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Solar radiation on domed roofs
Energy and Buildings 41 (2009) 1238–1245
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Solar radiation on domed roofs Ahmadreza K. Faghih, Mehdi N. Bahadori * School of Mechanical Engineering, Sharif University of Technology, Tehran, Iran
A R T I C L E I N F O
A B S T R A C T
Article history: Received 10 August 2008 Received in revised form 17 March 2009 Accepted 23 July 2009
Solar radiation received and absorbed by four domed roofs was estimated and compared with that of a flat roof. The domed roofs all had the same base areas, and equal to that of the flat roof. One of the roofs considered was the dome of the St. Peter’s Church in Rome. Compared with the other roofs considered, this dome had a higher aspect ratio. It was found that all domed roofs received more solar radiation than the flat roof. Considering glazed tiles to cover a selected dome in Iran and the dome of the St. Peter’s Church, it was found that the solar radiation absorbed by these roofs is reduced appreciably. In the case of the dome of St. Peter’s Church, the amount of radiation absorbed was roughly equal to that absorbed by the comparable flat roof in the warm months. In the case of the glazed reference dome located in Yazd, Iran (a city with very high solar radiation), the radiation absorbed was less than that of flat roof at all times. In addition to aesthetics, this may be a reason for employing glazed tiles to cover the domes of all mosques, shrines, and other large buildings in Iran. ß 2009 Elsevier B.V. All rights reserved.
Keywords: Domed roof Solar radiation Numerical method
1. Introduction Domed roofs have traditionally been used throughout the world to cover large areas. In Iranian architecture, they have played another important role of reducing the total heat gain from the roof, and providing a passive cooling effect for the buildings they served. Solar energy absorbed by a domed roof causes its temperature to rise above the ambient air temperature. Wind blowing over the dome increases the convection heat transfer to the ambient air. Furthermore, the heat loss from the roof is increased by thermal radiation to sky. The rest of the heat absorbed by the dome is conducted through the dome material, and is finally transferred to the inside air by convection, and to the interior walls by radiation. The geometry of these roofs causes the wind velocity to increase over them, resulting in an increase in the convection heat transfer coefficient. Furthermore, the heat transfer from these roofs is increased by the fact that their areas are greater than the comparable flat ones. Fig. 1 shows the pictures of several famous buildings with domed roofs. In addition to their structural applications, domed roofs have been employed in Iran for natural ventilation and passive cooling of buildings. In addition to mosques, shrines, and large halls, one can find such dome structures in bazaars, or market places, water cisterns, and small rooms [1]. A cross-section of a typical domed roof employed in such applications is shown in Fig. 2. The airflow over domed roofs is reported by the authors [2,3].
* Corresponding author. Tel.: +98 21 66 16 55 08; fax: +98 21 66 00 00 21. E-mail address: [email protected] (M.N. Bahadori). 0378-7788/$ – see front matter ß 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.enbuild.2009.07.022
The solar radiation absorbed by domed roofs constitutes the major heat gain of the buildings they cover. Solar radiation on domed and vaulted roofs was studied by a number of investigators [4–6]. Serpoushan and Yaghoubi [4] estimated the solar radiation on three-dimensional surfaces. However, they did not consider the radiation absorbed by the roof in their study. Runsheng et al. [5], as well as Gomez et al. [6], studied the solar radiation absorbed by domed and vaulted roofs. The purpose of this investigation was to estimate the solar radiation received by several domed roofs, and to determine the effect of glazed tiles covering these domes on the total solar radiation absorbed by such roofs. The results of this study will be useful in the thermal performance evaluation of domed roofs, and particularly for determining their passive cooling effects for the buildings they serve. They can also help architects concerned with energy conservation of large halls and buildings to consider domed roofs covered with glazed tiles for such structures.
2. Governing equations To estimate the solar radiation received by a domed roof, we divided the roof surface into several small surfaces, and considered the solar radiation received by each of these small surfaces. Each of these small surface elements was assumed to be flat. By integrating the solar radiation received by these small surfaces, we then estimated the total solar radiation received by the domed roof. Fig. 3 shows the geometry of the solar radiation received by a tilted surface [7]. In this figure various angles are defined as follows:
A.K. Faghih, M.N. Bahadori / Energy and Buildings 41 (2009) 1238–1245
b u uZ aS g gS
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Slope, the angle between the plane of the surface and the horizontal Angle of incidence, the angle between the beam radiation on a surface and the normal to the surface Zenith angle, the angle between the vertical and the line to the sun Solar altitude angle, the angle between the horizontal and the line to the sun Surface azimuth angle, the deviation of the projection on a horizontal plane of the normal to the surface from the local meridian Solar azimuth angle, the angular displacement from the south of the projection of beam radiation on the horizontal plane
The angle of incidence (u) is given by the following equation [7]: Cos u ¼ Sin d Sin f Cos b Sin d Cos f Sin b Cos g þ Cos d Cos f Cos b Cos v þ Cos d Sin f Sin b Cos g Cos v þ Cos d Sin b Sin g Sin v
(1)
Cos uZ ¼ Sin d Sin f þ Cos d Cos f Cos v
(2)
In these equations f is the latitude of the location and d is the declination angle (the angular position of the sun at solar noon), defined by n þ 284 (3) d ¼ 23:45 Sin 360 365 where n is the day number (n = 1 for January 1), and v is the hour angle (the angular displacement of the sun east or west of the local meridian due to rotation of the earth on its axis at 158 per hour), defined by (t is the hour of the day, with solar noon being at 12):
vðtÞ ¼ ðt 12Þ 15
(4)
Fig. 2. Cross-section of a typical domed roof, with air circulation in and over it [1].
radiation received by each of the small surface elements considered on the dome. This solar radiation can be estimated by knowing the clearness index for that hour and day for the location, and by employing the following equations [7]. In these equations various parameters are defined as follows:
Gsc
vs
To estimate the solar radiation received by a domed roof, located in a certain location with a known latitude and longitude, and at a given hour of the day and the day of the year, we need to know the
K¯ t Kt kt Ho H Io I Ib Id Ig It
rg
Fig. 1. Domed roof of (a) Taj Mahal in Agra, India, (b) St. Peter’s Church in Rome, Italy, (c) The Capitol Building in Washington, DC, USA, and (d) Sheikh Lotfollah Mosque in Isfahan, Iran.
