Moist Air - Specific Volume

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 (m³/kg)

V = total volume of moist air (m³)

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 of dry air (Pa)

R_a = 286.9 - the individual gas constant dry 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 the air can be expressed as

p_a = p - p_w                              (3b)

where

p = pressure in the moist air (Pa)

p_w = partial pressure water vapor (Pa)

Combining (3) and (2):

v_da =  R_a T / (p - p_w )                     (3c)

The ideal gas law can also be applied to the water vapor as

p_w V = m_w R_w T                        (4)

where

p_w = partial pressure of water vapor (Pa)

R_w = individual gas constant water vapor ( 461.4 J/kg K)

T = temperature of the moist air (K)

The mass of the water vapor can be calculated by using the humidity ratio and the mass of air as

m_w = x m_a                (5)

where

x = m_w / m_a = specific humidity or humidity ratio (kg_h2o /kg_{dry_air})

m_w = mass of water vapor (kg)

m_a = mass of dry air (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 (m³/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 (moist air) can be expressed as

v = (R_a T / p) ((1 + x R_w / R_a )/ (1 + x))                       (14)

(14) can used to express the inverse of the specific volume - the density - of moist air as

ρ = 1 / v

= (p / R_a T) (1 + x) / (1 + x R_w / R_a )                      (15)