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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
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
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