Lecture 11. Heat Engines (Ch. 4)

Heating – the transfer of energy to a system by thermal contact with a reservoir.Work – the transfer of energy to a system by a change in the external parameters (V, el.-mag. and grav. fields, etc.).

The main question we want to address: what are the limitations imposed by thermodynamic on the performance of heat engines?

A heat engine – any device that is capable of converting thermal energy (heating) into mechanical energy (work). We will consider an important class of such devices whose operation is cyclic.

Perpetual Motion Machines are Impossible

Perpetual Motion Machines of the first type – these designs seek to create the energy required for their operation out of nothing.

Perpetual Motion Machines of the second type - these designs extract the energy required for their operation in a manner that decreases the entropy of an isolated system.

violation of the First Law (energy conservation)

violation of the Second Law

hot reservoir TH

work

hea

t

Word of caution: for non-cyclic processes, 100% of heat can be transformed into work without violating the Second Law.

Example: an ideal gas expands isothermally being in thermal contact with a hot reservoir. Since U = const at T = const, all heat has been transformed into work.

impossible cyclicheat engine

Fundamental Difference between Heating and Work

- is the difference in the entropy transfer!

Transferring purely mechanical energy to or from a system does not (necessarily) change its entropy: S = 0 for reversible processes. For this reason, all forms of work are thermodynamically equivalent to each other - they are freely convertible into each other and, in particular, into mechanical work.

An ideal el. motor converts el. work into mech. work, an ideal el. generator converts mech. work into el. work.

Work can be completely converted into heat, but the inverse is not true. The transfer of energy by heating is accompanied with the entropy transfer

Thus, entropy enters the system with heating, but does not leave the system with the work. On the other hand, for a continuous operation of a heat engine, the net entropy change during a cycle must be zero!

How is it possible???

T

QSd

The Price Should be Paid...

entr

op

y

heat

work

hea

t

hot reservoir, TH

cold reservoir, TC

Thus, the only way to get rid of the accumulating entropy is to absorb more internal energy in the heating process than the amount converted to work, and to “flush” the entropy with the flow of the waste heat out of the system.

An engine can get rid of all the entropy received from the hot reservoir by transferring only part of the input thermal energy to the cold reservoir.

“Working substance” – the system that absorbs heat, expels waste energy, and does work (e.g., water vapor in the steam engine)

T

QSd

Essential parts of a heat engine (any continuously operating reversible device generating work from “heat”)

An essential ingredient: a temperature difference between hot and cold reservoirs.

Perfect Engines (no extra S generated)

H

HH T

QS

C

CC T

QS

HQ

CQ

Wentr

op

y

heat

work

hea

t

hot reservoir, TH

cold reservoir, TC

The condition of continuous operation:

CH SS C

C

H

H

T

Q

T

Q

HH

CC Q

T

TQ

The work generated during one cycle of a reversible process:

HH

CHCH Q

T

TTQQW

Carnot efficiency:

the highest possible value of the energy

conversion efficiency

11max H

C

H T

T

Q

We

Sadi Carnot

(to simplify equations, I’ll omit in Q throughout this lecture)

Consequences

Any difference TH –TC 0 can be exploited to generate mechanical energy.

The greater the TH –TC difference, the more efficient the engine.

Energy waste is inevitable.

Example: In a typical nuclear power plant, TH = 3000C (~570K), TC = 400C (~310K), and the maximum efficiency emax=0.45. If the plant generates 1000 MW of “work”, its waste heat production is at a rate

- more fuel is needed to get rid of the entropy then to generate useful power.

This creates the problem of waste heat - e.g., the waste heat produced by human activities in the LA basin exceeds 7% of the solar energy falling on the basin (~ 1kW/m2).

MW 122011

e

WWQQ HC

Real Engines

H

HH T

QS

C

CC T

QS

HQ

CQ

Wentr

op

y

heat

work

hea

t

hot reservoir, TH

cold reservoir, TC

Real heat engines have lower efficiencies because the processes within the devices are not perfectly reversible – the entropy will be generated by irreversible processes:

max1 eT

T

Q

We

H

C

H

e = emax only in the limit of reversible operation.

Some sources of irreversibility:

heat may flow directly between reservoirs;

not all temperature difference TH – TC may be available (temperature drop across thermal resistances in the path of the heat flow);

part of the work generated may be converted to heat by friction;

gas may expand irreversibly without doing work (as gas flow into vacuum).

ProblemThe temperature inside the engine of a helicopter is 20000C, the temperature of the exhaust gases is 9000C. The mass of the helicopter is M = 2103 kg, the heat of combustion of gasoline is Qcomb = 47103 kJ/kg, and the density of gasoline is = 0.8 g/cm3. What is the maximum height that the helicopter can reach by burning V = 1 liter of gasoline?

