OPTICAL ANALYSIS AND OPTIMIZATION OF 

LINE FOCUS SOLAR COLLECTORS 

P. BENDT 

A. RAeL 

H. W. GAUL 

K. A. REED 

f \ 

Solar Energy Research Institute 

1536 Cole Boulevard 

Golden, Colorado 80401 

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U.S Department of Energy 

Contract No. EG. 77.C.01.4042 

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FOREWORD 

This report was prepared under Contract No. EG-77-C-0 1-4042, SERI Task 

3432.30, by P. Bendt, A. Rabl, and H. W. Gaul of the Thermal Conversion 

Branch of the Solar Energy Research hstitute, and by K. A. Reed of 

Argonne National Laboratory. The authors thank F. Biggs, D. F. Grether, 

A. Hunt, K. Masterson, and G. W. Treadwell for helpful discussions. 

I ---- _- .- -- -- 

Frank Kreith 

Branch Chief 

Ther ma1 Conversion Branch 

Approved for: 

SOLAR ENERGY RESEARCH INSTITUTE 

Acting 

for Research 
SUMMARY 

The optical analysis of a solar concentrator is usually carried out by means of computer 

ray tracing, a microscopic method that provides an enormous amount of detailed 

information but obscures functional relationships. This paper describes a macroscopic 

approach that yields all the parameters needed for the optical design of line focus 

parabolic troughs in closed analytical form, requiring only minimal computation. 

The goal of the optical analysis developed in this report is to determine the flux at the 

receiver as a function of concentrator configuration, receiver size, width of sun, and 

optical errors (e.g., tracking, reflector contour). All causes of image spreading are 

quantified as angular standard deviation. Ray tracing with a real reflector and a real sun 

is shown to be equivalent to convoluting the angular acceptance function of a perfect 

concentrator with an effective radiation source. This effective source, in turn, is 

obtained by convoluting the distribution function of optical errors with the angular 

profile of the sun. The problem is reduced to two dimensions by projecting the threedimensional motion of the sun on the plane normal to the tracking axis. In this frame the 

apparent width of the sun increases as l/cos 8 with incidence angle 8. 

A formula is derived for the optimal geometric concentration ratio, maximizing net 

power output as a function of all relevant variables (all-day average insolation, optical 

errors, effective transmittance-absorptance, heat loss, and concentrator configuration). 

Graphical solution of this equation consists of finding the intersection between a universal curve and a straight line representing a critical intensity ratio. 

In the last section, which is written as a self-contained users guide, the results are 

summarized and illustrated by specific examples. 

LIST OF FIGURES 

Page 

Definition of Coordinates and Projected Incidence 

Angles (a) and (b) €3, for Two-Dim ensional 

Concentrators ...................................................... 4 

Projected Ray-Trace Diagram for Troughlike Reflector 

Is Independent of Elevation of Incident Ray from x-y 

Plane.......... .................................................... 5 

Position of Sun Relative to Line Focus Concentrator 

at Noon and in Morning or Afternoon .................................. 7 

Definition of Mirror Contour Error Angular 

Variablesdd andd% .............................................. 10 I I 

Equivalence Between (a) Imperfect Reflector with Point 

Source and (b) Perfect Reflector with Smeared Source ................... 14 

(a) Geometric Relations for the Calculation of the 

Angular Acceptance Function. .................................... 17 

(b) Angular Acceptance Function for 8 = 90' Trough 

with Cylindrical Receiver (Schematic) ............................. 19 

(c) Angular Acceptance Function for 8 = 45' Trough 

with Flat Receiver (Schematic) ................................... 19 

Intercept Factor Y Versus atot C for Different Rim 

Angles @ (Gaussian Approximation) for 

(a) Cylindrical Receiver. ............................................. 24 

(b) Flat Receiver.. .................................................. 25 

Intercept Factor Y Versus 1/C for 

(a) Circumsolar Scan Number 1 (Narrow) ............................... 26 

(b) Circumsolar Scan Number 11 (Wide) ................................ 27 

(c) Circumsolar Scan Number 16 (very wide). ........................... 28 

(d) Average over Circumsolar Scans 1 through 10 ........................ 29 

The Curve G (otot C) for Finding the Optimal Concentration 

Ratio for Different Rim Angles 0 for 

(a) Cylindrical Receiver. ............................................. 38 

(b) Flat Receiver. ................................................... 39 

Intercept Factor Y Versus Rim Angle for Parabolic Trough with 

(a) Cylindrical Receiver. ............................................. 45 

(b) Flat Receiver. ................................................... 46 
LIST OF TABLES 

Page 

2-1 Quantities Needed for Evaluation of Transverse Effects 

of Longitudinal Contour Errors in Parabolic Trough 

with East-West Tracking Axis.. ....................................... 11 

4-1 Parameters for the 16 Standard LBL Circumsolar Scans ................... 31 

5-1 Optimization of Concentration Ratio for Real Sun and for 

GaussianSun ....................................................... 36 

6-1 CollectorParameters. ................................................ 43 

6-2 Tracking Modes and Associated Solar Data. .............................. 47 

6-3 Optimization of Concentration Ratio for Parabolic Trough 

withEast-WestAxis: Worksheet ...................................... 49 

6-4 Calculation of Efficiency for Collector Optimized 

According to Table 6-3: Worksheet.. .................................. 52 

6-5 Sensitivity of Optimization to Change in Heat Loss 

ParameterqL ...................................................... 53 

6-6 Sensitivity to Circurnsolar Radiation, Exact Calculation 

and Gaussian Approximation. Intercept Factor Y as 

Function of aoptical and C (For 0 = go0) .............................. 54 

6-7 Calculation of Efficiency of Collector with C = 27.3 

(Optimized for East-W est Axis) If Operated with 

Horizontal North-South Axis.. ........................................ 56 

A-1 Average of Cosine Factors for Collector with East-West 

Tracking Axis ...................................................... 63 
NOMENCLATURE 

The optimization procedure proposed in this paper is based on typical all-day average 

values of insolation. All-day averages are designated by angular brackets < >* A 

subscript tc under the bracket indicates that the average is taken over an operating 

period from tc hours before until tc hours after solar noon. Subscripts I I and i designate 

angular variables measured parallel or transverse to the tracking axis. 

%un (') Angular profile of sun (w/m2 rad) for line focus geometry 

Beff (8) 

2 Effective source (W/m rad) = convolution of solar profile BSun(8) 

and distribution function E(0) of optical errors 

Geometric concentration ratio = ratio of aperture area over receiver 

surface area (For example, a trough of aperture width D and receiver 

tube diameter d has C = D/( ~d),) 

Optimal concentration ratio 

D Aperture width 

d Absorber diameter 

E(8) Distribution function of optical errors (rad-l) 

Angular acceptance function = fraction of rays incident on aperture 

at incidence angle 8 from optical axis that reach receiver 

G(aC) Function used for optimizing C 

Hd 

2 Daily total diffuse irradiation on horizontal surface (J/m ) 

Hh Daily total hemispherical irradiation on horizontal surface (~/m~) 

Ho 

2 Daily total extraterrestrial irradiation on horizontal surface (J/m ) 

b Beam component of solar irradiance (w/m2) as measured by 

pyrheliometer (also known as direct normal insolation) 

<Ib cos 8) Day-long average beam irradiance on collector aperture (including 

cosine factor) 

Id Diffuse component of solar irradiance, assumed to be isotropic (w/m2) 

I h Hemispherical irradiance on horizontal surface 

xiii 
Solar constant = 1353 w/m2 

Hh/Ho = clearness index 

That portion of Ib that would reach the receiver (W per rn2 of 

aperture area) if (P T a) were equal to one 

Cqloss = heat loss (w/m2) per receiver surface area 

( P r a) ¶in - gloss = power output of collector (W per rn2 of aperture 

area) 

That portion of $, prevented from reaching receiver because of 

shading of aperture by receiver 

Id/Hd = conversion from irradiance to daily irradiation for the diffuse 

component 

Ih/Hh = conversion from irradiance to daily irradiation for the 

hemispherical component 

Time of day (from solar noon) 

Collector cutoff time 

Sunset time 

- Id)), = critics1 intensity ratio 

Contribution of shading term to critical intensity ratio 

Absorptance of receiver 

qin/Ib = intercept factor 

Intercept factor if sun is approximated by Gaussian distribution 

Intercept factor if collector is misaligned; that is, with its optical 

axis pointing an angle 81 away from the sun 

Declination 

qnet/Ib = collector efficiency 

Optical efficiency = (P T U)Y 

Incidence angle 

xiv 
Qh Incidence angle on horizontal surface 

1 Geographic latitude 

h( Q)) Rim-angle-dependent contribution of longitudinal mirror errors to 

transverse beam spreading 

(P aL) Effective reflectance-transmittance-absorptance product of collector 

%on tour rms angular deviation of contour from design direction 

'displacement Equivalent rms angular spread which accounts for imperfect placement of receiver 

'spec ular rms spread of reflected beam due to imperfect specularity of 

reflector material ' 

'optical rms angular spread caused by all optical errors 

Os un rms angular width of sun in line focus geometry 

Otot Total rms beam spread 

t Transrni ttance of collector glazing, if any 

8 Rim angle 

Optimal rim angle 

2 rf/T = hour angle 

2 7rtc/T = collector cutoff angle 

Ws 2 rt,/T = sunset hour angle 
SECTION 1.0 

INTRODUCTION 

Traditionally the optical analysis of solar concentrators has been carried out by means of 

computer ray-trace programs [I-31. Ray tracing is a microscopic method which can 

provide an enormous amount of detailed information but obscures functional 

relationships. Recognition of functional relationships is of utmost importance for the 

development of simplified design procedures. This paper shows how all the relevant 

parameters for the optical design of line focus solar concentrators can be obtained by a 

simple macroscopic approach. It is shown in most cases that approximations are 

permissible whereby all quantities of interest can be determined from a few graphs. This 

paper deals explicitly with parabolic troughs, but much of the analysis is applicable to 

other line focusing collectors as well. 

