This article has permanently moved here from the Natural Frequency Journal, though you can still view a copy of the original at the Internet Archive.
The domed roof is a common architectural feature of many hot dry climates around the world. It is obviously a product of the available materials and constructional techniques used at the time, being a logical extension of the self-supporting arch and the vault. However, it has another key benefit related to the way it responds to incident solar radiation.
The arch, the vault and the dome are all important as they distribute forces only through compression. When constructed of solid non-compressible materials such as stone, concrete or mud-brick, they can carry great weight and span large distances. As a result, they are usually constructed of heavy materials with a high thermal mass.
If you consider a flat roof, its entire planar surface is always exposed to the sun throughout the day. As in Figure 2 below, there may be some times early in the morning and late in the afternoon when the parapet protects a small part of it, however the heat build-up due to incident solar radiation on the same surface will continue pretty much throughout the day.
Obviously the intensity of the radiation on the surface will be slow to build as the sun in the early morning strikes it from a relatively low angle. However as the sun rises in the sky, the angle comes closer to normal incidence. At its height around noon, the entire surface is subject to its full intensity, which could be as much as 900W/m2.
When subject to incident solar radiation, the outside surface of any material will heat up. Whilst the calculation of actual surface temperature is quite complex, the equivalent sol-air temperature [ASHRAE, 1974] is a good indicator of the overall effects and is a function of incident radiation, air temperature, solar absorption and the resistance of the external air film along the surface. Assuming an outside air temperature of around 32°C, a solar absorption of 0.6 and an outside air-film resistance of 0.045 m2K/W (for an exposed horizontal surface), this will result in an equivalent sol-air temperature of over 56°C.
Given the path of the sun in summer, the whole roof could be subject to relatively high levels of incident gains for up to 10 hours each day. Assuming a reasonably thick earthen roof with a thermal lag of around 8 hours, this heat energy will begin to be emitted from the inside surface of the roof in the late afternoon, building up to a peak around 8-9pm and continuing through until as late as 2am.
Traditionally flat roofs require a timber support structure beneath and, having both insulating properties and a relatively low emissivity, significantly reduce the radiative effect. A domed roof does not require such support, so it is basically just exposed thermal mass from which this heat would be readily emitted.
When a dome is exposed to solar radiation, only a small part of its surface area receives it directly at normal incidence. The rest of its surface is either self-shaded or receives the radiation at much greater incidence angles. More importantly, the areas exposed to the radiation change throughout the day as the sun moves through the sky, as shown in Figure 5 below.
As you can see in the animation in Figure 5, each facet receives a variable amount of solar radiation for no more than 5-6 hours. The area of greatest incidence can be clearly seen to travel across the surface from the east to the west side.
Thus, a much smaller area of the roof is exposed to the full intensity of the sun and, even then, for a much shorter amount of time. There is not the same slow build-up of sol-air temperature over the whole day as with a flat roof, so the peak heat flows are much less and occur from different parts of the inside surface of the dome at different times during the evening and night.
It can be inferred from this simple example that there is the possibility of shaping the form of a building such that its response to solar radiation can be controlled to some extent by the designer. In a highly overshadowed environment, such as an inner-city site, a dome may not be an optimum shape at all. Depending on the nature of overshadowing, much more complex shapes may be more appropriate if response to solar radiation is a major design consideration.
Thus, before any shape analysis can be done, it must be possible to determine both the exact nature of overshadowing on any site and the availability of solar radiation at different points within it. Then, to display the distribution of radiation over complex surfaces, a method for accurately calculating incident solar radiation for each facet must be used.
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