thermodynamics review/tutorial - ideal gas law - heat capacity

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PHYS-575/CSI-655PHYS-575/CSI-655Introduction to Atmospheric Physics and ChemistryIntroduction to Atmospheric Physics and Chemistry

Atmospheric Thermodynamics – Part 2Atmospheric Thermodynamics – Part 2

1. Thermodynamics Review/Tutorial - Ideal Gas Law - Heat Capacity - 1st & 2nd Laws of Thermodynamics - Adiabatic Processes - Energy Transport2. Hydrostatic Equilibrium3. Adiabatic Lapse Rate – DRY4. Adiabatic Lapse Rate - WET5. Static Stability6. SLT and the Atmosphere

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Term Paper TopicsTerm Paper TopicsPHYS-575PHYS-575

Aaron Arthur: Processes in Pluto’s AtmosphereAaron Arthur: Processes in Pluto’s Atmosphere

Heather Bloemhard: Upward directed lightning Heather Bloemhard: Upward directed lightning

Miguel Cervoni: Ice Ages, Global Warming and Future PredictionsMiguel Cervoni: Ice Ages, Global Warming and Future Predictions

Lisa Horne: TornadoesLisa Horne: Tornadoes

Mahmoud Lababidi: Aurora on JupiterMahmoud Lababidi: Aurora on Jupiter

David Voglozin: Ionosphere/Global Carbon BuildupDavid Voglozin: Ionosphere/Global Carbon Buildup

CSI-655CSI-655

Qunying Huang: Greenhouse Models and Long Term PredictionsQunying Huang: Greenhouse Models and Long Term Predictions

Tracey Jenkins: Exotic Solutions to Global Climate ModelsTracey Jenkins: Exotic Solutions to Global Climate Models

Jing Li: La Nina and Extreme WeatherJing Li: La Nina and Extreme Weather

Wenwen Li: Hurricane PredictionWenwen Li: Hurricane Prediction

Heather Miller: Comparisons of Atmospheric Transmission ModelsHeather Miller: Comparisons of Atmospheric Transmission Models

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Announcements: Feb. 25, 2008Announcements: Feb. 25, 2008 Office Hours (with appointment) Monday: 3:30-5:00 pm; Wednesday: 3:00-5:00pm (other times possible)

Homework #2: Due Today Homework #3: Due March 10 Problem 3.18 (a) though (j) Problems 3.19, 3.29, 3.30 (show all steps), 3.39

Feb. 18 – Tentative Term Paper Titles due March 17 – Exam #1 March 17 – Term Paper Abstract due

Instructor Travel (still somewhat tentative):Feb. 28-29 March 13-20April 15-16

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Role of Water in the AtmosphereRole of Water in the Atmosphere

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What is a Storm?What is a Storm?

http://www.noaanews.noaa.gov/stories2005/images/ivan091504-1515zb.jpg

1. Do all storms have the same cause?

2. Do all storms have the same ending?

3. Are there aspects that all storms have in common?

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How Long Can a Storm Last?How Long Can a Storm Last?

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Role of Water Vapor in Atmospheric Role of Water Vapor in Atmospheric ThermodynamicsThermodynamics

http://www.auf.asn.u/metimages/lapseprofile.gif

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4. Adiabatic Lapse Rate - Wet4. Adiabatic Lapse Rate - Wet

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Water Vapor in the Atmosphere:Water Vapor in the Atmosphere:The Wet (Moist) Adiabatic Lapse RateThe Wet (Moist) Adiabatic Lapse Rate

The Wet Adiabatic Lapse Rate is smaller than the DALR, because theeffective heat capacity of a wet atmosphere is larger than that of a dryatmosphere. The phase change of water is a heat reservoir.

Γd = -g/Cp = DALR

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Saturation ConditionsSaturation Conditions

At saturation, the flux of water moleculesinto and out of the atmosphere is equal.

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Moisture ParametersMoisture Parameters

d

v

m

mes = Saturation Partial Pressure

w = Mass Mixing Ratios

Where mv is the mass of water vapor in a given parcel, and md is the mass of dry air of the same parcel. This is usually expressed as gramsof water per kilogram of dry air. w typically varies from 1 to 20 g/kg.

Specific Humidity (typically a few %)

The amount of water vapor in the atmosphere may be expressed ina variety of ways, and depending upon the problem under consideration,some ways of quantifying water are more useful than others.

w

w

mm

mq

dv

v

1

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Moisture Parameters for SaturationMoisture Parameters for Saturation

d

vs

m

m

p

e

ep

ew s

s

ss 622.0622.0

)(

)(/

)('

'

TR

ep

TR

ew

d

s

v

s

d

vss

ws = Saturation Mixing Ratio

es = Saturation Partial Pressure

ρ’vs is the mass density of water

required to saturate air at a given T.p = total pressure

For Earth’s Atmosphere:

ss e

e

w

wRH 100100

Relative Humidity

The dew point, Td, is the temperature to which air must be cooled at constantpressure for it to become saturated with pure water.

