Solar Energy, Vol. 17, pp. 277-283. Pergamon Press 1975. Printed in Great Britain

ENHANCED SOLAR ENERGY COLLECTION USING REFLECTOR-SOLAR THERMAL

COLLECTOR COMBINATIONS

D. K. MCDANIELS and D. H. LOWNDES Physics Department, University of Oregon, Eugene, OR 97403, U.S.A.

H. MATHEW Coos Bay, Oregon

and

J. REYNOLDS and R. GRAY Architecture Department, University of Oregon, Eugene, OR 97403, U.S.A.

(Received 26 September 1974; in revised[otto 25 April 1975)

Abstract--The amount of direct light gathered by a combination of reflector plus flat-plate collector has been analyzed. The calculations were done allowing variable reflector and collector orientation angles, variables latitude, and arbitrary sun hour angle away from solar noon. The effects of reflection and transmission losses and of polarization of the incident light were included. Correction was also made for the finite size of the reflector. It was found that the optimum orientation has the collector plane almost perpendicular to the plane of the reflector. This optimum orientation is approximately independent of the sun's azimuthal dependence, The optimum reflector angle is found to be between 0 ° and 10 ° above the horizon for winter solar conditions. For typical winter operating conditions the enhancement in light gathering power for direct solar radiation is about a factor of 1.4-1.7. This results in an effective increase of 100% in the useful winter heat output from a practical reflector-coUector combination with a reflector angle of 0 °, over the useful heat output obtained with an optimally oriented simple flat-plate collector. An approximate calculation was also made of the overall enhancement in useful heat output for diffuse solar radiation; an increase by a factor of about 1.5 is predicted. Comparison with the preliminary analysis of the performance of the Coos Bay, Oregon solar house shows substantial agreement with the predictions of the present analysis.

INTRODUCTION

For the next few years the type of solar-energy collecting device which will undoubtedly be most commonly used is the simple fiat-plate solar thermal collector[l]. Further development work on other ideas[2-5] will begin to change this situation in the near future, but probably not in a dramatic way in practice for 10-20 years. The flat-plate collector has many advantages, an important one being its ability to collect both the direct and diffuse components of the incident solar radiation. In order to optimize for the direct component of the solar insolation, the collector should be oriented to have its flat surface normal to the sun's rays. On the other hand, the presence of a significant diffuse component of the solar radiation due to scattering[6] modifies the situation so that the precise collector orientation is not crucial. Experimental evidence reported in the literature [7, 8] shows that the fall off in solar radiation collected, as the collector orientation angle is varied away from the optimum angle calculated for a direct beam, is not very steep, at least for angular changes up to about 15-20 ° away from the optimum.

Concentrating devices have been utilized in many dif- ferent ways to produce an intense distribution of solar radiation over a small area. This idea forms the basis of several of the schemes [9, 10] for large-scale generation of power from solar energy. However, such devices have only rarely been used in combination with the simple plat-plate collector. H. Thomason, in Washington, D.C.,

reported a 30% increase in solar energy collected using a straight reflector [11]. Straight, non-specular mirrors were used previously by Sherman[12], Tabor[12] and others[13-15] to increase the amount of solar energy collected for their solar devices.

This study was motivated by the success of the Coos Bay, Oregon solar home of Henry Mathew which utilizes a reflector and fiat-plate collector combination. Prelimi- nary performance of the Mathew house indicated a con- siderable improvement over that expected from a simple fiat-plate collector. In the Mathew design a key departure from the usual was that the collector was oriented almost vertically. We decided to find the basis for this result by doing the required theoretical analysis for a reflector- collector system with a straight reflector, for which the mathematical analysis is not unduly complicated. The amount of light gathered by the reflector-collector combi- nation is compared to that of an optimally oriented collector alone. The theoretical analysis has been done in a general enough fashion to permit calculation of the solar intensity received at any latitude and solar time of the day and for any arbitrary orientation of the collector and reflector. Since reflectance losses from the transparent cover plates of the collector can be important at large angles of incidence the calculations include this correc- tion, taking into account correctly both the polarization of the incident light and the absorption losses in the glass cover plates. Enough numerical work is presented to

277 SE Vol. 17, No. 5--B

278 D. K. MCDANIELS et al.

establish the optimum collector orientation with respect to the reflector and to establish the optimum reflector orientation for winter heating needs. The performance of a fixed reflector-collector combination for diffuse radia- tion is also estimated.

