Page:Journal of the American Society of Mechanical Engineers, Volume 33.pdf/681

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

[40]

$\Sigma =\{H_{1}-q'+[C_{ps}(t-t_{1})]\}dW$

The change in sensible heat of 1 lb. of air and $W$ lb. of water vapor due to the temperature increment $-dt$ is

[41]

$\Sigma =-(C_{pa}+WC_{ps})dt$

Since the change is adiabatic these values may be related by the equation

[42]

$\{H_{1}-q'+[C_{ps}(t-t_{1})]\}dW-(C_{pa}+WC_{ps}dt=0$

[43]

$\int _{W}^{W'}\{H_{1}-q'+[C_{ps}(t-t_{1})]\}dW=\int ^{t'}(C_{pa}+WC_{ps})dt$

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.

[44]

$\{H_{1}-q'+[C_{ps}(t-t_{1})]\}dW=\{H_{1}-q'+[C_{ps}(t'-t_{1}))+(C_{ps}(t-t')]\}dW$

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