Treatment of optical properties in isolation is justified because, to an excellent 

approximation, the optical and the thermal behaviors of solar collectors are independent * of each other. In the optimization of the concentrator the thermal properties enter 

only through a single parameter q~, the heat loss per unit receiver surface. In Section 

2.0 the optical problem is reduced from three to two dimensions by projecting all rays 

onto the plane normal to the tracking axis. In this frame the apparent size of the sun 

increases with incidence angle 0 as l/cos 8. The effect of longitudinal mirror errors on 

this projection is evaluated; it increases with rim angle 8 but even at 0 = 90' it is 

small. In Section 3.0 ray tracing is shown to be equivalent to convoluting an effective 

source function with the angular acceptance function of the concentrator. It is 

convenient to calculate the angular acceptance function for perfect optics and to include 

errors in the effective source. The effective source is obtained by convoluting the solar 

profile with the distribution function of optical errors, and in most cases it can be 

replaced by a Gaussian distribution. This step, henceforth referred to as Gaussian 

approximation, is motivated by the central limit theorem of statistics.** It is attractive 

because it reduces the number of independent parameters. The difference between the 

intercept factors calculated with the real sun and with the Gaussian approxiination is 

evaluated in Section 4.0, using the circumsolar data supplied by the Lawrence Berkeley 

Laboratory [5]. Only for collectors with relatively low optical errors is the difference 

significant. In Section 5.0 the optimal concentration ratio, i.e., the ratio of aperture to 

receiver surface area, is calculated for thermal collectors by maximizing the 

efficiency. The optimum is broad enough to permit determination with the Gaussian 

approximation in all cases of interest. The optimization procedure consists of finding the 

* Warming of the receiver enclosure by direct absorption of solar radiation can reduce heat 

losses, but this effect can be treated as a shift in the optical efficiency (see Section 7.9 

of Ref. 14) and we shall assume it to be included in the effective @T@ product. 

** See, for example, Reference 4. 
intersection of a straight line, corresponding to a critical intensity ratio, with a universal 

curve. The optimal rim angle 0, is shown to be in the range 90' to 120' for a parabolic 

trough with cylindrical receiver and 45' to 60' for a flat receiver. Near the optimum 

the efficiency is so insensitive to @ that the choice of rim angle can be based entirely on 

other considerations, such as the structural strength and ease of manufacturing. 

Readers who are not interested in the technical details are advised to proceed directly to 

Section 6.0 where the results are summarized and illustrated by specific examples. First, 

the optimal concentration ratio of a parabolic trough with cylindrical receiver and 90' 

rim angle is calculated for specified values of optical errors and heat loss. (The 

optimization is based on a typical long-term average insolation value [6], integrated over 

time of day, as calculated in the Appendix.) The resulting peak and all-day average 

efficiencies are then calculated. The sensitivity of the optimization to changes in 

collector parameters and operating conditions is evaluated, particularly with regard to 

circumsolar radiation. 
SECTION 2.0 

TWO-DIMENSIONAL PROJECTION 

For the following discussion it is convenient to choose a coordinate system in which the z 

axis is placed along the tracking axis and the y axis along the axis of symmetry or optical 

axis, as shown in Figure 2-1. QL and 8,, are the projections of the incidence angle of the 

sun on the x-y plane and on the yz plane, respectively. With perfect tracking the 

misalignment angle BL vanishes and ell equals the incidence angle. 

2.1 PROJECTION OF RAY-TRACE DIAGRAM 

First, the law of specular reflection is stated in vector notation in terms of the following 

unit vectors: 

A 

i = direction of incident ray, 

fi = direction of normal of reflector surface, and 

? = direction of reflected ray. 

All three vectors point away from the surface. 

For given and the direction of the reflected ray is 

Next, a particular ray which hits the reflector with angle of incidence Qi is traced. The 

projections Bi, and Br Xy of the angles of incidence Qi and reflection er on the X-y 

plane, shown inYigure 24, are given by 

A A 

cos Bi,w = i 

xy '"w A =i,n,+i n =ion-i n Y Y z z and 

cos 8 = f.v fixy = rXnX + r n = i fi - r,n,. r?xY Y Y 

They are equal if n, vanishes. Thus, in any troughlike reflector aligned along the z axis, 

all incident rays with the same x-y projection (plane of the paper in Fig. 2-2) are 

represented by the same two-dimensional ray-trace diagram, no matter how large their 

elevation I from the x-y plane. 

Therefore, rays with the same x,y components but different z components need not be 

traced separately. If a planar ray entering with = (i i ,0) has been found to leave in x' Y the direction i = (s ,s ,O), then a nonplanar ray entering wlth 

x Y 

i; being arbitrary, has the same x-y projection and leaves with 


8' = (s, dm, sy my ii)* 

This implies that in troughlike reflectors the ray-trace diagram and, in particular, the 

focal length are independent of the elevation of the incident ray from the x-y plane 171 .* 

2.2 PROJECTION OF POSITION OF SUN 

When a line focus concentrator is employed as a solar collector, the elevation of the sun 

from the x-y plane does have an effect** on the width of the image on the receiver. 

Within a two-dimensional analysis this can be explained by the following argument. In 

the frame of the earth the sun moves in a circle, and it is this circular motion which 

breaks the translational symmetry of a line focus reflector. This is shown schematically 

in Figure 2-3. At noon the sun is in the x-y plane and the angular half-width As of the 

solar disc (note As <1) is 

with r = radius of the solar disc and R = distance from the earth to the sun. 

In the reference frame of the collector the apparent diurnal motion of the sun is a circle 

of radius R around the earth; therefore, away from solar noon, the projected angular 

half-width of the sun in the x-y plane is 

- r --= As 

*s,4 Sy COS ell ' 

where R is the projection xy 

of the sun-to-earth distance on the x-y plane. This is the projected angular width which 

must be used as input in the two-dimensional ray-trace diagram, and hence the width of 

the solar image on the receiver varies as l/cos GI,. For a concentrator with east-west 

axis, the effective angular width of the sun at four hours from noon will be twice as large 

as at noon. For tracking polar-mounted concentrators, on the other hand, 1 equals the 

solar declination 6, and this effect can probably be neglected because cos 6 is always 

larger than 0.92. For collectors with horizontal north-south tracking axes, however, this 

*This contrasts with linear refractive concentrators for which the focal lengths change 

with nonnormal incidence. 

**In troughs of finite length there is an obvious additional effect which is design 

dependent. This effect is the loss of radiation from the ends of the reflectors. It can be 

minimized by using end reflectors, long troughs, or polar mounts. 

image spread can cause serious problems at latitudes h far from the equator. Not only is 

the image enlarged by a factor 1 /cos( A + 8 ), but the sum mer-to-winter variation yields 

the widest image and hence the lowest intercept factor at a time when the insolation is 

also at its minimum. 

The variability of the projected solar size raises the question of which value should be 

chosen as the basis for concentrator design. An all-day average of the standard deviation 

as,,, weighted by the available beam insolation Ib cos el,, appears reasonable: 

dt Ib(t) cos ell 

<.:,> = t 

f C 

€3, = 8,, (t) is the angle of incidence at time-of-day t, and the collector is assumed to 

operate from tc hours before until tc hours after solar noon. Since it will become clear 

shortly that oJm is rather insensitive to any details of the averaging procedure, the 

next step is to consider equinox when GI, for the east-west case equals the hour angle 

and 

6' d w I~(o) /COS w 

<oiun> = ~kn, noon . dw Ib(u) cosw 

The long-term average dependence of Ib(w)/Ib(0) on hour angle is certainly bounded by 

one and cos w (for a more precise analysis see the Appendix). Choosing t, = 4 h as a 

typical value we find 

and 

2 

<a > = 1.41 a 2 , with 1 (w) = I (0) cos2 w. 

sun sun, noon b b 

This leads to the recommendation of 
<o;lm) = 1.5 sun, 2 noon 

as a reasonable rule for incorporating time-of-day effects into the projected size of the 

sun for line focus collectors with east-west tracking axes. 

2.3 PROJECTION OF OPTICAL ERRORS 

In this section the effect of local contour errors in the mirror are treated first, followed 

by a discussion of concentrator tracking errors. It is assumed that contour errors can be 

analyzed as if they manifested themselves as rotations but not displacements of 

elemental surface areas, giving rise to a rotation of the unit surface normal 6 at a point 

of reflection. The contour error can then be described in terms of two angular variables, 

dGL and dZI 1, which lead to rotations of fi in the planes perpendicular and parallel to the 

trough axis, respectively. As shown in Figure 2-4, dGL= dm@ and dz,, are orthogonal to 

n and to each other. The change in the unit normal is 

assuming fi lies in the x-y plane. The change in the reflected ray is found by 

differentiation of Eq. 2-1: 

The quantity sought is the change in the angle of reflection projected on the x-y plane, 

der?x~ (Figure 2-2), or, for brevity, do. Inverting the relation 

produces the desired result: 

Substituting the necessary components from Eqs. 2-1 and 2-14, setting ix = 0, and 

recognizing that i /i = tan el,, one obtains the final result = Y 

dQ = 2dwL+ 2n, tan dwl . (2-1 7) 

If the distributions of mirror contour errors dwl and du, are independent and described 

respectively by standard deviations acontourL and ocontour the distribution in 

projected angle will be described by the varlance 

The transverse angular spread resulting from longitudinal contour errors is seen to 

depend on the time of day (via and on the incidence point of radiation on the 

2 aperture (via n,); a rigorous ray-trace analysis of a parabolic trough should be based on 

Eq. 2-1 8. On the other hand, for the formalism developed in this paper it is desirable to 

replace this equation with a single effective contour error which is averaged over 

aperture and over time of day. Such averaging involves some approximation and 

2 arbitrariness; however, the resulting error is negligible in view of the smallness of n, and 

tan2 9, li We therefore propose the following simple rule: Replace nz with 

and tan Oil with <tan2 8, (> day, as listed in Table 2-1, to obtain an effective contour 

error 

The lack of perfect specularity of the reflector material 181 causes further beam 

spreading with an rms width aspec,l,r. 

