DERIVATION OF FORMULA [6] GIVING THE EQUATION OF THE ADIABATIC SATURATION LINE
86Assuming 1 lb. of pure air having the temperature containing lb. of moisture with the corresponding dew point hand vapor pressure having a resultant adiabatic saturation temperature of , assume also a moisture increment under adiabatic conditions resulting in a temperature increment of . This moisture increment is evidently evaporated at a vapor pressure corresponding to temperature and superheated to temperature . The temperature of the liquid is evidently constant at temperature , from principle C. The total heat of the vapor in the increment is , where is the total heat of steam corresponding to temperature and vapor pressure , and is the heat required to superheat from saturation temperature to dry-bulb temperature . The heat of the liquid evaporated, however, is corresponding to temperature of saturation .
87 The total heat interchange required to evaporate under these conditions is therefore
[40]
The change in sensible heat of 1 lb. of air and lb. of water vapor due to the temperature increment is
[41]
Since the change is adiabatic these values may be related by the equation
[42]
[43]
in which and </math>t_1</math> are variables corresponding to the variable while is a variable related to by the different equation. A constant corresponding to is while , may be taken approximately as a mean between its values at and at and as a mean between its values at and at .
The temperature of saturation is , and is the corresponding moisture content at saturation.
88It is not necessary, however, to solve this equation in this form as this relationship may be simplified.