Christian Kongsgaard

2021-08-13

Mean radiant temperature (MRT) is a measure of the average temperature of the surfaces that surround a particular point, with which it will exchange thermal radiation.^{1}

The amount of radiant heat lost or received by the human body is the sum of all radiant fluxes exchanged with the surrounding sources, MRT can therefore be calculated from the temperature of the surrounding surfaces and their positions with respect to the person.

Therefore, it is necessary to simulate those temperatures and calculate the angle factors between the person and the
surrounding surfaces. Most building materials have a high emittance ε, so all surfaces in the room or outdoors 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 factor.^{2}

Where:

- MRT is the mean radiant temperature in °C
- T
_{n}is the temperature of surface*n*in °C - F
_{p-1}is the angle factor between a person and a surface*n*

If relatively small temperature differences exist between the surfaces of the enclosure, the equation can be simplified to the following linear form

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

The MRT formula is rather straight forward to calculate indoors, but outdoors it becomes a little more complicated
because we have to take the sky and sun into consideration. The method we use to calculate the MRT on Compute is coming
from __this paper__

Where:

- MRT
_{i}is MRT for the hour*i*in °C - T
_{sky, i}is the sky temperature for the hour*i*in °C - T
_{ground, i}is the ambient air temperature for the hour*i*in °C - F
_{sky}is the view factor to the sky - F
_{ground}is the view factor to everything that is not the sky - ΔMRT
_{sun, i}is the MRT contribution from the sun for the hour*i*in °C

By applying this formula to every face center of mesh we are able to generate a spatial map of the MRT for every hour of the year.

Calculating the sky view factors can be done relatively easy with the help of Radiance. Compute already have
capabilities for computing sky view factor as a __standalone study__, but running the MRT simulation we pack that
simulation nicely with the other tasks.

The view factors are a value between 0 and 1, where 0 means that the analysis point cannot see the sky dome and 1 means that the point can see the complete sky dome. Since all view factors must add up to 1 for every point, we can simply subtract our sky view factor from 1 to get the ground view factor.

Using the same formula for sky temperature
as __EnergyPlus__
it is fairly easy to calculate the sky temperature for the whole year from an EPW file.

The ΔMRT_{sun, i} component from the outdoor MRT formula above comes
from __this paper__
.

Where:

- ERF is the effective radiant field in W/m2
- f
_{eff}is the fraction of body exposed to sun - h
_{r}is the heat transfer coefficient in W/m2K

The ERF is dependent on the solar radiation as well as the view factors. We therefore, run a solar radiation simulation to obtain solar radiation values for each point in our mesh.

Finally, we can add the three contributions to the MRT up and generate the MRT for our project: