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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
⇒𝑇 𝐴=𝑇 𝐵=𝑇 h
⇒𝑇𝐶=𝑇 𝐷=𝑇 𝑙
𝑝𝐴 ,𝑉 𝐴 ,𝑇 𝐴
𝑝𝐵 ,𝑉 𝐵 ,𝑇 𝐵
𝑝𝐶 ,𝑉 𝐶 ,𝑇𝐶
𝑝𝐷 ,𝑉 𝐷 ,𝑇 𝐷
Isotherm 1
Adiabat 1
Adiabat 2
Isotherm 2
Efficiency of a Carnot engine
𝑝𝐴 ,𝑉 𝐴 ,𝑇 h
𝑝𝐵 ,𝑉 𝐵 ,𝑇 h
𝑝𝐶 ,𝑉 𝐶 ,𝑇 𝑙
𝑝𝐷 ,𝑉 𝐷 ,𝑇 𝑙
⇒ Δ𝑄h=𝑅𝑇 h ln𝑉 𝐵
𝑉 𝐴p
V
⇒ Δ𝑄𝑙=𝑅𝑇 𝑙 ln𝑉 𝐷
𝑉 𝐶
⇒ Δ𝑄=0⇒𝑇h𝑉 𝐵
𝛾−1=𝑇 𝑙𝑉 𝐶𝛾−1
⇒ Δ𝑄=0⇒𝑇 𝑙𝑉 𝐷
𝛾−1=𝑇h𝑉 𝐴𝛾−1
Isotherm 1
Adiabat 1
Adiabat 2
Isotherm 2
Efficiency of a Carnot engine
Δ𝑄h=𝑅𝑇 h ln𝑉 𝐵
𝑉 𝐴
Δ𝑄𝑙=𝑅𝑇 𝑙 ln𝑉 𝐷
𝑉 𝐶
𝑇 h𝑉 𝐵𝛾− 1=𝑇 𝑙𝑉 𝐶
𝛾 −1
𝑇 𝑙𝑉 𝐷𝛾 −1=𝑇 h𝑉 𝐴
𝛾 −1
(1) From isotherm 1
(3) From isotherm 2
(2) From adiabat 1
(4) From adiabat 2
From the first law of thermodynamics: Δ𝑈=Δ𝑄+Δ𝑊
For the complete Carnot cycle since is a state variable
Efficiency of a Carnot engine
Δ𝑄h=𝑅𝑇 h ln𝑉 𝐵
𝑉 𝐴
Δ𝑄𝑙=𝑅𝑇 𝑙 ln𝑉 𝐷
𝑉 𝐶
𝑇 h𝑉 𝐵𝛾− 1=𝑇 𝑙𝑉 𝐶
𝛾 −1
𝑇 𝑙𝑉 𝐷𝛾 −1=𝑇 h𝑉 𝐴
𝛾 −1
(1) From isotherm 1
(3) From isotherm 2
(2) From adiabat 1
(4) From adiabat 2
From the first law of thermodynamics: Δ𝑈=Δ𝑄+Δ𝑊
For the complete Carnot cycle since is a state variable
⇒ Δ𝑄=− Δ𝑊
From (1) and (3): +
is the work done on the engine (system), let be the work done by the engine
⇒𝑊=− Δ𝑊=Δ𝑄
Efficiency of a Carnot engine
Δ𝑄h=𝑅𝑇 h ln𝑉 𝐵
𝑉 𝐴
Δ𝑄𝑙=𝑅𝑇 𝑙 ln𝑉 𝐷
𝑉 𝐶
𝑇 h𝑉 𝐵𝛾− 1=𝑇 𝑙𝑉 𝐶
𝛾 −1
𝑇 𝑙𝑉 𝐷𝛾 −1=𝑇 h𝑉 𝐴
𝛾 −1
(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.
