Short Notes: Entropy | Thermodynamics - Mechanical Engineering PDF Download

 Page 1


En trop y and Thermo dynamics
In tro duction
En trop y is a fundamen tal concept in thermo dynamics, serving as a measure of disorder
and pla ying a critical role in the second and third la ws of thermo dynamics. It is essen-
tial for analyzing the e?iciency and feasibilit y of engineering pro cesses. This do cumen t
pro vides a comprehensiv e o v erview of en trop y , including its definition, the Clausius in-
equalit y , en trop y c hanges in v arious systems, thermo dynamic relations, and applications,
with a fo cus on b oth closed and op en systems.
1 En trop y Definition
En trop y (S ) is a thermo dynamic prop ert y that quan tifies the degree of disorder or ran-
domness in a system. It serv es as a measure of the una v ailabilit y of a system’s energy to
do w ork.
1.1 Conceptual Understanding
• En trop y reflects the p ossibilit y of con v erting heat in to w ork; higher en trop y indi-
cates l ess a v ailabilit y for w ork.
• It measures disorder: a more disorganized system has higher en trop y .
• Examples of en trop y c hanges:
– Solid to liquid (e.g., ice melting): En trop y increases due to greater molecular
freedom.
– Liquid to solid (e.g., w ater freezing): En trop y decreases as molecules b ecome
more ordered.
– Gas to liquid (e.g., condensation): En trop y decreases due to reduced random-
ness.
– Liquid to gas (e.g., ev ap oration): En trop y increases as molecules gain freedom.
1.2 Mathematical Definition
En trop y c hange for a rev ersible pro cess is defined using the Clausius theorem:
dS =

dQ
T

rev
where:
1
Page 2


En trop y and Thermo dynamics
In tro duction
En trop y is a fundamen tal concept in thermo dynamics, serving as a measure of disorder
and pla ying a critical role in the second and third la ws of thermo dynamics. It is essen-
tial for analyzing the e?iciency and feasibilit y of engineering pro cesses. This do cumen t
pro vides a comprehensiv e o v erview of en trop y , including its definition, the Clausius in-
equalit y , en trop y c hanges in v arious systems, thermo dynamic relations, and applications,
with a fo cus on b oth closed and op en systems.
1 En trop y Definition
En trop y (S ) is a thermo dynamic prop ert y that quan tifies the degree of disorder or ran-
domness in a system. It serv es as a measure of the una v ailabilit y of a system’s energy to
do w ork.
1.1 Conceptual Understanding
• En trop y reflects the p ossibilit y of con v erting heat in to w ork; higher en trop y indi-
cates l ess a v ailabilit y for w ork.
• It measures disorder: a more disorganized system has higher en trop y .
• Examples of en trop y c hanges:
– Solid to liquid (e.g., ice melting): En trop y increases due to greater molecular
freedom.
– Liquid to solid (e.g., w ater freezing): En trop y decreases as molecules b ecome
more ordered.
– Gas to liquid (e.g., condensation): En trop y decreases due to reduced random-
ness.
– Liquid to gas (e.g., ev ap oration): En trop y increases as molecules gain freedom.
1.2 Mathematical Definition
En trop y c hange for a rev ersible pro cess is defined using the Clausius theorem:
dS =

dQ
T

rev
where:
1
• dS : Differen tial c hange in en trop y (J/K)
• dQ : Differen tial heat transfer (J)
• T : Absolute temp erature (K)
• Subscript rev: Rev ersible pro cess
F or a finite rev ersible pro cess:
?S =
Z
2
1

dQ
T

rev
2 Clausius Inequalit y
The Clausius inequalit y , deriv ed from the second la w of thermo dynamics, pro vides a
criterion for assessing the rev ersibilit y of thermo dynamic cycles.
2.1 Statemen t
The Clausius inequalit y states:
I
dQ
T
=0
where the cyclic in tegral is tak en o v er a complete cycle, and:
• dQ : Heat transfer at temp erature T
• T : Absolute temp erature at the system b oundary (K)
2.2 In terpretation for Cycles
• Rev ersible cycle:
H
dQ
T
=0 .
• Irrev ersible cycle:
H
dQ
T
<0 .
• Imp ossible cycle:
H
dQ
T
>0 .
This can b e summarized in a table:
System Rev ersible Irrev ersible
Heat Engine
H
dQ
T
=0
H
dQ
T
<0
Refrigeration
H
dQ
T
=0
H
dQ
T
<0
2.3 Application to a Carnot Cycle
F o r a rev ersible Carnot cycle op erating b et w een t w o reserv oirs at temp eraturesT
H
(hot)
and T
L
(cold):
Q
H
T
H
-
Q
L
T
L
=0
2
Page 3


