The Humidity Resource

[JohnS]

The Humidity Resource

This resource is compiled from two of my posts (#8 and #10) in a 2006 thread entitled:
Relative Humidity <> Absolute Humidity
It is intended as a resource and tutorial for any "moist air" problems involving water vapor pressure, relative humidity, specific humidity, absolute humidity, mixing ratio, etc. This resource contains a few minor updates from the original posts, and a section on station pressure following two worked examples.

For anyone looking for a chart for solving humidity problems, Google "psychrometric chart." There are free ones out there, I recommend the Carrier chart if you can find it.

For those who want formulas for programs, I will try. Most of this is pretty straightforward with the exception of what equation to use for saturated water vapor pressure.

Relative humidity is a ratio of actual water vapor pressure to saturated water vapor pressure. Saturated vapor pressure is the maximum that can exist at a given temperature; excess will condense out as dew or frost. If unsaturated air is cooled, at some temperature, it will become saturated and dew will form. This dewpoint is another way to describe the actual water vapor pressure.

Let ea = actual water vapor pressure, es = saturated water vapor pressure, RH = relative humidity, T = temperature, and Td = dewpoint temperature, then

RH = ea(T)/es(T) or ea(T) = RH * es(T)
ea(T) = es(Td)
RH = es(Td)/es(T)

If one knows the shape of the saturated vapor pressure curve, actual water vapor pressure and relative humidity can be calculated from temperature and dewpoint. More on the saturated vapor pressure calculation in a moment.

If ea(T) is known, other humidity properties of moist air can be calculated assuming ideal gas law. If total pressure is P, and water vapor pressure is ea, then pressure of dry air is P-ea. Mixing ratio, as kg of water vapor/kg of dry air, is given by
MR = 0.622*ea/(P-ea)
the constant is the ratio of molecular weights, water/air.

Specific humidity, kg of water vapor per kg of moist air mixture, is
MR/(1+MR).

We all know the ideal gas law, PV = nRT. But the number of moles, n, can be written as m/M where m is the mass of gas, and M is the molecular weight in g/mol. Then PV = (m/M)*RT.

Rearranging, the density m/V = P*M/(T*R). The absolute humidity is just the density of water vapor,
ea*M/(T*R)
where M = 18.02 g/mol, and R = 8.314472 Pa·m³/(mol·K), and T is absolute temperature in kelvins.

The key to all this is the es(T) curve. There are MANY equations to calculate or approximate it. See notes below on models and approximations. However, the approximation used by the National Weather Service for humidity calculations in surface observations is one due to Bolton,

es(t) = 611.2 Pa * exp(17.67*t/(243.5 + t))

where 611.2 Pa is the saturated vapor pressure at 0 °C, and
t is temperature in °C, not K.
(this fits steam table data to about 0.4% at 50 °C, and better at lower temperatures)
With this equation:
*Calculate es at T and Td
*Calculate relative humidity
*Calculate mixing ratio, specific humidity, or absolute humidity as desired from above.

Models and Approximations
As mentioned, there are MANY formulas to calculate es(T), and that makes this subject confusing.
I use "models" for formulations that have a thermodynamic basis and fit data well over a wide temperature range. The model formulas tend to be rather complex. Calculation is acceptable for a table (one time), but not for on-the-fly calculations in automated stations. "Approximations" are empirical fits (usually to a model) over a limited temperature range, but are faster to calculate. Also most approximations are much easier to "reverse," that is calculate the temperature at which a given saturation pressure occurs.

Whether models or approximations, there are always different formulas for vapor pressure over water and over ice, and for the models, both fit solid experimental data. Unfortunately, below 0 °C, meteorologists like to calculate relative humidity over supercooled water, not ice; however, scant data exists to justify the extrapolations they make to formulas only validated for water above 0 °C. The models extrapolate quite differently below 0 °C. (I have no clue which, if any, is right.)

