Aerodynamics for Airplanes - Alan Robock

Aerodynamics for Airplanes - Alan Robock

Second Law of Thermodynamics 1 2 T1 T2 Prof. Fred Remer University of North Dakota Hess Chapter 3 pp 32 37 Tsonis Chapter 5

pp 49 70 Bohren & Albrecht Chapter 4 pp 135-180 Wallace and Hobbs Chapter 3 pp 93-97 Prof. Fred Remer University of North Dakota

Reading Objectives Be able to state the Second Law of Thermodynamics in its various forms Be able to define heat engine Be able to draw the Carnot cycle and describe the changes in state variables, heat and work throughout the cycle Prof. Fred Remer University of North Dakota Objectives Be able to determine whether work is done by or on a system through a Carnot cycle Be able to define cyclic process Be able to define the term reversible process

Prof. Fred Remer University of North Dakota Objectives Be able to describe and give examples of natural, reversible, and impossible processes Be able to calculate the change of entropy of a system Prof. Fred Remer University of North Dakota First Law of Thermodynamics Energy cannot be created or destroyed. It can only be changed from one form into another. Rudolf Clausius 1850 Prof. Fred Remer

University of North Dakota First Law of Thermodynamics Conservation of Energy Says Nothing About Direction of Energy Transfer Prof. Fred Remer University of North Dakota Second Law of Thermodynamics Preferred (or Natural) Direction of Energy Transfer Spontaneous Process Hot dQ Cold Prof. Fred Remer University of North Dakota

Second Law of Thermodynamics Un-Natural Process Does not occur spontaneously Requires energy to transfer heat in opposite direction Hot dQ Cold Prof. Fred Remer University of North Dakota Second Law of Thermodynamics Heat passes from a warmer to colder body. Rudolf Clausius 1850 Prof. Fred Remer University of North Dakota

Second Law of Thermodynamics Duh.... Prof. Fred Remer University of North Dakota Carnot Cycle Nicolas Leonard Sadi Carnot French engineer and physicist A Reflection on the Motive Power of Heat (1824) Cyclic and Reversible Processes Prof. Fred Remer University of North Dakota Second Law of Thermodynamics

The foundations of the laws of thermodynamics are a result of the steam engine Prof. Fred Remer University of North Dakota Thomas Newcomen Steam Engine 1765 Prof. Fred Remer University of North Dakota Heat Engine Absorb Heat From Source Performs Mechanical Work Discards Some

Heat At Lower Temperature Prof. Fred Remer University of North Dakota Carnot Cycle The atmosphere is a giant heat engine Cold Hot Prof. Fred Remer University of North Dakota dQ Cold

Cyclic Process P V Prof. Fred Remer University of North Dakota Sequence of Processes That Leaves A Working Substance In The Same State In Which It Started Types of Cycles Otto Cycle Internal Combustion

Engine Prof. Fred Remer University of North Dakota Types of Cycles Diesel Cycle Internal Combustion Engine Prof. Fred Remer University of North Dakota Types of Cycles Rankin Cycle Prof. Fred Remer University of North Dakota

Types of Cycles Carnot Cycle Basis for All Others P 1 2 A B T1 D C T2

V Prof. Fred Remer University of North Dakota Carnot Cycle Heat Engine Performs work by transferring heat from warm reservoir to colder reservoir P 1 A Warm 2 B T1

D Cold C T2 V Prof. Fred Remer University of North Dakota Types of Cycles Carnot Cycle Heat Absorbed at Same Temperature as Heat Reservoir P

1 A Warm 2 B T1 D C T2 V Prof. Fred Remer University of North Dakota Types of Cycles Carnot Cycle Heat Rejected at Same Temperature as

Cold Reservoir P 1 A Warm 2 B T1 D Cold C T2 V Prof. Fred Remer University of North Dakota

Types of Cycles Carnot Cycle Assumes Warm and Cold Reservoirs Are Unaffected by Heat Transfer Large Source & Sink P 1 A Warm 2 B T1 D