En trop y and Thermo dynamics
In tro duction
En trop y is a fundamen tal concept in thermo dynamics, serving as a measure of disorder
and pla ying a critical role in the second and third la ws of thermo dynamics. It is essen-
tial for analyzing the e?iciency and feasibilit y of engineering pro cesses. This do cumen t
pro vides a comprehensiv e o v erview of en trop y , including its definition, the Clausius in-
equalit y , en trop y c hanges in v arious systems, thermo dynamic relations, and applications,
with a fo cus on b oth closed and op en systems.
1 En trop y Definition
En trop y (S ) is a thermo dynamic prop ert y that quan tifies the degree of disorder or ran-
domness in a system. It serv es as a measure of the una v ailabilit y of a system’s energy to
do w ork.
1.1 Conceptual Understanding
• En trop y reflects the p ossibilit y of con v erting heat in to w ork; higher en trop y indi-
cates l ess a v ailabilit y for w ork.
• It measures disorder: a more disorganized system has higher en trop y .
• Examples of en trop y c hanges:
– Solid to liquid (e.g., ice melting): En trop y increases due to greater molecular
freedom.
– Liquid to solid (e.g., w ater freezing): En trop y decreases as molecules b ecome
more ordered.
– Gas to liquid (e.g., condensation): En trop y decreases due to reduced random-
ness.
– Liquid to gas (e.g., ev ap oration): En trop y increases as molecules gain freedom.
1.2 Mathematical Definition
En trop y c hange for a rev ersible pro cess is defined using the Clausius theorem:
dS =

dQ
T

rev
where:
1
• dS : Differen tial c hange in en trop y (J/K)
• dQ : Differen tial heat transfer (J)
• T : Absolute temp erature (K)
• Subscript rev: Rev ersible pro cess
F or a finite rev ersible pro cess:
?S =
Z
2
1

dQ
T

rev
2 Clausius Inequalit y
The Clausius inequalit y , deriv ed from the second la w of thermo dynamics, pro vides a
criterion for assessing the rev ersibilit y of thermo dynamic cycles.
2.1 Statemen t
The Clausius inequalit y states:
I
dQ
T
=0
where the cyclic in tegral is tak en o v er a complete cycle, and:
• dQ : Heat transfer at temp erature T
• T : Absolute temp erature at the system b oundary (K)
2.2 In terpretation for Cycles
• Rev ersible cycle:
H
dQ
T
=0 .
• Irrev ersible cycle:
H
dQ
T
<0 .
• Imp ossible cycle:
H
dQ
T
>0 .
This can b e summarized in a table:
System Rev ersible Irrev ersible
Heat Engine
H
dQ
T
=0
H
dQ
T
<0
Refrigeration
H
dQ
T
=0
H
dQ
T
<0
2.3 Application to a Carnot Cycle
F o r a rev ersible Carnot cycle op erating b et w een t w o reserv oirs at temp eraturesT
H
(hot)
and T
L
(cold):
Q
H
T
H
-
Q
L
T
L
=0
2
Using Carnot e?iciency:
?
th
=1-
T
L
T
H
,
Q
L
Q
H
=
T
L
T
H
Th us:
Q
H
T
H
-
Q
L
T
L
=
Q
H
T
H
-
Q
H
·
T
L
T
H
T
L
=
Q
H
T
H
-
Q
H
T
H
=0
This confirms the rev ersibilit y of the Carnot cycle.
3 En trop y Change in Closed Systems
En trop y is a state function, meaning its c hange (?S ) dep ends only on the initial and
final states, not the path.
3.1 Rev ersible Pro cesses
F o r a rev ersible pro cess b et w een states A and B :
?S =S
B
-S
A
=
Z
B
A

dQ
T

rev
F o r a rev ersible isothermal pro cess (T = constan t):
?S =
Z
B
A
dQ
T
=
1
T
Z
B
A
dQ=
Q
T
If no heat is transferred (Q=0 ), ?S =0 , making the pro cess isen tropic.
3.2 Irrev ersible Pro cesses
F o r an y pro cess (rev ersible or irrev ersible):
?S =S
2
-S
1
=
Z
2
1
dQ
T
• Equalit y holds for rev ersible pro cesses.
• F or irrev ersible pro cesses: dS >

dQ
T


irrev
.
F or a cycle with a rev ersible path (2 to 1) and a p ossibly irrev ersible path (1 to 2):
I
dQ
T
=
Z
2
1
dQ
T
+
Z
1
2

