Original model

Equivalent temperature was not universally used in North America until the 21st century. Until the 1970s, the coldest parts of Canada reported the original Wind Chill Index, a three- or four-digit number with units of kilocalories/hour per square metre. Each individual calibrated the scale of numbers personally, through experience. The chart also provided general guidance to comfort and hazard through threshold values of the index, such as 1400, which was the threshold for frostbite.

The original formula for the index was:^[5][6]

$${\displaystyle WCI=\left(10{\sqrt {v}}-v+10.5\right)\cdot \left(33-T_{\mathrm {a} }\right),}$$ where:

• WCI = wind chill index, kg⋅cal/m²/h
• v = wind velocity, m/s
• T_a = air temperature, °C

North American and United Kingdom wind chill index

In November 2001, Canada, the United States, and the United Kingdom implemented a new wind chill index developed by scientists and medical experts on the Joint Action Group for Temperature Indices (JAG/TI).^[7][8][9] It is determined by iterating a model of skin temperature under various wind speeds and temperatures using standard engineering correlations of wind speed and heat transfer rate. Heat transfer was calculated for a bare face in wind, facing the wind, while walking into it at 1.4 m/s (5.0 km/h; 3.1 mph). The model corrects the officially measured wind speed to the wind speed at face height, assuming the person is in an open field.^[10] The results of this model may be approximated, to within one degree, from the following formulas.

The standard wind chill formula for Environment Canada is:^[3]

$${\displaystyle T_{\mathrm {wc} }=13.12+0.6215T_{\mathrm {a} }-11.37v^{+0.16}+0.3965T_{\mathrm {a} }v^{+0.16},}$$

where T_wc is the wind chill index, based on the Celsius temperature scale; T_a is the air temperature in degrees Celsius; and v is the wind speed at 10 m (33 ft) standard anemometer height, in kilometres per hour.^[11]

When the temperature is −20 °C (−4 °F) and the wind speed is 5 km/h (3 mph), the wind chill index is −24. If the temperature remains at −20 °C and the wind speed increases to 30 km/h (19 mph), the wind chill index falls to −33.

The equivalent formula in US customary units is:^[12][3]

$${\displaystyle T_{\mathrm {wc} }=35.74+0.6215T_{\mathrm {a} }-35.75v^{+0.16}+0.4275T_{\mathrm {a} }v^{+0.16},}$$

where T_wc is the wind chill index, based on the Fahrenheit scale; T_a is the air temperature in degrees Fahrenheit; and v is the wind speed in miles per hour.^[13]

Windchill temperature is defined only for temperatures at or below 10 °C (50 °F) and wind speeds above 4.8 km/h (3.0 mph).^[12]

As the air temperature falls, the chilling effect of any wind that is present increases. For example, a 16 km/h (10 mph) wind will lower the apparent temperature by a wider margin at an air temperature of −20 °C (−4 °F) than a wind of the same speed would if the air temperature were −10 °C (14 °F).

• 
  Celsius wind chill index
• 
  Comparison of old and new wind chill values at −15 °C (5 °F)
• 
  Wind chill calculator

The 2001 WCET is a steady-state calculation (except for the time-to-frostbite estimates).^[14] There are significant time-dependent aspects to wind chill because cooling is most rapid at the start of any exposure, when the skin is still warm.

Australian apparent temperature

The apparent temperature (AT), invented in the late 1970s, was designed to measure thermal sensation in indoor conditions. It was extended in the early 1980s to include the effect of sun and wind. The AT index used here is based on a mathematical model of an adult, walking outdoors, in the shade (Steadman 1994). The AT is defined as the temperature, at the reference humidity level, producing the same amount of discomfort as that experienced under the current ambient temperature and humidity.^[15]

The formula^[16] is:

$${\displaystyle \mathrm {AT} =T_{\mathrm {a} }+0.33e-0.7v-4.00,}$$

where:

• T_a is dry-bulb temperature (°C)
• e is water vapour pressure (hPa)
• v is wind speed (m/s) at an elevation of 10 m (33 ft)

The vapour pressure can be calculated from the temperature and relative humidity using the equation: $${\displaystyle e={\frac {\mathrm {RH} }{100}}\cdot 6.105\cdot \exp {\left({\frac {17.27\cdot T_{\mathrm {a} }}{237.7+T_{\mathrm {a} }}}\right)},}$$

where:

• T_a is dry-bulb temperature (°C)
• RH is relative humidity (%)
• exp represents the exponential function

The Australian formula includes the important factor of humidity and is somewhat more involved than the simpler North American model. The North American formula was designed to be applied at low temperatures (as low as −46 °C or −50 °F) when humidity levels are also low. The hot-weather version of the AT (1984) is used by the National Weather Service in the United States. In the United States, this simple version of the AT is known as the heat index.