Calculation of adiabatic saturation temperature

We will review how to calculate the adiabatic saturation temperature. The definition of this quantity was given in a previous post.

Problem definition

Given a gas-vapor mixture, if the inlet temperature and humidity are known, find the adiabatic saturation temperature.  The following equation is used:

From the inlet conditions we know $T_{\text{air}}$, Y and C. The main problem is that the humidity at saturation $Y_{\text{sat}}$ and the enthalpy of vaporization $\lambda$ depend on $T_{\text{as}}$. So an iterative calculation needs to be employed. Fortunately it can be easily implemented in Excel.

Example

This example comes from Seader's book. Air with inlet temperature 140 [F] and 12.5% relative humidity of water enters an process. Find the adiabatic saturation temperature of this current.

To solve this we also need enthalpy of vaporization and vapor pressure of water, and specific heat of both air and water, all as functions of temperature. All the required correlations can be found on Perry's book.

After a few trials by hand, the result is found using solver, and is equal to 87 [F]. The detailed calculations are on this file.

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).