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Lifting Condensation LevelLifting Condensation Level

The Lifting Condensation Level (LCL) is defined as the level to which anunsaturated (but moist) parcel of air can be lifted adiabatically before itbecomes saturated with pure water.

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Saturation of AirSaturation of Air

Air is Saturated if the abundance of water vapor (or any condensable) isat its maximum Vapor Partial Pressure.

In saturated air, evaporation is balanced by condensation. If water vapor is added to saturated air, droplets begin to condense and fall out.

Under equilibrium conditions at a fixed temperature, the maximum vapor partial pressure of water is given by its Saturated Vapor Pressure Curve.

Relative Humidity is the ratio of themeasured partial pressure of vaporto that in saturated air, multiplied by 100.

The relative humidity in clouds is typically about 102-107%, in other words, the clouds are Supersaturated.

http://apollo.lsc.vsc.edu/classes/met130/notes/chapter5/graphics/sat_vap_press.free.gif

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Saturation Vapor PressureSaturation Vapor Pressure::Clausius-Clapeyron Equation of StateClausius-Clapeyron Equation of State

Psv(T) = CL e-Ls/RT

Psv(T) = Saturation vapor pressure at temperature TCL = constant (depends upon condensable)Ls = Latent HeatR = Gas constant

Phase Diagram of Water

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Vertical Motion and CondensationVertical Motion and Condensation

Upward motion leads to cooling, via the FLT. Cooling increases therelative humidity. When the relative humidity exceeds 100%, thencondensation can occur.

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Adiabatic Motion of Moist ParcelAdiabatic Motion of Moist Parcel

As a parcel of air moves upwards, it expands and cools. The cooling leadsto an increase in the relative humidity. When the vapor pressure exceeds the saturation vapor pressure, then condensation can occurs.

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Saturation Profile and TemperatureSaturation Profile and Temperature

Amounts of water necessary for supersaturation, and thuscondensation.

Is it possible to have snow when the atmospheric temperature is below – 30oC?

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Water/Ice TransitionWater/Ice Transition

The saturation vapor pressure of water over ice is higher than thatover liquid water. This leads to small, but measurable change isthe relative humidity.

Water Triple Point

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Liquid/Ice TransitionLiquid/Ice Transition

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Wet (Moist) Adiabatic Lapse RateWet (Moist) Adiabatic Lapse Rate

Γd = -g/Cp = Dry Adiabatic Lapse Rate

In determining the moist adiabatic lapse rate, we must modify the first law ofthermodynamics to include the phasechange energy.

Let μs = mass of liquid water.

dQ = CpdT + gdz (FLT for a parcel)

dQ = – Lsdμs (Heat added from water condensation)

Here we assume that the water which condenses drops out of the parcel. Thusthis process is strictly irreversible.

Together this implies that the FLT becomes: CpdT + gdz + Lsdμs = 0

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Wet Lapse Rate - continuedWet Lapse Rate - continued

CpdT + gdz + Lsdμs = 0 (FLT for a saturated parcel)

The mass of water depends upon the degree of saturation:

μs = Є (es/p) and by the chain rule dμs/μs = des/es – dp/p

des = (des/dT) dT

(1/es) des/dT = Ls/RT2 (Differential form of Clausius-Clapeyron Eqn.)

dP = -gdz/RT (Hydrostatic Law)

This gives us dμs/μs = LsdT/RT2 + gdz/RT

Using this equation and the FLT form at the top of this page we get:

(Cp + Ls2μs/RT2) dT + g(1+Lsμs/RT) dz = 0

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Wet Lapse Rate - ContinuedWet Lapse Rate - Continued

ΓΓww = dT/dz = -(g/C = dT/dz = -(g/Cpp) ) ((1+L((1+Lssμμss/RT) / (1 + L/RT) / (1 + Lss22μμss/C/CppRTRT22)) ))

Note that when μs = 0, this reduces to Γd

The factor ((xx)) is always less or equal to1. So Γd < Γw

Thus, water acts as anagent to increase theeffective heat capacityof the atmosphere.

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Archimedes Principle:Archimedes Principle: The The upward forceupward force (buoyancy) (buoyancy) is equal to the weight of the is equal to the weight of the displaced air.displaced air.