Cold C T2 V Prof. Fred Remer University of North Dakota Carnot Cycle Idealized Heat Engine Operating at maximum efficiency No system works exactly like this Gives us a foundation for discovery Prof. Fred Remer University of North Dakota

P 1 2 A B T1 D C T2 V Carnot Cycle Isothermal Expansion

Heat Added = Work Done Volume Increases Pressure Decreases Prof. Fred Remer University of North Dakota Carnot Cycle Adiabatic Expansion Change of = Work Done Internal Energy Volume Increases Pressure Decrease Temperature Cools Prof. Fred Remer University of North Dakota Carnot Cycle

Isothermal Compression Heat Removed = Work Done Volume Decreases Pressure Increase Prof. Fred Remer University of North Dakota Carnot Cycle Adiabatic Compression = Work Done Change of Internal Energy Volume Decreases Pressure Increase Temperature Warms Prof. Fred Remer

University of North Dakota Carnot Cycle P Adiabat 1 Adiabat 2 A B T1 Isotherm D C T2 Isotherm V Prof. Fred Remer University of North Dakota Carnot Cycle P

Adiabat 1 Adiabat 2 A B Q1 Heat Added T1 Isotherm D C T2 Isotherm V Prof. Fred Remer University of North Dakota Isothermal Expansion

Carnot Cycle P Adiabat 1 Adiabat 2 A B cvdT = pdV Cooling T1 Isotherm D C T2 Isotherm V Prof. Fred Remer University of North Dakota

Adiabatic Expansion Carnot Cycle P Adiabat 1 Adiabat Q2 2 Heat Removed A B T1 Isotherm D C T2 Isotherm V Isothermal Compression Prof. Fred Remer University of North Dakota

Carnot Cycle P Adiabat 1 Adiabat 2 A B cvdT = pdV Warming T1 Isotherm D C T2 Isotherm V Adiabatic Compression Prof. Fred Remer

University of North Dakota Carnot Cycle Mechanical energy (work) is done as a result of heat transfer P 1 A Warm 2 B T1 D Cold

C T2 V Prof. Fred Remer University of North Dakota Carnot Cycle What if the cycle is reversed? Heat Taken From Colder Temperature and Deposited at Warmer Temperature P 1 A

2 Warm B T1 D Cold C T2 V Prof. Fred Remer University of North Dakota Carnot Cycle Refrigerator

Heat Taken From Colder Temperature and Deposited at Warmer Temperature Prof. Fred Remer University of North Dakota Carnot Cycle Refrigerator Work is required to transferring heat from cold reservoir to warm reservoir P 1 A 2

Warm B T1 D Cold C T2 V Prof. Fred Remer University of North Dakota Carnot Cycle Alternate Form of Second Law

Heat cannot of itself pass from a cold to a warm body. P 1 A 2 Warm B T1 D Cold C T2

V Prof. Fred Remer University of North Dakota Carnot Cycle Reversible Process each state of the system is in equilibrium process occurs slow enough state variables reach equilibrium Prof. Fred Remer University of North Dakota P 1

2 A B T1 D C T2 V Carnot Cycle Reversible Process reversal in direction returns substance & environment to original states

P 1 2 A B T1 D C T2 V Prof. Fred Remer University of North Dakota Carnot Cycle

Cyclic Process A process in which the Initial State is also the Final State P 1 2 A B T1 D C dQ dU dW Prof. Fred Remer

University of North Dakota T2 V Carnot Cycle Cyclic Process Internal Energy (U) is unchanged dU m c v dT P 1 2 A B

T1 D C T2 V Prof. Fred Remer University of North Dakota Carnot Cycle Cyclic Process An exact differential Intergral around a closed path is zero P 1

2 A B T1 D m C v dT 0 dU 0 Prof. Fred Remer University of North Dakota C T2 V Carnot Cycle

Cyclic Process Work done is not an exact differential Path dependent P 1 2 A B T1 D C dW p dV Prof. Fred Remer

University of North Dakota T2 V Carnot Cycle P W1 pdV 1 2 A B D T1 W1 C

T2 V Work done by system during expansion Prof. Fred Remer University of North Dakota Carnot Cycle P W2 pdV 1 2 A B