Solar constant, assumed 1353 W/m2 Sunset hour angle Monthly average clearness index Daily clearness index Hourly clearness index Daily solar radiation on a horizontal flat surface outside atmosphere (J/(m2 day)) Daily solar radiation on a horizontal flat surface (J/(m2 day)) Hourly solar radiation on a horizontal flat surface outside atmosphere (J/(m2 h)) Hourly solar radiation on a horizontal flat surface (J/(m2 h)) Beam solar radiation on a horizontal flat surface (J/(m2 h)) Diffuse solar radiation on a horizontal flat surface (J/(m2 h)) Ground-reflected solar radiation (J/(m2 h)) Total solar radiation on a tilted surface (J/(m2 h)) Ground reflection factor
Fig. 3. Geometry of solar radiation on a tilted surface [7].
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Ho ¼
Io ¼
360n Gsc 1 þ 0:033 365 p 2pvs Cos f Cos d Sin vs þ Sin f Sin d 360 24 3600
24 3600 360n Gsc 1 þ 0:033 2p 365 Cos f Cos dðSinðvðt 2 ÞÞ Sinðvðt 1 ÞÞÞ 2pðvðt 2 Þ vðt 1 ÞÞ Sin f Sin d þ 360
(5)
(6)
To estimate the monthly solar radiation received by each surface element, we assumed that 15th day of each month represents all days of that month. Therefore, we can consider the daily clearness index (Kt) of 15th day of the month to be equal to the monthly average clearness index (K¯ t ). We determine the hourly solar radiation received by each small surface element on the dome through the following equations [7]: H ¼ K t Ho
(7)
I ¼ rt H
(8)
rt ¼
p 24
ða þ b Cos vÞ
Cos v Cos vs Sin vs ðpvs =180Þ Cos vs
(9)
a ¼ 0:409 þ 0:5016 Sinðvs 60Þ
(10)
b ¼ 0:6609 0:4767 Sinðvs 60Þ
(11)
To estimate the diffuse and beam components of the solar radiation, we employed the hourly clearness index (kt) and the following equations [7]: kt ¼
I Io
kt 0:35 )
(12) Id ¼ 1 0:249kt I
0:35 kt 0:75 )
Id ¼ 1:557 1:84kt I
(13)
(15)
I b ¼ I Id
(16)
To estimate the radiation received by a surface element on the roof, considered as a tilted flat surface with a slope which can be determined from the geometry of the roof, we used the following equation [7]. This equation accounts for the diffuse radiation received from the sky, and that reflected by the ground surrounding the domed roof:
Rb ¼
1 þ Cos b 1 Cos b þ ðIb þ Id Þrg 2 2
Cos u Cos uZ
a. b. c. d.
A flat roof A hemispherical roof A conical roof The dome of a theological school in Yazd, Iran, which will be called the ‘‘reference dome’’ (Fig. 5) e. The dome of St. Peter’s Church in Rome, Italy
(14)
I kt 0:75 ) d ¼ 0:177 I
IT ¼ Ib Rb þ Id
Fig. 4. Models of the roofs considered in this study. (a) A flat roof, (b) a hemispherical roof, (c) a conical roof, (d) reference dome in Yazd, and (e) the dome of St. Peter’s Church in Rome, Italy which is scaled down to have the same base area as the other domes.
The dome of the St. Peter’s Church is not smooth, as can be seen in Fig. 6. Definitely, edges on this dome shade the rest of the dome, and affect the total solar radiation received by the dome. However, to be able to compare the solar radiation received by this dome with that of the other domes, we ignored this shading effect, and considered the dome to be smooth, similar to the other domes considered in this study. We assumed that all the domes to have the same base area, which is a circle with a radius of 300 cm. The dome of St. Peter’s
(17) (18)
For the surface elements being in the shade, where u is larger than 908, Rb is considered zero. 3. Solar radiation received by domed- and flat roofs To compare the solar radiation received by various roofs, we considered the roofs shown in Fig. 4. They are
Fig. 5. Model of the reference domed roof.
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Fig. 8. Comparison between total solar radiation on the flat and hemispherical roofs assumed to be located in the city of Tehran. The radiation is based on the unit area of the flat roof and hemispherical roof.
Fig. 6. Model of the dome of St. Peter’s Church.
Fig. 9. Solar radiation on the reference dome located in the city of Yazd for different mesh size.
Fig. 7. Network or the mesh arrangement considered for the reference dome.
Church was scaled down to have the same base area as the other domes considered in this study. All domes have a height of 300 cm, measured from the base, except for St. Peter’s Church which (after being scaled down) has a height of 441 cm. The reference dome has an area of 51 m2, and the dome of the St. Peter’s Church (after being scaled down) has an area of 74 m2. The flat roof considered for comparison has the same area as the base area of the reference dome. A model of the reference dome (d) was considered for air flow studies [2,3]. Except for the flat roof, all other roofs were divided into a network of small surface elements (meshed area). Fig. 7 shows the network considered for the reference dome. 4. Results The governing equations given above were written for each surface element, integrated for the entire day length, and finally summed up for all surface elements to find out total daily solar radiation (Ht). Daily direct (Hb), diffuse (Hd) and ground-reflected
(Hg) solar radiation were estimated for all the roofs considered in this study. We compared the results of our study with those of Serpoushan and Yagoubi’s [4]. Fig. 8 shows this comparison for flat- and hemispherical roofs, assumed to be located in the city of Tehran. Serpoushan and Yaghoubi employed the solar radiation model suggested by Daneshyar [8] in their study to estimate the total solar radiation received by flat- and hemispherical roofs, whereas, we employed the clearness index model, as described above. The differences seem to be acceptable, considering the fact that different solar radiation models were employed in the two studies. The maximum difference for the total solar radiation on flat roofs is 9.3%, occurring in the month of September, and for the hemispherical roof it is 8.9%, occurring in November. The solar radiation data of Yazd, a desert city in central Iran (with 31.54 N. latitude and 54.17 longitude, 1237.2 m above the sea level), was considered for the study. Table 1 gives the monthly average sky clearness index for different months, for the city of Yazd [9]. To find an optimized mesh size in our solar energy estimation, we considered the reference dome and the solar radiation data of the city of Yazd for the entire year. Fig. 9 shows the effect of mesh sizes on the total solar radiation received by this dome. From this figure we can see a minor difference between mesh size of 30 cm and 15 cm. To have an optimum CPU time, a mesh size of 30 cm was selected for all the domes considered in this study. Fig. 10 shows the total, direct, diffuse and ground-reflected solar radiation received by the reference dome, located in the city of Yazd.
Table 1 Monthly average sky clearness index for the city of Yazd [9]. Month
January
February
March
April
May
June
Clearness index
0.60
0.62
0.59
0.61
0.68
0.74
Month
July
August
September
October
November
December
Clearness index
0.73
0.77
0.76
0.74
0.65
0.64
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Fig. 10. Solar radiation on the reference dome located in the city of Yazd.