The The net forcenet force on a parcel is on a parcel is equal to the difference equal to the difference between weight of the air in between weight of the air in the parcel and the weight of the parcel and the weight of the displaced air.the displaced air.

5. Static Stability

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Vertical StabilityVertical Stability

dT/dz = -g/Cp = dry adiabatic lapse rate (neutrally stable)

dT/dz < -g/Cp Unstable

dT/dz > -g/Cp Stable

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Static StabilityStatic Stability

Γd = -g/Cp

Stable Unstable

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Stability and the Effects of CondensationStability and the Effects of Condensation

Moisture leads to conditional stability in the atmosphere.

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Analogs for StabilityAnalogs for Stability

Under stable atmospheric conditions, an air parcel that is displaced in the vertical direction will return to its original position.

Neutral stability occurs when the air parcel will remain at it’s displaced position without any additional forces acting on it.

For unstable conditions, an air parcel that is displaced in the vertical will continue to move in the direction of the displacement.

Conditional instability occurs when a significant displacement of the air parcel must occur before instability can occur.

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Regions of Convective InstabilityRegions of Convective Instability

Convective instability may occur in only a small portion of the vertical structure.Temperature inversions may inhibit convection.

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Atmospheric Oscillations: Atmospheric Oscillations: Gravity Waves in Stable AirGravity Waves in Stable Air

gF )( '

gdt

zd)( '

2

2' g

F

dt

zd

'

'

'2

2

g

T

TTdt

zd

'

'

2

2

1

11

T

TTg

dt

zd '

2

2

Consider the force on a parcel of air that has been displaced verticallyby a distance z’ from its equilibrium altitude. Assume that the air is dryand that displacements occur sufficiently slow that we can assume thatthey are adiabatic. Primed quantities will denote parcel variables.

By Archimedes Principle, the force on the parcel is the buoyancy forceminus the gravitational force. The net force is:

Acceleration: OR

Substituting from IGL: OR

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Atmospheric Oscillations - continuedAtmospheric Oscillations - continued

'0 zTT

'' zTT do

'' zTT d

If we assume a linear atmospheric temperature profile with rate of changewith altitude of Г, then the temperature profile may be written

The parcel moves adiabatically in the vertical, so its temperature is:

The equation of motion becomes: '2

'2

zT

g

dt

zdd

0'22

'2

zNdt

zd

dT

gN 2

Which gives:

Which can be written:

Brunt-Väisälä Frequency:

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Atmospheric Oscillations - continuedAtmospheric Oscillations - continued

The equation of motion for the parcel is 0'22

'2

zNdt

zd

dT

gN 2

Brunt-Väisälä Frequency:

If the air is stably stratified, i.e., Гd > Г,then the parcel will oscillate about itsstarting position with simple harmonic motion.

These are called buoyancy oscillations. Typical periods are about 15 minutes.

For winds of ~ 20 ms-1, the wavelength is ~10-20 km.

Here Гe = Г in notes

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Mountain (Lee) WavesMountain (Lee) Waves

Lee Waves

Observed from ground

dT

gN 2Buoyancy Oscillations:

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Lee Waves Observed from SpaceLee Waves Observed from Space

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Mountain WindsMountain Winds

Mountain regions display many interesting weather patterns. One example is the valley wind which originates on south-facing slopes (north-facing in the southern hemisphere). When the slopes and the neighboring air are heated the density of the air decreases, and the air ascends towards the top following the surface of the slope. At night the wind direction is reversed, and turns into a down-slope wind. If the valley floor is sloped, the air may move down or up the valley, as a canyon wind. Winds flowing down the leeward sides of mountains can be quite powerful:

Examples are the Foehn in the Alps in Europe, the Chinook in the Rocky Mountains, and the Zonda in the Andes. Examples of other local wind systems are the Mistral flowing down the Rhone valley into the Mediterranean Sea, the Scirocco, a southerly wind from Sahara blowing into the Mediterranean sea.

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Mountain Winds and ClimateMountain Winds and Climate

Hawaii

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Implications of the Second LawImplications of the Second Law

It is impossible for any process (engine), working in a It is impossible for any process (engine), working in a cycle, to completely convert surrounding heat to work.cycle, to completely convert surrounding heat to work.

Dissipation will always occur.Dissipation will always occur. Entropy will always increase.Entropy will always increase.

The Second Law of Thermodynamics states that it is impossible to completely convert heat energy into mechanical energy. Another way to put that is to say that the level of entropy (or tendency toward randomness) in a closed system is always either constant or increasing.

6. The Second Law of Thermodynamics

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Second Law of Thermodynamics Second Law of Thermodynamics and Atmospheric Processesand Atmospheric Processes

Entropy is the heat added (or subtracted) to a system divided by its temperature.

