# Uncertainty of measurement: relative humidity via dew point and air temperature

This article discusses the calibration of a relative humidity (RH) hygrometer by comparison with a condensation (“chilled mirror”) dewpoint hygrometer and a thermometer measuring air temperature (“dry-bulb temperature”, in humidity parlance). It focuses on estimation of the uncertainty to be associated with the measurement result.
The comparison between the RH hygrometer (Unit Under Test, or UUT) and the dewpoint hygrometer and thermometer (together forming the reference standard) is performed in  a temperature- and humidity-variable chamber. The dewpoint hygrometer measures dew-point (or frost-point, if below 0 °C) temperature, $Latex formula$, with an uncertainty of 0.1 °C (coverage factor k=2). A resistance thermometer is used to measure the air temperature, $Latex formula$, also with an uncertainty of 0.1 °C (k=2). (No correction is applied for self-heating of the resistance thermometer, as it was calibrated in air, that is, in similar conditions to those in which it is used.) The temperature uniformity of the chamber is specified by the manufacturer to be ± 0.3 °C. (We assume a coverage factor of k = √3.)
Measurements are performed at temperatures of 5 °C, 20 °C and 50 °C, and at relative humidities of 10 %rh, 50 %rh and 90 %rh.

First, we must be able to calculate relative humidity from measured values of dew point and air temperature. Relative humidity is defined as a ratio of water vapour pressures: $Latex formula$, where $Latex formula$ is the actual vapour pressure of water and $Latex formula$ is the saturation vapour pressure of water at the prevailing temperature [Beginner’s guide to humidity measurement, NPL Good Practice Guide No 124, p 17]. (Here we express RH from 0 to 1, not 0 %rh to 100 %rh.) We will use the Magnus formula to calculate water vapour pressures: $Latex formula$, where $Latex formula$ is saturation water vapour pressure (in Pa) at temperature $Latex formula$ (in °C), $Latex formula$ is used for the irrational number 2.718… (to distinguish it from the symbol for vapour pressure), and the constants are $Latex formula$ and $Latex formula$ [Guide to the measurement of humidity, Institute of Measurement and Control, 1996, p 53]. The Magnus formula has an uncertainty of less than 1.0 % (k=2) from -65 °C to 60 °C. We will not apply the water vapour enhancement factor to $Latex formula$ or $Latex formula$, to account for the presence of gases other than water vapour, as it would cancel in the ratio $Latex formula$.

How do we determine the actual water vapour pressure, $Latex formula$? It is the saturation water vapour pressure at the dew-point temperature $Latex formula$, by the definition of dew point [Beginner’s guide to humidity measurement, p 2]. So, applying the Magnus formula, $Latex formula$.

We may also need to calculate dew point from relative humidity and air temperature. To achieve this, first calculate vapour pressure $Latex formula$, then manipulate the Magnus formula to obtain $Latex formula$ [Guide to the measurement of humidity, Institute of Measurement and Control, 1996, p 54].

We will also need the sensitivity coefficients $Latex formula$ and $Latex formula$:

$Latex formula$ $Latex formula$

Evaluating the sensitivities at typical temperatures $Latex formula$ = -20 °Cfp to 50 °Cdp and $Latex formula$ = 5 °C to 50 °C, we see the familiar rule-of-thumb that $Latex formula$ or $Latex formula$, in other words, RH changes by approximately 6% of the value, for a change of 1 °C in dew point or air temperature. (The symbols °Cfp and °Cdp, for “degrees Celsius frostpoint” and “degrees Celsius dewpoint”, are commonly used to distinguish dew-point temperature, a measure of humidity, from air temperature.) In other words, if $Latex formula$ increases from 8 °Cdp to 9 °Cdp (at $Latex formula$ = 20 °C), RH changes from 0.46 (or 46 %rh) to 0.49 (or 49 %rh), an increase of 3 %rh, or 6% of value. If $Latex formula$ increases from 49 °C to 50 °C (at $Latex formula$ = 48 °Cdp), RH changes from 0.95 (95 %rh) to 0.90 (90 %rh), a decrease of 5 %rh, or 6% of value.

Here are the dew points, for the air temperature and relative humidity measurement points mentioned above:

 Air temp: 5 °C 20 °C 50 °C RH: 10 %rh -21.8 °Cfp -11.2 °Cfp 10.1 °Cdp 50 %rh -4.0 °Cfp 9.3 °Cdp 36.7 °Cdp 90 %rh 3.5 °Cdp 18.3 °Cdp 47.9 °Cdp

We will evaluate uncertainties for the three combinations indicated in bold in the table above.

We estimate hysteresis of the UUT, a capacitive RH sensor, to vary from zero at 10 %rh to 0.6 %rh (k=1) at 50 %rh to zero at 90 %rh.

5 °C, RH = 0.10 (10 %rh):

 Component Value u(k=1) Sensitivity u(RH) Ref std: Dewpoint -21.8 °Cfp 0.05 °C 0.010 0.0005 Air temperature 5.0 °C 0.05 °C -0.007 -0.0004 Chamber: Temperature gradient 0.17 °C -0.007 -0.0012 UUT: Hysteresis 0.000 RH 1 0.0000 Combined uncertainty (k=1) 0.0013 U(k=2) 0.003 (0.3 %rh)

20 °C, RH = 0.50 (50 %rh):

 Component Value u(k=1) Sensitivity u(RH) Ref std: Dewpoint 9.3 °Cdp 0.05 °C 0.034 0.0017 Air temperature 20.0 °C 0.05 °C -0.031 -0.0015 Chamber: Temperature gradient 0.17 °C -0.031 -0.0054 UUT: Hysteresis 0.006 RH 1 0.0060 Combined uncertainty (k=1) 0.0083 U(k=2) 0.017 (1.7 %rh)

50 °C, RH = 0.90 (90 %rh):

 Component Value u(k=1) Sensitivity u(RH) Ref std: Dewpoint 47.9 °Cdp 0.05 °C 0.046 0.0023 Air temperature 50.0 °C 0.05 °C -0.045 -0.0022 Chamber: Temperature gradient 0.17 °C -0.045 -0.0078 UUT: Hysteresis 0.000 RH 1 0.0000 Combined uncertainty (k=1) 0.0084 U(k=2) 0.017 (1.7 %rh)

(Contact the author at lmc-solutions.co.za.)

## 2 thoughts on “Uncertainty of measurement: relative humidity via dew point and air temperature”

1. Michele says:

Very very interesting article.
I have a question. In the computation of the uncertainty should also be considered the measurement errors of the dew point and air temperature.
For example, suppose that the error of the dew point is 0.1 ° C at 10% rh, and to immagine and a coverage factor k = 1.732, then u(k=1)=0.0577, senstivity=0.010 and u(rh) for this element =0,000577.
The same reasoning for the error related to air temperature.
Is it correct.