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Solar performance of hemispherical vault roofs
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Building and Environment 38 (2003) 1431 – 1438
www.elsevier.com/locate/buildenv
Solar performance of hemispherical vault roofs & Victor M. G&omez-Mu˜noza , Miguel Angel Porta-G&andarab;∗ , Christopher Heardc a Centro
Interdisciplinario de Ciencias Marinas, Av. Instituto Politecnico Nacional s/n, Playa El Conchalito, La Paz, Baja California Sur 23090, Mexico b Centro de Investigaciones Biol ogicas del Noroeste S.C., El Comitan La Paz, P.O. Box 128, Baja California Sur 23000, Mexico c Instituto Mexicano del Petr oleo, P.O. Box 14-805, DF 07730, Mexico Received 30 January 2003; received in revised form 17 June 2003; accepted 13 July 2003
Abstract In hot climates, the improvement of comfort by passive solar techniques is a very important issue. In many parts of the world such as the Middle East, vault roofs are widely used in construction. The solar and energy performance of a hemispherical vault roof is studied, including the auto-shading instant e7ect during several days for di7erent latitudes and throughout the year also. The results are compared with the standard horizontal 8at roo9ng used in the typical modern low-cost housing in Mexico. The hemispherical vault receives around 35% less energy than the 8at roof between the equinoxes, besides having other advantages such as a greater ceiling height, natural ventilation and illumination possibilities, and structural stability. ? 2003 Elsevier Ltd. All rights reserved. Keywords: Hemispherical vault roof; Shading; Solar control
1. Introduction Roofs are a building element that is exposed to solar radiation throughout the day. As such they can have a considerable in8uence on the thermal performance of buildings, where they form a large proportion of the building envelope surface area. Hemispherical vaults, domes or cupolas have probably been used in buildings since about 6000 years ago. Through the ages they have been used for large state or religious buildings mainly cathedrals, churches, and mosques, and also for buildings right down to single-family dwellings such as the Arab houses and Inuit igloos. Gadi [1] used a CFD program to study a new design of roof system to induce natural ventilation and cooling in summer, and provide heating in winter by combining a transparent dome with a pitched thermal roof. There is a considerable body of work on solar passive cooling. Nahar et al. [2] established that 50% of the heat load in a building in an arid area is from the roof, and various solar passive techniques, such as thermal insulation and painting the roof white, were analysed. Raeissi and Thaeri [3], by means of a mathematical model, concluded that for a popular ∗
Corresponding author. Tel.: +52-612-1238484; fax: +52-612-55070. E-mail address: [email protected] (M.A. Porta-G&andara).
0360-1323/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.buildenv.2003.07.005
house in Shiraz, important cooling load demand reductions may be obtained by using shaded-pond, pond and shaded roofs. Bahadori and Haghighat [4] studied natural ventilation due to wind e7ects in buildings employing domed roofs. The dome was assumed to have an opening at its crown. When compared with 8at roofs, the domed roofs always increased the air8ow rate through the building. A bibliographic search was carried out to 9nd work on the solar performance of domed roofs. It appears that very little has been published on the thermal or solar performance of an opaque hemispherical or domed roof. There are some papers on transparent domed skylights or roofs. For instance, Thompson [5] described a roughly dome-shaped 9ve-sided pyramid, which was claimed to be a solar hut; however, the author gave no analysis of its solar performance. In this work the solar incidence over a hemispherical vault roof is studied, including its auto-shading e7ect and then compared to a horizontal roof type ceiling. 2. Methodology The amount of solar beam radiation reaching the earth’s surface varies because of the change in the atmospheric
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V.M. Gomez-Mu˜noz et al. / Building and Environment 38 (2003) 1431 – 1438
conditions and the sun position (altitude and azimuth angles), both during the day and throughout the year. The total instantaneous solar radiation Gt (t) over a horizontal surface was separated into direct beam radiation Gb (t) and di7use Gd (t): Gt (t) = Gb (t) + Gd (t); where
1:5 t − 12 ; SDL 1:2 t − 12 ; Gb (t) = Gbm cos SDL
(1)
Gt (t) = Gtm cos
(2) (3)
where Gtm is the maximum total radiation; Gbm the maximum beam radiation; SDL the solar day length; Gd (t) is calculated from expression (1): Gd (t) = Gt (t) − Gb (t):
(4)
For the calculation of the solar radiation on a sloped surface Gst (t), it is necessary to consider a projection factor over the normal to the surface as follows: Gst (t) = Gsb (t) + Gd (t);
(5)
where the instantaneous solar beam radiation incident on a surface Gsb (t) depends on the solar time t of the day, the projection factor Rb of the surface and the sun incidence, in the form of the following expression: 1:5 12 − t Gsb (t) = I (t)Rb Gbm cos ; (6) SDL where I (t) is the sun incidence factor, one when sun reaches the surface, and zero in the contrary case Rb =
cos() ; cos(Z )
(7)
where is the angle of the solar beam radiation on a surface and z the zenith angle, angle between the angle of incidence on a horizontal surface [6]. The air-mass value at any particular time depends on the location (latitude), altitude, the time of day, and the day of the year. Air-mass values are higher when the sun is lower in the sky. When the sun is closer to the horizon, direct beam radiation must pass through a longer distance in the earth’s atmosphere than when the sun is overhead. The cosine function in expressions (2,3, and 6) represents the solar beam attenuation from minimum at noon to the maximum at sunrise and sunset [7]. In order to estimate the incident solar beam radiation on a hemispherical vault roof of radius R, the vault surface was 9tted by means of small 8at rectangles whose dimensions depend on the number of “parallels” np and “meridians” nm used to locate each rectangle on the vault surface. Each parallel is characterized by an angle , like earth latitude and each meridian by the corresponding azimuth y (South
Fig. 1. Parameterization of 8at rectangles to cover the dome: (a) Position of each rectangle in spherical coordinates: azimuth and elevation, and radius r of the corresponding parallel; (b) rectangle tilt angle as a function of angle.
