5105-140

Solar Thermal Power Systems Project

Parabolic Dish Systems Development

\

DOE/JPL-1060-77

Distribution Category UC-62

ASA-CR-174109 NAS,

1.26: 1741 09

I '18.5 0 0 0 50h b

Concentrator Optical Characterization

Using Computer Mathematical

Modelling and Point Source Testing

E.W. Dennison

S.L. John

G.F. Trentelman

LIBRARY COpy

A~~~ .•

LANGLEY RESEA~CH CENTER

LIBRARY, NASA

W,M:"TON, VIt'tGINIA

September 15, 1984

Prepared for

U.S. Department of Energy

Through an Agreement with

National Aeronautics and Space Administration

by

Jet Propulsion Laboratory

California Institute of Technology

Pasadena, California

JPL Publication 84-75

5105-140

Solar Thermal Power Systems Project

Parabolic Dish Systems Development

DOE/JPL-1 060-77

Distribution Category UC-62

Concentrator Optical Characterization

Using Computer Mathematical

Modelling and Point Source Testing

E.Wo Dennison

SoL. John

Go F0 Trentelman

September 15, 1984

Prepared for

u.s. Department of Energy

Through an Agreement with

National Aeronautics and Space Administration

by

Jet Propulsion Laboratory

California Institute of Technology

Pasadena, California

JPL Publication 84-75
Prepared by the let Propulsion Laboratory, California Institute of Technology,

for the U.S. Department of Energy through an agreement with the National

Aeronautics and Space Administration.

The lPL Solar Thermal Power Systems Project is sponsored by the U.S.

Department of Energy and is part of the Solar Thermal Program to develop lowcost solar thermal and electric power plants.

This report was prepared as an account of work sponsored by an agency of the

United States Government. Neither the United States Government nor any

agency thereof, nor any of their employees, makes any warranty, express or

implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process

disclosed, or represents that its use would not infringe privately owned rights.

Reference herein to any specific commercial product, process, or service by trade

name, trademark, manufacturer, or otherwise, does not necessarily constitute or

imply its endorsement, recommendation, or favoring by the United States

Government or any agency thereof. The views and opinions of authors

expressed herein do not necessarily state or reflect those of the United States

Government or any agency thereof.
ABSTRACT

The optical characteristics of a paraboloidal solar concentrator are

analyzed using the intercept factor curve (a format for image data) to

describe the results of a mathematical model and to represent reduced data

from experimental testing. This procedure makes it possible not only to test

an assembled concentrator, but also to evaluate single optical panels or to

conduct non-solar tests of an assembled concentrator.

The use of three-dimensional ray tracing computer programs to calculate

the mathematical model is described. These ray tracing programs can include

any type of optical configuration from simple paraboloids to arrays of

spherical facets and can be adapted to microcomputers or larger computers,

which can graphically display real-time comparison of calculated and measured

data.

The technique described herein has demonstrated that

(1) It is possible to create a model for predicting concentrator

optical performance from data obtained at various points of the

experimental testing process, and that

(2) The intercept factor curve is a powerful format for optical

performance representation whether it is derived from the

mathematical model or from experimental data.

iii
ACKNOWLEDGMENTS

The mathematical modelling described in this report was developed by

coauthors George F. Trentelman and Susan L. John and was funded partly through

a U.S. Department of Energy (DOE) summer research fellowship program and

support from the Northern Michigan University, Marquette, Michigan. George

(Fred) Trentelman is a member of the Physics Department at the University;

Susan John is currently with GTE-Sylvania in Mountain View, California.

The work described herein was conducted by the Jet Propulsion

Laboratory, California Institute of Technology, for the U.S. Department of

Energy through an agreement with the National Aeronautics and Space

Administration (NASA Task RE-152, Amendment 327; DOE/ALO/NASA Interagency

Agreement No. DE-AM04-80AL13137).

1V
D

f

T

X,Y, Z

Vectors

I

N

R

...............

I,N,R

x,y,z

Subscripts

m

p

s

x,y,z

NOMENCLATURE

diameter

focal length

distance from the optical ax~s

point coordinates

incident ray from a reflection point to a single point source

surface normal particular to the paraboloidal surface

reflected ray terminating at the receiving plane

unit vectors

unit vectors parallel to the coordinate axes

mirror coordinates relative to the vertex

coordinate of the reflected ray

coordinate of the point source relative to the vertex

components of a unit vector (direction cosine)

v

CONTENTS

I. INTRODUCTION • . • • • • • • • • • • • • • • • • • • • • • • • • • 1-1

I I • COMPUTER MODEL • • • • • • • • • • • • • • • • • • • • • • . • • • 2-1

III. ABERRATIONAL EFFECTS AND IMPERFECT CONCENTRATORS • • • • • • • • • 3-1

IV. RESULTS AND DISCUSSION • • • • • •• 4-1

A.

B.

C.

D.

FULL MIRROR COMPARISONS.

POINT SOURCE TESTING OF INCOMPLETE MIRRORS •

INTERCEPT FACTOR COMPARISONS WITH ONE PARAMETER.

SOLAR EXTRAPOLATION. • • • • • • • • • • • • • •

. • • 4-1

4-3

. • • • 4-5

• 4-6

V. CONCLUSIONS •• • • • • • • • • • • • • • • • • • • • • • • • • • 5-1

REFERENCES •••••••••••••••••••••••••••••• R-I

APPENDIX: DATA TABLES FOR FIGURES IN TEXT••••••••••••••• A-I

Figures

2-1. Vector Analysis for Three-Dimensional Ray Tracing •••••• 2-2

2-2. Computer-Generated Graphic Display of Paraboloidal

Concentrator and Reflected Rays ••••••••••••••• 2-5

3-1. Solar Source Intercept Factors for Paraboloidal

Concentrator with No Surface Error and for a

Perfect Optical System • • • • • • • • • • • • • • • • • • • 3-2

3-2. Randomly Perturbed Surface Normal on Concentrator

Surface. . . . . . . . . . . . . . . . . . . 3-4

4-1. Computed Model and Experimental Intercept Factors for a

Paraboloidal Concentrator • • •• • • • • • • • • 4-2

4-2. Model and Experimental Intercept Factors for a Partially

Complete Concentrator ••••••••• ••••••• • 4-4

vii
Figures

4-3.

