Formula Sheet: Entropy | Thermodynamics - Mechanical Engineering PDF Download

 Page 1


En trop y F orm ula Sheet
En trop y and Second La w of Thermo dynamics
• En trop y (S ) : A measure of the disorder or una v ailabilit y of energy in a system.
S =
Z
dQ
rev
T
(Units: JK
-1
)
where dQ
rev
is the rev ersible heat transfer, T is absolute temp erature (K ).
• Sp ecific En trop y ( s ) : En trop y p er unit mass.
s=
S
m
(Units: Jkg
-1
K )
• Second La w of Thermo dynamics : The total en trop y of an isolated system nev er
decreases.
?S
univ erse
=?S
system
+?S
surroundings
=0
• Clausius Inequalit y : F or an y thermo dynamic cycle.
I
dQ
T
=0
Equalit y holds for rev ersible pro cesses.
En trop y Change
• En trop y Change for a System :
?S =
Z
dQ
rev
T
• En trop y Change for Ideal Gases :
?s= c
v
ln

T
2
T
1

+Rln

v
2
v
1

?s= c
p
ln

T
2
T
1

-Rln

P
2
P
1

where c
v
is sp ecific heat at constan t v olume ( Jkg
-1
K ), c
p
is sp ecific heat at constan t
pressure, R is the gas constan t (Jkg
-1
K ), v is sp ecific v olume ( m
3
kg
-1
), P is pressure
(Pa ).
• En trop y Change for Incompressible Substances (Liquids/Solids) :
?s= cln

T
2
T
1

where c is the sp ecific heat ( Jkg
-1
K ).
1
Page 2


En trop y F orm ula Sheet
En trop y and Second La w of Thermo dynamics
• En trop y (S ) : A measure of the disorder or una v ailabilit y of energy in a system.
S =
Z
dQ
rev
T
(Units: JK
-1
)
where dQ
rev
is the rev ersible heat transfer, T is absolute temp erature (K ).
• Sp ecific En trop y ( s ) : En trop y p er unit mass.
s=
S
m
(Units: Jkg
-1
K )
• Second La w of Thermo dynamics : The total en trop y of an isolated system nev er
decreases.
?S
univ erse
=?S
system
+?S
surroundings
=0
• Clausius Inequalit y : F or an y thermo dynamic cycle.
I
dQ
T
=0
Equalit y holds for rev ersible pro cesses.
En trop y Change
• En trop y Change for a System :
?S =
Z
dQ
rev
T
• En trop y Change for Ideal Gases :
?s= c
v
ln

T
2
T
1

+Rln

v
2
v
1

?s= c
p
ln

T
2
T
1

-Rln

P
2
P
1

where c
v
is sp ecific heat at constan t v olume ( Jkg
-1
K ), c
p
is sp ecific heat at constan t
pressure, R is the gas constan t (Jkg
-1
K ), v is sp ecific v olume ( m
3
kg
-1
), P is pressure
(Pa ).
• En trop y Change for Incompressible Substances (Liquids/Solids) :
?s= cln

T
2
T
1

where c is the sp ecific heat ( Jkg
-1
K ).
1
En trop y Change in Sp ecific Pro cesses (Ideal Gas)
• Isothermal Pro cess (T = constan t) :
?s= Rln

v
2
v
1

=-Rln

P
2
P
1

• Isobaric Pro cess (P = constan t) :
?s= c
p
ln

T
2
T
1

• Iso c horic Pro cess (V = constan t) :
?s= c
v
ln

T
2
T
1

• A diabatic Rev ersible Pro cess (Isen tropic, ?s=0 ) :
T
2
T
1
=

v
1
v
2

?-1
,
T
2
T
1
=

P
2
P
1

?-1
?
where ? =
cp
cv
is the sp e cific heat ratio.
• P olytropic Pro cess (Pv
n
= constan t) :
?s= c
v

?-n
1-n

ln

T
2
T
1

where n is the p olytropic index.
En trop y Balance
• En trop y Balance for a System :
?S
system
=
Z
dQ
T
+S
gen
where S
gen
is en trop y genera tion (JK
-1
), S
gen
=0 (irrev ersible pro cesses).
• En trop y T ransfer : Via heat transfer.
S
transfer
=
Q
T
• En trop y Generation : Due to irrev ersibilities (friction, mixing, etc.).
S
gen
=?S
univ erse
=0
2
Page 3


