Solar Energy, Vol. 54, No. 2, pp. 115-123, 1995 Copyright 0 1995 Elsevier Science Ltd Printed in the USA. All rights reserved

0038-092x/95 $9.50 + .lO

0038-092X(94)00107-3

CONCENTRATION DISTRIBUTIONS IN CYLINDRICAL RECEIVER / PARABOLOIDAL DISH

CONCENTRATOR SYSTEMS

PETER D. JONES and LILI WANG Auburn University, Auburn, AL 36849-5341, U.S.A.

Abstract-Concentration distributions on a cylindrical receiver in a paraboloidal dish concentrator are computed for space applications (no atmosphere). A geometric optics method is applied which integrates over the solar disk and the concentrator projected surface, and maps analytically, in implicit closed form, through the concentrator and onto the receiver. Finite sunshape, concentrator surface errors, and pointing system zero-mean and constant offset errors are considered. Results define the section of the receiver surface which receives the majority of the concentrated flux, where the receiver’s aperture might be located. Results are given in terms of concentrator geometry, concentrator total system error tolerance, receiver geometry, and pointing offset error. In cases with pointing offset error (nonzero mean pointing error), circumferentially varying concentration distributions are shown.

1. INTRODUCXON

A commonly considered configuration for space-based solar power systems is a paraboloidal dish concentrator with a cylindrical receiver housing aligned with the axis of the paraboloid (Fig. 1). This configuration leaves the edges of the dish free and allows the con- centrator surface to be deployable, such as in the sun- flower concept (which was studied by NASA in the 1960s and is receiving renewed attention). This con- figuration has also been suggested by Overfelt et al. ( 1993) for use in a satellite recrystallization furnace, in which the cylindrical receiver geometry is dictated by a materials processing concept. In either a power system or a materials processing furnace application, the internal design of the receiver is dictated by the distribution of solar concentration over its cylindrical surface. This distribution is governed by: (a) the overall geometry of the paraboloid/cylinder system; (b) zero- mean error allowances for the slope and specularity of the concentrator surface, and for the sun-following pointing system; and (c) allowances for nonzero-mean errors (offset) in the pointing system. The goal of this paper is to determine the magnitude and distribution of concentration on a cylindrical receiver on the axis of a paraboloidal dish concentrator subject to these three influences, and to provide design information for selection of the size and location of an active ap- erture on the receiver.

The purpose of this work is to support simulation and design activities for the satellite recrystallization furnace concept referenced above, and also described by Lan et al. ( 1994). This concept involves melting of amorphous semiconductor material contained within a cylindrical ampoule, and recrystallization with a large crystal size. Processing in space reduces the effect of free convection in the melt, which otherwise limits attainable crystal size. Direct solar concentration pro- vides the rather significant processing power. Crystal

growth simulation studies of the melt inside the cylin- drical ampoule (receiver) require detailed heat transfer and fluid flow computations, and a packaged code is chosen for this application. The packaged code is ap- pended with a dedicated solar concentration subpro- gram for heat input, which is the subject of the present paper. This concentration subprogram must be able to compute instantaneous solar concentrations in order to provide continuous input to the surface of the cy- lindrical ampoule, in order to simulate the effects of the time-varying attitude of the recrystallization furnace during its orbital flight. Therefore, in order to optimize running time and code interfacing issues, this subpro- gram is based on an analysis which is dedicated to the geometry at hand, and which carries analysis as far as possible before resorting to discretization for a final numerical solution.

A geometric optics formulation is used in the pres- ent analysis, with numerical integration over the pro- jected areas ofboth the sun and the concentrator. Bun- dles of rays of solar heat flux from each solar element, in turn, are made incident with a unique incidence angle on each of the concentrator elements. The re- flection from each concentrator element is mapped analytically onto the surface of the cylindrical receiver. Tracing each bundle of rays in this manner allows shadowing and interference effects to be dealt with di- rectly. This technique can be reasonably applied be- cause the simple receiver geometry allows a closed form mapping from the concentrator to the receiver. For general receiver shapes, the analytically more complex technique of mapping the complete solar image through the concentrator might be necessary (Evans, 1977; Romero, 1992).

