Wet-bulb Temperature (WBT)

Wet-bulb temperature is the air temperature measured by a wet-bulb( cover the tip of the thermometer bulb with wetted cloth-muslin sleeve) that is freely exposed to the air and away from radiation and moisture.

OR
It is the temperature at which the water is evaporating into air @ given DBT & humidity ratio W. This process can bring the air to Saturation adiabatically @ the same temperature while the total pressure (partial pressure of water vapor + partial pressure of gas) is maintained constant.

Measuring WBT

As described above, the Wet-bulb temperature can be measured using a Sling Psychrometer with a wetted wick. And it is the easiest way to find it.

Calculating WBT from DBT & RH

We will see how we can calculate Wet-bulb temperature from Dry-bulb temperature(DBT) and Relative Humidity (RH).

Consider we have,
DBT = 75° F,
RH ($\Phi$) = 70.5%
measured @ 10’ ft Sea 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

$^{o}R = ^{o}F + 459.67$
$T_{DBT} = 75 + 459.67 = 534.67^{o}$

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 @ DBT = Saturation pressure
$p_{s} = p_{dbt}$
$\ln(p_{dbt}) = \dfrac{-10440.397}{534.67} + -11.29465$
$\quad+$ $-0.027022355*534.67 + 1.28904E-05*534.67^2$
$\quad+$ $-2.47807E-09*534.67^3$
$\quad+$ $6.5459673*\ln(534.67)$

$p_{s} = p_{dbt} = 0.43 psia$

Partial pressure of water vapour ($p_{w}$) from Relative Humidity

$\Phi = \dfrac{p_{w}}{p_{s}}$

Where,
$p_{w}$ - Partial pressure of water vapour (VP) in psia
$p_{s}$ - Saturation pressure in psia

$p_{w} = 0.705 * 0.43 = 0.303 \, lb/lb$

Humidity Ratio (W) from Partial pressure of Water vapour

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

Where,
$p_{w}$ - Partial pressure of water vapour (VP) in psia
$W$ - Humidity ratio in lb/lb
$p$ - Standard atmospheric pressure at given altitude in psia

$= \dfrac{0.621945 * 0.303} {14.691-0.303}$
$= 0.0131 \, lb/lb$

WBT from Humidity Ratio (W)

When the given temperature is above 32° F
$W =\dfrac{(1093-0.556*WBT)*W_{wbt}-0.240*(DBT-WBT)}{1093+0.444*DBT-WBT}$

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

$0.0131 =\dfrac{(1093-0.556*WBT)*W_{wbt}-0.240*(75-WBT)}{1093+0.444*75-WBT}$

By iterating the above equation for various values of WBT, we can find WBT.

Assume that WBT is always less than or equal to DBT; let us do the iteration as stated below.

Find the saturated pressure at WBT using the below equation.

And find the Humidity ratio at WBT using the below equation.

If WBT = 75° F; We get $p_{WBT}$ = 0.43 & $W_{WBT}$ = 0.0187, then W becomes 0.0188.
If WBT = 74° F; We get $p_{WBT}$ = 0.416 & $W_{WBT}$ = 0.0181, then W becomes 0.0179.
If WBT = 73° F; We get $p_{WBT}$ = 0.402 & $W_{WBT}$ = 0.0175, then W becomes 0.0170.
…
…
…
If WBT = 68° F; We get $p_{WBT}$ = 0.339 & $W_{WBT}$ = 0.0147, then W becomes 0.0131.

This 0.0131 is the same as the Humidity Ratio calculated in the previous step. So we decide the WBT = 68°F.