Francis Massen, in response to the posts,

Comment On The Northeast Heat Wave

Further Information On The Role Of Water Vapor In Measuring Heat By Francis Massen

has graciously prepared a write up of how to compute the moist enthalpy of surface air. Francis websites include http://meteo.lcd.lu/; http://computarium.lcd.lu/; and http://bmb.lcd.lu/

**Calculating Moist Enthalpy From Usual Meteorological Measurements By Francis Massen**

Abstract: This short article shows how to compute the moist enthalpy from usual meteorological measurements of dry temperature, air pressure and relative humidity. The result is used to add a plot of moist air enthalpy to the other near-live graphs shown by meteoLCD, the meteorological station of the LCD, Diekirch, Luxembourg

1. Sensible heat of dry air

The sensible heat of dry air is defined as H_{a} = Cp*T [ref. 3] with Cp usually taken as 1.005 when H_{a }is given in [kJ/kg] and temperature T in [°C].

Here we will use for Cp the following expression, valid for temperatures higher than 0 °C and lower than 60 °C, as given by PADFIELD [ref.2]

H_{a} = 1.007*T – 0.026 0 °C < T < 60 °C [eq.1]

2. Heat content of water vapor at temperature T

The heat content of water vapor is the sum of the latent heat of vaporization and the sensible heat of water vapor:

H_{v} = q*( L + 1.84*T) [ref. TET] [eq.2]

Where L = heat of vaporization = 2501 kJ/kg at 0°C

and 1.84*T = sensible heat of water vapor in kJ/kg

The sensible heat term of eq.3 (1.84*T) is very often considered negligible and omitted.** **

**Note:**L is a function of temperature, becoming slightly smaller with increasing T; for values between 0°C and 50°C one can use the linear interpolation L(T) = 2502 – 2.378*T computed by the author from a table with enthalpy values given by YHCHEN [ref.4]: The linear fit is excellent with R2 = 0.9998.

Combining eq.2 with L(T) gives:

**H _{v} = q*(2502 – 0.538*T)** with H

_{v}in kJ/kg and T in °C [eq.3]

3. Total enthalpy of moist air

Total enthalpy of moist air is the sum of H_{a} and H_{v}:

**H = H _{a} + H_{v} = (1.007*T -0.026) + q*(2502 – 0.538*T) **[eq.4]

with H in kJ/kg, T in °C and specific humidity q in kg/kg

The problem with this formula is that the **specific** humidity q is usually not measured by a standard meteorological equipment which commonly measures **relative** humidity.

4. Finding q from measured dry bulb temperature, relative humidity and atmospheric pressure

PIELKE [ref.3] and the AOMIP website [ref.1] give the following formula for the specific humidity q:

where e_{a} = vapor pressure in [Pa] and p_{a} = atmospheric pressure in [Pa].

Attention: p_{a} is the true air pressure, not the barometric pressure reduced to sea level!

Dividing numerator and denominator by e_{a} gives:

Relative humidity is the fraction of water vapor pressure to saturated water vapor pressure, usually multiplied by 100 to give a percent value:

RH = 100* e_{a}/e_{sat} → e_{a} = RH/100*e_{sat}

There are many different formulas relating e_{sat} to temperature. We will use the expression given in AOMIP [ref.1] and valid up to 40°C:

with saturated water vapor pressure e_{sat} in [Pa] and temperature T in °C.

Equations 4, 6 and 7 contain only T, RH and p_{a}, which are parameters measured by practically every standard weather station. Together they can be used to calculate the enthalpy of moist air by a single (albeit unwieldy) formula:

This expression is valid for temperatures 0°C < T < 40°C. Units: H[kJ/kg], T[°C], p_{a}[Pa]

5. A practical example

The author has used eq.8 in GNUPLOT to display near-live plots of the moist enthalpy at meteoLCD, Diekirch, Luxembourg (see http://meteo.lcd.lu/today_01.html). The following figure shows the situation for the week from 10 to 16^{th} July 2010. Sensible heat is shown by the blue bottom curve; the difference between the upper red curve ( = moist enthalpy) and the blue curve corresponds to the latent heat.

Technisolve Software has a website with an online moist air calculator, which is very handy for a quick validation check of individual values: http://www.coolit.co.za/airstate/airmoistobject.htm

References

[1] AOMIP: Atmospheric Forcing Data – Humidity

http://efdl.cims.nyu.edu/project_aomip/forcing_data/atmosphere/humidity.html

[2] PADFIELD, Tim: Conservation Physics

http://www.conservationphysics.org/atmcalc/atmoclc1.php

[3] PIELKE, Roger, Sr., WOLTER, Klaus: The July 2005 Denver Heat Wave: How unusual was it ?. National Weather Digest, vol.31, no. 1, July 2007

http://pielkeclimatesciencesci.files.wordpress.com/2009/10/r-313.pdf

[4] TET (The Engineering Toolbox)

http://www.engineeringtoolbox.com/enthalpy-moist-air-d_683.html

[5] YHCHEN: Calculation of Enthalpy Changes