In the following analysis, concentration on the re- ceiver surface is first calculated under the idealized conditions of collimated solar incidence, aligned with the axis of a perfect-surface concentrator. These results for an idealized sun and a perfect concentrator serve

115

116 P. D. JONES and L. WANG

concentrator

Fig. 1. General appearance of a paraboloidal dish/cylindrical receiver concentrator/receiver arrangement.

as a simple benchmark for analysis of more complex cases. An arbitrary incidence angle is then used to de- rive a general mapping of solar rays through the con- centrator and onto the receiver surface. This mapping is numerically integrated over the concentrator pro- jection and over a finite solar disk aligned with the concentrator/receiver axis to determine results for a more realistic sun and a perfect concentrator. Gaussian errors in the concentrator surface and pointing system are then included by convolving the solar disk shape (brightness distribution) with the concentrator error distribution to determine an effective sunshape, which provides results for a more realistic sun and a more realistic concentrator. Pointing offset (nonzero mean pointing error) effects are then determined by varying the location of this effective sunshape.

The general character of the results may be de- scribed as low concentration values over most of the receiver, with a band of high concentration near the level of the parabolic focal point. For zero pointing offset, the distribution is axisymmetric. This distribu- tion may be summarized by the axial location and width of the concentration band, and the mean con- centration within that band. The dependence on con- centrator/receiver design and operating parameters of the location, width, and mean strength of the concen- tration band is shown. For nonzero pointing offset, these results are asymmetric, and are shown as func- tions of the receiver angular coordinate. From this in- formation, the size and location of the heat-receiving aperture in a cylindrical receiver design may be spec- ified.

2. ANALYSIS

2.1 Collimated aligned incidence The geometry of a collimated bundle of rays with

an incidence angle parallel to the paraboloid/cylinder axis is shown in Fig. 2, where the concentrator is a perfect paraboloid of revolution described by z = ar* , with its focus on the axis at/= t a, a maximum radius D/2, and a reflectivity of unity. The receiver is a cyl- inder of radius r, and height h . The incident collimated heat flux which falls on the normal projection of an element of concentrator area is reflected toward a par- ticular element of receiver surface area, as shown in Fig. 3. Considering the one-to-one relationship of con- centrator points to receiver points inherent to the per- fectly paraboloidal concentrator under perfectly aligned collimated incidence, the concentration is defined

rd4dr co = - r, dh dz, ’

where C, is defined as the ratio between the heat flux incident on the receiver surface and the collimated heat flux incident on the projected concentrator area. It is clear by symmetry that dd, = d&. The axial location z, that the ray strikes on the cylinder is determined by algebraically finding the intersection with r, (see Fig. 2) on a straight line from a concentrator point (r, 4) to the focal point:

r - r, z, = - + arr,.

4ar (2)

Differentiating eqn (2) and solving eqn ( 1) yields the concentration:

cdzC) = (zarc)3 1 {4az, - 1 + [16a*(zz + r,‘)

- 8az, + l]‘/*}; ar,2 < z, 5 z,

C,(z,) = 0; zc > z0, (3)

where z, is the highest point on the cylinder which can be reached by reflected flux:

z, = $ (1 + g [(aD)’ - I]). (4) .

Note that the only variable in eqns (3) and (4) is the receiver coordinate z, . Figure 4 shows some results from eqns ( 3) and (4) for different rim angles, (Y, where a = r/2 + tan-‘[(aD - aD-‘)/2]. Figure 4 suggests that a rim angle of 90” gives the highest concentration peak, as is also concluded by Riveros and Oliva ( 1986 ) . Figure 4 shows a discontinuity in C,(zJ at z, = z,. This results from the assumption in this section of per- fectly aligned collimated incidence on a perfect con- centrator surface, an assumption which is progressively relaxed in the following sections.

focalpoint

Fig. 2. Coordinate system for analysis of a paraboloidal dish/ cylindrical receiver concentrator/receiver system.

Concentration distributions 117

incident collimated heat flux

of the collector

I element of the receiver corresponding to

i focal point the element of I I the collector I I I I

Fig. 3. Concentrator/receiver differential area elements for eqn ( 1).

