Specific humidity capacity

Specific humidity capacity ($c_{p-h}$) relates changes in humidity content in air (ω) to changes in vapor partial pressure ($P_{v}$) and is defined as $d\omega /dP_{v}$. It is analogous to specific heat capacity ($c_{p}$) used in heat transfer that relates changes in enthalpy to changes in temperature (dh/dT). This thermodynamic property of air is useful when using $\epsilon -NTU$ correlations for membrane based dehumidification mass transfer models. The introduction of the specific humidity capacity allows the definition of NTU for dehumidification to be entirely analogous to the definition of NTU for heat transfer since it incorporated the vapor partial pressure as the driving force for mass transfer (in the same way that temperature difference drives heat transfer). Using the specific humidity capacity is required when dealing with any form of vacuum membrane dehumidification, where humidity concentration difference across the membrane is not a suitable representation of the mass transfer driving force.

The ideal gas definition of the humidity ratio (water vapor in air) is given by:

$\omega ={\frac {m_{w}}{m_{a}}}={\frac {M_{w}P_{v}}{M_{a}P_{a}}}={\frac {0.62198P_{v}}{P_{a}}}$

Here, $m_{w}$ is the mass of water and $m_{a}$ is the mass for dry air. The ratio of $m_{w}/m_{a}$ can be approximated by knowing the molecular weight of water and air, represented by $M$, the water vapor partial pressure ($P_{v}$) and the air partial pressure ($P_{a}$). Using the ideal gas definition of humidity ratio and the fundamental definition of specific humidity capacity, a simplified expression for specific humidity capacity can be defined as follows:

$c_{p-h}={\frac {dw}{dP_{v}}}={\frac {M_{w}/M_{air}}{P_{air}}}={\frac {0.62198}{P_{air}}}={\frac {0.62198}{P_{total}-P_{v,inlet}}}$

Note that the air partial pressure is estimated using the total pressure (water vapor + all other gases in air) minus the water vapor partial pressure at the "inlet" of the dehumidification device ($P_{total}-P_{v,inlet}$).