The work done on lifting the helicopter: MgHW

H

C

H T

T

Q

We 1For the ideal Carnot cycle (the maximum efficiency):

VqQ combH The heat released in the gasoline combustion:

mK

K

m/skg

literkg/liter0.8kJ/kg2

9282273

11731

8.92000

110471

1

3

H

Ccomb

combH

C

T

T

Mg

VqH

VqT

TMgH

Thus,H

H

C QT

TW

1

Problem [ TH = f(t) ]

.(a) Calculate the heat extracted from the hot reservoir during this period.(b) What is the change of entropy of the hot reservoir during this period?(c) How much work did the engine do during this period?

HHH

CH

T

T

Q

W

Q

QQTe 11

12 TTcQ VH (a) (b)2

1ln1

2T

Tc

T

dTcS

T

dTc

T

QdS V

T

T H

HV

H

HV

H

H

(c)HVH dTcQ

HHV

HH T

TdTc

T

TQW 11 11

1

2112

1 ln11

2T

TTcTTcdTc

T

TW VVHV

T

T H

A reversible heat engine operates between two reservoirs, TC and TH.. The cold reservoir can be considered to have infinite mass, i.e., TC = T1 remains constant. However the hot reservoir consists of a finite amount of gas at constant volume ( moles with a specific heat capacity cV), thus TH decreases with time (initially, TH =T2, T2 > T1). After the heat engine has operated for some long period of time, the temperature TH is lowered to TC =T1

Note: if TH and/or TC vary in the process, we still can introduce an “instanteneous” efficiency:

HH

CH

Q

W

Q

QQTe

a) What is the temperature of the water?

b) How much heat has been absorbed by the block of ice in the process?

c) How much ice has been melted (the heat of fusion of ice is 333 J/g)?

d) How much work has been done by the engine?

(a) Because the block of ice is very large, we can assume its temperature to be constant. When work can no longer be extracted from the system, the efficiency of the cycle is zero: CTT

TTe

icewater

waterice

00

0/1

A reversible heat engine absorbs heat from the water and expels heat to the ice until work can no longer be extracted from the system. The heat capacity of water is 4.2 J/g·K. At the completion of the process:

Problem

Problem (cont.)

(c) The amount of melted ice: kg J/g 333

kJ 07.1

9.357

L

QM C

I

(d) The work : kJ 1.62kJ 9.357-100KKkJ/kg2.4kg1 CH QQW

(b) The heat absorbed by the block of ice:

kJkJ/kgkgK 9.357273

373ln2.41273

,1,1,

273

373

W

WWWI

T

T

WWWW

IC

WWWW

IWWWIWHIWC

H

CHIW

T

dTcmTdTcm

T

TQ

dTcmT

TdTcmTTeQTTeQ

Q

QQTTe

f

i

Given 1 kg of water at 1000C and a very large block of ice at 00C.

Carnot Cycle

1 – 2 isothermal expansion (in contact with TH)

2 – 3 isentropic expansion to TC

3 – 4 isothermal compression (in contact with TC)

4 – 1 isentropic compression to TH

(isentropic adiabatic+quasistatic)

Efficiency of Carnot cycle for an ideal gas:

(Pr. 4.5) H

C

T

Te 1max

T

S

TH TC

1

23

4

On the S -T diagram, the work done is the area of the loop:

entr

opy

cont

aine

d in

ga

s

PdVTdSdU 0

The heat consumed at TH (1 – 2) is the area surrounded by the broken line:

CHHH SSTQ S - entropy contained in gas

- is not very practical (too slow), but operates at the maximum efficiency allowed by the Second Law.

TC

P

V

TH

1

2

34

absorbsheat

rejects heat

ProblemProblem. Consider a heat engine working in a reversible cycle and using an ideal

gas with constant heat capacity cP as the working substance. The cycle

consists of two processes at constant pressure, joined by two adiabatic processes.

(a) Which temperature of TA, TB, TC, and TD is highest, and which is lowest?

(b) Find the efficiency of this engine in terms of P1 and P2 .

(c) Show that a Carnot engine with the same gas working between the highest and lowest temperatures has greater efficiency than this engine.

(a) From the equation of state for an ideal gas (PV=RT), we know that

P

CD

A B

V

P1

P2 DCAB TTTT

From the adiabatic equation : DACB TTTT

Thus DCBADDCBAB TTTTTTTTTT ,,,min,,,max

The Carnot heat engine operates at the maximum efficiency allowed by the Second Law. Other heat engines may have a lower efficiency even if the cycle is reversible (no friction, etc.)

Problem (cont.)

P

CD

A B

V

P1

P2

(b) The heat absorbed from the hot reservoir ABPAB TTCQ

The heat released into the cold reservoir DCPCD TTCQ

Thus, the efficiency AB

DC

AB

CDAB

TT

TT

Q

QQe

1

From the equation for an adiabatic process:

constPTconstPV 1/

1111

C

C

B

B

D

D

A

A

P

T

P

T

P

T

P

T

/1

21

/1/1

/1//

1

PPTT

PPTPPTe

AB

ADABCB

(c) max

1

2

1 111 eT

T

T

T

P

Pe

B

D

A

D

Stirling heat engine

The gasses used inside a Stirling engine never leave the engine. There are no exhaust valves that vent high-pressure gasses, as in a gasoline or diesel engine, and there are no explosions taking place. Because of this, Stirling engines are very quiet. The Stirling cycle uses an external heat source, which could be anything from gasoline to solar energy to the heat produced by decaying plants. Today, Stirling engines are used in some very specialized applications, like in submarines or auxiliary power generators, where quiet operation is important.

