**Equations describing the physical properties of moist air
**

**
**This is a formal presentation of the formulæ that describe those
properties of water vapour in air that concern conservators. The derivation of
these equations is explained concisely and the useful equations are emphasised
by

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Saturation vapour
pressure

Definition of
the Pascal

Concentration of
water vapour in space

Relative humidity

Dew point

Concentration of
water vapour in air

Enthalpy

The
psychrometer

In a closed container partly filled with water there will be some water
vapour in the space above the water. The concentration of water vapour depends
only on the temperature. It is not dependent on the amount of water and is only
very slightly influenced by the presence of air in the container.

The
water vapour exerts a pressure on the walls of the container. The empirical
equations given below give a good approximation to the saturation water vapour
pressure at temperatures within the limits of the earth's climate.

Saturation vapour pressure, p_{s}, in pascals:
**p**_{s}** = 610.78 *exp( t / ( t + 238.3 ) *17.2694 )
**where

The svp below freezing can be corrected after using the equation above, thus:

The relationship between vapour pressure and concentration is defined for any
gas by the equation:

p = nRT/V *p is the pressure in
Pa, V is the volume in cubic metres, T
is the temperature in degrees Kelvin (degrees Celsius + 273.16), n is the quantity of gas expressed in molar mass ( 0.018 kg in the case
of water ), R is the gas constant: 8.31
Joules/mol/m ^{3} *
To convert the
water vapour pressure to concentration in kg/m

**kg/m ^{3} = 0.002166 *p / ( t + 273.16 )
**

The Relative Humidity (RH) is the ratio of the actual water vapour pressure to the saturation water vapour pressure at the prevailing temperature.

**RH = p/p**_{s}

RH is usually expressed as a percentage rather than as a fraction.

The RH is a ratio. It does not define the water content of the air
unless the temperature is given. The reason RH is so much used in conservation
is that most organic materials have an equilibrium water content that is mainly
determined by the RH and is only slightly influenced by temperature.

Notice that air is not involved in the definition of RH. Airless space
can have a RH. Air is the transporter of water vapour in the atmosphere and in
air conditioning systems, so the phrase *"RH of the air"* is commonly used,
and only occasionally misleading. The independence of RH from atmospheric
pressure is not important on the ground, but it does have some relevance to
calculations concerning air transport of works of art and conservation by freeze
drying.

The water vapour content of air is often quoted as dew point. This is the
temperature to which the air must be cooled before dew condenses from it. At
this temperature the actual water vapour content of the air is equal to the
saturation water vapour pressure. The dew point is usually calculated from the
RH. First one calculates** p**_{s}, the saturation vapour pressure at
the ambient temperature. The actual water vapour pressure, **p**_{a},
is:

**p**_{a}**= p**_{s}** * RH% / 100**

The next step is to calculate the temperature at which **p**_{a}
would be the saturation vapour pressure. This means running backwards the
equation given above for deriving saturation vapour pressure from temperature:

Let **w = ln ( p**_{a}**/ 610.78 ) Dew point = w *238.3
/ ( 17.294 - w ) **This calculation is often used to judge the
probability of condensation on windows and within walls and roofs of humidified
buildings.

The dew point can also be measured directly by cooling a mirror until it fogs. The RH is then given by the ratio

**RH = 100 * p _{s
}**

Concentration of water vapour in air

It is sometimes convenient to quote water vapour concentration as
kg/kg of dry air. This is used in air conditioning calculations and is quoted on
psychrometric charts. The following calculations for water vapour concentration
in air apply at ground level.

Dry air has a molar mass of 0.029 kg. It is denser than water vapour, which
has a molar mass of 0.018 kg. Therefore, *humid air is lighter than dry
air*. If the total atmospheric pressure is P and the water vapour pressure is
p, the partial pressure of the dry air component is P - p** **. The weight
ratio of the two components, water vapour and dry air is:

kg water vapour / kg dry air = 0.018 *p / ( 0.029 *(P - p ) )

= 0.62 *p / (P - p )

At room temperature P - p** **is nearly equal to P, which at ground level
is close to 100,000 Pa, so, approximately:

**kg water vapour / kg dry air = 0.62 *10 ^{-5} *p
**

**Thermal properties of damp air **The heat
content, usually called the

The enthalpy of dry air is not known. Air at zero degrees celsius is
*defined* to have zero enthalpy. The enthalpy, in kJ/kg, at any
temperature, t, between 0 and 60C is approximately:

h = 1.007t - 0.026
*below zero: h = 1.005t *

The enthalpy of liquid water is also defined to be zero at zero degrees
celsius. To turn liquid water to vapour at the same temperature requires a very
considerable amount of heat energy: 2501 kJ/kg at 0C

At temperature t
the heat content of water vapour is:

h_{w} = 2501 + 1.84t
*Notice that water vapour, once generated, also requires more heat
than dry air to raise its temperature further: 1.84 kJ/kg.C against about 1
kJ/kg.C for dry air. *The enthalpy of moist air, in kJ/kg, is
therefore:

The final formula in this collection is the **psychrometric equation**.
The psychrometer is the nearest to an absolute method of measuring RH that the
conservator ever needs. It is more reliable than electronic devices, because it
depends on the calibration of thermometers or temperature sensors, which are
much more reliable than electrical RH sensors. The only limitation to the
psychrometer is that it is difficult to use in confined spaces (not because it
needs to be whirled around but because it releases water vapour).

The
psychrometer, or wet and dry bulb thermometer, responds to the RH of the air in
this way:

Unsaturated air evaporates water from the wet wick. The heat required to evaporate the water into the air stream is taken from the air stream, which cools in contact with the wet surface, thus cooling the thermometer beneath it. An equilibrium wet surface temperature is reached which is very roughly half way between ambient temperature and dew point temperature.

The air's potential to absorb water is proportional to the difference between
the mole fraction, m_{a}, of water vapour in the ambient air and the
mole fraction, m_{w}, of water vapour in the saturated air at the wet
surface. It is this capacity to carry away water vapour which drives the
temperature down to t_{w}, the wet thermometer temperature, from the
ambient temperature t_{a} :

( m_{w} - m_{a}) =
B( t_{a}- t_{w})

B is a constant, whose numerical value can
be derived theoretically by some rather complicated physics (see the reference
below).

The water vapour concentration is expressed here as mole
fraction in air, rather than as vapour pressure. Air is involved in the
psychrometric equation, because it brings the heat required to evaporate water
from the wet surface. The constant B is therefore dependent on total air
pressure, P. However the mole fraction, m, is simply the ratio of vapour
pressure p to total pressure P: p/P. The air pressure is the same for both
ambient air and air in contact with the wet surface, so the constant B can be
modified to a new value, A, which incorporates the pressure, allowing the molar
fractions to be replaced by the corresponding vapour pressures:

p_{w} - p_{a}= A* ( t_{a}- t_{w})

The relative humidity (as already defined) is the ratio of
p_{a}, the actual water vapour pressure of the air, to p_{s},
the saturation water vapour pressure at ambient temperature. **RH%
**= 100 *p_{a}/ p_{s }**= 100 *( p _{w} - (
t_{a}- t_{w})** *

*To check your program, take air at 20C and 15.7C wet bulb temperature. The
RH is 65%. The water vapour pressure is 1500 Pa. The water vapour concentration
in kg/m3 is 0.011, in kg/kg it is 0.009. The dew point is 13C.*

Index

Copyright Tim
Padfield, 1996