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, $\displaystyle t_d$ , with an uncertainty of 0.1 °C (coverage factor k=2). A resistance thermometer is used to measure the air temperature, $\displaystyle t$ , 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: $\displaystyle RH = \frac{e}{e_s}$ , where $\displaystyle e$ is the actual vapour pressure of water and $\displaystyle e_s$ 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: $\displaystyle e_s = 611.2 \ \mathrm{Pa} \cdot exp^{a \cdot t/(b + t)}$ , where $\displaystyle e_s$ is saturation water vapour pressure (in Pa) at temperature $\displaystyle t$ (in °C), $\displaystyle exp$ is used for the irrational number 2.718… (to distinguish it from the symbol for vapour pressure), and the constants are $\displaystyle a = \begin{cases} 17.62, & \text{if } t \ge 0 ^{\circ} \text{C} \\ 22.46, & \text{if }t < 0 ^{\circ} \text{C} \end{cases}$ and $\displaystyle b = \begin{cases} 243.12, & \text{if } t \ge 0 ^{\circ} \text{C} \\ 272.62, & \text{if }t < 0 ^{\circ} \text{C} \end{cases}$ [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 $\displaystyle e$ or $\displaystyle e_s$ , to account for the presence of gases other than water vapour, as it would cancel in the ratio $\displaystyle RH = \frac{e}{e_s}$ .

How do we determine the actual water vapour pressure, $\displaystyle e$ ? It is the saturation water vapour pressure at the dew-point temperature $\displaystyle t_d$ , by the definition of dew point [Beginner’s guide to humidity measurement, p 2]. So, applying the Magnus formula, $\displaystyle e = 611.2 \ \mathrm{Pa} \cdot exp^{a \cdot t_d/(b + t_d)}$ .

We may also need to calculate dew point from relative humidity and air temperature. To achieve this, first calculate vapour pressure $\displaystyle e = RH \cdot e_s(t)$ , then manipulate the Magnus formula to obtain $\displaystyle t_d = \frac{b \cdot ln(e/611.2 \ \mathrm{Pa})}{a - ln(e/611.2 \ \mathrm{Pa})}$ [Guide to the measurement of humidity, Institute of Measurement and Control, 1996, p 54].

We will also need the sensitivity coefficients $\displaystyle \frac{\partial RH}{\partial t_d}$ and $\displaystyle \frac{\partial RH}{\partial t}$ :

$\displaystyle \frac{\partial RH}{\partial t_d} = \frac{\partial RH}{\partial e} \frac{\partial e}{\partial t_d} = \frac{RH}{e} \cdot e \cdot \frac{a \cdot b}{(b+t_d)^2} = RH \cdot \frac{a \cdot b}{(b+t_d)^2}$ $\displaystyle \frac{\partial RH}{\partial t} = \frac{\partial RH}{\partial e_s} \frac{\partial e_s}{\partial t} = \frac{-RH}{e_s} \cdot e_s \cdot \frac{a \cdot b}{(b+t)^2} = -RH \cdot \frac{a \cdot b}{(b+t)^2}$

Evaluating the sensitivities at typical temperatures $\displaystyle t_d$ = -20 °Cfp to 50 °Cdp and $\displaystyle t$ = 5 °C to 50 °C, we see the familiar rule-of-thumb that $\displaystyle \frac{\partial RH}{\partial t_d} \approx 0.06 \ \mathrm{RH}$ or $\displaystyle \frac{\partial RH}{\partial t} \approx -0.06 \ \mathrm{RH}$ , 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 $\displaystyle t_d$ increases from 8 °Cdp to 9 °Cdp (at $\displaystyle t$ = 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 $\displaystyle t$ increases from 49 °C to 50 °C (at $\displaystyle t_d$ = 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”

Michele says:

November 23, 2014 at 7:04 pm

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.
Thanks in advance for your time.

Regards,

Michele Bolettieri (from Italy)

1. Hans Liedberg says:
  
  July 30, 2015 at 4:14 pm
  
  In principle, it is best to apply corrections for all known errors, rather than including them in the uncertainty budget. CCT Working Group 3 stated it well:
  “Corrections are applied where bias (systematic error) is known, including in most cases where the correction is less than the uncertainty. This is done to maximise the information retained in reported measurement results and to prevent the accumulation of significant bias.” [CCT/08-19, “Uncertainties in the realisation of the SPRT subranges of the ITS-90″]

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

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

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