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Solar thermal collector augmented by flat plate booster reflector: Optimum inclination of collector and reflector
Applied Energy 88 (2011) 1395–1404
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Solar thermal collector augmented by flat plate booster reflector: Optimum inclination of collector and reflector Hiroshi Tanaka ⇑ Department of Mechanical Engineering, Kurume National College of Technology, Komorino, Kurume, Fukuoka 830-8555, Japan
a r t i c l e
i n f o
Article history: Received 14 June 2010 Received in revised form 19 October 2010 Accepted 21 October 2010 Available online 20 November 2010 Keywords: Solar energy Solar thermal collector Flat plate reflector Collector–reflector Optimum inclination
a b s t r a c t In this report we present a theoretical analysis of a solar thermal collector with a flat plate top reflector. The top reflector extends from the upper edge of the collector, and can be inclined forwards or backwards from vertical according to the seasons. We theoretically predicted the daily solar radiation absorbed on an absorbing plate of the collector throughout the year, which varies considerably with the inclination of both the collector and reflector, and is slightly affected by the ratio of the reflector and collector length. We found the optimum inclination of the collector and reflector for each month at 30°N latitude. An increase in the daily solar radiation absorbed on the absorbing plate over a conventional solar thermal collector would average about 19%, 26% and 33% throughout the year by using the flat plate reflector when the ratio of reflector and collector length is 0.5, 1.0 and 2.0 and both the collector and reflector are adjusted to the proper inclination. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction For a solar thermal collector, booster reflectors can be a useful and inexpensive modification to increase the solar radiation incident on the solar thermal collector. Many studies have been done to investigate the effect of booster reflectors on solar thermal collectors [1–13]. Chiam [3], Garg and Hrishikesan [6] and Kostic et al. [12,13] studied a solar thermal collector with top and bottom reflectors. The top reflector extends from the upper edge of the collector and is inclined slightly from vertical, while the bottom reflector extends from the lower edge of the collector and is inclined slightly from horizontal. Garg and Hrishikesan [6] reported the optimum inclination of both the top and bottom reflectors on March, June and December when the collector inclination is horizontal or equal to the latitude where the collector is located. Kostic et al. [12] reported the optimum inclination of both the top and bottom reflectors throughout the year when the collector inclination is fixed at 45°. McDaniels and Lowndes [1], Rao et al. [7], Bollentin and Wilk [8] and Hussein et al. [9] studied the effect of the top reflector on the solar collector. McDaniels and Lowndes [1] reported the effect of the inclination of both the reflector and the collector on solar radiation absorbed onto the collector in winter. They also reported the effects of reflector size. Hussein et al. [9] reported the optimum inclination of the top reflector on 3 days (summer and winter sol⇑ Tel.: +81 942 35 9359; fax: +81 942 35 9321. E-mail address: [email protected] 0306-2619/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.apenergy.2010.10.032
stice and spring equinox days) when the collector inclination is 30°. Bollentin and Wilk [8] reported that the solar radiation incident on the collector varied with the inclination of reflector and collector, respectively. Rao et al. [7] investigated the effect of reflector inclination when the collector is set at horizontal. Further, Taha and Eldighidy [2] studied an off-south oriented solar collector and reflector system. The optimum inclination of a conventional solar thermal collector without booster reflector varies according to the seasons (or months), and can be easily determined. However, the optimum inclination for a solar thermal collector with a booster reflector may differ from that of a conventional solar collector. Further, the optimum inclination of the reflector would also vary with the seasons. So the optimum inclination of the collector as well as the optimum inclination of the booster reflector should be determined by considering the combination of these two inclinations to maximize solar radiation incident on the solar thermal collector. However, studies to determine these optimum inclinations have only been done in limited cases, and detailed analysis to determine the optimum inclination of both the collector and the reflector taking into account their combination has not been presented in spite of the fact that many studies have been done as mentioned above. Therefore, in this paper, the objective of the study is to theoretically determine the optimum inclination of the solar thermal collector as well as a flat plate booster reflector (top reflector only) throughout the year at 30°N latitude. We will also present a new graphical method to calculate solar radiation reflected by the flat plate top booster reflector and then absorbed onto the solar thermal collector.
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Nomenclatures Gdf, Gdr
diffuse and direct solar radiation on a horizontal surface, W/m2 I0 extra-terrestrial solar radiation, W/m2 Isc solar constant, W/m2 lc, lm length of collector and reflector, m nd number of day absorption of reflected solar radiation, W Qsun,re Qsun,df, Qsun,dr absorption of diffuse and direct solar radiation, W w width of collector and reflector, m
ac b /, u /0 , u0 hc hm
qm satm sg
absorptance of absorbing plate incident angle of sunrays to glass cover altitude and azimuth angle of the sun altitude and azimuth angle of reflected radiation angle of collector from horizontal angle of reflector from vertical reflectance of reflector transmittance of atmosphere transmittance of glass cover
Recently, we have performed numerical analysis of a tilted wick solar still with a flat plate top reflector. The tilted wick solar still consists of a glass cover and black wick cloth. We presented a geometrical model to calculate solar radiation reflected by a flat plate reflector and then absorbed onto the wick [14–16]. The geometrical model can be applied to the solar collector and reflector system since the tilted wick solar still is also a flat plate system as is the solar thermal collector. The proposed solar collector and reflector system is shown in Fig. 1. The solar collector consists of a glass cover and an absorbing plate, and a flat plate reflector is assumed to be made of highly reflective material. Direct and diffuse solar radiation and also the reflected solar radiation from the reflector are transmitted through the glass cover and then absorbed onto the absorbing plate. The reflector can be inclined forwards or backwards from vertical according to the seasons. In this paper, the inclination angle of the reflector was determined as positive when it was tilted for-
wards and negative when it was tilted backwards as shown in Fig. 1. In winter, the altitude angle of the sun decreases so a considerable amount of the reflected radiation from the vertical reflector would escape to the ground without hitting the collector. Therefore, the reflector should be inclined slightly forwards to absorb the reflected sunlight on the collector effectively as shown in Fig. 2a. On the other hand, the altitude angle of the sun increases in summer and the vertical reflector cannot effectively reflect sunrays to the collector. Therefore, the reflector should be inclined slightly toward the back as shown in Fig. 2b. Numerical models, such as vector analysis [2,4–6], graphical analysis [1,7–10] and two-dimensional analysis [12] have been proposed to calculate the solar radiation reflected by the inclined reflector and then absorbed onto the absorbing plate. In this paper, we introduce a new graphical analysis using geometrical models to calculate the absorption of solar radiation from an inclined reflector onto the absorbing plate. The geometrical models are similar to those for analyzing the tilted wick solar still with a flat plate reflector [14–16], and basically the same as the graphical analysis for solar collector reflector systems [1,7–10].
Fig. 1. Schematic diagram of a solar thermal collector and top reflector system that can be inclined backwards or forwards.
Fig. 2. Reflected sunlight from vertical and inclined reflectors on the absorbing plate in (a) winter and (b) summer.
2. Theoretical analysis 2.1. Solar thermal collector with inclined flat plate top reflector
H. Tanaka / Applied Energy 88 (2011) 1395–1404
To simplify the following calculations to determine the absorption of solar radiation on the absorbing plate of a solar collector, the walls of the solar collector are disregarded, since the height of the walls (10 mm) is negligible in relation to the collector’s length (1 m) and width (1 m). In this calculation, the solar collector and reflector are assumed to be facing due south to maximize the solar radiation on the absorbing plate. The design conditions and physical properties employed in this calculation are listed in Table 1. Practically, for a collector with heat losses, solar radiation at lower levels cannot be utilized. However, in the calculation, it was assumed that solar radiation can be absorbed and utilized even at lower levels.
2.2. Reflected radiation from a reflector inclined forward in winter Fig. 3 shows a schematic diagram of the shadows produced by the collector and the reflector inclined forward as well as the projection of the reflected sunlight from the inclined reflector on a horizontal surface caused by direct solar radiation. Here, the shadows of the glass cover and the absorbing plate of the collector would be exactly the same since the walls of collector are disregarded as mentioned above. lc is the length of the collector (ABCD), lm is the length of the reflector (ABEF), w is the width of both the collector and reflector, hc is the inclination of the collector from horizontal and hm is the inclination of the reflector from vertical. / and u are the altitude and azimuth angle of the sun.
