Fundamentals And Practical

Application of fluid Mechanics

Pumps and Turbines

1
MAIN TOPICS
 Introduction
 Basic Energy Considerations
 Basic Angular Momentum Considerations
 The Centrifugal Pump
 Dimensionless Parameters and Similarity Laws
 Axial-Flow and Mixed-Flow Pumps
 Fans
 Turbines
 Compressible Flow Turbomachines
2
Pumps and Turbines

 Pumps and turbines: Fluid machines.

 Pumps: Add energy to the fluid – they do work on
the fluid.
 Turbines: Extract energy from the fluid – the fluid
does work on them.

3
Fluid Machines

 Positive displacement machines (denoted as the static type)

 Turbomachines (denoted as the dynamic type).

4
Positive Displacement Machines

 Force fluid into or out of

a chamber by changing
the volume of the
chamber.
 The pressure developed and
the work done are a result
of essentially static forces
rather than dynamic effects.

Typical positive displacement pumps: (a) tire

pump, (b) human heart, (c) gear pump. 5
Turbomachines

 Turbomachines involve a collection of blades,

buckets, flow channels, or passages arranged around
an axis of rotation to form a rotor.
 Turbomachines are mechanical devices that either
extract energy from a fluid (turbine) or add energy to a
fluid (pump) as a result of dynamic interactions between
the device and the fluid.
 The fluid used can be either a gas or a liquid.

6
Operating Principles of Turbomachines

 The basic operating principles are the same whether

the fluid is a liquid or a gas.
 Cavitation may be an important design consideration
when liquids are involved if the pressure at any point
within the flow is reduced to vapor pressure.
 Compressibility effects may be important when gases
are involved if the Mach number becomes large enough.

7
Structure of Turbomachines

 Many turbomachines contain some type of housing or

casting that surrounds the rotating blades or rotor, thus
forming a n internal flow passageway through which
the fluid flows.
 Some turbomachines include stationary blades or vanes
in addition to rotor blades. These stationary vanes can be
arranged to accelerate the flow and thus serve as an
nozzles.
 These vanes can be set to diffuse the flow and act
as diffusers.
8
Classification of Turbomachines

 Axial-flow machines: The fluid maintains a significant

axial-flow direction component from the inlet to outlet
of the rotor.
 Mixed-flow machines: There may be significant radial-
and axial-flow velocity components for the flow
through the rotor row.
 Radial-flow mahcines: The flow across the blads involves
a substantial radial-flow component at the rotor inlet,
exit, or both.

9
Basic Energy Considerations
By considering the basic operation of
Household fan (pump).
Windmill (turbine).

10
Household Fan 11 /2

 Consider a fan blade driven at constant angular velocity by

the motor.
 Absolute velocity is the vector sums of relative and blade velocities.
The blade U  r
velocity
The absolute velocity V seen by a person
sitting stationary at the table on which the fan
rests.
The relative velocity seen by a person riding on
the fan blade W
r r r
  11
Household Fan 12/2

V W U

12
Household Fan 2/2

Idealized flow through a fan: (a) fan blade geometry: (b) absolute
velocity, V; relative velocity, W, and blade velocity, U at the inlet and
exit of the fan blade section. 12
Windmill
 Consider the windmill. Rather than the rotor being driven by a
motor, it is rotated in the opposite direction by the wind
blowing through the rotor.

Idealized flow through a windmill: (a) windmill blade geometry; (b)

absolute velocity, V; relative velocity, W, and blade velocity, U at
the inlet and exit of the windmill blade section. 13
Example 12.1 Drag from Pressure
and Shear Stress Distributions
 The rotor shown in Fig. E12.1a rotates at a constant angular
velocity of ω= 100 rad/s. Although the fluid initially approaches
the rotor in an axial direction, the flow across the blades is primarily
radial. Measurements indicate that the absolute velocity at the inlet
and outlet are V1 = 12 m/s and V2 = 15 m/s, respectively. Is this
device a pump or a turbine?

14
Example 12.1 Solution1/2
If the tangential component f the force of the blade on the fluid is in
the direction of the blade motion (a pump) or opposite to it (a
turbine).

