Moist Adiabatic Processes

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Moist Adiabatic Processes
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Moist Adiabatic Processes

• Parcel Theory
• Dry Adiabatic Process
• Moist Adiabatic Process
• Saturated Adiabatic Process
• Pseudo Adiabatic Process

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Moist Adiabatic Processes

• Reading
• Hess
• Section 4.6
• pp 51 - 58
• Equivalent Temp.
• pp 63

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Moist Adiabatic Processes

• Wallace Hobbs
• pp 84 87
• Tsonis
• pp 103 116
• Bohren Albrecht
• pp 287-291

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Moist Adiabatic Process

• Objectives
• Be able to identify parcel theory assumptions
• Be able to state the composition of air involved
  in dry adiabatic processes
• Be able to perform calculations using Poissons
  Equation for dry air

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Moist Adiabatic Process

• Objectives
• Be able to state the composition of air involved
  in moist adiabatic processes
• Be able to perform calculations using the
  equation for moist adiabatic processes

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Moist Adiabatic Process

• Objectives
• Be able to state the composition of an air parcel
  involved in saturated adiabatic processes
• Be able to describe the heat transfer in a parcel
  of air during saturated adiabatic ascent

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Moist Adiabatic Process

• Objectives
• Be able to state the composition of an air parcel
  involved in pseudo adiabatic processes
• Be able to describe the heat transfer in an air
  parcel during pseudo adiabatic ascent

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Moist Adiabatic Process

• Objectives
• Be able to state the definition of equivalent
  potential temperature
• Be able to identify conserved properties for dry
  and pseudo adiabatic ascent
• Be able to calculate equivalent potential
  temperature

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Moist Adiabatic Process

• Lets review dry adiabatic processes
• Parcel Theory
• Poissons Equation

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Parcel Theory

• Assumptions
• Thermally insulated from its environment
• Temperature changes adiabatically
• Always at the same pressure as the environment at
  that level

Tp,P
Te,P
w
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Parcel Theory

• Assumptions
• Hydrostatic equilibrium
• Moving slow enough that its kinetic energy is a
  negligible

Tp,P
Te,P
w
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Parcel Theory

• Types of Processes
• Dry Adiabatic
• Moist Adiabatic
• Saturated Adiabatic
• Pseudo Adiabatic

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Dry Adiabatic Process

• First Law of Thermodynamic

pda

• Adiabatic

• Work of expansion results in a temperature change
  of the parcel

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Dry Adiabatic Process

• Poissons Equation

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Dry Adiabatic Process

• Potential Temperature

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Dry Adiabatic Process

• Parcel Contains NO Moisture at All
• No Water Vapor
• No Liquid Water

Dry as a popcorn ......
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Dry Adiabatic Process

• Rising Parcel Cools at the Dry Adiabatic Lapse
  Rate
• Potential Temperature is Constant in the Parcel

q const
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Moist Adiabatic Process

• Moist Adiabatic Process?
• Dry adiabatic processes that involve water vapor

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Moist Adiabatic Process

• Air is a Mixture
• Dry Air
• Oxygen, Nitrogen, Argon, Carbon Dioxide, etc.
• Moisture
• Water Vapor

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Moist Adiabatic Process

• Lets look at a moist adiabatic process that
  involves only water vapor
• Parcel Remains Unsaturated

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Moist Adiabatic Process

• Moist Adiabatic (Unsaturated) Ascent
• Must include the contribution of water vapor to
  mixture

• Adjust R/cp to account for water vapor

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Moist Adiabatic Process

• Gas Constant for Mixture

md mass of dry air mv mass of water vapor m
total mass of air Rd Gas Constant of Dry
Air Rv Gas Constant of Water Vapor
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Moist Adiabatic Process

• Substitute

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Moist Adiabatic Process

• Remember

q specific humidity

• Substitute

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Moist Adiabatic Process

• Pull out an Rd

• Remember

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Moist Adiabatic Process

• Rearrange

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Moist Adiabatic Process

• Similarly, evaluate cp for the mixture
• The heat absorbed to raise the temperature during
  an isobaric process

