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Simplified Model of a Mixture Involving Gases and a Vapor

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A simplification will be considered with respect to a mixture of ideal gases that is in contact with a solid or liquid phase of one of the components. For example, a mixture of air and water vapor in contact with liquid water or ice, such as is encountered in air conditioning or in drying. Or consider the condensation of water from the atmosphere when it cools on a summer day.

This problem and a number of similar problems can be analyzed quite simply and with considerable accuracy if the following assumptions are made:

1. The solid or liquid phase contains no dissolved gases.
2. The gaseous phase can be treated as a mixture of ideal gases.
3. When the mixture and the condensed phase are at a given pressure and temperature, the equilibrium between the condensed phase and its vapor is not influenced by the presence of the other component. This means that when equilibrium is achieved, the partial pressure of the vapor will be equal to the saturation pressure corresponding to the temperature of the mixture.

Since this approach is used extensively and with considerable accuracy, attention will be given to the terms that have been defined and the type of problems for which this approach is valid and relevant. In the discussion this will be referred to as a gas-vapor mixture.

The dew point of a gas-vapor mixture is the temperature at which the vapor condenses or solidifies when it is cooled at constant pressure. This is shown on the T-s diagram for the vapor shown in the following figure:



Suppose that the temperature of the gaseous mixture and the partial pressure of the vapor in the mixture are such that the vapor is initially superheated at state 1. If the mixture is cooled at constant pressure, the partial pressure of the vapor remains constant until point 2 is reached, and then condensation begins. The temperature at state 2 is the dew-point temperature. Line 1 to 3 indicates that if the mixture is cooled at constant volume the condensation begins at point 3, which is slightly lower than the dew-point temperature.

If the vapor is at the saturation pressure and temperature, the mixture is referred to as a saturated mixture, and for an air-water vapor mixture, the term saturated air is used.

The relative humidity φ is defined as the ratio of the mole fraction of the vapor in the mixture to the mole fraction of vapor in a saturated mixture at the same temperature and total pressure. Since the vapor is considered an ideal gas, the definition reduces to the ratio of the partial pressure of the vapor as it exists in the mixture, P_υ, to the saturation pressure of the vapor at the same temperature, P_g:

φ =

Pυ Pg

In terms of the numbers on the T–s diagram, the relative humidity φ would be:

φ =

P1 P4

Since the vapor is considered to be an ideal gas, the relative humidity can also be defined in terms of specific volume or density:

φ =

Pυ Pg

=

ρυ ρg

=

υυ υg

The humidity ratio ω of an air–water vapor mixture is defined as the ratio of the mass of water vapor m_υ to the mass of dry air m_a. The term dry air is used to emphasize that this refers only to air and not to the water vapor. The term specific humidity is used synonymously with humidity ratio.

ω =

mυ ma

This definition is identical for any other gas–vapor mixture, and the subscript a refers to the gas, exclusive of the vapor. Since the vapor and mixture are both considered to be ideal gases, a very useful expression for the humidity ratio in terms of partial pressures and molecular weights can be developed. From the expressions:

mυ = PυV RυT = PυVMυ T

and

ma = PaV RυT = PaVMa T

the following may be obtained:

ω =

PυV/RυT PaV/RaT

=

RaPυ RυPa

=

MυPυ MaPa

For an air–water vapor mixture, this reduces to:

ω = 0.622

Pυ Pa

The degree of saturation is defined as the ratio of the actual humidity ratio to the humidity ratio of a saturated mixture at the same temperature and total pressure.

An expression for the relation between the relative humidity φ and the humidity ratio ω can be found by solving Eq1 and Eq2 for P_υ and equating them:

φ =	ωPa 0.622Pg

Regarding the nature of the process that occurs when a gas–vapor mixture is cooled at constant pressure, suppose that the vapor is initially superheated at state 1 in the following figure:



As the mixture is cooled at constant pressure, the partial pressure of the vapor remains constant until the dew point is reached at point 2, where the vapor in the mixture is saturated. The initial condensate is at state 4 and is in equilibrium with the vapor at state 2. As the temperature is lowered further, more of the vapor condenses, which lowers the partial pressure of the vapor in the mixture. The vapor that remains in the mixture is always saturated, and the liquid or solid is in equilibrium with it. For example, when the temperature is reduced to T₃, the vapor in the mixture is at state 3, and its partial pressure is the saturation pressure corresponding to T₃. The liquid in equilibrium with it is at state 5.