Chapter 3
One Dimensional Flow
in
Turbo-Machines
VELOCITY DIAGRAMS FOR AN AXIAL TURBINE
STAGE
cU
Turbine stage velocity diagrams
VELOCITY DIAGRAMS FOR A RADIAL
CENTRIFUGAL COMPRESSOR STAGE
VELOCITY DIAGRAMS
&
ENERGY TRANSFER
C W  U
Where C & W are the absolute and relative
velocity vectors of flow
U is the peripheral velocity.
U
b
a
C
UCu = 1/2
W
2
(C
2
+U
–
2
W)
VELOCITY DIAGRAMS FOR A RADIAL
CENTRIFUGAL COMPRESSOR STAGE
ENERGY TRANSFER
Euler turbine equation
 w  U 2Cu 2  U1Cu1

2
C2
2
 C1
2

2
U2
2
2
2
 U1
2

2
1
C
C
 h 2  h1 

2
2
h1
2
W1

2
2
U1

2
 h2
2
W2

2
2
U2

2
2
W2
2
 W1
2
 cons tan t  I
= RELATIVE TOTAL ENTHALPY OR ROTHALPY
COMPONENTS EFFICIENCY ROTOR (Turbine)
TURBINE ROTOR EXPANSION
ENERGY TRANSFER
Euler turbine equation for isentropic flow
C C
W W
U U
 w isT 


 w isC
2
2
2
2
2s
2
1
2
2
2
2s
2
1
2
2s
2
1
2
1
C
C
 h 2s  h1 

2
2
2
1
2
1
2
2
2
2
W
U
W
U
h1 

 h2 

2
2
2
2
2
2
W2s U 2
 h 2s 

 cons tan t  I
2
2
COMPONENTS EFFICIENCY
The ratio between the component
exit kinetic energy
to
the corresponding exit kinetic energy in
case of isentropic flow
based on
Relative velocities in case of rotors
Absolute velocities in case of stators
COMPONENTS EFFICIENCY
STATOR
The energy equation for the process in a
nozzle or a stator
h 00  h 01  h 01s  constant
The efficiency of the nozzles or the inlet stator
2
2
2
h 0  C0 2  h1  C1 2  h1s  C1s 2
2
h 00  h1
C1 2
n  s1 

2
h 00  h1s
C1s 2
COMPONENTS EFFICIENCY
NOZZLE (STATOR S1)
h-s diagram for expansion in a nozzle (stator S1)
C12 2
2
2
s1 
 C1 C1s
2
C1s 2
COMPONENTS EFFICIENCY
DIFFUSER (RECOVERY ELEMENT) (STATOR S2)
h-s diagram for compression in a diffuser (stator S2)
2
3
2
3s
C
d  s 2 
C
2
 C23 C23s
2
COMPONENTS EFFICIENCY ROTOR (Turbine)
TURBINE ROTOR EXPANSION
r 
W22 2
W22s 2
 W22 W22s
COMPONENTS EFFICIENCY ROTOR (impeller)
COMPRESSOR ROTOR (impeller) COMPRESSION
r 
2
W2
W22s
2
2
 W22 W22s
THE ENERGY LOSS COEFFICIENT
For the nozzle
C / 2C
 s1 1  s1 
2
C1s 2
2
1s
2
1
2
2
2
2
 (h1  h1s ) (C / 2)
2
1s
For the rotor
W / 2 W
 r  1  r 
2
W2 s 2
2
2s
h-s FOR A TURBINE STAGE
h-s FOR A TURBINE STAGE
TURBINE STAGE EFFICIENCY
The total-to-total stage efficiency
tt   W 
 Ws 
1
h 01  h 0 2

h 01  h 02s
 c12 c 22 
h1  h 2     
2 2


 c12 c 22 
h1  h 2s     
2 2
The positive for turbines & the negative for compressor
2
(C1
2
2
2
2
2
 C 2 )  ( U1  U 2 )  ( W1  W2 )
tt 
2
2
2
2
2
2
(C1 / S1  C 2 )  ( U1  U 2 )  ( W1  W2 / r )
For axial flow turbines:
2w
tt  2
2
2
2
C1 / S1  C2  W2 / r  W1
h-s FOR A COMPRESSOR STAGE
h-s FOR A COMPRESSOR STAGE
COMPRESSOR STAGE EFFICIENCY
The total-to-total stage efficiency
 c32S c12 
h3SS  h1     
2
2
hO 3SS  h1



