# Rational Psychrometric Formulae/Appendix No. 4

APPENDIX No. 4
DERIVATION OF FORMULA [6] GIVING THE EQUATION OF THE ADIABATIC SATURATION LINE
$r'(W'-W)=c_{pa}(t-t')+c_{ps}W(t-t')$
86Assuming 1 lb. of pure air having the temperature $t$ containing $W$ lb. of moisture with the corresponding dew point hand vapor pressure $e_{1}$ having a resultant adiabatic saturation temperature of $t'$, assume also a moisture increment $dW$ under adiabatic conditions resulting in a temperature increment of $-dt$. This moisture increment $dW$ is evidently evaporated at a vapor pressure $e_{1}$ corresponding to temperature $t_{1}$ and superheated to temperature $t$. The temperature of the liquid is evidently constant at temperature $t'$, from principle C. The total heat of the vapor in the increment is $H_{1}dW+C_{ps}(1-t_{1})dW$, where $H_{1}$ is the total heat of steam corresponding to temperature $t_{1}$ and vapor pressure $e_{l}$, and $C_{ps}(t-t_{1})dW$ is the heat required to superheat from saturation temperature $t_{1}$ to dry-bulb temperature $t$. The heat of the liquid evaporated, however, is $q'dW$ corresponding to temperature of saturation $t'$.
87 The total heat interchange required to evaporate $dW$ under these conditions is therefore
$\Sigma =\{H_{1}-q'+[C_{ps}(t-t_{1})]\}dW$
The change in sensible heat of 1 lb. of air and $W$ lb. of water vapor due to the temperature increment $-dt$ is
$\Sigma =-(C_{pa}+WC_{ps})dt$
Since the change is adiabatic these values may be related by the equation
$\{H_{1}-q'+[C_{ps}(t-t_{1})]\}dW-(C_{pa}+WC_{ps}dt=0$
$\int _{W}^{W'}\{H_{1}-q'+[C_{ps}(t-t_{1})]\}dW=\int ^{t'}(C_{pa}+WC_{ps})dt$
in which $H_{1}$ and </math>t_1</math> are variables corresponding to the variable $W$ while $t$ is a variable related to $W$ by the different equation. A constant corresponding to $t'$ is $q'$ while $C_{ps}$, may be taken approximately as a mean between its values at $t_{1}$ and at $t'$ and $C_{ps}$ as a mean between its values at $t$ and at $t'$. The temperature of saturation is $t'$, and $W'$ 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.
$\{H_{1}-q'+[C_{ps}(t-t_{1})]\}dW=\{H_{1}-q'+[C_{ps}(t'-t_{1}))+(C_{ps}(t-t')]\}dW$
$H_{1}-q'+C_{ps}(t'-t_{1})]=H'-q'=r'$
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 $r'$ WITH VALUES OF $r'$ COMPUTED FROM THE TOTAL HEAT AT DIFFERENT TEMPERATURES $t_{1}$ :
| $t'=80{\text{deg., }}r'=1046.7$ |||||||
| $t'$ | $t_{1}$ | $C_{ps}$ | $H_{1}$ | $q'$ | $r'$ (Computed) | $r'$ (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 |
| $t'=100{\text{deg., }}r'=1035.6$ |||||||
| 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 |
| $t'=120{\text{deg., }}r'=1024.4$ |||||||
| 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 |
| | | $r'=H_{1}-q'+[C_{pa}(t'-t_{1})]$ | $r'=H_{1}-q'+[C_{pa}(t'-t_{1})]$ | $r'=H_{1}-q'+[C_{pa}(t'-t_{1})]$ | $r'=H_{1}-q'+[C_{pa}(t'-t_{1})]$ | $r'=H_{1}-q'+[C_{pa}(t'-t_{1})]$ |
| | or | $H'=H_{1}+[C_{ps}(t'-t_{1})]$ | $H'=H_{1}+[C_{ps}(t'-t_{1})]$ | $H'=H_{1}+[C_{ps}(t'-t_{1})]$ | $H'=H_{1}+[C_{ps}(t'-t_{1})]$ | $H'=H_{1}+[C_{ps}(t'-t_{1})]$ |
Hence substituting in equation [43]
$r'\int _{W}^{W'}dW+C_{ps}\int _{W}^{W'}(t-t')dW=C_{pa}\int _{t'}^{t}dt+C_{ps}\int _{t'}^{t}Wdt$
$r'(W'-W)+C_{pa}\int _{W}^{W'}\left[\int _{t'}^{t}dt\right]dW=C_{pa}(t-t')+C_{ps}W\int _{t'}^{t}dt+C_{ps}\int _{W}^{W'}\left[\int _{t'}^{t}dt\right]dW$
$r'(W'-W)=C_{pa}(t-t')+C_{ps}W(t-t')$
$t'$
$q'$
$\Sigma =C_{pa}t'+r'W'$
91The total heat under any other adiabatic condition, where temperature is $t$ and moisture $W$, is
$\Sigma =C_{pa}t+[(H_{1}-q_{1})+C_{ps}(t'-t_{1})]W$
which is substantially equivalent to
$\Sigma =C_{pa}t+r'W+C_{ps}(t-t')W$
Therefore since the change is adiabatic we may equate [47] and [49].
$C_{pa}t+r'W+C_{ps}(t-t')W=C_{pa}t'+r'W'$
$C_{pa}(t-t')+C_{ps}(t-t')W=r'(W'-W)$
where
    $(t-t')$ = the true wet-bulb depression
    $(W'-W)$ = 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 $W$
    $C_{pa}$ = mean speciﬁc heat of air at constant pressure between temperature $t$ and $t'$
    $C_{ps}$ = speciﬁc heat of steam at constant pressure between $t$ and $t'$
    $r'$ = latent heat of evaporation at wet-bulb temperature $t'$
This is identical with equation [20] obtained by the differential method.