HOW TO MEASURE THE OPTICAL 

QUALITY OF FOCUSSING SOLAR 

COLLECTORS WITHOUT LASER 

RAY TRACING 

SoOn~r Energy Research Institute 

'153E Cole Boulevard 

Golclen, Colarado 60401 

A Div~sion of Midwest Research Instif.ute 

Prepared for the 

U.S. Department of Ecergy 

Contract No. EG.77C.01.4042 
HOW TO MEASllRE OPTICAL QUALI'IY OF FOCUSSING SOLAR COLLECTORS 

WITHOUT LASER lW TRACING 

P. Bendt 

8. Gaul 

A. Rabl 

Solar Energy Research Institute 

1536 Cole Boulevard 

Golden, Colorado 80401 

Designers and manufacturers need a tool for 

evaluating the optical quality of solar concentrators. In addition to the optical efficiency no, it is important to have a neasure of optical errors and of losses due to 

reflected radiation dssing the receiver. 

This paper describes a novel alternative to 

the laser ray trace technique. The new 

method does not require any equipment beyond 

what is used for measuring collector efffciency; it could therefore become part of 

routine collector testing. The total optical errors resulting from imperfect specularity and from inaccuracies in reflector 

position or slope are characterized by an 

angular standard deviation aopti al, the rms 

deviation of the reflected rays &om the design direction. The method is based on the 

fact that the off-axis performance of a concentrator depends on Ooptical. An angular 

scan is performed; i.e., the collector output is measured as a function of misaligument angle over the entire range of angles 

for which there is measurable output (typically a fev degrees). This test should be 

carried out on a very clear day, with rtceiver close to ambient temperature (if the 

latter condition cannot be satisfied, appropriate corrections are necessary). The parameter Q is then determined by a 

leasc squ~?&??it betvcen the measured and 

the calculated angular scan. We tested the 

met hod on a parabolic trough collector usanufactirred by Bexcel but it is suitable for 

parabolic dishes as well. The method appears to be accurate enough to determine 

aoptical within about 10Z. 

Evaluation of the optical quality of a solar 

concentrator is important: to the designer 

to tell him whether a collector needs improvement, and to the manufacturer ro ensure 

proper quality control- The nethods t5a.t 

are available or have been proposed for 

measuring the contour accuracy of soLar concentrztors require either laser ray tracing 

or flux mapping at the receiver su:rfact!. 

Both approaches provide very acxurate sry 

sults but are costly: the eqti.:l.pmenr: ... :I 

specialized and expensive and dem;l.rrds ;a gc~o~:l 

deal of time and/or expertise (1 ,;I.) . 

The question arf ses, therefore, uh.ethr?r' t?~e 

instantaneous ef f iciencp rneasurent!nts tti:. t: 

are performed as part of a st:anda~rc;lized pel-... 

formance evaluation (3) could ~~t.~mehow l>.t! 

used to deternine Uoptica,, !:he I:na alngul;. 1: 

beam spread caused by opt cca~ Impa!:rf ec~iJ:,oni; a 

This paper shows that this can tn~:le!ed 'be a.r:-. 

complished by misaligning the col lect:cr 1: 

slightly away from the sun and mail.ai;uriing tile 

efficiency for several values o:E' the d,:;... 

alignment angle. The optical err::lrr Qo.pt:,icJL'. 

is then extracted by finding the t:heortetic.~i ficurve which best fits these m:i.al;ali,gnmetrr: 

data. Thus, the determination at Uoeri c.i, 

could become part of the srandarcl, test. p~';~..: 

cedures for concent ra tins solar ::.ol let=. !::oK.!I , 

The method is suitable for both pi:rr:~tov~s I[.-.:. :: 

and thermal collectors. 