hO 2  h1
 c22 c12 
h2  h1     
2 2
WS 

tt 
W 

(C22  C12 / S1 )  ( U 22  U12 )  ( W22 / r  W12 )  C23 (1  1 / S 2 )
tt 
(C22  C12 )  ( U 22  U12 )  ( W22  W12 )
For axial flow compressors:
(C  C / S1 )  ( W / r  W )  C (1  1 / S 2 )
tt 
2W
2
2
2
1
2
2
2
1
2
3
TURBINE STAGE EFFICIENCY (continue)
The static to static turbine efficiency:
If the steam velocity at turbine inlet and exit is equal, c1
= c2, then the turbine total to total efficiency can be
written as:
 t .t .
h1  h 2

 s .s .
h1  h 2 s
which is called static to static turbine efficiency .
It must be mentioned here that the total to total efficient
(t.t.t) is used for single turbine but the static to static turbine
efficiency general used in multistage machine.
TURBINE STAGE EFFICIENCY (continue)
When the turbine exit velocity is totally wasted (i.e.
there is no recovery), the total back pressure equals
the exit static pressure. Putting C2 = 0 in the last
relation, gets the turbine total - to - static
efficiency
2w
ts  2
C1 / S1  W22 / r  W12

2w
C12 / S1
(For impulse , where h1  h 2s , W12  W22s )
Note that in compressors C2 is generally large and
the recovery compnent (S2) is generally present.
Therfore the term compressor total - to - static
efficiency is not used.
THE DEGREE OF REACTION
THE DEGREE OF REACTION
The degree of reaction is a measure of the enthalpy (or pressure)
drop or rise in the rotor to the total enthalpy (or pressure) drop or
rise in the stage. There are different definitions of the degree of
reaction, the most important are:.
1.
R = hr / hst = hr / (hr + hs)
where hr , hs & hst are the actual enthalpy drops (or
rises) through the rotor, stator and stage respectively.
From the relation
 h  h  h  h  (C 2  C 2 ) 2
st
00 02
0 2
0
2
h  h  h  h  h  (C 2  C 2 ) 2
r
01 02 1 2
1
2
 (U 2  U 2 ) 2  (W 2  W 2 ) 2  (C 2  C 2 ) 2
1
2
1
2
1
2
h
for C  C
0
2
THE DEGREE OF REACTION (continue)
Then the degree of reaction is,
1
1
2
2
2
2
R  [( U1  U 2 )  ( W1  W2 )] / [( U12  U 22 )  ( W12  W22 )  (C12  C 22 )]
2
2
For axial machines U1 = U2 and
R  ( W22  W12 ) / [( W22  W12 )  (C12  C22 )]
It has to be noticed that this expression is really correct
for repeated stages where for a single stage C2 must be
replaced by Co.
THE DEGREE OF REACTION (continue)
2. Sometimes the degree of reaction is based on the isentropic
enthalpy drop (or rise),
Rs = hr,s / hst,s = hr,s / (hr,s + hs,s)
where hr,s , hs,s & hst,s are the values of isentropic enthalpy
drop (or rise) in the rotor, stator and stage respectively.
The above definition has the advantage that it could also be
applied directly to hydraulic machines in the form:
Rs = Pr / Pst = Pr / (Pr + Ps)
In terms of the velocities and for repeated stages this degree of
reaction is:
1
1
R  [( W22 / r  W12 )  ( U12  U 22 )] / [( W22 / r  W12 )  ( U12  U 22 )  (C12 / s  C 22 )]
2
2
THE DEGREE OF REACTION (continue)
3. For axial machines, specially pumps and compressors, a
kinematic degree of reaction is usually used. This can be obtained
by assuming constant axial velocity through the stage, i.e.,
Ca1 = Ca2
hence, using relation (3.6), the kinematic degree of reaction will be
( Wu22  Wu21 )
( W22  W12 )
R 