Table 2-1. QUANTITIES NEEDED FOR EVALUATION OF 

TRANSVERSE EFFECTS OF LONGITUDINAL CONTOUR 

EBRORS IN PARABOLIC TROUGH WITH EAST-WBST 

TRACKING AXIS 

Average Over Aperture Average Over Time of Day 

Rim Angle <":>aper t ure Cutoff Time 

8 =I- 0 t~ <tan 'll>day 2 

(degrees) 2 tan (@Dl (h) 
Conceptually, concentrator alignment and tracking errors must be handled differently 

from mirror contour errors. Any given moment a concentrator is pointing in some 

direction, albeit with a transverse angular errorA)l. It is only when an entire field of 

concentrators or a long time average is considered that a distribution in tracking error is 

obtained, described by a standard deviation utrackin Similarly, errors resulting from 

displacement of the receiver can be characterized gY an equivalent angular standard 

deviation adisplacement when averaged over a large collector field. 

The total effective rms optical error 0, tical resulting from all of these effects is 

obtained by adding quadratically the indivi&al standard deviations: 

2 - 2 2 - 2 + + 2 

"optical 'contourL %pecularL + '(@) (4 'contour (1 'sPecular 11 1 

with 

For collectors in which reflector and receiver do not move as a unit, the term "tracking 

must be multiplied by a factor of two; this is the case for Fresnel reflectors. Note also 

that the rule (Eq. 2-20) for adding standard deviations holds regardless of the detailed 

shape of the individual error distributions; in particular, they do not have to be Gaussian. 
SECTION 3.0 

CALCULATION OF FLUX AT RECEIVER 

3.1 EPFECTMi SOURCE 

In a real solar concentrator, rays are incident from a range of directions, covering the 

solar disc and possibly the circumsolar region, and are reflected by an imperfect reflector surface that causes further angular dispersion. From the point of view of the 

receiver it does not matter whet her the angular deviation of a ray from the design direction originates at the radiation source or at the reflector. This is illustrated in Figure 3- 

1. In Figure 3-la a ray from a point source S strikes the reflector at a point R and would 

reach point Q of the receiver if the reflector were perfect. A real reflector differs from 

the design slope by an error Bslop, and thus the reflected ray reaches the receiver at Qf, 

an angle 2~slope away from Q. The same reflected ray would have resulted from a 

perfect reflector if the incident ray had come from S1, an angle 2eslO away from the 

point source S, as shown in Figure 3-lb. In general the distribution of !he slope errors is 

nearly Gaussian, and the corresponding flux distribution at the receiver is indicated by 

the curves in Figures 3-la and b. 

The angular distribution of radiation from a real source like the sun is given by the 

functional dependence of brightness Bsource (Gin) on incidence angle. Solar brightness 2 data is usually reported as radial distribution Bradial(8) in W/m sr, 8 being measured 

from the center of the solar disc. For line focus systems it is convenient to transform 

the radial distribution to a linear one according to 

#) 

'linear('L) = J -00 Bradial('). with 8 =J-H . 2 In the remainder of this paper only the linear brightness function (in W/m rad) is 

considered, and the subscripts for longitudinal and transverse are dropped. 

If the mirror slope errors are characterized by a normalized distribution function 

E(29s10pe) in units of rad-l, then the reflected intensity in the direction 8 (measured 

from the design direction) is 

where (8 - ein)/2 = Bslope is the slope error and Bsource (8. ,, ) dQin is the intensity of 

radiation coming from an angular region of width dein around gin. Integrating over dGin 

produces the equivalent effective source: 

The limits of integration can be extended to infinity because in practice the distribution 

will be negligible outside a range of a few degrees. 

In a real collector there will be several statistically independent sources of optical 

error: lack of perfect specularity [8], macroscopic surface deviations in position and 

slope, displacement of the receiver, and tracking errors. Averaged over time and over 

the entire collector or array of collectors, all of these errors can be assumed to be 

approximately Gaussian.* Even if the distributions for individual optical errors are not 

quite Gaussian, the central limit theorem of statistics [a implies that the distribution 

resulting from their convolution can be expected to be nearly Gaussian (at least as long 

as the distribution is not dominated by a single non-Gaussian component). This is a most 

reasonable assumption; it is supported by the limited data which are available [91, and it 

is usually made in solar systems studies. The distribution function E of optical errors in 

Eq. 3-2 should, of course, include the convolution of all relevant optical errors. 

Since the convolution of two Gaussians with zero mean and standard deviations al and a2 

2 is again a Gaussian distribution, with standard deviation given by o2 = CJ + o2 sun and 

mirror errors can be replaced by an effective source 

2' 

The standard deviation ooptical accounts for all optical errors and has been calculated in 

Eqs, 2-19, 2-20, and 2-21: 

For collectors in which the reflector and receiver do not move as a unit, otracking must 

be multiplied by two. 

In case the Gaussian model for the optical error is not acceptable, Eq, 3-3 must be 

replaced by the convolution of the appropriate error distribution functions. Certain 

reflector materials (for example, alzac) are characterized by the sum of two Gaussian 

distributions of relative weights R1 and R2 and widths ospecular and aspecular 2. This 

type of material (which does not appear to be very practical for focusing solar collectors) 

can be dealt with by replacing the single Gaussian distribution in Eq. 3-3 with a sum of 

two Gaussian distributions with weights R1 and R2 and widths ooptical and ooptical (corresponding to each of the two nonspecular components) [8,91. 

'Tracking errors may differ from this model when the tracking system has a significant 

tolerance band; for this case, a flat top with sloping shoulders may be more appropriate. 
3.2 ANGULAR ACCEPTANCE FUNCTION 

In the framework of this paper, aberrations due to off-axis incidence on a perfect reflector are not included among errors but are accounted for by means of the angular 

acceptance function. 

The angular acceptance function f(0) is defined as the fraction of rays incident on the 

collector aperture at an angle 8 (from the optical axis) which reaches the receiver. This 

function depends on the configuration of reflector and receiver. For example, an 

untruncated compound parabolic concentrator (CPC) [a of acceptance half angle 8, is 

characterized by the angular acceptance function 

f,,(8) = 

1 for lel~e, 

0 for 191>Qa ; 

in other words, all rays within 8, are accepted, all rays outside are rejected. 

For a parabolic trough of rim angle 8 with cylindrical receiver, the angular acceptance 

function is more complicated and can be calculated from the geometric relations in 

Fig. 3-2a. The focal length is f; the aperture width, D; and the receiver tube diameter, 

d. The geometric concentration C is 

Light rays incident at point P = (x,y) of the reflector hit the receiver, provided their 

angle of incidence 8 (measured from the optical axis) satisfies 

191 4 ex, with sin 0, = d/2 - d . (3-7) 

2 dx +(f - Y12 2f [l + (+y] 

191 = €I, is the angle at which they reach the receiver tube tangentially. ex decreases 

with x. Therefore, with x = D/2, Eq. 3-7 yields the largest angle for which all 

incident rays are accepted. This angle 8 , which has sometimes been called the 

acceptance half angle of a parabolic trough [ , can be written in terms of rim angle and 

concentration as 

71 

sin @ sin Q1 = - 

'rr C 

For the angular acceptance function this implies that fpT (8) = 1 for lei<@,. For 9 Y incidence angles larger than el but smaller than €32, given by 

d sin €I2 z - 2f ' 

only the central section of the aperture is effective, from -xe to xe, with 

d 

% = 2f(zf sin e - lI2, 

and the angular acceptance equals 2xe/D. For angles larger than e2 only direct hits are 

accepted, but this region will not be of interest in this paper. It is convenient to express 

everything in terms of rim angle and concentration by means of the relation 

D 8 - = tan - 4f 2 

Neglecting complications or inaccuracies that may arise for very low concentrations, 

very large incidence angles, or very small rim angles (cases which are not of interest for 

solar energy applications), the angular acceptance function for the parabolic trough with 

cylindrical receiver, rim angle 0, and concentration C can be summarized as* 

/I for 191 < el 

(2t~dem/2)-i)112 fore1<le<e2 fET,cylindrical(e) = { Cot - 2 

o for 181 > e, 

(3-12) 

with 

and 

C is assumed large enough (C 2 3) to justify replacing sin 8 with 8 for I 91 < e2. 

For the parabolic trough with a flat one-sided receiver, the angular acceptance function 

can be derived in a similar manner. The result is 

*The addition of a glass tube, placed concentrically around a tubular receiver, has no 

effect on the angular acceptance function [lo]. 
- sin 8 cos 8 1 for 101 < 9, - C 

fPT, f lat-one-sided 

2 8 for < 191 <e2 =-tanC 2 

(0 for 101 >% . (3-14) 

This function is unaffected by the addition of a CPC second-stage concentrator of 

acceptance half angle 8 because such a CPC only prevents rays with 1 el >e2 from 

reaching the receiver. 

For future reference we note that the angular acceptance function depends only on the 

product C8, not on C and 8 separately. The angular acceptance functions are plotted 

schematically in Figs. 3-2b and c for the cylindrical and flat receivers. 

