Passive optical solar tracking system

Passive optical solar tracking system Ross McCluney

An optical system consisting of a convex reflective sphere or cylinder and a collimating lens is proposed to produce a collimated beam of solar radiation whose direction of propagation varies only slightly over the course of a day. Called a passive optical solar tracking (POST) system, a spherical version of the proposed

device has limited usefulness for daylighting applications due to the inverse relationship which is shown to exist between collimated beam directional stability and contained flux. A cylindrical version of the system exhibits greater promise for solar illumination of building interiors. Other applications for the device are proposed for the fields of solar research, education, and artistic design.

1.

Introduction

There are several proposals for beam daylighting systems.'- 7 Such systems can deliver useful levels of illumination to the interior spaces of buildings. Questions of durability and cost-effectiveness, however,

have so far prevented widespread utilization of these systems. At the center of these problems lies the relatively high cost of providing large-aperture tracking optical systems

to collect,concentrate, and direct the radiation to where it is needed. Attempts to avoid this cost have lead to a search for simpler more durable systems, hopefully without moving parts and using inexpensive optical components. Beam daylighting using inverted reflective venetian blinds on south-facing windows to bounce sunlight deep

within a room is one proposed solution to this problem.8 -10 This technique, however, suffers from sensitivity to the sun's movement through the sky and the corresponding movement of the beam projected into the

building's interior. Compound holographic concentrators have also been proposed, 1 1-14 as have a variety of nonimaging nontracking concentrators. 15'1 6 A wide-angle lens with an

image collapsing subreflector was recently proposed that shows promise for meeting the needs outlined above.'7 A set of segmented Fresnel reflectors placed

in two opposing arrays has been used on the Civil/ Mineral Engineering Building of the University of Minnesota's Minneapolis Campus to provide interior illumination of a pedestrian walkway two levels below

grade.18 A related optical system capable of producing a moderately collimated beam of solar radiation that moves only slightly as the sun moves through the sky is

herein proposed. Sunlight glinting from shiny convex surfaces such as

automobile windows appears very bright (indicating high source luminance) and is sometimes rather insensitive to the angular positions of both the sun and the observer. Apart from some interesting optical properties of this glint (which are explored in this paper), practical advantage can be taken of the effect to project a beam of solar radiation along a relatively well-collimated path whose propagation direction does not change appreciably as the sun sweeps through the sky. The key element of the proposed optical system is a reflective sphere used as a convex mirror to deflect incident solar radiation into a collimating lens. Since this

optical system has no moving parts, the effect is one of tracking the sun by purely passive means without recourse to auxiliary power. As we shall see, the principal price one pays for this capability is loss of flux, since it is not possible to achieve

both high projected flux and angular beam stability at the same time. In spite of this limitation, there appear to be some useful applications of the technique in the fields of art, architectural design, scientific research, and

The author is with Florida Solar Energy Center, 300 State Road 401, Cape Canaveral, Florida 32920. Received 4 May 1983. 0003-6935/83/213433-07$01.00/0. ©1983 Optical Society of America.

education. A cylindrical version of the device shows greater promise for projecting usable quantities of luminous flux into buildings. The purpose of this paper is to describe the optical system in some detail, to develop the principal design equations, and to describe some possible uses of the system. 1 November 1983 / Vol. 22, No. 21 / APPLIED OPTICS

3433

Measurements of the (y,z) coordinates of the focal positions on this plot fit the following equations very closely: R

Y =2

I f

Fg1PsvOtal

z

T

Fig. 1. Passive Optical Solar Tracker (POST) operating principle. II.

=

T

cos-

2

(7)

,

R sin-Ay

(8)

Since sin2 (4I/2) + cos2 (T/2) = 1, these coordinates are connected by the equation

Principle of Operation

Z2= R2_

A diagram illustrating the operation of the device, which we shall call the Passive Optical Solar Tracker

4y2.

(9)

z

(POST), is shown in Fig. 1. As the sun moves through

the sky, its virtual image through the reflective sphere always remains inside the sphere. If the focal length f of the collimating lens is fairly large compared to the radius of the sphere, the motion of the sun's image inside the sphere will produce relatively small changes in the direction of the collimated beam. The longer the focal length of this lens and the smaller the diameter of the sphere, the smaller will be the motion of the beam and its contained radiative flux. A goal of this paper is to develop an equation for the inverse relationship which exists between the magnitude of the projected flux and the magnitude of beam movement.

z,

R

Yo

111. Image Location

Using the coordinate system shown in Fig. 2, we will consider first those rays incident on the sphere in the x = 0 plane. Incident and reflected rays are described by the equations

Fig. 2.

z-

= mi(y - Yo),

(1)

Z-

= mr(y - Yo),

(2)

1400

yo = R sinO, zo = R cosO,

(3)

1600

mi = tan(a + 2() = cot(O- ,3)

(4)

R

mr = tan(a) = cot( + ).

