Long wave radiation equilibrium between building, sky and ground
The building establishes long wave radiation equilibrium with the sky and the surrounding ground. The building itself is seen as a long wave radiator but the sky and the ground as well. The radiation emitted by the building and the ground is called long wave emission, the radiation emitted by the sky is named atmospheric counter radiation or sky radiation.
The geometry of the building is known. It usually has plane, sloped and oriented surfaces. The sky can be modeled as an half-infinite spheric room or as a flat plane and the ground is regarded as an flat plane perpendicular to vertical building surfaces.
The long wave radiation balance is determined by the radiation equilibrium between building, sky and ground. The influencing parameters are the emission coefficients of the building component and of the surrounding ground (albedo). The emission coefficient of the sky plays no role. The surface of the sky is much bigger compared to the building surface.
For radiation calculations after Boltzmann's law the surface temperatures of all radiators must be known. This is easily done for the building; its surface temperature is part of the numerical solution, but this is more complicated for the ground and the sky. For sake of simplification, the ground temperature and the sky temperature are usually set equal to the outdoor air temperature. This in not problematic for the ground emission but for the sky emission a model has to be applied that corrects for air humidity and cloudiness.
Ground emission modeling
We assume that the building surface, the ground and the sky can be treated as planes in radiation equilibrium. The best way to approach the physics of radiation equilibrium is to start with a simple model: with two parallel planes. Considering the emission and the reflection (reflection of one plane depends on emission and reflection of the opposite one) of both planes, the single flux of each plane to the other one can be easily derived.
The total or resulting flux between the two planes is given by the sum of the single flux of each plane. Rearranging the terms leads to the equation below that contains an expression for the radiation exchange coefficient. The emission coefficients of both planes are parameters.
Using the formula above for flux between parallel planes we can describe the radiation balance between planes that enclose an arbitrary angle. Usually the wall inclination is 90° for vertical building elements and a horizontal ground. It is evident that the transformation factor from parallel planes to non-parallel planes is given by the sinus of the wall inclination divided by 2. Using that, the equation below in the box can be written.
One question remaining concerns the ground temperature. Since it is usually not part of the numerical solution it must be known or estimated. Using the air temperature, which is part of meteorological data files, the calculated ground emission can be compared with data from a meteorological station when available. For the location Germany-middle the calculated data fit very well indicating than the meteorological data were calculated in the same way but using an emission coefficient of 0.9. This, on the other hand, is important to know when using meteorological data for simulation.
Sky emission modeling
At first sight the sky emission modeling seems to be very similar to the ground emission modeling. The sky can be regarded as an plane opposite to the ground and just the sinus of the wall inclination angle changes to cosinus. Problems are caused by the unknown parameters emission coefficient and temperature of the sky.
For simplified approaches, the sky can be seen as an half-infinite spheric room with an area very much larger than the building's surface area. An emission coefficient of εsky = 1 would follow for this model. The problem of the unknown sky temperature remains unsolved.
A new model for the long wave sky emission
Modeling of the long wave sky emission or atmospheric counter-radiation is part of meteorological research. Different models have been developed and applied to account for the influences of humidity and cloudiness. Since both components, humidity and cloudiness together with temperature are usually given in climatic data files we strive for a model that can reproduce atmospheric counter-radiation data from that input.
For the clear sky (no clouds) the model of Angstroem is applied. It describes the humidity-dependence of the sky emission coefficient by a power function as written below with water vapor pressure as exponent. This formula doesn't account for cloudiness.
For overcast sky, the formula is expanded by two further terms. The first term describes the increase of the emission coefficient with cloudiness by a linear relationship. The maximum alternates with the season, two maximum values for summer and winter have been determined. The transition from summer to winter is described by a sinus function of the year time. The second term is similar to the model of Feussner who considered a quadratic relationship that is here reduced to a linear one since the first term was introduced.
The arguments of the model are the cloudiness, the vapor pressure in the air and the year time. The parameters given below have been determined by comparison of the calculated sky radiation data with data available from meteorological stations, here Germany-North, Germany-Middle and Germany-South. Despite an unique set of parameters has been used for different locations the agreement between meteorological data and the fitted curves is satisfying. This is an indication that the model could be generally applicable. For many meteorological stations no sky radiation is available. By applying the model below it should be possible to generate more accurate data.
Fitted atmospheric counter-radiation data for Germany-North
Fitted atmospheric counter-radiation data for Germany-Middle
Fitted atmospheric counter-radiation data for Germany-South
Literature
Angstroem, A. K. (1924). Solar and terrestrial radiation. Quarterly Journal of the Royal Meteorological
Society 50, 121–125.
Feussner, K. and P. Dubois (1930). Truebungsfaktor, precipitable water, Staub. Gerlands Beitraege zur
Geophysik 27, 132–175.