Table 1 Design conditions and physical properties. w = 1 m, lc = 1 m, ac = 0.9, qm = 0.8 sg(b) [17]: sg(b) = 2.642cos b 2.163cos2 b 0.320cos3 b + 0.719cos4 b
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The shadow and reflected projection of the inclined reflector (ABEF) on a horizontal surface are shown as A00 B00 E00 F00 and A0 B0 E0 F0 , respectively. All of the reflected sunrays from the reflector cannot hit the collector, and some portion of the reflected sunrays will escape to the ground. Assuming that radiation having the same azimuth and altitude angle as the reflected radiation from the reflector hits the whole surface of the collector, the shadow of the collector would be an area shown as A0 B0 CD. The portion of the reflected radiation from the reflector which would be absorbed on the absorbing plate can be determined as the overlapping area of the reflected projection of the reflector (A0 B0 E0 F0 ) and the shadow of the collector (A0 B0 CD) shown as a trapezoid A0 B0 GF0 , and the residue shown as a triangle B0 E0 G would escape to the ground. Since the reflected radiation from the inclined reflector would be concentrated or diluted, the intensity of the reflected radiation from the inclined reflector on a horizontal surface can be determined as Gdr l4/l1, where Gdr is the direct solar radiation on a horizontal surface. Therefore, the solar radiation reflected from the inclined reflector and absorbed on the absorbing plate, Qsun,re, can be determined as the product of the intensity of the reflected radiation, Gdr l4/l1, the overlapping area of the reflected projection of the reflector and the shadow of the collector shown as A0 B0 GF0 , reflectance of the reflector, qm, transmittance of the glass cover, sg(b), and absorptance of the absorbing plate, ac, and this may be expressed as
Q sun;re ¼ Gdr
l4 1 sg ðbÞqm ac l1 w ðl2 þ l3 Þ 2 l1
ð1Þ
To calculate Eq. (1), the lengths of l1–l4 and the angles of x1 and x2 have to be determined. Therefore, isometric and side views of an inclined flat plate reflector (ABEF), which is directly placed on a horizontal surface, are shown in Fig. 4a and b. The shadow (ABE00 F00 ) and the reflected projection (ABE0 F0 ) of the reflector on a horizontal surface shown in Fig. 4a would be in exact accordance with those
Fig. 3. The shadow of the collector and the shadow and reflected projection of the forwardly inclined reflector. l1 = lmcos hm{tan(x3 2hm) + tan hm}, l2 = l1tan x1, l3 = l1tan x2, l4 = lm(cos hmcos u/tan / sin hm), l5 = lmcos hmsin|u|/tan /, l6 = (l7 + l9)tan x1, l7 = lcsin hctan(x3 2hm), l8 = lcsin hccos u/tan /, l9 = lcsin hctan hm, tan x1 = l5/l4, tan x2 = l6/ (lccos hc l7), tan x3 = cos u/tan /.
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Fig. 5. The overlapping area of the reflected projection of the forwardly inclined reflector and the shadow of the collector when the overlapping area is (a) a trapezoid and (b) a triangle in winter.
Fig. 4. (a) Isometric and (b) side views of a forwardly inclined reflector placed directly on a horizontal surface. l1 = lmcos hm{tan(x3 2hm) + tan hm}, l4 = lm(cos hmcos u/tan / sin hm), l5 = lmcos hmsin|u|/tan /, l10 = lmcos hmcos u/tan /, l11 = lmcos hm, tan x1 = l5/l4, tan x3 = l10/l11 = cos u/tan /.
shown as A00 B00 E00 F00 and A0 B0 E0 F0 in Fig. 3. Since the incident angle and reflected angle of the sunrays for the reflector would be the same as x3 hm shown in Fig. 4b, the length of the reflected projection of the inclined reflector, l1, can be determined as follows with the angle x3 (=tan1(cos u/tan /))
l1 ¼ lm cos hm ftanðx3 2hm Þ þ tan hm g
ð2Þ
By drawing an additional vertical line from point E to a horizontal surface (point M) in Fig. 4a, the length of l4 and l5 can be determined as follows:
l4 ¼ lm ðcos hm cos u= tan / sin hm Þ
ð3Þ
l5 ¼ lm cos hm sin juj= tan /
ð4Þ
Therefore, Qsun,re can be determined with lengths of l1–l11 and angles of x1–x3 shown in Figs. 3 and 4. When Qsun,re is calculated, there are three exceptions as follows: 1. When the length l7 > lccos hc, Qsun,re would be zero. 2. When l1 + l7 > lccos hc as shown in Fig. 5a, the overlapping area of the reflected projection of the reflector and the shadow of the collector would be a trapezoid shown as A0 B0 CH, and Qsun,re may be expressed by substituting the overlapping area in Eq. (1) (l1{w 1/2(l2 + l3)}) with
1 ðlc cos hc l7 Þ w ðlc cos hc l7 Þðtan x1 þ tan x2 Þ 2
ð5Þ
3. When the overlapping area of the reflected projection and the shadow forms a triangle A0 B0 I as shown in Fig. 5b, Qsun,re may be expressed by substituting the overlapping area in Eq. (1) with
1 2 sinðp=2 x1 Þ sinðp=2 x2 Þ w 2 sinðx1 þ x2 Þ
ð6Þ
The incident angle of sunlight from the reflector to the glass cover, b, in Eq. (1) can be expressed as
cos b ¼ sin /0 cos hc þ cos /0 sin hc cos u0
ð7Þ
where u0 and /0 are the azimuth and altitude angle of the reflected sunlight from the inclined reflector, and can be expressed as
/0 ¼ p=2 ðx3 2hm Þ
ð8Þ
u0 ¼ p x1
ð9Þ
2.3. Reflected radiation from a backwardly inclined reflector in summer Solar radiation reflected by a backwardly inclined reflector and then absorbed onto the absorbing plate can be calculated with a similar geometrical model to that of a forwardly inclined reflector. Diagrams for the collector and reflector system with the reflector inclined backward are shown in Figs. 6 and 7. The absorption of reflected solar radiation on the absorbing plate, Qsun,re, can be calculated using Eq. (1). However, the lengths l1, l4, l6 and l7 and angle x2 should be changed as shown in Figs. 6 and 7 to
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Fig. 6. The shadow of the collector and the shadow and reflected projection of the backwardly inclined reflector. l1 = lmcos hm{tan(x3 + 2hm) tan hm}, l2 = l1tan x1, l3 = l1tan x2, l4 = lm(sin hm + cos hmcos u/tan /), l5 = lmcos hmsin|u|/tan /, l6 = l7tan x1, l7 = lcsin hc{tan(x3 + 2hm) tan hm}, l8 = lcsin hccos u/tan /, l9 = lcsin hctanhm, tan x1 = l5/ l4, tan x2 = l6/{lccos hc (l7 + l9)}, tan x3 = cos u/tan /.
l1 ¼ lm cos hm ftanðx3 þ 2hm Þ tan hm g
ð10Þ
l4 ¼ lm ðsin hm þ cos hm cos u= tan /Þ
ð11Þ
l6 ¼ l7 tan x1
ð12Þ
l7 ¼ lc sin hc ftanðx3 þ 2hm Þ tan hm g
ð13Þ
x2 ¼ tan
1
l6 lc cos hc ðl7 þ l9 Þ
ð14Þ
The condition for the first exception (l7 > lccos hc) should be changed to l7 + l9 > lccos hc. The condition for the second exception (l1 + l7 > lccos hc) should be changed to l1 + l7 + l9 > lccos hc, and Eq. (5) to calculate the overlapping area of the second exception should be also changed as shown in Fig. 8 to
1 flc cos hc ðl7 þ l9 Þg w flc cos hc ðl7 þ l9 Þgðtan x1 þ tan x2 Þ 2 ð15Þ The azimuth and altitude angle of the reflected sunlight from the 0 0 backwardly inclined reflector, u and / used to calculate incident angle b in Eq. (7) may be expressed with u shown in Fig. 7a as
/0 ¼ tan1
lm cos hm cos u l1 þ lm sin hm
u0 ¼ p u u ¼ tan1
l2 l1 þ lm sin hm
ð16Þ ð17Þ ð18Þ
2.4. Direct and diffuse radiation The direct solar radiation absorbed on the absorbing plate, Qsun,dr, can be determined as the product of the direct solar radia-
tion on a horizontal surface, the shadow area of the collector on a horizontal surface (shown as A00 B00 CD) in Figs. 3 and 6, transmittance of the glass cover and absorptance of the absorbing plate, and this may be expressed as
Q sun;dr ¼ Gdr sg ðbÞac wlc ðcos hc þ sin hc cos u= tan /Þ
ð19Þ
cos b ¼ sin / cos hc þ cos / sin hc cos u
ð20Þ
Diffuse solar radiation absorbed on the absorbing plate, Qsun,df, can be determined assuming that diffuse radiation comes uniformly from all directions in the sky dome, and may be expressed as
Q sun;df ¼ Gdf ðsg Þdf ac wlc
ð21Þ
where Gdf is the diffuse solar radiation on a horizontal surface, and (sg)df is a function of inclination of collector, hc, and is calculated by integrating the transmittance of the glass cover for diffuse radiation from all directions in the sky dome. This may be expressed as
ðsg Þdf ¼ 2:03 105 h2c 2:05 103 hc þ 0:667; hc ½ ð22Þ 2.5. Effect of shadow of reflector During the months of April–August, the sun moves north, that is, azimuth angle of the sun, u, is larger than p/2 as shown in Fig. 9, in the morning and evening and the reflector obstructs the sunrays and shades the collector. The shadows of the collector and the reflector caused by the direct solar radiation on a horizontal surface when the sun moves north is shown in Fig. 9. The reflector shades the collector, and the shaded area is shown as A0 B0 GF0 . Therefore, Qsun,re and Qsun,dr when the sun moves north can be determined as
Q sun;re ¼ 0
ð23Þ
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Fig. 9. The shadows of the collector and the reflector when the sun moves north.
