Analysis of Second Law & Reversible Cyclic Machines

P M V SubbaraoProfessor

Mechanical Engineering Department

Methods to Recognize Practicable Good Innovations…..

Kelvin Planks postulate

“It is impossible to construct a heat engine which produces no effect other than the extraction of heat

from a single source and the production of an equivalent amount of work”

Clausius postulate

“The Clausius statement: It is impossible to construct a heat pump produces no effect other than the transfer

of heat from a cooler body to a hotter body.”

Statements of Second Law of Thermodynamics

http://www.humanthermodynamics.com/2nd-Law-Variations.html

Discussion of Statements of Second Law

• Both are negative statements.

• They cannot be proved.

• They will remain correct till they are disproved.

• Violation of Kelvin Planks statement leads to violation of Clasius statement and vice versa.

• Can a heat engine be reversed to work as heat pump or refrigerator?

• If yes, what will be the COP of this reversed engine?

• Can a reversed heat pump perform same as forward engine between same reservoirs?

Reversible Heat Pump

First law:WLP = QLP - QHP

QHP

QLP

WHP

HTR (Sink)

LTR (Source)

rev = QHP/WHP

WRHP

QRHP

QRLP

First law:WRHP = QRHP - QRHP

Definition of Reversible Heat Pump

A Reversed heat pump works as a Heat Engine.

If A Heat Pump & Reversed heat pump are working between same reservoirs,

HEHP WW LELP QQ HEHP QQ

HE

HERHP Q

W

HP

HPRHP Q

W

HP

HPhp W

Q

HPRHP

1

Consequences of Second Law

• The performance of a reversed heat pumps is same as a heat engine.

• The performance of reversed heat engine is same as a heat pump.

• Heat engine and Reversed heat pump follow Kelvin-Plank statement.

• Heat pump and Reversed heat engine follow Clausius statement.

• All reversible heat engines working between same reservoirs should equally perform.

• It is impossible to construct a reversible heat engine better than another reversible machine working between same reservoirs.

• All reversible heat pumps working between same reservoirs should equally perform.

• It is impossible to construct a reversible Heat pump better than another reversible machine working between same reservoirs.

A Compound Reversible Machine

QHP

QLP

QHE

E Wnet

QHE = QHP

QLE

LTR (Source)

HTR (Sink)

QLE = QLP

Both Reversible Pump and Engine having same performance

rev = 1/rev

Perpetual Motion Machine III

Liberal Market & Innovation

• All innovations will perform equally, if each innovation is a reversible heat engine.

• All innovations will perform equally, if each innovation is a reversible heat pump.

• Innovation of reversible machines will lead to innovation of PMM –III.

• There is no scope for further innovation after first innovation.

• No need to have many ideas for a given need…

• What is this PMM-III?

Models for Reversible Machines

A Blue Print for Construction of Reversible Machine!!!!

Famous Models for Reversible Machines

• The Stirling Cycle: Reverend Robert Stirling patented a hot air engine in 1816 called “The Economiser”.

• The Carnot Cycle :1824 : Réflexions sur la puissance motrice du feu et sur les machines propres à développer cette puissance which includes his description of the "Carnot cycle".

• The Regenerative Cycle

The Reversible Cycles : Carnot Cycle

• The first model (1824) for reversible machine is the Carnot cycle.

• This consists of two reversible isothermal processes and two reversible adiabatic processes.

• Hence Carnot Cycle is a Reversible Cycle.• This mode can be used to construct either a heat engine or

a heat pump.

pv Diagram : Gaseous (Single Phase) substance executing a Carnot Cycle

• 1 – 2 : Reversible Isothermal heat addition

• 2 – 3 : Reversible Adiabatic Expansion

• 3 – 4 : Reversible Isothermal Heat Rejection.

• 4 – 1 : Reversible Adiabatic Compression.

Carnot Gas Engine : Crank-Slider Mechanism

Boiler

Condenser

LTR

Turbine

HTR

Compressor

4

12

3

Carnot Engine using Phase Change Substance

pv Diagram

1 – 2 : Boiler: Isothermal Heating : T2 = T1

No work transfer, change in kinetic and potential energies are negligible

QCV

outinCV hmhmQ

ssuming a single fluid entering and leaving…

inoutboiler hhmQ

23 yyTm high

CVout

outin

inCV WgzVhmgzVhmQ

22

2-3 : Turbine :Reversible Adiabatic Process

No heat transfer. Change in kinetic and potential energies are negligible

2

3

T

CVoutin Whmhm

4

3

23 vdpmhhmhhmW outinturbine

CVout

outin

inCV WgzVhmgzVhmQ

22

3 – 4 : Condenser : Isothermal Cooling : T3= T4

No work transfer, change in kinetic and potential energies are negligible

QCV

outinCV hmhmQ

ssuming a single fluid entering and leaving…

41 yyTmhhmQ lowinoutCondenser

CVout

outin

inCV WgzVhmgzVhmQ

22

Compressor : Reversible Adiabatic Compression Process

2

1

41 vdpmhhmW compressor

CVout

outin

inCV WgzVhmgzVhmQ

22

SSSF: Conservation of massoutin mm

First Law :

No heat transfer, change in kinetic and potential energies are negligible

Analysis of Cycle

• A Cycles is a Control Mass : Constant Mass Flow Rate

• First law: qi = wi

• qb+qc = wt+wc

wnet = qnet = h2-h1 + (h4-h1) = Thigh (y2-y1) + Tlow(y4-y1)

Thigh (y2-y1) = Thigh yboiler & Tlow (y4-y1) = Tlow ycondenser

wnet = qnet = h2-h1 + (h4-h1) = Thigh yboiler + Tlow ycondenser

qboiler = Thigh(y2-y1) = Thigh yboiler

• Efficiency of the cycle = net work/heat input

boilerhigh

condenserlowboilerhigh

b

netCarnot yT

yTyT

q

w

y is a change in a variable of a working fluid. • Different working fluid will have different values of y at same Temperatures.•However, the efficiencies of all reversible cycles operating between same reservoirs should have same efficiency!!•The magnitude of y should be same at hot and cold reservoir conditions.

bhigh

clow

b

netCarnot yT

yT

q

w

1

yyy cb

yT

yT

q

w

high

low

b

netCarnot

1

high

low

b

netCarnot T

T

q

w 1

high

low

b

cb

b

netCarnot T

T

q

qq

q

w

1

Higher the temperature of heat addition, higher will be the efficiency.Lower the temperature of heat rejection, higher will be the efficiency.Efficiency of a Reversible Engine is independent of work fluid !!!!

The Original Problem To be solved by Carnot

• What is the maximum work possible from a kg of steam?

• Is this also independent of substance ?

condenserlowboilerhighnet yTyTw

yyy cb

clowbhighnet yTyTw

yTTw lowhighnet

y is a change in property of a working fluid and depends on substance!!!How to achieve required temperature with a given substance?

The Size of A Carnot Engine

• What decides the size (capital cost) of an engine?

Work done per unit change in volume of a substance.

Mean Effective Pressure.

v

wMEP net

minmax vv

wMEP net

A mathematical model for an engine is said to be feasible iff both size and efficiency are reasonable !!!!

The Stirling engine and Stirling cycle

The Stirling Cycle

Ideal Regenerative Cycle