referenced), like an earth longitude (Fig. 1a). The radius r( ) of a parallel located by an angle is calculated from: r( ) = R cos( ) (Fig. 1b), then the sides of each rectangle Lp and Lm are given by Lp =
R ; 2np
(8)
Lm =
r( ) : nm
(9)
The instantaneous total solar radiation TSR v (t), per square meter (W=m2 ) over the hemispherical vault surface, is calculated by the addition of the radiation over all the rectangles: TSR v (t) =
1 Gst (t)Ar ; Av
(10)
where Ar =Lp •Lm is the rectangle area; Gst (t) is the radiation on the rectangle of azimuth ( ), and slope () (Fig. 1b); and Av = 2R 2 is the vault area. The total solar radiation over a 8at roof per square meter TSR f (t) is calculated using Eq. (2).
V.M. Gomez-Mu˜noz et al. / Building and Environment 38 (2003) 1431 – 1438
1433
2.1. Hemispherical vault roof performance The total solar energy incident on the hemispherical vault roof Ev (n) over a solar day (n) in (MJ=m2 ) was calculated by numeric integration of TSR v (t) over solar time in seconds, and similarly, the energy incident on the 8at roof Ef (n) was calculated by integration of TSR f (t). Then, the energy ratio percentage Er (n), between a hemispherical vault roof and the 8at roof, was calculated by Er (n) = 1 −
Ev (n) ; Ef (n)
(11)
where Ev (n) is the total solar energy incident on the vault roof during a day (n) and Ef (n) the total solar energy incident on the 8at roof during a day (n). This Er (n) represents the energy performance of the hemispherical vault with respect to the 8at roof, and indicates the percentage of the change of the energy received by the hemispherical vault compared to the 8at roof.
3. Results In order to evaluate the solar performance of the hemispherical vault roof, a computer program was developed to analyse each of the rectangular planes which covered the entire surface of the hemispherical vault roof. A 40 × 60 rectangle grid was selected and occurrence of solar incidence was determined for a given time, day and latitude for each grid point’s rectangle. 3.1. Auto-shading hemispherical vault performance One particular characteristic of the dome-type roof is the auto-shading e7ect, which occurs most of the day except possibly at noon (Fig. 2). Fig. 3 shows the evolution of the percentage of the irradiated and shaded hemispherical vault area (calculated by adding the illuminated rectangles areas), with respect to the total dome surface, during the solar day, for the equinox (n = 80), the autumn and winter solstices (n = 172 and 355) for northern latitudes, 0◦ , 20◦ and 40◦ , respectively. The lighter line indicates the illuminated percentage of the total dome area and the darker line is its complement, i.e. the shaded percentage of dome area. For any day of the year and for any latitude, 50% of the dome receives direct beam solar radiation at sunrise, reaching the maximum solar incidence and minimum auto-shading at noon. After noon the reverse process occurs. The area between the auto-shaded and incidence percentage curves represents the strength of the solar luminance over the dome. For instance, when the area is smallest, the auto-shaded advantage of the dome is maximum, which occurs in winter. For latitudes north of the equator, the maximum irradiation is achieved during the summer solstice.
Fig. 2. Hemispheric vault shading simulation at 10 am solar time for march 21 at 24◦ N Latitude. Lighter colour shows the solar radiation over the surface and darker colour represent the shadow.
Only on the equator, the maximum irradiation is on the equinoxes. The mean value of the shaded fraction of the hemispherical vault area is calculated for each solar day during the year, representing the shading performance of the hemispherical vault roof for 0◦ , 10◦ , 20◦ , 30◦ , 40◦ and 50◦ North latitudes (Fig. 4). Analogous results are obtained in southern latitudes with 6 months di7erence. At higher latitudes, the percentage of the hemispherical vault area goes from 45% to 32%, meanwhile at equatorial latitudes varies from 29% to 26%. The auto-shading vault performance is better for higher latitudes, especially during the wintertime and in equatorial latitudes is about the same for summer and winter, with the minimum at the equinoxes. The dip in each curve around summer solstice changes for latitudes above the tropic. 3.2. Comparative solar performance between =at roof and hemispherical vault roof The roof-covered area in a dwelling is the same for a dome and for a 8at roof. A hemispherical vault roof of 3 m in diameter has a surface area of 56 m2 . Meanwhile, the corresponding area for the 8at roof is 26 m2 , which is equal to the base area of the hemispherical vault. The instantaneous solar performance of a 8at roof and a hemispherical vault roof (from expressions 2 and 10, respectively) along the solstices and equinoxes for latitude of 24◦ N is shown in Fig. 5. As an actual example, averages of observed daily maximums of beam and total solar radiation (W=m2 ) in La Paz City, Mexico, for the following months were used: March (649, 820), June (556, 763), September (611, 796) and December (522, 654).