Fig. 7. (a) Isometric and (b) side views of backwardly inclined reflector placed directly on a horizontal surface. l1 = lmcos hm{tan(x3 + 2hm) tan hm}, l4 = lm(sin hm + cos hmcos u/tan /), l5 = lmcos hmsin|u|/tan /, l10 = lmcos hmcos u/ tan /, l11 = lmcos hm, tan x1 = l5/l4, tan x3 = l10/l11 = cos u/tan /.
Daily solar radiation absorbed on the absorbing plate, MJ/m2day
25 June May, July 20
Apr., Aug Mar., Sep. Jan., Nov.
15 Feb., Oct.
10
Dec.
5
0
0
10
20
30
40
50
Collector inclination, θc Fig. 10. The daily solar radiation absorbed on the absorbing plate without a reflector varying with collector inclination hc throughout the year at 30°N.
where the term
cosðp uÞ sin hm wlm cos hm tan / 1 2 sin jp uj cosðp uÞ lm cos hm cos hm sin hm 2 tan / tan /
ð25Þ
represents the effect of the shadow of the reflector on the absorbing plate. Fig. 8. The overlapping area of the reflected projection of the backwardly inclined reflector and the shadow of the collector when the overlapping area is a trapezoid in summer.
Q sun;dr
cosðp uÞ ¼ Gdr sg ðbÞac wlc cos hc sin hc tan / cosðp uÞ sin hm wlm cos hm tan / 1 2 sin jp uj cosðp uÞ lm cos hm cos hm sin hm 2 tan / tan / ð24Þ
2.6. Direct and diffuse solar radiation on a horizontal surface Direct and diffuse solar radiation on a horizontal surface, Gdr and Gdf, was calculated with Bouguer’s equation and Berlage’s equation respectively [18] as 1= sin /
Gdr ¼ I0 sin /satm
ð26Þ
sin / =ð1 1:4 ln satm Þ Gdf ¼ 0:5 I0 sin / 1 s1= atm
ð27Þ
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I0 ¼ Isc ½1 þ 0:033 cosf360ðnd 2Þ=365g
ð28Þ
The daily solar radiation absorbed on the absorbing plate of a solar thermal collector without a reflector varying with collector inclination hc throughout the year at 30°N is shown in Fig. 10. Daily solar radiation increases with an increase in inclination hc in winter, and decreases with an increase in hc in summer, since the solar altitude angle is high in summer and low in winter. In the 4 months of March, April, August and September, daily solar radiation has a gentle peak around hc = 20–30°. The daily solar radiation in five pairs of months (January and November, February and October, March and September, April and August, and May and July) would be almost the same, since the loci of the sun would be similar in each set of months. Isometric diagrams of the daily solar radiation (MJ/m2 day) absorbed on the absorbing plate of a solar thermal collector with an inclined reflector and a varying collector inclination hc and reflector inclination hm with a ratio of the reflector and collector length,
where I0 is extra-terrestrial solar radiation, satm is transmittance of atmosphere, Isc is solar constant (=1370 W/m2) and nd is the number of day during a year (nd = 1 on January 1 and 365 on December 31). In this calculation, satm was assumed to be 0.7. 3. Results In this paper, the daily solar radiation absorbed on the absorbing plate was calculated at 23rd day for each month since the equinox and solstice days are around 23rd. The azimuth and altitude angle of the sun and direct and diffuse solar radiation on a horizontal surface were calculated with 600 s steps from sunrise to sunset to determine the solar radiation absorbed on the absorbing plate for each day.
(a)
50 17 45 16 40 18 35 30 25 20 15 17 10 16 5 0 15 0 5 10 50 45 40 35 30 25 20 15 10 5 0
Jan. 18 17 16 15 14
15
20
25
18 19 20
30
19 20
23
Apr. 21 24 23
24
50 45 40 35 30 25 20 15 10 5 0 0
22
-5
-10 -15 -20 -25 -30
17 18 19 20
16 17 18 19
21
20 21 22
22 23
23 25
24
24
July -5
-10 -15 -20 -25 -30
50 18 45 19 40 20 35 30 21 25 20 15 20 10 19 5 18 0 0 5 10
Oct. 20
19 18
17
15
20
16
25
50 18 45 19 21 40 35 20 30 25 20 15 20 10 5 19 18 0 0 5 10 15 50 45 40 35 30 25 20 15 10 5 0
18
21 22
0
θc
19
30
17 18 19 20 21
Feb. 20
19 18 17
20
16
25
30
50 45 40 35 30 25 20 15 10 5 0
16 17 18 19 20 21 23 22
22
23 24
24
May 25
0
-5
50 21 45 40 22 35 30 25 23 20 15 22 10 5 21 0 0 5 10
25
-10 -15 -20 -25 -30
15 14
20
25
30
20
25
30
15 16 17 18 19 20 21 22 23
23
25
24
24
June 0
-5
50 45 40 35 14 30 25 20 15 10 5 13 0 0
16
18
15
22
50 17 45 16 40 18 35 30 Nov. 25 20 15 17 10 16 5 15 0 15 0 5 10 15
17
19
20
21
50 20 45 21 40 22 35 30 25 20 15 10 22 5 21 0 0 5
18
21 Mar.
16
16 17 18 19 20
50 17 18 19 45 18 20 40 21 19 35 22 30 20 23 25 20 21 15 Aug. 22 10 23 24 5 24 23 0 0 -5 -10 -15 -20 -25 -30
19
19 20
-10 -15 -20 -25 -30 18 19
20
Sep. 21
19
20 18 19
10
15
20
25
30
15 17 16
Dec.
17
16
15
15
14 13
14
5
10
15
20
25
30
θm Fig. 11. Isometric diagrams of daily solar radiation (MJ/m2 day) absorbed on the absorbing plate of the collector with an inclined reflector varying with collector inclination hc and reflector inclination hm throughout the year at 30°N when the ratio of reflector and collector length, lm/lc = (a) 1.0 and (b) 3.0.
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(b)
50 17 17 45 16 19 40 16 18 35 18 30 25 20 15 17 17 10 5 15 1616 15 0 0 5 10 15
Jan. 20
18
17 19 16 18 15 14
20
17
25
30
50 18 22 45 19 40 21 20 35 30 25 Feb. 20 22 15 10 21 5 20 0 0 5 10 15
50 18 19 45 18 20 40 19 21 35 22 23 30 20 24 25 Apr. 20 25 21 15 22 10 23 5 24 0 25 0 -5 -10 -15 -20 -25 -30
50 45 40 35 30 25 20 15 10 5 0
50 45 40 35 30 25 20 15 10 5 0
50 45 40 35 30 25 20 15 10 5 0
18 19 20 21 22 23
17
16 17 18 19
24 25
July 26
0
θc
20 19
-5
20 21 22 23 24 25 26
-10 -15 -20 -25 -30
50 22 18 45 19 40 20 21 35 30 Oct. 25 20 22 15 10 21 5 20 0 0 5 10 15
22
21
20 19 18
20
25
30
18 17 19 20 21 22 23
20 19 18
20
25
30
17 18 19 20 21 22 23 25 24
25
May 26
-5
21
16
24
0
21
22
26
-10 -15 -20 -25 -30
50 45 21 22 40 35 23 24 30 25 20 15 10 Mar. 5 23 0 0 5 10 50 45 40 35 30 25 20 15 10 5 0
19 22
21 20
23
19
22 21 20
18
15
20
25
30
15
17 18 19 20 21 22
16 17 18 19 20 21 22 23 24
23 24 25
25
June 26
0
-5
26
-10 -15 -20 -25 -30
50 45 21 22 18 40 21 19 35 23 22 23 30 20 24 24 25 Aug. 20 21 15 25 22 10 23 5 Sep. 24 0 23 0 -5 -10 -15 -20 -25 -30 0 5 17
18 19 20
50 17 45 19 40 16 35 18 30 25 20 15 17 10 5 15 16 0 0 5 10 15
20
Nov. 20
19 18 17
20
25
30
50 45 40 35 14 30 25 20 15 10 5 13 0 0
18 22
21 20 19
23
22 21 20 19
10
15
20
18
25
30
16 18 17
15
19
Dec. 18 16
17 16
15 14
5
15
10
15
20
25
30
θm Fig. 11 (continued)
lm/lc = (a) 1.0 and (b) 3.0 throughout the year at 30°N is shown in Fig. 11. The daily solar radiation absorbed on the absorbing plate for each month was calculated at 1° steps for both inclinations hc and hm, so 30 50 = 1500 calculations were done to investigate the effect of both inclinations in each month at each ratio lm/lc. Here, the reflector is inclined forwards from September to March (hm is positive), and backwards from April to August (hm is negative). The optimum combination of the inclinations hc and hm, that maximize the daily solar radiation, would vary considerably according to month, and be slightly affected by the ratio lm/lc. The optimum collector inclination hc is lower in summer and higher in winter for the same reasons as in the results for a collector without the reflector (Fig. 10) as mentioned above. The absolute value of optimum reflector inclination |hm| is lower in spring and autumn, and higher in summer and winter, since the solar altitude angle would be larger in summer and lower in winter than in spring and autumn.
