Humidity ratio (or) Specific humidity

As per its definition, the humidity ratio or Specific humidity (W) is equal to

$W = \dfrac{M_{w}} {M_{da}}$

Where $M_{w}$ is the mass of water vapor and $M_{da}$ is the mass of dry air.

Finding the mass of water vapor directly from atmospheric air is not the best way to estimate the Humidity ratio. So instead, the Humidity ratio will be calculated from Dry bulb and wet bulb temperatures.

W in terms of Partial Pressure

Refer Dalton’s law for details.

Dry air has 28.9647 g/mol &
Water vapour has 18.01528 g/mol
$\dfrac{18.01528}{28.9647} \approx 0.621945$
$p_{da} * V = n_{da} * RT$
$p_{w} * V = n_{w} * RT$
$n = n_{da} + n_{w}$
$p = p_{da} + p_{w}$
$(p_{da} + p_{w}) * V = (n_{da} + n_{w}) * RT$

Where,
$p_{da}$ & $p_{w}$ are partial pressure of dry air & water vapour
$n_{da}$ & $n_{w}$ are molecular mass of dry air & water vapour

So the Humidity ratio (W) can be written as

$W = 0.621945 * \dfrac{x_{w}} {x_{da}}$

Where, $x_{da}$ & $x_{w}$ are mol fraction of dry air & water vapour

$W = 0.621945 * \dfrac{p_{w}} {p - p_{w}}$

W in terms of DBT & WBT

I am not giving a detailed explanation about this equation in this article; instead, I will do this in another article.

The below equation is derived, considering the total pressure remains constant in the atmosphere (i.e., adiabatically), the Enthalpy is raised from the temperature at which the liquid water evaporates into the air (refer definition for WBT) to saturation at the same temperature.

When the given temperature is above 32° F

Where,
$W$ - Humidity ratio in lb/lb
WBT - Wet bulb temperature in °F
DBT - Dry bulb temperature in °F
$W_{wbt}$ - Humidity ratio at saturation corresponding to WBT

Calculation for W

Given,
DBT = 75° F
WBT = 68° F
measured @ 10 ft seal level.

Then,

Standard atmospheric pressure

$p =$ $14.696*(1 - 6.8754 * 10^{-6} * Z)^{5.2559}$

Where, $z$ - altitude in ft
$= 14.696*(1 - 6.8754 * 10^{-6} * 10)^{5.2559} = 14.691 \,psia$

Temperatures in Rankine scale

$^{o}R = ^{o}F + 459.67$

$T_{DBT} = 75 + 459.67 = 534.67^{o}$
$T_{WBT} = 68 + 459.67 = 527.67^{o}R$

Saturation pressure @ given temperature

When the given temperature is between 32° F to 392° F

$\ln(p_{t}) = \dfrac{C_{8}}{t} + C_{9}$
$\quad\quad+$ $C_{10}*t + C_{11}*t^2 + C_{12}*t^3$
$\quad\quad+$ $C_{13}*\ln(t)$

Where,
t - Given temperature in °R
C8, C9, …, C13 are constants and their values are -10440.397, -11.29465, -0.027022355, 1.28904E-05, -2.47807E-09, 6.5459673 respectively

Saturation pressure @ WBT
$\ln(p_{wbt}) = \dfrac{-10440.397}{527.67} + -11.29465$
$\quad+$ $-0.027022355*527.67 + 1.28904E-05*527.67^2$
$\quad+$ $-2.47807E-09*527.67^3$
$\quad+$ $6.5459673*\ln(527.67)$

$p_{wbt} = 0.3392 psia$

Humidity ratio at given temperature

At the given temperature, humidity ratio is calculated as below

$W_{wbt} = \dfrac{0.621945 * p_{wbt}} {p-p_{wbt}}$

Where,
$W_{wbt}$ - Humidity ratio at saturation corresponding to WBT
$p_{wbt}$ - Saturated pressure at given WBT in psia
$p$ - Standard atmospheric pressure at given altitude in psia

Humidity ratio @ saturation corresponding to WBT
$W_{wbt} = \dfrac{0.621945 * 0.3392} {14.691-0.3392}$
$W_{wbt} = 0.0147$

Humidity ratio / Specific humidity

$W = \dfrac{(1093-0.556*68)*0.0147-0.240*(75-68)}{1093 + 0.444*75-68}$

$W = 0.0131\,lb/lb$