Ab initio computations of the mean radiant temperature of indoor spaces

Abstract

Convenient thermal comfort in indoor spaces is a standard requirement that is commonly asked and expected by occupants. Although the notion of thermal comfort seems to be understandable, its monitoring and maintenance are not easy. In 1970, Fanger defined six parameters to quantify thermal comfort. Among these parameters, the so-called mean radiant temperature characterizes the temperature state of the room envelope and strongly influences the apparent temperature perceived by the occupants. The mean radiant temperature can be measured or computed. For its computation, Fanger's classical equation is frequently used. Unfortunately, this equation holds only for absolutely black surfaces that are free of reflections, but such surfaces do not exist in practice. Real non-black surfaces are accompanied by varying degrees of heat reflections; consequently, with such surfaces, Fanger's equation can provide only compromised values. So far, nobody has improved Fanger's equation to include reflections of low-emissive room envelopes. In this paper, the generalized equation is derived to compute the mean radiant temperature of room envelopes with arbitrary emissivities. The equation is derived based on the so-called algebraic radiosity method and uses the entire matrix of view factors, while Fanger's equation uses only one row of that matrix. The classical Fanger equation and the new generalized equation have been applied to a common living room with variable surface emissivities, and the results have been compared. Such a comparison enables quantification of the influence of heat reflections on mean radiant temperatures. Both equations show similar temperatures for emissivities in the range between 1 and 0.9, but with surfaces of lower emissivity, they yield different results due to non-negligible heat reflections. When the emissivities of room surfaces approach 0.8, the temperature differences reach 0.6 degrees C. When the emissivities are close to 0.6, the temperature difference is 1.6 degrees C, and at emissivities 0.1, a large temperature difference appears, reaching 8.3 degrees C. This fact has direct consequences for measuring temperatures with radiometers and thermocouples. Measurements with thermocouples that are attached to surfaces are almost insensitive to heat reflections, whereas measurements with radiometers placed apart from the surfaces suffer from heat reflections.