Mean radiant temperature

The mean radiant temperature (MRT) is defined as the uniform temperature of an imaginary enclosure in which the radiant heat transfer from the human body is equal to the radiant heat transfer in the actual non-uniform enclosure.^[1]

MRT is a concept arising from the fact that the net exchange of radiant energy between two objects is approximately proportional to their temperature difference multiplied by their ability to emit and absorb heat (emissivity). It is simply the area weighted mean temperature of all the objects surrounding the body. This is valid as long as the absolute temperatures of objects in question are large compared to the temperature differences, allowing linearization of the Stefan-Boltzmann Law in the relevant temperature range.

MRT also has a strong influence on thermophysiological comfort indexes such as physiological equivalent temperature (PET) or predicted mean vote (PMV).^[2]

What we experience and feel relating to thermal comfort in a building is related to the influence of both the air temperature and the temperature of surfaces in that space. The mean radiant temperature is expressed as this surface temperature and is controlled by enclosure performances. Maintaining a balance between the operative temperature and the mean radiant temperature can create a more comfortable space.^[3] This is done with effective design of the building, interior and with the use of high temperature radiant cooling and low temperature radiant heating.^[4]

Calculation

There are different ways to estimate the mean radiant temperature, either applying its definition and using equations to calculate it, or measuring it with particular thermometers or sensors.

Since the amount of radiant heat lost or received by human body is the algebraic sum of all radiant fluxes exchanged by its exposed parts with the surrounding sources, MRT can be calculated from the measured temperature of surrounding walls and surfaces and their positions with respect to the person. Therefore, it is necessary to measure those temperatures and the angle factors between the person and the surrounding surfaces.^[1] Most building materials have a high emittance ε, so all surfaces in the room can be assumed to be black. Because the sum of the angle factors is unity, the fourth power of MRT equals the mean value of the surrounding surface temperatures to the fourth power, weighted by the respective angle factors.

The following equation is used:^[1][5]

$MRT^4 = T_1^4 F_{p-1} + T_2^4 F_{p-2} + ... + T_n^4 F_{p-n}$

where:

$MRT$   is Mean Radiant Temperature;

$T_n$         is the temperature of surface "n", in Kelvins;

$F_{p-n}$    is the angle factor between a person and surface "n".

If relatively small temperature differences exist between the surfaces of the enclosure, the equation can be simplified to the following linear form:^[1][5]

$MRT = T_1 F_{p-1} + T_2 F_{p-2} + ... + T_n F_{p-n}$

This linear formula tends to give a lower value of MRT, but in many cases the difference is small.^[1]

In general, angle factors are difficult to determine, and they normally depend on the position and orientation of the person.Furthermore, this method becomes complex and time consuming as the number of surfaces increases and they have elaborate shapes. For this reason, an easier way to determine the MRT is by measuring it with a particular thermometer.

Measurement

The MRT can be measured using a black-globe thermometer. The black-globe thermometer consists of a black globe in the center of which is placed a temperature sensor such as the bulb of a mercury thermometer, a thermocouple or a resistance probe. The globe can in theory have any diameter but as the formulae used in the calculation of the mean radiant temperature depend on the diameter of the globe, a diameter of 0.15 metres (5.9 in), specified for use with these formulae, is generally recommended. The smaller the diameter of the globe, the greater the effect of the air temperature and air velocity, thus causing a reduction in the accuracy of the measurement of the mean radiant temperature. So that the external surface of the globe absorbs the radiation from the walls of the enclosure, the surface of the globe shall be darkened, either by the means of an electro-chemical coating or, more generally, by means of a layer of matte black paint. ^[1] This thermometer actually measures the globe temperature (GT), tending towards thermal balance under the effect of convection and radiation coming from the different heat sources in the enclosure. Thanks to this principle, knowing GT allows the mean radiant temperature MRT to be determined.^[1] According to ISO 7726 Standard, the equation that is used most frequently (forced convection) is the following:

$MRT = \left[ \left(GT+273 \right)^4 + \frac{1,1 \cdot 10^8 \cdot v_a^{0,6}} {\varepsilon \cdot D^{0,4}}(GT - T_a) \right]^{1/4} - 273$

where:

$MRT$ is the mean radiant temperature (°C);

$GT$      is the globe temperature (°C);

$v_a$         is the air velocity at the level of the globe (m/s);

$\varepsilon$          is the emissivity of the globe (no dimension);

$D$         is the diameter of the globe (m);

$T_a$        is air temperature (°C);

And for the standard globe (D = 0.15 m, $\varepsilon$ = 0.95):

$MRT = \left[ \left(GT+273 \right)^4 + 2,5 \cdot 10^8 \cdot v_a^{0,6}(GT - T_a) \right]^{1/4} - 273$

The measurement is affected by air movement because the measured GT depends on both convection and radiation transfer. By effectively increasing the size of the thermometer bulb, the convection transfer coefficient is reduced and the effect of radiation is proportionally increased. Because of local convective air currents GT typically lies between the air temperature and MRT. The faster the air moves over the globe thermometer, the closer GT approaches the air temperature.

Moreover, since the MRT is defined with respect to the human body, the shape of the sensor is also a factor. The spherical shape of the globe thermometer gives a reasonable approximation of a seated person; for people who are standing, the globe, in a radiant nonuniform environment, overestimates the radiation from floor or ceiling, so an ellipsoid sensor gives a better approximation.^[5]

There are several other precautions to be taken when using a black-globe thermometer, depending on the conditions of the measurement. Furthermore, there are different measuring methods, such as the two-sphere radiometer and the constant-air-temperature sensor.^[1]

See also

• Thermal comfort
• Heating, Ventilation and Air-Conditioning
• ASHRAE
• Glossary of HVAC

References

1. "ISO 7726. Ergonomics of the thermal environment - Instrument for measuring physical quantities". Geneva, Switzerland: International Organization for Standardization. November 1998.
2. Fanger, P.O. (1970). Thermal Comfort: Analysis and Applications in Environmental Engineering. New York: McGraw Hill.
3. Matzarakis, Andreas. Estimation and Calculation of the Mean Radiant Temperature within Urban Structures.
4. Mclntyre and Griffiths, D.A. and I.D. (1972). Subject Response to Radiant and Convective Environments.
5. 2009 ASHRAE Handbook Fundamentals, ASHRAE, Inc., Atlanta, GA.

External links

• http://ergo.human.cornell.edu/DEA3500allnotes.html
• Free online thermal comfort and mean radiant temperature calculation tool developed by CBE at UC Berkeley