3.7: Apparent Temperature Indices

Warm-blooded (homeothermic) animals including humans generate heat internally via metabolism of the food we eat with the oxygen we breathe. But we also rely on heat transfer with the environment to help maintain an internal core temperature of about 37°C (= 98.6°F). (Our skin is normally cooler — about 33.9°C = 93°F). Heat transfer occurs both via sensible heat fluxes (temperature difference between air and our skin or lungs) and latent heat fluxes (evaporation of moisture from our lungs and of perspiration from our skin).

The temperature we “feel” on our skin depends on the air temperature and wind speed (as they both control the bulk heat transfer between our skin and the environment) and on humidity (is it affects how rapidly perspiration evaporates to cool us).

Define a reference state as being a person walking at speed M_o = 4.8 km h^–1 through calm, moderately dry air. The actual air temperature is defined to be the temperature we “feel” for this reference state.

The apparent temperature is the temperature of a reference state that feels the same as it does for non-reference conditions. For example, faster winds in winter make the temperature feel colder (wind chill) than the actual air temperature, while higher humidities in summer make the air feel warmer (humidex or heat index).

3.7.1. Wind-Chill Temperature

The wind-chill temperature index is a measure of how cold the air feels to your exposed face. The official formula, as revised in 2001 by the USA and Canada, for wind chill in °C is:

\(\ \begin{align}T_{\text {wind chill }}=\left(a \cdot T_{\text {air}}+T_{1}\right)+\left(b \cdot T_{\text {air}}-T_{2}\right) \cdot\left(\dfrac{M}{M_{o}}\right)^{0.16} for\ M>M_{o}\tag{3.64a}\end{align}\)

or

\(\ \begin{align}T_{\text {wind chill}}=T_{\text {air}}
\ for M \leq M_{o}\tag{3.64b}\end{align}\)

where a = 0.62, b = 0.51, T₁ = 13.1°C, and T₂ = 14.6°C. M is the wind speed measured at the official anemometer height of 10 m. For M < M_o, the wind chill equals the actual air temperature. This index applies to non-rainy air.

Figure 3.12 and Table 3-3 show that faster winds and colder temperatures make us “feel” colder. The data used to create eq. (3.64) was from volunteers in Canada who sat in refrigerated wind tunnels, wearing warm coats with only their face exposed.

Table 3-3. The wind-chill-index temperature (°C).
Wind Speed		Actual Air Temperature (°C)
km· h^-1	m·s^-1	–40	–30	–20	–10	0	10
60	16.7	–64	–50	–36	–23	–9	5
50	13.9	–63	–49	–35	–22	–8	6
40	11.0	–61	–48	–34	–21	–7	6
30	8.3	–58	–46	–33	–20	–6	7
20	5.6	–56	–43	–31	–18	–5	8
10	2.8	–51	–39	–27	–15	–3	9
0	0	–40	–30	–20	–10	0	10

Figure 3.12 For any wind speed M and actual air temperature T read the wind-chill temperature index (°C) from the curves in this graph.

At wind chills colder than –27°C, exposed skin can freeze in 10 to 30 minutes. At wind chills colder than –48°C: WARNING, exposed skin freezes in 2 to 5 min. At wind chills colder than –55°C: DANGER, exposed skin freezes in less than 2 minutes. In this danger zone is an increased risk of frostbite (fingers, toes, ears and nose numb or white), and hypothermia (drop in core body temperature).

3.7.2. Humidex and Heat Index

On hot days you feel warmer than the actual air temperature when the air is more humid, but you feel cooler when the air is drier due to evaporation of your perspiration. In extremely humid cases the air is so uncomfortable that there is the danger of heat stress. Two apparent temperatures that indicate this are humidex and heat index.

