# Fanger’s comfort equation

The comfort equation developed by Fanger (1972) was established from experimental work with North American subjects and allows the calculation (by computer or, with difficulty, manually using diagrams) of the combination of activity, clothing and environmental factors that will produce thermal comfort (see section 4.1). It was subsequently checked by applying it to Danish subjects and excellent correlation found. Fanger concludes his equation is appropriate within the temperate zone for adults, regardless of sex. The thermal balance of the body for comfort is expressed in simple terms by

H — Ed- Esw — Eie — L = R + C (4.9)

Where H is the internal rate of bodily heat production (related to the activity), Јd is the heat loss by vapour diffusion through the skin (insensible perspiration and not subject to thermo­regulatory control), Esv/ is the heat lost by the evaporation of sweat, Eie is the latent heat loss by respiration, L is the sensible heat loss by respiration, R is the radiation loss from a clothed person and C is the similar convective loss. Fanger’s comfort equation then defines comfort for the case when a function of the relevant variables equals zero, namely:

(4.10)

Where /c] is the insulating value of the clothing, ra is the ambient dry-bulb temperature and <|)a the relative humidity. Fanger finds the effect of relative humidity is not great for persons in comfort balance and that for such people a change in humidity from 0 per cent to 100 per cent can be compensated by a temperature decrease of about 1.5°C to 3.0°C. Clothing and activity are very significant: an increase in clothing from 0 clo to 1.5 clo corresponds to a decrease of 8°C in the dry-bulb for sedentary work (115 W) but to 19°C for more strenuous activity (345 W).

Using his comfort equation Fanger (1972) has derived an index of thermal sensation, making it possible to predict a mean comfort response (predicted mean vote, abbreviated as PMV), on a standard scale from -3 to +3, in a large group of persons for any combination of the four environmental variables, activity and clothing. Tables and diagrams are given to simplify the intrinsically complicated mathematical approach to this. The predicted percentage dissatisfied (abbreviated PPD) for an indoor climate is perhaps more meaningful. The PPD is determined from the PMV for several positions in a room. After measurement of the environmental variables the lowest possible percentage of dissatisfied persons (abbreviated LPPD) attainable by altering the temperature can be established. To quote Fanger: ‘The magnitude of the LPPD is an expression for the non-uniformity of the thermal environment and is therefore suitable for characterising the heating or air conditioning system…’. Table 4.4 is based on Fanger’s work and quotes some dry-bulb temperatures and associated air velocities in an environment of 50 per cent relative humidity, with the

Table 4.4 Temperature and air velocity for zero PMV at various clo-values for sedentary workers

 Clo-value 0 0.5 1.0 1.25 1.5 Temp (°C) 29 28 27 26 25 24 24 22 22 Velocity 0.5 to 0.5 to 0.2 to 0 to 0.5 to 0.2 to 1.0 to 0.1 to 0.5 to (m s-1) 0.15 1.0 0.3 0.1 1.0 0.3 1.5 0.15 1.0

Mean radiant temperature equal to the dry-bulb, when the PMV passes through zero for sedentary workers and therefore indicates when they are most comfortable.

In experiments with sedentary subjects clothed at 0.6 clo those expressing ‘warm dissatisfaction’ and those expressing ‘cold dissatisfaction’ gave equal votes at 25.6°C, equal to the optimum temperature predicted by the comfort equation, implying that, for a given activity and clothing there is one comfortable temperature for all people, regardless of age, sex, etc., as outlined earlier. A further important conclusion of Fanger’s work is that a value of less than 5 per cent for the PPD value is not achievable for similarly clothed people engaged in the same activity, no matter how perfect the environmental system: a few complaints must not necessarily be interpreted as an indication that the system is defective or the controls wrongly set. The system should be set up to suit the comfort of the majority and not subsequently tampered with to cater for the few who will invariably complain, see also BS EN ISO 7730 (1995).

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