10 20 30 40 50 60 70 80 90 1000
10
20
30
40
50
60
70
80
90
p
V
Isotherm, pV = constant = NkBT
Adiabat, pV = constant
v = 10:1:100;t = 100;r = 8.314;gamma = 1.67;p = r*t./v;k = (10^(gamma-1)).*r*t;pa = k./v.^gamma;plot(v,pa,v,p)
Carnot cycle v = 10:1:100;th = 100;tl = 50;r = 8.314;gamma = 1.67;p1 = r*th./v;k1 = (30^(gamma-1)).*r*th;pa1 = k1./v.^gamma;p2 = r*tl./v;k2 = (30^(gamma-1)).*r*tl;pa2 = k2./v.^gamma;plot(v,p1,v,pa1,v,p2,v,pa2)
Carnot cycle
p
V
Qh
Th
Isotherm 1
Adiabat 1
Adiabat 2
Isotherm 2
pA, VA, TA
pB, VB, TB
pC, VC, TC
pD, VD, TD
Carnot cycle
p
V
Qh
pA, VA, TA
pB, VB, TB
pC, VC, TC
pD, VD, TD
Th
Carnot cycle
Isotherm 1
Adiabat 1
Adiabat 2
Isotherm 2
p
V
Qh
pA, VA, TA
pB, VB, TB
pC, VC, TC
pD, VD, TD
Tl
Ql
Carnot cycle
Isotherm 1
Adiabat 1
Adiabat 2
Isotherm 2
p
V
Qh
pA, VA, TA
pB, VB, TB
pC, VC, TC
pD, VD, TD Ql
Tl
Carnot cycle
Isotherm 1
Adiabat 1
Adiabat 2
Isotherm 2
Why such a strange engine?
Will discuss in class
p
V
Efficiency of a Carnot engine
βπ π΄=π π΅=π h
βππΆ=π π·=π π
ππ΄ ,π π΄ ,π π΄
ππ΅ ,π π΅ ,π π΅
ππΆ ,π πΆ ,ππΆ
ππ· ,π π· ,π π·
Isotherm 1
Adiabat 1
Adiabat 2
Isotherm 2
Efficiency of a Carnot engine
ππ΄ ,π π΄ ,π h
ππ΅ ,π π΅ ,π h
ππΆ ,π πΆ ,π π
ππ· ,π π· ,π π
β Ξπh=π π h lnπ π΅
π π΄p
V
β Ξππ=π π π lnπ π·
π πΆ
β Ξπ=0βπhπ π΅
πΎβ1=π ππ πΆπΎβ1
β Ξπ=0βπ ππ π·
πΎβ1=πhπ π΄πΎβ1
Isotherm 1
Adiabat 1
Adiabat 2
Isotherm 2
Efficiency of a Carnot engine
Ξπh=π π h lnπ π΅
π π΄
Ξππ=π π π lnπ π·
π πΆ
π hπ π΅πΎβ 1=π ππ πΆ
πΎ β1
π ππ π·πΎ β1=π hπ π΄
πΎ β1
(1) From isotherm 1
(3) From isotherm 2
(2) From adiabat 1
(4) From adiabat 2
From the first law of thermodynamics: Ξπ=Ξπ+Ξπ
For the complete Carnot cycle since is a state variable
Efficiency of a Carnot engine
Ξπh=π π h lnπ π΅
π π΄
Ξππ=π π π lnπ π·
π πΆ
π hπ π΅πΎβ 1=π ππ πΆ
πΎ β1
π ππ π·πΎ β1=π hπ π΄
πΎ β1
(1) From isotherm 1
(3) From isotherm 2
(2) From adiabat 1
(4) From adiabat 2
From the first law of thermodynamics: Ξπ=Ξπ+Ξπ
For the complete Carnot cycle since is a state variable
β Ξπ=β Ξπ
From (1) and (3): +
is the work done on the engine (system), let be the work done by the engine
βπ=β Ξπ=Ξπ
Efficiency of a Carnot engine
Ξπh=π π h lnπ π΅
π π΄
Ξππ=π π π lnπ π·
π πΆ
π hπ π΅πΎβ 1=π ππ πΆ
πΎ β1
π ππ π·πΎ β1=π hπ π΄
πΎ β1
(1) From isotherm 1
(3) From isotherm 2
(2) From adiabat 1
(4) From adiabat 2
From (1) and (3): +
Efficiency is defined as: OutputInput
Output is the work done by the engine i.e. and input is the heat absorbed by the engine i.e.