I won't list all the formulas, but you can Google terms below, along with water vapor pressure and find them. A quick survey of models:
*Goff-Gratch: Originally 1945, updated in 1946, 1957, 1965. The standard of WMO (WMO No.49) and basis of Smithsonian Meteorological Tables.
*Hyland-Wexler: 1983, used by ASHRAE as the standard in heating/air conditioning calculations for several years (now they use a steam table formulation). Several weather equipment manufacturers use Hyland-Wexler or approximations to it.
*Sonntag, 1994 (can't find out much but a mention, and the formula)
*IAPWS Formulation, 1997: These are the guys who develop official Steam Tables for power plants, etc. Goff-Gratch and Hyland-Wexler claim validity to 100 °C. IAPWS is valid to the critical point of water (374 °C). Only it should be used above 100 °C
NOTE: See update on models in Post #4

A quick survey of approximations:
*The original Magnus equation dates to 1844; the slightly modified form above was by Murray, 1967, with the coefficients by Bolton, 1980. Many people have fit the Magnus form to different temperature ranges by tweaking coefficients. It fits well over about a 50 °C range, generally.
*Bogel equation: Bogel added a fourth parameter to Magnus equation which improves fit and range.
*Flatau, et al, developed 6th and 8th order polynomial fits. The 6th order fits -50 °C to +50 °C to about 0.01%, the 8th order fits -85 °C to +70 °C, but fit is about 0.1%, near the maximum temperature.

Arden Buck has a good 1981 paper on fitting Magnus and Bogel coefficients to different temperature ranges. It is available on the web as a pdf. The Magnus and Bogel forms can explicitly be solved in reverse; given actual water vapor pressure, dewpoint can be calculated. The higher-order Flatau method requires numerical iteration to get dewpoint.


Examples

I was asked to add one or more examples to demonstrate use of the above equations and added two. The first is based on data from Pontiac, MI, a NWS office (and airport) in the northern Detroit suburbs, and closest NWS office to me.

Their 2:00 pm report (2008-03-26) gave temp 43 °F, dew point 25 °F, RH = 49%. Because they are 298 m above sea level, "normal" barometric pressure is about 97.8 kPa, today's is about 98.3 kPa. (That computation needs to be the subject of another post; this one will be complex enough).

NWS uses a dog's breakfast of units in their reports. If you want to get agreement with their figures, you have to know which are source data and which are rounded. For temperature use the °F data. It needs to be converted to °C and you need two or more decimals because water vapor pressure is a strong function of temperature. 43/25 convert to 6.1111 °C/ -3.8889 °C. (The data is NOT that accurate, but it is that precise in calculation.)

Evaluating ea from the dewpoint, 0.4588 kPa.
Evaluating es at temperature, 0.9420 kPa, so RH is 48.7%, which agrees with published number of 49%

The mixing ratio is 0.622*(0.4588 kPa)/(98.3 kPa - 0.4588 kPa) = 0.00292 kg H2O/kg dry air

SH = MR/(1+MR)= 0.00291 kg H2O/kg moist air

AH (remember T is in kelvins) is ea*M/(T*R) = 3.56 g/m³ H2O.

For second example, we'll use some place warm, specifically Miami, where it is drier based on RH but wetter on any absolute measure. Temperature is 76 °F, dew point 54 °F, RH = 46%.

Converting the temperatures, we get 24.444 °C, 12.222 °C
ea(dew point) = 1.422 kPa
es(temp) = 3.0639 kPa, RH = 46.4%, again agrees with published figure.

Miami airport is 2 m above sea level. We ignore altitude correction, but local pressure is 102.4 kPa.(Pressure of the dry air is 102.4 - 1.4 kPa = 101 kPa, approx.)

Mixing ratio, using ea, is 0.00876 kg H2O/kg dry air
SH is 0.00868 kg H2O/kg moist air
AH is 10.36 g/m³

Well, I am over allowed length. The station pressure section will be post #2.

[JohnS]

Re: The Humidity Resource

Station Pressure

Mixing ratio and specific humidity require knowledge of the local barometric pressure, NOT reduced to sea level. It is best to have your own barometer calibrated to absolute pressure. However, you can calculate it from your elevation, and the altimeter setting information your local airport provides to pilots. The error is probably tolerable if the airport is within 100 miles (160 km) of your location, and 1000 feet (300 m) of your elevation. The error will be VERY small to 10 or 20% of those values.

Altimeter setting is a correction to standard barometric pressure (101.325 kPa) that will make a plane's altimeter read the official elevation of the runway, while on the runway. It is a slightly different algorithm than the reduction to sea level used for weather forecasts. In a METAR (report of local conditions to pilots), in the US, it is coded "A" and is reported in inches of mercury. 101.325 kPa = 29.9213 in Hg, but only two decimals are retained and the decimal point suppressed (A2992).

If
AS = altimeter setting
Po = standard pressure, 101.325 kPa or 29.9213 in Hg
Ps = station pressure
z = geopotential height about sea level, then

(AS/Po)^N = (Ps/Po)^N + 0.0065*z/288.15, where N = 0.190284)
This equation is valid for -5000 m <= z <= 11000 m. In the US, AS is only used for assigned altitudes below 18000 ft (5490 m), and is set equal to Po above that.

