The Heat Index

The heat index is used frequently to represent how hot it feels. It is based on a study by Steadman (1979). The index is a complicated function dependent on many variables including effective wind speed, human activity, clothing cover – a total of 20 factors. In practice, an approximate equation has been developed that makes assumptions about most of these factors and represents the heat index as a function only of the temperature and relative humidity. See this description. The equation is: \[HI=\Sigma_{i=1}^9 c_iF_i\] where \(c_i=\{-42.379, 2.04901523, 10.14333127, -0.22475541, -6.83783\times 10^{-3}, -5.481717\times 10^{-2}, 1.22874\times 10^{-3},\)
\(\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 8.5282\times 10^{-4}, -1.99\times 10^{-6}\}\)
and \(F_i=\{1, T_f, R, T_F*R, T_F^2, R^2, T_F^2R, T_FR^2, T_F^2R^2\}\) with \(T_F\) the temperature in deg.F and \(R\) the relative humidity in percent. Steadman discussed the resulting temperature as an “apparent” temperature or “sultriness”, but it has become known as the heat index.1 The intended range for Steadman’s index was temperature in the range 20–50\(^{\circ}\)C and dew point temperature \(<31^{\circ}\)C, and his tables are limited to values of the apparent temperature less than about \(50^{\circ}\)C or about \(122^{\circ}\)F.

Code is included here that duplicates the algorithm used by the NWS to calculate the heat index. It is also the basis for the function Ranadu::heatIndex() that is a new addition to the Ranadu package. The actual code has some additional modifications for special regions (like low temperature or high humidity for specific temperatures), to match the NWS algorithm.

The Wet-Bulb Temperature

The wet-bulb temperature is the temperature that would apply to a wet thermometer if fully ventilated by sufficient airflow. Evaporation cools the thermometer below the air temperature if the relative humidity is less than 100%. It provides an indicator of the ability of the human body to remove heat and maintain body temperature. If the wet-bulb temperature exceeds the body temperature, conditions will be extremely hazardous; it has been estimated that wet-bulb temperature exceeding \(95^{\circ}\)F is a survival threshold.

Correspondence to the Heat Index

Because there is no similar threshold for the heat index, TWB may be worth further consideration to represent when conditions become intolerably dangerous.

Figure 1 shows the relationship between the heat index (HI) and the wet-bulb temperature (TWB) for air temperature \(T\) from 25 to 35\(^{\circ}\)C and dewpoint temperatures TDP from T-15 to T. There is a reasonable correlation, starting with TWB\(\approx 70^{\circ}\)F for HI\(\approx 80^{\circ}\)F and extending to TWB\(\approx 100^{\circ}\)F for HI\(\approx 200^{\circ}\)F. The red points in this plot correspond to AT=\(30^{\circ}\)C; this shows the important dependence of both HI and TWB on humidity, as expected. A wet-bulb temperature equal to the body temperature corresponds to a value of the heat index of about \(185^{\circ}\)F. A possible interpretation of HI would be the equivalent dry temperature producing the same sensation as the actual conditions if the humidity were 16 hPa (as assumed for HI). This is represented in this plot by adding green dots for TWB obtained for T=HI and TDP=\(14^{\circ}\)C (corresponding to 16 hPa vapor pressure as cited by Steadman for a reference vapor pressure). Another set of points, shown as dark-green, is calculated similarly for TDP=25\(^{\circ}\)C, selected because it gives a good match to the upper limits of TWB. These temperatures will be called TBT for perceived body temperature, and are shown in units of \(^{\circ}\)F. The two results for TBT span the values of TWB, so they provide some support for further exploration of TWB as another measure that might be substituted for the heat index.

Figure 1: Correspondence between wet-bulb temperature (T~WB~, deg F) and heat index(HI, deg F) for a set of temperatures from 25C to 35C in steps of 1C and for dewpoint temperatures from AT-15 to AT in steps of 1C. Values of T~BT~ (deg F) corresponding to HI are also shown, as green points. The black dashed line denotes the maximum value of apparent temperature in the Steadman paper.