2.2 Collimated misaligned incidence In treating incident rays which are not aligned with

the paraboloid/cylinder axis (because they originate from a point displaced from the center of the sun and/or there exists a pointing offset error) the direction of the reflected ray from the concentrator is not known a priori as in the aligned case. Figure 5 shows the ge- ometry for an incident ray misaligned by 8, where 0 is defined to lie in the 4 = 0 plane. Resolution of the reflected direction of a ray striking the concentrator at a point (r, 4) using Snell’s laws (Z-i = -Z-i, ii X F = fi X ;) yields:

r, =

rm =

r, =

r& + r& + r,&

(4a2r2 - 1 )sin e cos do - 4ar cos e 4a2r2 + 1

sin e sin 4

cos e + 2ar(r, - sin e cos 4) (5)

(Note that rr, r9, and r, are dimensionless, and rf + r: + ri = 1.) This reflected direction may or may not intersect the receiver surface. If r, > r 1 sin 4 1 and (h -ar2)tanB+r,lcosq51 >rlcos~#~I forr>r,andti < rr / 2, then the ray will strike the cylinder directly (in other words, (z,, &) for this ( r, 4) lies in the shadow ofthe cylinder). If (r+/q) > [rJ(r* - rf)“‘], the re- flected ray will not intersect the receiver [or, there is no possible (z,, 4,) for this (r, +)] and will eventually

reflect out of the concentrator. If the reflected ray does intersect the receiver, it does so after travelling a dis- tance b from the reflection point, where

b=- rr, + Vrlrfrf + ri(rf - r2)

rf + ri (6)

The intersection point on the receiver is then defined by

120 r 0” a0

El ._ 5 E $40 8

c

0 ( 1.6

z, = ar2 + br,

(7)

D/d=1 0

0.9 1.0

receiver axial location, 2,/f

Fig. 4. Concentration distribution due to collimated, aligned incidence on a perfect concentrator for a range of rim angle,

at Dfd = 10.

118 P. D. JONFS and L. WANG

~definedin+Oplane ’ r

Fig. 5. Definition of ray directions for reflections from mis- aligned incidence.

Equation (7) is the misaligned equivalent of eqn (2), mapping a concentrator point to a receiver point. The concentration at each receiver point for misaligned incidence is defined by the differential areas on the concentrator projection and on the receiver,

C’ = r cos Bdqbdr

rCd&dzC ’ (8)

which may be written in terms of functional derivatives by expressing d& and dzC as total derivatives of the orthogonal coordinates d4 and dr:

C’(zc, &, 8) = rcosf3

r 1 * 84, az, ~ ak az, 1 (9)

’ a+ ar ar a+ The partial derivatives in eqn (9) may be derived

from eqns (5), (6), and (7) in a straightforward man- ner, and the final expressions are listed in the Appendix. The partial derivatives in eqn (9) are functions of r, 4, and 0, and due to the form of eqns (5), (6), and (7), it is not possible to explicitly invert z,( r, q5,O) and &(r, q5, 0). Therefore, calculating the concentration for any point (z,, 4,) on the receiver from incidence misaligned by 19 involves a search over r and 4. This search is aided by noting that z, is predominantly a function of r, while 4, is predominantly a function of 4. The search must also check the nonintersection cri- teria. Results of concentration for misaligned colli- mated incidence are similar in character to the results of Fig. 4, although the peak concentration varies slightly with &, and the z, location of the peak concentration varies significantly with 4, (i.e., the concentration band becomes skewed in zC). Equation (9) provides the concentration at any point (z,, 4,) [defined by eqn (7)] for collimated incidence misaligned by 8.

Equation (9)) with the relations in the text and ap- pendix defining its terms, represents an implicit ana- lytical expression for the concentration distribution on a cylindrical receiver due to collimated incidence from an arbitrary direction on a segment of a paraboloidal

concentrator. Final application of this expression re- quires only integration over a representative sunshape and over the concentrator projected surface, as shown in the following sections. Gaussian surface and pointing errors may be incorporated into the effective sunshape. The final integration is the only part of the process which, practically speaking, must be discretized for numerical solution (note that surface curvature effects are included analytically, and need not be discretized) . The foregoing method of analysis should be applicable to other regularly defined concentrator/receiver ge- ometries, and should yield similar results. For instance, the numerator in eqn ( 8 ) is similar to the “first integral of the flux density” derived by Jeter ( 1986, 1987) for a parabolic trough case, from which a closed form con- centration ratio is derived.