The optimum inclination for the collector hc and reflector hm at ratios lm/lc = 1.0, 2.0 and 3.0 throughout the year is shown in Fig. 12. The optimum collector inclination hc is almost the same for any ratio of lm/lc. The optimum reflector inclination hm would vary with the ratio lm/lc, and would increase with an increase in the ratio lm/lc. The reason is as follows. In winter with a forwardly inclined reflector (Fig. 3), when l1 + l7 > lccos hc, the reflected radiation shown as A0 B0 E0 F0 exceeds the collector’s shadow (A0 B0 CD), and a part of or all of the reflected radiation from the reflector would escape to the ground without hitting the absorbing plate. By increasing the reflector inclination hm, the lengths l1 and l7 can be shortened, but the amount of reflected radiation from reflector shown as A00 B00 E00 F00 would decrease. So a smaller inclination hm is preferable as long as the reflected projection does not exceed the collector’s shadow. On the other hand, in summer with a backwardly inclined reflector (Fig. 6), the amount of reflected radiation can be increased by increasing reflector tilt angle (or decreasing
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30 θc (lm /lc=1.0)
40
Daily solar radiation absorbed on the absorbing plate, MJ/m2day
Optimum inclination of collector θc and reflector θ m
50
(lm /lc=2.0) (lm/lc=3.0)
30
θm (lm/lc=1.0) (lm/lc=2.0)
20
(lm/lc=3.0)
10
0
-10
-20
Summer
Autumn 20 Average
1
2
3
4
5
6
7
8
9
10
11
Winter
15
10
5
0*: θc is fixed at 30 o throughout the year 0
-30
Spring
25
0*
0
1
2
3
lm/lc
12
Month Fig. 13. Daily solar radiation absorbed on the absorbing plate varying with the ratio of reflector length to collector length, lm/lc for four seasons at 30°N.
Fig. 12. Optimum inclination of collector hc and reflector hm at the ratio of reflector and collector length, lm/lc = 1.0, 2.0 and 3.0 throughout the year at 30°N.
hm since hm is negative in summer as shown in Fig. 1), but the lengths l1 and l7 would also be elongated with an increase in reflector tilt angle as shown in Fig. 6. So a larger reflector tilt angle (or smaller hm) is preferable as long as the reflected projection does not exceed the collector’s shadow conversely to winter. The optimum collector inclination hc and reflector inclination hm throughout the year are listed in Table 2. Similar analysis with Fig. 11 was done for lm/lc = 0.5 and 2.0 to determine the optimum inclinations of hc and hm for each month. lm/lc = 0 shows the collector without a reflector. Here, to facilitate ease of setting, the optimum inclinations hc and hm are assumed to be set at 5° steps. The decline of daily solar radiation absorbed on the absorbing plate by setting the inclination hc and hm to the values listed in Table 2 compared with daily solar radiation with hc and hm shown in Fig. 12 is less than 1%. Therefore, the solar thermal collector with an inclined reflector can receive near the maximum solar radiation by setting the inclination hc and hm listed in Table 2 on each month according to the ratio lm/lc at 30°N. Daily solar radiation absorbed on the absorbing plate varying with the ratio lm/lc for four seasons (spring (February–April), summer (May–July), autumn (August–October) and winter (November–January)) and the average values for 12 months at 30°N is shown in Fig. 13. Here, both inclinations hc and hm are assumed to be set at the optimum angle listed in Table 2 for each month. lm/lc = 0 shows the results for the collector without a reflector in which the inclination hc is set to optimum as listed in Table 2, and lm/lc = 0 shows the results for a collector without a reflector in which the inclination hc is fixed at 30° throughout the year.
The daily solar radiation is highest in summer and lowest in winter, and in spring and autumn and the average values are almost the same. The daily solar radiation can be increased by using an inclined flat plate reflector and adjusting the inclinations hc and hm to the proper values. As a result, compared with the conventional solar thermal collector (lm/lc = 0: collector without reflector and hc is fixed at 30°), the average daily solar radiation throughout the year can be increased about 6% by adjusting the collector inclination hc to proper values on each month (lm/lc = 0), and about 19%, 26% and 33% by using an inclined flat plate reflector and adjusting both the inclinations hc and hm to proper values for each month at lm/lc = 0.5, 1.0 and 2.0.
4. Conclusions We have performed numerical analysis of a solar thermal collector with a flat plate top reflector, which extends from the upper edge of the collector. In this paper, the collector was assumed to be a square shape (w = lc = 1 m). The numerical analysis to calculate the solar radiation absorbed on the absorbing plate would be valid when the collector have rectangular shape and both the widths of the collector and the reflector are the same as w. However, the optimum collector and reflector inclinations would be different if the collector do not have square shape. We have determined the optimum inclination of both the collector and reflector throughout the year at 30°N latitude and investigated the effect of the size of the reflector on the daily solar radiation absorbed on the absorbing
Table 2 Optimum inclinations of collector and reflector throughout the year at 5° steps. lm/lc = 0 shows the collector without reflector. Month
December January, November February, October March, September April, August May, July June
lm/lc = 0
lm/lc = 0.5
hc
hc
hm
lm/lc = 1.0 hc
hm
lm/lc = 2.0 hc
hm
lm/lc = 3.0 hc
hm
50 50 45 30 15 5 0
50 45 35 25 10 0 0
25 20 10 0 15 25 25
50 45 35 25 5 0 0
25 20 10 0 15 20 25
45 45 35 25 0 0 0
25 25 15 5 10 15 20
50 45 40 25 5 0 0
30 25 20 10 5 15 15
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H. Tanaka / Applied Energy 88 (2011) 1395–1404
plate of the collector. The results of this work are summarized as follows: (1) The solar radiation absorbed on the absorbing plate of the collector can be increased by using a flat plate top reflector, which is inclined forwards in winter and backwards in summer, and by setting the inclination angle of the reflector at less than 30° throughout the year. (2) The optimum collector inclination is lower in summer and higher in winter. (3) The optimum collector inclination hc is slightly affected by the ratio of reflector and collector lengths, lm/lc, while the optimum reflector inclination hm is considerably affected by the ratio lm/lc, and would increase with an increase in the ratio lm/lc. (4) An increase in the daily solar radiation absorbed on the absorbing plate over a conventional solar thermal collector would average about 19%, 26% and 33% throughout the year by using a flat plate reflector at a ratio of lm/lc = 0.5, 1.0 and 2.0 and adjusting the inclination of both the collector and reflector to the proper angle.
References [1] McDaniels DK, Lowndes DH. Enhanced solar energy collection using reflectorsolar thermal collector combinations. Solar Energy 1975;17:277–83. [2] Taha IS, Eldighidy SM. Effect of off-south orientation on optimum conditions for maximum solar energy absorbed by flat plate collector augmented by plane reflector. Solar Energy 1980;25:373–9.