4-4.

Projected Solar Source Intecept Factors for a

Paraboloidal Concentrator. •• ••.• • • •

Solar Source Intercept Factors for Concentrators

with Hypothetical Surface Errors • • • • • • • •

viii

4-7

4-8
SECTION I

INTRODUCTION

This report presents an analysis of solar energy concentrators as

optical elements. A substantial amount of experimental and mathematical

analyses have already been performed on solar concentrators, but these

analyses have generally considered the concentrator as a subsystem component

of an overall energy production system. For this reason, characteristics of

the concentrator optics have not been systematically analyzed in a manner that

is independent of the application.

Any discussion of the operation of a solar concentrator requires a

definition of "performance" and a quantitative measure of performance. The

basic purpose of a solar concentrator is to image (or focus) solar energy from

a large entrance aperture into a small receiver aperture where the radiant

energy is converted into thermal energy. The thermal energy is in turn

converted to electrical energy by way of a heat engine. The operating

efficiency of the thermal-to-electrical energy conversion unit is directly

related to the receiver temperature. The temperature of the receiver

increases with the mean energy flux density passing into the receiver

aperture. The mean energy flux density increases with the image forming

quality of the concentrator, i.e., the smaller the image (with respect to the

size of the concentrator entrance aperture), the higher the flux density.

Herein, the optical performance of a solar concentrator is defined as "the

degree to which the concentrator can form a high flux density solar image."

Because this report covers only optical imaging, surface reflectance and

entrance aperture shading are not included in this definition of performance

although these factors do have a significant effect on the overall operating

performance of a solar concentrator.

The term "concentration ratio," or the ratio of the concentrator aperture

area to the receiver aperture area, has often been used to define the optical

performance of a solar concentrator. This definition has the disadvantage of

being both incomplete and subject to misinterpretation because there is no

precise definition of the fraction of the focal plane image that passes through

the receiver aperture. For example, a 13% change in the receiver aperture

radius (corresponding to a 30% change in the concentration ratio) would give a

1% change in the total input energy.

There are many ways to describe the image forming quality of an optical

system. For solar concentrators, a very practical and generally accepted

description is the focal plane "intercept factor" curve, which is the same as

the standard optical term "encircled energy." These data can be expressed as

a table of measured or calculated points that are presented in graphical

form. The curve indicates the fract~on of the total image energy passing

through an aperture with a radius that is specified by a linear distance from

the optical axis in the focal plane or by an angle with the vertex at the

center of the concentrator. The latter parameter is dimensionless; as a

consequence, the results can be applied to any concentrator regardless of

linear dimensions.

1-1
A practical way to measure the image forming characteristics of a

reflecting panel or an assembled concentrator is use of a fixed point source

of light. A high intensity spotlight at a distance of 1000 ft is ideal in a

moderately dark environment, but good measurements can also be made with a

high intensity light source at a distance of several focal lengths from the

concentrator vertex. The size of the image from an extended source is

determined by both the size of the source and the optical imperfections. In

this case, it is generally difficult, if not impossible, to make a

quantitative determination of the optical imperfections. The final

confirmation of concentrator performance must be made with the sun as a

source, but for preliminary testing the sun has the disadvantage of both high

levels of radiant energy and constant motion.

The intercept factor curve can be directly measured with a photometer

and a series of circular apertures. For small panels, a Fresnel lens can be

used to image the energy passing through the aperture onto the photometer.

For large panels, or an assembled concentrator, an imaging photometer can be

used to measure the energy falling outside the aperture of a white annular

target. With a series of water-cooled apertures and a cold-water cavity

calorimeter, the intercept factor curve can be measured when the concentrator

is pointing at the sun. The intercept factor data can also be obtained by the

use of a photometer or radiometer that scans the image. Each aperture

measurement is divided by the total image measurement to obtain the intercept

factor for that aperture. For ray tracing calculations, each point is

determined by dividing the number of rays that fall within a specified circle

by the total number of rays falling on the concentrator.

An intercept factor curve can describe the image from a point source at

a finite or infinite distance, or it can describe the image from the sun. It

is also an ideal format for representing both the measured and calculated

image data. It can be used to directly determine the collection efficiency of

an existing thermal receiver or determine the relationship between receiver

temperature and the optimum receiver aperture radius.

Prototype concentrator systems have been built on the basis of overall

system analyses and subsequently tested under operating conditions. Difficulties arise when total performance does not meet expectations and there is

no direct way to systematically isolate the problems. These problems may

result from any or all of the concentrator components: mirror geometry and

surface properties, mirror mounting and tracking instabilities, receiver

configuration and location, etc.

Because the optical performance of the concentrator is paramount to the

success of the energy conversion, a detailed systematic study was made of the

optics of a prototype concentrator. l This study was based on an interactive

lThe paraboloidal concentrator used in this analysis for experimental testing

is the l2-m-diameter Parabolic Dish Concentrator No.1 (PDC-l), which is

described in detail in a companion report entitled Development and Testing of

Parabolic Dish Concentrator No.1 by E.W. Dennison and T.O. Thostesen, Jet

Propulsion Laboratory, 5105-143, November 1984.

1-2
development of a mathematical model and experimental testing procedures.

The experimental procedures used a point source of light at a finite distance

and were performed under static mirror conditions. Complete and incomplete

mirrors were incorporated into the testing to determine concentrator characteristics early in the manufacturing process and to evaluate quality control

procedures.

Previous analyses (Refs. 1 through 4), for example, have incorporated

all of the system components mentioned into a single comprehensive framework,

hence making it difficult to isolate specific effects. Moreover, these

analyses are specific to paraboloidal concentrators. The mathematical

approach used by these authors emphasizes geometric considerations and

utilizes cone optics (an analysis that incorporates the solar disk as an

intrinsic part of the computations) as a procedure. This approach is burdened

with the problems of transcendental equations that are common to many

geometric optical problems. The references cited tend to minimize the

requirement for extensive use of computers and their associated expense, and

provide procedures for circumventing direct numerical analyses.