Q total heat qd heat (per mass) of dry air md
mass of dry air qv heat (per mass) of water
vapor mv mass of water vapor
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Moist Adiabatic Process

• Remember

• Substitute

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Moist Adiabatic Process

• Divide by total mass (m)

• Specific humidity (q)

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Moist Adiabatic Process

• Substitute

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Moist Adiabatic Process

• Divide by dT

• But

cp specific heat at constant pressure
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Moist Adiabatic Proces

• Substitute

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Moist Adiabatic Process

• Pull out cpd

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Moist Adiabatic Process

• Rearrange

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Moist Adiabatic Process
cpd specific heat of dry air at constant
pressure 1004 J kg-1 K-1
cpv specific heat of water vapor at constant
pressure 1870 J kg-1 K-1
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Moist Adiabatic Process

• Evaluate

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Moist Adiabatic Process

• Now let look at Poissons Equation

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Moist Adiabatic Process

• Simplify using polynomial division

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Moist Adiabatic Process

• Poissons Equation for moist, unsaturated air

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Moist Adiabatic Process

• Potential Temperature for Moist, Unsaturated Air
• Use 1000 mb As Reference Pressure

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Moist Adiabatic Process

• Little difference between
• Dry adiabatic process

• Moist, unsaturated adiabatic process

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Moist Adiabatic Process

• Rising Air Expands and Cools Adiabatically
• Temperature Decreases
• Mixing Ratio (w) is Constant
• Only depends on mass
• At Saturation
• ws w
• Condensation
• Latent Heat Release

w ws
T decr. w const.
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Saturated Adiabatic Process

• Now lets consider condensation

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Saturated Adiabatic Process

• Two Methods to Describe the Ascent
• Saturated Adiabatic Process
• Pseudoadiabatic Process

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Saturated Adiabatic Process

• Latent Heat Release Will
• 1. Warm Air
• Heterogeneous System
• Dry Air
• Water Vapor
• Condensation Particles

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Saturated Adiabatic Process

• Latent Heat Release
• 2. Do Work
• Heating Causes Expansion

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Saturated Adiabatic Process

• Water Droplets Remain in Parcel
• Latent Heat Remains within Parcel

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Saturated Adiabatic Process

• Reversible
• Adiabatic
• Isentropic

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Saturated Adiabatic Process

• Mass of System
• Dry Air (md)
• Constant
• Total Water (mt)
• Vapor (mv)
• Liquid Water (mw)

md
mt mt mt
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Saturated Adiabatic Process

• Condensation
• Water Vapor to Liquid Water

dmv
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Saturated Adiabatic Process

• First Law of Thermodynamics

• Latent Heat Release

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Saturated Adiabatic Process

• Ideal Gas Law

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Saturated Adiabatic Process

• Divide by T

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Saturated Adiabatic Process

• Lets evaluate cp, R and p

• Specific Heat

• Latent Heat Warms
• Dry Air
• Water Vapor
• Liquid Water

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Saturated Adiabatic Process

• Gas Constant

• Dry Air
• Water Vapor

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Saturated Adiabatic Process

• Pressure

• Dry Air
• Water Vapor

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Saturated Adiabatic Process

• Yikes!
• Saturated Adiabatic Process
• Latent Heat of Condensation Warms
• Dry Air
• Water Vapor
• Liquid Water

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Saturated Adiabatic Process

• Assumptions
• Difference Between Moist Adabatic and Dry
  Adiabatic Processes is Small

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Saturated Adiabatic Process

• Assumptions
• Can neglect heating of water vapor
• Pressure of dry air is the total pressure

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Saturated Adiabatic Process

• Equation for Saturated Adiabatic Process
• Ignore Heating of Water Vapor
• Small Error
• Liquid Water Remains with Rising Parcel

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Pseudoadiabatic Process

• Water Droplets Fall Out of Parcel
• Latent Heat Remains within Parcel
• Latent Heat Warms Only Dry Air

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Pseudoadiabatic Process

• Irreversible
• Pseudoadiabatic
• Change in Entropy

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Pseudoadiabatic Process

• What does that do to our equation?

• No water mass to heat!