Fi gure 3- 2b. ANGULAR ACCEPTANCE FUNCTION Figure 3-2c. ANGULAR ACCEPTANCE FUNCT 

FOR $ = 90' TROUGH WITH FOR @ = 45O TROUGH WITH 

CYLINDRICAL RECEIVER FLAT RECEIVER (SCHEMATIC 

(SCHEMATIC). 

ION 

> - 
3.3 FLUX AT RECEIVER AND INTERCEPT FACTOR 

The effective source function ~~~~.(8) of Eq. 3-3 gives the intensity of radiation (w/rn2 

rad) coming from the direction 8; it accounts correctly for the shape of the sun and for 

all optical errors. The angular acceptance function f(8) states how much of this 

radiation is transmitted to the receiver.* The total flux intercepted by the receiver is 

obtained by multiplying these two functions and integrating over all incidence angles: 

The receiver size enters through the concentration ratio C in the angular acceptance 

function. 

Dividing Eq. 3-15 by the total incident flux 

one obtains the intercept factor 

which is useful because it is independent of the intensity. 

This formulation is equivalent to a detailed computer ray-trace program. It is much 

simpler and faster, requiring at most a double integration. (A further integration may be 

needed to convolute non-Gaussian optical errors.) The relevant par am eters and their 

interrelation are clearly identified. In many cases approximations can be made to the 

point where the explicit result can be presented in graphical form. This is discussed in 

the following section. 

Time-ofday effects car: be treated exactly by evaluating qin of Eq. 3-15 for each hour of 

the day using the projected sun shape discussed in Section 2.0. In most cases, however, a 

single calculation with an effective average sun shape for the whole day (see Section 2.2) 

is sufficiently accurate. 

*f(@) is defined as a purely geometrical quantity. Absorption losses will be accounted for 

later by a multiplicative factor, the transmittance-reflectance-absorptance product 

(P ta)* 
For collectors with cylindrical receivers the derivative of Y with respect to concentration 

C can be used to calculate the distribution of incidence angles on the receiver. Consider 

the radiation intercepted by a virtual receiver, smaller than the real receiver and 

corresponding to a concentration C, placed inside the real receiver which has 

concentration C. One can show that there is a one-to-one relationship, 

cosp =d - c2/c2 cos 0, (3-18) 

between the incidence angle fl of a ray on the actual receiver surface and the 

concentration c < C of the virtual receiver which the same ray would hit tangentially; 8 

is the incidence angle of the ray on the collector aperture. From this it follows that the 

unnorm alized probability distribution of incidence angles on the receiver surface is 

given by 

This point is of interest because the absorptance of most surfaces decreases at large 

incidence angles. For example, if the absorptance a of a flat surface is given as a 

function a@) of incidence angle 6, then the effective absorptance of this surface, when 

used on a cylindrical absorber tube, is 

where 8 is the angle of incidence on the collector aperture. 

For collectors with flat receivers the derivative of Y with respect to C yields the spatial 

flux distribution on the receiver. The incidence angle distribution can be obtained from 

the derivative of Y with respect to rim angle 0. 

Thus far tracking errors have been assumed to be averaged over time, since the collector 

is, on tin average, aligned correctly. The effect of misalignment on the instantaneous 

efficiency can also be calculated. If the optical axis of the collector points an angle 

away from the design direction (center of the source), the origin of the angular 

acceptance function is shifted by and the intercept factor is given by a 

straightforward modification of Eq. 3-17: 

SECTION 4.0 

APPROXIMATION OF SUN SHAPE BY GAUSSIAN DISTRIBUTION 

When the optical errors are large compared to the width of the sun, Eq. 3-3 is insensitive 

to details of the sun shape. In this case it is convenient to approximate the sun shape by 

a Gaussian distribution: 

with the variance 

2 - Lde 8 B sun, real (9) 

OSU~ = rw 

of the real sun. (Note that alinear and Oradial for linear and radial brightness - distributions are related by alinear - uradial/v2: In this sense the sun appears to be 

narrower for a line focus collector than for a point focus collector.) 

The resulting effective source is also a Gaussian distribution: 

(9) = Ib 

Be f f ,~auss a 

tot 

6 

=tot 

with width 

2 + 2 1 /2 0 totEkoptical 'sun) 

The intercept factor Y is defined as the ratio of the flux reaching the receiver over the 

incident beam irradiance Ib: 

Since the angular acceptance is a function of C8 only, YGau, depends only on the 

product atot C, not on C and otot separately. For the parabolic trough the resulting 

integrals do not seem to be expressible in closed form; YGau, has been evaluated 

numerically and plotted versus utot C for several values of rim angle, for the cylindrical 

receiver in Figure 4-la and for the flat one-sided receiver in Figure 4-lb. 

To evaluate the error introduced by using the Gaussian sun instead of the real sun, the 

intercept factor Y is compared in Figure 4-2a through c as calculated with the measured 






solar profile (solid line) and as calculated with the corresponding Gaussian shape (dotted 

line). A cylindrical receiver and 90' rim angle have been assumed. Curves are shown for 

three error sizes, oOptical = 5, 10, and 20 mrad, and for three solar profiles, 

corresponding to clear sky, haze, and heavy haze. These profiles are numbers 1, 11, and 

16 of the standard circumsolar profiles provided by the Lawrence Berkeley Laboratory 

[a. The insolation values and rms widths for the 16 profiles are listed in Table 4-1. The 

broad sun shapes seem to be associated only with times when the beam component is so 

low (Ib I 500 w/rn2) that little energy can be collected; thus only the narrow and medium 

shapes need to be considered [ll]. This suggests the use of an average intercept factor 

for all circumsolar scans with high insolation. The unweighted average over scans one 

through ten (b >500 w/m2) is plotted in Figure 4-Zd, with the same notation as in 

Figures 4-2a through c. The Gaussian sun shape (dashed line in Figure 4-2d) is based on 

the standard deviation 

which is the average of the standard deviations for scans one through ten. As for optical 

errors, current plans [la for advanced parabolic trough collectors suitable for mass 

production call for a value of ooptical around 7 mrad. Whether greater optical accuracy 

is compatible with the low cost requirements of solar applications remains uncertain. 

Figures 4-2 a-d show that the Gaussian sun approximation is excellent for optical errors 

beyond 20 mrad, regardless of sun shape. For narrow sun shapes the approximation is 

good to I%, even for the rather small optical error of 5 mrad. More detailed analysis 

seems to be necessary for the case of intermediate solar widths and very good optical 

quality uoptical < 5 mrad. 

In any case it is worth noting that the Gaussian approximation consistently 

underestimates the radiation reaching the receiver for any practical value of C. Usually 

C will be chosen such that on a clear day approximately 95% of the incident rays reach 

the receiver; i.e., losses due to spillover amount to about 5%. (The Gaussian 

approximation would overestimate the collected flux only if the concentration ratio were 

much smaller than this value.) Therefore, a simple upper bound on the loss of 

performance due to circumsolar radiation is obtained by considering the change in atot 

caused by an increase in aSun. This is illustrated by the examples in Section 6. 

Furthermore, the Gaussian approximation is useful for determining the optimal 

concentration ratio, as shown in Section 5.0. 
Table 4-1. PARAMETERS FOR THE 16 STANDARD LBL CIRCUMSOLAR SCANS 

Gsun 

Linear 

Geometry 

(m rad) 

Solar 

Times 

(h) 

'disk 

(4.80 mrad) 

Location (w/rn2) 

Ib 

(pyrheliom eter) 

(w/m2) 

Data 

- Set 

Circumsolar 

Date Ratio 

Aug. 7 

1976 

Albuquer- 947.8 

que 

Aug. 25 

1976 

Fort Hood 708.9 

Nov. 20 

1976 

Fort Hood 894.8 

Nov. 22 

1976 

Fort Hood 775.8 

Albuquer- 919.5 

que 

Dec. 29 

1976 

Fort Hood 7 14.9 

Jan. 25 

1977 

A1 buquer- 736.6 

que 

Dec. 29 

1976 

Fort Hood 705.4 

Albuquer- 699.8 

que 

Jan. 25 

1977 

Albuquer- 517.3 

que 

Dec. 14 

1976 

Albuquer- 340.6 

q ue 

Dec. 29 

1976 

Fort Hood 2 17.5 

Jan. 25 

1977 

Albuquer- 164.0 

que 

Dec. 29 

1976 

Fort Hood 168.1 

Fort Hood 40.5 

Jan. 25 

1977 

A1 buquer- 29.5 

que 

SECTION 5.0 

OPTIMAL CONCENTRATION RATIO 

Concentration entails necessarily the loss of some diffuse and possibly the loss of some 

circumsolar radiation. If maximization of absorbed energy were the goal, C = 1 would be 

chosen; however, in thermal collectors heat losses increase with receiver size and 

therefore the optimal concentration ratio, the one maximizing delivered energy, will be 

greater than one. In this section a formula is derived for determining the optimal 

concentration ratio.* 

Heat losses can be assumed to scale as 1 /C: 

This statement is exact if the aperture area is varied while the receiver size is kept 

fixed. If the aperture is held constant, variation of receiver size may cause slight 

deviations from the scaling rule (Eq. 5-1) because convective heat transfer may depend 

weakly on absolute size. For the derivation of a general procedure for finding the 

optimal concentration ratio, the scaling rule is the most reasonable choice. 

In line focus collectors two additional effects should be considered: the collection of 

diffuse sky radiation, and the shading of the aperture by the receiver. Typical levels of 

diffuse radiation Id range from 100 to 200 w/rn2, and a fraction 1/C of this is collected 

[I :** 

For example, with C = 25, diffuse radiation amounts to 4 to 8 w/rn2, which is small 

compared to typical heat losses qloss on the order of 100 w/m2, but it is not negligible. 

