# Specific Volume of Moist Air

## Specific volume is defined as the total volume of humid air per mass unit of dry air

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When discussing specific volume of moist air its necessary to differentiate between

- specific volume of moist air per
**mass unit of dry air**, or - specific volume of moist air per
**mass unit of dry air and water vapor**

### Specific Volume of Moist Air per Mass Unit of Dry Air

Specific volume is defined as the total volume of dry air and water vapor mixture per kg of dry air (SI-units). The specific volume can be expressed as:

v_{da}= V / m_{a}(1)

where

v_{da}= specific volume of moist air per mass unit of dry air (m^{3}/kg)

V= total volume of moist air (m^{3})

m_{a}= mass of dry air (kg)

When dry air and water vapor with the same temperature occupies the same volume the equation for an ideal gas can be applied.

p_{a}V = m_{a}R_{a}T(2)

where

p_{a}= partial pressure air (Pa)

R_{a}= 286.9 - the individual gas constant air (J/kg.K)

T= temperature of the moist air (K)

Combining (1) and (2):

v_{da}= R_{a}T / p_{a}(3)

The partial pressure of air can be expressed as:

p_{a}= p - p_{w}(3b)

where

p= pressure in the humid air (Pa)

p_{w}= partial pressure water vapor (Pa)

Combining (3) and (2b):

v_{da}= R_{a}T / (p - p_{w})(3c)

The ideal gas law can also be applied for the water vapor:

p_{w}V = m_{w}R_{w}T(4)

where

p_{w}= partial pressure water vapor (Pa)

R_{w}= 455 - the individual gas constant water vapor (J/kg.K)

T= temperature of the moist air (K)

The mass of water vapor can be expressed with the humidity ratio and the mass of air:

m_{w}= x m_{a}(5)

where

x= specific humidity or humidity ratio (kg/kg)

Combining (4) and (5):

p_{w}V = x m_{a}R_{w}T(6)

Combining (6) with (1):

v_{da}= x R_{w}T / p_{w}(7)

Transforming (7):

p_{w}= x R_{w}T /v_{da}(8)

Combining (8) with (3c):

v_{da}= R_{a}T / (p - (x R_{w}T /v_{da}))(9)

(9) can be transformed to express the __specific volume of moist air per unit mass of dry air__ as:

v_{da}= (1 + x R_{w}/ R_{a}) R_{a}T / p(10)

### Specific Volume of Moist Air per unit Mass of Dry Air and Water Vapor

Specific volume is defined as the total volume of dry air and water vapor mixture per kg of dry air and water vapor (SI-units). The specific volume can be expressed as:

v = V / m_{a}+ m_{w}(11)

where

v= specific volume of moist air per mass unit of dry air and water vapor (m^{3}/kg)

Combining with (5):

v = V / m_{a}(1 + x)(12)

Combining with (1):

v = v_{da}/ (1 + x)(13)

and the __specific volume of moist air per unit mass of dry air and water vapor__ can be expressed as:

v = (R_{a}T / p) [(1 + x R_{w}/ R_{a})/ (1 + x)](14)

(14) can used to express the density of moist air.

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