D W2 T1 C T2 V Work done on system during compression Prof. Fred Remer University of North Dakota Carnot Cycle P W W1 W2 1

2 A B D pdV T1 W C T2 V Net work done by system Prof. Fred Remer University of North Dakota

Carnot Cycle Cyclic Process Net work done by system is the net heat absorbed P 1 2 A dQ dU dW B T1

D C dQ dW Prof. Fred Remer University of North Dakota T2 V Carnot Cycle Net work done by system W = Q = Q1 - Q2 P 1 2 Q

1 A B D T1 Q2 C T2 V Prof. Fred Remer University of North Dakota Carnot Cycle Efficiency of a System Cannot Convert All Heat Available Into Work P

1 2 Q 1 A B D T1 Q2 C T2 V Prof. Fred Remer University of North Dakota Carnot Cycle Problems Net work done Path dependent

Not mathematically elegant Difficult to make quick calcualtions P 1 A 2 D QNet B C T2 V

QNet pdV Prof. Fred Remer University of North Dakota T1 Carnot Cycle Is there a way to describe the heat change of a system without having to deal with work done? YES!!!! but it will take some work! Prof. Fred Remer University of North Dakota Carnot Cycle Before we begin...

It can be shown from the First Law & the Ideal Gas Law 1 Final TFinal V 1 Initial TInitial V PFinal V Final PInitial V Poissons Equation For an adiabatic process where = cp/cv Prof. Fred Remer

University of North Dakota Initial Carnot Cycle Lets examine each leg of the cycle P 1 A Q1 2 B D Q2 T1 C

T2 V Prof. Fred Remer University of North Dakota Carnot Cycle From A to B isothermal expansion temperature is constant (T1) so.. P 1 A Q1 2

B T1 T2 V PAVA = PBVB Prof. Fred Remer University of North Dakota Carnot Cycle From A to B isothermal expansion work done (W1) is heat added (Q1) P 1 A

Q 1 2 B T1 T2 V B Q1 p dV A Prof. Fred Remer University of North Dakota Carnot Cycle From A to B P

heat added (Q1) for 1 mole 1 Q 1 2 A B B T2 Q1 p dV A Ideal Gas Law Prof. Fred Remer University of North Dakota

R*T p V T1 V So... dV Q1 R T1 A V * B Carnot Cycle From A to B heat added (Q1) for 1 mole

VB Q1 R T1 ln VA Prof. Fred Remer University of North Dakota A Q1 2 B T1 T2 dV * Q1 R T1 A V B *

P 1 V Carnot Cycle From B to C adiabatic expansion (2) PCVC = PBVB T2 VC 1 T1VB 1 Prof. Fred Remer University of North Dakota P

1 2 B T1 C T2 V Carnot Cycle From C to D isothermal compression work done (W2) is heat removed (Q2) P 1

2 T1 D Q 2 C T2 V C Q 2 p dV D Prof. Fred Remer University of North Dakota Carnot Cycle From C to D P

heat removed (Q2) for 1 mole 1 2 T1 D C Q 2 p dV Q 2 C T2 D Ideal Gas Law Prof. Fred Remer University of North Dakota R*T

p V V So... dV Q 2 R T2 D V * C Carnot Cycle From C to D heat added (Q1) for 1 mole dV *

Q 2 R T2 D V C VC Q 2 R T2 ln VD * Prof. Fred Remer University of North Dakota P 1 2 T1 D Q 2 C T2 V

Carnot Cycle From D to A adiabatic compression (2) P 1 2 A T1 D PDVD = PAVA 1 D

T2 V Prof. Fred Remer University of North Dakota 1 A T1V T2 V Carnot Cycle For the adiabatic processes 1 C T2 V 1 D

T2 V 1 B T1V P 1 A T1V 1 2 A B D

Take the ratio VC VD Prof. Fred Remer University of North Dakota 1 VB VA T1 C T2 V

1 or VC VB VD VA Remember this!! Carnot Cycle For the isothermal processes VB Q1 R T1 ln VA *

