```Adiabatic Processes
1000 mb
How can the first law really help me forecast thunderstorms?
Thermodynamics
M. D. Eastin
Outline:
 Review of The First Law of Thermodynamics
 Poisson’s Relation
 Applications
 Potential Temperature
 Applications
 Applications
Thermodynamics
M. D. Eastin
First Law of Thermodynamics
Statement of Energy Balance / Conservation:
• Energy in = Energy out
• Heat in = Heat out
dq  cvdT  pdα
Heating
Sensible heating
Latent heating
Evaporational cooling
Thermodynamics
Change in
Internal Energy
Work Done
Expansion
Compression
M. D. Eastin
Forms of the First Law of Thermodynamics
For a gas of mass m
For unit mass
dQ  dU  dW
dq  du  dw
dQ  dU  pdV
dq  du  pd
dQ  cvdT  pdV
dq  cvdT  pd
dQ  cpdT  Vdp
dq  cpdT  dp
cp  cv  nR*
cp  c v  R d
where:
p = pressure
V = volume
T = temperature
α = specific volume
U = internal energy
W = work
Q or q = heat energy
n = number of moles
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)
Thermodynamics
M. D. Eastin
Types of Processes
Isothermal 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)
• Transformations without the exchange of heat between the environment
and the system (dQ = 0 or dq = 0)
Thermodynamics
M. D. Eastin
Basic Idea:
• No heat is added to or taken from the system
which we assume to be an air parcel
dq  cvdT  pdα  0
Parcel
dq  cpdT  dp  0
• 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
Thermodynamics
M. D. Eastin
P-V Diagrams:
Isobar
p
i
Isochor
Isotherm
f
V
Thermodynamics
M. D. Eastin
Poisson’s* Relation
A Relationship between Temperature and Pressure:
• Begin with:
cpdT  dp
• Substitute for “α” using
the Ideal Gas Law
and rearrange:
• Integrate the equation:
pα  R d T
dT R d dp

T
cp p
Tfinal

Tinitial
* NOT pronounced like “Poison”
Thermodynamics
of the First Law
Rd
dT

T
cp
p final

p intital
dp
p
See: http://en.wikipedia.org/wiki/Simeon_Poisson
M. D. Eastin
Poisson’s Relation
A Relationship between Pressure and Temperature:
• After Integrating the equation:
Tfinal
Rd
pfinal
ln

ln
Tinitial
cp
pinitial
• After some simple algebra:
 p final 
Tfinal

 
Tinitial
 pinitial 
Rd
cp
Tfinal
 p final 

 Tinitial
 pinitial 
Rd
cp
• Relates the initial conditions of temperature and pressure to
the final temperature and pressure
Thermodynamics
M. D. Eastin
Applications of Poisson’s Relation
Tfinal
 p final 

 Tinitial
 pinitial 
Rd
cp
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?
Thermodynamics
M. D. Eastin
Applications of Poisson’s Relation
Tfinal
 p final 

 Tinitial
 pinitial 
Rd
cp
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
pfinal = 770 mb
Rd = 287 J / kg K
cp = 1004 J / kg K
Tinitial = -40ºC = 233K
Tfinal = ???
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 cruising altitude of 30,000 feet (300 mb) is -40ºC, what is
the temperature inside the cabin?
pinitial = 300 mb
pfinal = 770 mb
Rd = 287 J / kg K
cp = 1004 J / kg K
Tinitial = -40ºC = 233K
Tfinal
 770mb
 233K

 300mb
287
1004
Tfinal  305K
Tfinal  32C
Thermodynamics
M. D. Eastin
Applications of Poisson’s Relation
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
300 mb
Tfinal
 p final 

 Tinitial
 pinitial 
Thermodynamics
Rd
cp
600 mb
-37oC
2oC
M. D. Eastin
Applications of Poisson’s Relation
Comparing Temperatures at different Altitudes:
Are they relatively warmer or cooler?
Tfinal
 p final 

 Tinitial
 pinitial 
Rd
cp
pinitial = 300 mb
pfinal = 600 mb
Tinitial = -37ºC = 236 K
Tfinal = 288 K = 15ºC
300 mb
-37oC
600 mb
2oC 15oC
Note: We could we have chosen
to expand the 600 mb parcel
to 300 mb for the comparison
Thermodynamics
M. D. Eastin
Potential Temperature
Special 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
 1000mb

θ  Tinitial
 pinitial 
 p0 
θ  T 
 p
Rd
Rd
cp
cp
1000 mb
where: p0 = 1000 mb
Thermodynamics
M. D. Eastin
Applications of Potential Temperature
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:
Air Parcel #2:
p = 900 mb
T = 21ºC
p = 700 mb
T = 0.6ºC
 p0 
θ  T 
 p
Rd
cp
Which parcel is relatively colder? warmer?
Thermodynamics
M. D. Eastin
Applications of Potential Temperature
Comparing Temperatures at different Altitudes:
Air Parcel #1:
p = 900 mb
T = 21ºC = 294 K
 1000mb
θ  294K

 900mb 
0.286
θ  303K
Air Parcel #2:
p = 700 mb
T = 0.6ºC = 273.6 K
 1000mb
θ  273.6K

 700mb 
0.286
θ  303K
The parcels have the same potential temperature!
Are we measuring the same air parcel at two different levels?
Thermodynamics
M. D. Eastin
Applications of Potential Temperature
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
Constant θ
Thermodynamics
M. D. Eastin
Applications of Potential Temperature
Potential Temperature as an Air Parcel Tracer:
• Therefore, under dry adiabatic conditions, potential
temperature can be used as a tracer of air motions
Constant θ
Thermodynamics
Constant θ
• Track air parcels moving up and down (thermals)
• Track air parcels moving horizontally (advection)
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):
dp
  g
dz
Hydrostatic
Relation
(more on this later)
• 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
cpdT  dp
Thermodynamics
of the First Law
z
Lapse Rate?
T
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:
Thermodynamics
cpdT  dp
dT R d dp

T
cp p
1 dT R d 1 dp

T dz c p p dz
dp
  g
dz
 T Rd g
dT

dz
p cp
M. D. Eastin
How does Temperature change with Height for a Rising Thermal?
• Substitute for “ρ” using
the Ideal Gas Law
and cancel terms:
 T Rd g
dT

dz
p cp
p   R dT
dT
g

dz
cp
• We have arrived at the Dry Adiabatic Lapse Rate (Γd):
dT
g
d 
 
  9.8C / km
dz
cp
Thermodynamics
M. D. Eastin
Application of the Dry Adiabatic Lapse Rate
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?
dT
  9.8C / km
dz
Tfinal  (9.8C / km) dz  Tinitial
Tfinal   9.8 * 0.8  25
Tfinal  17.2C
Mixed Layer
Constant θ
Thermodynamics
M. D. Eastin
Summary:
• Review of The First Law of Thermodynamics
• Poisson’s Relation
• Applications
• Potential Temperature
• Applications
• Applications
Thermodynamics
M. D. Eastin
References
Petty, 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.
Thermodynamics
M. D. Eastin
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