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, , with an uncertainty of 0.1 °C (coverage factor k=2). A resistance thermometer is used to measure the air temperature, , 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: , where is the actual vapour pressure of water and 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: , where is saturation water vapour pressure (in Pa) at temperature (in °C), is used for the irrational number 2.718… (to distinguish it from the symbol for vapour pressure), and the constants are and [*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 or , to account for the presence of gases other than water vapour, as it would cancel in the ratio .

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

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

We will also need the sensitivity coefficients and :

Evaluating the sensitivities at typical temperatures = -20 °Cfp to 50 °Cdp and = 5 °C to 50 °C, we see the familiar rule-of-thumb that or , 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 increases from 8 °Cdp to 9 °Cdp (at = 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 increases from 49 °C to 50 °C (at = 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.)

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)

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″]