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V.M. Gomez-Mu˜noz et al. / Building and Environment 38 (2003) 1431 – 1438
Latitude 0°N
Latitude 40°N
100
100
75
75
75
50
50
50
25
25
25
0
Partial vault area (%)
Latitude 20°N
100
6 8 10 12 14 16 18
0
0
6 8 10 12 14 16 18
100
100
100
75
75
75
50
50
50
25
25
25
0
6 8 10 12 14 16 18
0
0
6 8 10 12 14 16 18
100
100
100
75
75
75
50
50
50
25
25
25
0
0
6 8 10 12 14 16 18
0
6 8 10 12 14 16 18
Day 80
6 8 10 12 14 16 18
Day 172
6 8 10 12 14 16 18
Day 355
6 8 10 12 14 16 18
Solar time (h) Fig. 3. Percentage of the radiated and shaded hemispherical vault area, as regarding the total hemispherical vault surface, along the solar day, for the equinox (n = 80), the autumn and winter solstices (n = 172 and 355) for northern latitudes, 0◦ , 20◦ and 40◦ , respectively. The lighter line indicates the partial illuminated hemispherical vault area and the darker line is the complement-shaded partial vault area.
Latitude 0°N
45 40
40
35
35
30
30
Partial vault area (%)
25
34
80
126 172 218 264 310 356
Latitude 20°N
45
25
40
35
35
30
30 34
80
126 172 218 264 310 356
Latitude 40°N
45
25
40
35
35
30
30 34
80
126 172 218 264 310 356
Day
80
25
126 172 218 264 310 356
Latitude 20°N
34
80
126 172 218 264 310 356
Latitude 40°N
45
40
25
34
45
40
25
Latitude 0°N
45
34
80
126 172 218 264 310 356
Day
Fig. 4. Daily mean value of the shaded partial vault area along the year of a hemispherical vault roof for: 0◦ , 10◦ , 20◦ , 30◦ , 40◦ and 50◦ N latitudes.
The solar radiation during the day is less for the hemispherical vault than the corresponding 8at roof, mainly around noon, when the sun’s rays strike nearly perpendicular
to the horizontal. Solar radiation over these two roofs is coincident near sunrise and sunset. The oblique sunrays on the 8at roof after and before sunrise and sunset
V.M. Gomez-Mu˜noz et al. / Building and Environment 38 (2003) 1431 – 1438
1435
Latitude 24° N
Total Solar Radiation (W/m2)
Day 80, Ef = 21.2, Ev = 15.8
Day 172, Ef = 22.2, Ev = 16.6
800
800
600
600
400
400
200
200
0
6
8
10
12
14
16
18
0
6
8
10
12
14
16
18
Vault Flat roof
Day 264, Ef = 20.6, Ev = 15.5
Day 356, Ef = 14.9, Ev = 12.5
800
800
600
600
400
400
200
200
0
6
8
10
12
14
16
18
0
6
8
10
12
14
16
18
Solar time (h) Fig. 5. Instantaneous total solar radiation for 24◦ N during solstices and equinoxes of a hemispherical vault and a 8at roof of equivalent basis areas. Ef and Ev represent the total solar radiation (MJ=m2 ) over the 8at and the vaulted roofs respectively of the corresponding day.
respectively improve the performance of this roof over the dome for a few minutes. Other than during these short time periods, the hemispherical vault roof has a better performance than that of a 8at roof for solar radiation gain. During the summer solstice this bene9t lasts more than 10 h. The daily total solar energy was always greater on the 8at roof (Ef ) than over the vaulted (Ev ). There were slight variations of the energy on each roof during the year, excepting the winter solstice when the lower maximum mean radiation and the solar day length were considerably lower than the rest of the year (Fig. 5). In order to show the energy performance in (MJ=m2 ) of these two shelter construction elements during the year for several latitudes, a constant solar radiation of 1000 W=m2 was used for each day. The left column in Fig. 6 shows the daily-received energy from the sun over the two kinds of roof for 0◦ , 10◦ , and 20◦ N. The energy received over the 8at roof is always higher than that of the domed roof, with an important increment attained for the summer at higher latitudes. The energy received for the hemispherical vault roof is almost the same for these latitudes. The improvement of the domed roof was calculated from Eq. (11) (right column of Fig. 6). This improvement was always over 25%, with maximums at the equator during the equinoxes and a minimum in the summer solstice. The extreme values were almost coincident during summer when the latitude increases, creating a 8at zone in the graph around the summer solstice, representing a constant improvement
of the domed roof greater than 35% over the 8at roof. The former e7ect was caused by the improvement of the performance of the hemispherical vault roof at greater latitudes during summer days. Similarly, the energy performance above the tropic is shown in Fig. 7, the left column shows the daily-received energy from the sun over the two kinds of roof for 30◦ , 40◦ , and 50◦ N latitudes. The solar energy performance for the hemispherical vault roof is better than the 8at roof every day of the year, excepting for a few days above 40◦ N latitude and for several days around the winter solstice at 50◦ N. In the right column of Fig. 7, it may be observed that the local minimum around the summer solstice disappears above the tropic, due to a greater eQciency of the domed roof over the 8at one during those days, although the percentages are lower than below the tropic. The 8attened shape of the percentage curve around the summer solstice became rounded with increasing latitude. During wintertime, the percentages are negative, that is to say, the 8at roof performance is better than the vaulted roof during those days when the sun rays are too oblique and the 8at roof is less a7ected by these. At the Tropic of Cancer (23:45◦ N), the energy ratio is completely 8at during summer time (Fig. 8), representing a constant improvement of the domed roof over the 8at one, of about 36%. For the rest of the seasons, an abrupt reduction of the improvement was obtained, reaching the minimum value at winter solstice, of around 22%.