(5)

Coordinate system used for determining image location.

with

We shall choose the y axis to be the optical axis of the focusing lens. We shall use the symbol 'I for the angle a + 23 between the incident solar ray and the y axis as shown in Fig. 2. For the reflected ray to be parallel to the y axis, a must be zero. Thus T = 20, 3 = 900 - o, mr = 0, and mi = tanI

= tan2f3 = tan2(90'

-

o)

x N

(6)

with 00 = 90° - 1. Rays reflected slightly above and below this ray (those for which 0 = 0 0 + AO for some AO)will intersect

at the approximate location of the virtual image inside the sphere. The incident rays and projections of the reflected rays inside the sphere are plotted in Fig. 3 for angles 0 0 ranging from 10 to 70° in increments of 10° and for AO= 20. (Corresponding angles T range from 160 to 200 in increments of 200.) 3434

APPLIED OPTICS / Vol. 22, No. 21 / 1 November 1983

0

0

R/2

R

Y-Axis-

Fig. 3. Plot of incident rays and projections of reflected rays inside the sphere for various angles made by the sun with respect to the +y axis.

Rays incident on the sphere at points not in the x = 0 plane but which are reflected parallel to the y axis will lie in a plane containing the y axis and rotated about it by an angle we shall call

(see Fig. 4).

z

The point

(X0 ,YOZo) of intersection with the sphere will possess spherical coordinates (R,O,O)that are related to Cartesian coordinates by Xo = R

sinO cos

Yo = R sin6 sing

Zo = R cosO. (10)

x

Projecting this point onto the X-Z plane parallel to the Y axis yields the following expression for 5: tanb = Xo/Zo = tanO coso.

(11)

The coordinates (y,z') of the image in the rotated y-z' plane are given by Eqs. (7) and (8), repeated here for convenience with I being the angle between the incident ray and the y axis in the y-z' plane with z replaced by z': R

'

T (hours)

a. (degrees)

Os (degrees)

x (R)

y (R)

z (R)

z'= R sin-. 2

(13)

0 1 2 3 4 5 6 7

76.4 71.1 60.5 48.6 36.4 24.3 12.4 1.1

90 42.2 19.8 7.9 4.9 11.2 (-) 18.5 (-) 26.5 (-)

0 0.154 0.303 0.443 0.549 0.583 0.572 0.526

0.393 0.390 0.382 0.369 0.365 0.383 0.405 0.425

0.618 0.606 0.570 0.508 0.406 0.268 0.133 0.011

x = R sin 2 sinb, 2

R

T

2

2

y =-COS-,

z = R sin - cos6. 2

(14) (15)

TableII. WinterSolstice (16)

'I is the angle in the y-z' plane of the incident sunlight with respect to the y axis. The cosine of T is equal to the scalar (dot) product of a unit radius vector from the sphere's center pointing toward the sun and a unit vector along the y axis. Thus cost

= sinG sino,

angles 0 and X, respectively.

a, (degrees)

As (degrees)

X (R)

y (R)

Z (R)

0 1 2 3 4 5

32.5 30.7 24.8 18.4 9.1 -1.4

90 73.8 59.0 46.4 35.7 26.5

0 0.125 0.246 0.356 0.452 -0.526

0.480 0.478 0.471 0.459 0.444 0.425

0.280 0.268 0.231 0.172 0.089 0.014

solar noon, and D is the number of days since 31 December. Choosing a mid-U.S. latitude ( = 350) and the summer and winter solstices (D = 151 and 334), the limits of the annual excursions of the sun (the solstices) can be specified:

Image Motion

Assuming the +y axis to be pointing south (in the northern hemisphere) we may obtain solar coordinates (0,O) as functions of the time of day T and day of the year D as follows: Let a = 900 - 0 be the solar altitude angle. Then a = sin'1[cosX cosd cos(15T) + sinX sind],

(18)

0 = 900 - sin-'sin(15T) cosd/cosa],

(19)

where t0.43378 sin[O.9863(D - 81)])

T (hours)

(17)

where 0 is now the solar zenith angle, and 0 is its azimuth angle. Equations (11), (14), and (15)-(17) provide the Cartesian coordinates (xy,z) of images of the sun inside the sphere as functions of the solar zenith and azimuth

1

TableI. SummerSolstice

(12)

Since x = z' sin3 and z = z' cosa from Fig. 4, we have

d = tan

Plot of the x-z plane showing angle 6.

Y = - cos 2, 2 2

the coordinates of the sun's image

IV.

Fig. 4.

(20)

is the approximate solar declination angle, Xis the latitude of the site, T is the time in hours before and after

a, = sin'1[0.757 cosi5T + (0.215 or -0.220)],

(21)

0 = 900 - sin1[(sinl5T)0.926/cosa].