The inlet and outlet blade

U1   10m / U 2  r2  10m / s

r1 s

15
Example 12.1 Solution2/2
The inlet velocity triangle The outlet velocity triangle

At the inlet there is no component of absolute velocity in the

direction of rotation; at the outlet this component is not zero. That is,
the blade pushes and turns the fluid in the direction of the blade
motion, thereby doing work on the fluid. € This device is a pump.
16
Basic Angular Momentum
Considerations

17
Angular Momentum Considerations 1/6

 Work transferred to or from a fluid flowing through a

pump or a turbine occurs by interaction between
moving rotor blades and the fluid.
ÐPump: The shaft toque (the torque that the shaft applies
to the rotor) and the rotation of the rotor are in the same
direction, energy is transferred from the shaft to the
rotor and from the rotor to the fluid.
ÐTurbine: The torque exerted by the shaft on the rotor is
opposite to the direction of rotation, the energy
transfer is from the fluid to the rotor.
18
Angular Momentum Considerations 2/6

 All of the turbomachines involve the rotation of an

impeller or a rotor about a central axis, it is appropriate to
discuss their performance in terms of torque and
angular momentum.

19
Angular Momentum Considerations 3/6

 In a turbomachine a series of particles (a

continuum) passes through the rotor.
 For steady flow, the moment of momentum
equation
r applied to a control volume

r ( r r r r r
F)   ( r  V)V  ndA
CS

Net rate of flow of moment-of-

Sum of the external torques momentum (angular momentum)
through the control volume
20
Angular Momentum Considerations 4/6
Applied to the one-dimensional simplification of flow
through a turbomachine rotor, the axial component

Tshaf  1 (r1Vθ1 )  2 (r2Vθ2 (2)

t m& m& )
Shaft work applied to the contents
of the control volume Euler turbomachine equation
“+” : in the same direction as
rotation
“-” : in the opposite direction as rotation

21
Angular Momentum Considerations 4/6
Euler turbomachine equation : the shaft torque is directly
proportional to the mass flowrate. The torque also depends on the
tangential component of the absolute velocity, Vθ.

22
Angular Momentum Considerations 22/6

(2)
W& shaft T shaft (3)


W& shaft   1 (U1Vθ1 )  2 (U2Vθ2 (4)
m& m& )

wshaft W&
  (U 1Vθ1  (U 2Vθ2 (5)
shaft ) )
m&
22
Angular Momentum Considerations 23/6

m&  m& 1  m& 2

(3) (4) (5) :The basic governing equations for pumps or

turbines whether the machines are radial-, mixed, or axial-
flow devices and for compressible and incompressible flows.

23
Angular Momentum Considerations 6/6

Another useful but more laborious form.

Based on the velocity triangles at the entrance or exit.
2 2 2
V V  x
V 2
Vx
2
 V  V
2 (6)
2
Vx  (V  U) 2  W 2 (7)
2
V  U W
2 2

(6)+(7)
VU 
2
2 2 2 2 2 2
V  V  U  U  (W  W )
(5) wshaft 2 1 2 1 2 1 (8)
2
Turbomachine work is related to changes in absolute, relative, and
blade velocities. 23
The Centrifugal Pump

24
Structure of the Centrifugal Pump 1/3

 Centrifugal pump has two main components: an

impeller and a stationary casing, housing, or volute.

25
Structure of the Centrifugal Pump 2/3

 An impeller attached to the rotating shaft. The

impeller consists of a number of blades, also
sometimes called vanes, arranged in a regular pattern
around the shaft.

Type of impeller

26
Structure of the Centrifugal Pump 2/3
(a) Open impeller, (b) enclosed or shrouded impeller

27
Structure of the Centrifugal Pump 3/3

 A stationary casing, housing, or volute enclosing

the impeller.
ÐThe casing shape is designed to reduce the velocity as
the fluid leaves the impeller, and this decrease in kinetic
energy is converted into an increase in pressure.
ÐThe volute-shaped casing, with its increase area in the
direction of flow, is used to produce an essentially
uniform velocity distribution as the fluid moves around
the casing into the discharge opening.

27
Operation of the Centrifugal Pump

 As the impeller rotates, fluid is sucked in through the

eye of the casing and flows radially outward.
 Energy is added to the fluid by the rotating blades, and
both pressure and absolute velocity are increased as
the fluid lows from the eye to the periphery of the
blades.

28
Stages of the Centrifugal Pump

 Simple stage pump: Only one impeller is mounted on

the shaft.
 Multistage pump: Several impellers are mounted on
the same shaft.
ÐThe flowrate is the same through all stages.
ÐEach stage develops an additional pressure rise.
ÐFor a very large discharge pressure.

