Heat EnginesProduce work from heat in a cyclical process operated

between 2 or more T’sExamples: Carnot, Rankine, Diesel, Otto, Ericsson, Stirling, Brayton, …

Working fluid (“the engine”) absorbs heat from a high-temperature heat “reservoir”, performs net work, and releases heat to a low-

temperature heat “reservoir”.

TH

TL

E

qin(+)

qout (–)

wnet (–)

Conditions/Limitations:Kelvin’s 2nd Law Expression (see before): No cyclical engine process is possible in which the sole result is the absorption of heat from a reservoir and its complete conversion into work:

qout 0

1st Law: 0 = Utotal = qtotal + wtotal = qin + qout + wnet :

–wnet = wnet = qnet = qin – qout > 0 qin > qout

Note: qin = qin, qout = qout.

Efficiency:

1 q

q1

q

q q

in

out

in

outin

in

net

q

w η

input(s) heat

output worknet

Q: Are there any assumptions that are needed regarding the working material?

High

Low

1 © Prof. Zvi C. Koren 20.07.2010

1796–1832 (Paris)

Sadi Nicolas Léonard Carnot

http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Carnot_Sadi.html

In 1812, at age 16 the minimum age possible, Carnot entered the École Polytechnique. Carnot graduated from the École Polytechnique in 1814

but, before he graduated, Carnot and other students from the École Polytechnique fought unsuccessfully with Napoleon to defend Vincennes. This

skirmish against the Allies was fought just outside Paris, to the east of the city.

The problem occupying Carnot, a military engineer, was how to design good steam engines. Steam power already had many uses - draining

water from mines, excavating ports and rivers, forging iron, grinding grain, and spinning and weaving cloth - but it was inefficient. The import

into France of advanced engines after the war with Britain showed Carnot how far French design had fallen behind. It irked him particularly that

the British had progressed so far through the genius of a few engineers who lacked formal scientific education. British engineers had also

accumulated and published reliable data about the efficiency of many types of engines under actual running conditions; and they vigorously

argued the merits of low- and high-pressure engines and of single-cylinder and multi-cylinder engines. His publication, Réflexions, was an

attempt by Carnot to answer two fundamental questions, firstly whether there was an upper limit to the power of heat, and secondly

whether there was a better means than steam to produce this power. He died of cholera at the age of 36.

Carnot Engine & Cycle (1824)

(continued)2 © Prof. Zvi C. Koren 20.07.2010

Carnot Engine & Cycle (continued)

Cycle of 4 reversible steps & operating between 2 constant temps.

wq +U =NameStep

(for an ideal gas, [CV])

–nRT1ln(V2/V1)–w1 > 00Isothermal

Expansion1

U20nCV(T2 – T1)Adiabatic

Expansion2

–nRT2ln(V4/V3)–w3 < 00Isothermal

Compression3

U4 = – U20nCV(T1 – T2)Adiabatic

Compression4

nRℓn[(V1/V2)T1(V3/V4)

T2]–wcycle0CycleTotal

Note: Not every reversible heat engine is a Carnot engine, but if it operates between 2

constant T’s, then it’s a Carnot. Prove it yourself later!

The idealized Carnot Engine is the most efficient of all heat engines!

Nike Problem

T1 = TH

T2 = TL

3 © Prof. Zvi C. Koren 20.07.2010

Carnot Engine & Cycle (continued)

TH, T1

TL, T2

E

qin (+), q1

qout (–), q2

wnet (–)

wnetw1

w2

w3

w4

wnet,total,cycle = w1 + w2 + w3 + w4

(h=hot, c=cold)

4 © Prof. Zvi C. Koren 20.07.2010

Efficiency of the Carnot Engine

For any heat engine (from before): 1 q

q1

q

q q

in

out

in

outin

in

net

q

w η

For Carnot engine: Carnot = f(other parameters)?; wnet,total,cycle = w1+w2+w3+w4

Use an ideal gas as the working material to simplify the mathematics since the nature of the working fluid is unimportant:

For the adiabatic steps, 2 & 4:

Recall: dVV

nRT PdV dVp dTCn exV dw dU (conditions?)

i

f

i

fVV

V

VR

T

TC

V

dVR

T

dTC nn (conditions?)

2

3

1

2V

V

VR

T

TC nn For 2: .

4

1

2

1V

V

VR

T

TC nn For 4: .

3

4

2

1

V

V

V

V

From before (see table): 1

221net

T

4

3

T

2

1net

V

VTTnR w

V

V

V

VnR w

21

nn

1 T

T1 η

T

T1

H

LCarnot

1

2 in

net

q

w η

1

21in,1

V

VnRT q n

Every reversible heat engine operating between 2 constant T’s is a Carnot one with this efficiency!!!

(Proved later)5 © Prof. Zvi C. Koren 20.07.2010

Interim Summary

1 q

q1

in

out

in

net

EngineHeat q

w η

1 T

T1

H

L Carnotη

One can also more generally prove that for the Carnot cycle

in

out

H

L

q

q

T

T

from the Clausius equationT

dq dS rev

without assuming any specific working fluid.