The procedures presented here also rely on computed results. With

current microcomputing technology, an exact and general formulation can be

utilized that is neither disproportionately time-consuming nor expensive and

can be made portable enough for use at test site locations. In addition, the

procedures are not limited to paraboloidal configurations or single mirror

systems. Facetted mirrors, spherical mirrors, and mirrors of special design

may be modelled and analyzed with equal facility.

1-3

SECTION II

COMPUTER MODEL

The computer model developed to simulate the concentrator is based on

three-dimensional ray tracing. The law of reflection is formulated as a set

of vector equations involving the incident rays, reflected rays, and the normal

to the concentrator surface at the reflection points. The reflected ray direction is computed for a specified source location, a particular optical surface,

and a specified reflection point on the concentrator. Iterating the computation for an array of reflection points and normal vectors suitably describing

the surface produces an array of reflected rays.

The points of intersection of these rays with the receiving surface

forms a model of the optical image. The coordinates of the intersection

points are calculated explicitly, and therefore may be analyzed in the same

way that experimental data are analyzed. The number of reflected rays per

unit area for prescribed small areas of the image represents the intensity in

that area. For intercept factor calculations, the image plane is divided into

concentric apertures centered about the optical axis.

In the interest of having a methodical approach to the analysis, the

modelling proceeded in a sequence of three distinct steps. The first modelled

a perfect geometric concentrator 'free from surface irregularity and illuminated

by a point source. The second introduced variable degrees of surface irregularity into the system, and the third integrated the results of the first two

and used as a source an array of points that simulate the finite solar disk.

The vector nature of the three-dimensional reflection problem is

illustrated in Figure 2-1 where two reflection points are shown. The rays

11 and 12 are vectors~repres~nting incident rays from a single point

source. The vectors Nl and N2 are the surfac~ normals particular to the

paraboloidal surface with a focal length f. Rl and R2 are the resulting

reflected rays terminating at the receiving plane.

As indicated in the figure, the law of reflection requires that the

angle of incidence and reflection be equal in each case:

~2i = ~2r

(1)

(2)

and that the incident, normal, and reflected rays be coplanar. Algorithms

suitable for iterative programming can be developed by writing the law of

reflection as a combination of vector and scalar products of the vectors

involved.

2-1
REFERENCE

PLANE

z

APERTURE

RADIUS

MIRROR EQUATION:

SURFACE NORMAL:

""Y ""

FOCAL

PLANE

A A

y+Z

LAW OF REFLECTION: A R A A A A A '" R =-= U •N) N - N x( I x N)

IRI

Figure 2-1. Vector Analysis for Three-Dimensional Ray Tracing

2-2
The first step is determining an analytic form for the surface normals.

This is accomplished by writing the functional form of the reflecting surface

as an equipotential surface and taking the negative gradient of the equation.

For the paraboloidal surface with the surface equation usually written as:

(3)

(where the subscript m denotes mirror coordinates relative to the vertex and f

1S the focal length), the equivalent equipotential is:

o (4)

Application of the negative gradien~ operator

-v (5)

to g(Xm,Ym,Zm) yields the surface normal Nin terms of the coordinate

location Xm,Ym,Zm of the reflection point as:

N (6)

For a paraboloid this becomes explicitly:

(7)

The latter equation specifies the components of the surface normal vector for

any reflection point.

The incident ray vector from any reflection point to the source becomes

I (8)

where Xs ' Ys ' and Zs are the coordinates of the point source relative to

the vertex.

2-3
With this formulation, both the incident ray and the surface normal are

numerically specified. These two vectors are reduced to unit vectors by

dividing them by their respective magnitudes:

'"N N/ INI

'"1=1/111.

(9)

(10)

For a point source at infinity, the incident unit vector can also be described

as:

'"I = sino. cos S x + sin 0. sin Sy + cos 0. Z ( 11)

where 0. is the radial angle from the optical axis and S ~s the azimuthal angle

from the X axis.

The law of reflection may be written in terms of the scalar (DOT) product

(I . N) and the vector (CROSS) product (I x N) to give the associated

reflected ray unit vector as:

'"R (I • N) N Nx (I x N) •

This unit vector has three components numerically specified in terms of the x,

y, and z directions. These components specify the direction of the reflected

ray relative to the coordinate axes.

The vector equation represents three linearly independent component

equations that may be written as a matrix multiplication:

.....

R

x

R

y

R

z

(N2 .....2 N2) '" '" ..... '" N - '" 2 N N 2 N N I

x Y z x Y x z x

'" '" (N 2 "'2 N2) '" '" '" 2 N N N - 2 N N I ( 13) x Y Y x z Y z Y '" ..... 2 N N (N 2 ..... 2 N2) '" 2 N N N I

x z Y z z x y z

As a result, the reflection process is viewed mathematically as a transform of

the incident ray into the reflected ray with the elements of the

transformation depending only on the surface normal components.

The matrix formulation adapts well to iterative computation, and the use

of component multiplication eliminates the need for overt use of trigonometric

functions. A computer graphic display of the model analogous to Figure 2-1 is

shown in Figure 2-2. In the latter figure, the number of zones and the number

of reflection points circumscribing each zone are variable.

2-4
Figure 2-2. Computer-Generated Graphic Display of Paraboloidal

Concentrator and Reflected Rays

2-5
The final step in determining the intercept factor is to compute the

coordinates of the reflected ray (designated by the subscript p) as it

intercepts the target or focal plane. These coordinates can be found from the

general equation:

x - X Y - Y Z - Z

P m P m P m . R R R

x Y z

In practice this becomes two equations:

(14)

and

X

P

(z

p

R

x - Z )*- +

m

Rz

X

m

(15)

R

X = (Z - z )*-l + y

ppm R m

z

The coordinates Xp ' Yp ' and Zp therefore describe an image point whose

distance from the opt~cal axis is:

T= ~l\.p ~,... Ip

(16)

( 17)

and the ray will be inside the aperture if T is less than the radius of the

aperture. The intercept factor is the fraction of the total number of rays

striking the concentrator that fall inside the specified focal plane aperture.

