Chapter 7

1
Chapter 7 Moist Air

• Humidity Variables
• The water vapor content of moist air can be
  expressed by a number of different variables.
• First is specific humidity, defined as the ratio
  of the mass of water vapor, mv, to the total
  mass, m mv md, of moist air, expressed as q
  mv/m (q in the text).
• Another humidity variable is the mixing ratio,
  defined as the ratio of the mass of water vapor,
  mv, to the mass of the dry air, md, containing
  the vapor, expressed asw mv/md. Various
  symbols are used (r is a common one)

2
Chapter 7 Moist Air

• Humidity Variables
• The specific gas constant for moist air comes
  from the mixture of components as
• where md/m (m mv)/m 1 q. Then
• This can also be taken (in text) as defining the
  (variable) mean molecular weight of moist air,
  but it is usually viewed as modifying the gas
  constant, as presented here.

3
Chapter 7 Moist Air

• Humidity Variables
• Obviously, specific humidity and mixing ratio are
  related to each other. The relationships are
• Given that both q and w are quite small
  (typically lt 0.05), we often assume q ? w, i.e.,
  they are interchangeable.
• The mixing ratio is directly related to the air
  and vapor pressures

4
Chapter 7 Moist Air

• Humidity Variables
• Expanding this we have
• Because, usually, e ltlt p, we can take
• to be the case.
• As you go higher in the atmosphere, it gets
  colder, so the amount of moisture (vapor
  pressure) decreases as the air pressure
  decreases. This keeps the assumption valid.

5
Chapter 7 Moist Air

• Humidity Variables
• If the air is saturated with respect to water, we
  get the saturation mixing ratio
• If the saturation is with respect to ice, we have
• Both of these are subject to the approximate form

6
Chapter 7 Moist Air

• Humidity Variables
• It is inconvenient to use the variable moist
  air gas constant, Rm.
• We overcome this inconvenience by putting
  everything into the equation of state and
  rearranging the interpretation. Thus,
• Here we have associated the moisture effect with
  T.
• Define virtual temperature, Tv, as the
  temperature of dry air having the same p and ? as
  moist air (Tv gt T).

7
Chapter 7 Moist Air

• Humidity Variables
• Another humidity variable is the relative
  humidity, defined as the ratio of the mass of
  water vapor contained in air to the mass
  corresponding to saturation with respect to
  water, or with respect to ice.
• If we use r to indicate relative humidity, then
• are the respective values for water and ice. In
  as much as we are dealing with ideal gases, these
  expressions (using partial pressures) define
  relative humidity.

8
Chapter 7 Moist Air

• Humidity Variables
• To derive the mixing ratio, w, as a function of
  rw and wsw we use
• We continue to manipulate to get

9
Chapter 7 Moist Air

• Humidity Variables
• Further simplifying, we get
• So the humidity is defined by partial pressures,
  but can be approximated as the ratio of the
  mixing ratio to the saturation mixing ratio.
• The same relations hold for vapor over ice.
• Usually rw is obtained experimentally, p and T
  are known, wsw is obtained from a table and q and
  w are calculated.

10
Chapter 7 Moist Air

• Humidity Variables
• rw and ri are commonly given in percentage and q
  and w are given in g/kg.
• It is customary to express RH at T lt 0C with
  respect to water (rather than ice). This is done
  because
• Most hygrometers, which respond to RH, indicate
  RH with respect to water at all temperatures
• The majority of the cold portion of clouds
  (between 0C and ?20C, at least) contain, at
  least in part, supercooled water
• Supersaturation with respect to ice can easily
  occur, but not so with respect to water.

11
Chapter 7 Moist Air

• Heat Capacities of Moist Air
• The heat absorbed, ?Q, by a unit mass of moist
  air at constant pressure to cause a temperature
  increase dT, will have to heat both the dry air
  and water vapor portion, so
• Here the subscripted qs are the heats and q is
  the specific humidity. Divide by dT and recall
  ?Q cpdT at p const
• We can also make the approximation that w ? q and
  have

12
Chapter 7 Moist Air

• Heat Capacities of Moist Air
• The other heat capacities and constants related
  to them can be determined in a similar manner.
  We have

13
Chapter 7 Moist Air

• The Dew (Frost) Point
• In every closed system consisting of moist air, q
  and w remain constant, masses dont change in a
  closed system.
• This is not true for equilibrium vapor pressure
  or RH.
• Both are strongly dependent on T.
• Consider mass of moist air cooling isobarically
  (contracts)
• q, w, and e remain constant, rw will increase due
  to decrease in esw.
• As cooling continues, e esw saturation reached
  , rw 1.
• The temperature of saturation called the dew
  point, Tdew.

14
Chapter 7 Moist Air

• The Dew (Frost) Point
• If saturation is reached with respect to ice we
  are at the frost point temperature, Tf.
• Dew point temperature is a new variable that can
  be used to characterize the humidity of the air.
• Additional assumption pressure can change
  (rising or subsiding air) AND humidity may change
  (e.g., turbulent diffusion of vapor from a water
  source or rain falling through the air mass).
• To find the relation between Td, w, and p, we
  have to apply the C-C equation for the
  equilibrium curve.

15
Chapter 7 Moist Air

• The Dew (Frost) Point
• By definition, at Tdew, e esw(Tdew).
• Start by logarithmically differentiating the
  approximation e pw/? to get
• We can use the C-C equation in the following way
• where e is the vapor pressure of the air mass at
  T, and Tdew corresponds to e over the saturation
  curve.
• Tdew and e are humidity parameters giving the
  same info.

16
Chapter 7 Moist Air

• The Dew (Frost) Point
• If we solve for Tdew, we get
• Expressing this as a relative variation we have
• where we have used Tdew 270K indicating that
  the relative increase in Tdew is 5 of the sum
  of the relative increases in w and p.

17
Chapter 7 Moist Air

• The Dew (Frost) Point
• If we integrate the C-C between Tdew and T, we
  get
• If we solve for (T Tdew), and substitute for
  constants, we get (using lv 2.501 ? 106 J kg-1)
• For the frost point we get (using lf 2.8345 ?
  106 J kg-1)

18
Chapter 7 Moist Air

• The Dew (Frost) Point
• The figure shows the relationships between T and
  e during a process.
• The process starts at P at temperature T and
  vapor pressure e.
• Isobarically cool the air to Q where T Tdew and
  e is on the saturation curve.
• The integration we performed was between points Q
  and R.

19
Chapter 7 Moist Air

• The Dew (Frost) Point
• This figure shows the relation between Tdew, Tf ,
  and triple point.
• Starting at P and isobarically cooling the air,
  we pass thru Tf before reaching Tdew.
• Once the Tdew is reached, condensation begins
  (requires solid surface or CN).
• Without surface or CN no condensation occurs and
  the air becomes supersaturated.

20
Chapter 7 Moist Air

• The Dew (Frost) Point
• Atmosphere has abundant CN, so condensation is
  not a problem (only small supersaturations).
• With respect to ice, if a suitable surface is
  present freezing or sublimation will proceed as
  soon as the water or the vapor reaches the
  equilibrium curve.
• Ice Nuclei favor the appearance of ice crystals,
  but only activate at temperatures well below the
  equilibrium curve.
• Spontaneous nucleation of ice does not take place
  with either small supercooling of water or
  supersaturation of vapor.