THE MATHEW SOLAR HOUSE

Experiments were conducted in early 1965 on an 8 × 16 ft collector of the Thomason design, with no reflector and a single plastic cover. It was installed next to an existing house, with the collector oriented at 700 to the horizontal and tied into an existing thermal storage sys- tem, using a 2000gal. water tank. This first collector provided adequate heat, but was unable to withstand the prevailing climatic conditions. Problems associated with the construction of the collecting trough, with corrosion of the top feed pipe, and with overheating of the plastic cover in the summer, caused tests to be made with other designs. Tests were then made on several types of collec- tors in late fall to establish several critical parameters. The crucial experimental criterion used was the time required for a drop of water to evaporate when placed on the hot collector plate. The final experimental tests were performed early in December 1965. A 75 ° tilted collector with a horizontal reflector was tested with 1,2 and 3 cover plates; the collector with one cover was hottest. The measurements were made at noon on a cold, clear day. The absorbing plate was a piece of black steel, had 1-5 in. of fiberglass insulation behind it, and the first cover plate was spaced 3/8 in. above the black absorber. Under the same weather conditions, the evaporation times were measured for the collector tilted at 45 °, 60 °, 90 o and 110 °. The fastest evaporation times (hottest temperature) were achieved with the collector in the vertical position and the reflector horizontal.

Design and construction of the solar house was started the same year, with all work performed by H. Mathew. Construction of the house itself was finished in June 1967, and the solar heating system was ready for use early in 1968. The house is one story, 89× 37ft, with the long dimension running east to west. Only the west half is lived in and heated; the east half is a combination garage, workshop and greenhouse. The collector is 80 ft long and 5 ft high, tilted at 82 ° to the horizontal. The 100 × 200 ft roof in front of the collector was sloped at 8 ° and covered with aluminum foil pressed into hot roofing cement. The collector was tilted at 82 ° , rather than oriented vertically, in order to provide more heating in fall and spring, to compensate for trees on low hills around the house, and to improve the appearance of the house by softening the angles to look more like a conventional peaked roof. The reflector was built with an 8 ° slope to allow a 3 ft high truss to be used to span the 40 ft between the outer walls, to improve roof drainage, and to give the appearance of a conventional peaked roof.

A careful analysis of the performance of the Coos Bay solar house is now underway. However, a preliminary estimate of its performance can be obtained by comparing its actual annual electrical consumption to that estimated for a typical Pacific Northwest home of its size. Data for the year 1970 are most complete and will be utilized here.

In the year 1970, space heating for a house of 1600 ft: is estimated as (16 -+ 3) × 103 kWh. Adding in a basic electri- cal consumption of (11-1) × l0 s kWh for other needs gives (27 -+ 3.2) × 103 kWh for total estimated consumption. The actual consumption was 14.2× 103kWh giving a solar contribution of about (12.8- 3.2) × 103 kWh. This number can now be compared with that expected for a 400ft 2 collector oriented at 60 ° (for near-optimum winter heating operation of a simple fiat-plate collector without reflec- tor). Utilizing data typical for the Willamette Valley in Oregon [13] we obtain (8 -+ 2) x 103 kWh which would have been collected by a 30% efficient collector tilted at 60 °. Hence, the reflector-collector combination has resulted in an improvement of 1.6-+0.6 over that expected for a straight solar collector. An improved analysis is underway and will be reported on at a later date.