This can be solved for Ps/Po = [(AS/Po)^N - 0.0065*z/288.15]^(1/N)


(the value for Po may be taken in inches Hg with altimeter setting and in kilopascals with station pressure).

As an example, the published altitude of my local airport is 298 m, and the METAR is
METAR KPTK 121853Z 28013KT 10SM FEW049 25/09 A2991 RMK AO2 SLP121 T02500089

The altimeter setting is 29.91 in Hg
Ps/101.325 kPa = [(29.91/29.9213)^0.190284 -0.0065*298/288.15]^5.255303
Ps = 97.759 kPa

NOTE: For station pressure at your station, use your elevation above sea level, not the airport's.

[JohnS]

Re: The Humidity Resource

Accuracy and Enhancement Factor

The Bolton equation above is claimed to have accuracy of 0.1% from -30 °C to +35 °C. The error deteriorates relatively rapidly outside those limits. This covers most practical situations and it is the approximation used by NWS for all surface observations.

However, the biggest accuracy limit is enhancement effect. All of the water vapor equations referenced above are for a two-phase system, water and water vapor (no air). Moist air is not an ideal gas, and holds a little more water than the two-phase system. At standard atmospheric pressure and temperaturs from -20 °C to +30 °C, it holds about 0.4% more. This is known as the enhancement factor and varies with pressure and temperature.

The 1981 Arden Buck paper gives several models for enhancement factor, ranging from a constant, to a pressure term to multiple pressure and temperature terms. With only pressure terms, relative humidity calculations can ignore enhancement effect, but mixing ratio, specific and absolute humidity should consider it. With more elaborate corrections, the enhancement factor can be calculated to better than 0.05% over a wide temperature and pressure range. See the Buck paper for details:
http://www.public.iastate.edu/~bkh/t...n_buck_sat.pdf

With full correction of enhancement factor, it would make sense to use the Bogel equation in place of the Bolton equation. Buck gives coefficients for the Bogel equation for vapor pressure over water (ew4) and ice (ei3). In 1996, Buck suggested slightly different coefficients over water. See Appendix 1 of the owners manual for a chilled-mirror hygrometer made by Buck:
http://www.hygrometers.com/wp-conten...al-2009-12.pdf

Finally, note that all of Buck's approximations are fit to the Wexler model developed around 1977. The Wexler model is not in perfect agreement with the 1997 IAPWS model (steam tables). I have not seen a definitive comparison, but I note ASHRAE, which has a huge interest in heating/air conditioning/humidity calculations, has moved from the Hyland-Wexler equations to IAWPS. I have not seen redoes of the Bolton or Bogel equations to get best fit parameters to IAPWS. The differences are small, but there are differences.

[JohnS]

Update on the models section of the first post. Since that is near max character limit, I'm adding it here.

Older models such as Goff-Gratch, Wexler, Hyland-Wexler were fitted to data based on the IPTS-68 temperature scale or earlier. Any modern temperature sensors of sufficient accuracy for this to matter are based on ITS-90 temperature scale. This causes minor errors to these older models, and they probably should not be used unless updated. In a paper by NIST, Wexler is shown to a systemic error growing to about 0.1% at 100 °C, primarily due to temperature scale differences. The 1995 IAPWS-Scientific and 1997 IAPWS-Industrial models reflect ITS-90 scale as do the Sonntag (1994) model, which they also recommend. Hardy (1998) has updated the coefficients of the Wexler model to ITS-90; however, the Sonntag model is slightly simpler and has equally good performance 0 - 100 °C. If you need the accuracy of a model, you should probably use one of the IAPWS formulations or Sonntag. For the approximations (Bolton, or Bogel), it won't matter much.

Also note that in 2008 IAPWS issued a new low temperature vapor pressure over ice equation which replaces the auxiliary equation suggested in the 1995 formulation; it has one addition term, and different coefficients. It is valid from 50 K to 273.16 K.

There is still not a particularly authoritative reference for vapor pressure over supercooled water, mostly extrapolations.

2017-08-18 Update: Not sure when this recommendation was made but WMO no longer recommends any of the Goff-Gratch models for extrapolation to supercooled water below -50 °C. It suggests any of Wexler, Hyland-Wexler, or Sonntag. Since Hardy's formulation is Wexler's updated to ITS-90, it should probably be included. Hardy's equations are particularly useful in that he also fit a reverse model to recover dewpoint (or frostpoint over ice) from the saturated water vapor pressure.