En trop y F orm ula Sheet
En trop y and Second La w of Thermo dynamics
• En trop y (S ) : A measure of the disorder or una v ailabilit y of energy in a system.
S =
Z
dQ
rev
T
(Units: JK
-1
)
where dQ
rev
is the rev ersible heat transfer, T is absolute temp erature (K ).
• Sp ecific En trop y ( s ) : En trop y p er unit mass.
s=
S
m
(Units: Jkg
-1
K )
• Second La w of Thermo dynamics : The total en trop y of an isolated system nev er
decreases.
?S
univ erse
=?S
system
+?S
surroundings
=0
• Clausius Inequalit y : F or an y thermo dynamic cycle.
I
dQ
T
=0
Equalit y holds for rev ersible pro cesses.
En trop y Change
• En trop y Change for a System :
?S =
Z
dQ
rev
T
• En trop y Change for Ideal Gases :
?s= c
v
ln

T
2
T
1

+Rln

v
2
v
1

?s= c
p
ln

T
2
T
1

-Rln

P
2
P
1

where c
v
is sp ecific heat at constan t v olume ( Jkg
-1
K ), c
p
is sp ecific heat at constan t
pressure, R is the gas constan t (Jkg
-1
K ), v is sp ecific v olume ( m
3
kg
-1
), P is pressure
(Pa ).
• En trop y Change for Incompressible Substances (Liquids/Solids) :
?s= cln

T
2
T
1

where c is the sp ecific heat ( Jkg
-1
K ).
1
En trop y Change in Sp ecific Pro cesses (Ideal Gas)
• Isothermal Pro cess (T = constan t) :
?s= Rln

v
2
v
1

=-Rln

P
2
P
1

• Isobaric Pro cess (P = constan t) :
?s= c
p
ln

T
2
T
1

• Iso c horic Pro cess (V = constan t) :
?s= c
v
ln

T
2
T
1

• A diabatic Rev ersible Pro cess (Isen tropic, ?s=0 ) :
T
2
T
1
=

v
1
v
2

?-1
,
T
2
T
1
=

P
2
P
1

?-1
?
where ? =
cp
cv
is the sp e cific heat ratio.
• P olytropic Pro cess (Pv
n
= constan t) :
?s= c
v

?-n
1-n

ln

T
2
T
1

where n is the p olytropic index.
En trop y Balance
• En trop y Balance for a System :
?S
system
=
Z
dQ
T
+S
gen
where S
gen
is en trop y genera tion (JK
-1
), S
gen
=0 (irrev ersible pro cesses).
• En trop y T ransfer : Via heat transfer.
S
transfer
=
Q
T
• En trop y Generation : Due to irrev ersibilities (friction, mixing, etc.).
S
gen
=?S
univ erse
=0
2
T-ds Relations
• T-ds Equations : Relate en trop y to other thermo dynamic prop erties.
Tds= du+Pdv
Tds= dh-vdP
where u is sp ecific in ternal energy ( Jkg
-1
), h is sp ecific en thalp y ( Jkg
-1
), v is sp ecific
v olume (m
3
kg
-1
).
• F or Ideal Gases :
ds=
c
v
T
dT +
R
v
dv
ds=
c
p
T
dT-
R
P
dP
Key N otes
• Use SI units: S in JK
-1
, s in Jkg
-1
K , T in K , P in Pa , v in m
3
kg
-1
.
• Standard v alues: R
air
˜ 287Jkg
-1
K , c
p, air
˜ 1005Jkg
-1
K , c
v, air
˜ 718Jkg
-1
K ,
?
air
˜1.4 .
• Isen tropic pro cess: ?s=0 (rev ersible adiabatic).
• En trop y increases in irrev ersible pro cesses due to S
gen
>0 .
• F or phase c hanges (e.g., b oiling), ?s =
h
fg
T sat
, where h
fg
is laten t heat (Jkg
-1
), T
sat
is
saturation temp erature (K ).
3