βπ ( efficiency )= πΞπh
=Ξπh+ Ξπ π
Ξπh
=1+Ξππ
Ξπh
Efficiency of a Carnot engine
Ξπh=π π h lnπ π΅
π π΄
Ξππ=π π π lnπ π·
π πΆ
π hπ π΅πΎβ 1=π ππ πΆ
πΎ β1
π ππ π·πΎ β1=π hπ π΄
πΎ β1
(1) From isotherm 1
(3) From isotherm 2
(2) From adiabat 1
(4) From adiabat 2
βπ ( efficiency )= πΞπh
=Ξπh+ Ξπ π
Ξπh
=1+Ξππ
Ξπh
βπ=1+π π π ln
π π·
π πΆ
π π h lnπ π΅
π π΄
=1βπ π
πh
lnπ π·
π πΆ
lnπ π΄
π π΅
(from (1) and (3))
(2) βπ h
π π
=(π πΆ
π π΅)πΎ β1
and (4) βπ h
π π
=(π π·
π π΄)πΎβ 1
βπ πΆ
π π΅
=π π·
π π΄
βπ π΄
π π΅
=π π·
π πΆ
Efficiency of a Carnot engine
βπ=1βπ π
π h
0. There is a game
Laws of thermodynamics
1.You can never win
2. You cannot break even, either
3. You cannot quit the game
π h
π π
Carnot
πh=|Ξπh|
π π=|Ξπ π|=β Ξπ π
π=Ξπh+ Ξππ=πhβππ
Carnot engine: Schematic representation
π h
π π
Carnot
πh
ππ
π=πhβππ
Carnot engine: Schematic representation
π h
π π
Carnot
πh
ππ
π=πhβππ
Carnot engine is reversible
π h
π π
Carnot
πh
ππ
π=πhβππ
Carnot engine is reversible (refrigerator)
Carnotβs theorem
Of all heat engines working between two given temperatures, none is more efficient than a Carnot engine
reversible
π h
π π
Carnot
πh
ππ
π=πhβππ
Carnot engine is reversible (refrigerator)
R
π β²=πhβ² βπ π
β²
πhβ²
π πβ²
Adjust the cycles so that
π h
π π
Carnot
πh
ππ
π=πhβππ=πhβ² βππ
β²
Carnot engine is reversible (refrigerator)
R
πhβ²
π πβ²
π h
π π
Carnot
πh
ππ
π=πhβππ=πhβ² βππ
β²
Carnot engine is reversible (refrigerator)
R
πhβ²
π πβ²
If then:
ππhβ² >
ππh
βπhβ² <πhβπhβπh
β² >0
Also:
βπhβπhβ² =π πβππ
β² >0
π h
π π
Carnot
πh
ππ
π=πhβππ=πhβ² βππ
β²
Is this possible?
R
πhβ²
π πβ²
βπh>πhβ² πππππ>ππ
β²
βπhβπhβ² =π πβππ
β² >0
πhβπhβ²
ππβππβ²
The Second Law of Thermodynamics
β’Clausiusβ statement: It is impossible to construct a device that operates in a cycle and whose sole effect is to transfer heat from a cooler body to a hotter body.
βπ β²β―οΏ½ π
Carnotβs theorem
Of all heat engines working between two given temperatures, none is more efficient than a Carnot engine
reversible
π πππ<π πππ£
βπ πππ<ππππ£
π h
π π
Carnot
πh
ππ
π=πhβππ=πhβ² βππ
β²
For reversible engines
R
πhβ²
π πβ²
βπ β²β―οΏ½ πππππβ―οΏ½ πβ²βπ=πβ²
Carnotβs theorem
Of all heat engines working between two given temperatures, none is more efficient than a Carnot engine
All reversible engines working between two temperatures have the same efficiency as
The Second Law of Thermodynamics
β’Clausiusβ statement: It is impossible to construct a device that operates in a cycle and whose sole effect is to transfer heat from a cooler body to a hotter body.
β’Kelvin-Planck statement: It is impossible to construct a device that operates in a cycle and produces no other effect than the performance of work and the exchange of heat from a single reservoir.
π h
π π
Carnot
πh
ππ
π=πhβ² =πhβππ
Carnot refrigerator and Kelvin violator
Kelvin violator
πhβ²
βπhβππ=πhβ²
βπhβπhβ² =π π>0
π h
π π
Carnot
πh
ππ
π=πhβππ
Carnot engine and Claussius violator
Claussius violator
ππ
ππ