(22)

Values of a, 0s, x, y, and z calculated from the above formulas are presented for several hours T before and after solar noon in Tables I (summer solstice) and II (winter solstice). From the data given in these tables it is clear that movement in the +y direction which produces defocus or reduced collimation of the beam emerging from the lens is limited, ranging from 0.36 to 0.42 R in summer and from 0.42 to 0.48 R in winter. Horizontal (+x) movement is at most 0.53 R, for a total range approximately equal to the sphere radius. Vertical movement is less, being 0.6 R total. 1 November 1983 / Vol. 22, No. 21 / APPLIED OPTICS

3435

V.

Image Size

An approximate expression for the size of the image may be derived as follows. Taking the angular diameter of the solar disk to be half a degree and noting that solar radiation travels distance I (shown in the diagram in Fig.

5) to the sun's virtual image following reflection, the image diameter in the z direction (Az) will be approximately given by Az = 0.507rl/1800 = 0.00871.

(XYZ)

,.-\(xyz)

(23)

Taking (X,Y,Z) to be the coordinates of points on the surface of the sphere, = vR 2 _-X2 _z 2 - y (24) The point (X,Y,Z) of interest on the sphere has coordinatesX = x and Z = z. Substituting for x, y, and z from Eqs. (7)-(9) into Eq. (24) yields the equality = y so that Az = 0.008 7 y.

Fig. 5.

Drawing showing geometry of sun's image in the sphere.

4) = rATLi sin2 b = 7r2pTL, D2 + 4f2

2

(30)

or

(25)

4) = 1.9 X 10- 4pTL

Using the maximum and minimum values for y calcu-

D

(31)

lated earlier, we see that the image diameter Az ranges

Using the mean value fory of 0.41 R and assuming the

from 3.17 X 10-3 to 3.7 X 10-3 R in summer and from 3.7 X 10-3 to 4.2 X 10-3 R in winter.

value 0.8 for both the reflectivity of the sphere and the transmittance of the lens, Eq. (31) becomes

Assuming the image to be approximately circular with

41)= 2.0 X 10- 5 L,

radius Az/2, its area will be given by A = 4 (AZ)2 = 4.7 X 10- 5 y 2 . 4

VI.

(26)

The apparent luminance of the solar image in the sphere will be related to that L, of the solar disk by LI = pL8,, where p is the reflectivity of the sphere for the angle ,3 of incidence To simplify flux calculations and for a first approximation, the collimating lens will be treated as if it were always centered at the position (x,z) of the solar image and a distance y from the center of the sphere, y being selected to place the image always at the focus of the lens. Thus Y+ ,

(27)

with f being the focal length of the lens. The solid angle Q subtended by the lens at the image will be given by -

cos)

=

2r [I - cos (tan-

Defining F = fID to be the f/No. of the lens, Eq. (32)

-j]

vJ (98)

4) = 2.0 X 10-5 L

image.

Using the symbols A for the area of the sun's image and T for the effective transmittance of the lens, the luminous flux 4' contained in the projected beam will be given approximately by T

ffL

1

cosOdwda.

(29)

The luminance of the image will be very nearly constant over the two ranges of integration, permitting removal of LI from the integrand.

Following evaluation

of the integrals and substituting for A from Eq. (26) we have 3436

APPLIED OPTICS/ Vol. 22, No. 21 / 1 November 1983

R2 2

1 + 4F

(33)

In cases for which F exceeds -1.5 the denominator of Eq. (33) can be replaced with 4F2 to a good approximation, resulting in 4) = 5.1 X 10- 6 L (R)2

(34)

Equation (34) indicates that the ratio of sphere radius to lens f/No. is the principal parameter governing flux magnitude with this optical system. On a clear day the apparent luminance of the solar disk can be of the order of 1.8 X 109 cd/M2 (or nits) (di-

rect normal irradiance of 900 W/m multiplied by luminous efficacy of 120 lm/W and divided by the solid angle subtended by the solar disk). Thus Eq. (34) can be written as

where 8 is the half-angle subtended by the lens at the

4- =

(32)

becomes

Projected Flux

0 = 2r(1

R2 D 2

D2 + 4f2

4) _ 9885 ( )

(35)

For a sphere radius of 0.25 m and an f/2 lens, the projected flux will be of the order of 154 lm according to Eq. (35). VII.

Collimated Beam Movement

For the purpose of relating changes in the direction of the projected beam to movements in the image inside the sphere we shall assume the lens always to be centered at the point (O,y,O). The direction of the projected beam is that of the ray from the solar image passing through this point. Projecting this ray onto the x-y and y-z planes and defining the angles which these

90

z "I-, 1---1

80

(0, y, z)

70

60

y

50

4

(X y, Z)

0 40

x Fig. 8.

I

30

projections make with the x and z axes as Ox and , respectively, we have

I

20

Cylindrical case coordinate system.

