ThermodynamicsM. D. Eastin Adiabatic Processes How can the first law really help me forecast...

Thermodynamics M. D. Eastin

Adiabatic Processes

How can the first law really help me forecast thunderstorms?

1000 mb1000 mb


Outline:

Review of The First Law of Thermodynamics Adiabatic Processes Poisson’s Relation

Applications Potential Temperature

Applications Dry Adiabatic Lapse Rate

Applications

Adiabatic Processes


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


Forms of the First Law of Thermodynamics

For a gas of mass m For unit mass

dW dUdQ pdV dUdQ

pdV dTcdQ v

Vdp dTcdQ p

dw du dq pd du dq

pd dTcdq v

dp dTcdq p

where: p = pressure U = internal energyV = volume W = workT = temperature Q or q = heat energyα = specific volume 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)

nRcc *vp Rcc dvp


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)


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


Adiabatic ProcessesP-V Diagrams:

p

V

f

i

Isotherm

Isobar

Adiabat

Isochor


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

p

dp

c

R

T

dT

p

d

final

intital

final

initial

p

pp

d

T

T p

dp

c

R

T

dT

Adiabatic Form of the First Law


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

p

pln

c

R

T

Tln

p

dc

R

initial

final

initial

final

p

p

T

T

p

dc

R

initial

finalinitialfinal p

pT T


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?

p

dc

R

initial

finalinitialfinal p

pT T



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 = ???

p

dc

R

initial

finalinitialfinal p

pT T


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 300mb

770mbK 233 T

K 305 Tfinal

C32 Tfinal


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

-37-37ooCC300 mb300 mb

22ooCC600 mb600 mb


p

dc

R

initial

finalinitialfinal p

pT T



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

-37-37ooCC300 mb300 mb

22ooCC600 mb600 mb


p

dc

R

initial

finalinitialfinal p

pT T

1515ooCC


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 mb1000 mb

p

dc

R

initialinitial p

1000mbT θ

p

dc

R

0

p

pT θ



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

p

dc

R

0

p

pT θ



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?


286.0

900mb

1000mb294K θ

K303 θ

286.0

700mb

1000mb273.6K θ

K303 θ


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


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)


Co

ns

tan

t θ

Constant θ


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

dz

dp g

HydrostaticRelation

(more on this later)

dp dTcp Adiabatic Form of the First Law

TT

zzDry Adiabatic Dry Adiabatic Lapse Rate?Lapse Rate?



• 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:


dz

dp g

dp dTcp

p

dp

c

R

T

dT

p

d

dz

dp

p

1

c

R

dz

dT

T

1

p

d

p

d

c

g

p

TR

dz

dT



• Substitute for “ρ” using the Ideal Gas Law and cancel terms:

• We have arrived at the Dry Adiabatic Lapse Rate (Γd):


TRp dp

d

c

g

p

TR

dz

dT

pc

g

dz

dT

kmCd /8.9c

g

dz

dT

p


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 /8.9 dz

dT initialfinal T dz)/8.9( T kmC

258.0*8.9 Tfinal

C17.2 Tfinal


Summary:

• Review of The First Law of Thermodynamics• Adiabatic Processes• Poisson’s Relation

• Applications• Potential Temperature

• Applications• Dry Adiabatic Lapse Rate

• Applications

Adiabatic Processes


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.

ThermodynamicsM. D. Eastin Adiabatic Processes How can the first law really help me forecast...

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