1436
V.M. Gomez-Mu˜noz et al. / Building and Environment 38 (2003) 1431 – 1438 Lat 0°N
30
Lat 0°N
40 35
25
30 20
25 80
172
264
Lat 10° N
30 25 20 15
80
172
20
356
Energy ratio (%)
Energy (MJ/m²)
15
264
356
Lat 20° N
30
80
172
264
356
264
356
264
356
Lat 10°N
40 35 30 25 20
80
172
Lat 20°N
40 35
25
30 20 15
25 80
172
264
20
356
Vault roof Flat roof
80
172
Day
Fig. 6. Comparison of 8at and domed roof along the year for latitudes below the Tropic of Cancer. Daily solar energy received (MJ=m2 ) for each roof type (left column) and percentage of improvement of the hemispherical vaulted roof over the 8at one (right column).
Lat 30° N
20
30 25
0
20
-20 80
172
264
Lat 40° N 35 30 25 20 15
80
172
-40
356
Energy ratio (%)
Energy (MJ/m²)
15
264
356
Lat 50° N
80
172
264
356
264
356
264
356
Lat 40°N
40 20 0 -20 -40
80
172
Lat 50° N
40
35
20
30 25
0
20
-20
15
Lat 30° N
40
35
80
172
264
-40
356
Vault roof Flat roof
80
172
Day
Fig. 7. Comparison of 8at and domed roof along the year for latitudes above the Tropic of Cancer. Daily solar energy received (MJ=m2 ) for each roof type (left column) and percentage of improvement of the hemispherical vaulted roof over the 8at one (right column).
4. Concluding remarks The hemispherical vault roof has some solar advantages over other types of roofs: (i) its auto-shading property, which diminishes the received solar energy, especially when
the sun is out of the zenith. (ii) At any time, the solar energy strikes the dome normally at only one point on the surface. (iii) The exposed surface of the dome is the largest among roofs with regular geometry (8at, tilted, gable, hipped, pavilion hipped, barrel vault, etc.), for the same height and
V.M. Gomez-Mu˜noz et al. / Building and Environment 38 (2003) 1431 – 1438
1437
Lat 23.45° N
40
35
35
30
30
Ef 25
25
20
20
Energy ratio (%)
Energy (MJ/m²)
Er
Ev 15
34
80
126
172
218
264
310
356
Day
Fig. 8. Energy (Ef and Ev ) and energy ratio (Er ) performance at the Tropic of Cancer of the 8at and hemispherical vaulted roof along the year.
equivalent base area, with the result that the intensity of solar radiation is spread over a larger area and heat transmission to the interior is reduced. (iv) Most of the day, part of the roof is shaded from the sun, at that time it can act as a radiator, absorbing heat from the sunlit part of the roof and the internal air, and transmitting it to the cooler outside air in the roof’s shade. This e7ect is particularly e7ective for roofs domed in the form of a hemisphere since at least part of the roof is always shaded except at noon when the sun is directly overhead. The former properties are better understood as a function of solar elevation angle, since the solar radiation received by the hemispherical vault roof is independent of the azimuth. The solar elevation angle is the only one that a7ects the instantaneous and daily solar performance of the dome. This angle is a function of declination, latitude and solar time. Therefore, the changes of the solar performance of a dome are solely a consequence of the solar elevation angle. From the former point of view, latitudes and seasons obviously play an important role in the dome performance. The best performance of the hemispherical vault is during the equinoxes with lower and similar solar energy received during solstices. The results demonstrate that the solar performance of the hemispheric vault roof is better near equatorial latitudes. At these latitudes, the received energy is always less than for higher latitudes and it is also less 8uctuating. When the sun passes near the zenith, during summer time for northern latitudes, the solar performance of a dome is better than a 8at roof of equivalent base area. At noon, the dome’s performance is always better than that of a 8at roof.
The hemispherical vault roof has particular architectural characteristics due its geometry. It has the maximum structural stability because there are no corners or edges where stresses occur. The structure’s weight is also evenly distributed and presses directly down, eliminating any spreading forces on the foundation. Undoubtedly, the dome gives much better comfort than other roof types, giving a higher ceiling for the shelter and a greater indoor air volume. The higher ceiling reduces the view factor for radiative heat transfer between it and the heads of the occupants of the space, and so reduces any uncomfortable e7ects due to high interior surface temperatures. It is also possible to provide fenestrations in the dome to increase the air8ow and natural lighting inside the dwelling. These advantages together with the possibility of using a wide range of construction materials under purely compressive forces explain the widespread traditional use of domes in hot climates. Acknowledgements The authors appreciate the 9nancial support from the National Council of Science and Technology of Mexico (CONACyT), Grant No. G35167-U. References [1] Gadi MB. A novel roof-integrated cooling and heating system. International Journal of Ambient Energy 2000;21(4):203–11. [2] Nahar NM, Sharma P, Purohit MM. Studies of solar passive cooling techniques for arid areas. Energy Conversion & Management 1999;40:89–95.
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V.M. Gomez-Mu˜noz et al. / Building and Environment 38 (2003) 1431 – 1438
[3] Raeissi S, Thaeri M. Cooling load reductions of buildings using passive roof options. Renewable Energy 1996;7(3): 301–13. [4] Bahadori MN, Haghighat F. Passive cooling in hot, arid regions in developing countries by employing domed roofs and reducing the temperature of internal surfaces. Building and Environment 1985;20(2):103–13.
[5] Thompson WH. Concept of an earthquake-proof hut for Mexico (and other countries). Renewable Energy 1995;6(8):977–81. [6] DuQe JA, Beckman WA. Solar engineering of thermal processes. New York: Wiley; 1991. p. 1–141. [7] Porta-G&andara, Rubio E, Fern&andez JL. Economic feasibility of passive ambient comfort in Baja California Dwellings. Building and Environment 2002;37:993–1001.