2.3 Solar incidence Solar incidence (as opposed to collimated inci-

dence) implies a range of incidence angles dictated by the finite size of the sun and a variation in solar inten- sity over the disk of the sun; the combination of these factors is known as the sunshape, and is approximated to be axisymmetric. The sunshape is defined by the normalized solar intensity distribution s(p) and the half angle subtended by the solar disk /3,, subject to the condition

Zr 80 ss &Y)sin Pdj3d$ = 1, (10) 0 0

where (8, fi) defines a point on the solar disk, in spherical coordinates from an origin on the concentrator/n3 ceiver, and the $ = 0 reference is in the plane of the C$ = 0 reference. Each point (8, #) on the solar disk may be considered to be a source of collimated inten- sity, with a unique angle of misalignment. The total concentration at a point (z,, 9,) is found by integrating the concentration from each point on the solar disk,

~:(r,V,8),PlsinBdBd~. (11)

Note that since the local misalignment angle 0 is defined in the C$ = 0 plane, the auxiliary variables 4’ = C#J - ti and c#& = & - # must be defined. Equation ( 11) ex- presses the total concentration on a point ( z, , I$,) from a sunshape defined by ,!?(/I) and @,, centered about the concentrator/receiver axis. Since the system of the solar disk and the concentrator/receiver is axisym- metric, eqn ( 11) gives the same result for all 4,.

The procedure for integrating eqn ( 11) is straight- forward and consists of two nested loops. The first loop is over the solar disk. Incident solar heat flux from each element in a mesh over the solar disk is considered to be collimated, at an incidence angle defined by the element location (and by the gross solar misalignment

Concentration distributions 119

for computations in the following section), and at a strength defined by the integral of the sunshape over the element. The second loop is over the concentrator’s projected surface, and consists of a regular two-di- mensional mesh in r and 4. The receiver surface is discretized into a regular mesh in z, and 4,. For each concentrator element, the receiver concentration magnitude and location are calculated from eqns (7) and (9)) and this concentration is distributed into the appropriate receiver mesh element. As the second loop is closed, the concentration distributions over the entire receiver due to each concentrator element in turn are summed, yielding the receiver concentration distri- bution due to the entire concentrator surface for one solar element. This procedure is completed for each solar mesh element, closing the first loop, and the re- sultant receiver concentration distributions due to each solar element are summed to arrive at the total receiver concentration distribution. In order to verify this pro- cedure, comparison is made to eqn (3) which is an analytical solution defining the limit of eqn ( 11) as @, + 0. It is found that the results are most sensitive to the solar mesh size and the regular mesh in r, and fairly insensitive to the mesh in 4. A solar mesh of 64 elements (evenly distributed to maintain similar ele- mental aspect ratio), 200 points in r, and 10 points in d, are found necessary for results within 1% of eqn ( 3 ) over the practical range of concentrator and receiver parameters. The mesh in z, and 4, is chosen simply to yield smooth results. The procedure is also exercised for variable mesh sizes for fiO # 0 (and for gross solar misalignment as described in the following section), and shows that the above finite mesh sizes are sufficient for results within 1% of asymptotic values.

The general character of concentration results for solar incidence differs from collimated incidence results in that the very sharp concentration peak is smoothed, and instead the sharp drop at z, is more gradual, as shown by comparison of the collimated and usY = 0 curves in Fig. 6. Concentration results from eqn ( 1 I ) show little sensitivity to which particular normalized solar intensity model is chosen. Concentration distri- bution results for a uniform solar disk (Jeter, 1986),

receiver axial location, 2,/f

Fig. 6. Concentration distribution due to solar incidence with zero-mean pointing error for a range of concentrator system error standard deviation, at D/d = i0 and a rim angle df90”.

Distribution for collimated incidence is also shown.

m9 = 0; B > PO, (12)

where &, = 4.65 mrad, an empirical solar disk model (Jefferies, 19851,

s(p) = !$? o [I -0.342(# @<PO

m4 = 0; P > PO, (13)

and a Gaussian model (Harris and Duff, 198 1))

S(P) = &w(T) t (14)

vary by only about 3%. The uniform model generally gives higher concentrations, the empirical model gives lower values, and the Gaussian model is in between. Note that in eqn ( 14) all /I < ?r are considered. A rea- sonable value for u, is 2.5 mrad (Harris and Duff, 198 1; Giiven and Bannerot, 1986).

2.4 Solar incidence with concentrator errors The degree of perfection of the concentrator surface,

apart from its reflectivity, is expressed by errors in sur- face slope and specularity. Another significant error source is the accuracy of the pointing system. These may all be regarded as zero-mean errors with Gaussian probability density functions (Romero, 1992):

tan’/3 p(B)=&exp --F ,

[ I (15)

v SY

where and cf = c$,,,~ + g& + c&int represents the total system error. Treadwell and Grandjean ( 1982) suggest that usy between 4 and 12 mrad can be achieved.

