SECOND LAW OF THERMODYNAMICS

Heat Engine• Transforms heat partly into work• Working substance undergoes : heat

exchange, change in volume or change in phase

• For those operating in a cycle, ∆U=0 hence, net heat = net work• Interacts with: Hot reservoir which supplies heat QH at

constant temperature TH

Cold reservoir, which absorbs heat Qc at

constant temperature TC

Thermal Efficiency of a Heat Engine

Let: QH = heat absorbed /cycle

Qc = heat released / cycle

Q = net heat / cycle W = net work done by substance/cycleThen in a cycle,

∆U=0 and Q = W = QH + QC = H CQ Q

= 1H C

HT

H H

CQ QWe

Q Q

Q

Q

This is 100 % if 0. From experience, this is not possible

CQ

Engine Statement of the Second Law:

“No heat engine, reversible or irreversible, operating in a cycle, can take in heat from its surrounding and convert it all into work.”

Kelvin (1851)

Exercises

13/551.What is the efficiency of an electric generator that produces 1.17 kW h /kg of coal burned? The heat of combustion of coal is 6.71 x 106 J/kg.

16/551The efficiency of an engine is 0.21. For every 1 kJ of heat absorbed by the engine, how much

a. net work is done by itb. How much is released by it?

21/551An engine works at 30 % efficiency. The engine raises a 5 kg-crate from rest to a vertical height of 10 m at which point the crate has a speed of 4 m/s. How much heat input is required for this engine?

Gasoline Engine

Operations in a GASOLINE ENGINE: The Otto Cycle

1.Intake StrokeIntake valve opens exhaust valve closes, piston descends,

Air-gasoline vapor enters through the intake valve. As piston descends, volume increases from V to rV (where r = compression ratio)

2. Compression Stroke Intake valve closes. Mixture is compressed adiabatically to volume V. Mixture is ignited by spark plug

3. Power StrokeHeated gas expands adiabatically back to rV, pushing the piston. (W > 0)4. Exhaust StrokeExhaust valve opens, combustion gases rush out. Cylinder is ready for the next cycle.

H

C

H c

H

Along cd:

Along eb:

v d c

v b e

v d c v b e d c b e

v d c d c

Q nC T T

Q nC T T

nC T T nC T T T T T TQ Qe

Q nC T T T T

11 1 1

1

111 1

For the adiabatic processes bc and de:

b c c b cb

c b

e e d e dd

T V T V T rV T V

T T r

T V T V T rV T V

1

11 1

1 1 1 1

1 11

d e

e be b b e

e b e b

T T r

T T rT r T r T Te

T r T r T T r r

Thermal efficiency, 1

11Ter

et increased by increasing r, but this causes increase in temperature, mixture explodes spontaneously during

compression. This is pre-ignition detonation, heard as a knocking

sound can damage the engine. For high- octane gasoline, maximum practical r =10Note: Cycle assumes air-fuel mixture behaves as ideal

gas.Neglects friction, turbulence, heat loss through cylinder walls, incomplete combustionActual efficiencies in typical gas engines is only about 20 %

Exercises

1. The Otto-cycle engine in a Toyota Cecilia GT has a compression ratio r = 9.5a. What is the ideal efficiency of the engine?

Use = 1.4b. Other sports cars have engines with a

slightly higher compression ratio, r = 10.

How much increase in the ideal efficiency results from this increase in the

compression ratio?

2. For a gas with = 1.4, what compression ratio r must an Otto cycle have to achieve an ideal efficiency of 60 %?

The Diesel Engine

Strokes in a Diesel Engine

1. Intake stroke Air enters the cylinder

2. Compression Stroke Adiabatically compressed to a very high temperature Fuel injected at the end of stroke burns without ignition spark plug

4. Power Stroke First part is combustion at constant Pressure Last part is adiabatic expansion

5. Exhaust Stroke Combustion gases cooled at constant volume

Note:• No pre-ignition (no fuel yet during compression)• r is much higher (15 to 20)• Theoretical efficiency is about 0.65 to 0.7

• Diesel engine typically heavier than gas engines, often harder to stop; fuel-injection system requires expensive high-precision machine; requires no carburator or ignition system

Carnot Engine

•Conversion of heat to work, an irreversible process•Heat flow through a finite ∆T, an irreversible process• For maximum engine efficiency, irreversible processes must

be avoided.• Every process that involves heat transfer must be isothermal

at either TH or TC

•Any process in which the temperature of the working substance changes must be adiabatic

• These conditions are assumed in a Carnot cycle.

Along ab: ln bh ha

VQ nRT

V Along cd: ln d

c cc

VQ nRT

V

h c

h

Q Qe

Q

ln

1ln

dc

c

bh

a

VnRT

VV

nRTV

ln

1ln

dc

c

bh

a

VT

VV

TV

11 1Along bc: b c

h c cbc h

V TT V T V

V T

111Along da: a c

h a c dd h

V TT V T V

V T

ln

1

ln

dc

cT

dh

c

VT

Ve

VT

V

1 1b c a

c h d

V T V

V T V

ln ln lnb c d

c d c

V V V

V V V

b c

c d

V V

V V

1 C

H

Te

T

Exercises

24/552An ideal engine has an efficiency of 0.725 and uses gas from a hot reservoir at a temperature of 622 K. What is the temperature of the cold reservoir to which it exhausts heat?