THEORETICAL ANALYSIS

Before making an explicit mathematical analysis, it is useful to first discuss qualitatively what is expected. For a simple flat-plate collector the amount of direct solar radiation received is

dPL = LW{Io. h}T(/3). (1)

Thus, ~ L - cos/3 and is a maximum when /3 = 0, i.e. when the collector is oriented normal to the incident solar flux. This will occur when the tilt angle of the collector, Or is equal to 0~. If a horizontal reflector is added, the intensity of the beam reflected into the collector will be maximized when Or = ~"- 0~. For approximately equal contributions from the reflected radiation and the solar radiation directly incident upon the collector, optimum performance will occur when 0~r is about halfway be- tween 0~ and 7r - 0~, or when 07 = ~r/2. This qualitatively explains the optimal vertical orientation of the collector for a horizontal reflector and is verified quantitatively in the explicit calculations below.

The total solar flux incident upon the solar thermal collector absorber plate is the sum of the directly col- lected intensity and that from the reflector,

d~ = RW{Io. ?},0T(0h) + LW{Io. h}T(/3). (2)

The transmission factors for m collector plates are given explicitly [16] as,

l - p s Tm(s,p)=F,{l+(2__~_l)ps}+Fp{ l l - p , + (2 m - l)p,, /

(3)

where ps and pp are the reflected intensities for light of 2 different polarizations at a single interface,

= J'sin (/3"-/3')~ 2 P~ I sin (/3"+/3')J

= ~ tan ( / 3 " - / 3 ' ) 2 P" [tan (/3"+/3')} " (4)

In order to calculate the direct contribution from Eq. (2), the following angular relations are needed:

Enhanced solar energy collection

cos/3 = sin ~ sin (A - 0r) + cos ~ cos (h - 0r) cos to, (5)

cos 0~ = sin $ sin h + cos ~ cos ;t cos to, (6)

= 23.45 sin {2~r (284 + n)/365}. (7)

The calculation of the reflected intensity as given by the first term on the right hand side of Eq. (2) is considerably more involved. The basic geometry involved is outlined in Fig. 1. The angles involved in the reflection from the reflector are detailed in Fig. 2. The angle of incidence OR ~ of the incident solar radiation with respect to the normal

to the reflector plane is

cos 0R' = sin ~ sin (A - OR) + cos ~ cos (A - OR) COS to,

~8)

in complete analogy with Eq. (5). The angle 0b between the normal to the collector surface and the reflected beam is obtained from

cos 0b = sin OR ~ sin (0r - OR) COS 4~R ~

- cos 0R ~ cos (0r - 0R). (9)

The azimuthal angle 6R ~ can be obtained simply by calculating cos e in Fig. 2 in both the reflector and earth corrdinate systems and then setting a = ~r/2, with the result that

sin 0R ~ cos ~R' = sin (A - 0R) cos ~ cos to

- c o s (h - 0R) sin ~. (10)

/ , v !

Fig. 1. Reflector-collector geometry for calculation of the effective reflector length ratio R(t)/L and enhancement factor P described by Eq. (14) in the text. 0R' and tbR j are the zenith and azimuthal angles, respectively, of a reflected ray measured as

shown on the figure.

N

- T - k

: J f - - -~

Fig. 2. Coordinate systems for the reflector. Io is the direction of the incident solar radiation, described by a zenith angle 8R ~ and azimuthal angle ~R' with respect to the normal ~ to the plane of the reflector. Q is an arbitrary unit vector making angles a and ~ with r

and Io, respectively.

