Formula Sheet: Second Law | Thermodynamics - Mechanical Engineering PDF Download

 Page 1


F ormula Sheet: Second Law of Thermodynamics
Introduction to the Second Law
• Definition : The second law of thermodynamics states that the total entrop y
of an isolated system can never decrease over time; it either increases or
remains con stant in reversible processes.
• K ey Statem ents :
– K elvin-Planck : No heat engine can convert all heat input into work;
some heat must be rejected.
– Clausius : Heat cannot flow spontaneously from a colder to a hotter
body without external work.
Entrop y
• Entrop y Cha nge (Gener al) :
dS =
dQ
rev
T
whereS = entrop y ,Q
rev
= reversible heat tr ansfer ,T = absolute temper ature
(K).
• Entrop y Cha nge for Ideal Gas :
?S =m

c
p
ln
T
2
T
1
-R ln
P
2
P
1

or
?S =m

c
v
ln
T
2
T
1
+R ln
V
2
V
1

wherem = mass,c
p
,c
v
= specific heats, R = gas constant,P,V,T = pressure,
volume, temper ature.
• Entrop y Gene r ation (Irreversible Processes) :
S
gen
=?S
system
+?S
surroundings
=0
• Clausius Ine quality :
I
dQ
T
=0
(Equality for revers ible cycles, inequality for irreversible cycles.)
1
Page 2


F ormula Sheet: Second Law of Thermodynamics
Introduction to the Second Law
• Definition : The second law of thermodynamics states that the total entrop y
of an isolated system can never decrease over time; it either increases or
remains con stant in reversible processes.
• K ey Statem ents :
– K elvin-Planck : No heat engine can convert all heat input into work;
some heat must be rejected.
– Clausius : Heat cannot flow spontaneously from a colder to a hotter
body without external work.
Entrop y
• Entrop y Cha nge (Gener al) :
dS =
dQ
rev
T
whereS = entrop y ,Q
rev
= reversible heat tr ansfer ,T = absolute temper ature
(K).
• Entrop y Cha nge for Ideal Gas :
?S =m

c
p
ln
T
2
T
1
-R ln
P
2
P
1

or
?S =m

c
v
ln
T
2
T
1
+R ln
V
2
V
1

wherem = mass,c
p
,c
v
= specific heats, R = gas constant,P,V,T = pressure,
volume, temper ature.
• Entrop y Gene r ation (Irreversible Processes) :
S
gen
=?S
system
+?S
surroundings
=0
• Clausius Ine quality :
I
dQ
T
=0
(Equality for revers ible cycles, inequality for irreversible cycles.)
1
Carnot Cycle
• Carnot Effi ciency (Heat Engine) :
?
Carnot
=1-
T
C
T
H
whereT
H
= temper ature of hot reservoir ,T
C
= temper ature of cold reservoir
(in K elvin).
• Carnot C oefficient of Performance (Refriger ator) :
COP
ref
=
T
C
T
H
-T
C
• Carnot C OP (Heat Pump) :
COP
hp
=
T
H
T
H
-T
C
Thermodynamic Processes
• Reversible Isothermal Process (Ideal Gas) :
?S =
Q
T
=mR ln
V
2
V
1
• Reversible A diabatic Process (Isentropic) :
?S =0
• Isobaric P rocess :
?S =mc
p
ln
T
2
T
1
• Isochoric P rocess :
?S =mc
v
ln
T
2
T
1
A vailability and Irreversibility
• A vailable W ork (Exergy) :
A=W
max
=(U-U
0
)-T
0
(S-S
0
)+P
0
(V -V
0
)
whereU,S,V = internal energy , entrop y , volume of system;U
0
,S
0
,V
0
,T
0
,P
0
= refer ence state properties.
• Irreversibility :
I =T
0
S
gen
whereI = lost work due to irreversibilities.
2
Page 3


F ormula Sheet: Second Law of Thermodynamics
Introduction to the Second Law
• Definition : The second law of thermodynamics states that the total entrop y
of an isolated system can never decrease over time; it either increases or
remains con stant in reversible processes.
• K ey Statem ents :
– K elvin-Planck : No heat engine can convert all heat input into work;
some heat must be rejected.
– Clausius : Heat cannot flow spontaneously from a colder to a hotter
body without external work.
Entrop y
• Entrop y Cha nge (Gener al) :
dS =
dQ
rev
T
whereS = entrop y ,Q
rev
= reversible heat tr ansfer ,T = absolute temper ature
(K).
• Entrop y Cha nge for Ideal Gas :
?S =m

c
p
ln
T
2
T
1
-R ln
P
2
P
1

or
?S =m

c
v
ln
T
2
T
1
+R ln
V
2
V
1

wherem = mass,c
p
,c
v
= specific heats, R = gas constant,P,V,T = pressure,
volume, temper ature.
• Entrop y Gene r ation (Irreversible Processes) :
S
gen
=?S
system
+?S
surroundings
=0
• Clausius Ine quality :
I
dQ
T
=0
(Equality for revers ible cycles, inequality for irreversible cycles.)
1
Carnot Cycle
• Carnot Effi ciency (Heat Engine) :
?
Carnot
=1-
T
C
T
H
whereT
H
= temper ature of hot reservoir ,T
C
= temper ature of cold reservoir
(in K elvin).
• Carnot C oefficient of Performance (Refriger ator) :
COP
ref
=
T
C
T
H
-T
C
• Carnot C OP (Heat Pump) :
COP
hp
=
T
H
T
H
-T
C
Thermodynamic Processes
• Reversible Isothermal Process (Ideal Gas) :
?S =
Q
T
=mR ln
V
2
V
1
• Reversible A diabatic Process (Isentropic) :
?S =0
• Isobaric P rocess :
?S =mc
p
ln
T
2
T
1
• Isochoric P rocess :
?S =mc
v
ln
T
2
T
1
A vailability and Irreversibility
• A vailable W ork (Exergy) :
A=W
max
=(U-U
0
)-T
0
(S-S
0
)+P
0
(V -V
0
)
whereU,S,V = internal energy , entrop y , volume of system;U
0
,S
0
,V
0
,T
0
,P
0
= refer ence state properties.
• Irreversibility :
I =T
0
S
gen
whereI = lost work due to irreversibilities.
2
T-S Diagr am
• Heat Tr ansfer :
Q=
Z
TdS
(Area under the curve on a T-S diagr am represents heat tr ansfer .)
• W ork in Rankine/Br a yton Cycles : Determined b y integr ating over the cy-
cle on T- S or P-v diagr ams.
Applications
• Used to evaluate the efficiency of heat engines, refriger ators, and heat pumps.
• Helps anal yze irreversibilities in real processes (e.g., friction, heat loss).
• Guides design of thermodynamic cycles (e.g., Rankine, Br a yton) to maxi-
mize work output.
3