[3] Chiam HF. Stationary reflector-augmented flat-plate collectors. Solar Energy 1982;29:65–9. [4] Arata AA, Geddes RW. Combined collector–reflector systems. Energy 1986;11:621–30. [5] Dang A. Collector, collector–reflector systems – an analytical and practical study. Energy Convers Manage 1986;26:33–9. [6] Garg HP, Hrishikesan DS. Enhancement of solar energy on flat-plate collector by plane booster mirrors. Solar Energy 1988;40:295–307. [7] Rao AVN, Chalam RV, Subramanyam S, Rao TLS. Energy contribution by booster mirrors. Energy Convers Manage 1993;34:309–26. [8] Bollentin JW, Wilk RD. Modeling the solar irradiation on flat plate collectors augmented with planar reflectors. Solar Energy 1995;55:343–54. [9] Hussein HMS, Ahmad GE, Mohamad MA. Optimization of operational and design parameters of plane reflector-tilted flat plate solar collector systems. Energy 2000;25:529–42. [10] Pucar MDJ, Despic AR. The enhancement of energy gain of solar collectors and photovoltaic panels by the reflection of solar beams. Energy 2002;27:205–23. [11] Hellstrom B, Adsten M, Nostell P, Karlsson B, Wackelgard E. The impact of optical and thermal properties on the performance of flat plate solar collectors. Renew Energy 2003;28:331–44. [12] Kostic LT, Pavlovic TM, Pavlovic ZT. Optimal design of orientation of PV/T collector with reflectors. Appl Energy 2010;87:3023–9. [13] Kostic LT, Pavlovic TM, Pavlovic ZT. Influence of reflectance from flat aluminum concentrators on energy efficiency of PV/Thermal collector. Appl Energy 2010;87:410–6. [14] Tanaka H, Nakatake Y. Improvement of the tilted wick solar still by using a flat plate reflector. Desalination 2007;216:139–46. [15] Tanaka H, Nakatake Y. Increase in distillate productivity by inclining the flat plate external reflector of a tilted-wick solar still in winter. Solar Energy 2009;83:785–9. [16] Tanaka H. Tilted wick solar still with external flat plate reflector: optimum inclination of still and reflector. Desalination 2009;249:411–5. [17] Tanaka H, Nosoko T, Nagata T. A highly productive basin-type – multiple-effect coupled solar still. Desalination 2000;130:279–93. [18] Japan Solar Energy Soc. Solar energy utilization handbook. Tokyo: Onkodo Press; 1985. p. 1,7,20.
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Applied Energy journal homepage: www.elsevier.com/locate/apenergy
Solar thermal collector augmented by flat plate booster reflector: Optimum inclination of collector and reflector Hiroshi Tanaka ⇑ Department of Mechanical Engineering, Kurume National College of Technology, Komorino, Kurume, Fukuoka 830-8555, Japan
a r t i c l e
i n f o
Article history: Received 14 June 2010 Received in revised form 19 October 2010 Accepted 21 October 2010 Available online 20 November 2010 Keywords: Solar energy Solar thermal collector Flat plate reflector Collector–reflector Optimum inclination
a b s t r a c t In this report we present a theoretical analysis of a solar thermal collector with a flat plate top reflector. The top reflector extends from the upper edge of the collector, and can be inclined forwards or backwards from vertical according to the seasons. We theoretically predicted the daily solar radiation absorbed on an absorbing plate of the collector throughout the year, which varies considerably with the inclination of both the collector and reflector, and is slightly affected by the ratio of the reflector and collector length. We found the optimum inclination of the collector and reflector for each month at 30°N latitude. An increase in the daily solar radiation absorbed on the absorbing plate over a conventional solar thermal collector would average about 19%, 26% and 33% throughout the year by using the flat plate reflector when the ratio of reflector and collector length is 0.5, 1.0 and 2.0 and both the collector and reflector are adjusted to the proper inclination. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction For a solar thermal collector, booster reflectors can be a useful and inexpensive modification to increase the solar radiation incident on the solar thermal collector. Many studies have been done to investigate the effect of booster reflectors on solar thermal collectors [1–13]. Chiam [3], Garg and Hrishikesan [6] and Kostic et al. [12,13] studied a solar thermal collector with top and bottom reflectors. The top reflector extends from the upper edge of the collector and is inclined slightly from vertical, while the bottom reflector extends from the lower edge of the collector and is inclined slightly from horizontal. Garg and Hrishikesan [6] reported the optimum inclination of both the top and bottom reflectors on March, June and December when the collector inclination is horizontal or equal to the latitude where the collector is located. Kostic et al. [12] reported the optimum inclination of both the top and bottom reflectors throughout the year when the collector inclination is fixed at 45°. McDaniels and Lowndes [1], Rao et al. [7], Bollentin and Wilk [8] and Hussein et al. [9] studied the effect of the top reflector on the solar collector. McDaniels and Lowndes [1] reported the effect of the inclination of both the reflector and the collector on solar radiation absorbed onto the collector in winter. They also reported the effects of reflector size. Hussein et al. [9] reported the optimum inclination of the top reflector on 3 days (summer and winter sol⇑ Tel.: +81 942 35 9359; fax: +81 942 35 9321. E-mail address: [email protected] 0306-2619/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.apenergy.2010.10.032
stice and spring equinox days) when the collector inclination is 30°. Bollentin and Wilk [8] reported that the solar radiation incident on the collector varied with the inclination of reflector and collector, respectively. Rao et al. [7] investigated the effect of reflector inclination when the collector is set at horizontal. Further, Taha and Eldighidy [2] studied an off-south oriented solar collector and reflector system. The optimum inclination of a conventional solar thermal collector without booster reflector varies according to the seasons (or months), and can be easily determined. However, the optimum inclination for a solar thermal collector with a booster reflector may differ from that of a conventional solar collector. Further, the optimum inclination of the reflector would also vary with the seasons. So the optimum inclination of the collector as well as the optimum inclination of the booster reflector should be determined by considering the combination of these two inclinations to maximize solar radiation incident on the solar thermal collector. However, studies to determine these optimum inclinations have only been done in limited cases, and detailed analysis to determine the optimum inclination of both the collector and the reflector taking into account their combination has not been presented in spite of the fact that many studies have been done as mentioned above. Therefore, in this paper, the objective of the study is to theoretically determine the optimum inclination of the solar thermal collector as well as a flat plate booster reflector (top reflector only) throughout the year at 30°N latitude. We will also present a new graphical method to calculate solar radiation reflected by the flat plate top booster reflector and then absorbed onto the solar thermal collector.
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Nomenclatures Gdf, Gdr
diffuse and direct solar radiation on a horizontal surface, W/m2 I0 extra-terrestrial solar radiation, W/m2 Isc solar constant, W/m2 lc, lm length of collector and reflector, m nd number of day absorption of reflected solar radiation, W Qsun,re Qsun,df, Qsun,dr absorption of diffuse and direct solar radiation, W w width of collector and reflector, m
ac b /, u /0 , u0 hc hm
qm satm sg
absorptance of absorbing plate incident angle of sunrays to glass cover altitude and azimuth angle of the sun altitude and azimuth angle of reflected radiation angle of collector from horizontal angle of reflector from vertical reflectance of reflector transmittance of atmosphere transmittance of glass cover
Recently, we have performed numerical analysis of a tilted wick solar still with a flat plate top reflector. The tilted wick solar still consists of a glass cover and black wick cloth. We presented a geometrical model to calculate solar radiation reflected by a flat plate reflector and then absorbed onto the wick [14–16]. The geometrical model can be applied to the solar collector and reflector system since the tilted wick solar still is also a flat plate system as is the solar thermal collector. The proposed solar collector and reflector system is shown in Fig. 1. The solar collector consists of a glass cover and an absorbing plate, and a flat plate reflector is assumed to be made of highly reflective material. Direct and diffuse solar radiation and also the reflected solar radiation from the reflector are transmitted through the glass cover and then absorbed onto the absorbing plate. The reflector can be inclined forwards or backwards from vertical according to the seasons. In this paper, the inclination angle of the reflector was determined as positive when it was tilted for-
wards and negative when it was tilted backwards as shown in Fig. 1. In winter, the altitude angle of the sun decreases so a considerable amount of the reflected radiation from the vertical reflector would escape to the ground without hitting the collector. Therefore, the reflector should be inclined slightly forwards to absorb the reflected sunlight on the collector effectively as shown in Fig. 2a. On the other hand, the altitude angle of the sun increases in summer and the vertical reflector cannot effectively reflect sunrays to the collector. Therefore, the reflector should be inclined slightly toward the back as shown in Fig. 2b. Numerical models, such as vector analysis [2,4–6], graphical analysis [1,7–10] and two-dimensional analysis [12] have been proposed to calculate the solar radiation reflected by the inclined reflector and then absorbed onto the absorbing plate. In this paper, we introduce a new graphical analysis using geometrical models to calculate the absorption of solar radiation from an inclined reflector onto the absorbing plate. The geometrical models are similar to those for analyzing the tilted wick solar still with a flat plate reflector [14–16], and basically the same as the graphical analysis for solar collector reflector systems [1,7–10].
Fig. 1. Schematic diagram of a solar thermal collector and top reflector system that can be inclined backwards or forwards.
Fig. 2. Reflected sunlight from vertical and inclined reflectors on the absorbing plate in (a) winter and (b) summer.