279

Finally, we need to obtain expressions for the effective reflector length R(t) and the enhancement factor P for the solar flux collected by the reflector-collector combina- tion. R (t) is easily found from Fig. 1 using the law of sines,

R(t) = cos tkR ~ sin v (11)

L cos OR' cos r/'

where

v = ? + 0R i - 2

L sin(0r - 0R) tan 3' - - tan (0~ - OR) cos 6R i

Y (12a)

tan , / = cos (0r - 0R) tan 6R'. (12b)

Note that the effective reflector length is defined such that a ray striking the far edge of the reflector, at a distance R from the lower edge of the collector, will just enter the top of the collector. The enhancement P is calculated by comparing the light collected by the reflector-collector combination with that collected by an optimally oriented fixed collector, i.e. with a simple collector oriented at 0r ° = 0~, with 0j chosen for t = 0 (solar noon). Thus the enhancement will be given by dividing the total collected flux for the reflector-collector combination by

IoLto cos {/3(0°)} * T{/3(0r°)}, (13)

with /3(0 °) evaluated using Eq. (5). The generalized equation for the enhancement P of energy collection for the reflector-collector combination is then

R(t) cos 0R i T(Ob) cos/3 T(/3) P = ---L---- t5 +

cos{/3(0°)} T{/3(0r°)} cos{/3(0r°)} T{/3(0f)}"

~14) Finally, it is useful to calculate the extra width A needed

on each end of the reflector in order to reflect rays into the collector in the early morning and late afternoon. As illustrated by Fig. 1,

so that

A= W+x sin 6R' (15)

s in s in ~ = cos (0r - OR) tan 4~R ~ -~

P

COS 17 COS OR'

= tan ~R'{COS (0r -- 0R) +---ff-~.R (t)] (16)

As will be seen below in the discussion of numerical results, R(t)/L often gets very large, as the sun's azimuthal angle increases. For any practical situation, however, a finite reflector length Ro must be utilized. The correction for a finite reflector length is easily made by multiplying the first term on the right in Eq. (14) by Ro/R(t) whenever R ( t ) > Ro, and leaving it unchanged when R(t) <- Ro.

280 D. K. MCDANIELS et al.

The correction for finite reflector width is more compli- cated. For present purposes it will be neglected.

DISCUSSION

Numerical calculations have been carried out for the all-winter "average" day, n = 41 in Eq. (7), corresponding to 0, = 59"9 o at solar noon, t = 0. The enhancement calcu- lated from Eq. (14) at solar noon is plotted in Fig. 3 as a function of collector tilt angle for several values of the reflector angle OR and for two values of the reflector specular reflectivity,/5. The value t5 = 0.9 is slightly less than that of freshly prepared silver of aluminum mirrors covered with a protective coating of SiO, while the value

= 0.7 allows us to model in a simple way the effect of weathering on a reflector surface.

The maximum enhancement factor, relative to a simple fiat-plate collector oriented at its optimum angle (07- = 60°), varies from about 1.15-1.85 (for ~ = 0.9) or 1.05-1.65 (for ~ = 0.7), as the reflector angle varies from 200 below the horizontal to 20 o above the horizontal, In each case, however, the optimum performance for the reflector- collector combinat ion is obtained when 0T -- 0, = 90 °, i.e. when the collector is oriented perpendicular to the reflec- tor (or very slightly smaller values of 0r - 0~ for the lower reflectivity reflector).

From a practical point of view it is important to note that the enhancement curve for a given reflector angle is reasonably flat near its maximum. For example, at 08 = - 1 0 ° the dropoff in light transmitted into the collector is less that 3% in moving the collector from Or = 700 to 0T = 90 °. Beyond this +--10 ° range in 0T the dropoff be- comes more severe.

This same relation between the collector and reflector angles holds approximately true as the sun hour angle varies, as shown in Fig. 4 for the cases of t = 2 and t -- 4. For large azimuthal angles the maximum enhancement occurs when the collector is tilted slightly less than 900 from the reflector plane. As for t = 0, the peak in the enhancement factor is sufficiently broad that a difference

t = 4 . 0 _ '2.5 - - ( 0 i : 80 .8 °, ~ i =58.00)

~ : 0.7 0 E Unres t r i c ted

la2 I-- i R_.~ z r f.. z - ~ - "

= 2 z , ; ">,

/7,, / 7/ \ ~'?OR= +10 *

0 5 ] = I I I I I I I I I = = I ~ l 0 50 I00 150

t =2.0 (WinterAvg. Day)/ ~-- (8 i =65.8". ~b i = 32.0") --t 7 : 0 . 7 , ~ Unrestricted~

