# Rational Psychrometric Formulae/Appendix No. 4

APPENDIX No. 4

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]

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.

[44]

89This may also be demonstrated approximately for the range of temperatures under discussion by computation from the values given in the steam tables of Marks and Davis, as in Table 6:

TABLE 6 COMPARISON OF ACTUAL VALUES OF WITH VALUES OF COMPUTED FROM THE TOTAL HEAT AT DIFFERENT TEMPERATURES | ||||||

(Computed) | ||||||

80 | 70 | 0.44365 | 1090.2 | 48.03 | 1046 | .70 |

80 | 60 | 0.44356 | 1085.9 | 48.03 | 1046 | .74 |

80 | 50 | 0.44347 | 2081.4 | 48.03 | 1046 | .78 |

80 | 40 | 0.44347 | 1076.9 | 48.03 | 1046 | .5 |

100 | 90 | 1099.2 | 0.44401 | 67.97 | 1035 | .67 |

100 | 80 | 0.44392 | 1094.8 | 67.97 | 1035 | .7 |

100 | 70 | 0.44383 | 1090.8 | 67.97 | 1035 | .64 |

100 | 60 | 0.44374 | 1085.9 | 67.97 | 1035 | .67 |

100 | 50 | 0.44365 | 1081.4 | 67.97 | 1035 | .81 |

100 | 40 | 0.44356 | 1076.9 | 67.97 | 1035 | .54 |

120 | 110 | 0.44419 | 1108.0 | 87.91 | 1024 | .4 |

120 | 100 | 0.44410 | 1103.6 | 87.91 | 1024 | .57 |

120 | 90 | 0.44401 | 1099.2 | 87.91 | 1024 | .52 |

120 | 80 | 0.44392 | 1094.4 | 87.91 | 1024 | .64 |

120 | 70 | 0.44383 | 1090.3 | 87.91 | 1024 | .58 |

120 | 60 | 0.44374 | 1085.9 | 87.91 | 1024 | .50 |

120 | 50 | 0.44365 | 1081.4 | 87.91 | 1024 | .54 |

120 | 40 | 0.44356 | 1076.9 | 87.91 | 1024 | .47 |

or |

Hence substituting in equation [43]

[46]

91The total heat under any other adiabatic condition, where temperature is and moisture , is

[50]

which is substantially equivalent to

[51]

Therefore since the change is adiabatic we may equate [47] and [49].

[52]

where

- = the true wet-bulb depression
- = the moisture absorbed per lb. of pure air when it is adiabatically saturated from an initial dry-bulb temperature to and an initial moisture content
- = mean speciﬁc heat of air at constant pressure between temperature and
- = speciﬁc heat of steam at constant pressure between and
- = latent heat of evaporation at wet-bulb temperature

This is identical with equation [20] obtained by the differential method.