Mole Fraction Humid Air

Overview

Engineering reference data for Mole Fraction Humid Air in air psychrometrics.

Key Formulas

Humidity Ratio

$\omega = 0.622 \frac{P_v}{P_a}$

Mass of water vapor per mass of dry air.

Relative Humidity

$\phi = \frac{P_v}{P_{vs}} \times 100\%$

Ratio of actual to saturation vapor pressure.

Wet Bulb Temperature

$T_{wb} = T_{db} - \frac{P_{vs} - P_v}{\gamma}$

Temperature measured by wet-bulb thermometer.

Enthalpy of Moist Air

$h = c_p T + \omega h_g$

Sensible + latent heat per unit mass of dry air.

Variables

Mole Fraction Relationships

The mole fraction of water vapor in moist air is defined as:

$x_w = \frac{n_w}{n_a + n_w}$

Similarly, the mole fraction of dry air can be expressed as:

$x_a = \frac{n_a}{n_a + n_w} = 1 - x_w$

These mole fractions must sum to unity:

$x_a + x_w = 1$

Conversion to Humidity Ratio

The mole fraction relates to the humidity ratio ($\omega$) through the molecular weights of water vapor ($M_w = 18.015 \text{ g/mol}$) and dry air ($M_a = 28.965 \text{ g/mol}$):

$x_w = \frac{\omega / M_w}{(\omega / M_w) + (1 / M_a)}$

Rearranging to find mole fraction from humidity ratio:

$x_w = \frac{\omega \cdot M_a}{\omega \cdot M_a + M_w}$

Saturation Mole Fraction

At saturation conditions, the mole fraction of water vapor becomes:

$x_{w,sat} = \frac{p_{ws}}{p}$

where $p_{ws}$ is the saturation pressure of water vapor and $p$ is the total atmospheric pressure. The relative humidity ($\phi$) can then be expressed as:

$\phi = \frac{x_w}{x_{w,sat}} \cdot \frac{1 - x_{w,sat}}{1 - x_w}$

For low humidity conditions where $x_w \ll 1$, this simplifies to approximately:

$\phi \approx \frac{x_w}{x_{w,sat}}$

Partial Pressure Relationship

Since mole fraction is directly proportional to partial pressure (Dalton's Law):

$x_w = \frac{p_w}{p}$

where $p_w$ is the partial pressure of water vapor and $p$ is the total pressure of the moist air mixture.

References

• Original Source