The Second Law of Thermodynamics - Web Space - OIT...• The Isentropic process involves no...
Transcript of The Second Law of Thermodynamics - Web Space - OIT...• The Isentropic process involves no...
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The Second Law of Thermodynamics
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Note: for a reversible process, on p-v diagram
and If the volume is constant,
revWvPd =∫2
1
02
1
=∫ vPd
1
2
wpdv =∫2
1
P
V
W=0
21 vv =
2
1
P
V
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• Similarly, for any reversible adiabatic process, Q=0
• For any other reversible process
0== ∫TdsQrev
∫=2
1
TdsQrev
T
S
1
2
∫=2
1
TdsQrev
1S
2S
21 SS =0=Q
T
S
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• Clausius Inequality:
0≤∫ TQδ
• Clausius inequality is valid for all thermodynamic cycles, reversible or irreversible.
for internally reversible cycle
for irreversible cycle
0=⎟⎠⎞
⎜⎝⎛∫
revTQδ
0<⎟⎠⎞
⎜⎝⎛∫ T
Qδ
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represents a property changerevT
Q⎟⎠⎞δ
∫ ⎟⎠⎞=−=∆
⎟⎠⎞=
2
112
rev
rev
TQSSS
TQdS
δ
δ
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• Consider a cycle composed of a reversible and an irreversible process:
irreversibleprocess
internallyreversibleprocess
1
2
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• The quantity represents the entropy change of the system.
• For a reversible process, entropy transfer with heat
• For an irreversible process
• Some entropy is generated during an irreversible process and is always positive quantity. Its value depends on the process, and thus it is not a property of the system.
12 sss −=∆
∫=∆2
1 TQs δ
gensys sTQs +=∆ ∫
2
1
δ
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Entropy Change for Pure Substances
• The entropy of a pure substance is determined from the tables, just as for any other property.
T
s
1
1
TP
11 Tatss f≈
1 Saturated liquid-vapor mixture
2
2
xT
fg2f2 sxss +=
2
2
PT
3s =from superheatedtable
2
3
fgf xsss +=( )12 ssmS −=∆
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• T-s diagram of properties:
)( 12
2
1
SSTQ
TdsQ
TdsQ
rev
rev
rev
−=
=
=
∫
δ
Constantpressure lines
=revQArea underthe process
1 2
T
s
• Entropy of a fixed mass can be changed by:1) Heat transfer2) Irreversibilities
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• The entropy of a fixed mass will not change during a process that is internally reversible and adiabatic.
• For a reversible, adiabatic process (called Isentropic process)
210 sss =⇒=∆
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• If a steam turbine is reversible, and the turbine is insulated (thus the process is reversible and adiabatic)
21
0then
ssor
s
=
=∆T
s
1
2
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• The Tds relations: differential form of thefirst law:
second law:
dUwQ revrev =−δδ
TdsQrev =δ
duPdvTdsdUVPdTdS
+=+=
vdPdhTdsvdPPdvdudh
PvuhPdvduTds
−=++=
+=+=
First Tds relation
Second Tds relation
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• Entropy changes for Liquids and solids:Tds=du+pdv
ds=
• Liquids and solids are incompressible substances
Tpdv
Tdu
+
0≅dv
1
22
112 ln
TTc
TdTcss
TcdT
Tduds
==−
==
∫
( )
kgKkJ
ccc vp ==
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Isentropic process for liquids and solids:
the isentropic process of an incompressible substance is also isothermal.
121
212 ln0 TT
TTcss =⇒==−
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Note: Tds equations are derived by considering an internally reversible process. An entropy change obtained by integrating these equations is the change for any process.
“Entropy is a property and the change in entropy between any two states is independent of the details of the process linking the states.”
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Tds equations are used to evaluate the entropy change between two states of an ideal gas.
dvTv
Tdhds
dvTP
Tduds
−=
+=
Entropy change of an Ideal gas
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For an ideal gas:
PdPR
TdTcds
vdvR
TdTcds
RccRTPv
dTcdhdTcdu
P
v
vp
p
v
−=
+=
+==
==
,
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On integration these equations give, respectively
To integrate these relations, we must know the temperature dependence of the specific heats.
1
212
1
212
ln
ln
2
1
2
1
PPR
TdTcss
vvR
TdTcss
T
TP
T
Tv
−=−
+=−
∫
∫
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Using Ideal gas Tables:
Define
Where is the specific entropy at a temperature T and a pressure of 1 atm.
Note: The specific entropy is set to zero at the state where the temperature is 0 K and the pressure is 1 atm.
dTTTc
TsT
po ∫=0
)()(
)(Tso
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Note: Because So depends only on the temperature, it can be tabulated versus temperature, like h and u.