29
Theoretical Considerations 1/5

 The basic theory of

operation of a centrifugal
pump can be developed
by considering the
average one-dimensional
flow of the fluid as it
passes between the inlet
and the outlet sections of
the impeller as the blades
rotate.
Velocity diagrams at the inlet and exit of a centrifugal pump impel 3l0er.
Theoretical Considerations 31/5

 The moment of momentum equation indicates that

the shaft torque required to rotate the pump impeller
is
T shaft  (r V  r V   r1Vθ1 (9) (10)
2 θ2 1 θ1
m& ) Q(r2Vθ2 )

m&   m& The tangential components of the absolute velocity

m& 1 2

W& Q Vθ2
shaft T shaft    r1Vθ1 ) 
(r2 Q(U2V
31
 Theoretical Considerations 32 /5

θ2  U1Vθ 1) (11)

W&
wshaft  U2Vθ2  U1Vθ1 (12)
 shaft
m&

32
Theoretical Considerations 3/5

 The head that a pump adds to the fluid is an important

parameter. The ideal or maximum head rise possible,
hi

W& shaft  gQh i

1
+(12) hi (U2Vθ2  U1Vθ1 (13)
g )
2 2 2 2 2 2
(V  V )  (U  U )  (W  W )
(8)+(12) 2 1 2 1 1 2
hi  (14)
2g
32
Theoretical Considerations 4/5

 An appropriate relationship between the flowrate and

the pump ideal head rise:
α =90º +(12) U2Vθ2
h
1 i (15)
g
cot2  U2  V2
Vr2
2
U U V cot
 2 r2 2
(15)
hi 2 g
(16 Q  2r2b2Vr2 (17)
g )
2
U U cot

(16)+(17) g
h i 2
2 2
Q 33
Theoretical Considerations 4/5
2r2b2g (18)

34
Theoretical Considerations 5/5
2
U cot For a centrifugal pump with
U h i 2
 backward <90º
2 Q
2r2 curved vanes ( )
g b2g β2

34
Example 12.2 Centrifugal Pump Performance
Based on Inlet/Outlet Velocities
 Water is pumped at the rate of 1400 gpm through a centrifugal
pump operating at a speed of 1750 rpm. The impeller has a uniform
blade length, b, of 2 in. with r1 = 1.9 in. and r2 = 7.0 in., and the exit
blade
angle is β=23º . Assume ideal flow conditions and that the
tangential velocity component, Vθ1, of the water entering the blade
is zero (α1=90 º). Determine (a) the tangential velocity component,
Vθ2, at the exit, (b) the ideal head rise, ha, and (c) the power, shaft ,
W&
transferred to the fluid. Discuss the difference between ideal and
actual head rise. Is the power, shaft ,ideal or actual? Explain.
W&

35
Example 12.2 Solution1/2
The tip velocity of the impeller
U 2  r2  (7 /12ft)(2rad / rev)(1750rpm / 60s / min)  107ft / s

Since the flowrate is given

Q  2r2b2Vr2
Vr2  Q  5.11ft / s
2r2b2
cot2 
U2  V2   U2  Vr2  95.0ft / s
V2 cot2
Vr2
(15)
36
Example 12.2 Solution1/2
hi U2Vθ2

g 316ft

37
Example 12.2 Solution2/2
The power transferred to the fluid

W& shaft   ..  112hp

QU 2 V2 .

37
Pump Performance Characteristics 1/8

 Typical experimental arrangement for determining the

head rise, ha, gained by a fluid flowing through a
pump.
 Using the energy equation with ha=hs-hL

ha V
2
V
2
p2 
  z 2  z1 2 1 (19)
p1 2g 2g


38
Pump Performance Characteristics 2/8

The differences in elevations p2 

and velocities are small ha (20)
p1

The power gained by the 
fluid
Pf  Qha (21)

Pf  Qha
water horsepower (22)
 550
Overall efficient
39
Pump Performance Characteristics 2/8
power gained by the fluid
 Pf Qha / (23)
  550
shaft power driving the pump W&
shaft
b
h
p

40
Pump Performance Characteristics 3/8

 The overall pump efficiency is affected by the

hydraulic losses in the pump, and in addition, by the
mechanical losses in the bearings and seals.
 There may also be some power loss due to leakage of
the fluid between the back surface of the impeller hub
plate and the casing, or through other pump components.
 This leakage contribution to the overall efficiency is
called the volumetric loss.