The theory underlying this t achn!I.c ue :Is die!... 

scribed i.n Section 2. Secti:on 2; pr'eserir: i; 

the test results for a para balic t:-n:ougla cct.:. 

lector &.th cylindrical receiver ., aa:ntlfa.c: 

tured by Bexcel. The daca abt:ained wj.1::: 

this collector indicates that the ac8=:lra,r:jr 

of this nethod is acceptable (or! :he orc!r!r: 

of flOX). 

Since the theory of focus sin.^ sola r c2:l.lr!l:.-. 

tors has been described elsewnerri! (4,5), u 

brief summary suffices at thf s ~oi nt., ai~r:l 

for siinplicity we consider the :..!ne :iloc:i~s 

case. Th.e crucial concepts (ire ':.he ztngu].ii I:' 

acceptance function and the intercept: fat:.- 

tor. The angular acceptance func !.ion f (0. ; 

is defined as that fractiar; of :I unlfo!!n 

beam of parallel rays incident or1 rhe ape:-- 

ture at an angle 9, f~om the ol;~t ica; ax..:; 

that reach the receiver if the c1~tfc:s a:::? 

perfect. For the exainole 2f a. parsbal .c:. 

trough with cylindrical receiver,, f (eL) ::; 

given by 
I 1 for I ell < ell 

( 0 for 1 ell > e12. 

7 rim .a.ngle, and 

C - geomc;!t:ric concentration ratio. 

The tntercepie factor ~(9 ) is that fraction 

:)f' !::.:Is f'rom t:he sun that? reach the receiver 

of .:I co:ll.ecti,~r. with real optical errors vhen 

!:he 1ptic:al a.:rds is misaligned by an angle 

[?:om the c!en:er of the sun. The intercept: factor 1.13 the convolution of the nor 

coal. 1.2 nd angu.!;~r distribution of the sun, the 

til,st:~.:lbution ::lf optical errors, and the angnl.a\r, ac'cepta nce Function for perfect op- 

:I 'Rle orcier of carrying out the convo- 

:L!.ic:l.c~n is1 immiaterlal; hence the problem can 

lye t: reated as an equivalent source (d:efined 

cis :o,mro.Lut i,o.n of real source and c~ptical 

c?rro!.s) !:hat is incident on a perfect cow 

centl-ator. 

In a salair concentrator several statifittically indepe-nden!: factors contribute to the op: 

lt?ca . error: contour errors, tracking ercar !; (when averaged over time), defo~mation 

iand d$splacem~snt ~f the receiver, and lack 

10; l:lc?rrfect swo~cularity (6). 

I?acb error tgne can be characterized by its 

nrm aogu:Lar width (one-sided deviation from 

!he dasi,p ~:lreczion). The rms width for 

tl-e :otnl. ootI.ca1 error is obtained bly addi~e :he squairtis of the individual widths: 

(1- = $,p:: icai contour + d;specalar 

+ c? + b [2-21 

displacement tracking' 

( :I is nultiplied by two beca,use of c r.Il'FOU7,,) :;r,a./.~ . 

iVc'ti! chat this rule for combining standard 

~ierr.i;~tbons is valLd regardless of the shape 

of : 1e individual error distributions; they 

t:c ~:l:l.cI be Gausrsian, boxlike, or anything else 

!s:: rlc !: al.1 3:i.st ributions under disc:usaion 

'~UY-:C zerc~ mean. The tstal beam wfbth 0 is 

c:lt~tri...xed by adding :he rms width of the sun 

a 8:. :::::. :di ;lg; iO 

Measurements of reflector surfaces have 

shown that the distributions corresponding 

t:o acontour and uspecular can be treated as 

G;aussian. The other terms may or may not be 

Gaussian. However, when many statistically 

independent distributions are convoluted , t:he result is nearly Gaussian unless a single non-Gaussian contribution dominates (7). 

In the case of focussing solar collectors, 

the Gaussian contour errors appear to be the 

I.argest, and a Gaussian approximation for 

the total optical error is reasonable. 