Shading also scales with C; the portion of the beam irradiance Ib which is prevented from 

reaching the receiver because of shading can be written as 

*Similar considerations apply to photovoltaic concentrators where, per unit area, receiver 

cost is much higher than reflector cost. Here the optimal concentration ratio minimizes 

cost for delivered energy. 

**Since diffuse radiation is collected along the length of the receiver, this formula is quite 

insensitive to anisotropy of sky radiation. 
where XS depends on the geometry. 

An example is a parabolic trough with an aperture width of D = 200 cm and a cylindrical 

receiver of d = 2.5 cm diameter, surrounded by a glass envelope of dglass = 5 cm outer 

diameter * for which 

c=-- * - 25, and 

7Td 

qs had i ng = 11.25 w/m2 if Ib = 900 w/m2. 

The design goal is to maximize the net flux collected by the receiver: 

The factor (Wa) accounts for the effective transmittance and absorptance of the 

collector, and f and Bef are the angular acceptance (Eq. 3-12) and effective source (Eq. 

3-3), respectively. (P a!, Id, Xg and q~ enter only in the combination 

which shall be referred to as critical intensity-ratio. 

The optimal concentration ratio Co is then found by setting the derivative of qnet with 

respect to C equal to zero: 

*Rays passing through the glass but missing the receiver on their first pass deviate so 

much from the correct direction as to miss the receiver altogether. 
Even though for small values of croptical the Gaussian approximation for the sun may 

result in significant underprediction for the absorbed flux, it may be acceptable for 

finding Co because the net power qnet, being near its maximum, is a slowly varying 

function of C. The Gaussian approximation is attractive because Co depends only on 

(ottot C) and X and can therefore be obtained from a single graph. To test the validity of 

this assumption, the exact value Co,ex,t is determined for a wide range of parameters 

for a real sun and the corresponding Co,Gaus. for a Gaussian sun. The net power qnet 

(for the real circumsolar profile) is then calculated for these two concentration values 

and the results are compared. The discrepancy* 

is shown in Table 5-1 for a,ptical = 5, 10, 20 mrad; qL = 1000, 2000, . . ., 8000 W per m 2 

of receiver surface; and three solar shapes (narrow, scan No. 1; wide, scan No. 11; very 

wide, scan No. 16). 

Table 5-1 shows not only the closeness of the approximation for Co provided by the 

Gaussian sun shape, but also how much Co varies with sun shape. For narrow sun shapes 

Co,Gauss approximates Co,real within 4%, even for small optical errors; and the change 

in qnet associated with such a change in Co is less than a tenth of a percent and 

negligible. Even for wide sun shapes the difference in qnet optimized for the real sun 

and for the Gaussian sun is only a few percent.** The very wide sun shape has such a low 

insolation level that the collectible energy is negligible, and for large a or high heat loss 

no energy can be collected at all. More important is the fact that the difference 

between Co,real and Co,Ga,, for the narrow sun shape is of the same order of 

magnitude as the change in Co between different sun shapes. Furthermore, the accuracy 

of a determination of Co is limited by the uncertainty o~o~~~~~~

typically on the order 

of 10%. Therefore, ColGauss, as obtained by the Gaussian approximation for the narrow 

sun shape, provides an excellent compromise for the optimal concentration value under 

all reasonable operating conditions. 

For a Gaussian sun of total flux Ib (in w/rn2), the effective source is 

(9) = Ib 

Bef f ,~auss 0 tot fi (- 2Zot)- 

*In this comparison we have set XS = 0 = Id and (P Ta) = 1 because their precise values do 

not matter. 

**h going from a narrow to a wide sun shape, Co can either increase or decrease; this 

depends on the relative importance of reducing gloss or enhancing Y. 

Inserting this into Eq. 5-6 and taking the derivative with respect to C produces the 

condition for the optimal concentration ratio Co as 

The integral is a function of (otot C) only. It is convenient to define a function 

because in terms of G and the critical intensity ratio X of Eq. 5-5 the optimization 

condition reads 

The function G has been evaluated numerically and is plotted versus otot C for different 

rim angles in Figure 5-la for a cylindrical receiver and in Figure 5-lb for a one-sided flat 

receiver. To find the optimal concentration ratio Co, one draws a horizontal line 

corresponding to the value of otot X; the intersection of this line with the curve has the 

abscissa otot Coo 



SECTION 6.0 

EXAMPLES/USERS GUIDE 

Demonstrated in this section is how the formulas and graphs developed in this paper can 

be used to design and optimize a parabolic trough concentrator, to calculate its thermal 

performance, and to evaluate its sensitivity to circumsolar radiation. The optimization 

is based on typical average insolation data [61 rather than on peak radiation at solar noon, 

as discussed in the Appendix. 

6.1 OPTIMIZATION PHILOSOPHY 

The performance of any solar energy system improves if the collector efficiency is 

increased, all other variables being constant. Therefore, the performance of the 

collector should be maximized independent of the rest of the system if such a step does 

not significantly increase the collector cost. The variables which affect collector 

performance fall into several groups: 

(1) operating conditions (insolation, tracking mode, operating temperature, flow 

rate, etc.); 

(2) properties of materials (reflectance, absorptance, etc.); 

(3) receiver design parameters (absorber shape, width of gap between absorber 

surface and glazing, etc.); and 

(4) concentrator geometry (concentration ratio C and rim angle@). 

The operating conditions may vary from installation to installation. On the other hand, 

cost reduction by mass production requires a certain amount of standardization. Thus it 

is preferable to design a solar collector which is approximately optimal for a range of 

typical operating conditions rather than for a specific application. The examples in this 

paper show that the optimum is sufficiently broad to permit much standardization 

without significant performance penalty. 

If several candidate materials are available, the optimal choice has to be made by 

examining cost and performance for each case. As for the receiver design, one first has 

to select the generic type (flat or cylindrical, evacuated or nonevacuated). For an 

evacuated receiver a cylindrical absorber with a concentric glass envelope is probably 

the most reasonable choice, and the spacing between the absorber and glass envelope 

should be as small as is practical. For a nonevacuated receiver the spacing between 

absorber surface and glazing will usually be optimized if the corresponding Raleigh 

number is approximately 1500; in other words the spacing should be made as large as 

possible without initiating convection. 
Once the receiver (type and size) and the materials have been chosen, one has to address 

the remaining set of variables, group (4). The concentration ratio C, i.e., the relative 

size of aperture and receiver, is particularly important. As C is increased, the heat loss 

per aperture decreases but so does the fraction of the incident solar radiation that is 

intercepted by the receiver. At the optimal concentration ratio the incremental heat 

loss equals the incremental loss of intercepted solar radiation. The optimization of C is 

carried out most conveniently by keeping the receiver size fized while varying the 

aperture. Only a single number, the heat loss rate qL (in ~/m" per receiver surface 

area is needed to characterize the thermal properties of the receiver. -9 

For the optimization in this paper a standard set of operating conditions, variables in 

group (11, is assumed as a starting point. The selection of values for the variables in 

groups (2) and (3) is not addressed in this paper; rather a set of values is assumed as 

input. 

Once the variables in groups (1) to (3) have been chosen, the optimization of 

concentration ratio and rim angle can be accomplished by the design procedure developed 

in this paper. In practice the mathematical optimum is not alway the most desirable 

design because certain components, e.g., reflector sheets, may be available only in 

discrete sizes. Nonetheless, knowledge of the optimum is valuable as a guide in the 

selection of a practical design. 

6.2 COLLECTOR PARAMETERS 

Consider a long east-west-mounted parabolic trough reflector with a cylindrical 

receiver. The receiver has a selective coating and a glass envelope around it, as 

appropriate for operation in the 20OoC to 300°C range 1131. It is assumed that the 

collector is characterized by the parameters listed in Table 6-1; they represent typical 

values for state-of-the-art technology. The entries under the contributions to beam 

spreading (optical error) are based on data reported by Sandia Laboratories for realistic 

materials and fabrication techniques [12,13]. The reflectance (p = 0.85) is typical of 

clean aluminum or dirty silver reflectors. The transmittance (r = 0.88) and the 

absorptance (a= 0.94) are reasonable values when incidence angle effects are taken into 

account [14,15]. The product (Pza) will be reduced by dirt, but compensating 

improvement with antireflection coatings is possible. Present operating experience is 

insufficient to evaluate with confidence the effects of long-term environmental 

degradation, but preliminary data [17,18] indicate that dirt on a reflector reduces the 

specular reflectance by about 0.05 to 0.2 (depending on the cleaning cycle) with little 

change in ospecular. A value of 0 -70 for the all-day average reflec tance-transmi ttanceabsorptance product <(P ta)) appears to be realistic. In any case only the product, not 

the individual factors, matters for the present purpose. At noon it will be higher and 

(' ' ')noon = 0.73 is assumed [la; the change in (P r a) with incidence angle should be 

measured, of course. 

In many concentrating collectors part of the aperture is shaded by the receiver; this 

effect can be accounted for by a shading correction (XS in this paper). In the case of a 

cylindrical receiver with glazing, an equivalent correction is needed for that fraction of 
Table 6-1. COLLECTOR PARAMETERS 

Parameter Value 

'con tourl 

'tracking 

dglass 

dabsor ber 

2.5 mrad 

2.5 mrad 

2.0 mrad 

2.0 mrad 

2.0 mrad 

2.0 mrad 

the beam radiation incident on the aperture that passes through the glass envelope but 

misses the receiver. Due to refraction* it deviates so much from the design direction 

that it misses the receiver altogether on its return from the reflector. To find a typical 

value of XS for the present configuration, assume a receiver tube of 2.5-cm outer 

diameter (dabsorber), surrounded by a glass tube of 5-cm outer diameter (d lass)* (With 

reasonable glass thickness this leaves an air gap of 1.0 cm, approximate P y optimal in 

terms of heat transfer because the corresponding Rayleigh number is just below the onset 

of convection.) XS is given by the formula 

- 

Xs = dglass dabsorber 

Tdabsor ber 

and takes the value 0.318 in the present case (the factor of -;r inserted for consistent 

normalization to absorber surface area). 