Stirling engine – a simple, practical heat engine using a gas as working substance. It’s more practical than Carnot, though its efficiency is pretty close to the Carnot maximum efficiency. The Stirling engine contains a fixed amount of gas which is transferred back and forth between a "cold" and and a "hot" end of a long cylinder. The "displacer piston" moves the gas between the two ends and the "power piston" changes the internal volume as the gas expands and contracts.

T

41

2 3

V2 V1

T1

T2

Efficiency of Stirling Engine

T

41

2 3

V2 V1

T1

T2

1-2 002

312121212 WTTnRTTCQ V

2-3 23231

22223 0ln

2

1

2

1

WQV

VnRT

V

dVnRTPdVQ

V

V

V

V

3-4 002

334212134 WTTnRTTCQ V

4-1 23232

11141 0ln

1

2

1

2

WQV

VnRT

V

dVnRTPdVQ

V

V

V

V

max1212

2

1212

12212

4123

2312 1

/ln2

3

/ln

/ln23

1

eVVTT

T

VVTT

VVTTT

QQ

QQ

QQ

Q

e CH

H

In the Stirling heat engine, a working substance, which may be assumed to be an ideal monatomic gas, at initial volume V1 and temperature T1 takes in “heat” at

constant volume until its temperature is T2, and then expands isothermally until its

volume is V2. It gives out “heat” at constant volume until its temperature is again T1

and then returns to its initial state by isothermal contraction. Calculate the efficiency and compare with a Carnot engine operating between the same two temperatures.

Internal Combustion Engines (Otto cycle)

Otto cycle. Working substance – a mixture of air and vaporized gasoline. No hot reservoir – thermal energy is produced by burning fuel.

1

2

1

2

1 11T

T

V

Ve

1

2

V

V- the compression ratio

0 – 3 intake (fuel+air is pulled into the cylinder by the retreating piston)

3 – 4 isentropic compression 4 – 1 isochoric heating 1 – 2 isentropic expansion 2 – 3 – 0 exhaust

The efficiency: (Pr. 4.18) = 1+2/f - the adiabatic exponent

For typical numbers V1/V2 ~8 , ~ 7/5 e = 0.56, (in reality, e = 0.2 – 0.3)

(even an “ideal” efficiency is smaller than the second law limit 1-T3/T1)

2V- minimum cylinder volume

- maximum cylinder volume

1V

P

V

3

4

1

2

igni

tion

exha

ust

expansion

work done by gascompressionwork done on gas

V2 V1

Patm intake/exhaust

0

- engines where the fuel is burned inside the engine cylinder as opposed to that where the fuel is burned outside the cylinder (e.g., the Stirling engine). More economical than ideal-gas engines because a small engine can generate a considerable power.

Otto cycle (cont.)

P

V

3

4

1

2

igni

tion

exha

ust

expansion

work done by gascompressionwork done on gas

V2 V1

Patm intake/exhaust

0

S

V

34

1 2

V2 V1

S2

S1

H

C

Q

Qe 1

CQHQ

0Q

0Q

41 TTCQ VH 023 TTCQ VH

1

1

2

1

2

11

22

1

4

2

3

11

22

41

32

1

2

41

32 1111

1

111

V

V

T

T

PV

PV

PPPP

PV

PV

PP

PP

V

V

TT

TTe

VC

R 1

VCR

V

Ve

/

1

21

Refrigerators

H

HH T

QS

C

CC T

QS

HQ

CQ

W

heat

work

hea

t

hot reservoir, TH

cold reservoir, TC

The purpose of a refrigerator is to make thermal energy flow from cold to hot. The coefficient of performance for a fridge:

entr

op

y

1/

1

CHCH

CC

QQQQ

Q

W

QCOP

C

H

C

H

T

T

Q

Q

CH

C

TT

TCOPCOP

max

COP is the largest when TH and TC are close to each other!

For a typical kitchen fridge TH ~300K, TC~ 250K COP ~ 6 (for each J of el. energy, the coolant can suck as much as 6 J of heat from the inside of the freezer).

A fridge that cools something from RT to LHe temperature (TC~ 4K) would have to be much less efficient.

Example:

A “perfect” heat engine with e = 0.4 is used as a refrigerator (the heat reservoirs remain the same). How much heat QC can be transferred in one cycle from the cold reservoir to the hot one if the supplied in one cycle work is W =10 kJ?

H

CH

H Q

QQ

Q

We

H

C

H

C

CH

cc

QQ

QQ

QQ

Q

W

QCOP

1

eQ

Q

H

C 1 5.14.0

6.01

11

1

e

e

e

eCOP

kJ 15cycle) one(in COPWQC