2'heref ore, the equivalent source (convolut:ion of sun and optical errors) is given by 

(e -et) 

ct e eisun,ear i I 

exP(-ef/2<ptica1 1 

X -a [2-4 I 

optical 

where B (el) is the linear b~ightness di"s~~&$?%g of the sun (in ~/m' rad) 

clbtained from the circumsolar scans of the 

L,awrence Berkeley Laboratory (8). In t ems 

of the radial scans, Bradial(e), measured by 

Lawrence Berkeley Laboratory, the linear 

brightness distribution is 

To obtain the dimensionless intercept factor, Eq. [2-41 must be normalized by the 

beam irradiance Ib of the sun (measured by a 

pyrheliometer): PO 

For simplicity we have replaced all integration limits by infinity, assuming that the 

distributions are negligible beyond a few 

degrees. The intercept factor corresponding 

to a aisalignment angle em is 

0) a C dBl f (em-oL) Beffective(el)/~b~ 

To standardize the tests described in this 

paper, it is best to take data only when the 

sicp is very clear. The rms width of the sun 

under such conditions is 

0 sun = 2.6 k 0.1 mrad 

for line focus geometry. This is narrow 

enough to justify a Gaussian approxbation 

e- en for the equivalent source, Eq. [2-4], 

7rovided the optical errors are on the order 

of 5 mad or larger. Since this is the case 

for tha current generation of line focus 

-;arabolic collectors, we have rmde this apcroximation for the data analysis in :'.is 

paper. The resulting intercept factor is 
with the total width a of Eq. 12-31. 

he to reflection and absorption losses the 

radiation fncident on the collector is attenuated by a factor (Pta] where 

P reflectance of ref lector; 

r = transmittance of receiver glazing, if 

any; and 

a = absorptance of absorber. 

(The parentheses indicate that the factor is 

an effective transmittsnce-reflectance-absorptance product, including secondary effects such as multiple reflections.) When 

the absorber surface is at ambient temperature the heat loss is zero and the efficiency equals the optical efficiency: 

If the heat loss is nor zero, an appropriate 

correction must be applied. Variation of 

(P'fa) with 0 is sufficiently small to be 

negligible fo? the present purpaee. 

Figure 2-1 shows schematically what the angular scan looks like for a parabolic trough 

with cylindrical receiver, rim angle $ = 

90°, and concentration ratio C-25, for three 

values of the optical error: 0, 5, and 10 

mrad. From these curves one sees that this 

test is most sensftive to data taken around 

the curved portion of the graph; data corresponding to the halfway point, on the 

other hand, do not provide any information 

On aopticalm 

Fig. 2-1. Interce~t factor vs. misalignmene 

aogle for u = 0, 5, and 10 mraa. optical 

Since Y(8 ) depends on a It is clear tha4 an 

angular mscan of n(e ) versus 0 cont3i.n~ 

enough iclf omation to 2etermine l%th ( #'K~x) 

and g, at: least in principl~. :Cn order co 

determine whether this mechod is accuqa~ze 

enough to be useful in practice, we decided 

to test ii parabolic trough collector Nmlfactured by Bexcel (9). The test setup Ls 

descrfbed in another publication (10). '1*!112 

collector has a cylindrical rece l.ver coqt:ad 

with black chrome. The heat shietlLd and 1:sceiver glazing originally suppliecl by Bexc~el 

were removed for this test, so (Pta) is g:Lr 

ply Pa The reflector is made of' an alyrainum honeycomb substrate, coated with PIE<- 

163, an aluminized second-surface acr):L lc 

film manufactured by the 3M C'c~rporatioo. 

The rim angle of the collector ds 4 = 12!", 

and the geometric concentration r31 io is ,C: 

20.9. Tile tracking axis is horizontal :n the east-west direction and the testEl YE!IT~ 

carried out at solar noon, so that the 4ongirudinal incidence angle e,, vanishes. 

A typical scan of rl (0 ) versus it is shovn 

in 1. 3-1. ~osit?ioe? and aegal::he valut~s 

of 8 have been included on the same! si.tle 

becazse of symmetry. Plotting im+e, and -4 

together has the advantage of pointing art? 

any systemtic error in the zero olignnertt:.. 