Fig. 13. Solar radiation on a flat horizontal roof located in the city of Rome.
Fig. 11. Solar radiation on a flat horizontal roof located in the city of Yazd.
Fig. 14. Ratio of total solar radiation received by the reference dome and the dome of St. Peter’s Church to that of the flat horizontal roof located in the cities of Yazd and Rome.
Fig. 12. Solar radiation on the dome of St. Peter’s Church located in the city of Rome.
Fig. 11 shows the same values for a flat horizontal roof located in the same city. Figs. 12 and 13 are similar to Figs. 10 and 11, but for the dome of St. Peter’s Church, and a horizontal flat roof, located in the city of Rome, Italy. For the city of Rome, (with 41.89 N. latitude and 12.48 longitude), no data for its monthly average clearness index was available to the authors. However, to be able to estimate the solar radiation received by the dome located in this city, the clearness index of the city of Athens, Greece, which is believed to be very similar to that of Rome, was employed. Table 2 gives the monthly average sky clearness index for the city of Athens [7]. To estimate the total solar radiation received by the surface elements on all domed roofs, a ground reflection factor of 0.2 was assumed.
Table 3 shows the maximum total daily solar radiation (Ht) occurring in the month of June, received by various roofs, located in the cities of Yazd and Rome. This table also includes the percentages of the beam, diffuse and ground-reflected solar radiation. Figs. 10 and 11, as well as Table 3, show that for both flat roof and the reference dome located in the city of Yazd the main part of the solar radiation is beam radiation. This is around 70% for the reference dome and about 77% for the flat roof. The diffuse component of solar radiation is almost the same for both roofs. Figs. 12 and 13, as well as Table 3, show similar results, assuming the flat roof and the domed roof (St. Peter’s) to be located in the city of Rome. For these roofs, the direct and the diffuse components of the solar radiation in month of June are nearly equal to each other. The ground-reflected solar radiation received by the domed roofs, located in both cities of Rome and Yazd, is very small. Fig. 14 shows the ratio of the total solar radiation received by the reference dome over that received by the flat horizontal roof (R), located in the city of Yazd, as well as the ratio of the radiation received by the dome of St. Peter’s Church to that of a flat horizontal roof located in the city of Rome. Fig. 15 shows the total solar radiation received by various roofs (shown in Fig. 4), when these roofs are all assumed to be located in the city of Yazd. Fig. 16 is the same as Fig. 15, but per unit area of the roof itself. It can be seen from these two figures that the total solar radiation received by the domed roofs of various geometries is higher than the flat
Table 2 Monthly average sky clearness index for the city of Athens [7]. Month
January
February
March
April
May
June
Clearness index
0.40
0.43
0.48
0.51
0.57
0.59
Month
July
August
September
October
November
December
Clearness index
0.61
0.60
0.57
0.52
0.46
0.40
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Fig. 15. Total solar radiation (in MJ/day) received by various roofs, assumed to be all located in the city of Yazd.
Fig. 17. The nodes considered on the reference dome.
Fig. 16. Total solar radiation (in MJ/(m2 day)) received by the unit area of various roofs, assumed to be all located in the city of Yazd.
horizontal roof of the same base area. However, when considered on the unit area of the roof itself, then the flat roof receives the highest value of the total solar radiation. Fig. 14 shows the ratio (R) for the reference dome located in the city of Yazd is lower than the same ratio for the of dome of St. Peter’s Church located in the city of Rome. The reason is that the dome of St. Peter’s Church has a larger surface area than the reference dome, and the fact that Yazd is a desert city, with a higher clearness index than the city of Rome. For both domes, the ratio (R) reduces during the warmer months of the year. It may be concluded from this figure that employing domed roofs with lower aspect ratio (for example the reference dome as compared with the dome of St. Peter’s Church) has more beneficial passive cooling effects in the desert regions of the world than other locations with lower solar radiation. This may be a major reason for finding such dome structures in the traditional Iranian architecture. Figs. 15 and 16 show that the total solar radiation received by various domes located in the city of Yazd depends greatly on their
surface areas. For the roof models considered in this study, the hemispherical dome with 56 m2 roof area receives more total solar radiation than the reference dome with 51 m2 roof area, and the conical roof with 40 m2 roof area, and the flat roof of 28.3 m2 area. To see the solar radiation received by different points on domed roofs at different hours of the day, we considered several nodes on south-north and east-west longitudes of the reference dome. Fig. 17 shows these nodes on the reference dome. Figs. 18 and 19 show the total solar radiation received by these nodes at the hours 8:30, 12 and 15:30, solar time, on June 15th, in the city of Yazd. These figures are helpful to find which locations of a domed roof has a higher temperature during any time of the year, and then determine any thermal stresses which may develop in various domes. 5. Solar radiation absorbed by domed roofs For thermal performance evaluation of buildings with domed roofs, it is necessary to determine the heat gain through these roofs. Almost all of the domed roofs employed in large buildings in Iran are covered with glazed tiles. This is for aesthetic reason, and for the reduction of heat gain through these roofs. The solar radiation
Fig. 18. Solar radiation received by different nodes on a south-north longitude of the reference dome located in Yazd, on June 15th.
Table 3 Maximum Ht (in June) for flat roof, reference and St. Peter’s dome. City
Dome
Max. Ht (MJ/day)
Hb
Hd
Hg
Yazd
Reference dome Flat roof
110.6 85.4
70.1% 77.1%
23.7% 22.9%
6.2% 0%
Rome
St. Peter’s dome Flat roof
121.5 69.15
43.3% 54%
47.4% 46%
9.3% 0%
A.K. Faghih, M.N. Bahadori / Energy and Buildings 41 (2009) 1238–1245
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Fig. 19. Solar radiation received by different nodes on an east-west longitude of the reference dome located in Yazd, on June 15th.
Table 4 Maximum received and absorbed Ht (in June) for reference and St. Peter’s dome.