2-6
SECTION III

ABERRATIONAL EFFECTS AND IMPERFECT CONCENTRATORS

Because the optical source of primary interest for solar concentrators

is the sun, a practical optical model must include sources of light that are

at least one solar radius from the optical axis. The mean half angle of the

solar disk is 4.65 mrad, but for convenience a value of 5 mrad has been used

for the calculations in this report. For the purpose of the ray tracing

model, the points on the sun can be either the angular distance from the

optical axis or the X,Y,Z coordinates with respect to the concentrator

vertex. (The Z distance of the sun is approximately 1.5 x lOll m.)

For a perfect paraboloid, off-axis source points are not formed into

point images. These aberrations increase with the angular distance of the

point source from the optical axis, and the composite effect of these

aberrations is significant. To demonstrate this, an intercept factor curve

was calculated for a paraboloidal concentrator with a focal length/diameter

(f/D) of 0.5 and a circular source of 5 mrad radius. The results are shown 1n

Figure 3-1, and the data are given in Table 3-1. 2 For comparison, the

intercept factor curve for a perfect optical system (no aberrations) with the

same source is also included. The intercept factor is shown as a function of

the aperture radius divided by the focal length. This dimensionless parameter

is the tangent of the aperture radius angle as viewed from the vertex.

Because the angles are small and to clarify the interpretation of the curves,

the angle is given in milliradians (mrad).

The intercept factor curve for a perfect paraboloid is the upper limit

of optical performance. Physical imperfections that extend the intercept

factor must be accounted for in any acceptable model. The model described

here assumes the uncertainties to be of two categories: systematic and random.

The systematic uncertainties include errors in the geometric form, focal

length, and tracking. These errors can be modelled by specifying a different

geometry for the mirror and by offsetting the solar source from the optical

axis. However, for the PDC-l concentrator, these errors were detected and

corrected by other direct methods and were eliminated as a major concern in

the final calculations.

Of more relevance to the model is a technique for simulating the random

surface irregularities that are an inherent part of any concentrator. Because

the sources of these irregularities are diverse and usually cannot be

specifically isolated, a computational scheme based on random surface errors

was developed and incorporated into the model. The test of such a modelling

technique is comparison with experimental data.

The model is based on the concept that the surface normals are randomly

perturbed, and these perturbations cause errors in the concentrator image.

2Data tables are given in the Appendix.

3-1
2

0.9

0.8

0.7 -

0 0.6 IU

«

u..

Ia.. 0.5 w

U -

w

IZ 0.4

2 - PERFECT OPTICAL SYSTEM

SOURCE RADIUS 5 mrad

0.3

0.2

o. 1

--f- t

1 - PERFECT PARABOLOIDAL MIRROR_ /D - f 0.50, SIGMA 0.00 mrad -

SOURCE RAD IUS 5 mrad ::

=

--

0.0

o 2 4 6 8 10 12 14

- f16

NORMALIZED APERTURE RADIUS, mrad

Figure 3-1. Solar Source Intercept Factors for Paraboloidal

Concentrator with No Surface Error and for a

Perfect Optical System

3-2
Because the concentrator has a large number of small independent surface

irregularities, the model must be calculated for a large number of reflection

points. The perturbations introduced into the model surface normals were

randomly selected but modulated by a Gaussian distribution.

The process is illustrated in Figure 3-2. In the figure, N i~ the

surface normal prescribed by the geometry of the concentrator, and N' is the ..... ..... perturbed normal ...... Nt is inclined relative to N by an angle ¢ and then allowed

to precess about N with a precession angle o. The inclination angle ¢ is

chosen at random from a Gaussian distribution according to the equation:

RDMI (14)

RDMI is a random number in the range O<RDMI~I. The standard deviation,

sigma, is determined empirically by comparison with experimental data and

becomes the surface quality factor of the concentrator.

For computational purposes, the equation is inverted to:

a ,J2 [In( l/a J27i) - In(RDMl)] 1/2 •

The precession angle is selected from a uniform distribution with 0 =

(RDM2) • (2rr) and O«RDM2)~I.

(15)

"The random numbers RDMI and RDM2 are obtained by separate selections

from a random number generator to avoid any coupling of the calculations.

Establishing the angles ¢ and 0 per~its the components of the perturbed normal

to be written as those of a vector Nt in a coordinate system that has been

rotated relative to the vertex axes. A rotation transformation is required to

establish these components in the X, y, z coordinate system.

3-3
Figure 3-2. Randomly Perturbed Surface Normal on Concentrator Surface

3-4
SECTION IV

RESULTS AND DISCUSSION

As a test of the computer model and of the general procedure,

comparisons were made between the model predictions and experimental

measurements for a paraboloidal concentrator. This comparison was designed

for three specific objectives: (1) to determine if the modelling procedure

had the credibility and sensitivity required for a practical analytic device,

(2) to determine a surface quality factor representative of a specific

concentrator, and (3) to present an accurate extrapolation from the point

source test data to the solar performance.

The concentrator under study had nominal values of 6.00 m for the focal

length and 0.50 for the focal length/diameter ratio (f/D), and for testing

purposes was illuminated with a point source. Experimentally measured

intercept factors were compared with those predicted by the computer model.

The experimental procedures developed for using point sources at finite

distances permitted direct methodical comparison of intercept factors without

the necessity of relying on solar models in the analysis. The sun, as a

finite source, enlarges the image and obscures the image defects resulting

from concentrator imperfections. The use of point sources permits an accurate

analysis of the concentrator and allows the experiment to be well defined.

The effect of the finite solar source can be calculated after the optical

characteristics of the concentrator have been determined.

A. FULL MIRROR COMPARISONS

Experimental data for the complete concentrator were gathered with the

point source (a high quality spotlight) on the optical axis 900 m (nominally

150 focal lengths) from the mirror vertex. The image plane was established as

the location of minimum image size and was located 6.03 m from the vertex,

very near the nominal focal plane. The intercept factors were determined by

measuring the amount of light falling on a series of white apertures. The

measurements were made with an imaging photometer mounted at the concentrator

vertex.