Building and Environment 38 (2003) 1431 – 1438
www.elsevier.com/locate/buildenv
Solar performance of hemispherical vault roofs & Victor M. G&omez-Mu˜noza , Miguel Angel Porta-G&andarab;∗ , Christopher Heardc a Centro
Interdisciplinario de Ciencias Marinas, Av. Instituto Politecnico Nacional s/n, Playa El Conchalito, La Paz, Baja California Sur 23090, Mexico b Centro de Investigaciones Biol ogicas del Noroeste S.C., El Comitan La Paz, P.O. Box 128, Baja California Sur 23000, Mexico c Instituto Mexicano del Petr oleo, P.O. Box 14-805, DF 07730, Mexico Received 30 January 2003; received in revised form 17 June 2003; accepted 13 July 2003
Abstract In hot climates, the improvement of comfort by passive solar techniques is a very important issue. In many parts of the world such as the Middle East, vault roofs are widely used in construction. The solar and energy performance of a hemispherical vault roof is studied, including the auto-shading instant e7ect during several days for di7erent latitudes and throughout the year also. The results are compared with the standard horizontal 8at roo9ng used in the typical modern low-cost housing in Mexico. The hemispherical vault receives around 35% less energy than the 8at roof between the equinoxes, besides having other advantages such as a greater ceiling height, natural ventilation and illumination possibilities, and structural stability. ? 2003 Elsevier Ltd. All rights reserved. Keywords: Hemispherical vault roof; Shading; Solar control
1. Introduction Roofs are a building element that is exposed to solar radiation throughout the day. As such they can have a considerable in8uence on the thermal performance of buildings, where they form a large proportion of the building envelope surface area. Hemispherical vaults, domes or cupolas have probably been used in buildings since about 6000 years ago. Through the ages they have been used for large state or religious buildings mainly cathedrals, churches, and mosques, and also for buildings right down to single-family dwellings such as the Arab houses and Inuit igloos. Gadi [1] used a CFD program to study a new design of roof system to induce natural ventilation and cooling in summer, and provide heating in winter by combining a transparent dome with a pitched thermal roof. There is a considerable body of work on solar passive cooling. Nahar et al. [2] established that 50% of the heat load in a building in an arid area is from the roof, and various solar passive techniques, such as thermal insulation and painting the roof white, were analysed. Raeissi and Thaeri [3], by means of a mathematical model, concluded that for a popular ∗
Corresponding author. Tel.: +52-612-1238484; fax: +52-612-55070. E-mail address: [email protected] (M.A. Porta-G&andara).
0360-1323/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.buildenv.2003.07.005
house in Shiraz, important cooling load demand reductions may be obtained by using shaded-pond, pond and shaded roofs. Bahadori and Haghighat [4] studied natural ventilation due to wind e7ects in buildings employing domed roofs. The dome was assumed to have an opening at its crown. When compared with 8at roofs, the domed roofs always increased the air8ow rate through the building. A bibliographic search was carried out to 9nd work on the solar performance of domed roofs. It appears that very little has been published on the thermal or solar performance of an opaque hemispherical or domed roof. There are some papers on transparent domed skylights or roofs. For instance, Thompson [5] described a roughly dome-shaped 9ve-sided pyramid, which was claimed to be a solar hut; however, the author gave no analysis of its solar performance. In this work the solar incidence over a hemispherical vault roof is studied, including its auto-shading e7ect and then compared to a horizontal roof type ceiling. 2. Methodology The amount of solar beam radiation reaching the earth’s surface varies because of the change in the atmospheric
1432
V.M. Gomez-Mu˜noz et al. / Building and Environment 38 (2003) 1431 – 1438
conditions and the sun position (altitude and azimuth angles), both during the day and throughout the year. The total instantaneous solar radiation Gt (t) over a horizontal surface was separated into direct beam radiation Gb (t) and di7use Gd (t): Gt (t) = Gb (t) + Gd (t); where
1:5 t − 12 ; SDL 1:2 t − 12 ; Gb (t) = Gbm cos SDL
(1)
Gt (t) = Gtm cos
(2) (3)
where Gtm is the maximum total radiation; Gbm the maximum beam radiation; SDL the solar day length; Gd (t) is calculated from expression (1): Gd (t) = Gt (t) − Gb (t):
(4)
For the calculation of the solar radiation on a sloped surface Gst (t), it is necessary to consider a projection factor over the normal to the surface as follows: Gst (t) = Gsb (t) + Gd (t);
(5)
where the instantaneous solar beam radiation incident on a surface Gsb (t) depends on the solar time t of the day, the projection factor Rb of the surface and the sun incidence, in the form of the following expression: 1:5 12 − t Gsb (t) = I (t)Rb Gbm cos ; (6) SDL where I (t) is the sun incidence factor, one when sun reaches the surface, and zero in the contrary case Rb =
cos() ; cos(Z )
(7)
where is the angle of the solar beam radiation on a surface and z the zenith angle, angle between the angle of incidence on a horizontal surface [6]. The air-mass value at any particular time depends on the location (latitude), altitude, the time of day, and the day of the year. Air-mass values are higher when the sun is lower in the sky. When the sun is closer to the horizon, direct beam radiation must pass through a longer distance in the earth’s atmosphere than when the sun is overhead. The cosine function in expressions (2,3, and 6) represents the solar beam attenuation from minimum at noon to the maximum at sunrise and sunset [7]. In order to estimate the incident solar beam radiation on a hemispherical vault roof of radius R, the vault surface was 9tted by means of small 8at rectangles whose dimensions depend on the number of “parallels” np and “meridians” nm used to locate each rectangle on the vault surface. Each parallel is characterized by an angle , like earth latitude and each meridian by the corresponding azimuth y (South
Fig. 1. Parameterization of 8at rectangles to cover the dome: (a) Position of each rectangle in spherical coordinates: azimuth and elevation, and radius r of the corresponding parallel; (b) rectangle tilt angle as a function of angle.