[ =-

%=+1o° I I I I I I I I 1 1 i l l I I I

50 I00 150 TILT ANGLE (Degrees)

Fig. 4. Enhancement factor P vs 0T for several reflector orientations, for t = 2.0 and t = 4.0 hr. To simulate an average winter day, n = 41 was chosen with the corresponding zenith and azimuthal values as shown. The reflector reflectivity used is # = 0.7, for R/L = 2 and R/L =4. No restriction was placed on A/L (A = extra length at each end of the reflector, see text) for this calculation. In each case the maximum of P occurs when the

collector makes an angle of about 95 ° with the reflector.

in 0T of --+10 ° for the optimum value is not critical, though larger errors in 0T do produce a serious dropoff in the enhancement due to the reflector.

As the reflector angle moves upwards, and as the sun's azimuthal angle gets larger, the effective reflector length needed to insure that reflected light strikes the collector over its entire length increases dramatically. This can be seen in Figs. 3 and 4 and is strikingly illustrated in Fig. 5, which shows that the reflector length becomes unreasona- bly long for large hour angles, when the sun is low in the sky, particularly for OR < 0 °. However, for OR/> 0 ° (reflec- tor horizontal or tilted slightly below the horizon) R [L = 2 is entirely sufficient for the middle 5 hr of the solar day, and R/L --~ 3 suffices for the middle 8 hr of the solar day. The effect upon the enhancement factor P of restricting the reflector length to reasonable values is shown in Fig. 6, where the resulting enhancements for R/L = 2 and

t . . . . I . . . . I . . . . I . . . . I . . . . I . . . . z.5 0i :600 (a) -- e i :60" (b)

j5:0.9 ~ =0.7

"~ 2.0 o R • io.

• 1.5

i 1.0 IZ" ' ,

50 IOO 150 TILT ANGLE (DEC-~EES)

:o. f 7;?

, , , I , , , , I , , , L L 50 I00 150

TILT ANGLE (DEGREES)

Fig. 3. Winter enhancement (for Oj = 60 °) vs Or, for -20 ° ~< 0, ~< 20 °. (a) # = 0.9. (b) # = 0.7. The solid lines show the enhancement calculated with no restriction upon the needed reflector length. The dashed lines show the effect of limiting the reflector length, for OR = - 2 0 ° (R/L ~<3) and OR = - 1 0 ° (R/L ~<2). A reflector length of R/L = 2 is sufficient for the other reflector angles for tilt angles corresponding to the maximum enhancement (see Fig. 5).

Note the suppressed ordinate.

9

8

7

6

4

5

2

I

- I i . . . . . I&O 288

"t'-3 e't '= 4

i I i I i I

-20 -IO O I0 2 0 3 0 REFLECTOR ANGLE (DEGREES)

Fig. 5. Effective reflector length needed vs reflector angle, for various hours of the solar day. The calculation was done for the

"average" winter day (10 Feb.) at latitude 45 °.

Enhanced solar energy collection 281

2.00

1.7~

1.50

~) 2 .00

I"/ '5

~,~ ~ 1 . 5 0

--~ 125

~uJ2.00

1"75 I I..50

125

I.O0

W I N T E R CONDIT IONS (n=41) , u , i , i , i , i ,

" t = O

• , ~ , o 9

" / ' = 2

• f ~ ' ~ " / ' = 4

- 5 0 - 2 0 - I 0 0 I0 2 0 5 0 R E F L E C T O R ANGLE (DEGREES)

Fig. 6. Illustrates the effects of restricting the reflector length R upon the maximum enhancement factor P and optimum reflector orientation 0,. The effect is shown for R/L = 2 and 4, and for t = 0, 2 and 4. for the "average" winter day (10 Feb.). AlL is

unrestricted.