The set of equations below approximates Steadman’s temperature-humidity index of sultriness (i.e., a heat index):

\(\ \begin{align}T_{\text {heat index}}\left(^{\circ} \mathrm{C}\right)=T_{R}+\left[T-T_{R}\right] \cdot\left(\dfrac{R H \cdot e_{s}}{100 \cdot e_{R}}\right)^{p}\tag{3.65a}\end{align}\)

where e_R = 1.6 kPa is reference vapor pressure, and

\(\ \begin{align} T_{R}\left(^{\circ} \mathrm{C}\right)=0.8841 \cdot T+\left(0.19^{\circ} \mathrm{C}\right)\tag{3.65b}\end{align}\)

\(\ \begin{align}p=\left(0.0196^{\circ} \mathrm{C}^{-1}\right) \cdot T+0.9031\tag{3.65c}\end{align}\)

\(\ \begin{align}e_{s}(\mathrm{kPa})=0.611 \cdot \exp \left[5423\left(\dfrac{1}{273.15}-\dfrac{1}{(T+273.15)}\right)\right]\tag{3.65d}\end{align}\)

The two input variables are T (dry bulb temperature in °C), and RH (the relative humidity, ranging from 0 for dry air to 100 for saturated air). Also, T_R (°C), and p are parameters, and e_s is the saturation vapor pressure, discussed in the Water Vapor chapter. Eqs. (3.65) assume that you are wearing a normal amount of clothing for warm weather, are in the shade or indoors, and a gentle breeze is blowing.

Table 3-4. Heat-index apparent temperature (°C).
Rel. Hum. (%)	Actual Air Temperature (°C)
	20	25	30	35	40	45	50
100	21	29	41	61
90	21	29	39	57
80	21	28	37	52
70	20	27	35	48
60	20	26	34	45	62
50	19	25	32	41	55
40	19	24	30	38	49	66
30	19	24	29	36	44	56
20	18	23	28	33	40	48	59
10	18	23	27	32	37	42	48
0	18	22	27	31	36	40	44

Table 3-5. Humidex apparent temperature (°C)
Td (°C)	Actual Air Temperature T (°C)
	20	25	30	35	40	45	50
50							118
45						96	101
40					77	82	87
35				62	67	72	77
30			49	54	59	64	69
25		37	42	47	52	57	62
20	28	33	38	43	48	53	58
15	24	29	34	39	44	49	54
10	21	26	31	36	41	46	51
5	19	24	29	34	39	44	49
0	18	23	28	33	38	43	48
–5	17	22	27	32	37	42	47
–10	16	21	26	31	36	41	46

The dividing line between feeling warmer vs. feeling cooler is highlighted with the bold, underlined heat-index temperatures in Table 3-4.

In Canada, a humidex is defined as

\(\ \begin{align}T_{\text {humidex}}\left(^{\circ} C\right)=T\left(^{\circ} C\right)+a \cdot(e-b)\tag{3.66a}\end{align}\)

where T is air temperature, a = 5.555 (°C kPa^–1), b = 1 kPa, and

\(\ \begin{align}e(\mathrm{kPa})=0.611 \cdot \exp \left[5418\left(\dfrac{1}{273.16}-\dfrac{1}{\left[T_{d}\left(^{\circ} \mathrm{C}\right)+273.16\right]}\right)\right]\tag{3.66b}\end{align}\)

T_d is dew-point temperature, a humidity variable discussed in the Water Vapor chapter.

Humidex is also an indicator of summer discomfort due to heat and humidity (Table 3-3). Values above 40°C are uncomfortable, and values above 45°C are dangerous. Heat stroke is likely for humidex ≥ 54°C. This table also shows that for dry air (T_d ≤ 5°C) the air feels cooler than the actual air temperature.

Sample Application

Use the equations to find the heat index and humidex for an air temperature of 38°C and a relative humidity of 75% (which corresponds to a dew-point temperature of about 33°C).

Find the Answer

Given: T = 38°C , RH = 75% , T_d = 33°C

Find: T_{heat index} = ? °C , T_humidex = ? °C

For heat index, use eqs. (3.65):

T_R = 0.8841 · (38) + 0.19 = 33.8°C (3.65b)

p = 0.0196 · (38) + 0.9031 = 1.65 (3.65c)

e_s = 0.611·exp[5423·( {1/273.15} – {1/(38+273.15)})] = 6.9 kPa (3.65d)

T_{heat index} = 33.8 + [38–33.8]·(0.75·6.9/1.6)^1.65 = 62.9°C (3.65a)

For humidex, use eqs. (3.66):

e = 0.611·exp[5418·( {1/273.16} – {1/(33+273.16)})] = 5.18 kPa (3.66b)

T_humidex = 38 + 5.555·(5.18–1) = 61.2°C (3.66a)

Check: Units are reasonable. Values agree with extrapolation of Tables 3-4 and 3-5.

Exposition: These values are in the danger zone, meaning that people are likely to suffer heat stroke. The humidex and heat index values are nearly equal for this case.