Fig. 20. The solar radiation received and absorbed by the reference dome, located in the city of Yazd, when covered with ordinary material, and with glazed tiles.
absorbed by a surface depends on the color, the texture and the angle of incidence (u). The following equation was used to determine the solar radiation absorbed by a surface, when the beam is at angle u [7] (in this equation, an is the absorptance at u = 0):
a 2 ¼ 1 1:5879 103 u þ 2:7314 104 u 2:3026 an 3 4 5 105 u þ 9:0244 107 u 1:8000 108 u 6
þ 1:7734 1010 u 6:9937 1013 u
7
(19)
The solar radiation absorbed by the reference dome, located in the city of Yazd, and that absorbed by the dome of the St. Peter’s Church, located in the city of Rome, were estimated for two cases of an being 0.8 and 0.4, representing regular roofing material for domes and glazed tiles, respectively. Figs. 20 and 21 show the results. Solar radiation absorbed by glazed tiles also depends on their colors. However, because most of the glazed tiles employed on the domes of
Fig. 21. The solar radiation received and absorbed by the dome of St. Peter’s Church located in Rome, when covered with ordinary material, and when it can be covered with glazed tiles.
Dome
Max. Ht (MJ/day)
Absorbed Ht an = 0.8
Absorbed Ht an = 0.4
Reference dome in Yazd St. Peter’s dome in Rome
110.6 121.5
75.4% 75.8%
40.6% 45.7%
mosque, shrines and other large buildings in Iran are in light colors, we considered an absorptance of 0.4 for all these tiles. Table 4 shows the total radiation received and absorbed by the reference dome, located in the city of Yazd, and the dome of St. Peter’s Church, located in the city of Rome, during the month of June. This table shows that the percentage of the absorbed solar radiation for the reference dome in the city of Yazd in June is less than St. Peter’s. The ratio of the solar radiation absorbed by the reference dome, and that of the St. Peter’s Church, covered with glazed tiles, over the flat horizontal roofs located in the respected cities, is shown in Fig. 22. For the flat roofs, an absorptance of 0.8 was assumed. A comparison between Figs. 14 and 22 shows that glazed tiles can be helpful in reducing the total solar radiation absorbed, particularly in warm months of the year. Fig. 22 shows that covering the reference dome in the city of Yazd with glazed tiles reduces the absorbed solar radiation even below that of the flat roof (ratio < 1) during all months of the year. This is an important result for the passive cooling effects of domed roofs in Iran and other desert countries, employing glazed tiles on the domes. For the dome of St. Peter’s Church in Rome, the solar radiation absorbed by this dome, if it were covered by glazed tile, is reduced to that of flat horizontal roofs in warm months. 6. Thermal radiation to sky Thermal radiation exchange with sky is an important factor in passive cooling of buildings in desert regions, employing domed
Fig. 22. Ratio of total solar radiation absorbed by the domed roofs covered with glazed tiles to a horizontal flat roof of the same base area. The buildings are located in the cities of Yazd and Rome.
A.K. Faghih, M.N. Bahadori / Energy and Buildings 41 (2009) 1238–1245
roofs. The thermal radiation exchange with sky may be determined through the following equation [10]: 1 þ Cos b 4 4 (20) T Þ Isky ¼ es ðTroo f sky 2 T sky ¼ e0:25 sky T a
(21)
esky ¼ 0:74 þ 0:006T d p
(22)
In these equations various parameters are defined as follows: e Surface emittance
s
Stefan–Boltzmann constant
Troof
Roof temperature, in Kelvin
Tsky
Sky temperature, in Kelvin
Ta
Ambient air temperature, in Kelvin
Tdp
Ambient air dew point temperature, in Centigrade
esky
Sky emissivity
Thermal radiation exchange with sky, the convection heat transfer between the dome surface and the ambient air, and the heat conducted through the dome material depend on the temperature of the surface of the dome. The dome surface temperature is not constant at any one location on the dome and at any one time. To evaluate the domed roof surface temperature at any one location and at any one time, we need to consider, in addition to the above heat transfer modes, the convection heat transfer between the inside surface of the dome with the inside air, and thermal radiation exchange between this surface and all interior surfaces that it can ‘‘see’’. This requires a complete thermal analysis of the dome and the building it covers. This analysis will not be considered in this study, but will be reported in another paper. 7. Conclusions Direct, diffuse, as well as ground-reflected, solar radiation was estimated for several domed roofs, a flat roof, and a roof of conical shape, assumed to be located in the city of Yazd, Iran, and the dome of the St. Peter’s Church, located in the city of Rome, Italy. The results show that maximum solar radiation received by these roofs, located in the northern hemisphere, occurs in June, and it is independent of the geographical locations and the climatic
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conditions. Main portion of the radiation received by the domes located in the city of Yazd was beam radiation, whereas for the dome located in Rome, the direct and diffuse components of the solar radiation were almost equal to each other. It was also found that the domed roofs receive more solar radiation than the flat roofs of equal base areas. Based on the unit areas of the dome itself, the flat roofs received more solar radiation. The domes with higher aspect ratios, for example, that of St. Peter’s Church, received more solar radiation than the ones with lower aspect ratios, when considered to be located in the same city. The solar radiation absorbed by domed roofs can be reduced appreciably by covering them with glazed tiles. This is particularly important in locations with very high solar radiation. For the city of Yazd, which receives very high solar radiation throughout the year, the domes covered with glazed tiles absorb less solar radiation than the comparable flat roofs. This is important in domed roofs, providing passive cooling for the buildings they cover. The results of this investigation become useful in thermal performance, and passive cooling evaluation of domed roof buildings. Acknowledgement The authors wish to acknowledge with thanks the financial support of the Iranian National Science Foundation. References [1] M.N. Bahadori, Passive cooling systems in Iranian architecture, Scientific American 238 (February) (1978) 144–154. [2] A.K. Faghih, M.N. Bahadori, Experimental investigation of air flow over domed roofs, Iranian Journal of Science and Technology; Transaction B: Engineering 33 (B3) (2009) 207–216. [3] A.K. Faghih, M.N. Bahadori, Three dimensional numerical investigation of air flow over domed roofs, submitted for publication. [4] S. Serpoushan, M. Yaghoubi, Solar energy calculation on 3D surfaces, Iran Energy Journal (13) (2002) 3–21 (in Persian). [5] T.I. Runsheng, A. Meir, Y. Etzion, An analysis of absorbed radiation by domed and vaulted roofs as compared with flat roofs, Energy and Building 35 (2003) 539–548. [6] V.M. Gomez, M.A. Porta, C. Heard, Solar performance of hemispherical vaulted roofs, Building and Environment 38 (2003) 1431–1438. [7] J.A. Duffie, W.A. Beckman, Solar Engineering of Thermal Processes, 3rd edition, Wiley, 2006. [8] M. Daneshyar, Solar radiation statistics for Iran, Solar Energy 21 (1978) 345–349. [9] M.N. Bahadori, S. Mirhosseini, Sky Clearness Index for Iranian Cities, Paper No. 1120 ISES 2005, Solar World Congress, Orlando, FL, USA, 2005. [10] P. Berdahl, R. Fromberg, The thermal radiance of clear skies, Solar Energy 29 (1982) 299–314.