6 © Prof. Zvi C. Koren 20.07.2010

T

Sis

entr

ope

isentrope

Carnot Cycle Diagrams

P

V

qnet

T-SP-V

2wnet 4

2

4

wnet = w1 + w2 + w3 + w4

wi,rev = –PdV

qnet = qin + qout = qin – qout

S = đqrev/T S = qrev,T/T

qrev,T = TS

1

TH,1isotherm

3isotherm

TL,2

1

3

7 © Prof. Zvi C. Koren 20.07.2010

T1

T2

T3

V

P

H

LCarnot

T

T1 η

The greater the difference between the two T’s

the greater the enclosed area, the greater the efficiency

The Geometry of Efficiency

Consider two different

Carnot cycles:

one operating between

T1 and T3 and another

between T1 and T2.

8 © Prof. Zvi C. Koren 20.07.2010

1 q

q1

in

out

in

net

EngineHeat q

w η

Problems 2 – Entropy & Heat Engines:

3-8.

There are five basic parts to any refrigerator (or air-

conditioning system):

• Compressor (B)

• Heat-exchanging pipes – serpentine or coiled set of pipes

outside the unit (D)

• Expansion valve (C)

• Heat-exchanging pipes – serpentine or coiled set of pipes

inside the unit (A)

• Refrigerant – liquid that evaporates inside the

refrigerator to create the cold temperatures. Many

industrial installations use pure ammonia as the

refrigerant. Pure ammonia evaporates at -27oF

(-32oC), but is toxic if it leaks out. Home

refrigerators use non-toxic CFC’s

(chlorofluorocarbons), also known as Freons

developed by Du Pont. CFC-12

(dichlorodifluoromethane) has about the same

boiling point as ammonia.

Refrigerators & Air Conditioners

(continued)

D

Adapted from: http://home.howstuffworks.com/refrigerator2.htm & http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/refrig.html#c1

9 © Prof. Zvi C. Koren 20.07.2010

D

1. The compressor (B) compresses the refrigerant gas. This

raises the refrigerant's pressure and temperature (orange), so

the heat-exchanging coils outside the refrigerator (D) allow the

refrigerant to dissipate the heat of pressurization.

2. As it cools, the highly pressurized refrigerant condenses into

liquid form (purple) still outside of the fridge, dumps heat out

to the room (TH), (g) (ℓ) + Hv , condensation, and flows through the expansion valve.

3. When it flows through the expansion valve, the liquid

refrigerant is allowed to move from a high-pressure zone to a

low-pressure zone, so it expands and evaporates (light blue).

In evaporating, vaporization, it absorbs heat from the fridge

itself, making it cold: (ℓ) + Hv (g) .4. The coils inside the refrigerator allow the refrigerant to

absorb heat, making the inside of the refrigerator cold.

5. The cold refrigerant gas is sucked up by the compressor, and

the cycle repeats.

By the way, if you have ever turned your car off on a hot summer day when you have had the air conditioner running, you may have heard a hissing noise under the hood. That noise is the sound of high-pressure liquid refrigerant flowing through the expansion valve.

D

(TH)

(TL)

The basic mechanism of a refrigerator requires a phase change and works like this:

qout

qin

(g)

(ℓ)

(ℓ)

(g)

Adapted from: http://home.howstuffworks.com/refrigerator2.htm & http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/refrig.html#c1

10 © Prof. Zvi C. Koren 20.07.2010

Carnot Engine

TH,1

TL,2

wnet

Engine

TH,1

TL,2

Heat Pump:pumps heat to high T

q1, qin (+)

q2, qout (–)

q1, qout (–)

q2, qin (+)

1 T

T1 η

H,1

L,2

Engine .TT

T

w

q β ,CP

L,2H,1

L,2

net

2RefRef

L,2H,1

H,1

net

1

H.P.H.PumpTT

T

w

q β ,CP

'

Coefficients of Performance (CoP) (values not limited):

Refrigerator & Heat Pump

Refrigerator:removes heat from low T

Efficiency (limited):

NikeProblems

“by” “on”

Carnot Refrigerator & Heat Pump

wnet

Note: From 1st Law:

wnet = qH,1 – qL,2

11 © Prof. Zvi C. Koren 20.07.2010

TH,1

TL,2

wnet

q1

q2

q1

q2

E R

A Net of Nothing

An ideal Carnot Engine driving an ideal Carnot Refrigerator (or Heat Pump)

Every reversible heat engine operating between 2 constant T’s is a Carnot one with the Carnot efficiency!!!

12 © Prof. Zvi C. Koren 20.07.2010

The Carnot Engine is the most efficient of all heat engines!

Carnot is It!