VC Q 2 R T2 ln VD * Take the ratio Prof. Fred Remer University of North Dakota P 1 A Q1 D

2 B T1 Q 2 C T2 V Carnot Cycle For the isothermal processes Q1 T1 ln( VB / VA ) Q 2 T2 ln( VC / VD ) But .... VC VB VD VA Prof. Fred Remer University of North Dakota

P 1 A Q1 D 2 B T1 Q 2 C T2 V Carnot Cycle For the cyclic process P Q1 T1

Q 2 T2 1 A Q1 D 2 B T1 Q 2 C T2 V A lot of work just to say ... Prof. Fred Remer University of North Dakota

Carnot Cycle For a cyclic process Q1 T1 Q 2 T2 P 1 A Q1 D 2 B T1

Q 2 C T2 V The ratio of the heat absorbed to heat rejected depends only on the initial and final temperature Prof. Fred Remer University of North Dakota Carnot Cycle Thats great for steam engines, but what about the atmosphere? Prof. Fred Remer University of North Dakota Carnot Cycle

Patience, grasshopper! Prof. Fred Remer University of North Dakota Carnot Cycle Isothermal Process Heat absorbed or released Amount of heat depends on temperature VB Q1 R T1 ln VA * Prof. Fred Remer University of North Dakota

1 P A Q1 D 2 B T1 Q 2 C T2 V VC

Q 2 R T2 ln VD * Carnot Cycle But ... P 1 A Q1 T1 Q 2 T2 or Q1 Q 2

T1 T2 Q1 D 2 B T1 Q 2 C T2 V The ratio Q/T is the same regardless of the isotherm chosen Prof. Fred Remer University of North Dakota

Carnot Cycle The ratio Q T is a measure of the difference between the two adiabats. P 1 A Q1 D 2 B

T1 Q 2 C T2 It is the same for any two adiabats in a cyclic process Prof. Fred Remer University of North Dakota V Entropy Difference in entropy between adiabats dQ dS T For a

reversible process Prof. Fred Remer University of North Dakota P 1 A Q1 D 2 B T1 Q 2 C T2 V Entropy

The path dependent integral dW dQ pdV P 1 2 dQ T1 T2 no longer depends on path Prof. Fred Remer University of North Dakota

V Entropy An integrating factor (T) makes the differential exact P Q1 T1 Q2 T2 V dQ dS T

Prof. Fred Remer University of North Dakota Q1 Q 2 T1 T2 Entropy For the cyclic & reversible process P Q1 T1 dq c p dT dp Q2 1st Law

V dq dT c p dp T T T Ideal Gas Law Prof. Fred Remer University of North Dakota T2 p R T RT p Entropy

Substitute for dq dT dp c p R T T p dq c p d(ln T ) R d(ln p) T Integrate Prof. Fred Remer University of North Dakota dq T c p d(ln T) R d(ln p) Entropy

Over the closed path P Q1 T1 c p d(ln T ) 0 Q2 V R d(ln p) 0 So .... Prof. Fred Remer University of North Dakota

T2 dq T 0 Entropy Function of state T, P Not path dependent P Q1 T1 Q2 T2 V

Prof. Fred Remer University of North Dakota Entropy For adiabatic processes P 1 T1 dq ds T T2 V But ...

dq 0 Prof. Fred Remer University of North Dakota 2 So ... ds 0 Entropy For adiabatic processes ds 0 No change in entropy Isentropic

Prof. Fred Remer University of North Dakota P 1 2 T1 T2 V Entropy Potential temperature () is conserved for adiabatic processes! How is potential temperature related to entropy? 1000 T

p Prof. Fred Remer University of North Dakota R Cp Entropy Logarithmically Differentiate 1000 T p R cp R ln ln T (ln1000 ln p)

cp c p d ln c p d ln T Rd ln p Prof. Fred Remer University of North Dakota Entropy Wait a second ... c p d ln c p d ln T Rd ln p This looks like ... dq c p d(ln T ) R d(ln p) T Lets combine them! Prof. Fred Remer University of North Dakota