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Pseudoadiabatic Process

• Divide both sides by mass of dry air

w mixing ratio
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Pseudoadiabatic Process

• Equation for pseudoadiabatic processes
• Neglects heating of water vapor
• Neglects heating of water mass
• Latent heat release only warms dry air

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Pseudoadiabatic Process

• Not a bad integration except for.
• Left hand side
• lv f(T)
• w f(T)

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Pseudoadiabatic Process

• Assumptions
• lv varies little with temperature (T)
• Valid assumption

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Pseudoadiabatic Process

• Assumptions
• Which varies more during a pseudoadiabatic
  process?
• dw or T?
• Lets evaluate!

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Pseudoadiabatic Process

• Lets look at

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Pseudoadiabatic Process

• Which term is more important?

?

• Lets evaluate using a thermodynamic diagram

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Pseudoadiabatic Process

• Look at the change in w and T along a
  pseudoadiabat

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Pseudoadiabatic Process
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Pseudoadiabatic Process
2.8
3.9
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Pseudoadiabatic Process
267
273
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Pseudoadiabatic Process

• dw gtgt dT, so

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Pseudoadiabatic Process

• Plug back into the pseudoadiabatic equation

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Pseudoadiabatic Process

• Remember from long ago

• So

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Pseudoadiabatic Process

• Also remember that

• So

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Pseudoadiabatic Process

• Lets integrate!

ws mixing ratio at q qe potential
temperature after all water vapor has condensed
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Pseudoadiabatic Process

• Remember dT is smaller than dw

ws mixing ratio at q qe potential
temperature after all water vapor has condensed
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Pseudoadiabatic Process

• Integrate and evaluate the limits

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Pseudoadiabatic Process

• Take the inverse natural logarithm

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Pseudoadiabatic Process

• Rearrange terms

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Equivalent Potential Temperature (qe )

• The potential temperature a parcel of air would
  have if all of its water vapor were condensed and
  the latent heat released warmed only the dry air.

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Equivalent Potential Temperature (qe )
ws mixing ratio at once air has become
saturated by adiabatic cooling or ... mixing
ratio at dew point
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Equivalent Potential Temperature (qe )
lv latent heat of vaporization cp
specific heat of dry air T temperature q
potential temperature of air with temperature T
and pressure p
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Equivalent Potential Temperature (qe )

• A measure of the total energy of a parcel of air
• Internal Energy
• Energy from Latent Heat

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Equivalent Potential Temperature (qe )
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Equivalent Potential Temperature (qe )

• Conserved for
• Dry Adiabatic Processes
• Pseudoadiabatic Processes

qe constant
q constant
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Equivalent Potential Temperature (qe )

• Conserved for
• Dry Adiabatic Processes
• Pseudoadiabatic Processes

qe constant
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Equivalent Potential Temperature (qe )

• As the air becomes drier (ws 0), qe approaches q

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Equivalent Potential Temperature (qe )
q
qe
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Review

• Potential Temperature (q )
• The temperature a parcel of air would have if it
  were expanded or compressed adiabatically from
  its existing pressure and temperature to a
  standard pressure of 1000 mb.

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Review

• Wet-Bulb Temperature (Tw)
• The temperature to which air is cooled by
  evaporating water into it at constant pressure
  until the air is saturated

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Review

• Wet Bulb Potential Temperature (qw )
• The wet bulb temperature the air would have if it
  were expanded or compressed adiabatically from
  its existing pressure and wet bulb temperature to
  a standard pressure of 1000 mb.

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Review

• Virtual Temperature (Tv)
• The temperature dry air must have in order to
  have the same density as moist air at the same
  pressure

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Review

• Virtual Potential Temperature (qv )
• The virtual temperature the air would have if it
  were expanded or compressed adiabatically from
  its existing pressure and virtual temperature to
  a standard pressure of 1000 mb.

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Review

• Equivalent Temperature (Te )
• The temperature a parcel of air would have if all
  of its water vapor were condensed and the latent
  heat released warmed only the dry air.

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Review

• Equivalent Potential Temperature (qe )
• The potential temperature a parcel of air would
  have if all of its water vapor were condensed and
  the latent heat released warmed only the dry air.

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