*4.3. Mean Radiant and Operative Temperatures*

Mean radiant temperature (MRT) expresses the influence of surface temperatures in the room on occupant comfort. The area-weighted method shown in Equation (2) is a simple way to calculate MRT, but it does not reflect the geometric position, posture, and orientation of the occupant, ceiling height, or radiant asymmetry [29]. In Equation (3), the calculation of mean radiant temperature from surrounding surfaces considers the surface temperatures of the surrounding elements and the angle factor. The angle factor is a function of shape, size, and the position concerning the occupant standing or being seated. The surfaces of the room are assumed as black, with high emissivity and no reflection. In this case, the angle factors weight the enclosing surface temperatures to the fourth power [28].

$$T\_{mr} = \frac{T\_1 A\_1 + T\_2 A\_2 + \dots + T\_N A\_N}{A\_1 + A\_2 + \dots + A\_N} \, \tag{2}$$

$$\overline{T}\_{mr}^4 = T\_1^4 F\_{p-1} + T\_2^4 F\_{p-2} + \dots + T\_N^4 F\_{p-N} \, \, \, \, \, \tag{3}$$

where

*Tmr* = mean radiant temperature, ◦C,

*TN* = surface temperature of surface *N*, ◦C (calculated or measured),

*AN* = area of surface,

*Fp-N* = is the angle factor between the person and surface N.

The angle factors quantify the amount of radiation energy that leaves the human body and reaches each surface. They were calculated according to Figures B.2 to B.5 in [28]. If the difference between the indoor surface temperatures is relatively small (<10 ◦C), Equation (4) can be used.

$$\overline{T}\_{mr} = T\_1 F\_{p-1} + T\_2 F\_{p-2} + \dots + T\_N F\_{p-N} \tag{4}$$

The MRT is calculated as the average value of the surrounding temperatures weighted according to the angle factors. If the temperature difference between indoor surfaces is below 10 ◦C, then the MRT error calculated with Equation (4) will be less than 0.2 ◦C [28]. Equation (5) shows the formula to calculate the angle factor [43].

$$F\_{p-N} = F\_{\max} \left( 1 - e^{-(a/c)\tau} \right) \left( 1 - e^{-(b/c)\gamma} \right),\tag{5}$$

where

γ = *A*+*B(a*/*c)*, τ = *C*+*D(b*/*c)* + *E(a*/*c)*.

Parameters *a*, *b*, and *c*, defined in Figure 14, are related to dimensions and distances between the occupant and the envelope. Table 13 shows the parameters *A*, *B*, *C*, and *D* to calculate angle factors for seated persons and walls, floors, and ceilings.

**Figure 14.** Geometry of the office space and occupant's position and dimensions to calculate the angle factor.

**Table 13.** Parameters for calculating angle factors 1.


Figure 14 illustrates the dimensions and geometry of the office space and the different surfaces considered to calculate the MRT for a seated person. The facing direction was ignored for simplification. The temperature of each rectangle (1 to 21) was measured to calculate the MRT. Due to small differences, only five temperatures have been taken into account. T1 = T6 = T1 = T14 = T15 = T16 = T17 = Tceiling; T3 = T3 = T4 = T4 = T9 = T11 = T12 = T18 = T19 = T20 = T21 = Tfloor; T7 = T5 = T5 = Twall; T8 = TWFG; T2 = T2 = TWFG\_TP.

A rough approximation to obtain the operative temperature may be to use the arithmetic average of the mean radiant temperature (MRT) of the heated space and dry-bulb air temperature if air velocity is less than 0.2 m/s and MRT is less than 50 ◦C. In cases where the air velocity is between 0.2 and 1 m/s, or where the difference between mean radiant and air temperature is above 4 ◦C, the ASHRAE 55 provides a formula, shown in Equation (6), to calculate operative temperature [27].

$$T\_{op} = A \, T\_a \, + \, (1 - A) \, T\_{\text{urr } \prime} \, \tag{6}$$

where

*Top* = operative temperature (◦C),