I tanO =

10

x

ff

tank 8 =

~~z * .

(36)

Letting A0 and AO, be the maximum values of Ox and O over a year's time we have

0 .1

.2

.3

.4

.5

2

.6 .7.8.91.0

3

4

5

6 7 8910

R/f

Fig. 6. Plot of beam instability parameter AO,vs the ratio Rif.

AO,,= tan-1(1.06R/f)

(37)

AO,8 = tan'1(0.62R/f).

(38)

0

Since A x exceeds A0 by nearly a factor of 2, we shall

II I /zII _"/

LI

/

-

l

take A0x to be the primary indicator of beam movement or instability. Its dependence on the ratio R/f is plotted in Fig. 6. SolvingEq. (37) for R/f and substituting into Eq. (35) yields the following approximate relationship between projected beam flux 4' and beam direction instability parameter A0x:

Z

1/

500

4) = 8798D2 tan 2 (AOx). 8

(39)

0

Plotting 4'vs A x for several values of the lens diameter

D yields the family of curves shown in Fig. 7. Assuming perfect collimation and projected beam area equal to r(D/2) 2 , the illuminance of the beam in

74 C 100 E

lux will be given by E = 11,000 tan 2 (A0X). Vil.

.5. 0 -0

Cylindrical Case

An improvement in the projected beam flux level can

be effected by replacing the spherical mirror and lens with cylindrical ones. In this case Figs. 1-3 are still applicable as are Eqs. (7)-(9). For rays incident on the cylinder and not parallel to

Q AAAP

the x = 0 plane, the projections of incident and reflected rays onto the x = 0 plane will still obey Eqs. (1)-(9), with T now being the projection of the altitude angle onto

1

5

the x = 0 plane. From the drawing shown in Fig. 8, cotT = y/z, and from Eq. (10), y/z = tan0 sink, so

that

cotT = tanG sink.

10

Fig. 7.

30 40 50 20 AngularBeam Motion q, In Degrees

60

70

Plot of projected flux vs angular beam motion for several values of the lens diameter D.

(40)

The Cartesian coordinates (y,z) of the linear image of the sun in the cylinder are given by Eqs. (7) and (8), with 4! as given in Eq. (40).

The vertical height Az of the sun's lineal imagein the cylinder is given by Eq. (25), with an approximate av1 November 1983

/

Vol. 22, No. 21

/

APPLIED OPTICS

3437

H/2

0 Rh -H/2

I

Fig. 9.

Cylindrica a case geometry.

erage seasonal value for Az of 3.7 X 10- 3 R, where R is the radius of curvature of the reflective cylinder. To estimate the projected flux 4' contained in the beam emanating from the cylindrical lens, we use the construction of Fig. 9. When the sun is directly ,south in the northern hemisphere (solar noon) (north in the southern hemisphere), the element of flux d34' from an element of rectangular source of (assumed) uniform luminance LI (the sun's image in the cylinder) of length dl and height Az received by an element of a rectangular

receiver of length d l' and height dh a distance R away is approximated by d 34) = LjAz

SH12 coS2G

H/2 R 2

dhdldl',

(41)

where is the angle made by a ray between elemental areas of the two surfaces and their normals, f is the focal length of the lens, cosO= fR, and R is the distance between the elements, given by R2 = f 2 + h 2 . Evaluation of the integral over h yields d24

=

2J2 H/zLj

2

[4f2(f2 + H /4)

+ 2f3 H tan

(2f)]

(42)

The total flux from a line source of length L received by

rectangular receiver of length L and height H a distance f away would normally be the double integral of the above expression over 1 and 1'. In the present case, however, the source luminance LI is zero when is not equal to 1'. Thus, we have 4) = 2f2 HAzL [4f 2(f 2 +H 2 /4) + 2f3Hta

(2)]

JO

or = 2f2HAzL [4f2(f 2 + H2)

+

tan

(2fJ

(43)

Evaluating this expression for a cylinder of radius 0.1 m, a cylindrical lens of height H = 0.5 m, a focal length of 4 m, Az = 3.7 X 10-4, and L = 1.2 X 109 cd/M 2 yields

the result 4'/L = 13,839lm/m. For a 10-m long building, this system can project of the order of 140,000lm into a north-facing aperture of height 0.5 m. If this flux can be distributed over a 100-M2 area using appropriate

interior optical elements, the spatially averaged illuminance over this area will be of the order of 1400 lx 3438

APPLIED OPTICS / Vol. 22, No. 21 / 1 November 1983

(140-fc). This is a very respectable illuminance level. The 4-m focal length of the cylindrical lens coupled with

an estimated 0.6R vertical movement in the sun's lineal image can be expected to produce a maximum vertical change in projected beam direction of tan-1(0.15R). For R = 0.1 m, this is