If there would be a SuperEngine that is more efficient than a

Carnot engine, an impossibility as shown below, then that

would be due to one (or both) of the following factors:

1. qin in the SuperEngine is less than in Carnot:

The overall effect would be the spontaneous flow of heat from

a cold body to a hot one, which obviously contradicts nature

and Clausius’s expression of the 2nd Law. (See next slide.)

2. wnet is greater in the SuperEngine:

This would result in the net perpetual production of work, i.e. energy, and, e.g., perpetual mobile (פרפטואום מובילה)machines, which are forbidden by the 1st Law (and/or the 2nd

Law). Also, this would result in transferring disordered q into

ordered w.

LoutHinnetnet

Hin

net

EngineHeat qq q wq

w η

, :recallFirst

For a Refrigerator: wnet = qoutH – qinL > 0

Correct

Note the correct scale of vector

magnitudes

?

13 © Prof. Zvi C. Koren 20.07.2010

Carnot’s Theorem:For two given heat reservoirs,

no engine can have a higher thermal efficiency than a Carnot engine.

LoutHinnetnetHinnetEngineHeat qq q wqw η ,

TH

TL

CarnotFridge

CarnotEngine

TH

TL

wnet

qinH

qoutL

QinH

qinL

qoutH

QoutL

SuperEngine

Proof:

wnet

qNET extracted from cold reservoir:

qinL – QoutL =

(qH – wnet) – (QH – wnet) =

qH – QH > 0 (above)

qNET delivered to hot reservoir:

qH – QH = qNET.

Thus, the only result of this

SuperEngine/Fridge combination is

the transfer of heat from the cold

to the hot reservoirs. This is not

allowed according to Clausius’s

2nd Law expression.

If, for the same work output wnet, Super > Carnot, then QinH < qinH.

Let the SuperEngine drive a Carnot refrigerator, and from before: wnet = qH – qL > 0

Also, recall that since all steps in Carnot are reverible, the Fridge is an exactly reversed

engine, so that their absolute qH values are equal as are their absolute qL values.

14 © Prof. Zvi C. Koren 20.07.2010

For a Refrigerator: wnet = qoutH – qinL > 0

Recall:

Combination of

Carnot EnginesTH

TL

TM

qM

Engine 1

qH

Engine 2

qM

qL

Heat rejected by the 1st engine is absorbed

by the 2nd

For each cycle, Ucycle = 0:

wnet,1 = qH – qM

wnet,2 = qM – qL

wNET = qH – qL = qNET

TH

V

P

TM

TL

What would be the

TS-diagram for the

two-engine combo? Nike

A “Bicycle” Built for Two

H

L

H

L

H

NET

T

T1

q

q1

q

w η total

Nike

wnet,1

wnet,2

wNET

TH

qH

TL

qL

ENGINE

This is equivalent to just one

engine, absorbing qH from TH

and releasing qL to TL:

2121 ηηηη

15 © Prof. Zvi C. Koren 20.07.2010

Problems from Thermo 1:

36-39.

Final Thoughts About Carnot

“Carnot” Knowledge

http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/carnot.html#c1

When the second law of thermodynamics states that not all the supplied heat in a heat engine can be used to do work, the Carnot efficiency sets the limiting value on the fraction of the heat which can be so used.

In order to approach the Carnot efficiency, the processes involved in the heat engine cycle must be reversible and involve no change in entropy. This means that the Carnot cycle is an idealization, since no real engine processes are reversible and all real physical processes involve some increase in entropy.

The most efficient heat engine is the Carnot engine.BUT don't bother installing a Carnot engine in your car;

while it would increase your gas mileage,you would be passed on the highway by pedestrians.

OT

16 © Prof. Zvi C. Koren 20.07.2010Carnot Problems 1-6 :תרגילי קרנו.

Non-Carnot Engine: Example 1

P

V

TH

TL

Nike Problems

Prove that the efficiency of this cycle is (a) as given below, and that (b) < Carnot.

H

L

V

1

2

V

1

2

LH

V

1

2

LH

V

1

2

T

T ,

1R

C

V

V

1R

C

V

V

1

TTR

C

V

V

TTR

C

V

V

1 η

x

x

xx

n

n

n

n

H

L

T

T

Possible Processes:(isothermal, isobaric, isochoric; adiabatic, isentropicrev+ad) (expansion, compression)

(isobaric, isochoric) (heating, cooling)

12

34

Recall: wnet /qin = 1 – qout/qin = 1 – something

Q’s: Why isn’t this a Carnot engine? Are there 2 constant-T heat reservoirs here?Any adiabats here? What would be the TS-Diagram?

17 © Prof. Zvi C. Koren 20.07.2010

ideal gas, [CV], reversible steps:NameStep

wq

1

2

3

4

What would be the PV-diagram for this engine?

What is the efficiency of this engine if all steps are reversible?

(Solution for is admittedly a bit long …)

S

T

18 © Prof. Zvi C. Koren 20.07.2010

Non-Carnot Engine: Example 2

T1

T2

T3

T4

Carnot Problems 7-9 :תרגילי קרנו.