The subsequent analysis permitted relatively direct comparison with the

model results. In the ray trace modelling procedures, the intensity is

interpreted as the percentage of rays falling within a circular aperture ~n

the image plane.

The computer model was used in an interactive mode to yield the

intercept factor curve shown in Figure 4-1 (data shown in Table 4-1), and

established a surface error sigma value of 16.25 mrad. Simultaneously, the

real focal length was confirmed to be 6.00 m. The RMS (root mean square)

deviation of the curve from the data is approximately 1%. It should be noted,

however, that the model curve is not an attempt to fit the data parametrically,

4-1
0.9

0.8

0.7

PDC-I

SURFACE ERROR DETERMINATION

20 24 28 32 36 40

-

=

=

=

=

:

==-

=

=:

SURFACE SIGMA: 16.25 mrad =:::

o INDICATES EXPERIMENTAL DATA FROM POINT SOURCE=

-r-r-rr--j j --r"-T -,-

16

2 - COMPARISON--

SOURCE DISTANCE: 900 m

SURFACE SIGMA: 0.00 mrad

3 - COMPARISON -- SOURCE DISTANCE: 1.5 x 101lm

EXPERIMENTAL PARAMETERS

SOURCE DISTANCE: 900 m

IMAGE PLANE: 6.03 m

MIRROR DIA: 12.00 m

- FOCAL LENGTH: 6.0 m (NOMINAL)

MODEL PARAMETERS

POINT SOURCE

IMAGE PLANE: 6.03 m FOR CURVES 1 & 2

(6.00 FOR CURVE 3)

MIRROR DIA: 12.00 m

FOCAL LENGTH: 6.00 m

CURVES

-- 1 -DETERMINES SURFACE ERROR

SOURCE DISTANCE: 900 m

SURFACE SIGMA: 16.25 mrad

12

L

3

8

2

4

o. 1

0.3

0.5

0.2

0.0

o

0.4

o 0.6

Iu

«

u..

le..

u.J

U

u.J

IZ

NORMALIZED APERTURE RADIUS, mrad

Figure 4-1. Computed Model and Experimental Intercept Factors for a

Paraboloidal Concentrator. (Image derived from a point

source 900 m from the vertex.)

4-2
but to establish a surface error number that represents the experimental

intercept factors. The criteria for the best representation was based on

subjective judgment rather than a numeric RMS deviation.

At the beginning of the testing program, there was some concern about

the propriety of using a point source at a finite distance because a

paraboloidal concentrator gives a well defined image only for an infinitely

distant point source. In practice, this did not present a problem because an

accurate model could simulate any test configuration.

The comparisons between numerous model intercept factors and those for

the experiment did demonstrate that the surface error was the only unknown

parameter. For that reason, the nominal values for the focal length, mirror

diameter, image plane distance, and source distance could be used directly in

the final model.

The development of the various computer models necessary to establish

the comparative intercept factors provided valuable information concerning the

sensitivity of the model to the physical parameters. In particular, the

intercept factor is relatively insensitive to the f/D ratio.

Precise measurement of the experimental image plane position and source

distance are crucial to the comparisons, as is a reasonably accurate nominal

value for the focal length. The most sensitive factor in the modelling is the

relative focal point/image plane distance because of the low f/D ratio of the

concentrator. The size of this image changes rapidly with small displacements

of the intercept plane away from the focal point.

Variation of other parameters in addition to the surface error to obtain

reasonable model-experiment agreement can be done, but involves looking for

subtle changes in the shape of the intercept factor curve as well as in its

overall magnitude. For this reason, such comparison depends heavily on the

precision and quantity of experimental data. The excellent agreement of the

computed data with the experimental data indicated that changes to the nominal

parameters were unnecessary.

B. POINT SOURCE TESTING OF INCOMPLETE MIRRORS

The feasibility of providing mirror quality information during the

manufacturing of the concentrator panels was investigated by performing point

source testing with only four of the twelve panels. The testing procedures

were similar to those used for the complete concentrator, i.e., a point source

at 900 m was used to illuminate the mirror, and the image was photometrically

measured to determine the intercept factor.

The experimental and computer model data are shown in Figure 4-2 and

Table 4-2. The best comparison for the four-panel system data yielded a

surface quality sigma value of 14.0 mrad as the aggregate for the four

panels. The computer model treated the data as though it were for a complete

concentrator (permissible because of the rotational symmetry) in order to

determine which of the physical parameters required adjustment. Combinations

4-3
,.. '"'" 1.0 tel:: -- -- - :=t

±-

0.9 t+- :t0.8

--'

c::: OJ 0

I- + U -L « 0.6 + + u.

I- - a... r: w L

U 0.5 0:::

w .p- II Z FOUR-PANEL TEST

.p- 0.4 -+ '---+ f 6.00 m 0 EXP. DATA

0.3 ,+ :r 1---' , f/D 0.50 -MODEL

-+ _i- -t + SIGMA /4.0 mrad t- ~. --,

-, t- 0.2 -+- j-

,.

0.0

o

~c::L ++ r- '.- -=R=+

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36

NORMALIZED APERTURE RADIUS, mrad

Figure 4-2. Model and Experimental Intercept Factors for a Partially

Complete Concentrator. (Image derived from a point

source 900 m from mirror vertex.)
of focal length, image plane, diameter, and surface error were modelled. The

study provided insight into the sensitivity of the modelling procedures to

parametric changes, as well as quantitative data concerning the concentrator.

Two different approaches to the parametric search yielded independent,

reasonable representations of the experimental intercept factors. The first

made the assumption that the effective focal length was 6.0 m, the focal

length/diameter ratio was 0.50, and the point source at 900 m was close enough

to infinity to assume imaging at 6.0 m. No acceptable intercept factor was

found for these parameters. An acceptable intercept factor agreement was

obtained by changing the f/D ratio to 0.54 and by using a surface error sigma

of 12.7 mrad. However, this would imply a mirror diameter of 11.1 m instead

of the actual 12 m. Moreover, modelling a point source at infinity with these

parameters did not give an intercept factor similar to that of the 900-m data.