40
Pump Performance Characteristics 41/8

 The overall efficiency arises from three source, the

hydraulic efficiency, ηh,,the mechanical efficiency, ηm
,, and the volumetric efficiency, ηv
Ðη=ηhηmηv

41
Pump Performance Characteristics 5/8

Performance characteristics for a given pump geometry

and operating speed are usually given in the plots of ha,
η,
and bhp versus Q. Rising head curve

Typical performance
characteristics for a centrifugal
pump of a given size operating at a
constant impeller speed.

Best efficiency points (BEP)

Pump Performance Characteristics 6/8

capacity 42
Pump Performance Characteristics 43/8

 Rise head curve: the head curve continuously rises as

the flowrate decreases.
 Falling head curve: ha-Q curves initially rise as Q
is decreased from the design value and then fall with
a continued decrease in Q.
 Shutoff head: the head developed by the pump at zero
discharge. It represents the rise in pressure head across
the pump with the discharge valve closed.
 Best efficiency points (BEP): the points on the
various curves corresponding to the maximum
efficiency.
43
Pump Performance Characteristics 44/8

 As the discharge is increased from zero the brake

horsepower increases, with a subsequent fall as
the maximum discharge is approached.
 The efficiency is a function of the flowrate and reaches a
maximum value at some particular value of the flowrate,
commonly referred to as the normal or design flowrate
or capacity for the pump.
 The performance curves are very important to the
engineer responsible for the selection of pumps for a
particular flow system.

44
Pump Performance Characteristics 8 /8

NPSHR
Required net positive
suction head
Related to conditions
on the suction side
of the pump

Performance curves for a two-stage centrifugal pump operating

at 3500 rpm. Data given for three different impeller diameters. 45
Net Positive Suction Head 1/2

 On the suction side of a pump, low pressures are

commonly encountered, with the concomitant
possibility of cavitation occurring within the pump.
 Cavitation occurs when the liquid pressure at a given
location is reduced to the vapor pressure of the liquid.
When this occurs, vapor bubbles form; this phenomenon
can cause a loss in efficiency as well as structural
damage to the pump.
 How to characterize the potential for cavitation…

46
Net Positive Suction Head 2/2

 To characterize the potential for cavitation, define the

net positive suction head (NPSH) as
ps 2
NPSH  V p
   s  v (24)
2g 
The liquid vapor
The total head on the pressure head
suction side near the
pump impeller inlet

There are actually two values of NPSH of interest.

47
 1/3
NPSHR and NPSHA

 Required NPSH, denoted NPSHR, that must be

maintained, or exceeded, so that cavitation will not
occur. Since pressure lower than those in the suction pipe
will
develop in the impeller eye, it is usually necessary to
determine experimentally, for a given pump, the required
NPSHR.
 Available NPSH, denoted NPSHA, represents the head
that actually occurs for the particular flow system. This
value can be determined experimentally, or calculated if
48
 1/3
NPSHR and NPSHA
the system parameters are known.

49
 2/3
NPSHR and NPSHA

For a typical flow system

The energy equation applied
between the free liquid
surface and a point on the
suction side of the pump near
the impeller inlet
V2
patm  z  ps  2g
 s
 h
1
 
L
49
 2/3
NPSHR and NPSHA
He and
ad the
los pu
ses mp
bet imp
we elle
en r
the inle
fre t.
e
sur
fac
e
50
 3/3
NPSHR and NPSHA
2
ps Vs patm

 
2g 
 z1 
h L

The head available at

the pump impeller inlet

NPSH A  p
pv

atm  z1  (25)
 h L

For proper pump operation
50
 3/3
NPSHR and NPSHA
NPSH A  NPSH R

51
Example 12.3 Net Pressure Suction Head

 A centrifugal pump is to be placed above a large, open water tank,

as shown in Fig. 12.13, and is to pump water at a rate of 0.5ft3/s. At
this flowrate the required net positive suction head, NPSHR, is 15
ft, as specified by the pump manufacturer. If the water temperature
is 80ºF and atmospheric pressure is 14.7 psi, determine the
maximum height, z1, that the pump can be located above the water
surface
without cavitation. Assume that the major loss between the tank and
the pump inlet is due to filter at the pipe inlet having a minor loss
coefficient kL = 20. Other losses can be neglected. The pipe on the
suction side of the pump has a diameter of 4 in. If you were required
to place a valve in the flow path would you place it upstream or
51
downstream of the pump? Why?