referenced), like an earth longitude (Fig. 1a). The radius r( ) of a parallel located by an angle is calculated from: r( ) = R cos( ) (Fig. 1b), then the sides of each rectangle Lp and Lm are given by Lp =
R ; 2np
(8)
Lm =
r( ) : nm
(9)
The instantaneous total solar radiation TSR v (t), per square meter (W=m2 ) over the hemispherical vault surface, is calculated by the addition of the radiation over all the rectangles: TSR v (t) =
1 Gst (t)Ar ; Av
(10)
where Ar =Lp •Lm is the rectangle area; Gst (t) is the radiation on the rectangle of azimuth ( ), and slope () (Fig. 1b); and Av = 2R 2 is the vault area. The total solar radiation over a 8at roof per square meter TSR f (t) is calculated using Eq. (2).
V.M. Gomez-Mu˜noz et al. / Building and Environment 38 (2003) 1431 – 1438
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2.1. Hemispherical vault roof performance The total solar energy incident on the hemispherical vault roof Ev (n) over a solar day (n) in (MJ=m2 ) was calculated by numeric integration of TSR v (t) over solar time in seconds, and similarly, the energy incident on the 8at roof Ef (n) was calculated by integration of TSR f (t). Then, the energy ratio percentage Er (n), between a hemispherical vault roof and the 8at roof, was calculated by Er (n) = 1 −
Ev (n) ; Ef (n)
(11)
where Ev (n) is the total solar energy incident on the vault roof during a day (n) and Ef (n) the total solar energy incident on the 8at roof during a day (n). This Er (n) represents the energy performance of the hemispherical vault with respect to the 8at roof, and indicates the percentage of the change of the energy received by the hemispherical vault compared to the 8at roof.
3. Results In order to evaluate the solar performance of the hemispherical vault roof, a computer program was developed to analyse each of the rectangular planes which covered the entire surface of the hemispherical vault roof. A 40 × 60 rectangle grid was selected and occurrence of solar incidence was determined for a given time, day and latitude for each grid point’s rectangle. 3.1. Auto-shading hemispherical vault performance One particular characteristic of the dome-type roof is the auto-shading e7ect, which occurs most of the day except possibly at noon (Fig. 2). Fig. 3 shows the evolution of the percentage of the irradiated and shaded hemispherical vault area (calculated by adding the illuminated rectangles areas), with respect to the total dome surface, during the solar day, for the equinox (n = 80), the autumn and winter solstices (n = 172 and 355) for northern latitudes, 0◦ , 20◦ and 40◦ , respectively. The lighter line indicates the illuminated percentage of the total dome area and the darker line is its complement, i.e. the shaded percentage of dome area. For any day of the year and for any latitude, 50% of the dome receives direct beam solar radiation at sunrise, reaching the maximum solar incidence and minimum auto-shading at noon. After noon the reverse process occurs. The area between the auto-shaded and incidence percentage curves represents the strength of the solar luminance over the dome. For instance, when the area is smallest, the auto-shaded advantage of the dome is maximum, which occurs in winter. For latitudes north of the equator, the maximum irradiation is achieved during the summer solstice.
Fig. 2. Hemispheric vault shading simulation at 10 am solar time for march 21 at 24◦ N Latitude. Lighter colour shows the solar radiation over the surface and darker colour represent the shadow.
Only on the equator, the maximum irradiation is on the equinoxes. The mean value of the shaded fraction of the hemispherical vault area is calculated for each solar day during the year, representing the shading performance of the hemispherical vault roof for 0◦ , 10◦ , 20◦ , 30◦ , 40◦ and 50◦ North latitudes (Fig. 4). Analogous results are obtained in southern latitudes with 6 months di7erence. At higher latitudes, the percentage of the hemispherical vault area goes from 45% to 32%, meanwhile at equatorial latitudes varies from 29% to 26%. The auto-shading vault performance is better for higher latitudes, especially during the wintertime and in equatorial latitudes is about the same for summer and winter, with the minimum at the equinoxes. The dip in each curve around summer solstice changes for latitudes above the tropic. 3.2. Comparative solar performance between =at roof and hemispherical vault roof The roof-covered area in a dwelling is the same for a dome and for a 8at roof. A hemispherical vault roof of 3 m in diameter has a surface area of 56 m2 . Meanwhile, the corresponding area for the 8at roof is 26 m2 , which is equal to the base area of the hemispherical vault. The instantaneous solar performance of a 8at roof and a hemispherical vault roof (from expressions 2 and 10, respectively) along the solstices and equinoxes for latitude of 24◦ N is shown in Fig. 5. As an actual example, averages of observed daily maximums of beam and total solar radiation (W=m2 ) in La Paz City, Mexico, for the following months were used: March (649, 820), June (556, 763), September (611, 796) and December (522, 654).