2. Theoretical analysis 2.1. Solar thermal collector with inclined flat plate top reflector
H. Tanaka / Applied Energy 88 (2011) 1395–1404
To simplify the following calculations to determine the absorption of solar radiation on the absorbing plate of a solar collector, the walls of the solar collector are disregarded, since the height of the walls (10 mm) is negligible in relation to the collector’s length (1 m) and width (1 m). In this calculation, the solar collector and reflector are assumed to be facing due south to maximize the solar radiation on the absorbing plate. The design conditions and physical properties employed in this calculation are listed in Table 1. Practically, for a collector with heat losses, solar radiation at lower levels cannot be utilized. However, in the calculation, it was assumed that solar radiation can be absorbed and utilized even at lower levels.
2.2. Reflected radiation from a reflector inclined forward in winter Fig. 3 shows a schematic diagram of the shadows produced by the collector and the reflector inclined forward as well as the projection of the reflected sunlight from the inclined reflector on a horizontal surface caused by direct solar radiation. Here, the shadows of the glass cover and the absorbing plate of the collector would be exactly the same since the walls of collector are disregarded as mentioned above. lc is the length of the collector (ABCD), lm is the length of the reflector (ABEF), w is the width of both the collector and reflector, hc is the inclination of the collector from horizontal and hm is the inclination of the reflector from vertical. / and u are the altitude and azimuth angle of the sun.
Table 1 Design conditions and physical properties. w = 1 m, lc = 1 m, ac = 0.9, qm = 0.8 sg(b) [17]: sg(b) = 2.642cos b 2.163cos2 b 0.320cos3 b + 0.719cos4 b
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The shadow and reflected projection of the inclined reflector (ABEF) on a horizontal surface are shown as A00 B00 E00 F00 and A0 B0 E0 F0 , respectively. All of the reflected sunrays from the reflector cannot hit the collector, and some portion of the reflected sunrays will escape to the ground. Assuming that radiation having the same azimuth and altitude angle as the reflected radiation from the reflector hits the whole surface of the collector, the shadow of the collector would be an area shown as A0 B0 CD. The portion of the reflected radiation from the reflector which would be absorbed on the absorbing plate can be determined as the overlapping area of the reflected projection of the reflector (A0 B0 E0 F0 ) and the shadow of the collector (A0 B0 CD) shown as a trapezoid A0 B0 GF0 , and the residue shown as a triangle B0 E0 G would escape to the ground. Since the reflected radiation from the inclined reflector would be concentrated or diluted, the intensity of the reflected radiation from the inclined reflector on a horizontal surface can be determined as Gdr l4/l1, where Gdr is the direct solar radiation on a horizontal surface. Therefore, the solar radiation reflected from the inclined reflector and absorbed on the absorbing plate, Qsun,re, can be determined as the product of the intensity of the reflected radiation, Gdr l4/l1, the overlapping area of the reflected projection of the reflector and the shadow of the collector shown as A0 B0 GF0 , reflectance of the reflector, qm, transmittance of the glass cover, sg(b), and absorptance of the absorbing plate, ac, and this may be expressed as
Q sun;re ¼ Gdr
l4 1 sg ðbÞqm ac l1 w ðl2 þ l3 Þ 2 l1
ð1Þ
To calculate Eq. (1), the lengths of l1–l4 and the angles of x1 and x2 have to be determined. Therefore, isometric and side views of an inclined flat plate reflector (ABEF), which is directly placed on a horizontal surface, are shown in Fig. 4a and b. The shadow (ABE00 F00 ) and the reflected projection (ABE0 F0 ) of the reflector on a horizontal surface shown in Fig. 4a would be in exact accordance with those
Fig. 3. The shadow of the collector and the shadow and reflected projection of the forwardly inclined reflector. l1 = lmcos hm{tan(x3 2hm) + tan hm}, l2 = l1tan x1, l3 = l1tan x2, l4 = lm(cos hmcos u/tan / sin hm), l5 = lmcos hmsin|u|/tan /, l6 = (l7 + l9)tan x1, l7 = lcsin hctan(x3 2hm), l8 = lcsin hccos u/tan /, l9 = lcsin hctan hm, tan x1 = l5/l4, tan x2 = l6/ (lccos hc l7), tan x3 = cos u/tan /.
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Fig. 5. The overlapping area of the reflected projection of the forwardly inclined reflector and the shadow of the collector when the overlapping area is (a) a trapezoid and (b) a triangle in winter.
Fig. 4. (a) Isometric and (b) side views of a forwardly inclined reflector placed directly on a horizontal surface. l1 = lmcos hm{tan(x3 2hm) + tan hm}, l4 = lm(cos hmcos u/tan / sin hm), l5 = lmcos hmsin|u|/tan /, l10 = lmcos hmcos u/tan /, l11 = lmcos hm, tan x1 = l5/l4, tan x3 = l10/l11 = cos u/tan /.
shown as A00 B00 E00 F00 and A0 B0 E0 F0 in Fig. 3. Since the incident angle and reflected angle of the sunrays for the reflector would be the same as x3 hm shown in Fig. 4b, the length of the reflected projection of the inclined reflector, l1, can be determined as follows with the angle x3 (=tan1(cos u/tan /))
l1 ¼ lm cos hm ftanðx3 2hm Þ þ tan hm g
ð2Þ
By drawing an additional vertical line from point E to a horizontal surface (point M) in Fig. 4a, the length of l4 and l5 can be determined as follows:
l4 ¼ lm ðcos hm cos u= tan / sin hm Þ
ð3Þ
l5 ¼ lm cos hm sin juj= tan /
ð4Þ
Therefore, Qsun,re can be determined with lengths of l1–l11 and angles of x1–x3 shown in Figs. 3 and 4. When Qsun,re is calculated, there are three exceptions as follows: 1. When the length l7 > lccos hc, Qsun,re would be zero. 2. When l1 + l7 > lccos hc as shown in Fig. 5a, the overlapping area of the reflected projection of the reflector and the shadow of the collector would be a trapezoid shown as A0 B0 CH, and Qsun,re may be expressed by substituting the overlapping area in Eq. (1) (l1{w 1/2(l2 + l3)}) with
1 ðlc cos hc l7 Þ w ðlc cos hc l7 Þðtan x1 þ tan x2 Þ 2
ð5Þ
3. When the overlapping area of the reflected projection and the shadow forms a triangle A0 B0 I as shown in Fig. 5b, Qsun,re may be expressed by substituting the overlapping area in Eq. (1) with
1 2 sinðp=2 x1 Þ sinðp=2 x2 Þ w 2 sinðx1 þ x2 Þ
ð6Þ
The incident angle of sunlight from the reflector to the glass cover, b, in Eq. (1) can be expressed as
cos b ¼ sin /0 cos hc þ cos /0 sin hc cos u0
ð7Þ
where u0 and /0 are the azimuth and altitude angle of the reflected sunlight from the inclined reflector, and can be expressed as
/0 ¼ p=2 ðx3 2hm Þ
ð8Þ
u0 ¼ p x1
ð9Þ
2.3. Reflected radiation from a backwardly inclined reflector in summer Solar radiation reflected by a backwardly inclined reflector and then absorbed onto the absorbing plate can be calculated with a similar geometrical model to that of a forwardly inclined reflector. Diagrams for the collector and reflector system with the reflector inclined backward are shown in Figs. 6 and 7. The absorption of reflected solar radiation on the absorbing plate, Qsun,re, can be calculated using Eq. (1). However, the lengths l1, l4, l6 and l7 and angle x2 should be changed as shown in Figs. 6 and 7 to
H. Tanaka / Applied Energy 88 (2011) 1395–1404
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Fig. 6. The shadow of the collector and the shadow and reflected projection of the backwardly inclined reflector. l1 = lmcos hm{tan(x3 + 2hm) tan hm}, l2 = l1tan x1, l3 = l1tan x2, l4 = lm(sin hm + cos hmcos u/tan /), l5 = lmcos hmsin|u|/tan /, l6 = l7tan x1, l7 = lcsin hc{tan(x3 + 2hm) tan hm}, l8 = lcsin hccos u/tan /, l9 = lcsin hctanhm, tan x1 = l5/ l4, tan x2 = l6/{lccos hc (l7 + l9)}, tan x3 = cos u/tan /.
