Chapter 7

1
Chapter 7 Moist Air

• The Dew (Frost) Point
• As isobaric cooling proceeds from P, sublimation
  will not generally occur at F.
• Between F and D, air will be supersaturated with
  respect to ice, but only condense water at D.
• Tf and Tdew only indicate the point where
  condensation or sublimation can occur. They do
  not guarantee that such will occur.

2
Chapter 7 Moist Air

• Equivalent and Wet-Bulb Temperatures
• Consider a closed system consisting of dry air,
  water vapor (moist air), and water (or ice).
• The enthalpy of the system can be written as
• where we have made use of lv(T) hv hw and mt
  mv mw. We can substitute h cpT for vapor
  and liquid phase.

3
Chapter 7 Moist Air

• Equivalent and Wet-Bulb Temperatures
• Consider two states of the system linked by an
  isenthalpic process (?H 0). Each state may be
  represented by a form of the previous expression.
• where md, mt, and const are the same in both
  states. Then, since ?H 0,

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Chapter 7 Moist Air

• Equivalent and Wet-Bulb Temperatures
• We can rewrite this as
• The denominator on each side is a constant for
  any (closed) system. Each of the 2 sides of the
  equation is a function of the state of the system
  only, i.e., the expression on either side of the
  equation is an invariant for an isenthalpic
  process.

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Chapter 7 Moist Air

• Equivalent and Wet-Bulb Temperatures
• Divide both numerator and denominator in the
  quotients by md and use the fact that wt (mv
  mw)/md mt/md to get
• Note that (cpd wtcw) is constant for a given
  system, but will vary for different systems
  according to their total water value (liquid
  vapor).

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Chapter 7 Moist Air

• Equivalent and Wet-Bulb Temperatures
• We can simplify this expression first by
  neglecting the heat capacity of the water (wt ?
  w) yielding,
• Now the denominator is no longer constant.
  Consider wcw as small compared with cpd and make
  lv a constant, we get

7
Chapter 7 Moist Air

• Equivalent and Wet-Bulb Temperatures
• In this expression cp can be taken as cpd, or
  approximated, to a better degree of accuracy, as
  where the mixing ratio is an
  average value of w.
• Consider the physical process that links 2
  specific states (T, w) unsaturated moist air
  water and (T, w) saturated or unsaturated
  moist air without water of the system to which
  our approximate equation applies.
• We have dry air with w gm of water vapor and (w
  w) gm of liquid water (may or may not be
  droplets in suspension).

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Chapter 7 Moist Air

• Equivalent and Wet-Bulb Temperatures
• Assume w gt w and that saturation is not reached
  at any time (except eventually when the final
  state is achieved).
• Liquid water evaporates so the mixing ratio
  increases from w to w.
• As water evaporates, it takes up heat of
  vaporization from moist air and water (system is
  adiabatically isolated).
• Cooling results reducing temperature from T to
  T.
• At any instant, the system state differs finitely
  from saturation process is spontaneous and
  irreversible.

9
Chapter 7 Moist Air

• Equivalent and Wet-Bulb Temperatures
• Equivalent temperature (isobaric equivalent
  temperature) Tei, is defined as the temperature
  moist air would reach if it were completely dried
  by condensation of all its water vapor.
• Water is withdrawn in a continuous fashion,
  process is performed isobarically, and system is
  thermally isolated.
• The formulation assumes no liquid water initially
  and starts from an infinitesimal variation dH,
  condensing

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Chapter 7 Moist Air

• Equivalent and Wet-Bulb Temperatures
• In this expression, dmv is considered negative,
  i.e., we are condensing a mass dmv of vapor to
  liquid.
• Having condensed this infinitesimal amount of
  liquid, we remove it before condensing any more
  vapor.
• The enthalpy of the system will decrease by
  hwdmv, but T is not affected.
• For the next infinitesimal condensation, the
  equation is valid with a new value of mv.
• Our equation describes the process with mv as a
  variable.

11
Chapter 7 Moist Air

• Equivalent and Wet-Bulb Temperatures
• Dividing through by md, and rewriting the
  differentials as differences, we get
• Assuming the latent heat to be constant, we get
• Assuming Ti T, Tf Tei, wi w, and wf 0, we
  get
• Leading to

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Chapter 7 Moist Air

• Equivalent and Wet-Bulb Temperatures
• Next we consider the Wet-Bulb Temperature,
  defined by the following process, the reverse of
  Tei.
• We are following a process that goes from any
  mixing ratio, w, to saturation, wsw.
• Closed, adiabatic system with unsaturated moist
  air and liquid water, temperature T, and total
  mixing ratio, wt
• Water will evaporate drawing heat from moist air
• Change in T related to change in wt due to
  evaporation
• When saturation is reached, the temperature is
  the isobaric wet-bulb temperature, Tw.

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Chapter 7 Moist Air

• Equivalent and Wet-Bulb Temperatures
• Examples
• A sling (wet-bulb) psychrometer used to calculate
  RH
• Evaporative cooling of air by rain falling
  through unsaturated region below cloud base
• Have the same equation to start with, same
  assumptions.
• Here we have wt instead of w because we have
  liquid water.
• Assuming md gtgt mt ( mv mw), Tf Tw (the
  limiting temperature), Ti T, wi w, wf wsw,
  and wtcw ltlt cpd, we get

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Chapter 7 Moist Air

• Equivalent and Wet-Bulb Temperatures
• Note that Tei can be approximated (assuming wcw lt
  cpd) as
• We see that there is a relation between
  equivalent and wet-bulb temperatures, to wit
• The definition of wet-bulb temperature also lends
  itself to the defining of a relationship between
  Tw and Tdew using the approximate form for w ?
  ?e/p and cp cpd wtcw,

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Chapter 7 Moist Air

• Equivalent and Wet-Bulb Temperatures
• In general the relationships between e and T can
  be given by
• The adiabatic isobaric process we are discussing
  occurs along a straight line with slope
  .
• Extending the line to increasing T will intersect
  at Tei, where e 0.