*Note parenthetically that refraction by a glass tube concentric with the absorber has no 

other effect on any of the calculations of this paper, as shown in Reference 10. 
A crucial parameter is qL, the heat loss per unit absorber surface area*. It must be 

interpreted as an average along the entire collector which is to be optimized. All the 

thermal properties of the collector enter only through this parameter, and, for the 

purpose of this paper, the detailed thermal properties are irrelevant; in particular, the 

precise value of ernissivity and heat extraction efficiency, and the temperature 

nonuniformities are of no concern. The reader who is interested in calculating qL is 

referred to the standard techniques in the heat transfer literature.** A rough and simple 

estimate for qL can be obtained by extrapolating the flat-plate top heat loss coefficient 

in Fig. 7.4.4 of Reference 14. If collector test data are available, it is preferable to take 

for qL the measured heat loss per unit of aperture area, multiplied by the geometric 

concentration ratio; such data have been reported, for example, in Reference 13. The 

value q~ = 2000 w/m2 in Table 6-1 is typical for the type of collector under 

consideration. 

6.3 CHOICE OF RIM ANGLE 

For cylindrical receivers the rim angle Q) will be in the range of 80' to 120'. 

Figures 6-la and b show the optimum to be so broad that the choice of rim angle can be 

determined by other considerations such as mechanical strength and ease of manufacturing. 

In the literature two simple arguments have been given for determining oO. One 

argument is based on minimizing, for a given aperture area, the average path length from 

reflector to receiver, and it yields o0 = 120' [31. The other argument [7,19] assumes 

complete capture of all rays within a prescribed angular range and results in 8, = 90°. 

Neither of these arguments accounts for the detailed angular distribution of the radiation 

source, and hence it is not surprising that there is disagreement with the exact 

optimization. For the following discussion a 90' rim angle is assumed as a practical 

value. 

6.4 OPTIMIZATION OF CONCENTRATION RATIO 

The appropriate insolation values for optimization of the concentration ratio with 

respect to all-day average performance are calculated in the Appendix. They depend on 

incidence angle and hence on tracking mode, and they are listed in Table 6-2 for the most 

important configurations. The all-day average quantities are designated by angular 

brackets. They are based on an assumed operating time of 8 h/day. For other collection 

periods they can be recalculated by the methods in the Appendix. 

*The heat loss term qL is defined with respect to receiver rather than aperture area 

because conceptually this optimization procedure for the concentration ratio keeps the 

receiver fixed while varying the aperture. 

**See, for example, Reference 16. 



The steps of the calculation are entered iqto the worksheet in Table 6-3. First one 

computes the optical error derived in Section 2.0: 

+L + 'tracking odisplacement ' 

The coefficient h(0) represents the rim-angle-dependent contribution of longitudinal 

mirror errors to transverse beam spreading. It depends weakly on incidence angle and 

hence on tracking mode. h (8) is given by Eq. 2-21 and Table 2-1. However, some 

simplification is permissible because this term is so small that its precise variation with 

0 has negligible influence on the choice of optimal concentration ratio and rim angle. 

For the cases of greatest interest one can use the following approximation 

0 for norm1 incidence 

0 for polar mount all-day average 

0 for 0 < 45' for east-west tracking axis all-day average (6-3) 

0.1 for 80' 5 8 5 11 O0 for east-west tracking axis all-day average. 

Hence, for the present example the all-day average optical error is as follows: 

For the effective sun width the average asun,noon,average = 4.1 mrad over circumsolar 

scans 1 through 10 (see Table 4-1) which appears to be representative of typical sky 

conditions. It is to be enlarged according to Eq. 2-1 2 to account for time-of-day 

variation: 

<'SU>EW =m Osun,noon,average = 5.0 mrad. 

The resulting total rms beam spread is given by 

- 2 2 1/2 

<'tot>~W- (<'optical>EW+ <'sun)EW) 

= 8.0 mrad. 

The average critical intensity ratio is considered next: 
Table 6-3. OPTIMIZATION OF CONCENTRATION RATIO FOR PARABOLIC 

TROUGH WITH EAST-WEST AXIS: WORKSHEET 

Para meter Value Reference 

0.1 Eq. 6-3 

<boptical EW) 6.3 mrad Eq. 6-2 and 6-4 

<~o~>Ew 8.0 rnrad Eq. 6-6 

<x>~~ 4.37 Eq. 6-1 and 6-8 

<%t)EW <x>~~ = G(<otot>EW C) 35.0 mrad Eq. 6-9 

<~tot)Ew Co 218 mrad Fig. 5-la 

C0 27.3 Eq. 6-11 

When the necessary items from Tables 6-1 and 6-2 are entered, it is found that 

If in Fig. 5-la a straight line is drawn parallel to the abscissa (otot C axis), corresponding 

to the ordinate value 

this line intersects the 0 = 90' curve at 

otot C = 218 mrad . (6-1 0) 

This is the product of <otot> EW and the optimal concentration ratio Co for this 

collector. Thus the optimal concentration ratio which maximizes all-day efficiency is 

- - 218 mrad = 27-3 

'0 8 mrad 

The efficiency has a broad maximum about Co and it is rather insensitive to the exact 

value of Co. Therefore, a range of concentration values from 25 to 30 can be 

recommended for this collector. Note that throughout this paper the geometric 

concentration is defined as the ratio of aperture area over receiver surface area. For a 

parabolic trough of aperture width D and absorber tube diameter dabsorber, the 

concentration is therefore 
For an absorber tube diameter dabsorber = 2.5 cm, the optimal aperture width is 

= 'o dabsorber = 214 cm. 

6.5 CALCULATION OF INTERCEPT FACTOR AND OPERATING EFFICIENCY 

To calculate the efficiency it is important to know the intercept factor Y. It is defined 

as the fraction of the rays incident on the aperture that are intercepted by the receiver,* 

and depends on Oopticalt sun shape, and concentration ratio C. The intercept factor is 

plotted versus 1 /C for several values of cr,ptical and for several sun shapes in Figure 4- 

2. Of particular interest is Figure 4-26 for the effective average sun shape. The solid 

lines give the result for the real sun shapes while the dotted lines are based on a Gaussian 

sun with the same rms width as the real sun. The assumption of a Gaussian sun shape has 

the advantage of reducing Y to a function of the quantity (otot C), thus facilitating 

graphical representation, as in Figure 4-1. The closeness of the solid and dotted lines in 

Figure 4-2 in the region of interest, i.e., C 130, implies that the error in Y resulting from 

the Gaussian approximation is at most 1%. For example, if aoptical = 10 mrad, the exact 

result for C = 25 and for the average sun shape, as indicated by the solid line in 

Figure 4-2d, is 

The dotted line in Figure 4-2d (or equivalently Figure 4-la) corresponding to the Gaussian 

approximation yields 

These numbers for the intercept factor are derived from Figure 4-2d and hence represent 

average values. The instantaneous intercept factor for extremely clear sky conditions at 

solar noon can be obtained in the same manner from Figure 4-2a for the narrow sun shape 

(circumsolar scan number 1 of Table 4-1). 

In this section the efficiency 

*Yis defined as a purely geometric quantity without regard to absorption or shading. 

50 
is calculated for peak and for all-day average conditions. The steps of the calculation 

are entered in the worksheet of Table 6-4. The efficiency can be expressed in terms of 

(PTa), Y, X, and C by 

The product 7, = (P a) Y is called optical efficiency. To evaluate the all-day average 

operating efficiericy <v>, the parameters in Eq. 6-17 are based on the average sun shape 

and the average insolation level <Ib cos WEW = 665 w/rn2, as in Section 6.3. With the 

resulting values 

<Y>M = 0.965 (read from Figure 4-la, with <atot >C = 218 mrad) 

c0 = 27.3, and 

* the average efficiency is 

*The only effect which has not been included is the spillover of radiation from collector 

ends in relatively short collectors. This end effect is installation-dependent and 

negligible for well-designed large collector fields. For short collectors (for example, test 

modules without end reflectors) this effect must be included by multiplying the intercept 

factor Y in Eq. 6-17 by an additional factor 

where f = focal length, f= trough length, D = trough width, and 8 = incidence angle. This 

is discussed in Ref. 23. 
Table 6-4. CALCULATION OF EFFICIENCY FOR COLLECTOR 

OPTIMIZED ACCORDING TO TABLE 6-3: WORKSHEET 

-- 

Parameter Value Reference 

27.3 

0.70 

218 rnrad 

0.965 

4.37 

0.563 

Eq. 6-11 

Table 6-1 

Eq. 6-10 

Fig. 4-la 

Eq. 6-8 

Eq. 6-18 

'tot ,noon 

C atot,noon o 

Ynoon 

Xnoon 

(P ' a)noon 

qnoon 

6.85 mrad 

187 rnrad 

0.982 

3.40 

0.73 

0.63 

Fig. 4-la 

Eq. 6-20 

Table 6-1 

Eq. 6-21 

For extremely clear sky (narrow sun shape) and a typical peak insolation value Ib 

,p ak of 

865 w/m2, the results for this collector are 

~sun,narrow,noon = 2.7 mrad, from Table 4-1, data set #l; 

2 2 

Otot,noon = ('optical + 'sun,narraw,noon )'I2 = 6.85 rnrad; (6-19) 

'noon = 0.982, from Fig. 4-2a; 

and 

The peak efficiency is 

for this collector. 