A nonlinear least-squares fit to t 'hestr &.t:a 

yields the values 

~lpa = 0.69 1 

and 

a = 6.5 mad. [:3-2 : 

(In the test procedure, Fn.iet {rind aoutl.t!t: 

temperatures of the fluid were meiasl;ured :, a.~t;l 

only the product F"lo of heat extrUa.ction E!I!... 

Fig. 3-1. Test data or optizai li!i ficfieqcp 

vs. misalignment angle. = josirive 

angles ; x = negative angles ; soli6: line = 

jest fit. 
~!ic:i.er$ and oytical efficiency could l~e det:ertai. -I . reflector is deformed; its curvature tends 

tcr be less than the design shape either be 

cause of the manufacturing proceas or because ef weight induced sag. A value of 4 

mrad for the associated beam spread may be 

realistic. In viev of the difficulty of 

'"he thdnr(etica1 scan corresponding to these 

;xl~-;~mee)erri is show by the solid line. To 

ryai,.aade the accuracy of this method, the 

3c:1n1 14s repe.sted several times, and Q'Pa 

ind 11 were ta lculated for each scan. The 

-enu. ts are 

noasuring a is la ent directly, and in 

v:i.w of the fact 0% laser rav trace data for 

the collector tested at SERI, one can invoke 

the Sandia data only for a qualitative comparison. To this extent our results are 

certainly consistent with the Sandia data. 

17 pa. --. -- (mrad:) 

Beffective ( BL) Equivalent source obtained 

error distribution 

as convolution of Bsun 

(W/m 9 

rad ) 

Cl~tr 5a~rnole s -andard deviation for the three 

Or neesvrement I is 0.2 mrad, and gives some 

Ln~i lratl on of the accuracy and reproducibil- 

;tqr. 'Towev~br, a sample of three is not 

La:?- enough for any reliable statistics, 

lncl 6-6! :c)nse I-,irativel y estimate the ac~curacy 

a$ b'1.5 mrad, ~.e., LOX relative error. 

radial ( 3, Angular brightness distribution of sun for point focus geometry (w/m2 sterad) 

'1) Angular profile of sun fox 

line focus geometry (WIm' 

rad) 

En ?!~te#e tests the misalignment ang1.e was 

rmon t3:o1$ed quite accurately, hence the trackinrz f8::tor doe11 not contribute significantly 

to the beam spread and Geometric concentration ratio (for example, a trough 

of aperture width D and receiver tube diameter d has 

Angular acceptance function 

for perfect optics I.ss~inir$g an r-us vidth of 2.6 mrad for the 

sun, ohe therefore obtains from Eq. [2--31 the 

aloe leal error Beam irradiance as me sured B by pyrheliometer (W/m ) 

"c,ptil:i3,1 = (6.5'-~.6~) 'I2 mrad 13-51 Absorptance of receiver 

Intercept factor if collector is misaligned; that is, 

with its optical axis 

pointing an angle em away 

1nE~~rtJnately we did not have a from the sun lastsr ray 

tr,lt:lr bpparar:l.ls for an independent evaluatl,:n oh Ithe optical error. However. meas- qnet/Ib = collector cfficiency 

Optical efficiency = (Pra)Y 

urr,mlraqs of c~,;~~~,~~, and ~contou ha& been 

zets~J5ctejd by La ndia hboratories for reflectors3 also maru~factured by Hexcel of a simie but different focal length (1,11, 

il!. le cont:our error ranged from 1.8 mrad 

be f J re erivir>rmental exposure to 2.2 mad 

aft!??- three nc-nths in an enviromaenta:L test 

2h.a~~ 5ed. va Lue of depends 

mrr ~>/ch on the PEK-I&"EY%'~~~ bonded 

to the! substri~te; a value of 1 mrad i:s possl5i.r (12). 