Kmole.K
KJ ln)(
Kg.KKJ ln)(
;
,
1
21212
1
21212
1200
122
1
PPRssssor
PPRssss
then
ssTdTc
TdTc
TdTc
Also
uoo
oo
ooT
p
T
p
T
Tp
−−=−
−−=−
−=−= ∫∫∫
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Entropy change of an Incompressible substance:for incompressible substance, specific heat depends solely on temperature, and
When the specific heat is constant:
Cconstant ible,incompress ln1
212 ←=−
TTCss
ibleincompress )(
)()(
)(
2
1
12 ←=−
=+=
+=
=
∫T
T TdTTCSS
TdTTCdv
TP
TdTTCds
dvTP
Tduds
dTTCdu
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Isentropic Processes of an Ideal Gas
For internally reversible process:
For Isentropic process(Reversible, adiabatic process):1
22
112
1
22
112
ln)(
ln)(
VVR
TdTTcss
PPR
TdTTcss
v
p
−=−
−=−
∫
∫
012 =−ss
1
22
1
1
22
1
ln)(0
ln)(0
VVR
TdTTc
PPR
TdTTc
v
p
−=
−=
∫
∫
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kk
PP
TT
similarly1
1
2
constants1
2
,−
=⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛1
2
1
constants1
2
1
2
1
1
2
2
1
1
2
1
2
1
2
1
2
1
2
1
2
lnln
lnlnln
lnln0
lnln0
−
=
−
⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛=
⎟⎟⎠
⎞⎜⎜⎝
⎛=−=⇒
+=
−=
k
k
cR
v
v
p
VV
TT
VV
TT
VV
VV
cR
TT
VVR
TTc
PPR
TTc
v
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Integrate:
heats specificconstant th wi gas, ideal process Isentropic Constant
Constant )ln(Constant lnConstant lnln
Constant lnln
lnln Constant
1
1
←=⇒
=×
=−=+
=+⇒=
+=
−
−
k
k
kv
v
PvvPv
RvPv
vcR
RPvRTVPuse
vRTc
TdvP
Tduds
PdvduTds
+=
+=
vdvR
TdTcv +=0
Ideal gas, Isentropic Process(PV=RT, ds=0)
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Rss
PP
Rss
PP
PPRss
PPRssss
PPR
TdTcss
oo
oo
oo
oo
T
Tp
12
1
2
12
1
2
1
212
12
1
21212
1
212
exp
ln
ln
0ss : process Isentropic
ln
ln2
1
−=
⎟⎟⎠
⎞⎜⎜⎝
⎛ −=
=−
=−
−−=−
−=− ∫
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Relative pressure and relative volume:
The quantity is a function of temperature only and is given the symbol Pr, define as the relative pressure, and is tabulated for air and other ideal gases
)/exp()/exp(
1
2
1
2
RsRs
PP
o
o
=
)/exp( Rso
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Note: Pr is not truly pressure, and also Pr should not be confused with the reduced pressure of compressibility chart.
)T(PP and )T(PP where
2rr1rr
1
2
21
1
2
==
=r
r
PP
PP Isentropic Process
(s1=s2) Ideal gas variable specific heats
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Similarly a relation between specific volumes and temperatures and for two states having the same entropy can also be developed
The ratio is the relative volume
or
1
1
2
2
1
2
⎟⎟⎠
⎞⎜⎜⎝
⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛
=
RTPP
RT
vv
rr
vPRT
=
1
2
1
2 ←=r
r
vv
vv Isentropic process,
Ideal gas variable specific heats
⎟⎟⎠
⎞⎜⎜⎝
⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛
=
1
2
1
2
1
2
RTPPRT
vv
r
r
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In general Isentropic process for ideal gas: (s2=s1)
re temperatuoffunction a isheat specific
heat specificconstant
1
2
1
2
1
2
1
2
←
⎪⎪⎭
⎪⎪⎬
⎫
=
=
←=
vv
vv
PP
PP
CPv
r
r
r
r
k
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General, reversible (Polytropic) process for an Ideal gas:Pvn=c, where n is constant
∫=
⎟⎟⎠
⎞⎜⎜⎝
⎛=
=
2
1
2
1
1
2
2211
PdvW
vv
PP
vPvPn
nn
12211
2
1 −−
== ∫ nvPvPdv
vc
n
1
2Slope=-n
P
Log VP
V
cPvn =1
2
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T
S
n=1
n=k constant)(v =±∞=nn=0 (P=constant)
n=1
1<n<k
±∞=n
Constant pressure
n=1 (T=constant)Isothermal process
n=k (S=constant)Isentropic process
V=constant
n=0
P
V
Polytropic process on P-v and T-s diagrams
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Isentropic Efficiencies of Steady-Flow Devices
• The Isentropic process involves no irreversibilitiesand serves as the ideal process for adiabatic devices.
• The actual process is irreversible and the actual device performance is less than the ideal case.
• The more closely the actual process approximates the idealized isentropic process, the better the device will perform.
• We define the efficiency (isentropic efficiency) of these devices as a measure of deviation of actual processes from the idealized one.
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1) Isentropic efficiency of Turbines
00
21
21t
t
100η
workIsentropic work turbineActualη
<−−
==
=
s
a
s
a
hhhh
ww
T
s
1
1P
2P
s2a2
21 hhw −=&
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2) Isentropic efficiency of Compressors and Pumps
12
12pc
pc
ηor η
ηor η
hhhh
WW
a
s
a
s
−−
=
=
T
s
T
s
a2s2
1
s2a2
2P
1P
1
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3) Isentropic efficiency of Nozzles
s
a
aa
s
a
hhhh
Vhh
VV
21
21N
22
21
22
22
N
N
η
2 :Note
η
exit Nozzleat K.E. Isentropicexit Nozzleat E. K. Actualη
−−
≅
+=
=
=
T
s
1
1P
2P
s2a2