dQ
T

rev
=0
Since
R
1
2

dQ
T


rev
=S
1
-S
2
, w e get:
Z
2
1
dQ
T
=S
2
-S
1
=?S
3
Page 4


En trop y and Thermo dynamics
In tro duction
En trop y is a fundamen tal concept in thermo dynamics, serving as a measure of disorder
and pla ying a critical role in the second and third la ws of thermo dynamics. It is essen-
tial for analyzing the e?iciency and feasibilit y of engineering pro cesses. This do cumen t
pro vides a comprehensiv e o v erview of en trop y , including its definition, the Clausius in-
equalit y , en trop y c hanges in v arious systems, thermo dynamic relations, and applications,
with a fo cus on b oth closed and op en systems.
1 En trop y Definition
En trop y (S ) is a thermo dynamic prop ert y that quan tifies the degree of disorder or ran-
domness in a system. It serv es as a measure of the una v ailabilit y of a system’s energy to
do w ork.
1.1 Conceptual Understanding
• En trop y reflects the p ossibilit y of con v erting heat in to w ork; higher en trop y indi-
cates l ess a v ailabilit y for w ork.
• It measures disorder: a more disorganized system has higher en trop y .
• Examples of en trop y c hanges:
– Solid to liquid (e.g., ice melting): En trop y increases due to greater molecular
freedom.
– Liquid to solid (e.g., w ater freezing): En trop y decreases as molecules b ecome
more ordered.
– Gas to liquid (e.g., condensation): En trop y decreases due to reduced random-
ness.
– Liquid to gas (e.g., ev ap oration): En trop y increases as molecules gain freedom.
1.2 Mathematical Definition
En trop y c hange for a rev ersible pro cess is defined using the Clausius theorem:
dS =

dQ
T

rev
where:
1
• dS : Differen tial c hange in en trop y (J/K)
• dQ : Differen tial heat transfer (J)
• T : Absolute temp erature (K)
• Subscript rev: Rev ersible pro cess
F or a finite rev ersible pro cess:
?S =
Z
2
1

dQ
T

rev
2 Clausius Inequalit y
The Clausius inequalit y , deriv ed from the second la w of thermo dynamics, pro vides a
criterion for assessing the rev ersibilit y of thermo dynamic cycles.
2.1 Statemen t
The Clausius inequalit y states:
I
dQ
T
=0
where the cyclic in tegral is tak en o v er a complete cycle, and:
• dQ : Heat transfer at temp erature T
• T : Absolute temp erature at the system b oundary (K)
2.2 In terpretation for Cycles
• Rev ersible cycle:
H
dQ
T
=0 .
• Irrev ersible cycle:
H
dQ
T
<0 .
• Imp ossible cycle:
H
dQ
T
>0 .
This can b e summarized in a table:
System Rev ersible Irrev ersible
Heat Engine
H
dQ
T
=0
H
dQ
T
<0
Refrigeration
H
dQ
T
=0
H
dQ
T
<0
2.3 Application to a Carnot Cycle
F o r a rev ersible Carnot cycle op erating b et w een t w o reserv oirs at temp eraturesT
H
(hot)
and T
L
(cold):
Q
H
T
H
-
Q
L
T
L
=0
2
Using Carnot e?iciency:
?
th
=1-
T
L
T
H
,
Q
L
Q
H
=
T
L
T
H
Th us:
Q
H
T
H
-
Q
L
T
L
=
Q
H
T
H
-
Q
H
·
T
L
T
H
T
L
=
Q
H
T
H
-
Q
H
T
H
=0
This confirms the rev ersibilit y of the Carnot cycle.
3 En trop y Change in Closed Systems
En trop y is a state function, meaning its c hange (?S ) dep ends only on the initial and
final states, not the path.
3.1 Rev ersible Pro cesses
F o r a rev ersible pro cess b et w een states A and B :
?S =S
B
-S
A
=
Z
B
A

dQ
T

rev
F o r a rev ersible isothermal pro cess (T = constan t):
?S =
Z
B
A
dQ
T
=
1
T
Z
B
A
dQ=
Q
T
If no heat is transferred (Q=0 ), ?S =0 , making the pro cess isen tropic.
3.2 Irrev ersible Pro cesses
F o r an y pro cess (rev ersible or irrev ersible):
?S =S
2
-S
1
=
Z
2
1
dQ
T
• Equalit y holds for rev ersible pro cesses.
• F or irrev ersible pro cesses: dS >

dQ
T


irrev
.
F or a cycle with a rev ersible path (2 to 1) and a p ossibly irrev ersible path (1 to 2):
I
dQ
T
=
Z
2
1
dQ
T
+
Z
1
2