l1 ¼ lm cos hm ftanðx3 þ 2hm Þ tan hm g
ð10Þ
l4 ¼ lm ðsin hm þ cos hm cos u= tan /Þ
ð11Þ
l6 ¼ l7 tan x1
ð12Þ
l7 ¼ lc sin hc ftanðx3 þ 2hm Þ tan hm g
ð13Þ
x2 ¼ tan
1
l6 lc cos hc ðl7 þ l9 Þ
ð14Þ
The condition for the first exception (l7 > lccos hc) should be changed to l7 + l9 > lccos hc. The condition for the second exception (l1 + l7 > lccos hc) should be changed to l1 + l7 + l9 > lccos hc, and Eq. (5) to calculate the overlapping area of the second exception should be also changed as shown in Fig. 8 to
1 flc cos hc ðl7 þ l9 Þg w flc cos hc ðl7 þ l9 Þgðtan x1 þ tan x2 Þ 2 ð15Þ The azimuth and altitude angle of the reflected sunlight from the 0 0 backwardly inclined reflector, u and / used to calculate incident angle b in Eq. (7) may be expressed with u shown in Fig. 7a as
/0 ¼ tan1
lm cos hm cos u l1 þ lm sin hm
u0 ¼ p u u ¼ tan1
l2 l1 þ lm sin hm
ð16Þ ð17Þ ð18Þ
2.4. Direct and diffuse radiation The direct solar radiation absorbed on the absorbing plate, Qsun,dr, can be determined as the product of the direct solar radia-
tion on a horizontal surface, the shadow area of the collector on a horizontal surface (shown as A00 B00 CD) in Figs. 3 and 6, transmittance of the glass cover and absorptance of the absorbing plate, and this may be expressed as
Q sun;dr ¼ Gdr sg ðbÞac wlc ðcos hc þ sin hc cos u= tan /Þ
ð19Þ
cos b ¼ sin / cos hc þ cos / sin hc cos u
ð20Þ
Diffuse solar radiation absorbed on the absorbing plate, Qsun,df, can be determined assuming that diffuse radiation comes uniformly from all directions in the sky dome, and may be expressed as
Q sun;df ¼ Gdf ðsg Þdf ac wlc
ð21Þ
where Gdf is the diffuse solar radiation on a horizontal surface, and (sg)df is a function of inclination of collector, hc, and is calculated by integrating the transmittance of the glass cover for diffuse radiation from all directions in the sky dome. This may be expressed as
ðsg Þdf ¼ 2:03 105 h2c 2:05 103 hc þ 0:667; hc ½ ð22Þ 2.5. Effect of shadow of reflector During the months of April–August, the sun moves north, that is, azimuth angle of the sun, u, is larger than p/2 as shown in Fig. 9, in the morning and evening and the reflector obstructs the sunrays and shades the collector. The shadows of the collector and the reflector caused by the direct solar radiation on a horizontal surface when the sun moves north is shown in Fig. 9. The reflector shades the collector, and the shaded area is shown as A0 B0 GF0 . Therefore, Qsun,re and Qsun,dr when the sun moves north can be determined as
Q sun;re ¼ 0
ð23Þ
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Fig. 9. The shadows of the collector and the reflector when the sun moves north.
Fig. 7. (a) Isometric and (b) side views of backwardly inclined reflector placed directly on a horizontal surface. l1 = lmcos hm{tan(x3 + 2hm) tan hm}, l4 = lm(sin hm + cos hmcos u/tan /), l5 = lmcos hmsin|u|/tan /, l10 = lmcos hmcos u/ tan /, l11 = lmcos hm, tan x1 = l5/l4, tan x3 = l10/l11 = cos u/tan /.
Daily solar radiation absorbed on the absorbing plate, MJ/m2day
25 June May, July 20
Apr., Aug Mar., Sep. Jan., Nov.
15 Feb., Oct.
10
Dec.
5
0
0
10
20
30
40
50
Collector inclination, θc Fig. 10. The daily solar radiation absorbed on the absorbing plate without a reflector varying with collector inclination hc throughout the year at 30°N.
where the term
cosðp uÞ sin hm wlm cos hm tan / 1 2 sin jp uj cosðp uÞ lm cos hm cos hm sin hm 2 tan / tan /
ð25Þ
represents the effect of the shadow of the reflector on the absorbing plate. Fig. 8. The overlapping area of the reflected projection of the backwardly inclined reflector and the shadow of the collector when the overlapping area is a trapezoid in summer.
Q sun;dr
cosðp uÞ ¼ Gdr sg ðbÞac wlc cos hc sin hc tan / cosðp uÞ sin hm wlm cos hm tan / 1 2 sin jp uj cosðp uÞ lm cos hm cos hm sin hm 2 tan / tan / ð24Þ
2.6. Direct and diffuse solar radiation on a horizontal surface Direct and diffuse solar radiation on a horizontal surface, Gdr and Gdf, was calculated with Bouguer’s equation and Berlage’s equation respectively [18] as 1= sin /
Gdr ¼ I0 sin /satm
ð26Þ
sin / =ð1 1:4 ln satm Þ Gdf ¼ 0:5 I0 sin / 1 s1= atm
ð27Þ
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H. Tanaka / Applied Energy 88 (2011) 1395–1404
I0 ¼ Isc ½1 þ 0:033 cosf360ðnd 2Þ=365g
ð28Þ
The daily solar radiation absorbed on the absorbing plate of a solar thermal collector without a reflector varying with collector inclination hc throughout the year at 30°N is shown in Fig. 10. Daily solar radiation increases with an increase in inclination hc in winter, and decreases with an increase in hc in summer, since the solar altitude angle is high in summer and low in winter. In the 4 months of March, April, August and September, daily solar radiation has a gentle peak around hc = 20–30°. The daily solar radiation in five pairs of months (January and November, February and October, March and September, April and August, and May and July) would be almost the same, since the loci of the sun would be similar in each set of months. Isometric diagrams of the daily solar radiation (MJ/m2 day) absorbed on the absorbing plate of a solar thermal collector with an inclined reflector and a varying collector inclination hc and reflector inclination hm with a ratio of the reflector and collector length,
where I0 is extra-terrestrial solar radiation, satm is transmittance of atmosphere, Isc is solar constant (=1370 W/m2) and nd is the number of day during a year (nd = 1 on January 1 and 365 on December 31). In this calculation, satm was assumed to be 0.7. 3. Results In this paper, the daily solar radiation absorbed on the absorbing plate was calculated at 23rd day for each month since the equinox and solstice days are around 23rd. The azimuth and altitude angle of the sun and direct and diffuse solar radiation on a horizontal surface were calculated with 600 s steps from sunrise to sunset to determine the solar radiation absorbed on the absorbing plate for each day.
(a)
50 17 45 16 40 18 35 30 25 20 15 17 10 16 5 0 15 0 5 10 50 45 40 35 30 25 20 15 10 5 0
Jan. 18 17 16 15 14
15
20
25
18 19 20
30
19 20
23
Apr. 21 24 23
24
50 45 40 35 30 25 20 15 10 5 0 0
22
-5
-10 -15 -20 -25 -30
17 18 19 20
16 17 18 19
21
20 21 22
22 23
23 25
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July -5
-10 -15 -20 -25 -30
50 18 45 19 40 20 35 30 21 25 20 15 20 10 19 5 18 0 0 5 10
Oct. 20
19 18
17
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20
16
25
50 18 45 19 21 40 35 20 30 25 20 15 20 10 5 19 18 0 0 5 10 15 50 45 40 35 30 25 20 15 10 5 0
18
21 22
0
θc
19
30
17 18 19 20 21
Feb. 20
19 18 17
20
16
25
30
50 45 40 35 30 25 20 15 10 5 0
16 17 18 19 20 21 23 22
22
23 24
24
May 25
0
-5
50 21 45 40 22 35 30 25 23 20 15 22 10 5 21 0 0 5 10
25
-10 -15 -20 -25 -30
15 14
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30
20
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30
15 16 17 18 19 20 21 22 23
23
25
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24
June 0
-5
50 45 40 35 14 30 25 20 15 10 5 13 0 0
16
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50 17 45 16 40 18 35 30 Nov. 25 20 15 17 10 16 5 15 0 15 0 5 10 15
17
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50 20 45 21 40 22 35 30 25 20 15 10 22 5 21 0 0 5
18
21 Mar.
16
16 17 18 19 20
50 17 18 19 45 18 20 40 21 19 35 22 30 20 23 25 20 21 15 Aug. 22 10 23 24 5 24 23 0 0 -5 -10 -15 -20 -25 -30
19
19 20
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20
Sep. 21
19
20 18 19
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Dec.
17
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14 13
14
5
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30
θm Fig. 11. Isometric diagrams of daily solar radiation (MJ/m2 day) absorbed on the absorbing plate of the collector with an inclined reflector varying with collector inclination hc and reflector inclination hm throughout the year at 30°N when the ratio of reflector and collector length, lm/lc = (a) 1.0 and (b) 3.0.