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Solar radiation on domed roofs Ahmadreza K. Faghih, Mehdi N. Bahadori * School of Mechanical Engineering, Sharif University of Technology, Tehran, Iran
A R T I C L E I N F O
A B S T R A C T
Article history: Received 10 August 2008 Received in revised form 17 March 2009 Accepted 23 July 2009
Solar radiation received and absorbed by four domed roofs was estimated and compared with that of a flat roof. The domed roofs all had the same base areas, and equal to that of the flat roof. One of the roofs considered was the dome of the St. Peter’s Church in Rome. Compared with the other roofs considered, this dome had a higher aspect ratio. It was found that all domed roofs received more solar radiation than the flat roof. Considering glazed tiles to cover a selected dome in Iran and the dome of the St. Peter’s Church, it was found that the solar radiation absorbed by these roofs is reduced appreciably. In the case of the dome of St. Peter’s Church, the amount of radiation absorbed was roughly equal to that absorbed by the comparable flat roof in the warm months. In the case of the glazed reference dome located in Yazd, Iran (a city with very high solar radiation), the radiation absorbed was less than that of flat roof at all times. In addition to aesthetics, this may be a reason for employing glazed tiles to cover the domes of all mosques, shrines, and other large buildings in Iran. ß 2009 Elsevier B.V. All rights reserved.
Keywords: Domed roof Solar radiation Numerical method
1. Introduction Domed roofs have traditionally been used throughout the world to cover large areas. In Iranian architecture, they have played another important role of reducing the total heat gain from the roof, and providing a passive cooling effect for the buildings they served. Solar energy absorbed by a domed roof causes its temperature to rise above the ambient air temperature. Wind blowing over the dome increases the convection heat transfer to the ambient air. Furthermore, the heat loss from the roof is increased by thermal radiation to sky. The rest of the heat absorbed by the dome is conducted through the dome material, and is finally transferred to the inside air by convection, and to the interior walls by radiation. The geometry of these roofs causes the wind velocity to increase over them, resulting in an increase in the convection heat transfer coefficient. Furthermore, the heat transfer from these roofs is increased by the fact that their areas are greater than the comparable flat ones. Fig. 1 shows the pictures of several famous buildings with domed roofs. In addition to their structural applications, domed roofs have been employed in Iran for natural ventilation and passive cooling of buildings. In addition to mosques, shrines, and large halls, one can find such dome structures in bazaars, or market places, water cisterns, and small rooms [1]. A cross-section of a typical domed roof employed in such applications is shown in Fig. 2. The airflow over domed roofs is reported by the authors [2,3].
* Corresponding author. Tel.: +98 21 66 16 55 08; fax: +98 21 66 00 00 21. E-mail address: [email protected] (M.N. Bahadori). 0378-7788/$ – see front matter ß 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.enbuild.2009.07.022
The solar radiation absorbed by domed roofs constitutes the major heat gain of the buildings they cover. Solar radiation on domed and vaulted roofs was studied by a number of investigators [4–6]. Serpoushan and Yaghoubi [4] estimated the solar radiation on three-dimensional surfaces. However, they did not consider the radiation absorbed by the roof in their study. Runsheng et al. [5], as well as Gomez et al. [6], studied the solar radiation absorbed by domed and vaulted roofs. The purpose of this investigation was to estimate the solar radiation received by several domed roofs, and to determine the effect of glazed tiles covering these domes on the total solar radiation absorbed by such roofs. The results of this study will be useful in the thermal performance evaluation of domed roofs, and particularly for determining their passive cooling effects for the buildings they serve. They can also help architects concerned with energy conservation of large halls and buildings to consider domed roofs covered with glazed tiles for such structures.
2. Governing equations To estimate the solar radiation received by a domed roof, we divided the roof surface into several small surfaces, and considered the solar radiation received by each of these small surfaces. Each of these small surface elements was assumed to be flat. By integrating the solar radiation received by these small surfaces, we then estimated the total solar radiation received by the domed roof. Fig. 3 shows the geometry of the solar radiation received by a tilted surface [7]. In this figure various angles are defined as follows:
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b u uZ aS g gS
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Slope, the angle between the plane of the surface and the horizontal Angle of incidence, the angle between the beam radiation on a surface and the normal to the surface Zenith angle, the angle between the vertical and the line to the sun Solar altitude angle, the angle between the horizontal and the line to the sun Surface azimuth angle, the deviation of the projection on a horizontal plane of the normal to the surface from the local meridian Solar azimuth angle, the angular displacement from the south of the projection of beam radiation on the horizontal plane
The angle of incidence (u) is given by the following equation [7]: Cos u ¼ Sin d Sin f Cos b Sin d Cos f Sin b Cos g þ Cos d Cos f Cos b Cos v þ Cos d Sin f Sin b Cos g Cos v þ Cos d Sin b Sin g Sin v
(1)
Cos uZ ¼ Sin d Sin f þ Cos d Cos f Cos v
(2)
In these equations f is the latitude of the location and d is the declination angle (the angular position of the sun at solar noon), defined by n þ 284 (3) d ¼ 23:45 Sin 360 365 where n is the day number (n = 1 for January 1), and v is the hour angle (the angular displacement of the sun east or west of the local meridian due to rotation of the earth on its axis at 158 per hour), defined by (t is the hour of the day, with solar noon being at 12):
vðtÞ ¼ ðt 12Þ 15
(4)
Fig. 2. Cross-section of a typical domed roof, with air circulation in and over it [1].
radiation received by each of the small surface elements considered on the dome. This solar radiation can be estimated by knowing the clearness index for that hour and day for the location, and by employing the following equations [7]. In these equations various parameters are defined as follows:
Gsc
vs
To estimate the solar radiation received by a domed roof, located in a certain location with a known latitude and longitude, and at a given hour of the day and the day of the year, we need to know the
K¯ t Kt kt Ho H Io I Ib Id Ig It
rg
Fig. 1. Domed roof of (a) Taj Mahal in Agra, India, (b) St. Peter’s Church in Rome, Italy, (c) The Capitol Building in Washington, DC, USA, and (d) Sheikh Lotfollah Mosque in Isfahan, Iran.
