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

$$ \mathrm{physiology}\left(\mathrm{HI},\,\frac{\text{1.6 kPa}}{p^*(\mathrm{HI})}\right) = \mathrm{physiology}(T_a,\,\mathrm{RH}) \, , $$

where “physiology” in this equation is a function that returns the physiological state, Ta 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

$$ C_c \frac{dT_c}{dt} = Q-Q_v - \frac{T_c-T_a}{R_a} - \frac{p_c-p_a}{Z_a} \, , $$

where Cc represents the heat capacity of the core, Tc = 310 K and Ta are the temperatures of the core and air; pc = \(\phi\)salt p*(Tc) and pa are the vapor pressures of the core and air; \(\phi\)salt is the effective relative humidity of sweat; p* is the saturation vapor pressure; Ra is the resistance to heat flow through the boundary layer at the surface of the skin; Za are the resistance to mass (i.e., sweat) flow through that boundary layer; Q is the metabolic heat flux generated in the core; and Qv 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 dTc/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 dTc/dt.

Otherwise, if dTc/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 dTc/dt = 0 at a normal core temperature of Tc = 310 K.

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

$$ \begin{align*} 0 &= Q-Q_v - \dfrac{T_c-T_s}{R_s} \\ 0 &= \dfrac{T_c-T_s}{R_s} - \dfrac{T_s-T_a}{R_a} - \dfrac{p_s-p_a}{Z_a} \\ p_s &= \min\left[\dfrac{Z_s p_a+Z_a p_c}{Z_s+Z_a},\phi_\text{salt}p^*(T_s)\right] \, , \end{align*} $$

where variables with a subscript “s” denote properties of the shell, which represents the skin and, at colder temperatures, also clothing. Here, Ts and ps are the temperature and vapor pressure of the shell’s outer surface, Rs is the shell’s resistance to heat flow; and Zs is the shell’s resistance to mass (i.e., sweat) flow. Setting Tc = 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 Qv 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 Rs, 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 Rs.