dQ
T

rev
=0
Since
R
1
2

dQ
T


rev
=S
1
-S
2
, w e get:
Z
2
1
dQ
T
=S
2
-S
1
=?S
3
4 En trop y Change in Op en Systems
In op en systems, mass flo w across the system b oundaries in tro duces additional en trop y
transfer terms.
4.1 En trop y Balance
The en trop y balance for an op en system is:
dS
dt
=
X
? m
in
s
in
-
X
? m
out
s
out
+
X ?
Q
k
T
k
+
?
S
gen
where:
•
dS
dt
: Rate of c hange of en trop y in the system (J/K·s)
• ? m
in
, ? m
out
: Mass flo w rates in and out (kg/s)
• s
in
,s
out
: Sp ecific en trop y of the inlet and outlet streams (J/kg·K)
•
?
Q
k
: Rate of heat transfer at temp erature T
k
(W)
•
?
S
gen
: Rate of en trop y generation (J/K·s, alw a ys=0 )
4.2 Ste ady-State Systems
F or a s teady-state op en system (
dS
dt
=0 ):
X
? m
in
s
in
-
X
? m
out
s
out
+
X ?
Q
k
T
k
+
?
S
gen
=0
This equation is useful for analyzing devices lik e turbines, compressors, and heat exc hang-
ers.
5 Ther mo dynamic Relations
En trop y can b e related to other thermo dynamic prop erties using fundamen tal equations.
5.1 E nergy Equation
The di fferen tial form of the first la w for a closed system is:
dQ-dW =dU
Substitute dQ=TdS (for a rev ersible pro cess) and dW =PdV :
TdS-PdV =dU ? TdS =dU +PdV
4
Page 5


En trop y and Thermo dynamics
In tro duction
En trop y is a fundamen tal concept in thermo dynamics, serving as a measure of disorder
and pla ying a critical role in the second and third la ws of thermo dynamics. It is essen-
tial for analyzing the e?iciency and feasibilit y of engineering pro cesses. This do cumen t
pro vides a comprehensiv e o v erview of en trop y , including its definition, the Clausius in-
equalit y , en trop y c hanges in v arious systems, thermo dynamic relations, and applications,
with a fo cus on b oth closed and op en systems.
1 En trop y Definition
En trop y (S ) is a thermo dynamic prop ert y that quan tifies the degree of disorder or ran-
domness in a system. It serv es as a measure of the una v ailabilit y of a system’s energy to
do w ork.
1.1 Conceptual Understanding
• En trop y reflects the p ossibilit y of con v erting heat in to w ork; higher en trop y indi-
cates l ess a v ailabilit y for w ork.
• It measures disorder: a more disorganized system has higher en trop y .
• Examples of en trop y c hanges:
– Solid to liquid (e.g., ice melting): En trop y increases due to greater molecular
freedom.
– Liquid to solid (e.g., w ater freezing): En trop y decreases as molecules b ecome
more ordered.
– Gas to liquid (e.g., condensation): En trop y decreases due to reduced random-
ness.
– Liquid to gas (e.g., ev ap oration): En trop y increases as molecules gain freedom.
1.2 Mathematical Definition
En trop y c hange for a rev ersible pro cess is defined using the Clausius theorem:
dS =

dQ
T

rev
where:
1
• dS : Differen tial c hange in en trop y (J/K)
• dQ : Differen tial heat transfer (J)
• T : Absolute temp erature (K)
• Subscript rev: Rev ersible pro cess
F or a finite rev ersible pro cess:
?S =
Z
2
1

dQ
T

rev
2 Clausius Inequalit y
The Clausius inequalit y , deriv ed from the second la w of thermo dynamics, pro vides a
criterion for assessing the rev ersibilit y of thermo dynamic cycles.
2.1 Statemen t
The Clausius inequalit y states:
I
dQ
T
=0
where the cyclic in tegral is tak en o v er a complete cycle, and:
• dQ : Heat transfer at temp erature T
• T : Absolute temp erature at the system b oundary (K)
2.2 In terpretation for Cycles
• Rev ersible cycle:
H
dQ
T
=0 .
• Irrev ersible cycle:
H
dQ
T
<0 .
• Imp ossible cycle:
H
dQ
T
>0 .
This can b e summarized in a table:
System Rev ersible Irrev ersible
Heat Engine
H
dQ
T
=0
H
dQ
T
<0
Refrigeration
H
dQ
T
=0
H
dQ
T
<0
2.3 Application to a Carnot Cycle
F o r a rev ersible Carnot cycle op erating b et w een t w o reserv oirs at temp eraturesT
H
(hot)
and T
L
(cold):
Q
H
T
H
-
Q
L
T
L
=0
2
Using Carnot e?iciency:
?
th
=1-
T
L
T
H
,
Q
L
Q
H
=
T
L
T
H
Th us:
Q
H
T
H
-
Q
L
T
L
=
Q
H
T
H
-
Q
H
·
T
L
T
H
T
L
=
Q
H
T
H
-
Q
H
T
H
=0
This confirms the rev ersibilit y of the Carnot cycle.
3 En trop y Change in Closed Systems
En trop y is a state function, meaning its c hange (?S ) dep ends only on the initial and
final states, not the path.
3.1 Rev ersible Pro cesses
F o r a rev ersible pro cess b et w een states A and B :
?S =S
B
-S
A
=
Z
B
A