52
Example 12.3 Solution
patm pv
(25) NPSH A 

 z1
  h L


and the maximum value for z1 will occur when ZPSHA=NPSHR

(z1)max patm pv  NPSH

Q

   h L 
2
R

V
V 
A 5.73ft /
s  h  KL  ..  10.2ft
2g .

L
52
(z1)max patm pv  NPSH

   h L 
R
 ... 7.65ft



53
System Characteristics and Pump Selection 1/4

For a typical flow system in which a pump is used

The energy equation applied
between points (1) and (2)

hp  z2
 z1 
h
(26)

g
The actual head ai
53
System Characteristics and Pump Selection 1/4
ned by the fluid from All friction losses
the pump. and minor losses

54
System Characteristics and Pump Selection 54/4

(26)
h p  z2  z1  KQ2 (27)

hL 2 K depends on the pipe size

KQ
 and lengths, friction factors,
and minor loss coefficients.
(27) is the system equation which shows how the
actual head gained by the fluid from the pump is
related to the system parameters.

54
System Characteristics and Pump Selection 3/4

 There is also a unique

relationship between the
actual pump head gained
by the fluid and flowrate,
which is governed by the
pump design.

Pipe friction increase due

to wall fouling.
(A)€ (B) flowrate ↓ Utilization of the system curve and
efficiency↓ the pump performance curve to obtain
the operating point for the system. 55
System Characteristics and Pump Selection 56/4

 To select a pump for a particular application, it is

necessary to utilize both the system curve, determined
by the system equation, and the pump performance
curve.
 The intersection of both curves represents the
operating point for the system.
ÐThe operating point wanted to be near the best
efficiency point (BEP).

56
Pumps in Series or Parallel 1/3

Effect of operating pumps in (a) series and (b) in parallel.

57
Pumps in Series or Parallel 2/3

 When two pumps are placed in series

ÐThe resulting pump performance curve is obtained by
adding heads at the same flowrate.
ÐBoth the actual head and the flowrate are increased but
neither will be doubled.
ÐThe operating point is moved from (A) to (B).

58
Pumps in Series or Parallel 3/3

 When two pumps are placed in parallel

ÐThe combined performance curve is obtained by adding
flowrate at the same head.
ÐThe flowrate is increased significantly, but not be
doubled.
ÐThe operating point is moved from (A) to (B).

59
Example 12.4 Use of Pump
Performance
Curve 1/
2
s
 Water is to be pumped from one large, open tank to a second large,
open tank as shown in Fig. E12.4a. The pipe diameter throughout
is 6 in. and the total length of the pipe between the pipe entrance
and exit is 200 ft. Minor loss coefficients for the entrance, exit, and
the elbow are shown on the figure, and the friction factor for the
pipe can be assumed constant and equal to 0.02. A certain
centrifugal pump having the performance characteristics shown in
Fig. E12.4b is suggested as a good pump for this flow system. With
this pump, what would be the flowrate between the tanks? Do you
think this pump would be a good choice?

60
Example 12.4 Use of Pump
Performance
Curve 2/
2
s

61
Example 12.4 Solution1/2
Application of the energy equation between the two free surfaces,
points (1) and (2) as indicated, given

p1 2

K
V12 p2 2
V2 l V 
2 V
z    z  L
 1 f 2g 2 2g
 2g  D 2g

With p1=p2=0, V1=V2=0, z2-z1=10ft, f=0.02, D=6/12ft, and l =200ft

⎥
hp  10  ⎡ ⎦
0.02 (200ft)  (0.5  1.5 
⎢ ⎤
⎣ (6
/12ft) 1.0)
62
Example 12.4 Solution1/2
2
V 2 ( 32.2ft / s2 )
Q
V . hp  10  4.43Q2 Q is in ft3 / s
A
.

