0.2 meters. To achieve different f-Numbers the focal length of the parabolic dishes, as well as the distance from the apex of the parabolic dish to the flat circular target were modified accordingly as shown in Figure 4.

 Figure 4. 3D view of the five parabolic dishes modeled in Tonatiuh to study the variation with the f-Number of the radius of the spot of concentrating sunlight on the concentrator’s target.

To overcome a bug on the version of Tonatiuh we were using (release 1.2.1), a back surface was added to the flat circular target to intersect the photons that were hitting the target in the back coming directly from the Sun. This target back surface was modeled as a flat parabolic dish of radius 0.202, centered at the center of the target, parallel to it and separated 0.01 meters from it.

Each one of the five parabolic dishes shown in Figure 4 where modeled in Tonatiuh independently. For each one of the five parabolic dish geometries, the program was run tracing between 50 to 65 million and the corresponding Tonatiuh file with information, in local coordinates, about the photons that hit the side of the target oriented towards the parabolic dish mirror was saved. Then, the following Mathematica script was run to calculate the size of the spot of concentrating sunlight on the target:

SetDirectory[NotebookDirectory []]

ByteOrdering -> +1];

photonMap = Partition[rawData[[2 ;; Length[rawData]]], 7];

Clear[rawData]

rSpot = Max[Abs[Min[photonMap[[All, 2]]]], Abs[Max[photonMap[[All, 2]]]],

Abs[Min[photonMap[[All, 4]]]], Abs[Max[photonMap[[All, 4]]]]]

For each one of the five parabolic dishes simulated, Table 1 shows the f-Number, the parabolic dish radius, the focal length, the radius of the spot of concentrating sunlight on the target plane as predicted by theory, the radius of the same spot of concentrating sunlight as estimated by Tonatiuh, and the error of the Tonatiuh estimate in percentage. As it can be seen, for all cases analyzed the error is relatively low, not exceeding 0.35 %.

Table 1. Comparison of theoretical results and Tonatiuh estimates

 f-Number Rdish(m) f(m) Rspot(expected) Rspot(Tonatiuh) Error(%) 0.4 5 4 0.0596501 0.0594546 -0.33 6.04548 5 6.04548 0.0467596 0.0467158 -0.09 1.2 5 12 0.0636923 0.0635592 -0.21 2.0 5 20 0.097656 0.097319 -0.35 2.8 5 28 0.133581 0.133421 -0.12 3.5 5 36 0.170103 0.169819 -0.17

 “The size of the focal spot of an ideal parabolic dish solar concentrator” by Manuel J. Blanco is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported License. This work is registered at Safe Creative. Rights Info.