Projection of incidence angle on plane perpendicular 

to tracking axis 

Projection of incidence angle on plane of tracking 

axis and optical axis 

Angle from center of sun to 

plane of symmetry of collector 

ZF :!~~ebe val~ics of acontour and as lPculat dr-? dpplicnble lo the Hexcel co?lector 

t,?3:ld it SER:, they would imply that the 

:IJIC -i&itton crd, splacement, caused bg dis- 

? l.pS:-zlan t of :h> recelver anci def omations 

3i 1e parahala, is fairly large, on the 

3r1i ?- or i zrad. Visual inspection of the 

;,:I, 1.. inage 41 the receiver shows that the 

Zf fective reflectancetransmi ttance-absorptance 

product of collector 

r;ns angular deviation cf 

contour from design direction 

"contour 
~displrccmc~t Equivalent rms angular 

spread that accounts for 

imperfect placement of receiver 

u 

specular ms spread of reflected 

beam due to imperfect specularity of reflector material 

'optical rms angular spread caused by all optical errors 

u sun nns angular width of sun in 

line focus geometry 

a Total rms beam spread 

Transmi ttaace of collector 

glazing, if any 

f+ Rim angle 

(1) Butler, B. L.; Pettit, R. B. "Optical 

Evaluation Techniques for Reflecting Solar 

Concentrators." Society of Photo-optical 

Instrumentation Engineers (SPIE). Vol. 114, 

O~tics Applied to Solar Energy Conversion: 

p. 43; 1977. 

(2) Biggs, F. ; Vittitoe, C. N. The HeLioa 

Yodel for the Optical Behavior of Reflecting 

Solar Concentrators. Albuquerque, NM: Sandia Labs .; SAW 76-0347. 

(3) American Society of Heating, Refrigeratinn. and Air-conditioning Engineers, Inc. AS= Standard, Method of Testing to Determine the Thermal Performance of Solar Collectors. New York, NY: ASHRAE; 1978. 

(4) \, Bendt. -- ... P.: Rabl, A.; Reed, K.A.; Gaul, 

8. W. op;icai Analysis and Optimization of 

Line Focus Solar Collectors. Golden, CO: 

Solar Energy Research Institute; SEXI/TR-34 - 092. To be published. 

(5) Bendt, P.; Rabl, A. "Optical hl,ysf !3 

of Parabolic Trough and Dish R~~f'lectors. " 

In preparation. 

(6) Pettft, B. B. "Charactetizat:ion of the 

Reflectet Beam Profile of Solar "4irror ?ktterials. Solar Energy. vol. 19: p. 732,; 

1977. 

(7) Cramer. K. Mathematical !;!ethods: of 

Statistics - Princeton, NJ: Princeton &J:- 

versity Press; 1947. 

(8) Grether, D.; Bunt, A. Desc:ription, of 

the LBL Reduced Data Base. Berakleley, CP,: 

Lavrence Berkeley Laboratory; August 9, 

1977. 

Grether, D. F.; Hunt, A.; Wahlig, H. "Ciz,-.. 

cumsolar Radiation: Correlations with Sala r 

Radiation." Berkeley, CA: Lawrence Berke-s 

ley Labs.; Oct. 11, 1977. 

(9) Dudley, V. E.; Workhoven, 8.. Y. zlm~ mam Reuort : Concentrat in9 c-l-- r-l''--+*"- 

(10) Gaul, 8. W.; Rabl, A. "Inc:idence! hrgle Modifier and Average Optical E: f f ici enc::? 

of Parabolic Trough Collectors." [SES Corrferencc. May 1979; Atlanta, SA. T'O be &IF 

lished as a SERI report. 

(11 ) Allred, R. E. "2nviromenr::a 1 E Ff ec.t :I 

on Solar Reflector Structures. " !:SES Cowfetence. May 1979; Atlanta, :A. To be pu.i)-- 

liahed as a Sandfa Labs repart:.