Solar constant, assumed 1353 W/m2 Sunset hour angle Monthly average clearness index Daily clearness index Hourly clearness index Daily solar radiation on a horizontal flat surface outside atmosphere (J/(m2 day)) Daily solar radiation on a horizontal flat surface (J/(m2 day)) Hourly solar radiation on a horizontal flat surface outside atmosphere (J/(m2 h)) Hourly solar radiation on a horizontal flat surface (J/(m2 h)) Beam solar radiation on a horizontal flat surface (J/(m2 h)) Diffuse solar radiation on a horizontal flat surface (J/(m2 h)) Ground-reflected solar radiation (J/(m2 h)) Total solar radiation on a tilted surface (J/(m2 h)) Ground reflection factor
Fig. 3. Geometry of solar radiation on a tilted surface [7].
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Ho ¼
Io ¼
360n Gsc 1 þ 0:033 365 p 2pvs Cos f Cos d Sin vs þ Sin f Sin d 360 24 3600
24 3600 360n Gsc 1 þ 0:033 2p 365 Cos f Cos dðSinðvðt 2 ÞÞ Sinðvðt 1 ÞÞÞ 2pðvðt 2 Þ vðt 1 ÞÞ Sin f Sin d þ 360
(5)
(6)
To estimate the monthly solar radiation received by each surface element, we assumed that 15th day of each month represents all days of that month. Therefore, we can consider the daily clearness index (Kt) of 15th day of the month to be equal to the monthly average clearness index (K¯ t ). We determine the hourly solar radiation received by each small surface element on the dome through the following equations [7]: H ¼ K t Ho
(7)
I ¼ rt H
(8)
rt ¼
p 24
ða þ b Cos vÞ
Cos v Cos vs Sin vs ðpvs =180Þ Cos vs
(9)
a ¼ 0:409 þ 0:5016 Sinðvs 60Þ
(10)
b ¼ 0:6609 0:4767 Sinðvs 60Þ
(11)
To estimate the diffuse and beam components of the solar radiation, we employed the hourly clearness index (kt) and the following equations [7]: kt ¼
I Io
kt 0:35 )
(12) Id ¼ 1 0:249kt I
0:35 kt 0:75 )
Id ¼ 1:557 1:84kt I
(13)
(15)
I b ¼ I Id
(16)
To estimate the radiation received by a surface element on the roof, considered as a tilted flat surface with a slope which can be determined from the geometry of the roof, we used the following equation [7]. This equation accounts for the diffuse radiation received from the sky, and that reflected by the ground surrounding the domed roof:
Rb ¼
1 þ Cos b 1 Cos b þ ðIb þ Id Þrg 2 2
Cos u Cos uZ
a. b. c. d.
A flat roof A hemispherical roof A conical roof The dome of a theological school in Yazd, Iran, which will be called the ‘‘reference dome’’ (Fig. 5) e. The dome of St. Peter’s Church in Rome, Italy
(14)
I kt 0:75 ) d ¼ 0:177 I
IT ¼ Ib Rb þ Id
Fig. 4. Models of the roofs considered in this study. (a) A flat roof, (b) a hemispherical roof, (c) a conical roof, (d) reference dome in Yazd, and (e) the dome of St. Peter’s Church in Rome, Italy which is scaled down to have the same base area as the other domes.
The dome of the St. Peter’s Church is not smooth, as can be seen in Fig. 6. Definitely, edges on this dome shade the rest of the dome, and affect the total solar radiation received by the dome. However, to be able to compare the solar radiation received by this dome with that of the other domes, we ignored this shading effect, and considered the dome to be smooth, similar to the other domes considered in this study. We assumed that all the domes to have the same base area, which is a circle with a radius of 300 cm. The dome of St. Peter’s
(17) (18)
For the surface elements being in the shade, where u is larger than 908, Rb is considered zero. 3. Solar radiation received by domed- and flat roofs To compare the solar radiation received by various roofs, we considered the roofs shown in Fig. 4. They are
Fig. 5. Model of the reference domed roof.
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Fig. 8. Comparison between total solar radiation on the flat and hemispherical roofs assumed to be located in the city of Tehran. The radiation is based on the unit area of the flat roof and hemispherical roof.
Fig. 6. Model of the dome of St. Peter’s Church.
Fig. 9. Solar radiation on the reference dome located in the city of Yazd for different mesh size.
Fig. 7. Network or the mesh arrangement considered for the reference dome.
Church was scaled down to have the same base area as the other domes considered in this study. All domes have a height of 300 cm, measured from the base, except for St. Peter’s Church which (after being scaled down) has a height of 441 cm. The reference dome has an area of 51 m2, and the dome of the St. Peter’s Church (after being scaled down) has an area of 74 m2. The flat roof considered for comparison has the same area as the base area of the reference dome. A model of the reference dome (d) was considered for air flow studies [2,3]. Except for the flat roof, all other roofs were divided into a network of small surface elements (meshed area). Fig. 7 shows the network considered for the reference dome. 4. Results The governing equations given above were written for each surface element, integrated for the entire day length, and finally summed up for all surface elements to find out total daily solar radiation (Ht). Daily direct (Hb), diffuse (Hd) and ground-reflected
(Hg) solar radiation were estimated for all the roofs considered in this study. We compared the results of our study with those of Serpoushan and Yagoubi’s [4]. Fig. 8 shows this comparison for flat- and hemispherical roofs, assumed to be located in the city of Tehran. Serpoushan and Yaghoubi employed the solar radiation model suggested by Daneshyar [8] in their study to estimate the total solar radiation received by flat- and hemispherical roofs, whereas, we employed the clearness index model, as described above. The differences seem to be acceptable, considering the fact that different solar radiation models were employed in the two studies. The maximum difference for the total solar radiation on flat roofs is 9.3%, occurring in the month of September, and for the hemispherical roof it is 8.9%, occurring in November. The solar radiation data of Yazd, a desert city in central Iran (with 31.54 N. latitude and 54.17 longitude, 1237.2 m above the sea level), was considered for the study. Table 1 gives the monthly average sky clearness index for different months, for the city of Yazd [9]. To find an optimized mesh size in our solar energy estimation, we considered the reference dome and the solar radiation data of the city of Yazd for the entire year. Fig. 9 shows the effect of mesh sizes on the total solar radiation received by this dome. From this figure we can see a minor difference between mesh size of 30 cm and 15 cm. To have an optimum CPU time, a mesh size of 30 cm was selected for all the domes considered in this study. Fig. 10 shows the total, direct, diffuse and ground-reflected solar radiation received by the reference dome, located in the city of Yazd.