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V.M. Gomez-Mu˜noz et al. / Building and Environment 38 (2003) 1431 – 1438
Latitude 0°N
Latitude 40°N
100
100
75
75
75
50
50
50
25
25
25
0
Partial vault area (%)
Latitude 20°N
100
6 8 10 12 14 16 18
0
0
6 8 10 12 14 16 18
100
100
100
75
75
75
50
50
50
25
25
25
0
6 8 10 12 14 16 18
0
0
6 8 10 12 14 16 18
100
100
100
75
75
75
50
50
50
25
25
25
0
0
6 8 10 12 14 16 18
0
6 8 10 12 14 16 18
Day 80
6 8 10 12 14 16 18
Day 172
6 8 10 12 14 16 18
Day 355
6 8 10 12 14 16 18
Solar time (h) Fig. 3. Percentage of the radiated and shaded hemispherical vault area, as regarding the total hemispherical vault surface, along the solar day, for the equinox (n = 80), the autumn and winter solstices (n = 172 and 355) for northern latitudes, 0◦ , 20◦ and 40◦ , respectively. The lighter line indicates the partial illuminated hemispherical vault area and the darker line is the complement-shaded partial vault area.
Latitude 0°N
45 40
40
35
35
30
30
Partial vault area (%)
25
34
80
126 172 218 264 310 356
Latitude 20°N
45
25
40
35
35
30
30 34
80
126 172 218 264 310 356
Latitude 40°N
45
25
40
35
35
30
30 34
80
126 172 218 264 310 356
Day
80
25
126 172 218 264 310 356
Latitude 20°N
34
80
126 172 218 264 310 356
Latitude 40°N
45
40
25
34
45
40
25
Latitude 0°N
45
34
80
126 172 218 264 310 356
Day
Fig. 4. Daily mean value of the shaded partial vault area along the year of a hemispherical vault roof for: 0◦ , 10◦ , 20◦ , 30◦ , 40◦ and 50◦ N latitudes.
The solar radiation during the day is less for the hemispherical vault than the corresponding 8at roof, mainly around noon, when the sun’s rays strike nearly perpendicular
to the horizontal. Solar radiation over these two roofs is coincident near sunrise and sunset. The oblique sunrays on the 8at roof after and before sunrise and sunset
V.M. Gomez-Mu˜noz et al. / Building and Environment 38 (2003) 1431 – 1438
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Latitude 24° N
Total Solar Radiation (W/m2)
Day 80, Ef = 21.2, Ev = 15.8
Day 172, Ef = 22.2, Ev = 16.6
800
800
600
600
400
400
200
200
0
6
8
10
12
14
16
18
0
6
8
10
12
14
16
18
Vault Flat roof
Day 264, Ef = 20.6, Ev = 15.5
Day 356, Ef = 14.9, Ev = 12.5
800
800
600
600
400
400
200
200
0
6
8
10
12
14
16
18
0
6
8
10
12
14
16
18
Solar time (h) Fig. 5. Instantaneous total solar radiation for 24◦ N during solstices and equinoxes of a hemispherical vault and a 8at roof of equivalent basis areas. Ef and Ev represent the total solar radiation (MJ=m2 ) over the 8at and the vaulted roofs respectively of the corresponding day.
respectively improve the performance of this roof over the dome for a few minutes. Other than during these short time periods, the hemispherical vault roof has a better performance than that of a 8at roof for solar radiation gain. During the summer solstice this bene9t lasts more than 10 h. The daily total solar energy was always greater on the 8at roof (Ef ) than over the vaulted (Ev ). There were slight variations of the energy on each roof during the year, excepting the winter solstice when the lower maximum mean radiation and the solar day length were considerably lower than the rest of the year (Fig. 5). In order to show the energy performance in (MJ=m2 ) of these two shelter construction elements during the year for several latitudes, a constant solar radiation of 1000 W=m2 was used for each day. The left column in Fig. 6 shows the daily-received energy from the sun over the two kinds of roof for 0◦ , 10◦ , and 20◦ N. The energy received over the 8at roof is always higher than that of the domed roof, with an important increment attained for the summer at higher latitudes. The energy received for the hemispherical vault roof is almost the same for these latitudes. The improvement of the domed roof was calculated from Eq. (11) (right column of Fig. 6). This improvement was always over 25%, with maximums at the equator during the equinoxes and a minimum in the summer solstice. The extreme values were almost coincident during summer when the latitude increases, creating a 8at zone in the graph around the summer solstice, representing a constant improvement
of the domed roof greater than 35% over the 8at roof. The former e7ect was caused by the improvement of the performance of the hemispherical vault roof at greater latitudes during summer days. Similarly, the energy performance above the tropic is shown in Fig. 7, the left column shows the daily-received energy from the sun over the two kinds of roof for 30◦ , 40◦ , and 50◦ N latitudes. The solar energy performance for the hemispherical vault roof is better than the 8at roof every day of the year, excepting for a few days above 40◦ N latitude and for several days around the winter solstice at 50◦ N. In the right column of Fig. 7, it may be observed that the local minimum around the summer solstice disappears above the tropic, due to a greater eQciency of the domed roof over the 8at one during those days, although the percentages are lower than below the tropic. The 8attened shape of the percentage curve around the summer solstice became rounded with increasing latitude. During wintertime, the percentages are negative, that is to say, the 8at roof performance is better than the vaulted roof during those days when the sun rays are too oblique and the 8at roof is less a7ected by these. At the Tropic of Cancer (23:45◦ N), the energy ratio is completely 8at during summer time (Fig. 8), representing a constant improvement of the domed roof over the 8at one, of about 36%. For the rest of the seasons, an abrupt reduction of the improvement was obtained, reaching the minimum value at winter solstice, of around 22%.
