How can the first law really help me forecast thunderstorms?
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Transcript of How can the first law really help me forecast thunderstorms?
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Thermodynamics M. D. Eastin
Adiabatic Processes
How can the first law really help me forecast thunderstorms?
1000 mb
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Thermodynamics M. D. Eastin
Outline:
Review of The First Law of Thermodynamics Adiabatic Processes Poisson’s Relation
Applications Potential Temperature
Applications Dry Adiabatic Lapse Rate
Applications
Adiabatic Processes
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Thermodynamics M. D. Eastin
First Law of Thermodynamics
pdα dTcdq v
Statement of Energy Balance / Conservation:
• Energy in = Energy out• Heat in = Heat out
HeatingSensible heating Latent heating
Evaporational cooling Radiational heating Radiational cooling
Change in Internal Energy
Work DoneExpansion
Compression
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Thermodynamics M. D. Eastin
Forms of the First Law of Thermodynamics For a gas of mass m Per unit mass
dW dUdQ pdV dUdQ
pdVdT mCdQ v
Vdp dTCmdQ p
dw du dq pd du dq
pd dTcdq v
dp dTcdq p
where: p = pressure U = internal energy n = number of molesV = volume W = work α = specific volumeT = temperature Q = heat energy m = mass
Cv = specific heat at constant volume (717 J kg-1 K-1)Cp = specific heat at constant pressure (1004 J kg-1 K-1)Rd = gas constant for dry air (287 J kg-1 K-1)R* = universal gas constant (8.3143 J K-1 mol-1)
nRCC *vp Rcc dvp
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Thermodynamics M. D. Eastin
Types of ProcessesIsothermal Processes:
• Transformations at constant temperature (dT = 0)
Isochoric Processes:
• Transformations at constant volume (dV = 0 or dα = 0)
Isobaric Processes:
• Transformations at constant pressure (dp = 0)
Adiabatic processes:
• Transformations without the exchange of heat between the environment and the system (dQ = 0 or dq = 0)
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Thermodynamics M. D. Eastin
Adiabatic ProcessesBasic Idea:
• No heat is added to or taken from the system which we assume to be an air parcel
• Changes in temperature result from either expansion or contraction
• Many atmospheric processes are “dry adiabatic”• We shall see that dry adiabatic process play
a large role in deep convective processes
• Vertical motions• Thermals
Parcel0pdα dTcdq v
0dp dTcdq p
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Thermodynamics M. D. Eastin
Adiabatic ProcessesP-V Diagrams:
p
V
f
i
Isotherm
Isobar
Adiabat
Isochor
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Thermodynamics M. D. Eastin
Poisson’s* RelationA Relationship between Temperature and Pressure:
• Begin with:
• Substitute for “α” using the Ideal Gas Law and rearrange:
• Integrate the equation:
* NOT pronounced like “Poison”
dp dTcp
See: http://en.wikipedia.org/wiki/Simeon_Poisson
TRpα d
pdp
cR
TdT
p
d
final
intital
final
initial
p
pp
dT
T pdp
cR
TdT
Adiabatic Form of the First Law
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Thermodynamics M. D. Eastin
Poisson’s RelationA Relationship between Pressure and Temperature:
• After Integrating the equation:
• After some simple algebra:
• Relates the initial conditions of temperature and pressure to the final temperature and pressure
initial
final
p
d
initial
final
pp
cR
TT
lnln
pd
cR
initial
final
initial
final
pp
TT
pd
cR
initial
finalinitialfinal p
pT T
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Thermodynamics M. D. Eastin
Applications of Poisson’s Relation
Example: Cabin Pressurization
• Most jet aircraft are pressurized to 8,000 ft (or 770 mb). If the outside air temperature at a cruising altitude of 30,000 feet (300 mb) is -40ºC, what is the temperature inside the cabin?
pd
cR
initial
finalinitialfinal p
pT T
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Thermodynamics M. D. Eastin
Applications of Poisson’s Relation
Example: Cabin Pressurization
• Most jet aircraft are pressurized to 8,000 ft (or 770 mb). If the outside air temperature at a cruising altitude of 30,000 feet (300 mb) is -40ºC, what is the temperature inside the cabin?
pinitial = 300 mb Rd = 287 J / kg Kpfinal = 770 mb cp = 1004 J / kg K
Tinitial = -40ºC = 233KTfinal = ???
pd
cR
initial
finalinitialfinal p
pT T
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Thermodynamics M. D. Eastin
Applications of Poisson’s RelationExample: Cabin Pressurization
• Most jet aircraft are pressurized to 8,000 ft (or 770 mb). If the outside air temperature at cruising altitude of 30,000 feet (300 mb) is -40ºC, what is the temperature inside the cabin?
pinitial = 300 mb Rd = 287 J / kg Kpfinal = 770 mb cp = 1004 J / kg K
Tinitial = -40ºC = 233K1004
287
final 300mb770mbK 233 T
K 305 T final
C32 T final
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Thermodynamics M. D. Eastin
Comparing Temperatures at different Altitudes:
Are they relatively warmer or cooler?