Table 1 Monthly average sky clearness index for the city of Yazd [9]. Month
January
February
March
April
May
June
Clearness index
0.60
0.62
0.59
0.61
0.68
0.74
Month
July
August
September
October
November
December
Clearness index
0.73
0.77
0.76
0.74
0.65
0.64
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Fig. 10. Solar radiation on the reference dome located in the city of Yazd.
Fig. 13. Solar radiation on a flat horizontal roof located in the city of Rome.
Fig. 11. Solar radiation on a flat horizontal roof located in the city of Yazd.
Fig. 14. Ratio of total solar radiation received by the reference dome and the dome of St. Peter’s Church to that of the flat horizontal roof located in the cities of Yazd and Rome.
Fig. 12. Solar radiation on the dome of St. Peter’s Church located in the city of Rome.
Fig. 11 shows the same values for a flat horizontal roof located in the same city. Figs. 12 and 13 are similar to Figs. 10 and 11, but for the dome of St. Peter’s Church, and a horizontal flat roof, located in the city of Rome, Italy. For the city of Rome, (with 41.89 N. latitude and 12.48 longitude), no data for its monthly average clearness index was available to the authors. However, to be able to estimate the solar radiation received by the dome located in this city, the clearness index of the city of Athens, Greece, which is believed to be very similar to that of Rome, was employed. Table 2 gives the monthly average sky clearness index for the city of Athens [7]. To estimate the total solar radiation received by the surface elements on all domed roofs, a ground reflection factor of 0.2 was assumed.
Table 3 shows the maximum total daily solar radiation (Ht) occurring in the month of June, received by various roofs, located in the cities of Yazd and Rome. This table also includes the percentages of the beam, diffuse and ground-reflected solar radiation. Figs. 10 and 11, as well as Table 3, show that for both flat roof and the reference dome located in the city of Yazd the main part of the solar radiation is beam radiation. This is around 70% for the reference dome and about 77% for the flat roof. The diffuse component of solar radiation is almost the same for both roofs. Figs. 12 and 13, as well as Table 3, show similar results, assuming the flat roof and the domed roof (St. Peter’s) to be located in the city of Rome. For these roofs, the direct and the diffuse components of the solar radiation in month of June are nearly equal to each other. The ground-reflected solar radiation received by the domed roofs, located in both cities of Rome and Yazd, is very small. Fig. 14 shows the ratio of the total solar radiation received by the reference dome over that received by the flat horizontal roof (R), located in the city of Yazd, as well as the ratio of the radiation received by the dome of St. Peter’s Church to that of a flat horizontal roof located in the city of Rome. Fig. 15 shows the total solar radiation received by various roofs (shown in Fig. 4), when these roofs are all assumed to be located in the city of Yazd. Fig. 16 is the same as Fig. 15, but per unit area of the roof itself. It can be seen from these two figures that the total solar radiation received by the domed roofs of various geometries is higher than the flat
Table 2 Monthly average sky clearness index for the city of Athens [7]. Month
January
February
March
April
May
June
Clearness index
0.40
0.43
0.48
0.51
0.57
0.59
Month
July
August
September
October
November
December
Clearness index
0.61
0.60
0.57
0.52
0.46
0.40
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Fig. 15. Total solar radiation (in MJ/day) received by various roofs, assumed to be all located in the city of Yazd.
Fig. 17. The nodes considered on the reference dome.
Fig. 16. Total solar radiation (in MJ/(m2 day)) received by the unit area of various roofs, assumed to be all located in the city of Yazd.
horizontal roof of the same base area. However, when considered on the unit area of the roof itself, then the flat roof receives the highest value of the total solar radiation. Fig. 14 shows the ratio (R) for the reference dome located in the city of Yazd is lower than the same ratio for the of dome of St. Peter’s Church located in the city of Rome. The reason is that the dome of St. Peter’s Church has a larger surface area than the reference dome, and the fact that Yazd is a desert city, with a higher clearness index than the city of Rome. For both domes, the ratio (R) reduces during the warmer months of the year. It may be concluded from this figure that employing domed roofs with lower aspect ratio (for example the reference dome as compared with the dome of St. Peter’s Church) has more beneficial passive cooling effects in the desert regions of the world than other locations with lower solar radiation. This may be a major reason for finding such dome structures in the traditional Iranian architecture. Figs. 15 and 16 show that the total solar radiation received by various domes located in the city of Yazd depends greatly on their
surface areas. For the roof models considered in this study, the hemispherical dome with 56 m2 roof area receives more total solar radiation than the reference dome with 51 m2 roof area, and the conical roof with 40 m2 roof area, and the flat roof of 28.3 m2 area. To see the solar radiation received by different points on domed roofs at different hours of the day, we considered several nodes on south-north and east-west longitudes of the reference dome. Fig. 17 shows these nodes on the reference dome. Figs. 18 and 19 show the total solar radiation received by these nodes at the hours 8:30, 12 and 15:30, solar time, on June 15th, in the city of Yazd. These figures are helpful to find which locations of a domed roof has a higher temperature during any time of the year, and then determine any thermal stresses which may develop in various domes. 5. Solar radiation absorbed by domed roofs For thermal performance evaluation of buildings with domed roofs, it is necessary to determine the heat gain through these roofs. Almost all of the domed roofs employed in large buildings in Iran are covered with glazed tiles. This is for aesthetic reason, and for the reduction of heat gain through these roofs. The solar radiation
Fig. 18. Solar radiation received by different nodes on a south-north longitude of the reference dome located in Yazd, on June 15th.
Table 3 Maximum Ht (in June) for flat roof, reference and St. Peter’s dome. City
Dome
Max. Ht (MJ/day)
Hb
Hd
Hg
Yazd
Reference dome Flat roof
110.6 85.4
70.1% 77.1%
23.7% 22.9%
6.2% 0%
Rome
St. Peter’s dome Flat roof
121.5 69.15
43.3% 54%
47.4% 46%
9.3% 0%
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Fig. 19. Solar radiation received by different nodes on an east-west longitude of the reference dome located in Yazd, on June 15th.
Table 4 Maximum received and absorbed Ht (in June) for reference and St. Peter’s dome.
Fig. 20. The solar radiation received and absorbed by the reference dome, located in the city of Yazd, when covered with ordinary material, and with glazed tiles.
absorbed by a surface depends on the color, the texture and the angle of incidence (u). The following equation was used to determine the solar radiation absorbed by a surface, when the beam is at angle u [7] (in this equation, an is the absorptance at u = 0):
a 2 ¼ 1 1:5879 103 u þ 2:7314 104 u 2:3026 an 3 4 5 105 u þ 9:0244 107 u 1:8000 108 u 6
þ 1:7734 1010 u 6:9937 1013 u
7
(19)
The solar radiation absorbed by the reference dome, located in the city of Yazd, and that absorbed by the dome of the St. Peter’s Church, located in the city of Rome, were estimated for two cases of an being 0.8 and 0.4, representing regular roofing material for domes and glazed tiles, respectively. Figs. 20 and 21 show the results. Solar radiation absorbed by glazed tiles also depends on their colors. However, because most of the glazed tiles employed on the domes of
Fig. 21. The solar radiation received and absorbed by the dome of St. Peter’s Church located in Rome, when covered with ordinary material, and when it can be covered with glazed tiles.