The second model was constructed to conform as closely as possible to

the actual experimental measurements. The focal length and f/D were

reestablished at 6.00 m and 0.50, respectively, but the image plane was set at

6.03 m as was actually measured instead of at the previously assumed 6.0 m.

These parameters yielded the best comparison and the sigma of 14.0 mrad.

These parameters, when modelled for a point source at infinity, give an

intercept factor showing the same relationship to the 900-m data as the one

shown in Figure 4-1 for the complete concentrator. This is the relationship

expected of correct modelling.

These studies of the four-panel assembly reinforce the importance of

accurately determining the image plane/focal plane distance for point source

testing and of accurately measuring the test configuration. This work also

provides the basis for further investigation of techniques for analyzing

concentrator optical elements early in their construction.

The difference between the surface quality of the four-panel system and

that of the full mirror is not large, and such an increase in surface

irregularity is not unexpected as additional panels are installed. Because

the computer model can be easily tailored to specific parameters and the

experimental measurements can be made within reasonable physical constraints,

this modelling method is a promising mode of quality control. Further study

of this application is clearly merited.

C. INTERCEPT FACTOR COMPARISONS WITH ONE PARAMETER

The ability to obtain good model-experiment intercept factor comparisons

using only one parameter, sigma, as the adjustable parameter should not be

surprising because the computation of the surface error actually incorporates

two variables: (1) the tilt angle of the perturbed normal relative to the

ideal value and (2) the precession angle of the tilted normal. Both

parameters are called independently from a random number generator. The first

assumes a Gaussian modulation of the random numbers and has a standard

deviation available as an adjustable parameter. The second assumes a uniform

distribution of the precession angles. As a result, the second parameter has

4-5
no standard deviation and does not appear explicitly. The inclusion of the

two independent parameters does introduce, however, both azimuthal and radial

effects of surface irregularities and, thus, gives a reasonable representation

of the actual optical surface.

D. SOLAR EXTRAPOLATION

The model can project the concentrator solar performance as a direct

extrapolation of the test results. Assuming constant geometric factors for

the mirror and the surface quality factor to be sigma = 16.25 mrad, intercept

factors for a solar source were modelled. For this specific extrapolation,

the simplest solar model, i.e., that of a uniformly illuminated disk, was used.

The solar model was given rotational symmetry about the optic axis, an

angular radius of 5.0 mrad, and a source distance of 1.5 x lOll m. This was

accomplished by assuming that the source rays are on a line extending from the

optical axis outward to 5.0 mrad in 1.0 mrad increments. Each point was the

source of 3565 rays distributed uniformly over the surface of the concentrator

for a total of 21,390 rays. The rays are intercepted at the focal plane

(6.00 m). The resulting solar source intercept factor is shown in Figure

4-3. The corresponding data are given in Table 4-3.

Because the surface error (16.25 mrad) is significantly greater than the

solar radius of 5 mrad and therefore dominates the image degradation, the

intercept factor of the sun is not greatly different from that of a single

point source on axis.

It should be noted with caution that if the solar radius is attributed a

sigma value of as = 5 mrad and combined with the surface error am = 16.25 mrad

to give a total of aT = (a~ + a~)1/2 = 17.0 mrad, a point source illuminating

a mirror with a surface error of am = 17.0 mrad gives an intercept factor essentially identical to that of the solar source with am = 16.25 mrad. Although this

is not unexpected when am» as' caution must be used in its general application.

The same procedure using am = 14.7 mrad and as = 5 mrad does not produce acceptable results even though the intercept factors are similar. Conversely, when

am«as , the intercept factor is primarily determined by the flD ratio of the concentrator and the angular size of the sun and is not dependent on the surface

errors.

For comparison, Figure 4-4 shows the intercept factors for a 5-mrad,

uniformly radiating solar disk and hypothetical mirror surfaces with sigma

values of 11, 13, 15, and 17 mrad.

In the intercept region of 85 to 95%, the curves indicate that an

uncertainty of 2 mrad in the initial determination of the surface error could

lead to errors as large as 10% in the projected solar performance. However,

the quality of the experimental data available indicates that the surface

error can be ascertained to within ~0.25 mrad. The intercept factor curves

show that the latter value would translate to a projected intercept factor

uncertainty of 2% or less. Improved data acquisition methods would probably

reduce this uncertainty.

4-6
0.9

0.8

t--

3

",

0.4

CURVES

1 - UNIFORM SOLAR DISK

RADIUS: 5.00 mrad

. I- MIRROR SIGMA: 16.25 mrad - lI- 2 _ POINT SOURCE (ON AXIS)

MIRROR SIGMA: 16.25 mrad

. 3 - UNIFORM SOLAR DISK

- RADIUS: 5.00 mrad

MIRROR SIGMA: 0.00 mrad

0.7

0.6

o

Iu

«, 0.5 u..

I0-

W

U

w

IZ

0.3

0.2 I

o. 1

.+-

. - f

D

--. MODELPDC-l PROJECTIONS ttt~-. ~=f~ MODEL PARAMETERS

6.00 m,IMAGE PLANE 6.00 m

11 12.00 m,SOURCE DIST: 1.5 x 10 m

.+-+-+---i::ttti:f-'=t-··

+-

0.0

o 4 8 12 16 20 24 28 32 36

NORMALIZED APERTURE RADIUS, mrad

Figure 4-3. Projected Solar Source Intercept Factors for a

Paraboloidal Concentrator

4-7
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36

1 ..1..+- -+-

-

-

::j: 2- ,

-r t , +

:T+ +- L

f - - ,-c. .

