Physics

Physics of the heatindex function

The heat index is the air temperature at a reference vapor pressure of 1.6 kPa that induces the same physiological response as the actual air temperature and humidity. Therefore, the definition of the heat index HI can be expressed mathematically as

where “physiology” in this equation is a function that returns the physiological state, T_a is the air temperature, RH is the relative humidity, and p* is the saturation vapor pressure as a function of air temperature. In addition to this definition of the heat index, there is a mathematical model of thermoregulation that is used to calculation the “physiology” function above.

To calculate the physiological state, we first check if the human is in a state of hyperthermia.

For the hyperthermic and lethal regions, we use

To check that, the relevant equation is equation (18) of Lu and Romps (2022), which is

where C_c represents the heat capacity of the core, T_c = 310 K and T_a are the temperatures of the core and air; p_c = \(\phi\)_salt p*(T_c) and p_a are the vapor pressures of the core and air; \(\phi\)_salt is the effective relative humidity of sweat; p* is the saturation vapor pressure; R_a is the resistance to heat flow through the boundary layer at the surface of the skin; Z_a are the resistance to mass (i.e., sweat) flow through that boundary layer; Q is the metabolic heat flux generated in the core; and Q_v is the respiratory heat flux from the core to the air. In this equation, the skin’s temperature and vapor pressure are assumed to be the same as in the core, made possible by a very high skin blood flow. If dT_c/dt is positive in this equation, then hyperthermia is unavoidable and the rate of warming defines the physiological state. The heat index then equals the air temperature (in the presence 1.6 kPa of water vapor) that would give the same dT_c/dt.

Otherwise, if dT_c/dt is negative, then thermoregulation can be achieved with a lower skin blood flow, i.e., with skin that resists the flow of heat and water. The physiological state is then defined by the skin’s resistance to heat flow that achieves dT_c/dt = 0 at a normal core temperature of T_c = 310 K.

Here, the relevant equations are equations (14-16) of Lu and Romps (2022), which are

where variables with a subscript “s” denote properties of the shell, which represents the skin and, at colder temperatures, also clothing. Here, T_s and p_s are the temperature and vapor pressure of the shell’s outer surface, R_s is the shell’s resistance to heat flow; and Z_s is the shell’s resistance to mass (i.e., sweat) flow. Setting T_c = 310 K, these equations describe a steady-state solution to thermoregulation with a normothermic core temperature. The first equation describes the core’s heat balance, which has metabolic heat production Q balanced by ventilation Q_v and the flow of heat from core to shell. The second equation describes the heat balance of the shell, which is affected by the flow of heat from core to shell, the flow of heat from shell to air, and the cooling from evaporation of sweat. The third equation is the solution for the vapor pressure of the partially wetted shell. The minimum function was introduced by Lu and Romps (2022) and ensures that the vapor pressure does not exceed the saturation vapor pressure of sweat; in effect, this allows excess sweat to drip off the shell. Together with some auxiliary equations, these equations can be solved to find R_s, which defines the physiological state. The heat index then equals the air temperature (in the presence 1.6 kPa of water vapor) that would give the same R_s.

Physics of the wetbulb function

The wetbulb function calculate four different temperatures using the accurate Rankine-Kirchhoff framework: thermodynamic wet-bulb (the default), psychrometric (a.k.a., ventilated or aspirated) wet-bulb, thermodynamic ice-bulb, and psychrometric ice-bulb. The most commonly used of these is the thermodynamic wet-bulb temperature, which is usually denoted by T, so we retain that notation here. To denote the psychrometric wet-bulb temperature, we use T_pw. Likewise, we use T_i and T_pi to denote the thermodynamic and psychrometric ice-bulb temperatures.

The air’s thermodynamic wet-bulb temperature is the temperature of liquid water that is invariant with respect to isobaric thermodynamic equilibration with a parcel of the air. In other words, the thermodynamic wet-bulb temperature T_w is the temperature that yields the following outcome:

Note that the liquid water and the air parcel end up equilibrated (they are at a common temperature T_w and the air parcel is saturated), the process is isobaric (the pressure is held constant at p), and, most notably, the temperature of the liquid water does not change (its temperature remains at T_w). Note that the masses of the parcels of liquid water and air do not, in general, remain the same through this equilibration process; for RH < 1 the parcel of liquid water gives mass to the air parcel, and vice versa for RH > 1; for RH = 1, there is no exchange of mass and T_w = T.

The air’s psychrometric wet-bulb temperature is the temperature of liquid water that is invariant with respect to an isobaric exchange of heat and moisture with a stream of the air. In other words, the psychrometric wet-bulb temperature T_pw is the temperature that yields the following outcome:

As in the case of the thermodynamic wet-bulb temperature, this process is isobaric (the pressure is held constant at p) and the temperature of the liquid water does not change (its temperature remains at T_pw), but the liquid water and the stream of air do not reach equilibration (the downstream air will have a temperature T’ between T and T_pw and a relative humidity RH’ between RH and 1).

The ice-bulb versions of the above are the same, but with “liquid water” replaced with “solid water” or “ice.” The derivation of equations for all of these bulb temperatures using the Rankine-Kirchhoff framework can be found in Romps (2025).