Dome
Max. Ht (MJ/day)
Absorbed Ht an = 0.8
Absorbed Ht an = 0.4
Reference dome in Yazd St. Peter’s dome in Rome
110.6 121.5
75.4% 75.8%
40.6% 45.7%
mosque, shrines and other large buildings in Iran are in light colors, we considered an absorptance of 0.4 for all these tiles. Table 4 shows the total radiation received and absorbed by the reference dome, located in the city of Yazd, and the dome of St. Peter’s Church, located in the city of Rome, during the month of June. This table shows that the percentage of the absorbed solar radiation for the reference dome in the city of Yazd in June is less than St. Peter’s. The ratio of the solar radiation absorbed by the reference dome, and that of the St. Peter’s Church, covered with glazed tiles, over the flat horizontal roofs located in the respected cities, is shown in Fig. 22. For the flat roofs, an absorptance of 0.8 was assumed. A comparison between Figs. 14 and 22 shows that glazed tiles can be helpful in reducing the total solar radiation absorbed, particularly in warm months of the year. Fig. 22 shows that covering the reference dome in the city of Yazd with glazed tiles reduces the absorbed solar radiation even below that of the flat roof (ratio < 1) during all months of the year. This is an important result for the passive cooling effects of domed roofs in Iran and other desert countries, employing glazed tiles on the domes. For the dome of St. Peter’s Church in Rome, the solar radiation absorbed by this dome, if it were covered by glazed tile, is reduced to that of flat horizontal roofs in warm months. 6. Thermal radiation to sky Thermal radiation exchange with sky is an important factor in passive cooling of buildings in desert regions, employing domed
Fig. 22. Ratio of total solar radiation absorbed by the domed roofs covered with glazed tiles to a horizontal flat roof of the same base area. The buildings are located in the cities of Yazd and Rome.
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roofs. The thermal radiation exchange with sky may be determined through the following equation [10]: 1 þ Cos b 4 4 (20) T Þ Isky ¼ es ðTroo f sky 2 T sky ¼ e0:25 sky T a
(21)
esky ¼ 0:74 þ 0:006T d p
(22)
In these equations various parameters are defined as follows: e Surface emittance
s
Stefan–Boltzmann constant
Troof
Roof temperature, in Kelvin
Tsky
Sky temperature, in Kelvin
Ta
Ambient air temperature, in Kelvin
Tdp
Ambient air dew point temperature, in Centigrade
esky
Sky emissivity
Thermal radiation exchange with sky, the convection heat transfer between the dome surface and the ambient air, and the heat conducted through the dome material depend on the temperature of the surface of the dome. The dome surface temperature is not constant at any one location on the dome and at any one time. To evaluate the domed roof surface temperature at any one location and at any one time, we need to consider, in addition to the above heat transfer modes, the convection heat transfer between the inside surface of the dome with the inside air, and thermal radiation exchange between this surface and all interior surfaces that it can ‘‘see’’. This requires a complete thermal analysis of the dome and the building it covers. This analysis will not be considered in this study, but will be reported in another paper. 7. Conclusions Direct, diffuse, as well as ground-reflected, solar radiation was estimated for several domed roofs, a flat roof, and a roof of conical shape, assumed to be located in the city of Yazd, Iran, and the dome of the St. Peter’s Church, located in the city of Rome, Italy. The results show that maximum solar radiation received by these roofs, located in the northern hemisphere, occurs in June, and it is independent of the geographical locations and the climatic
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conditions. Main portion of the radiation received by the domes located in the city of Yazd was beam radiation, whereas for the dome located in Rome, the direct and diffuse components of the solar radiation were almost equal to each other. It was also found that the domed roofs receive more solar radiation than the flat roofs of equal base areas. Based on the unit areas of the dome itself, the flat roofs received more solar radiation. The domes with higher aspect ratios, for example, that of St. Peter’s Church, received more solar radiation than the ones with lower aspect ratios, when considered to be located in the same city. The solar radiation absorbed by domed roofs can be reduced appreciably by covering them with glazed tiles. This is particularly important in locations with very high solar radiation. For the city of Yazd, which receives very high solar radiation throughout the year, the domes covered with glazed tiles absorb less solar radiation than the comparable flat roofs. This is important in domed roofs, providing passive cooling for the buildings they cover. The results of this investigation become useful in thermal performance, and passive cooling evaluation of domed roof buildings. Acknowledgement The authors wish to acknowledge with thanks the financial support of the Iranian National Science Foundation. References [1] M.N. Bahadori, Passive cooling systems in Iranian architecture, Scientific American 238 (February) (1978) 144–154. [2] A.K. Faghih, M.N. Bahadori, Experimental investigation of air flow over domed roofs, Iranian Journal of Science and Technology; Transaction B: Engineering 33 (B3) (2009) 207–216. [3] A.K. Faghih, M.N. Bahadori, Three dimensional numerical investigation of air flow over domed roofs, submitted for publication. [4] S. Serpoushan, M. Yaghoubi, Solar energy calculation on 3D surfaces, Iran Energy Journal (13) (2002) 3–21 (in Persian). [5] T.I. Runsheng, A. Meir, Y. Etzion, An analysis of absorbed radiation by domed and vaulted roofs as compared with flat roofs, Energy and Building 35 (2003) 539–548. [6] V.M. Gomez, M.A. Porta, C. Heard, Solar performance of hemispherical vaulted roofs, Building and Environment 38 (2003) 1431–1438. [7] J.A. Duffie, W.A. Beckman, Solar Engineering of Thermal Processes, 3rd edition, Wiley, 2006. [8] M. Daneshyar, Solar radiation statistics for Iran, Solar Energy 21 (1978) 345–349. [9] M.N. Bahadori, S. Mirhosseini, Sky Clearness Index for Iranian Cities, Paper No. 1120 ISES 2005, Solar World Congress, Orlando, FL, USA, 2005. [10] P. Berdahl, R. Fromberg, The thermal radiance of clear skies, Solar Energy 29 (1982) 299–314.