• Bring the two parcels to the same level• Compress 300 mb air to 600 mb
-37oC300 mb
2oC600 mb
Applications of Poisson’s Relation
pd
cR
initial
finalinitialfinal p
pT T
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Thermodynamics M. D. Eastin
Comparing Temperatures at different Altitudes:
Are they relatively warmer or cooler?
pinitial = 300 mbpfinal = 600 mbTinitial = -37ºC = 236 K
Tfinal = 288 K = 15ºC
Note: We could we have chosen to expand the 600 mb parcel to 300 mb for the comparison
-37oC300 mb
2oC600 mb
Applications of Poisson’s Relation
pd
cR
initial
finalinitialfinal p
pT T
15oC
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Thermodynamics M. D. Eastin
Potential TemperatureSpecial form of Poisson’s Relation:
Compress all air parcels to 1000 mb• Provides a “standard”• Avoids using an arbitrary pressure level
• Define Tfinal = θ• θ is the potential temperature
where: p0 = 1000 mb
1000 mb
pd
cR
initialinitial p
1000mbT θ
pd
cR
0
ppT θ
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Thermodynamics M. D. Eastin
Comparing Temperatures at different Altitudes:
An aircraft flies over the same location at two different altitudes and makes measurements of pressure and temperature within air parcels at each altitude:
Air parcel #1: p = 900 mbT = 21ºC
Air Parcel #2: p = 700 mbT = 0.6ºC
Which parcel is relatively colder? warmer?
Applications of Potential Temperature
pd
cR
0
ppT θ
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Thermodynamics M. D. Eastin
Comparing Temperatures at different Altitudes:
Air Parcel #1: p = 900 mbT = 21ºC = 294 K
Air Parcel #2: p = 700 mbT = 0.6ºC = 273.6 K
The parcels have the same potential temperature! Are we measuring the same air parcel at two different levels? MAYBE
Applications of Potential Temperature
286.0
900mb1000mb294K θ
K303 θ
286.0
700mb1000mb273.6K θ
K303 θ
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Thermodynamics M. D. Eastin
Potential Temperature Conservation:
• Air parcels undergoing adiabatic transformations maintain a constant potential temperature (θ)
• During adiabatic ascent (expansion) the parcel’s temperature must decrease in order to preserve the parcel’s potential temperature • During adiabatic descent (compression) the parcel’s temperature must increase in order to preserve the parcel’s potential temperature
Applications of Potential Temperature
Constant θ
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Thermodynamics M. D. Eastin
Potential Temperature as an Air Parcel Tracer:
• Therefore, under dry adiabatic conditions, potential temperature can be used as a tracer of air motions
• Track air parcels moving up and down (thermals)• Track air parcels moving horizontally (advection)
Applications of Potential Temperature
Con
stan
t θ
Constant θ
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Thermodynamics M. D. Eastin
How does Temperature change with Height for a Rising Thermal?• Potential temperature is a function of pressure and temperature: θ(p,T)• We know the relationship between pressure (p) and altitude (z):
• We can use this hydrostatic relation and the adiabatic form of the first law to obtain a relationship between temperature and height when potential temperature is conserved (dry adiabatic lapse rate)
Dry Adiabatic Lapse Rate
gdzdp
HydrostaticRelation
dp dTcp Adiabatic Form of the First Law T
zDry Adiabatic Lapse Rate?
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Thermodynamics M. D. Eastin
How does Temperature change with Height for a Rising Thermal?• Begin with the first law:
• Substitute for “α” using the Ideal Gas Law and rearrange:
• Divide each side by “dz”:
• Substitute for “dp/dz” using the hydrostatic relation and re-arrange:
Dry Adiabatic Lapse Rate
gdzdp
dp dTcp
pdp
cR
TdT
p
d
dzdp
p1
cR
dzdT
T1
p
d
p
d
cg
pTR
dzdT
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Thermodynamics M. D. Eastin
How does Temperature change with Height for a Rising Thermal?• Substitute for “ρ” using the Ideal Gas Law and cancel terms:
• We have arrived at the Dry Adiabatic Lapse Rate (Γd):
Dry Adiabatic Lapse Rate
TRp dp
d
cg
pTR
dzdT
pcg
dzdT
kmCcg
dzdT
pd /8.9
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Thermodynamics M. D. Eastin
Example: Temperature Change within a Rising Thermal• A parcel originating at the surface (z = 0 m, T = 25ºC) rises to the top of the mixed boundary layer (z = 800 m). What is the parcel’s new air temperature?
Application of the Dry Adiabatic Lapse Rate
Constant θ
Mixed Layer
kmC dzdT /8.9 initialfinal Tdz kmC( T )/8.9
258.0*8.9 T final
C17.2 T final
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Thermodynamics M. D. Eastin
Summary:
• Review of The First Law of Thermodynamics• Adiabatic Processes• Poisson’s Relation
• Applications• Potential Temperature
• Applications• Dry Adiabatic Lapse Rate
• Applications
Adiabatic Processes
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Thermodynamics M. D. Eastin
ReferencesPetty, G. W., 2008: A First Course in Atmospheric Thermodynamics, Sundog Publishing, 336 pp.
Tsonis, A. A., 2007: An Introduction to Atmospheric Thermodynamics, Cambridge Press, 197 pp. Wallace, J. M., and P. V. Hobbs, 1977: Atmospheric Science: An Introductory Survey, Academic Press, New York, 467 pp.