This completes the calculations of optimal concentration ratio and operating efficiency. 

Next we consider the sensitivity of these results to changes in collector parameters and 

changes in operating conditions. 
6.6 SENSITIVITY OF OPTIMIZATION TO CHANGES IN COLLECTOR PARAMETERS 

Once a collector has been optimized for operation at a certain temperature, the 

deviation from optimal performance at different temperatures can be calculated. 

Optimization of concentration ratio is assumed for a heat loss parameter qL = 2000 

W/m 2 as above, and the efficiency of this collector at lower or higher temperatures 

corresponding to qL = 1000 w/m2 and qL = 3000 w/m2 is then calculated. 

These efficiencies are compared with those resulting from correct optimization for the 

new heat loss levels. Table 6-5 lists the respective efficiencies and concentration ratios 

for several values of the beam spread otot calculated by using the Gaussian 

approximation, 

The central column with q~ = 2000 w/m2 contains only one entry: I(C = ... ) = .... For 

the other two heat loss levels the top line lists for C as optimized at q~ = 2000 w/m2, 

and the bottom line lists v(Cf) with C1 optimized for the new heat loss level. 

Again the optimum is rather broad. For example, if a concentrator with aoptical = 10 

mrad is optimized for operation at a temperature corresponding to a heat loss qL = 2000 

w/m2, it will perform with q= 0.6109 at half the heat loss. If it is optimized for 

2 operation at qL = 1000 W/m instead, the efficiency will be t) = 0.6 156, a gain of only 

half a percentage point. Therefore, it makes sense to market a single collector for a 

fairly wide range of operating temperatures and insolation levels rather than to try to 

optimize for each new application. 

Table 6-5. SENSITIVITY OF OPTI[MIZATION TO CHANGE IN 

HEAT LOSS PARAMETER q~ 

6.7 SENSITIVITY TO CIRCUMSOLAR RADIATION 

The variation in efficiency with changing levels of circumsolar radiation can be 

evaluated by means of Figure 4-2. Only the inter,cept factor needs to be considered. 

Some examples are listed in Table 6-6. For each value of aoptical and C, three numbers 
Table 6-6. SENSITIVITY TO CIRCUMSOLAR RADIATION, EXACT CALCULATION 

AND GAUSSIAN APPROXIMATION. INTERCEPT FACTOR Y AS 

FUNCTION OF UOPnCAL AND C (FOR (D = 

- -- - - --- . ..- -- 

Concentration Ratio C 

Oop ti cal 10 25 40 

a~ach entry contains 

for the average over circumsolar scans 1 through 10, for scan 1, and for scan 11 of 

Table 4-1. 
are given (plus three more in parentheses). The first states the intercept factor averaged 

over circumsolar scans 1 through 10 which seems to correspond to normal operating 

conditions. (The scan numbers refer to the standard circumsolar profiles in Table 4-1.) 

The second entry corresponds to the narrow sun shape, scan 1, and the last entry to the 

wide sun shape, scan 11. The adjacent numbers in parentheses are obtained with the 

Gaussian approximation for the sun shape. 

The following conclusions can be drawn. For concentration below 10 or for large optical 

errors, aoptical- > 20 mrad, the collector performance is practically insensitive to 

circumsolar radiation under normal operating conditions, and the Gaussian approximation 

is quite acceptable for calculating the intercept factor. For the important case of 

concentrations around 25 and optical errors in the 5- to 10-mrad range, the intercept 

factor variation appears to be fairly small, about - + I%, under normal conditions. 

For the wider circumsolar scans the intercept decreases significantly, for example, by 

4% for scan 11, but whether this is important in practice can only be decided when more 

information becomes available about the frequency distribution of the standard 

circurnsolar profiles listed in Table 4-1. 

6.8 OPERATION WITH NORTH-SOUTH AXIS 

Calculation of yearly energy delivery [20,21,22] shows that in midlatitudes (h z 35') an 

aperture tracking about the horizontal north-south axis receives approximately 15% more 

energy than one tracking about the east-west axis. Polar axis tracking approaches within 

4% the radiation availability of a two-axis tracker, surpassing the horizontal east-west 

axis by about 30%. Despite its higher collection potential, the polar axis mount generally 

is believed to be impractical for large installations because of problems with wind 

loading and plumbing. Polar mount may, however, be desirable for small installations 

with relatively short collector modules, for example, for home heating. 

The horizontal north-south axis suffers from large seasonal variation in output, resulting 

from variation not only of insolation but also of optical efficiency at low incidence 

angles. In order to get a quantitative assessment of these effects, it is useful to evaluate 

the performance of the collector discussed above if it is operated with a horizontal 

north-south instead of east-west axis. For its concentration ratio Co = 27.3 is assumed, 

as optimized for east-west orientation. Table 6-7 lists all the steps of the calculations 

for the winter and summer solstices, in addition to the calculation for equinox. 

The second to the last line of Table 6-7 shows that the intercept factor is fairly constant 

around 0.97 during spring and summer but drops to 0.926 at the winter solstice. This 

factor, coupled with a decreased (P a) product and lower beam insolation per aperture, 

leads to a significant drop in average operating efficiency from about 0.60 during the 

summer period to 0.50 at the winter solstice. Whether such low efficiency at a period of 

low available insolation is acceptable depends on the load profile for each particular 

application. This situation does not change if the collector is optimized specifically for 

north-south orientation rather than for east-west orientation as we have assumed. 

Repeating the optimization procedure for the north-south orientation (see Table 6-4) it is 
found that the optimal concentration ratios differ only insignificantly: 26.6 for winter, 

27.5 for equinox, and 27.7 for summer solstice. Therefore, a single concentration ratio in 

the range of 25 to 30 is optimal regardless of the orientation of the tracking axis. 

Table 6-7. CALCULATION OF EFFICIENCY OF COLLECTOR WITH C = 27.3 

(OPTIMIZED FOR EAST-WEST AXIS) IF OPERATED WLTH 

HORIZONTAL NORTH-SOUTH AXIS 

Parameter Value Reference 

Winter Equinox Summer 

Table 6-1 

Table 6-2 

Eq. 2-21 

Eq. 6-2 

Eq. 6-6 

Table 6-2 

Table 6-2 

Eq. 6-7 

Fig. 5-la 

Fig. 4-la 
SECTION 7.0 

REFERENCES 

Biggs, F.; Vittitoe, C. N. The Helios Model for the Optical Behavior of Reflecting 

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

Schrenk, G. M. Analysis of Solar Reflectors: Mat hem atical Theory and Methodology 

for Simulation of Real Reflectors. December 1963; GMC-AO-EDR3693. 

Treadwell, G. W. Design Considerations for Parabolic Cylindrical Solar Collectors. 

Albuquerque, NM: Sandia Labs.; 1976; SAND 76-0082. 

Cramer, H. Mathematical Methods of Statistics. Princeton, NJ: Princeton 

University Press; 1947. 

Grether, D.; Hunt, A. Description of the LBL Reduced Data Base. Berkeley, CA: 

Lawrence Berkeley Laboratory; August 9, 1977. 

Collares-Pereira, M.; Rabl, A. "The Average Distribution of Solar Radiation: 

Correlations between Diffuse and Hemispherical and between Daily and Hourly 

Insolation Values." Solar Energy. Vol. 22: pp. 155-164; 1979. 

Rabl, A. "Comparison of Solar Concentrators." Solar Energy. Vol. 18: p. 93; 1976. 

Pettit, R. B. "Characterization of the Reflected Beam Profile of Solar Mirror 

Materials." Solar Energy. Vol. 19: p. 733; 1977. 

Butler, B. L.; Pettit, R. B. rTOptical Evaluation Techniques for Reflecting Solar 

Concentrators." Society of Photooptical Instrumentation Engineers (SPIE). 

Vol. 114, Optics Applied to Solar Energy Conversion: p. 43; 1977. 

Rabl, A. "A Note on the Optics of Glass TubesOTT Solar Energy. Vol. 19: p. 215; 

1977. 

Gret her, D. F.; Hunt, A.; Wahlig, M. llCircurnsolar Radiation: Correlations with 

Solar Radiation." Berkeley, CA: Lawrence Berkeley Labs.; Oct. 11, 1977. 

Treadwell, G. (Sandia Labs.) Private communication. 

Dudley, V. E.; Workhoven, R. M. Summary Report: Concentrating Solar Collector 

Test Results, Collector Module Test Facility. Albuquerque, NM: Sandia Labs.; May 

1978; SAND 78-0815. 

Duffie, J. A.; Beckman, W. A. Solar Energy Thermal Processes. New York, NY: 

John Wiley & Sons; 1974. 

Kreith, F.; Kreider, J. F. Principles of Solar Engineering. New York, NY: 

McGraw-Hill; 1978. 

Kreith, F. Principles of Heat Transfer. New York, NY: Intext Educational 

Publishers; 1973. 
17. Pettit, R. B.; Freese, J. M.; and Arvizu, D. E. "Specular Reflectance Loss of Solar 

Mirrors Due to Dirt Accumulation." Presented at IES "Testing Solar Energy 

Materials and Systems" Seminar; May 22-26, 1978. Washington, DC: National 

Bureau of Standards. 

18. Freese, J. M. Effects of Outdoor Exposure on the Solar Reflectance Properties of 

Silvered Glass Mirrors. Albuquerque, NM: Sandia Labs.; 1978; SAND 78-1649. 