2 U Cu
2U ( Wu 2  Wu1 )
Wu1  Wu 2
Cu1  Cu 2 Wu
C u

 1

 1
2U
2U
U
U
It is evident that this expression of reaction is easy to calculate
from the velocity diagrams directly.
Although that there are some differences in the three mentioned
expressions of the degree of reaction, but all of them give the
same concept which is, degree of reaction is A a ratio between
enthalpy change in the rotor and that change in the whole stage.
THE DEGREE OF REACTION (continue)
Therefore, the degree of reaction is an important
parameter which specifies classes of machines
with particular characteristics. This parameter is
specially important for axial machines, since if it is
fixed, the shape of velocity diagrams and blade
arrangements are also fixed.
This will be illustrated by examining axial flow
stages having different degrees of reaction. It will
be assumed that all stages will have the same
peripheral speed U, axial velocity Ca , and energy
transfer (work) w (i.e. same Cu) i.e. the same f
and . All these types of blade arrangements are
possible, although not all of practical use.
EFFECT OF DEGREE OF REACTION ON DESIGN OF AXIAL
TURBINE STAGES
Effect of degree of reaction on design of axial
turbine stages:
With stage R < 0
W2 < W1
and there will be diffusion action (rise of pressure)
inside the rotor. This is undesirable since it leads
to high energy losses and hence this type of
stages are not used and must be avoided.
However, this may occur in old designs of Steam
impulse stages, with holes in the disc to equalize
the pressure P1 and P2.
EFFECT OF DEGREE OF REACTION (continue)
1. Zero reaction stage, R = 0
For a zero reaction stage, there is no enthalpy drop in the rotor
and the total enthalpy drop takes place in the stator (nozzle). This
means that the rotor blades will only turn the flow from the
direction of W1 to the direction of W2 at exit, Fig. 3.13. Assuming
that the flow is isentropic, then the condition R=0 (or zero
enthalpy drop across the rotor) implies no change in pressure
across the rotor.
TWO-DIMENSIONAL CASCADES (continue)
R = 0 for isentropic process
R < 0 for adiabatic process
Pure impulse stage P1 = P2
R < 0 for adiabatic process
R = 0 for isentropic process
Negative reaction stage
EFFECT OF DEGREE OF REACTION (continue)
These stages with zero pressure drop in the rotor are usually
termed “impulse-type”. For non-isentropic flow, impulse stages will
have a negative degree of reaction as W2 is less than W1 due to
friction effects. In order to have zero reaction, there must be a
slight expansion (pressure drop) in the rotor so that W2 = W1.
However, stages with reaction R<0.1 are also termed impulse and
are widely used.
To obtain the maximum efficiency of an impulse stage, assume
there is no expansion in the rotor
hr = 0
and hst = hs1 = hN = ho – h1 = C12/2 – Co2/2
The stage efficiency then can be written as,
ts=w/(h0st,s+C22/2)
= s1 U (Cu1 - Cu2) / (C12/2)
(3.29)
From the velocity diagram Fig. 3.13,
Cu1 = C1 cos a1 = U + W1 cos b1
C = C cos a = U + W cos b
EFFECT OF DEGREE OF REACTION (continue)
Therefore,
ts = (2s1 U/ C12) (W1 cos b1 - W2 cos b2)
= 2s1 (U/C1) (W1 cosb1/ C1) (1 – (W2 / W1)(cosb2/ cosb1))
Putting: W1 cos b1 = C1 cos a1 - U = C1 (cos a1 – U / C1),
(U/ C1) =  = velocity ratio
(W2 / W1) = √r
ts = 2s1  (cos a1 - ) (1 – √r (cos b2 / cos b1)) (3.30)
For constant values of s1, r, a1, b1 and b2 it is easy to get
that the optimum value of ts for an impulse stage is reached
when the speed ratio:
opt = cos a1 / 2
(3.31)
This value satisfies the condition of axial exit flow from the
rotor, i.e. the velocity C2 is minimum. For that case the optimum
efficiency is given by:
ts = (cos2a1 / 2) s1 (1 – √r (cos b2 / cos b1))
(3.32)
EFFECT OF DEGREE OF REACTION (continue)
Considering ideal flow through the stage and
(cos b2 / cos b1) = -1
Then the total - to - static efficiency will be
ts = cos2a1
(3.33)
It is evident that a1 must be as small as possible in order
to have the maximum efficiency.
Practically, a1 ranges between 0 - 20° for impulse
turbines
(reaches
zero
for
Pelton
turbines).
Correspondingly the optimum speed ratio is in the range:
  0.45 – 0.5
Effect of degree of reaction on design of axial turbine stages (continue)
2. Stage with 50% reaction R = 0.5
This type of stages is widely used and gives:
• The enthalpy drop in the stage is equally divided between the
stator and rotor
• Velocity triangles are symmetric
• Blade angles of both stator and rotor are identical
• maximum efficiency
• similar rotor and stator blades gave desirable manufacturing
advantage led to the wide use 50% reaction.
TWO-DIMENSIONAL CASCADES (continue)
A 50 per cent reaction stage
Effect of degree of reaction on design of axial turbine stages (continue)
The efficiency of the 50% reaction stage is obtained by
considering equal enthalpy drop in the stator and rotor, i.e.:
hs,s = hr,s
Hence, we may write:
W22
2r