- c 3 -I- -- - t --,

. 4 "" --- .--4 ,

-- f

---L

-i-

- f 6.00 m SOURCE RADIUS 5.0 mrad

D 12.00 m SOURCE DISTANCE 1.5 x 1011 m

IMAGE PLANE 6.00 m -,

t L

-!---

f , , , : CURVES

1 - a- "

mrad 3 -IJ 15 mrad

2 -IJ 13 mrad 4 -IJ 17 mrad

+-,- - ---' - --+-+- .j:::+ -+--+- -+-

- ,

" -t- ,--- _-,-_L L+ _---L. ' , '_L.L=F= -+{i, , -+- - q:::p::;:

0.2

0.6

o. 1

0.3

0.4

0.8

0.5

0.0

o

1.0

0.9

e::: 0.7 o

IU

<{

u..

I0-

W

U

e:::

w

IZ

I

00

NORMALIZED APERTURE RADIUS, mrad

Figure 4-4. Solar Source Intercept Factors for Concentrators

with Hypothetical Surface Errors
It should be noted that the measurements were made under static mirror

conditions. During actual operation with the concentrator tracking the sun,

uncertainties in tracking position and the changing gravity load could affect

the total performance. Measurements of this type can be made with cold-water

calorimeters, but are generally time-consuming and limited to a few aperture

values.

4-9

SECTION V

CONCLUSIONS

The use of basic paraboloidal geometry with the flD ratio and a single

surface or slope error as the fundamental parameters to describe a solar

concentrator is cornmon to both this work and that of previous authors. It 1S

reassuring to note that these different approaches give similar results.

The problem with previous solar concentrator models is that they cannot

be used for simple testing of single optical panels or non-solar tests of

assembled concentrators. This problem has been resolved by the use of the

intercept factor curve with a point source of light at any distance from the

concentrator or with the sun as a source. The intercept factor curve can be

used to describe the results of a mathematical model of a concentrator or to

represent reduced data from experimental image measurements. Measurements can

be made by scanning a photodetector or flux mapper over the image or by the

use of an integrating photometer or calorimeter to measure the relative

intensity of the image falling inside a series of circular apertures. With

the sun as a source, these intercept factor curves can be used to evaluate the

performance of power conversion thermal receivers.

The use of ray tracing computer programs as described herein is both

powerful and practical. These programs can include any type of optical

configuration from simple paraboloids to arrays of spherical facets. These

programs can be adapted to microcomputers at an acceptable cost in operating

time. The sophisticated graphics displays now available on many

microcomputing systems can be used for real-time interactive comparison of

calculated and measured data.

When the optical testing of the JPL PDC-l solar concentrator began,

there was a clear need for a comprehensive method to handle both the

theoretical and experimental aspects of imaging characteristics of solar

concentrators. While the work described in this report is not definitive, it

does demonstrate that the use of ray tracing programs and intercept factor

curves can provide a practical way to fulfill this need.

5-1

REFERENCES

1. Wen, L., Huang, L., Poon, P., and Carley, W., "Comparative Study of

Solar Optics of Paraboloidal Concentrators," ASME, 79-WA/Sol-8, 1979.

2. Wen, L., "Effects of Optical Surface Properties on High Temperature

Solar Thermal Energy Conversion," Journal of Energy, Vol. 3, No.2, 1979.

3. O'Neill, M.J., and Hudson, S.L., "Optical Analysis of Paraboloidal Solar

Concentrators," Proceedings of the 1978 SPIE Annual Meeting, Vol. 2,

1978, pp. 855-862.

4. Dendt, P., and Rabl, A., "Optical Analysis of Point Focus Paraboloidal

Radiation Concentrators," Applied Optics, Vol. 20, 1981, pp. 674-683.

R-l

Model Computed Intercept Factor Data for Figure 3-1. • • • • A-3

Model Computed and Experimental Data for Figure 4-1.. • A-4

Model Computed and Experimental Data for Figure 4-2 ••••• A-8

Table 3-1.

Table 4-1.

Table 4-2.

Table 4-3.

APPENDIX

DATA TABLES FOR FIGURES IN TEXT

Model Computed Solar Source Projections for Figure 4-3

A-l

• A-10

Table 3-1. Model Computed Intercept Factor Data for Figure 3-1.

(Solar Source Intercept Factor; Perfect Paraboloidal Mirror; flD = 0.50;

Model Sigma = 0.00; Source Distance = 1.5 x lOll;

Source Radius = 5.0 mrad)

Intercept Factor

Normalized

Aperture Radius,

mrad

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

3.5

4.0

4.5

5.0

5.5

6.0

6.5

7.0

7.5

8.0

8.5

9.0

9.5

10.0

Perfect

Paraboloidal

Mirror

0.042

0.085

0.099

0.142

0.170

0.199

0.236

0.262

0.297

0.327

0.359

0.392

0.419

0.498

0.578

0.656

0.711

0.814

0.873

0.915

0.943

0.963

0.977

0.986

0.992

0.996

0.999

A-3

Perfect

Optical

System

0.014

0.026

0.040

0.058

0.078

0.102

0.130

0.160

0.194

0.230

0.270

0.314

0.360

0.490

0.640

0.810

1.000
(Date:

f/D Ratio

Table 4-1a. Model Computed Data for Figure 4-1

11/11/83; Geometry: Modified Parabola; Focal Length = 6.00;

= 0.500; # Rays = 3565; # Circles = 100; # Points/Circle = 36;

Plane Cut z; Value: 6.030; Sigma = 16.25;

Source Points: X = 0, Y = 0, Z = 900)

Normalized Aperture Intercept Points

Radius, mrad Factor Out

2.00 0.022 3485

4.00 0.105 3190

4.21 0.119 3142

6.00 0.225 2762

8.00 0.356 2297

8.42 0.394 2159

10.00 0.492 1812

12.00 0.624 1339

12.64 0.660 1212

14.00 0.721 995

16.00 0.810 679

16.85 0.831 601

18.00 0.863 488

20.00 0.912 314

21.06 0.932 244

22.00 0.948 186

24.00 0.964 127

25.27 0.972 100

26.00 0.976 86

28.00 0.985 53

29.49 0.990 35

30.00 0.992 29

31.59 0.994 20

32.00 0.996 16

33.70 0.998 6

34.00 0.998 6

36.00 0.999 4

37.91 0.999 4

38.00 0.999 3

40.00 0.999 3

42.00 0.999 2

42.12 0.999 2

44.00 1.000 0

A-4
(Date:

f/D Ratio

Table 4-lb. Model Computed Data for Figure 4-1

11/11/83; Geometry: Modified Parabola; Focal Length = 6.00;