19. Tabor, H. "Stationary Mirror Systems for Solar Collectors." Solar Energy. Vol. 2: 

p. 27; 1958. 

20. Treadwell, G. W. Low-temperature Performance Comparisons of Parabolic-trough 

and Flat-plate Collectors Based on Typical Meteorological Year. Albuquerque, 

NM: Sandia Labs.; 1978; SAND 78-0965. 

21. Dickinson, W. C. "Annual Available Radiation for Fixed and Tracking Collectors." 

Solar Energy. Vol. 21: p. 249; 1978. 

22. Collares-Pereira, M.; Rabl, A. Simple Procedure for Predicting Long-term Average 

Performance of Nonconcentrating and of Concentrating Solar Collectors. Argonne, 

IL: Argonne National Lab.; ANL-78-67. To be published in Solar Energy. 

23. Gaul, H. W.; Rabl, A. "Incidence Angle Modifier and Average Optical Efficiency of 

Parabolic Trough C~Uectors.~' Golden, CO: Solar Energy Research Institute; 1979; 

SERI/TR-34-246R. To be published in ASME Journal of Solar Engineering. Vol. 1; 

1980. 
APPENDIX 

TYPICAL INSOLATION VALUES 

FOR USE IN OPTIMIZATION 

Optimization of concentrating collectors requires the input of the available solar 

radiation. Since a collector should perform well for a wide range of weather conditions, 

an exact optimization involves integration of collector output over insolation data for a 

long time period. Exact determination of rim angle and concentration ratio, however, is 

not necessary because the optimum is rather broad. Therefore, one can base the 

optimization on a single insolation value and sun shape, representing the average over 

typical operating conditions; i.e ., the central eight to ten hours of a clear day. 

The difference between such a simplified procedure and an exact optimization involves 

the replacement of the average of a product by the product of averages. A schematic 

example shows why such a procedure should be an excellent approximation in many solar 

energy calculations. Two sinusoidally varying terms are considered: 

and 

The time average of the product yly2 over one period is 

The difference between <y1y2> and <ylxOr%> is at most (bl b2)/(ala2). For example, if the 

relative magnitude of the oscillations about the average is bl/al = 0.1 = b2/a2, then the 

difference is at most 1%. If the frequencies differ, if the terms are out of phase, or if bl 

and b2 represent random fluctuations uncorrelated with each other, the difference will 

be very small or vanish altogether. 

One could expand the exact long-term average qnet exact into a sum over multiple 

products similar to Eq. A-3. Some of the time variations of insolation and temperature 

are periodic (daily, yearly) while others are nearly random (clouds). The amplitudes of 

the variations are sufficiently small, typically less than 20%, and the terms are 

sufficiently uncorrelated to justify the approximation 
To determine the appropriate average level of solar radiation, the Liu and Jordan 

correlations in the improved version recently published can be used [6]. The fundamental 

correlation parameter of this insolation model is the clearness index Kh, defined as the 

ratio of terrestrial (Hh) over extraterrestrial (H,,) daily hemispherical irradiation on a 

horizontal surface: 

It is convenient to express time-of-day t and sunset time ts as hour angles in radians 

from solar noon: 

2 nt, 

0 = T 

= arccos (-tan 6 tan 1) , S 

where T = length of day = 86,400 s, A = latitude, and 8= solar declination. The conversion 

between average irradiance I (w/rn2) and daily irradiation H (~/rn~) is given by 

for the hemispherical component, and by 

for the diffuse component, all measured on the horizontal surface. The factors rd and rh 

have been determined empirically. They are functions of time-of-day wand of time of 

year (through ws) and are well represented by the following expressions: 

- ( '0'"-"S" ) 

'd - sin w, - us cos us 

and 

rh = (a + b cos u)rd, 

with 

a = 0.409 + 0.5016 sin(ws - 1.047) 

and 

b = 0.6609 - 0.4767 sin@ - 1.047). 
The beam irradiance at normal incidence is given by 

Ib = 1 (Ih -Id) , cos q., 

where 

cos eh = cos h cos 6 (COS w - cos ws) (A- 1 3) 

is the cosine of the incidence angle on the horizontal surface. In terms of Kh and Ho, Eq. 

A-12 can be written as 

a form which is useful because the ratio Hd/Hh of diffuse over hemispherical irradiation 

is well correlated with Kh (see Eq. 6 and Figure la of Reference 6). The extraterrestrial 

irradiation Ho is given by 

H, = T I. cos 1 cos 6 (sin us - ws cos. w,), 5'T (A- 1 5) 

where I, = solar constant = 1353 w/m2. This formula neglects the seasonal variation 

(+3%) - of the effective solar constant because only the year-round average is of interest. 

Combining Eqs. A-9 and A-13 with Eq. A-15, most terms cancel, resulting in the simple 

expression 

The analogous result for diffuse irradiance is 

Id = cos 4 (? I$, 1,) . 

(A- 16) 

For present purposes Id can be assumed to be isotropic and the brightness of ground and 

sky can be equated; hence, Eq. A-17 gives the diffuse irradiance on both the horizontal 

surface and the collector aperture for all tracking modes. 

The yearly irradiation average is closely represented by its value at equinox, when 

6 = 0 = cos ws, 

a = 0.6598, and 

b = 0.4226. 

(A- 18) 
Since concentrating collectors will operate only during periods of fairly high insolation, it 

is appropriate to take 

Kh = 0.75, (A- 1 9) 

corresponding to clear days. The associated ratio of diffuse to hemispherical irradiation 

is 

At equinox the incidence angle 8 on a collector with an east-west tracking axis equals 

the hour angle o; hence, the product of beam irradiance and the cosine factor is 

Ib(u) cos o = cos w (0.4298 + 0.4226 cos w) I, Kh. (A-2 1) 

At solar noon,8 = o = 0 and we find a typical peak beam irradiance of 

Averaging Eq. A-21 over an operating period 2tc yields 

<Ib COS B)w dw $.,(a) COS~ 

with 

The result is 

with 

sin o 

(coso) = C 

Wc 

and 
which are listed in Table A-l for various values of we. 

Table A-1. AVERAGE OF COSINE FACTORS FOR COLLECTOR WITH 

EAST-WEST TRACKING AXIS 

wc (degrees) 

sin o C 

<cos w> t = C 

1 cos o sin o 2 <COS 0) = - + C C 0.914 0.818 0.707 2 0.596 0.500 

C Oc 

Taking tc = 4 h as a typical cutoff time, 

is the average clear-day beam insolation per aperture area of a collector with an eastwest tracking axis. This is not to be taken as an exact number; a range from 630 to 700 

w/m2 is reasonable. 

The analogous results for the diffuse irradiance at equinox are 

and 

On a clear day, for ?, = 35', this becomes 

'd,noon = 190 w/m2 
and 

For a polar-mounted tracking collector the incidence angle 8 equals the declination 8 at 

all times. To obtain an average yearly value of the beam insolation, Eq. A-16 is 

evaluated at equinox but cos 8 is replaced by its year-round average 

With a cutoff time t, = 4 h and Kh = 0.75, this yields 
Optical Analysis and Optimization of Line Focus 

Solar Collectors 

1 September 1979 I 

3. Recipient's Accession No. 

5. Publication Date 

Document Controi 2. NTJS Accession No. 

Page I 

a1 Title and Subtitle 

1. sER1 Report No. 

TR-34-092 

7 Author(s) 

P. Bendt, A. Rabl, H. 14, Gaul, K. A. Reed 

9. Performing Organization Name and Address 

Solar Energy Research Institute 

1536 Cole Blvd. 

Golden, Colorado 80401 

1 Technical Report I 

8. Performing Organization Rept. No. 

10. ProjectlTaskiWork Un~t No. 

3432.30 

11. contract (C) or Grant (G) No. 

(C) 

12. Sponsaring Organization Name and Address 

i 

15. Supplementary Notes 

1 

13. Type of Report & Period Covered 

16. Abstract ILimlt: 200 words) The optical analysis of a solar concentrator is usually carried 

out by means of a computer ray tracing, a microscopic method that provides an enormous amount of detailed information but obscures functional relationships, This paper 

describes a macroscopic approach that yields all the parameters needed for the optica 

design of line focus parabolic troughs in closed analytical form, requiring only mini 

ma1 computation. The goal of the optical analysis is to determine the flux at the 

receiver as a function of concentrator configuration, receiver size, width of sun, an 

optical errors (e.g., tracking, reflector contour). All causes of image spreading are 

quantified as angular standard deviation. Ray tracing with a real reflector and a rea 

sun is shown to be equivalent to convoluting the angular acceptance function of a per 

fect concentrator with an effective radiation source. This effective source, in turn, 

of the sun. The problem is reduced to two dimensions by projecting the threemotion of the sun on the plane normal to the tracking axis. In this frame 

is obtained by convoluting the distribution function of optical errors with the angular 

width of the sun increases as l/cos €I with incidence angle 8. A formula 

graphical procedure are provided for finding the optimal geometric conmaximizing net power output. In the last section, which is written 

s a self-contained users guide, the results are illustrated by specific examples. 

I 

17. Document Analysis 

a. Descr~ptors Line Focus Collectors ; Solar Collectors ; Solar Concentrators ; Optical 

Microscopy; Radiation Flux; Gauss Function; Sun; Optimization; Concentration 

Ratio; Concentrating Collectors 

b. Identifiers/Open-Ended Terms 

c. UC Categories 

UC-62a, 62b, 62e 

1 I I 

F.~rr3 NO. 8200-t3 (6-79i 

'U. 5. GOVERNMENT PRINTING OFFICE: 1979-0-677-175/76 

18. Availability Statement 

NTIS, U. S. Dept. of Commerce 

5285 Port Royal Road 

79. NG. of Pages 

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