Collimated intensity from a point source which is incident on a perfect reflector will be reflected into a single direction defined by Snell’s laws. Collimated in- tensity incident on a reflector with a zero-mean surface error distribution will be reflected into a solid angle cone about the direction defined by Snell’s laws, dis- tributed according to eqn ( 15). Distributed source in- tensity, with an angular distribution defined by the sunshape, incident on a perfect reflector, will also be reflected into a solid cone about the direction defined by Snell’s laws and distributed according to the sun- shape [e.g., eqns (12), (13), or (14)]. Distributed source intensity incident on a reflector with zero-mean surface errors, the case considered here, will be reflected into a cone about the direction defined by Snell’s laws, distributed according to the convolution of the sun- shape and the error probability density function,

F(p) = L” c g(w)p(y)sin odwd(, (16)

120 P. D. JONES and L. WANG

where cos y = cos /3 cos w + sin /3 sin w cos( $J - 5). Equation ( 16 ) gives the same result for all 9. The con- volution of eqn ( 16) is known as the effective sunshape. The reflected cone from the effective sunshape incident on a perfect reflector is equivalent to the reflected cone of the normalized solar intensity incident on a reflector with zero-mean surface errors. Therefore, substituting F( 8) for s( /3) in eqn ( 11) results in the concentration for any point (z,, 4,) for a real sun incident on a re- flector with zero-mean surface errors. .

Calculations using effective sunshapes derived from constant, measured, and Gaussian sunshapes exhibit a concentration difference of only about 3% for small crW, and a smaller difference for larger usY (as also ob- served by Nicolas, 1987). Therefore, because of the convenience of the convolution, a Gaussian sunshape is employed in the following analysis, where

tan*/3 F(B)=& - exp - 2aHm ’ 1 I (17) S”l-lI

and a&,,, = al + u& . Concentration distributions for a concentrator with a 90” rim angle, for collimated incidence, solar incidence without errors, and solar in- cidence with a range of surface error levels are shown in Fig. 6. Distributions at both higher and lower rim angles show a more pronounced effect of surface errors.

2.5 Solar incidence with concentrator errors and pointing offset

Solar concentration with pointing offset was inves- tigated by Baz et al. ( 1984) and Treadwell and Grand- jean ( 1982), who noted that a large degradation in receiver flux results from only a very small offset angle ( - lo ) . Concentration effects due to a pointing offset 8 (to the center of the solar disk) are calculated in the present analysis by varying the local misalignment an- gles, so that

C(z,, &, 8) = i2- s,” WW’[zdr, do’, 0,

4l-(r, do’, e’), Usin BdLW, (18)

where cos 0’ = cos 8 cos /3 + sin 8 sin B cos( r - +). Equation ( 18) is numerically integrated in the same

manner as eqn ( 11). Pointing offset results in a con- centration distribution which is asymmetric in 4,. The face of the receiver which is tilted away from the sun receives a higher concentration than in the zero offset case, the face titled toward the sun also receives a slightly higher concentration, and the side faces receive lower concentrations. The point of maximum concen- tration on the receiver face away from the sun is at a higher z, than in the symmetric case, the maximum concentration point on the face towards the sun is lower, and the point on the sides is about the same. An example concentration distribution result is shown in Fig. 7. Overall, the effect of misalignment is to skew the heated band so that it is not at a uniform height

receiver axial location, 2,/i

Fig 7. Concentration distribution due to solar incidence with a pointing system offset error of 4Bo, at D/d = 10, a rim angle of 90”, and oaY = 8 mrad, showing variation with circumfer-

ence.

around the cylinder and to produce asymmetry in the strength of the heated band. There is also some loss in total power received as some of the solar mtensity is reflected out of the concentrator without striking the receiver.