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(b)
50 17 17 45 16 19 40 16 18 35 18 30 25 20 15 17 17 10 5 15 1616 15 0 0 5 10 15
Jan. 20
18
17 19 16 18 15 14
20
17
25
30
50 18 22 45 19 40 21 20 35 30 25 Feb. 20 22 15 10 21 5 20 0 0 5 10 15
50 18 19 45 18 20 40 19 21 35 22 23 30 20 24 25 Apr. 20 25 21 15 22 10 23 5 24 0 25 0 -5 -10 -15 -20 -25 -30
50 45 40 35 30 25 20 15 10 5 0
50 45 40 35 30 25 20 15 10 5 0
50 45 40 35 30 25 20 15 10 5 0
18 19 20 21 22 23
17
16 17 18 19
24 25
July 26
0
θc
20 19
-5
20 21 22 23 24 25 26
-10 -15 -20 -25 -30
50 22 18 45 19 40 20 21 35 30 Oct. 25 20 22 15 10 21 5 20 0 0 5 10 15
22
21
20 19 18
20
25
30
18 17 19 20 21 22 23
20 19 18
20
25
30
17 18 19 20 21 22 23 25 24
25
May 26
-5
21
16
24
0
21
22
26
-10 -15 -20 -25 -30
50 45 21 22 40 35 23 24 30 25 20 15 10 Mar. 5 23 0 0 5 10 50 45 40 35 30 25 20 15 10 5 0
19 22
21 20
23
19
22 21 20
18
15
20
25
30
15
17 18 19 20 21 22
16 17 18 19 20 21 22 23 24
23 24 25
25
June 26
0
-5
26
-10 -15 -20 -25 -30
50 45 21 22 18 40 21 19 35 23 22 23 30 20 24 24 25 Aug. 20 21 15 25 22 10 23 5 Sep. 24 0 23 0 -5 -10 -15 -20 -25 -30 0 5 17
18 19 20
50 17 45 19 40 16 35 18 30 25 20 15 17 10 5 15 16 0 0 5 10 15
20
Nov. 20
19 18 17
20
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50 45 40 35 14 30 25 20 15 10 5 13 0 0
18 22
21 20 19
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22 21 20 19
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15
19
Dec. 18 16
17 16
15 14
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θm Fig. 11 (continued)
lm/lc = (a) 1.0 and (b) 3.0 throughout the year at 30°N is shown in Fig. 11. The daily solar radiation absorbed on the absorbing plate for each month was calculated at 1° steps for both inclinations hc and hm, so 30 50 = 1500 calculations were done to investigate the effect of both inclinations in each month at each ratio lm/lc. Here, the reflector is inclined forwards from September to March (hm is positive), and backwards from April to August (hm is negative). The optimum combination of the inclinations hc and hm, that maximize the daily solar radiation, would vary considerably according to month, and be slightly affected by the ratio lm/lc. The optimum collector inclination hc is lower in summer and higher in winter for the same reasons as in the results for a collector without the reflector (Fig. 10) as mentioned above. The absolute value of optimum reflector inclination |hm| is lower in spring and autumn, and higher in summer and winter, since the solar altitude angle would be larger in summer and lower in winter than in spring and autumn.
The optimum inclination for the collector hc and reflector hm at ratios lm/lc = 1.0, 2.0 and 3.0 throughout the year is shown in Fig. 12. The optimum collector inclination hc is almost the same for any ratio of lm/lc. The optimum reflector inclination hm would vary with the ratio lm/lc, and would increase with an increase in the ratio lm/lc. The reason is as follows. In winter with a forwardly inclined reflector (Fig. 3), when l1 + l7 > lccos hc, the reflected radiation shown as A0 B0 E0 F0 exceeds the collector’s shadow (A0 B0 CD), and a part of or all of the reflected radiation from the reflector would escape to the ground without hitting the absorbing plate. By increasing the reflector inclination hm, the lengths l1 and l7 can be shortened, but the amount of reflected radiation from reflector shown as A00 B00 E00 F00 would decrease. So a smaller inclination hm is preferable as long as the reflected projection does not exceed the collector’s shadow. On the other hand, in summer with a backwardly inclined reflector (Fig. 6), the amount of reflected radiation can be increased by increasing reflector tilt angle (or decreasing
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30 θc (lm /lc=1.0)
40
Daily solar radiation absorbed on the absorbing plate, MJ/m2day
Optimum inclination of collector θc and reflector θ m
50
(lm /lc=2.0) (lm/lc=3.0)
30
θm (lm/lc=1.0) (lm/lc=2.0)
20
(lm/lc=3.0)
10
0
-10
-20
Summer
Autumn 20 Average
1
2
3
4
5
6
7
8
9
10
11
Winter
15
10
5
0*: θc is fixed at 30 o throughout the year 0
-30
Spring
25
0*
0
1
2
3
lm/lc
12
Month Fig. 13. Daily solar radiation absorbed on the absorbing plate varying with the ratio of reflector length to collector length, lm/lc for four seasons at 30°N.
Fig. 12. Optimum inclination of collector hc and reflector hm at the ratio of reflector and collector length, lm/lc = 1.0, 2.0 and 3.0 throughout the year at 30°N.
hm since hm is negative in summer as shown in Fig. 1), but the lengths l1 and l7 would also be elongated with an increase in reflector tilt angle as shown in Fig. 6. So a larger reflector tilt angle (or smaller hm) is preferable as long as the reflected projection does not exceed the collector’s shadow conversely to winter. The optimum collector inclination hc and reflector inclination hm throughout the year are listed in Table 2. Similar analysis with Fig. 11 was done for lm/lc = 0.5 and 2.0 to determine the optimum inclinations of hc and hm for each month. lm/lc = 0 shows the collector without a reflector. Here, to facilitate ease of setting, the optimum inclinations hc and hm are assumed to be set at 5° steps. The decline of daily solar radiation absorbed on the absorbing plate by setting the inclination hc and hm to the values listed in Table 2 compared with daily solar radiation with hc and hm shown in Fig. 12 is less than 1%. Therefore, the solar thermal collector with an inclined reflector can receive near the maximum solar radiation by setting the inclination hc and hm listed in Table 2 on each month according to the ratio lm/lc at 30°N. Daily solar radiation absorbed on the absorbing plate varying with the ratio lm/lc for four seasons (spring (February–April), summer (May–July), autumn (August–October) and winter (November–January)) and the average values for 12 months at 30°N is shown in Fig. 13. Here, both inclinations hc and hm are assumed to be set at the optimum angle listed in Table 2 for each month. lm/lc = 0 shows the results for the collector without a reflector in which the inclination hc is set to optimum as listed in Table 2, and lm/lc = 0 shows the results for a collector without a reflector in which the inclination hc is fixed at 30° throughout the year.
The daily solar radiation is highest in summer and lowest in winter, and in spring and autumn and the average values are almost the same. The daily solar radiation can be increased by using an inclined flat plate reflector and adjusting the inclinations hc and hm to the proper values. As a result, compared with the conventional solar thermal collector (lm/lc = 0: collector without reflector and hc is fixed at 30°), the average daily solar radiation throughout the year can be increased about 6% by adjusting the collector inclination hc to proper values on each month (lm/lc = 0), and about 19%, 26% and 33% by using an inclined flat plate reflector and adjusting both the inclinations hc and hm to proper values for each month at lm/lc = 0.5, 1.0 and 2.0.
4. Conclusions We have performed numerical analysis of a solar thermal collector with a flat plate top reflector, which extends from the upper edge of the collector. In this paper, the collector was assumed to be a square shape (w = lc = 1 m). The numerical analysis to calculate the solar radiation absorbed on the absorbing plate would be valid when the collector have rectangular shape and both the widths of the collector and the reflector are the same as w. However, the optimum collector and reflector inclinations would be different if the collector do not have square shape. We have determined the optimum inclination of both the collector and reflector throughout the year at 30°N latitude and investigated the effect of the size of the reflector on the daily solar radiation absorbed on the absorbing
Table 2 Optimum inclinations of collector and reflector throughout the year at 5° steps. lm/lc = 0 shows the collector without reflector. Month
December January, November February, October March, September April, August May, July June
lm/lc = 0
lm/lc = 0.5
hc
hc
hm
lm/lc = 1.0 hc
hm
lm/lc = 2.0 hc
hm
lm/lc = 3.0 hc
hm
50 50 45 30 15 5 0
50 45 35 25 10 0 0
25 20 10 0 15 25 25
50 45 35 25 5 0 0
25 20 10 0 15 20 25
45 45 35 25 0 0 0
25 25 15 5 10 15 20
50 45 40 25 5 0 0
30 25 20 10 5 15 15
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plate of the collector. The results of this work are summarized as follows: (1) The solar radiation absorbed on the absorbing plate of the collector can be increased by using a flat plate top reflector, which is inclined forwards in winter and backwards in summer, and by setting the inclination angle of the reflector at less than 30° throughout the year. (2) The optimum collector inclination is lower in summer and higher in winter. (3) The optimum collector inclination hc is slightly affected by the ratio of reflector and collector lengths, lm/lc, while the optimum reflector inclination hm is considerably affected by the ratio lm/lc, and would increase with an increase in the ratio lm/lc. (4) An increase in the daily solar radiation absorbed on the absorbing plate over a conventional solar thermal collector would average about 19%, 26% and 33% throughout the year by using a flat plate reflector at a ratio of lm/lc = 0.5, 1.0 and 2.0 and adjusting the inclination of both the collector and reflector to the proper angle.
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