Page:Journal of the American Society of Mechanical Engineers, Volume 33.pdf/672

APPENDIX No. 1

PROOF OF FORMULA [1]

${\text{Per cent humidity}}={\frac {e_{1}}{e'_{2}}}={\frac {D_{2}}{D'_{2}}}$

71⁠In this formula $e_{1}$ is the pressure of saturated vapor corresponding to the dew point $T_{1}$, and $e'_{2}$ is the vapor pressure at saturation corresponding to temperature $T_{2}$. It also follows that

$D_{2}=D'_{2}\times {\frac {e_{1}}{e'_{2}}}$

At constant pressure

[30]

${\frac {D_{2}}{D_{1}}}={\frac {T_{1}}{T'_{2}}}$, or $D_{2}={\frac {D_{1}T_{1}}{T_{2}}}$

Also if it is assumed ${\frac {pv}{T}}$ is constant, i.e., ${\frac {e}{DT}}$ is constant for saturated water vapor (not exactly correct),

${\frac {e_{1}}{D_{1}T_{1}}}={\frac {e'_{2}}{D'_{2}t_{2}}}$

Therefore

[31]

${\frac {e_{1}}{e'_{2}}}={\frac {D_{1}T_{1}}{D'_{2}t_{2}}}$

But substituting $D_{2}$ for its value in [30]

${\frac {e_{1}}{e'_{2}}}={\frac {D_{2}}{D'_{2}}}={\text{per cent of isothermal saturation}}$

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