Week 13. Isentropic Efficiency
Entropy Balance
Objectives
1. Derive the reversible steady-flow work relations
2. Develop the isentropic efficiencies for various steady-flow devices
Isentropic Efficiencies of Steady-Flow Devices
The isentropic process
• involves no irreversibilities and serves as the ideal process for adiabatic devices
• The ideal process that can serve as a suitable model for adiabatic steady-flow devices (e.g. turbine, compressors, nozzle)
Isentropic Efficiencies of Turbines
The ratio of the actual work output of the turbine to the work output that would be achieved if the process between the inlet state and the exit pressure were isentropic
1 2
1 2
Actual turbine work Isentropic turbine work
( & )
a T
s a
s
w w h h
h h
ke h pe h
η = =
≅ −
−
∆ << ∆ ∆ << ∆
∵
EX 1) Isentropic Efficiency of a Steam Turbine
Isentropic Efficiencies of Compressors and Pumps
The ratio of the work input required to raise the pressure of a gas to a specified value in an isentropic manner to the actual work input
( )
2 1
2 1
2 1
2 1
Isentropic compressor work Actual compressor work
( & )
s C
a
s a
s P
a a
w w h h
h h
ke h pe h
v P P w
w h h
η
η
= =
≅ −
−
∆ << ∆ ∆ << ∆
= ≅ −
−
∵
A realistic model process for compressors that are intentionally cooled during the
compression process is the reversible
isothermal process, defined as isothermal efficiency
a t
C w
= w η
EX 2) Effect of Efficiency on Compressor Power Input
Isentropic Efficiencies of Nozzles
The ratio of the actual kinetic energy of the fluid at the nozzle exit to the kinetic energy value at the exit of an isentropic nozzle of the same inlet state and exit pressure
s a
a a
s a N
h h
h h
h V h
V V
V V
2 1
2 1
2 2 2
1 2
1
2 2
2 2
if 2
exit nozzle
at KE Isentropic
exit nozzle
at KE Actual
−
≅ −
+
⇒ =
<
=
η =
EX 3) Effect of Efficiency on Nozzle Exit Velocity
Entropy Balance
Total Total Total Change in the
entropy entropy entropy total entropy entering leaving generated of the system S
inS
outS
genS
system
− + =
− + = ∆
= Entropy Balance
Increase of entropy principle for any system
the entropy change of a system during a process is equal to the net entropy transfer through the system boundary and the entropy generated within the system
final initial 2 1
V
When the properties of the system are not uniform V
where V is the volume of the system and is density.
system
S S S S S
S s m δ s d ρ
ρ
∆ = − = −
= ∫ = ∫
Entropy Change of a System
CV gen
k
i i e e
k
Q dS
m s m s S
T + − + = dt
∑ ɺ ∑ ɺ ∑ ɺ ɺ
Mechanisms of Entropy Transfer, S
inand S
outBy Heat Transfer
By Mass Flow 0
constant) (T
constant) (T
work 2 heat 1
heat
=
≠
≅
=
=
=
∫ ∑
S
T Q T
S Q
T S Q
k
δ
kmass
mass and mass mass
where is the cross-sectional area of the flow, and is the local velocity normal to
c
n c
A t
c
n c
S ms
S s V dA S s m S dt
A
V dA
ρ δ
∆
=
=
∫
=∫
=∫
ɺ ɺ
Summary
zzz
zzz 1)
2)
Summary
zzz 3)
4)
Whew
Q
Q
Q
Q
Summary
zzz
No. Don’t go.
Please!!!
I’ll be back!!
1)
2)
3) 4)
Summary
zzz 1)
2)
Entropy Generation, S
gen(kW/K)
form rate
in the or,
(kJ/K)
entropy
in change of
Rate
system generation
entropy of
Rate gen
mass and
heat by transfer
entropy net
of Rate
out in
entropy in Change
system generation
Entropy gen mass
and heat by
ansfer entropy tr Net
out in
ɺ
ɺ ɺ
dt S
S S
S
S S
S S
∆
= +
−
∆
= +
−
s m T S
S Q ɺ ɺ ɺ
ɺ
heat= ,
mass=
Entropy balance for any system undergoing any process
Closed Systems
(kJ/K)
1 2
system
gen
S S S
T S Q
k
k
+ = ∆ = −
∑
The entropy change of a closed system during a process is equal to the sum of the net entropy transferred through the system boundary by heat transfer and the entropy generated within the system boundaries
adiabatic process :
kk
Q
∑ T + S
gen= ∆ S
adiabatic system( )
gen system surroundings
system 2 1
surroundings
system surroundings :
where,
surr
surr
S S S S
S m s s
S Q
T +
= ∆ =∆ + ∆
∆ = −
∆ =
∑
Since no mass flow across its boundaries
Control Volumes
( )
gen 2 1 CV
CV gen
(kJ/K) the rate form
(kW/K)
k
i i e e
k
k
i i e e
k
Q m s m s S S S
T
Q dS
m s m s S
T dt
+ − + = −
+ − + =
∑ ∑ ∑
∑ ɺ ∑ ɺ ∑ ɺ ɺ
The rate of entropy change within the control volume during a process is equal to the sum of the rate of
entropy transfer through the control volume boundary by heat transfer, the net rate of entropy transfer into the control volume by mass flow, and the rate of entropy generation within the boundaries of the control volume as a result of irreversibilities
The general entropy balance relations
Control Volumes (Continue)
Steady-flow process
CV gen
steady-flow
k i i e e
k
Q dS
m s m s S
T + − + = dt
∑ ɺ ∑ ɺ ∑ ɺ ɺ
( )
( )
gen
gen
gen
0
steady-flow,single stream
steady-flow,single stream,adiabatic
k
e e i i
k
k
e i
k
e i
S m s m s Q
T
S m s s Q
T
S m s s
=
= − −
= − −
= −
∑ ∑ ∑
∑
ɺ ɺ ɺ ɺ
ɺ ɺ ɺ
ɺ ɺ
If the flow through the device is reversible and adiabatic, then the entropy remains constant, regardless of the changes in other properties