R/L = 4 are compared for t =0 , t = 2 and t =4. The effect of changing R/L from 4 to 2 is to decrease the maximum value of P by about 0.15 (= 10%). This conclu- sion is practically independent of the sun's hour angle. Another important consequence of restricting the reflec- tor length is that the optimum choice of reflector angle is narrowed considerably, and pushed toward positive val- ues of OR, i.e. toward a horizontal reflector. For R/L = 4, the optimum reflector angle is -15 ° at t = 0, -12 ° at t = 2, and about - 2 ° at t = 4hr. For R[L = 2, the optimum reflector angle is -5 ° at t = 0, -3 ° at t = 2, and +8 ° at t = 4 hr. The practical advantage of having the reflector slope slightly downwards (OR > 0 °, e.g. on a rooftop) would favor OR = 0. At this reflector angle, using t5 = 0.8, and assuming that about 50% of the solar flux is received between t = 0 and t = 2, a time averaged enhancement of about 1.60 is obtained. This value will be used in all further calculations involving the reflector-collector com- bination.

It has been assumed so far that the reflector extends indefinitely beyond either end of the collector. We define A(t) as the additional length of reflector which is needed beyond either end of the collector, in order to cover the entire width of the collector with reflected light, at any particular time of day t. Figure 7 shows that A(t)[L may also be restricted to relatively small values without seri- ously affecting the enhancement factors already calcu- lated, provided that OR/> 0 °. Again, AlL = 2 is sufficient for the middle 5.5 hr of the solar day, and AlL = 3 suffices for the middle 7 hr.

While winter solar conditions are the most important for space heating applications, it is of interest to see what reflector-collector performance can be expected under summer conditions. The results of this calculation, using

7

6

5

3

2

, i , t ~ l I I I I I ) i i i i 123 44.1

I •

_ " t ' = O ~ " ' e . . . • . , , . ~ • , . . • _ . •

) / , J , " . i , I . i . n J

- 2 0 - I 0 0 I 0 2 0

REFLECTOR ANGLE (DEGREES)

Fig. 7. Illustrates the extra length of reflector A(t) needed beyond each end of the solar collector, in order to cover the entire width W of the collector with reflected light at time t. The calculation

was done for the "average" winter day (10 Feb.).

0~ = 30 °, are shown in Fig. 8 for the same 5 reflector angles which were considered for winter optimization. The re- flector enhancement is not really important in the summer until the reflector is well above the horizontal plane. For these reflector angles the reflector contribution again causes a maximum enhancement when the reflector is oriented perpendicular to the collector. For reflector angles greater than about - 5 ° , no enhancement is obtained through the use of the reflector. The effective reflector length needed for non-optimal summer operation is less than 1.6L for all the curves represented in Fig. 8. The optimum orientation for the winter is clearly not the same as for summer conditions. The optimum orientation for winter conditions with 0~ = 60 ° has a summer perfor- mance which is below that of a simple collector, as shown in Fig. 9. The enhancement over a simple, fixed, flat plate-plate collector (with 0R = 60 ° for winter operation) is presented for two solar hour angles. A simple way to get more summer heat is to raise the reflector angle upwards,

2 0 0

r,-

1.50 (,)

.~ I.O0 W

0.50

I i i , I ~ i i , I i , , i

(o) el. 3o.

~:0.9

-go,

20"

I I I 5 1 0 1 I I I I i i I 0"000 I00

i i i i i i , ~ 1 i , i T -

(b) 0,3o •

/~'07

i i i i I J I t J I i i i h I 150 50 100 150

TILT ANGLE (DEGREES)

Fig. 8. Summer enhancement for 0, = 30 ° vs 0-r, for -20 ° <~ 0, ~< 20 °. (a) t~ = 0.9. (b) ~ = 0.7. (R/L) is <1.6 for all the curves shown.

