How is relative humidity determined from a wet and dry bulb readings?

Question

The question What is the second thermometer in the image from the Esperanza Antarctic temperature record? shows what might be a wet/dry bulb setup.

I've given one to a friend to measure humidity for fun (it's me who thinks it's fun). I'd learned about them in Earth Science class a zillion years ago, so when I happened to see one recently I grabbed it.

I understand the very basics of how it can indicate humidity; the drier the air, the faster water will evaporate into it and so the colder the wet bulb will be with respect to the dry bulb.

To get a numerical value for humidity with this set up you use the coarse table on the front (there's a finer table on the back of the box) and look up the dry bulb temperature and the dry minus wet difference to find an approximate humidity. For example If I look up 19 and 2 °C for those respectively (roughly what's shown in the image) I get about 81% relative humidity.

But suppose I hooked up a camera to a Raspberry Pi and imaged the two thermometers, processed the image and determined the two temperatures and wanted to calculate the relative humidity without interpolating the table.

Question: How is relative humidity determined from a wet and dry bulb readings? What are the steps and where's a good source for the equations and parameters involved?

Click for full size

Answer 1

Edit: 1 May 2021

The following procedure uses the less accurate method from page 455 onward from the scanned sections of the book pictured below, from the original answer.

The procedure is a multistage process ideally suited to either a spreadsheet or programming code.

The equations use SI units.

1. Calculate the saturated vapor pressure at the dry bulb temperature

This is done using the equation: $P''_{ws} \ = \ 0.61078\ e\ ^{17.27T_{dry}/(273.3+T_{dry})} \ \ kPa$

Where $T_{dry}$ is the dry bulb temperature.

2. Calculate the saturated vapor pressure at the wet bulb temperature

This is done using the equation: $P'_{ws} \ = \ 0.61078\ e\ ^{17.27T_{wet}/(273.3+T_{wet})} \ \ kPa$

Where $T_{wet}$ is the wet bulb temperature.

3. Calculate the moisture content of saturated air at the wet bulb temperature

Use the equation: $r_o \ = \ \frac{18.016 \ (1.0048)P'_{ws}} {28.9664 \ (P_{atm}-1.0048)P'_{ws}} \ \ kg/kg$

Where $P_{atm}$ is atmospheric pressure

4. Calculate the enthalpy of water vapor pressure at the wet bulb temperature

$H'_{wo} \ = \ -0.00000662 \ t_{wet}^3-0.000194 \ t_{wet}^2+1.8375 \ t_{wet}+2500.83 \ \ kJ/kg$

5. Calculate the enthalpy of liquid water at the wet bulb temperature

$H'_{wl} \ = \ 0.0000063 \ t_{wet}^3-0.000727 \ t_{wet}^2+4.2058 \ t_{wet}+0.03 \ \ kJ/kg$

6. Calculate the enthalpy of water vapor pressure at the dry bulb temperature

$H'_{wi} \ = \ -0.00000662 \ t_{dry}^3-0.000194 \ t_{dry}^2+1.8375 \ t_{dry}+2500.83 \ \ kJ/kg$

7. Calculate the moisture content of saturated air at the wet bulb temperature

$r \ = \ \frac{r_o*(H'_{wo}-H'_{wl})-(1.005 \ t_{dry}-1.005 \ t_{wet})}{(H'_{wi}-H'_{wl})} \ \ kg/kg$

8. Calculate the vapor pressure

$P_w \ = \ \frac{r \ P_{atm}} {1.0048 \ (18.016/28.9664+r)} \ \ kPa$

9. Calculate the relative humidity

$\phi \ = \ P_w/P''_{ws}$

---

Previous Answer - scans from relevant pages from reference book

I'm going to go old school with this one.

The following pictures are the relevant pages from: Environmental Engineering in South African Mines, The Mine Ventilation Society of South Africa, 1989, pp 451-455. They give the psychometric equations that are required to calculate relative humidity from wet & dry bulb temperatures.

For simplicity, I use the less accurate equations on p455, section 9.2.

As a slight diversion, the other way to measure wet & dry bulb temperatures is to use a whirling hygrometer. The wet bulb temperature is always read first because it will start to warm as soon as the hygrometer stops whirling.