Figure 1: Correspondence between wet-bulb temperature (TWB, deg F) and heat index(HI, deg F) for a set of temperatures from 25C to 35C in steps of 1C and for dewpoint temperatures from AT-15 to AT in steps of 1C. Values of TBT (deg F) corresponding to HI are also shown, as green points. The black dashed line denotes the maximum value of apparent temperature in the Steadman paper.

An Apparent Temperature Based on Wet-Bulb Temperature

A disadvantage of TWB for this purpose is that it does not have a good association with the perceived apparent temperature. An apparent temperature TA related to TWB can be defined as the temperature that, with a fixed low humidity, gives the same value of TWB; that temperature is specified implicitly by the equation TWB=wetbulbT(1000, TA, TDPR) where TDPR might be a reference dewpoint corresponding to, e.g., 20% relative humidity at the air temperature TA. Values of TA were determined iteratively from this equation and are shown in Fig. 2.

Figure 2: Correspondence between the proposed apparent temperature and the heat index for a set of temperatures from 25C to 35C in steps of 1C and for dewpoint temperatures from T-15 to T in steps of 1C. Points with $T_{WB}>85^{\circ}$F, which would be particularly hazardous, are plotted in orange or, for $>88^{\circ}$F, in red.

Figure 2: Correspondence between the proposed apparent temperature and the heat index for a set of temperatures from 25C to 35C in steps of 1C and for dewpoint temperatures from T-15 to T in steps of 1C. Points with \(T_{WB}>85^{\circ}\)F, which would be particularly hazardous, are plotted in orange or, for \(>88^{\circ}\)F, in red.

This figure suggests that TA, so defined, might be a useful alternative to the conventional heat index. It represents the air temperature that, if the relative humidity were 20%, would make the shaded skin feel the same in as the ambient conditions. It is dependent only on the temperature and humidity (and to only a minor extent on atmospheric pressure) and does not incorporate any of the physiological aspects of HI. It assumes only sufficient airflow to produce full ventilation of the perspiring skin and appropriate clothing to permit the required heat tranfer from a human body.

An Alternative Discomfort Index Based on Rate of Heat Loss

Assume that a given TWB represents the equilibrium temperature of a wet human body, fully ventilated. The heat transferred away from that body will then be proportional to the difference between THB=\(98.6^{\circ}\)F and TWB. A measure of comfort, determined by the cooling rate available to the human body to maintain a steady temperature, might be proportional to THB\(-\)TWB. A discomfort index might be defined as inversely proportional to that cooling rate, and it would have the useful property that it becomes infinite where the cooling rate is zero, at which point the human body cannot release any heat by evaporative cooling. For example, DI might be defines as proportional to 1/(THB\(-\)TWB) and then scaled to match HI through some assumed range of validity like 110–122\(^{\circ}\)F, to obtain the best match for high temperature where HI is most useful. This index would then have the desirable characteristic of approximating a perceived or apparent temperature. The result is shown in Fig. 3.

Figure 3: A possible discomfort index (DI) vs. the conventional heat index.

Figure 3: A possible discomfort index (DI) vs. the conventional heat index.

Conclusions

The apparent temperature TA developed here can be a useful complement to the heat index. It is based only to temperature and humidity and, to a minor extent, on atmospheric pressure. Values tend slightly higher than those from the heat index (Fig. 2) but are mostly well correlated to HI, although the sensitivity of TA to humidity is higher than for HI. The heat index and the proposed apparent temperature are both provided by a new Ranadu function called heatIndex(). TWB can also be useful if used in combination with these apparent temperatures because as it approaches the normal body temperature it will indicate an extreme hazard. As global temperatures increase those conditions may become more common.

  1. “Heat” in thermodynamics is thermal energy transferred between two material objects and is not an appropriate characterization of apparent temperature, but in common usage heat often represents a sensation of temperature.↩︎