3. RESULTS

The immediate computation results are generated as curves of C versus z&for a given 4, and 8 (e.g. Fig. 7), with the parameters cr, D/d, and a,. As each of these curves shows the same general character, the results may be summarized in terms of (a) the axial location of the maximum concentration on the re- ceiver; z=,,,,.,l/; (b) the width of the heated band, ex- pressed as the axial distance between the two points where the concentration is at 2% of its peak value, Sz,//, and (c) the axial integral of concentration within 62,:

cw,, 8) =jJ% ctz,, h, e)dz. (19)

Figure 8 shows the variation of the peak location and width of the heated band with concentrator rim angle for a concentrator/receiver having no pointing offset and D/d = 10. Both the peak location and heated band width increase in proportion to the focal length as the rim angle increases. Therefore, a deep concen- trator must have a correspondingly long collection area (aperture) on the receiver, and the aperture must be located higher with respect to/l Figure 8 shows results for several total system error standard deviations, and it is seen that increased system error increases the re- quired size of the receiver aperture. For a rim angle of 90”, increasing the system error standard deviation from 4 to 12 mrad results in a 12% increase in aperture size requirement. System error lowers the concentra- tion peak location slightly, so that a concentrator/re- ceiver with a higher error tolerance would have its ap erture at a lower height on the receiver. C” is found to be only a very weak function of (Y, as most of the

Fig. 8. The effect of concentrator rim angle on concentration peak location and concentration band width, for a range of system error standard deviation, at D/d = 10, with zero

pointing offset.

rim angle, a(deg.) diameter ratio, D/d

Fig. 10. The effect of concentrator to receiver diameter ratio on concentration (integrated over the concentration band), for a range of system error standard deviation, at a rim angle

of 90”, with zero pointing offset.

flux incident on the projected area of the concentrator will ultimately be reflected to some point on the re- ceiver.

Figure 9 shows peak location and heated band width as functions of Dfd for a rim angle of 90” and no pointing offset. Peak location moves higher as the con- centrator is enlarged in proportion to the receiver, al- though the change becomes slight for Dfd > 15. As also illustrated in Fig. 8, lower concentrator system error leads to a higher receiver peak concentration point. The heated band width is not monotonic with D/d, but shows a minimum for D/d - 12-15. The D/d for minimum heated band width increases with increasing system error. Systems with, both, very small and very large receivers tend to have larger receiver aperture requirements. Figure 10 shows the variation in C” with Df d. C” is a measure of the total concen- trated flux falling on the receiver, as opposed to peak concentration at only one point. As might be expected, smaller receivers realize higher concentrations. Higher levels of system error result in more widely distributed sunshapes and a greater tendency for rays to make multiple reflections and leave the concentrator without striking the receiver. However, as is seen in Fig. 10, concentrator system error does not have an overpow- ering effect on C”.

5 15 20 25

diameter ratio, D/d 09M) 0 180

receiver angular location, + (deg.) Fig. 9. The effect of concentrator to receiver diameter ratio on concentration peak location and concentration band width, for a range of system error standard deviation, at a rim angle

of 90”, with zero pointing offset.

Fig. 11. Circumferential variation in concentration peak lo- cation, for two pointing offset errors and a range of system . ^^^^

error standard deuatlon, at D/d = 10 and a nm angle of YO’.

Concentration distributions 121

Imposing a pointing offset on the concentrator/re- ceiver system results in variations in z,,,,,//and C” with &. These are shown in Figs. 11 and 12 for pointing offsets of two and four times the subtended half angle of the solar disk as seen from Earth for a rim angle of 90” and D/d = 10. (There is very little change in Gz&vith 4, and, therefore, these results are not shown.) Figure 11 shows that the height of the concentration peak varies by about 3% of/from the front of the receiver (4, = 0) around to the back (4, = 180” ) for 8 = 2@, , and by about 6% for 8 = 4/3,. This variation is shown to be roughly constant with usY. It is also found that these percentage variations are roughly constant with D/d. Figure 12 shows that the integrated concentration varies significantly with 4, in pointing offset cases, varying by about 10% from front to back for 8 = 2&, and by about 20% for 8 = 4@,. System error is seen to have little effect. The C” versus 4, variation is found to increase with D/d. At Df d = 30, the front to back variation in C” is about 20% for 8 = 20, and about 60% for 8 = 48,. Larger variations are also found for rim angles # 90”.

4. CONCLUSIONS

Solar concentrations on a cylindrical receiver in a paraboloidal concentrator are computed by deriving a geometric optics method which integrates over the solar

122 P. D. JONES and L. WANG

receiver angular location, $ (deg.)