W12
2

C12
2s1

2
C0
2
Since the enthalpy drop is equal in the rotor and stator, then it
could be assumed that r = s1. Furthermore, for repeated
stages C0 = C2 and this will mean that velocity diagram is
symmetrical, Fig. 3.13.
The total-to-static efficiency of this stage, without recovery of
the exit energy C22/2 , will be:
ts  2w (C12 s1  w 22 r  w12 )
Effect of degree of reaction on design of axial turbine stages (continue)
Since s1 = r = , W1 = C2 and C1 = W2, then:
ts 
2 U(C u 1  Cu 2 )

2
2C1
/ 
2
C2

putting:
Cu2 = W2 cos (180 - b2) – U = C1 cos a1 - U
Cu1 = C1 cos a1
2
U
(
2
C
cos
a

U
)
1
1
 ts 
2

(
2
cos
a


)
1

2C /  C cosa U   C 
2
2
1
1
1
 2 /  cosa    sin a 
2
1
2
1
2
a
Effect of degree of reaction on design of axial
turbine stages (continue)
If a1 and  are considered constant, then for
maximum efficiency, it is easy to get that:
opt = cos a1
It should be noted that, also for repeated
stages, where recovery of exit energy C22/2 is
possible the same result is obtained. In this case
the total-to-total efficiency of the stage is possible
the same result is obtained. In this case the totalto-total efficiency of the stage is written as:
Effect of degree of reaction on design of axial
turbine stages (continue)
2 U(C u 1  C u 2 )
tt 
C12 / s1  C 22  ( W22 / r  W12 )


tt 


2 U(C u 1  C u 2 )
2C
 ( 2 cos a1   )
1 /   ((cos a
2
1
2
2

1
  )  sin a1 )
/ C
2
2

Effect of degree of reaction on design of axial
turbine stages (continue)
And the optimum speed ratio is also given by:
opt = cosa1
As it is mentioned before, for impulse stages, the
above condition means that the exit velocity C2
must be axial (i.e. minimum).
For ideal stage, with s1 = r = 1, the optimum
stage efficiency from relation (3.35) will be:
ts ,opt 
2 cos a1
2
1  cos
2
a1

