# How does the temperature and humidity of air affect the water consumption of a humidifier?

We can model the water consumption $$\dot{m}_{w}$$ of a simple humidifier using the following equations:

\begin{align} \eta_{hum} & = {T_{db,i} - T_{db,o} \over T_{db,i} - T_{wb,i}} \\ h_{o} & = h_{i} \\ \dot{m}_{w} & = \dot{m}_{a}(w_{o} - w_{i}) \end{align}

where the subscripts $$i$$ and $$o$$ denote the entrance and exit of the humidifier respectively, $$\eta_{hum}$$ is the efficiency of the humidifier, $$T_{db}$$ and $$T_{wb}$$ are the dry and wet bulb temperatures of the air respectively, $$h$$ and $$w$$ are the enthalpy and humidity ratio of the air respectively, and $$\dot{m}_{w}$$ and $$\dot{m}_{a}$$ are the mass flow rate of the water and air respectively.

We can fix $$\eta_{hum}$$, $$\dot{m}_{a}$$, and the atmospheric air pressure to study just the effects of the temperature and humidity of the entering air on the water consumption.

I created this psychometric chart with four different air conditions to see if I could pinpoint one thermodynamic property of the entering air that would be responsible for the water consumption, but the story seems to be a bit more complicated than that. The relative humidity $$\phi$$ appears to be the property with the highest correlation with water consumption, but higher relative humidities don't always result in lower water consumptions.

What is the full story?