Entropy Combining equations dq ds c p d ln T Integrate s c p ln C Prof. Fred Remer University of North Dakota Entropy Meteorologically speaking ...

s c p ln C Entropy depends only on potential temperature! Dry adiabatic processes are isentropic! Prof. Fred Remer University of North Dakota Review Second Law of Thermodynamics It is impossible for any system to undergo a process in which it absorbs heat from a reservoir at a single temperature and converts the heat completely into mechanical work, with the system ending in the same state in which it began. Prof. Fred Remer University of North Dakota

Review Second Law of Thermodynamics It is impossible for any process to have as its sole result the transfer of heat from a cooler to a hotter body. Prof. Fred Remer University of North Dakota Second Law of Thermodynamics No process is possible in which the total entropy decreases, when all systems taking part in the process are included. dQ dS T Prof. Fred Remer University of North Dakota

Types of Processes Natural (or Irreversible) Impossible Reversible Prof. Fred Remer University of North Dakota Natural (or Irreversible) Process Processes That Proceed Spontaneously in One Direction But Not The Other Non-Equilibrium Process Equilibrium Only At End of Process Prof. Fred Remer

University of North Dakota Natural (or Irreversible) Process Examples Conversion of Work to Heat Through Friction Free Expansion of Gas Prof. Fred Remer University of North Dakota Natural (or Irreversible) Process Most Natural Processes Are Irreversible Analogy Water Wheel Water Flows from Higher Elevation to Lower Elevation

Work is Done Prof. Fred Remer University of North Dakota Natural (or Irreversible) Process Analogy Water Wheel Water Ends Up At A Lower Height Unable to Perform More Work Prof. Fred Remer University of North Dakota Natural (or Irreversible) Process Analogy Water Wheel

Irreversible Water Does Not Flow Back Up By Itself Prof. Fred Remer University of North Dakota Natural (or Irreversible) Process Similarly, some of the heat is not available to due work P 1 Warm

2 T1 Cold T2 V Prof. Fred Remer University of North Dakota Impossible Process Violate Second Law Cannot Occur Prof. Fred Remer University of North Dakota

Impossible Process Examples Free Compression of Air Conduction in which Cold Object Gets Colder and Warm Object Gets Warmer Prof. Fred Remer University of North Dakota Reversible Process System & Surroundings Already Close to Thermodynamic Equilibrium Changes of State Can Be Reversed Through Infinitesimal Changes Prof. Fred Remer University of North Dakota

Reversible Process System Always in Equilibrium Idealized Concept Changes Would Never Take Place Initially Small Changes Prof. Fred Remer University of North Dakota Entropy Carnot Cycle Idealize Engine Reversible Process Maximum Efficiency dQ dS

T dQ1 dQ 2 dS T1 T2 Prof. Fred Remer University of North Dakota P 1 dQ1 dQ2 2 T1 T2 V

Dry Adiabatic Process Isentropic No Change in Entropy Potential Temperature is Constant Reversible Substance and Environment return to original condition Prof. Fred Remer University of North Dakota Entropy Real World Dry Adiabatic Process Approximation to real

world conditions Not what really happens Mixing Prof. Fred Remer University of North Dakota Entropy A Quantitative Measure of Randomness or Disorder dQ dS T Prof. Fred Remer University of North Dakota Entropy Example

Conversion of Mechanical Energy Into Heat Increase in Disorder Random Molecular Motion Prof. Fred Remer University of North Dakota Entropy In The Atmosphere Processes Are Not Exactly Dry Adiabatic Process Prof. Fred Remer University of North Dakota Entropy

Most Natural Processes Are Irreversible Prof. Fred Remer University of North Dakota Entropy Would Require More Energy to Return System and Environment to Equilibrium Prof. Fred Remer University of North Dakota Entropy This implies an

increase in entropy dQ dS T Prof. Fred Remer University of North Dakota Entropy The entropy of the universe is ever increasing Conversion from more useful to less useful states Prof. Fred Remer University of North Dakota Prof. Fred Remer University of North Dakota

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