Effect of degree of reaction on design of axial
turbine stages (continue)
Comparing the last relation and the relation
(3.33) giving optimum value of for an ideal
impulse stage, it is clear that the 50% reaction
stage will have higher efficiency as the quantity (1
+ cos2a1) is less than 2 always.
It should be noted, as it has mentioned before,
that the impulse stage will give work
approximately double that of a 50% reaction
stage for the same U, Ca.
Effect of degree of reaction on design of axial turbine stages (continue)
3. Stage with 100% or more, reaction R ≥ 1.0
This type of reaction stages has a poor efficiency
because in this case there will be deceleration of flow in
the stator (C0 > C1). This, together with the necessity of
large deflection angle in the stator causes high energy
loss in the stage.
From velocity triangle for R=1 C1 = C2.
Generally, these types of stages are not used.
TWO-DIMENSIONAL CASCADES (continue)
Figure 6.12 A 100 per cent reaction stage
TWO-DIMENSIONAL CASCADES (continue)
Figure 6.13 Stage expansion with reaction more than 100 percent
5. Choice of Turbine Reaction
For multi-stage axial machines, the selection of
number of stages required for a given total
energy transfer depends large on the chosen
degree of reaction for each stage.
Since maximum peripheral speed is limited by
blade material stress considerations, the zero
reaction stage will transfer twice as much energy
per stage approximately as will the 50% reaction
stage. Therefore, the number of stages of zeroreaction multi-stage turbine is approximately half
the number of stages in a 50% reaction multistage machine for the same total energy transfer
in both machines.
Choice of turbine reaction (continue)
But since the zero-reaction stages have a
slightly lower efficiency than the 50% reaction
stages, a compromise is effected between:
• excessive number of stages, if all were 50%
reaction and
• lower efficiency if all stages were zero reaction.
It is a frequent practice specially in large
turbine to use zero reaction stages for the first
few stages and to follow it by 50% reaction
stages. In such cases the zero reaction stage is
very useful in nozzle control governing.
EFFECT OF REACTION ON AXIAL
COMPRESSOR AND PUMP STAGES
A similar study for the reaction effect on the shape of
velocity diagrams and stage blading can be carried out
for axial flow compressors and pumps. The comparison
will be made for stages having the same peripheral
speed, axial velocity and work i.e. the same f and y.
Figure 3.14 compares the velocity diagrams for three
stages with different reactions (50, 80, 120 %
REACTION).
The first stage, 50% REACTION has a symmetrically
velocity diagram with C2= W1 and C1= W2. This stage is
frequently used, specially in single stage axial
compressors for the same reasons explained before in
the case of 50% reaction multi-stage axial turbines.
EFFECT OF REACTION ON AXIAL COMPRESSOR
AND PUMP STAGES
50% REACTION
Fig. 3.14 Axial pump or compressor stages with constant
work and flow coefficients and different degrees of
reaction.
EFFECT OF REACTION ON AXIAL
COMPRESSOR AND PUMP STAGES
REACTION < 50%
80% REACTION: Velocity diagram, Fig. 3.14, is
for a stage with axial inlet velocity C1. In this stage
the degree of reaction is higher than 0.5 (and less
than 1) and it is usually used for single stage
pumps and fans having no inlet guide vanes. But,
since the absolute velocity at exit of rotor is not
axial, a stator f the rotor is necessary in which the
flow is diffused and delivered axially.
EFFECT OF REACTION ON AXIAL COMPRESSOR
AND PUMP STAGES
Fig. 3.14 Axial pump or compressor stages with constant
work and flow coefficients and different degrees of
reaction. 80% REACTION
EFFECT OF REACTION ON AXIAL
COMPRESSOR AND PUMP STAGES
120% REACTION: Velocity diagram, Fig. 3.14,
shows a stage with axial exit velocity C2 and a
reaction higher than 1. As the absolute velocity at
exit from the rotor is axial, a stator at exit is not
necessary. At inlet to the rotor, the flow has a whirl
component of velocity and hence an inlet guide
blades (stator) is necessary. This stage is
sometimes used in single stage fans with inlet
guide vanes for flow rate control.
EFFECT OF REACTION ON AXIAL COMPRESSOR
AND PUMP STAGES
Fig. 3.14 Axial pump or compressor stages with constant
work and flow coefficients and different degrees of
reaction. 120% REACTION
EFFECT OF REACTION ON AXIAL COMPRESSOR
AND PUMP STAGES
With regard to optimum stage efficiency of axial
compressors and pumps, similar procedure can be
followed as that was explained for turbine stages.
But, some limitations must be introduced as the
flow is decelerated through compressor and pump
passages and hence high flow turning can not be
carried out. Therefore, blades are characterized by
low deflection angles and in order to increase the
energy transfer in the stage, very high speed is
necessary (specially in case of compressors).
However, the maximum theoretical stage efficiency is
obtained with 50% reaction stage operation with a
flow coefficient of 0.5.
EFFECT OF REACTION ON AXIAL
COMPRESSOR AND PUMP STAGES
Actually, higher reaction may give higher actual
efficiency values as in that case the leaving
energy loss is much less. Therefore, the degree
of reaction in axial compressors and pumps is
usually 0.5 or higher. It has to be noticed also that
there is another limitation on the work of
compressor stages which is Mach number
considerations .