Application of film theory to evaporative cooling

We will look at evaporative cooling as an application of film theory. A solid, adiabatic, insulated wall is covered by a film of water, over which unsaturated humid air flows, as shown in Fig. 1.52. 

The humid air takes water vapour from the film, by which the film of water and the air are cooled, until a time and position constant temperature is reached. It will be constant over the whole film because the adjoining wall is adiabatic and therefore no heat can be added to it. This adiabatic permanent temperature is called the wet bulb temperature The resistance to mass transfer is only on the gas side. Once the permanent temperature has been reached water still evaporates in the unsaturated air flowing over it. As the temperature of the water film is constant, the enthalpy of vaporization required for the evaporation will be removed as heat from the air. Fig. 1.52 indicates how the temperature and partial pressure of the water vapour in the air changes at this permanent state. The wet bulb temperature is lower than the temperature of the humid air flowing over the water surface. Therefore a wet substance can be cooled down to its wet bulb temperature by evaporation. 

We wish to find out the magnitude of the wet bulb temperature. It is determined by the amount of water transferred from the water surface into the humid air. As this is diffusion through a semipermeable plane the amount of water (substance A) being transferred to the air at the phase boundary I between the water and the air, according to (1.195) is given by 

Here both the moisture content XAl at the water surface and the mass flux mA of the water being transferred are still unknowns. At the surface of the water saturation prevails and so the moisture content XAi, as taught by thermodynamics [1.33], is given by 

The energy balance is available as a further equation. In order to set this up we will consider a balance region on the gas side, depicted by the dotted lines in Fig. 1.53, from the gas at the surface of the film to a point y. The energy balance 

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