= 0.500; # Rays = 3565; # Circles = 100; # Points/Circle = 36;

Plane Cut Z; Value: 6.000; Sigma = 16.25i

' Source Points: X = 0, Y = 0, Z = 1.5 x 10 1)

Normalized Aperture Intercept Points

Radius, mrad Factor Out

2.0 0.036 3437

4.0 0.124 3122

6.0 0.247 2683

8.0 0.383 2199

10.0 0.515 1729

12.0 0.638 1289

14.0 0.734 947

16.0 0.817 651

18.0 0.873 451

20.0 0.914 305

22.0 0.949 182

24.0 0.969 110

26.0 0.983 59

28.0 0.990 35

30.0 0.996 13

32.0 0.998 7

34.0 0.999 3

36.0 0.999 3

38.0 0.999 1

40.0 1.000 0

A-5
(Date:

f/D Ratio

Table 4-1c. Model Computed Data for Figure 4-1

11/10/83; Geometry: Modified Parabola; Focal Length = 6.00;

= 0.500; # Rays = 3565; # Circles = 100; # Points/Circle = 36;

Plane Cut z; Value: 6.030; Sigma = 0.0000;

Source Points: X = 0, Y = 0, Z = 900)

Normalized Aperture Intercept Points

Radius, mrad Factor Out

0.50 0.061 3448

l.00 0.172 2952

l. 50 0.273 2592

2.00 0.364 2268

2.50 0.444 1980

3.00 0.525 1692

3.50 0.596 1440

4.00 0.657 1224

4.50 0.717 1008

5.00 0.778 792

5.50 0.828 612

6.00 0.879 432

6.50 0.929 252

7.00 0.980 72

7.50 1.000 0

A-6
Table 4-1d. Experimental Data for Figure 4-1

(Full Mirror Experimental Data Source: Point, Visible Source Distance:

900 m; Image Plane: 6.03 m)

Aperture Radius Normalized Aperture Intercept Factor,

in. mm Radius, mrad Measured

1.000 25.4 4.21 0.131

2.000 50.8 8.42 0.377

3.000 76.2 12.64 0.670

4.000 101.6 16.85 0.839

5.000 127.0 21.06 0.927

6.000 152.4 25.27 0.965

7.000 177.8 29.49 0.981

7.500 190.5 31.59 0.985

8.000 203.2 33.70 0.988

9.000 228.6 37.91 0.991

10.000 254.0 42.12 0.994

11.000 279.4 46.33 0.997

12.000 304.8 50.55 0.998

A-7
(Date:

f/D Ratio

Table 4-2a. Model Computed Data for Figure 4-2

11/16/83; Geometry~ Modified Parabola; Focal Length = 6.00;

0.500; # Rays = 3565; # Circles = 100; # Points/Circle = 36;

Plane Cut z; Value: 6.030; Sigma = 14.00;

Source Points: X = 0, Y = 0, Z = 900)

Normalized Aperture Intercept Points

Radius, mrad Factor Out

2.0 0.037 3433

4.0 0.164 2982

6.0 0.314 2446

8.0 0.484 1839

10.0 0.643 1271

12.0 0.753 880

14.0 0.840 569

16.0 0.908 328

18.0 0.949 180

20.0 0.965 125

22.0 0.979 76

24.0 0.987 45

26.0 0.993 24

28.0 0.998 7

30.0 0.999 4

32.0 0.999 3

34.0 0.999 3

36.0 0.999 1

40.0 1.000 0

A-8
Table 4-2b. Experimental Data for Figure 4-2

(Four Panel Experimental Data Source: Point, Visible Source Distance:

900 m; Image Plane: 6.03 m)

Aperture Radius Normalized Aperture Intercept Factor,

in. mm Radius, mrad Measured

1.000 25.4 4.23 0.168

1.500 38.1 6.35 0.350

2.000 50.8 8.47 0.527

2.500 63.5 10.58 0.695

3.000 76.2 12.70 0.812

3.500 88.9 14.82 0.881

4.000 101.6 16.93 0.925

4.500 114.3 19.05 0.946

5.000 127.0 21.17 0.965

5.500 139.7 23.28 0.977

6.000 152.4 25.40 0.986

6.500 165.1 27.52 0.990

7.000 177.8 29.63 0.994

7.500 190.5 31. 75 0.997

8.000 203.2 33.87 0.999

A-9
Table 4-3. Model Computed Solar Source Projections for Figure 4-3

(PDC-1 Solar Source Projection Data Summary; Model Parameters: Focal

Length = 6.00 m, Diameter = 12.00 m, Source Distance = 1.5 x lOll m, Image

Plane = 6.00 m; (1/10 )1 - Uniform Solar Disk, Radius = 5.0 mrad and

Surface Sigma = 16.25 mrad; (1/10 )2 - Point Source (on axis) and

Surface Sigma = 16.25 mrad; (1/10 )3 - Uniform Solar Disk and

Surfa~e Sigma = 0.00 mrad)

Normalized Aperture

Radius, mrad

Intercept Factor

(Curve 1)

Intercept Factor

(Curve 2)

Intercept Factor

(Curve 3)

1.0 0.099

2.0 0.027 0.036 0.262

3.0 0.419

4.0 0.097 0.124 0.578

5.0 0.711

6.0 0.221 0.247 0.873

7.0 0.943

8.0 0.353 0.383 0.977

9.0 0.992

10.0 0.479 0.515 0.999

12.0 0.600 0.638

14.0 0.700 0.734

16.0 0.785 0.817

18.0 0.852 0.873

20.0 0.900 0.914

22.0 0.935 0.949

24.0 0.960 0.969

26.0 0.974 0.983

28.0 0.984 0.990

30.0 0.991 0.996

32.0 0.995 0.998

34.0 0.998 0.999

36.0 0.999 0.999

38.0 0.999 0.999

40.0 0.999 1.000

42.0 0.999

44.0 1.000

A-10



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