The wet bulb temperature

I started to study drying of solids, so is a good opportunity to revisit this important concept. The following results were adapted mainly from Treybal's book.

The humidity

Absolute humidity (Y) is simply the ratio of water mass as vapor and the air mass, expressed here in terms of partial and total pressures. Based  on the equality of the molar and pressure ratio:

 

Where M are the respective molecular weights.

Measurement of wet bulb temperature

Imagine a wet cloth that is exposed to an air stream. If the air is not saturated, water will evaporate from the cloth. The energy required for evaporation comes initially from the rest of the water, decreasing its temperature. Now due to the difference in air and water temperatures, heat will flow from the air to the remaining water.

As more water evaporates the remaining water will keep cooling, increasing the heat transfer rate from the air, until an equilibrium is reached. At this point the heat flow from the air is just enough to sustain the water vaporization. The following relation is satisfied:

 

 Adiabatic cooling

Now, imagine another process, this time air flows over a liquid surface. Similar to the previous case, water evaporates, taking energy from the air in the form of heat.  If this process is carried without external heat flow, it is termed adiabatic.

Before, air temperature and humidity were assumed constant, as the cloth is small compared to the air flow. This time however, both the temperature and humidity of the air change appreciably. The air humidity increases until the air becomes saturated.

A heat balance for the process gives:

Where C is the heat capacity of the inlet humid air. This equation says that the heat lost by the air together with its starting humidity is equal to the heat needed to add the additional humidity. 

Relation between wet bulb temperature and adiabatic cooling

Now lets see the equations for both processes side by side:

It turns out that the following ratio, called the Lewis relation is approximately one for an air-water mixture. So the wet bulb temperature (Teq) is approximately equal to the adiabatic saturation temperature (Tas).