Humid air mathematics

This page summarises the equations implemented in the HumidAir utility class for psychrometric calculations. The correlations are based on the ASHRAE Handbook Fundamentals (2017), CoolProp and the IAPWS formulation for the saturation pressure of water.

Saturation pressure of water

For temperatures $T$ above the triple point the saturation vapour pressure $p_{ws}$ in pascal is given by the IAPWS equation of Wagner and Pruss (2002)

\[\ln\left(\frac{p_{ws}}{p_c}\right) = \frac{T_c}{T}\left(a_1\theta + a_2\theta^{3/2} + a_3\theta^3 + a_4\theta^{7/2} + a_5\theta^4 + a_6\theta^{15/2}\right)\]

where $\theta = 1 - T/T_c$, $T_c = 647.096\ \text{K}$ and $p_c = 22.064\ \text{MPa}$. Below the triple point a sublimation correlation is used.

Humidity ratio

The humidity ratio $W$ relates the mass of water vapour to the mass of dry air

\[W = \varepsilon \frac{p_w}{p - p_w}\]

where $\varepsilon = M_w/M_{da} \approx 0.621945$, $p$ is the total pressure and $p_w$ the partial pressure of water.

For a given relative humidity $\phi$, the partial pressure is $p_w = \phi p_{ws}$.

Dew point temperature

Given a humidity ratio, the dew point temperature $T_d$ is found by solving $p_{ws}(T_d) = p_w$. The HumidAir implementation uses a simple Newton iteration.

Specific enthalpy

On a dry-air basis the specific enthalpy $h$ in kJ/kg dry air is approximated by

\[h = 1.006\,t + W (2501 + 1.86\,t)\]

where $t$ is the temperature in degrees Celsius and $W$ is the humidity ratio.

Saturated specific heat

CoolProp provides a correlation for the saturated humid-air specific heat $c_{p,\text{sat}}$ at 1\,atm valid from 250\,K to 300\,K

\[c_{p,\text{sat}} = 2.146\,27073 \times 10^{3} - 3.289\,17768 \times 10^{1}T + 1.894\,71075 \times 10^{-1}T^2 \\ - 4.862\,90986 \times 10^{-4}T^3 + 4.695\,40143 \times 10^{-7}T^4.\]