dQ
T

rev
F o r a rev ersible isothermal pro cess (T = constan t):
?S =
Z
B
A
dQ
T
=
1
T
Z
B
A
dQ=
Q
T
If no heat is transferred (Q=0 ), ?S =0 , making the pro cess isen tropic.
3.2 Irrev ersible Pro cesses
F o r an y pro cess (rev ersible or irrev ersible):
?S =S
2
-S
1
=
Z
2
1
dQ
T
• Equalit y holds for rev ersible pro cesses.
• F or irrev ersible pro cesses: dS >

dQ
T


irrev
.
F or a cycle with a rev ersible path (2 to 1) and a p ossibly irrev ersible path (1 to 2):
I
dQ
T
=
Z
2
1
dQ
T
+
Z
1
2

dQ
T

rev
=0
Since
R
1
2

dQ
T


rev
=S
1
-S
2
, w e get:
Z
2
1
dQ
T
=S
2
-S
1
=?S
3
4 En trop y Change in Op en Systems
In op en systems, mass flo w across the system b oundaries in tro duces additional en trop y
transfer terms.
4.1 En trop y Balance
The en trop y balance for an op en system is:
dS
dt
=
X
? m
in
s
in
-
X
? m
out
s
out
+
X ?
Q
k
T
k
+
?
S
gen
where:
•
dS
dt
: Rate of c hange of en trop y in the system (J/K·s)
• ? m
in
, ? m
out
: Mass flo w rates in and out (kg/s)
• s
in
,s
out
: Sp ecific en trop y of the inlet and outlet streams (J/kg·K)
•
?
Q
k
: Rate of heat transfer at temp erature T
k
(W)
•
?
S
gen
: Rate of en trop y generation (J/K·s, alw a ys=0 )
4.2 Ste ady-State Systems
F or a s teady-state op en system (
dS
dt
=0 ):
X
? m
in
s
in
-
X
? m
out
s
out
+
X ?
Q
k
T
k
+
?
S
gen
=0
This equation is useful for analyzing devices lik e turbines, compressors, and heat exc hang-
ers.
5 Ther mo dynamic Relations
En trop y can b e related to other thermo dynamic prop erties using fundamen tal equations.
5.1 E nergy Equation
The di fferen tial form of the first la w for a closed system is:
dQ-dW =dU
Substitute dQ=TdS (for a rev ersible pro cess) and dW =PdV :
TdS-PdV =dU ? TdS =dU +PdV
4
5.2 En thalp y Relation
Using the definition of en thalp y ( H =U +PV ):
dH =dU +PdV +VdP
Substitute dU +PdV =TdS :
TdS =dH-VdP
6 Increase of En trop y Principle
The second la w states that the en trop y of the univ erse alw a ys increases, reflecting the
natural tendency to w ard disorder.
6.1 Statemen t
F or an isolated system, closed adiabatic system, or system plus surroundings:
?S =S
2
-S
1
=
Z
2
1
dQ
T
Define en trop y generation ( S
gen
):
?S =
Z
dQ
T
+S
gen
, S
gen
=0
• S
gen
=0 for rev ersible pro cesses.
• S
gen
>0 for irrev ersible pro cesses.
6.2 I mplications
• En trop y is non-conserv ativ e, unlik e energy , and the en trop y of the univ erse con tin-
uously increas es.
• The univ erse b ecomes more disorganized o v er time, approac hing a state of maxi-
m um en trop y (c haos).
• En trop y generation is due to irrev ersibilities (e.g., friction, heat transfer across a
finite temp era ture difference).
• Higher en trop y generation indicates greater irrev ersibilities, reducing the e?iciency
of devices, a s rev ersible systems are the most e?icien t.
7 Thi rd La w of Thermo dynamics
The t hird la w pro vides an absolute reference for en trop y measuremen ts.
5