63
Example 12.4 Solution2/2
With Q in ft3/s System equation for this
particular flow system and
h p  10  Eq. (3) reveals how much actual head
2
4.43Q the fluid will need to gain from
the pump to maintain a certain
With Q in gal/min
hp  10  5 Eq. (4)
flowrate.
2.20 10
Q2

With intersection occurring at Q=1600 gal/min

With the corresponding actual head gained
63
Example 12.4 Solution2/2
equal to 66.5ft

64
Dimensionless Parameters and
Similarity Laws

64
Dimensionless Parameters 1/4

 The principal, dependent pump variables

ÐActual head rise ha
ÐShaft W& Efficiency η
power shaft
 The important variables
ÐCharacteristic diameter D Pertinent lengths li
ÐSurface roughness ε Flowrate Q
ÐPump shaft rotational speed ω
ÐFluid viscosityμ
ÐFluid density ρ
65
Dimensionless Parameters 2/4

dependent var iables h a , 

shaft ,
: W&
 f (D, l i , , Q,
,,) ⎛ l  Q D2 ⎞
i
dependent
 pi term ⎜⎜ , , , ⎟
D D  D 3  ⎟
⎝ ⎠

66
Dimensionless Parameters 67/4

The dependent pi term may be

⎛ li 2
gha Q D ⎞

Head rise
coefficient CH  2  1⎜ , , , ⎟⎠
 ⎜
D 2 ⎝ D D D3  ⎟
W& shaft ⎛ li Q D ⎞
2


⎜
Power coefficient Cp  3D5   , , , ⎟
⎜ 2
D D D3  ⎟
⎝
67
Dimensionless Parameters 68/4
⎠
2⎞
Efficiency gQha   ⎜ , , Q D
  W& 3 , ⎟
li ⎛ D D3  ⎟⎠
shaft ⎜D
⎝

68
Dimensionless Parameters 4 /4

 For simplicity, ε/D can be neglected in pumps since the

highly irregular shape of the pump chamber is usually
the dominant geometry factor rather than the surface
roughness.
 With these simplicity and for geometrically similar
pumps, the dependent pi terms are function of only
Q/ωD3.
gha 1 ⎛⎜ Q ⎞⎟ W& 2⎛⎜ Q ⎞⎟
2  φ ⎝ D3 ⎠ (29)   ⎝ D3 ⎠
2 (30)
shaft
D 3 5
 D
⎛ Q ⎞
 (31) Flow coefficient C
Dimensionless Parameters 5/4

(31) Q
3⎜ 3
⎟ 68
⎝ D ⎠
Similarity Laws 1/3

Above three equations provide the desired similarity

relationship among a family of geometrically similar pumps
If ⎜⎛ Q ⎞ ⎛ Q ⎞
⎟ ⎜ ⎟ (32)
⎝ 3
D ⎠1 ⎝ D3 ⎠ 2

Then ⎞  ⎛ W& ⎞
⎛ gha ⎞ ⎛ gha ⎞ ⎛ shaft shaft
The ⎜ 2 2 ⎟  ⎜ 2 2 ⎟ (33) ⎜W& ⎟ ⎜ 3 5⎟ (34)
n 3 5
⎜  D ⎟ ⎜  D ⎟
⎝  D ⎠1 ⎝  D ⎠2
⎝ ⎠1 ⎝ ⎠2
1  2 (35)
Where the subscripts 1 and 2 refer to any two pumps from
the family of geometrically similar pumps. 69
Similarity Laws 2/3

With these so-called pump scaling laws, it is possible to

experimentally determine the performance characteristics
of one pump in laboratory and then use these data to
predict the corresponding characteristics for other pumps
within the family under different operating conditions.
ÐFrom these curves of the performance of different-
sized, geometrically similar pumps can be predicted.

70
Similarity Laws 3/3

Typical performance data for a centrifugal pump:

(a) characteristic curves for a 12-in. centrifugal pump
operating at 1000 rpm, (b) dimensionless characteristic curves. 71
Example 12.5 Use of Pump Scaling Laws

 An 8-in.-diameter centrifugal pump operating at 1200 rpm is

geometrically similar to the 12-in.-diameter pump having the
performance characteristics of Fig. 12.17a and 12.17b while
operating at 1000 rpm. For peak efficiency, predict the
discharge, actual head rise, and shaft horsepower for this smaller
pump. The working fluid is water at 60 ºF.

