Isentropic Processes
For an isentropic process s1= s2
P 
s 2  s1  s 2o  s1o  R ln 2   0
 P1 
P 
s 2o  s1o
 ln 2 
R
 P1 
 s 2o  s1o  exp(s 2o / R )
P2
 
 exp
o
P1
R
exp(
s


1 / R)
Define relative pressure Pr  exps o / R 
Pr 2
 P2 
 

P
 1  s  const Pr 1
For ideal gas
v2 T2  P1  T2  Pr1  T2 / Pr 2 
 
    
v1 T1  P2  T1  Pr 2  T1 / Pr1 
Define vr  T / Pr , so
v
 v2 
 
 r2
 v1  s  const vr 1
Values of Pr and Tr as a function of temperature for air are
tabulated in Table A-22
144
Isentropic Process for ideal gas with constant cv and cP
 T2 
 v2 
s 2  s1  cV ln   R ln   0
 T1 
 v1 
note cV  R /(k  1)
 1
T 
 v 
0  R
ln 2   ln 2 
 v1 
 k  1  T1 
1
k 1
T 
v 
0  ln 2   ln 2 
 T1 
 v1 
1


k

1
T 
v 
0  ln  2    2 
 T1 
 v1 


Take exponential of both sides
T 
exp(0)  1   2 
 T1 
1
k 1
v 
  2 
 v1 
 T2 
 v1 
 
  
s

const
T
 1  c  const  v2 
k 1
p
but
Pv v 
T2 P2 v2
substituting 2 2   1 

P1v1  v2 
T1 P1v1
k 1
145
k
this yields
 P2 
v 
 
  1  or P1v1k  P2 v2k
 P1  cs const
 v2 
const
p
Recall, for a polytropic compression or expansion process
Pvn= const, for the special case of an isentropic process
(adiabatic and reversible)  n= k
Combining the two equations yields
1
k
 v1   P2 
T 
       2 
 v2   P1 
 T1 
 T2 
P 
 
  2 
 T1  cs const
 P1 
const
1
k 1
k 1
k
p
146
Control volume entropy rate balance
Similar approach to that used to derive conservation of
energy
m 2
m 1
CONTROL
VOLUME
m 3
Q j
dSCV

  m i si   m e se  S gen
j Tj
i
e
dt
Rate of
Entropy
change
Rate of entropy transfer
Rate of
Entropy
production
If temperature in CV is not uniform Tj corresponds to the
temperature at different points on the control surface
where heat is transferred
For steady-state, one inlet and one outlet, isothermal CV
S gen
1  Q CV 
0 
  sin  sout 
m  T 
m
147
Isentropic efficiencies of Turbines and Compressors
Recall, for a turbine First Law (steady-state, neglecting
KE and PE effects and heat losses) yields
P1
Expansion (P2 < P1)
W
T
WCV
 h1  h2  0
P2
m
S gen
An entropy balance yields s2  s1 
0
m
(Wout)
For an actual turbine, irreversibilities are present, so
accessible states are such that s2 > s1
1
T
P1
2
P2
2s
s
The state labeled 2s on the T-s and h-s diagrams would be
attained only in the limit of no irreversibilities, i.e.,
internally reversible expansion ( S gen  0 ) and thus s2 = s1
148
The maximum theoretical amount of turbine work output
is obtained for an isentropic expansion
WCV
 h1  h2 s
m
Since h1 - h2 < h1 - h2s the actual work produced is less
than the ideal isentropic turbine produces
The difference is gauged by the isentropic turbine
efficiency defined by
WCV / m
h1  h2
t 

WCV / m s h1  h2 s
Note, t < 1
149
Recall, for a compressor First Law (steady-state,
neglecting KE and PE effects and heat losses) yields
P1
C
Compression (P2 > P1)
WCV
 h1  h2  0 (Win)
m
W
P2
An entropy balance yields
s2  s1 
S gen
m
0
For an actual compressor irreversibilities are always
present so s2 > s1
T
2
P2
2s
P1
1
s
The state labeled 2s on the T-s and h-s diagrams would be
attained only in the limit of no irreversibilities, i.e.,
internally reversible compression where S gen  0 and thus
s2 = s1
150
The minimum theoretical amount of compressor work
required corresponds to isentropic compression
WCV
 (h2 s  h1 )
m
Since h2 – h1 > h2s – h1 the actual work input is more than
the ideal isentropic compressor requires
The difference is gauged by the isentropic compressor
efficiency defined by
c

WCV / m s

WCV / m

h1  h2 s
h1  h2
Note, c < 1
151
Internally Reversible Steady-State Flow Work
For a single inlet and exit (1-inlet, 2-exit) CV at steadystate neglecting KE and PE effects conservation of energy
 V12  V22 
W CV Q CV
  g  z1  z 2 

 h1  h2   
m
m
2 

For an internally reversible process Q / m   Tds
WCV
 12 Tds  h1  h2 
m
Recall:
Tds  dh  vdp  12 Tds  h2  h1   12 vdp
WCV
 h2  h1   12 vdP  h1  h2 
m
For pumps, turbines, compressors when KE= PE= 0
 W CV 

   12 vdP
 m  int
rev
Pumps and compressors dP > 0  work done on system
Turbines dP < 0  work done by system
152
Total Heat
Transferred
Total Work
Liquids – liquid are incompressible, so v1=v2= v
2
 W CV 

    vdP  v( P2  P1 )
1
 m  int
rev
Gases - when each unit of gas through the CV undergoes
a polytropic process Pvn= const
1
2
dP
 WCV 
n 2

    vdP   (const) 1 1
m

1

 int
rev
Pn
For the special case of an ideal gas where Pv = RT
nRT1  T2 
 WCV 
  1




n  1  T1 
 m  int
rev
n  1,
153
recall, for polytropic process
 W CV 


m


 int
rev
T2  P2 
  
T1  P1 
n 1
n
n 1



nRT1   P2  n



 1
n  1   P1 



so
n 1
Recall: If the process is internally reversible and adiabatic
(isentropic) for constant cp and cv  Pvk= const
Substitute n= k in above equations to get work per unit
mass for isentropic process (implies k  const  f (T ) )
For the case of n=1: P1v1 = P2v2  T1=T2 (isothermal)
 vdP gives:
 W CV 

   RT ln P2 P1 
 m  int
rev
n=1
154