Density of Moist Air Virtual Temperature
If x be the humidity-mixing-ratio in a given volume V of moist air at temperature T and total pressure p and if e be the partial pressure of water vapour, the density of moist air is the sum of the density of water vapour and the density of dry air and is found as follows:
The density of water vapour as given by the equation of state for water vapour may be written in the form, pv = x/V = e e /RT (4.3.1)
where pv is the density and e the specific gravity of water vapour(= 0.622) and R the gas constant for dry air(= R*/Md) and the subscripts v and d refer to water vapour and dry air respectively.
Similarly, from the equation of state for dry air, the density of dry air pd is given by the relation,
Adding (4.3.1) and (4.3.2), we get for the density of moist air pm,
Pm = (1 + x)/V = p{1 -e(1- e)/p}/RT = p(1 - 3e/8p)/RT
where T* (= T/(1-3e/8p)) is called the virtual temperature of the moist air.
Thus, the virtual temperature is the temperature of the dry air which will have the same density as the moist air at the same pressure. The virtual temperature of moist air increases with increase of vapour pressure in the atmosphere.
The virtual temperature of moist air can also be expressed in terms of the humidity-mixing-ratio x, as shown below:
The equations of state of water vapour and dry air given in (4.3.1) and (4.3.2) may be written in the form e V = xRT/ e (p - e) V = RT
Adding the two relations above, we get pV =(1 + x/ e) RT (4.3.4)
Substituting the value of V from (4.3.4) in the expression for the density of moist air, we get
Pm = (1 + x)/V =(p/RT)(1 + x)/(1 + x/ e) = (p/RT*) (4.3.5)
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