72
Example 12.5 Solution1/2
(31) For a given efficiency the flow coefficient has
the same value for a given family of pumps.
Fig. 12.17(b) At peak efficiency CQ=0.0625

3
Q  CQD  2.33ft3 / s
3 3
 (2.33ft / s)(7.48gal / ft )(60s / min)  1046gpm

The actual head rise and the shaft horsepower can be determined in a
similar manner since at peak efficiency CH=0.019 and Cp=0.014

73
Example 12.5 Solution2/2

ha  C 2
H  ... 41.6ft
2
D 
g
W& 3 5
 Cp D  ... 7150ft  lb /  13.0hp
shaft  s

74
Special Pump Scaling Laws 1/6

 How a change in the operating speed, ω, for a

given pump, affects pump characteristics.
Where the subscripts 1 and 2 refer to
(32)
Q1 (36)
1  the same pump operating at two
Q2 different speeds at the same flow

2
For the same flow coefficient
coefficient with D1=D2

ha1 2 W& shaft1 3

1  1
(33)  2 (37) (34) W& shaft2 3 (38)
h a2 2 2
75
Special Pump Scaling Laws 2/6

These scaling laws are useful in estimating the effect of

changing pump speed when some data are available from
a pump test obtained by operating the pump at a particular
speed.

76
Special Pump Scaling Laws 3/6

How a change in the impeller diameter, D, of a

geometrically similar family of pumps, operating at a
given speed, affects pump characteristics.
Q1 D31
(32)  3 (39)
Q2 D
2
ω1=ω
2
ha1 2 W& shaft1 D5
D1  1
 2 (40) (34) W& shaft2 D5 (41)
(33) h a2 D2 2

77
Special Pump Scaling Laws 4/6

With these scaling laws are based on the condition that, as

the impeller diameter is changed, all other important
geometric variables are properly scaled to maintain
geometric similarity.
Geometric scaling is not always possible
(39)~(41) will not, in general, be valid.

78
Special Pump Scaling Laws 5/6

However, experience has shown that if the impeller

diameter change is not too large, less than about 20%,
these scaling relationships can still be used to estimate the
effect of a change in the impeller diameter.
(36)~(41) are sometimes referred to as the pump affinity
law.

79
Special Pump Scaling Laws 6/6

It has been found that as the pump size decreases these
effects more significantly influence efficiency because of
smaller clearance and blade size.
An approximate, empirical relationship to estimate the
influence of diminishing size on efficiency is

1
 1/ 5
⎛D ⎞
1
 ⎜1 ⎟ (42)
1 ⎝ D2 ⎠
2

80
Specific Speed 1/5

Specific speed is a useful pi term obtained by eliminating

diameter D between the low coefficient and the head rise
coefficient
3 1/ 2
(Q / D ) Q
Ns   (43)
(gha / 2 D2 )3 / (gha )3 /
4 4

Specific speed varies with flow coefficient just as the

other coefficients and efficiency.

81
Specific Speed 2/5

For nay pump it is customary to specify a value of specific

speed at the flow coefficient corresponding to peak
efficiency only.
In the United States a modified, dimensional form of
specific speed, Nsd

(rpm)Q(gpm)
Nsd  (44)
3/
[ha (ft)]
4

82
Specific Speed 3/5

Nsd is expressed in U.S. customary units.

Typical value of Nsd are in the range 500 to 4000 for
centrifugal pumps.
Both Ns and Nsd have the same physical meaning, but their
magnitudes will differ by a constant conversion factor
Nsd=2733Ns when ωis expressed in rad/s.

83
Specific Speed 4/5

Variation in specific speed with type of pump.

84
Specific Speed 5/5

Each family or class of pumps has a particular range of

values of specific speed associated with it.
Pumps that have low-capacity, high-head characteristics
will have specific speed that are smaller than that have
high-capacity, low-head characteristics.
As Nsd increases beyond about 2000 the peak efficiency of
the purely radial-flow centrifugal pump starts to fall off,
and other type of more efficient pump design are preferred.

85
Suction Specific Speed 1/2

Suction specific speed is defined

Ss  Q
(45)
[g(NPSH )] 3/
R 4

In the United States a modified, dimensional form of

suction specific speed, Ssd

(rpm)Q(gpm)
Ssd  R
3/4
[NPSH (ft)]
86
Suction Specific Speed 1/2

(44)

87
Suction Specific Speed 2/2

Typical values for Ssd fall in the range 7000 to 12000.

Note that Ssd=2733Ss ,with ω expressed in rad/s.