⇒𝜂 ( efficiency )= 𝑊Δ𝑄h
=Δ𝑄h+ Δ𝑄 𝑙
Δ𝑄h
=1+Δ𝑄𝑙
Δ𝑄h
Efficiency of a Carnot engine
Δ𝑄h=𝑅𝑇 h ln𝑉 𝐵
𝑉 𝐴
Δ𝑄𝑙=𝑅𝑇 𝑙 ln𝑉 𝐷
𝑉 𝐶
𝑇 h𝑉 𝐵𝛾− 1=𝑇 𝑙𝑉 𝐶
𝛾 −1
𝑇 𝑙𝑉 𝐷𝛾 −1=𝑇 h𝑉 𝐴
𝛾 −1
(1) From isotherm 1
(3) From isotherm 2
(2) From adiabat 1
(4) From adiabat 2
⇒𝜂 ( efficiency )= 𝑊Δ𝑄h
=Δ𝑄h+ Δ𝑄 𝑙
Δ𝑄h
=1+Δ𝑄𝑙
Δ𝑄h
⇒𝜂=1+𝑅𝑇 𝑙 ln
𝑉 𝐷
𝑉 𝐶
𝑅𝑇 h ln𝑉 𝐵
𝑉 𝐴
=1−𝑇 𝑙
𝑇h
ln𝑉 𝐷
𝑉 𝐶
ln𝑉 𝐴
𝑉 𝐵
(from (1) and (3))
(2) ⇒𝑇 h
𝑇 𝑙
=(𝑉 𝐶
𝑉 𝐵)𝛾 −1
and (4) ⇒𝑇 h
𝑇 𝑙
=(𝑉 𝐷
𝑉 𝐴)𝛾− 1
⇒𝑉 𝐶
𝑉 𝐵
=𝑉 𝐷
𝑉 𝐴
⇒𝑉 𝐴
𝑉 𝐵
=𝑉 𝐷
𝑉 𝐶
Efficiency of a Carnot engine
⇒𝜂=1−𝑇 𝑙
𝑇 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
𝑇 h
𝑇 𝑙
Carnot
𝑄h=|Δ𝑄h|
𝑄 𝑙=|Δ𝑄 𝑙|=− Δ𝑄 𝑙
𝑊=Δ𝑄h+ Δ𝑄𝑙=𝑄h−𝑄𝑙
Carnot engine: Schematic representation
𝑇 h
𝑇 𝑙
Carnot
𝑄h
𝑄𝑙
𝑊=𝑄h−𝑄𝑙
Carnot engine: Schematic representation
𝑇 h
𝑇 𝑙
Carnot
𝑄h
𝑄𝑙
𝑊=𝑄h−𝑄𝑙
Carnot engine is reversible
𝑇 h
𝑇 𝑙
Carnot
𝑄h
𝑄𝑙
𝑊=𝑄h−𝑄𝑙
Carnot engine is reversible (refrigerator)
Carnot’s theorem
Of all heat engines working between two given temperatures, none is more efficient than a Carnot engine
reversible
𝑇 h
𝑇 𝑙
Carnot
𝑄h
𝑄𝑙
𝑊=𝑄h−𝑄𝑙
Carnot engine is reversible (refrigerator)
R
𝑊 ′=𝑄h′ −𝑄 𝑙
′
𝑄h′
𝑄 𝑙′
Adjust the cycles so that
𝑇 h
𝑇 𝑙
Carnot
𝑄h
𝑄𝑙
𝑊=𝑄h−𝑄𝑙=𝑄h′ −𝑄𝑙
′
Carnot engine is reversible (refrigerator)
R
𝑄h′
𝑄 𝑙′
𝑇 h
𝑇 𝑙
Carnot
𝑄h
𝑄𝑙
𝑊=𝑄h−𝑄𝑙=𝑄h′ −𝑄𝑙
′
Carnot engine is reversible (refrigerator)
R
𝑄h′
𝑄 𝑙′
If then:
𝑊𝑄h′ >
𝑊𝑄h
⇒𝑄h′ <𝑄h⇒𝑄h−𝑄h
′ >0
Also:
⇒𝑄h−𝑄h′ =𝑄 𝑙−𝑄𝑙
′ >0
𝑇 h
𝑇 𝑙
Carnot
𝑄h
𝑄𝑙
𝑊=𝑄h−𝑄𝑙=𝑄h′ −𝑄𝑙
′
Is this possible?
R
𝑄h′
𝑄 𝑙′
⇒𝑄h>𝑄h′ 𝑎𝑛𝑑𝑄𝑙>𝑄𝑙
′
⇒𝑄h−𝑄h′ =𝑄 𝑙−𝑄𝑙
′ >0
𝑄h−𝑄h′
𝑄𝑙−𝑄𝑙′
The Second Law of Thermodynamics
•Clausius’ 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.
⇒𝜂 ′≯� 𝜂
Carnot’s theorem
Of all heat engines working between two given temperatures, none is more efficient than a Carnot engine
reversible
𝑊 𝑖𝑟𝑟<𝑊 𝑟𝑒𝑣
⇒𝜂 𝑖𝑟𝑟<𝜂𝑟𝑒𝑣
𝑇 h
𝑇 𝑙
Carnot
𝑄h
𝑄𝑙
𝑊=𝑄h−𝑄𝑙=𝑄h′ −𝑄𝑙
′
For reversible engines
R
𝑄h′
𝑄 𝑙′
⇒𝜂 ′≯� 𝜂𝑎𝑛𝑑𝜂≯� 𝜂′⇒𝜂=𝜂′
Carnot’s 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
•Clausius’ 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.
•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.
𝑇 h
𝑇 𝑙
Carnot
𝑄h
𝑄𝑙
𝑊=𝑄h′ =𝑄h−𝑄𝑙
Carnot refrigerator and Kelvin violator
Kelvin violator
𝑄h′
⇒𝑄h−𝑄𝑙=𝑄h′
⇒𝑄h−𝑄h′ =𝑄 𝑙>0
𝑇 h
𝑇 𝑙
Carnot
𝑄h
𝑄𝑙
𝑊=𝑄h−𝑄𝑙
Carnot engine and Claussius violator
Claussius violator
𝑄𝑙
𝑄𝑙