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Chapter 7 Moist Air

• Moist Air Adiabats
• Poissons equation for adiabatic ascent (descent)
  between (p, T) and (p, T) is
• Noting that ?m ?d(1 0.26q) varies with the
  moisture content, the adiabats passing through a
  point p, T will be different for different values
  of q (w).
• Since ?m ? ?d, the adiabats for moist air will be
  slightly less steep than those for dry air (T
  will vary slightly more slowly). HOWEVER

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Chapter 7 Moist Air

• Moist Air Adiabats
• The potential temperature for moist, unsaturated
  air is
• If we substitute T ? (1000/p)-?d for T we get
• If p 100 mb and q 0.01 kg/kg, we get ?m
  0.9983?.
• If p 900 mb and q 0.01 kg/kg, we get ?m
  0.9999?.
• This tells us we can treat moist ascent as dry
  ascent.
• Substitute Tv for T we have the virtual potential
  temp, ?v.

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Chapter 7 Moist Air

• Moist Air Adiabats
• This also leads to a statement of the moist
  (unsaturated) adiabatic lapse rate.
• We have to account for the moisture in the value
  of the specific heat capacity of moist air,
• We apply the binomial theorem to the denominator
  to give
• As with the potential temperature, this says that
  moist air cooling adiabatically, cools less
  rapidly than dry air.

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Chapter 7 Moist Air

• Saturation of Air by Adiabatic Ascent
• The equation for the saturation temperature at
  the LCL is,
• On the diagram, P is the initial state, arrow
  indicates adiabatic ascent starting from P.
• As T changes, esw changes according to arrow from
  S along saturation curve.
• Two arrows will meet at Ts (not shown).
• Ts is solved for iteratively.

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Chapter 7 Moist Air

• Reversible Saturated Adiabatic Ascent
• The system under consideration is a parcel of
  cloud that rises and expands adiabatically and
  reversibly, indicating that all condensed water
  is kept (a closed system).
• Since it is adiabatic and reversible, the process
  is isentropic.
• What we derive here is valid for an ice cloud as
  well.
• The expression for the entropy of this system is

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Chapter 7 Moist Air

• Reversible Saturated Adiabatic Ascent
• If we divide by md and note that the entropy S
  const,
• Where wt,w mt/md wsw mw/md and the
  subscript w indicates saturation values (except
  in mw).
• If we differentiate the expression, we get
• This equation describes the saturated adiabatic
  process.
• If T and pd are independent variables, wsw
  ?esw/pd and esw f (T), so the equation
  determines a curve in T, pd plane.

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Chapter 7 Moist Air

• Reversible Saturated Adiabatic Ascent
• We can make several assumptions to get an
  approximate and somewhat simpler form of the
  equation
• Assume wt,w lt cpd and esw lt pd. This yields
• Next assume that lv varies slowly with T, we
  differentiate
• Use of the approximate form is good to within a
  few percent.

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Chapter 7 Moist Air

• Pseudoadiabatic Ascent
• In saturated adiabatic ascent, the equations
  depend on wt.
• While the saturation vapor mixing ratio, wsw, is
  determined, the liquid water mixing ratio, mw/md
  is arbitrary.
• This creates the situation where on a T, p
  diagram where saturation is just reached, there
  will be an infinite number of reversible
  saturated adiabats passing through, each
  differing ever so slightly from each other,
  depending on the amount of liquid water in the
  parcel.
• This dilemma can be avoided with a simple
  assumption.

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Chapter 7 Moist Air

• Pseudoadiabatic Ascent
• We assume that all the condensed liquid water (or
  ice) falls out of the parcel as soon as it
  appears.
• This is a process of saturated expansion, but the
  system is now open and irreversible.
• It is called a pseudoadiabatic process and gives
  unique curves through each T, p point on a
  diagram.
• Assume that mw 0 and wt,w ww, yielding
• The cooling is slightly greater in
  pseudoadiabatic than in reversible expansion for
  the same change in pressure.

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Chapter 7 Moist Air

• Equivalent Potential Temperature
• Go back to our isentropic equation
• Keep latent heat as lv lv(T) and differentiate
• We can introduce a potential temperature of the
  form
• Logarithmically differentiating this we get

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Chapter 7 Moist Air

• Equivalent Potential Temperature
• Comparing this with our other differential we see
  that
• If we integrate this we have
• We can get rid of the constant by taking the
  value of ? to be equal to ?e when all vapor has
  condensed , wsw 0

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Chapter 7 Moist Air

• Equivalent Potential Temperature
• What we have is the representation of a
  reversible saturated adiabatic process where all
  the water vapor has condensed out (take parcel to
  a very cold temperature).
• ?e is the equivalent potential temperature
  (conserved).
• As the equation stands, it applies to saturated
  air.
• We can apply it to unsaturated moist air by
  lifting the parcel to the LCL, yielding

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Chapter 7 Moist Air

• Summary of Temperatures

29
Chapter 7 Moist Air

• Summary
• of Temperatures