282 D. K. MCDANIELS et al.

w" 0

W

"1" Z h i

2.0

1.5

I.O

0.5

O.O

1.5

I.O

0.5

O.O

i I I I I i I I I I I I I

,:3 ~/~/" "-"

I I I I 1 [ 1 1 1 1 1 1 1 d F M A M J d A S O N D d

MONTH { DAY=21)

Fig. 9. Enhancement factor P plotted as a function of time throughout the year. The abscissa labels time in terms of the twenty-first day of each month. A collector angle of 0T = 80 o and a reflector angle of 0 ° were assumed, with A = 45 °. The maximum value of R/L was 3.94 for 21 Dec. at t = 3 hr. Most of the winter

months required R/L to be less than 2.0.

if architecturally feasible. Since the total solar radiation received in the summer is considerably above that of the winter, this modification would not usually be needed.

It was seen above that an enhancement factor of about 1-6 was obtained with a horizontal reflector (and ~ = 0.8). In addition to the enhancement in the light collected, an equally important factor is the change in collector effi- ciency due to the increased solar radiation provided by the reflector. The efficiency of a simple fiat-plate collector can be written as

r/0 = a r o - h, ~_~_T (17)

In this equation, r0~o is the solar flux incident upon the solar collector's absorbing surface. For the reflector- collector combination, the effective solar flux incident upon the absorber surface is P~'o~o. The ratio of the useful heat output of the reflector-collector combination to that of the regular fiat-plate collector is then

q R = a(Pro~o) - h, A T = 1 + (P - 1) at-----2°. (18) qo a(rod~o) - h~AT rio

We have assumed that the water flow for the reflector- collector combination has been increased in order to maintain the same AT as for the simple solar collector. For poor solar insolation days with "0o = 0.3, and using a = 0.94, ro = 0.88, we obtain gR/go = 2.6. As the solar insolation increases the enhancement of the useful heat output ratio, given by Eq. (18), approaches P, the en- hancement in solar radiation collection.

The preliminary analysis of the performance of the Coos Bay, Oregon solar house is in substantial agreement with the above analysis. The Mathew house has OR = 8 ° and the angle between the reflector and the collector is about 106 °. Using ~ = 0.8, the time-averaged enhance- ment from Fig. 6 is about 1.4. Estimating that "0o for a

simple flat-plate collector will average about 0.3 for winter weather conditions in the Pacific Northwest gives an enhancement of the useful heat output for the direct component of about 2.1. The performance of a reflector- collector combination for diffuse radiation has been estimated numerically. From this analysis, a useful heat enhancement of the order of 1.5 can be expected. The relative amounts of diffuse and direct solar radiation in the Northwest can be estimated for the winter months using the available data on a flat horizontal surface. Using the standard approach [18], it is found that the direct compo- nent is about one-half of the total. This gives an overall enhancement of the useful heat output ratio of the order of 1.80, in agreement with the estimated enhancement of 60% found in the second Section, although the present uncertainties in the Mathew house experimental data are large.

The present calculations appear to support a number of conclusions about the use of a reflector-solar thermal collector combination. First, the optimum orientation of the system for direct solar radiation occurs when the collector is oriented approximately perpendicular to the plane of the reflector. This conclusion is nearly indepen- dent of the time of day, and is modified only slightly by the inclusion of reflectivity losses at the collector glazings and reflector surface. The enhancement vs collector tilt angle curve is fairly flat within -+ 10 ° of the maximum and then drops off sharply. Including the effects of collector and reflector reflectance losses in the calcula- tions results in a sharper peaking of the enhancement vs tilt angle curves, with a small reduction in the versus tilt angle curves, with a small reduction in the optimum enhancement.

It is important to take the finite size of the reflector into account. Corrections due to the finite length and width of the reflector are greatest when the reflector is tilted above the horizontal, and the sun's hour angle is large. For a reasonable reflector length such as R / L = 2--4, the op- timum reflector orientation is found to be a few degrees above horizontal for winter operation. Undoubtedly, for actual solar conditions which include scattered radiation, a value within a few degrees either way of horizontal will prove satisfactory.