Fii. 12. Circumferential variation in concentration (integrated over the concentration band), for two pointing offset errors and a range of system error standard deviation, at D/d = 10

and a rim angle of 90”.

disk and the concentrator projected surface, and fea- tures an implicit analytical mapping to the receiver. The method accounts for solar incidence by consid- ering the size of the solar disk and the intensity distri- bution across the solar disk, for zero-mean concentrator errors through derivation of effective sunshapes, and for pointing error offset by varying the location of the effective sun. Results show that the distribution of concentration on the receiver surface may be charac- terized by the location of the concentration peak on the receiver, and the width on either side of this peak in which concentration has a significant value.

As a fraction of the focal length, the concentration peak location varies with the rim angle of the concen- trator and the concentrator to receiver diameter ratio, and to a lesser extent with the standard deviation of concentrator surface and pointing system errors. The concentration peak location is constant around the circumference of the receiver if the concentrator/re- ceiver pointing system maintains a zero-mean pointing error, but varies circumferentially under conditions of pointing system offset.

As a fraction of focal length, the width of the con- centration band varies with rim angle, diameter ratio, and system error, but is not sensitive to pointing offset. The concentration at a circumferential point, inte- grated over the width of the concentration band, is significantly affected only by the diameter ratio and the pointing offset. Pointing offset can lead to a large circumferential variation in concentration. This vari- ation is mitigated by small D/d and a rim angle of 9o”.

Acknowledgments-This work was supported by the Auburn University Center for the Commercial Development of Soaos. represented by Dr. Tony Overfelt. Graphical assistance was provided by Mr. David McLeod.

NOMENCLATURE

a concentrator parabolic coefficient, m-’ C concentration due to the whole solar disk for a given

pointing offset

c’ concentration due to a solar disk element for a given misalignment angle

C, concentration due to collimated, aligned incidence C” mean concentration integrated over 62,//

D concentrator diameter, m d receiver diameter, m

&, Q, & coordinate system unit vectors ,/ concentrator focal point height, m ‘F h i

fi P

A

8

8

effective sunshape receiver height, m concentrator incident direction vector concentrator surface normal vector concentrator surface error probability density func- tion concentrator radial coordinate, m receiver radius, m components of reflection direction vector reflection direction vector normalized solar intensity distribution concentrator axial coordinate, m receiver axial coordinate, m concentrator rim angle polar angle of a point on the solar disk, from a receiver-to-sun polar axis subtended half angle of the solar disk (maximum P) overall misalignment angle between the receiver axis and the solar disk center local misalignment angle between the receiver axis and a solar disk point pointing system standard deviation solar intensity standard deviation concentrator slope standard deviation concentrator specularity standard deviation total sun plus system standard deviation concentrator system standard deviation concentrator angular coordinate receiver angular coordinate azimuthal angle of a point on the solar disk, about a receiver-to-sun polar axis

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APPENDIX

The partial derivatives in eqn (9) are expressed as follows [see also eqns (5), (6), and (7)]:

dr -2 = sin 8 cos $I dh

az, -=22artr,$+b$ 2 mr r ar (Al) h rrrc xr - r :

ab r,+r-+

ar ??&=I+

-=_ \Ir:rf + r2,rt - r$r*

a& (~2) ar r: + r$

(A3)

(rrr + r:rt + r’,rz - r$r*) 1 (A9) (rz + r:)’

az, ab ar __=-r,-_b’

a4 a4 a4

using the subordinate relations:

_=_-!+!N ah ab m a4 a)r,& m, d&

dr 1 - 4a2rz -’ = sin B sin I$ - db 1 + 4a2r2

h, 4ar sin 0 sin I$ z= 1 + 4a2r2

(A8)

a L = 2ar, - 2a sin 0 cos &J ar

(A4) + 8a*r( 4a2r2cos 0 + 4ar sin 0 cos #I - cos 0)

(I f 4a2r2)2

(A5)

(A6)

a, _ 4a(4n2r2cos 0 + 4ar sin 0 cos $J - cos 6) ar ( 1 + 4a2r2)*

rir2 - r&rf - r&r’ - r:rz db - Zrr,r, r2r2 + r$ra - rir2 r c -=- % (r: + r:)* r:r: + r2,rf - r$r*

(A7)

r(r$ - rf)Qrf + r2,rf - r$r* ab - r3r2 - r,r:rf + 2rJ2r2 z=- (r: + r:)*iFr: + r$rf - r$r’

(AlO)

(All)

A12)

A13)