Air is a mixture of different gases. Under normal environmental conditions the gases have an ideal behavior, i.e. each gas molecule can act independently from all others. Dalton’s law is valid :
The partial pressure p is defined as the pressure of a gas, if it would occupy alone the whole volume of the gas mixture.
Water in its gaseous phase (vapor) is also a component of air mixture. Under normal conditions it behaves like an ideal gas. With Dalton’s law p becomes:
Vapor Pressure Above Liquid
The concentration of water vapor in air is limited. There is a maximum partial pressure of vapor which depends on temperature. Air at high temperature can take more vapor than at low temperature.
This behavior can be explained as follows :
the molecules in a liquid are moving with different velocities (or energies) whereby the average
energy is proportional to the temperature of the liquid. With respect to energy, the water molecules show a statistical distribution as in Fig. 1. The molecules with energy lower than the binding energy of the liquid cannot leave the water surface. Those with higher energy can leave the water.
They evaporate and increase the vapor partial pressure in the air (Fig. 2). The opposite phenomenon happens with the water vapor molecules. Those with lower energy than the binding level of the liquid condensate on the water surface and decrease the vapor partial pressure in the air.
In a closed volume partly filled with water at temperature T (Fig.2) there is an equilibrium between evaporation and condensation. If there is a lack of water molecules in the moist region, more evaporation will occur and the vapor concentration will increase. In the opposite case more molecules will condense than evaporate and the vapor concentration will decrease.
The balance between evaporation and condensation leads to a vapor partial pressure (respectively concentration) which only depends on temperature.
A temperature rise will increase the energy of water molecules (Fig.1) and the balance will be shifted to higher vapor concentration. For equilibrium at temperature T the vapor concentration (or water partial pressure e or number of water molecules per m³) is the maximum concentration which can exist at this temperature and cannot be exceeded. A higher concentration would lead to condensation again and after a short time the old balance would be reached.
This vapor concentration is called saturated concentration or in terms of partial pressure
at temperature T.
The saturation pressure above water has an exponential dependence on T and is given in Tab.1.
Vapor Pressure Above Ice
Below 0.01 °C (triple point of water) water can exist in a liquid phase as well as in a solid phase (ice) whereby the liquid phase is not stable. For temperatures lower than 0.01 °C, in addition to vapor pressure above water there is also a vapor pressure above ice. (Tab 2.)
Consequently there are two saturation curves below 0.01 °C which are given in Fig.3 in a logarithmic scale. From -100 °C to 100 °C the saturation vapor pressure is changing over 8 orders of magnitude.
Fig.3 : Vapor saturation curves above ice and water. Below the triple point (0.01 °C) the curve splits into two graphs.
Real Gas Correction
Up to now water vapor has been regarded as ideal gas, i.e. water molecules act independently from each other in the air mixture.
In reality there is a small interaction between molecules which leads to a small increase of saturation vapor in air. This is described by an enhancement factor f(p,T).
For normal pressure the enhancement factor is to 1 and can be neglected. In this case, water vapor can be seen as an ideal gas.
Relative Humidity RH [%RH]
Tab.1 and Tab.2 give the values for saturation vapor pressure as a function of temperature. These values are maximum values and cannot be exceeded. Usually the partial vapor pressure is lower.
Mixing Ratio r [g/kg]
r is the mass of water to evaporate and mix with 1 kg dry air to perform a certain relative humidity or partial vapor pressure e.
Specific Enthalpy h [kJ/kg]
The enthalpy of 1 kg moist air with relative humidity RH and corresponding mixing ratio r at temperature T is the total energy you need
- to warm up dry air from 0 °C to T
- to evaporate the water (latent heat of water)
- to warm up the vapor from 0 °C up to T
The specific enthalpy is a relative quantity, i.e. only variations are of interest, not the absolute value. The variation of enthalpy is the measure of energy required to transform the moist air from one equilibrium state to another.
To warm up air from 20 °C to 25 °C and humidify the air from 40 %RH to 60 %RH 20.2 kJ/kg would be needed.
example 2 :
Warming up from 20 to 25 °C at constant relative humidity 40% requires only 10.3 kJ/kg.
example 3 :
Warming up from 20 to 25 °C at constant partial vapor pressure (i.e. e = const , r = const , Td = const), the relative humidity decreases from 40% to 29.5 %RH. This requires only 5.1 kJ/kg energy.
The Mollier diagram is a very useful tool to solve HVAC-problems graphically. It includes all humidity functions in one chart.
Fig. 4a : Mollier diagram: curves of constant relative humidity . The region below 100% (fog region) is not valid because condensation occurs.
Fig. 4b : Curves of constant enthalpy are added to Fig.4a . Also example 1 is included.