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V.M. Gomez-Mu˜noz et al. / Building and Environment 38 (2003) 1431 – 1438 Lat 0°N
30
Lat 0°N
40 35
25
30 20
25 80
172
264
Lat 10° N
30 25 20 15
80
172
20
356
Energy ratio (%)
Energy (MJ/m²)
15
264
356
Lat 20° N
30
80
172
264
356
264
356
264
356
Lat 10°N
40 35 30 25 20
80
172
Lat 20°N
40 35
25
30 20 15
25 80
172
264
20
356
Vault roof Flat roof
80
172
Day
Fig. 6. Comparison of 8at and domed roof along the year for latitudes below the Tropic of Cancer. Daily solar energy received (MJ=m2 ) for each roof type (left column) and percentage of improvement of the hemispherical vaulted roof over the 8at one (right column).
Lat 30° N
20
30 25
0
20
-20 80
172
264
Lat 40° N 35 30 25 20 15
80
172
-40
356
Energy ratio (%)
Energy (MJ/m²)
15
264
356
Lat 50° N
80
172
264
356
264
356
264
356
Lat 40°N
40 20 0 -20 -40
80
172
Lat 50° N
40
35
20
30 25
0
20
-20
15
Lat 30° N
40
35
80
172
264
-40
356
Vault roof Flat roof
80
172
Day
Fig. 7. Comparison of 8at and domed roof along the year for latitudes above the Tropic of Cancer. Daily solar energy received (MJ=m2 ) for each roof type (left column) and percentage of improvement of the hemispherical vaulted roof over the 8at one (right column).
4. Concluding remarks The hemispherical vault roof has some solar advantages over other types of roofs: (i) its auto-shading property, which diminishes the received solar energy, especially when
the sun is out of the zenith. (ii) At any time, the solar energy strikes the dome normally at only one point on the surface. (iii) The exposed surface of the dome is the largest among roofs with regular geometry (8at, tilted, gable, hipped, pavilion hipped, barrel vault, etc.), for the same height and
V.M. Gomez-Mu˜noz et al. / Building and Environment 38 (2003) 1431 – 1438
1437
Lat 23.45° N
40
35
35
30
30
Ef 25
25
20
20
Energy ratio (%)
Energy (MJ/m²)
Er
Ev 15
34
80
126
172
218
264
310
356
Day
Fig. 8. Energy (Ef and Ev ) and energy ratio (Er ) performance at the Tropic of Cancer of the 8at and hemispherical vaulted roof along the year.
equivalent base area, with the result that the intensity of solar radiation is spread over a larger area and heat transmission to the interior is reduced. (iv) Most of the day, part of the roof is shaded from the sun, at that time it can act as a radiator, absorbing heat from the sunlit part of the roof and the internal air, and transmitting it to the cooler outside air in the roof’s shade. This e7ect is particularly e7ective for roofs domed in the form of a hemisphere since at least part of the roof is always shaded except at noon when the sun is directly overhead. The former properties are better understood as a function of solar elevation angle, since the solar radiation received by the hemispherical vault roof is independent of the azimuth. The solar elevation angle is the only one that a7ects the instantaneous and daily solar performance of the dome. This angle is a function of declination, latitude and solar time. Therefore, the changes of the solar performance of a dome are solely a consequence of the solar elevation angle. From the former point of view, latitudes and seasons obviously play an important role in the dome performance. The best performance of the hemispherical vault is during the equinoxes with lower and similar solar energy received during solstices. The results demonstrate that the solar performance of the hemispheric vault roof is better near equatorial latitudes. At these latitudes, the received energy is always less than for higher latitudes and it is also less 8uctuating. When the sun passes near the zenith, during summer time for northern latitudes, the solar performance of a dome is better than a 8at roof of equivalent base area. At noon, the dome’s performance is always better than that of a 8at roof.
The hemispherical vault roof has particular architectural characteristics due its geometry. It has the maximum structural stability because there are no corners or edges where stresses occur. The structure’s weight is also evenly distributed and presses directly down, eliminating any spreading forces on the foundation. Undoubtedly, the dome gives much better comfort than other roof types, giving a higher ceiling for the shelter and a greater indoor air volume. The higher ceiling reduces the view factor for radiative heat transfer between it and the heads of the occupants of the space, and so reduces any uncomfortable e7ects due to high interior surface temperatures. It is also possible to provide fenestrations in the dome to increase the air8ow and natural lighting inside the dwelling. These advantages together with the possibility of using a wide range of construction materials under purely compressive forces explain the widespread traditional use of domes in hot climates. Acknowledgements The authors appreciate the 9nancial support from the National Council of Science and Technology of Mexico (CONACyT), Grant No. G35167-U. References [1] Gadi MB. A novel roof-integrated cooling and heating system. International Journal of Ambient Energy 2000;21(4):203–11. [2] Nahar NM, Sharma P, Purohit MM. Studies of solar passive cooling techniques for arid areas. Energy Conversion & Management 1999;40:89–95.
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[3] Raeissi S, Thaeri M. Cooling load reductions of buildings using passive roof options. Renewable Energy 1996;7(3): 301–13. [4] Bahadori MN, Haghighat F. Passive cooling in hot, arid regions in developing countries by employing domed roofs and reducing the temperature of internal surfaces. Building and Environment 1985;20(2):103–13.
[5] Thompson WH. Concept of an earthquake-proof hut for Mexico (and other countries). Renewable Energy 1995;6(8):977–81. [6] DuQe JA, Beckman WA. Solar engineering of thermal processes. New York: Wiley; 1991. p. 1–141. [7] Porta-G&andara, Rubio E, Fern&andez JL. Economic feasibility of passive ambient comfort in Baja California Dwellings. Building and Environment 2002;37:993–1001.