26/552In a certain steam engine, the boiler temperature is 127 0C and the cold reservoir is at 27 0C. While this engine does 8.34 kJ of work, what minimum amount of heat must be discharged into the cold reservoir?

28/552An engine operates between temperatures of 650 K and 350 K at 65 % of its maximum possible efficiency.a. What is the efficiency of this engine?b. If 6,300 J is exhausted to the low

temperature reservoir, hoe much work does the engine do?

18-40/590 Young and FreedmanA Carnot engine whose low-temperature reservoir is at 200 K has an efficiency of 40 %. An engineer is assigned the problem of increasing this to 50 %.a. By how many kelvins must TH be increased

if Tc is to remain constant?

b. By how many kelvins must Tc be decreased

if TH is to remain constant?

18-41/590 Young and Freedman

A heat engine takes 0.2 mol of a diatomic gas around the cycle shown. Process bc is adiabatic.a. Find the pressure and volume at points a, b and c.b. Calculate Q, W and U for each process and for the

cycle.c. What is the thermal efficiency of the heat engine?d. How does compare with a Carnot engine operating

between the same minimum and maximum temperatures?

48/473 CutnellA Carnot engine has an efficiency of 0.4. The Kelvin temperature of its hot reservoir is quadrupled, and the Kelvin temperature of its cold reservoir is doubled. What is the efficiency that results from these changes?

52/473 Cutnell The hot reservoir for a Carnot engine is at 890 K while

the cold reservoir is at 670 K. The heat input for this engine is 4800 J. The 670-K reservoir also serves as the hot reservoir for a second Carnot engine. The 2 nd engine uses the rejected heat of the first engine as input and extracts additional work from it. The rejected heat from engine 2 goes into a reservoir at 420 K. Find the total work delivered by the 2 engines.

The Refrigerator

• Heat engine operating in reverse• Extracts heat QC

from food,

rejects heat QH to the surrounding,

takes in work in the process.• Refrigerant is a low-boiling fluid. Example., freon (BP= - 30 0C)

• Compression to a high pressure allows freon to liquefy when cooled.

• At condenser coils, freon cools and liquefy, releasing heat QH to the Environment. At throttling valve, it

evaporates and expands adiabatically (temperature further drops). Cold freon expands into the evaporator (cooling coils inside the refrigerator) removing heat QC

from foods. • Air conditioners operate similarly. Cooling coil is inside

the house, and condenser coils outside

Refrigerator statement of the second Law.

“It is impossible to make a refrigerator operating in a cycle whose sole effect is the transfer of heat from a cooler object to a hotter object”

Rudolph Clausius

H

CC C

C H C

QQ QK

W Q Q Q Q

W

t

greater than 1tH H

KPt P

EFFICIENCY OF REFRIGERATOR

Expressed as coefficient of performance, K (or COP or CP)

If H = rate of heat removal P = power input to compressor =

Typical value is 2.5

Carnot Refrigerator

Carnot Refrigerator

H

c c

c

Q QK

W Q Q

1

c

H

c

H

Q

Q

Q

Q

1

C

H

C

H

T

TT

T

C

H C

TK

T T

Heat Pumps

• Uses work to heat Qc from the wintry outdoors (the cold

reservoir) flow up the temperature “hill” into a warm house (the hot reservoir).

• Works in reverse to the air conditioner and the refrigerator. That is, while the air conditioner cools the inside of the house and warms the outdoors, the heat pump warms the house and cools the outdoors.

hCoefficient of Performance = h

h c

TQ

W T T

Exercises

22/551How much heat does a heat pump with a COP of 3.00 deliver when supplied with 1 kJ of electricity?

38/552An ideal refrigerator keeps its contents at 00C and exhausts heat into the kitchen at 40 0C. For every 1 J of work done, a. How much heat is exhausted?b. How much is removed from the contents?

60/474 Cutnell A Carnot refrigerator is used in a kitchen in which the temperature is kept at 301 k. This ref uses 241 J of

heat to remove 2561 J of heat from the food inside. What is the temperature inside the refrigerator?

62/474 CutnellA Carnot air conditioner maintains the temperature in a house at 297 K on a day when the temperature outside is 311 K. What is the COP of the air conditioner?

64/474 CutnellHow long would a 3 kW space heater have to run to put into a kitchen the same amount of heat to a refrigerator (COP=3) does when it freezes 1.5 kg of water at 20 0C to ice at 0 0C?

Homework

40/473 Cutnell Due to a tune up, the efficiency of an automobile engine increases by 5 %. For an input heat of

1,300 J, how much more work does the engine produce after the tune than before?

42/473 Cutnell Engine 1 has an efficiency of 0.18 and requires

5,500 J of input heat to perform a certain amount of work. Engine 2 has an efficiency of 0.26 and performs the same amount of work. How much input heat does the second engine require?

44/473 CutnellDue to design changes, the efficiency of an engine increases from 0.23 to 0.42. for the same input heat QH, these changes increase the work done by the more efficient engine and reduce the amount of heat rejected to the cold reservoir. Find the ratio of the heat rejected for the improved engine to that of the original engine.

46/473 Cutnell5000 J of heat is put into a Carnot engine whose hot and cold reservoirs have temperatures of 500 and 200 K respectively. How much heat is converted into work?

44/552 textA reversible refrigerator has a COP = 3.0. How much work must be done to freeze 1 kg of liquid water at 0 0C?