For typical winter operating conditions, the enhance- ment in light gathering ability for the reflector-collector combination is about 1.4-1.7. This enhancement increases slightly with hour angle. The enhancement for diffuse radiation is estimated to be about 1.25-1.30. However, for practical purposes it is the useful heat output of the collector that matters. Assuming winter operation in a somewhat cloudy climate gives an enhancement of the useful heat output of about 1+2.7 ( P - 1 ) over that obtained with an optimally oriented simple fiat-plate col- lector.

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Enhanced solar energy collection 283

4. A. Rosenblatt, Electronics 99, April 4 (1974); an evaluation of the NSF photovoltaic program.

5. H. Tabor, Solar Energy 7, 189 (1963); H. Weingerger, Solar Energy 8, 45 (1964). Some of the earlier articles on the features and potential of using solar ponds for storing solar energy.

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8. H. G. Lorsch, Proc. Solar Heating and Cooling for Buildings Workshop, Washington, C.D., p. 1, March 21-23 (1973).

9. A. F. Hildebrandt, G. M. Hoas, W. R. Jenkins and J. P. Colaco, Am. Geophysical Union 53, 684 (1972).

10. A. B. Meinel and M. P. Meinel, Physics Today, 44, Feb. (1972); R. B. Pope and W. P. Schimmel, Jr., Sandia Laboratories Report No. 739129, Albuquerque, New Mexico.

11. H. E. Thomason, Popular Mechanics (1%5); H. E. Thoma- son, Solar Houses and Solar Models, Edmund Scientific Co., Barrington, New Jersey; Sun at Work, First Quarter, 24 (1962).

12. Sec a description of this in the publication Power for the People, by A. B. Meinel and M. P. Meinel, McDonnell Douglas Corp. and Hello Associates, Inc.

13. J, Reynolds, Solar Energy for Pacific Northwest Buildings, Center for Environmental Research, University of Oregon (1974). Contains details and brief description of the solar house of H. Mathew.

14. MIT Builds Solar Heated Houses, Architectural Record 105, 135 (1949). A very small reflector was used, with an overall enhancement estimated as 5%.

15. H. H. Safwat and A. F. Souka, Solar Energy 13, 105 (1970). 16. H. C. Hottel and B. B. Woertz, Trans. ASME 64, 91 (1942).

NOMENCLATURE q~. direct solar radiation incident upon the absorber plate

when no reflector is present L vertical length of the collector W horizontal width of the collector

Io incident solar radiation h normal to the collector surface /3 angle between the normal to the collector surface and -Io

T(/3) transmission factor through the collector cover plates (< 1) to hour angle of the sun in the earth's coordinate system

07 tilt angle of the collector with the horizontal plane 0, zenith angle of the sun (measured from the vertical) R effective length of the reflector (see Fig. 1)

mean reflectivity of the collector surface for solar radiation

A latitude of the collector 8 solar declination

0b angle between the normal to the collector surface and rhe reflected beam

T(O,) transmission factor through the collector cover plates for the reflected beam

f~ fraction of incident light polarized perpendicular to the plane of incidence

fp fraction of incident light polarized parallel to the plane of incidence

m number of cover plates /3" angle of incidence of light upon an air-to-glass interface

(either/3 or 0n) /3' angle of refraction as given by Shell's law OR angle between plane of reflector and horizontal with

positive angles meaning a reflector orientation downwards (below horizontal)

0R ~ angle between the normal to the reflector surface and -Io R(t) effective reflector length at arbitrary time t

t time of day away from noon a absorptivity of collector plate for solar radiation ~- transmission factor of cover plates

AT (Tou, + T~.)/2- To,where Tout = outlet water temperature, T~o = inlet water temperature and To = effective ambient temperature

h, overall heat transfer coefficient away from collector due to all heat loss mechanisms

qbo solar flux incident on simple flat-plate collector at 0~ q,~ useful heat output of the reflector-collector combination qo useful heat output of the simple fiat-plate collector

oriented at 0~