The Entropy of an isolated system increases when the system undergoes a spontaneous change.

Second Law of Thermodynamics

dS = dQ/T

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The Carnot CycleThe Carnot CycleThe First Law of Thermodynamics is a statement about conservation of energy.

The Second Law of Thermodynamics is concerned with the maximum fraction of a quantity of heat that can be converted into work. There is a theoretical limit to this conversion that was first demonstrated by Nicholas Carnot.

A cyclic process is a series of operations by which the state of a substance(called the working substance) changes, but is finally returned to its original state (in all respects).

If the volume changes during the cycle, then work is done (dW = PdV).

The net heat that is absorbed by the working substance is equal to the workdone in the cycle. If during one cycle a quantity of heat Q1 is absorbed anda quantity Q2 is rejected, then the net work done is Q1 – Q2.

The efficiency is: abosrbedHeat

enginebydoneWork

Q

QQ

_

___

1

21

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Carnot’s Ideal Heat EngineCarnot’s Ideal Heat Engine

1. AB Adiabatic CompressionWork done on substance

2. B C Isothermal Expansion Work done on environment3. C D Adiabatic Expansion

Work done on environment4. D A Isothermal Compression Work done on substance

Incremental work done: dW = PdVSo the area enclosed on theP-V diagram is the total Work.

T1>T2

Only by transferring heat from ahot to a cold body can work bedone in a cyclic process.

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Isotherms and AdiabatsIsotherms and Adiabats

Isothermal Process: T = constant, dT = 0

Adiabatic: dQ = 0

Isentropic: dS = 0

P-V diagram

T-S diagram

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Saturation Vapor Pressure:Saturation Vapor Pressure:The Clausius-Clapeyron EquationThe Clausius-Clapeyron Equation

2*2 1000

1

TR

ML

TR

L

dT

de

ewv

v

vs

s

By application of the ideas of acyclic process changing waterfrom a liquid to a gas, we can derive the differential form ofthe Clausius-Clapeyron equation:

In its integrated form:

RTLs

vCeTe /)(

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Water Vapor and the Carnot CycleWater Vapor and the Carnot Cycle

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Ambient Pressure and Boiling PointAmbient Pressure and Boiling PointWater boils at a temperature TB

such that the water vapor pressureat that temperature is equal to theambient air pressure, i.e.,

es(TB) = Patmos

The change in boiling point, TB,as a function of temperature isgiven by a form of the Clausius-Clapeyon equation:

v

B

atmos

B

L

T

p

T )( 12

Because α2 > α1, TB increases with increasing patmos. Thus if the ambient atmospheric pressure is less than sea level, the TB will be lower.

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Generalized Statement of the Second Law of Generalized Statement of the Second Law of ThermodynamicsThermodynamics

For a reversible transformation there is no change in the entropy of the universe (system + surroundings).

The entropy of the universe increases as a result of irreversible transformations.

If the system is reversible, no dissipation occurs.

T

QS

“The Second Law of Thermodynamics cannot be proved. It is believed becauseit leads to deductions that are in accord with observations and experience.”

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Questions for DiscussionQuestions for Discussion

1.1. How does one define energy, apart from what How does one define energy, apart from what it does or is capable of doing?it does or is capable of doing?

2.2. What is Thermodynamics?What is Thermodynamics?

3.3. Why is Thermodynamics relevant to Why is Thermodynamics relevant to atmospheric science?atmospheric science?

4.4. Why is Thermodynamics a good starting point Why is Thermodynamics a good starting point for discussing atmospheric science?for discussing atmospheric science?

5.5. What causes energy transport?What causes energy transport?

6.6. Is it possible to perform work with an Is it possible to perform work with an isothermal system?isothermal system?

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Questions for DiscussionQuestions for Discussion

7.7. Why is entropy an important concept in Why is entropy an important concept in atmospheric physics?atmospheric physics?

8.8. Does an atmospheric “parcel” really exist? Does an atmospheric “parcel” really exist? 9.9. Is the atmosphere in thermal equilibrium? Is the atmosphere in thermal equilibrium? 10.10. Is the atmosphere in dynamical equilibrium?Is the atmosphere in dynamical equilibrium?11.11. What is the difference between steady state and What is the difference between steady state and

equilibrium?equilibrium?12.12. In what ways are the Earth’s atmosphere like a In what ways are the Earth’s atmosphere like a

heat engine?heat engine?13.13. Why is it impossible to prove the Second Law of Why is it impossible to prove the Second Law of

Thermodynamics?Thermodynamics?

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Pseudoadiabatic Pseudoadiabatic ChartChart

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Normand’s RuleNormand’s Rule