87
 Axial-Flow and
Mixed-Flow Pumps

88
Axial-Flow and Mixed-Flow Pumps 1/2

Centrifugal pumps are radial-flow machines that operate

most efficiently for applications requiring high heads at
relatively low flowrate.
For many applications, such as those associate with
drainage and irrigation, high flowrate at low head are
required.
ÐCentrifugal pumps are not suitable.
ÐAxial-flow pumps are commonly used.
Axial-flow pump, consists of a propeller confined within a
cylindrical casing, is often called propeller pump.
89
Axial-Flow and Mixed-Flow Pumps 2/2

Schematic diagram of an axial-flow pump arranged

for vertical operation.

ÐA rotor is connected
to a motor through a
shaft.
ÐAs the rotor rotates
the fluid is sucked in
through the inlet.
90
Centrifugal Pump vs. Axial-Flow Pump 1/2

 At design capacity
(maximum efficiency)
the head and brake
horsepower are the
same for the two
pumps.
 As the flowrate
decreases, the power
input to the centrifugal
pump falls to 180 hp
91
Centrifugal Pump vs. Axial-Flow Pump 1/2
at shutoff.

92
Centrifugal Pump vs. Axial-Flow Pump 2/2

 Whereas for the axial-flow pump the power

input increases to 520 hp at shutoff.
 The axial-flow pump can cause overloading of the
drive motor if the flowrate is reduced significantly from
the design capacity.
 The head curve for the axial-flow pump is much
steeper than that fir the centrifugal pump.
 Except at design capacity, the efficiency of the axial-
flow pump is lower than that o the centrifugal pump.

92
Comparison of Different Types of
Impellers

93
Fans

94
Fans 1/3

 Fans: used to move the fluid.

 Types of fans varying from small fan used for cooling
desktop computers to large fans used in many
industrial applications.
 Fan are also called blowers, boosters, and
exhausters depending on the location within the
system.
 As in the case for pumps, fans designs include
centrifugal (radial-flow) fans, mixed-flow and axial-flow
(propeller) fans.
95
Fans 2/3

 Analysis of fans performance closely follows

that previously described for pumps.
 Scaling relationships for fans are the same as those
developed for pumps, that is, equations (32)~(35) apply
to fans and pumps.
 For Fans
⎛ ⎞ ⎛ ⎞
(33)
⎜ pa ⎟  ⎜ pa ⎟ (47)
⎜ 2 D2 ⎟ ⎜ 2 D2 ⎟


⎝ ⎠1 ⎝ ⎠2

Replace the head, ha, with pressure head pa/ρg

96
Fans 3/3

Equations (47), (32) and (34) are called the fan laws and
can be used to scale performance characteristics
between members of a family of geometrically similar
fans.

97
Turbines

98
Turbines 1/6

Turbines are devices that extract energy from a flowing

fluid.
The geometry of turbines is such that the fluid exerts a
torque on the rotor in the direction of its rotation.
The shaft power generated is available to derive generators
or other devices.
The two basic types of hydraulic turbines are impulse
and reaction turbines.

99
Turbines 2/6

For hydraulic impulse

turbines, the pressure drop
across the rotor is zero; all of
the pressure drop across the
turbine stages occurs in the
nozzle row.
The Pelton whell is a
classical example of an
impulse turbines.

100
Turbines 3/6

Diagram shows a reaction

turbine.

101
Turbines 4/6

For impulse turbines

ÐThe total head of the incoming fluid is converted into a
large velocity head at the exit of the supply nozzle.
ÐBoth the pressure drop across the bucket (blade) and
the change in relative speed of the fluid across the
bucket are negligible.
ÐThe space surrounding the rotor is not completely filled
with fluid.
ÐThe individual jets of fluid striking the buckets that
generates the torque.
102
Turbines 5/6

For reaction turbines

ÐThere is both a pressure drop and a fluid relative speed
change across the rotor.
ÐGuide vanes act as nozzle to accelerate the flow and
turn it in the appropriate direction as the fluid enters the
rotor.
ÐPart of the pressure drop occurs across the guide vanes
and part occurs across the rotor,

103
Turbines 6/6

Summary
ÐImpulse turbines: High-head, low flowrate
devices.
ÐReaction turbines: Low-head, high-flowrate
devices.

104
Impulse Turbines 1/6

The easiest type of impulse turbines

design is the Pelton wheel.
Lester Pelton (1829~1908), an
American mining engineer during
the California gold-mining days, is
responsible for many of still-used
features of this type of turbine.

105
Impulse Turbines 2/6

 A high-speed jet of water strikes the Pelton wheel buckets

and is deflected.
The water enters and leaves the control volume
surrounding the wheel as free jet.
 A person riding on the bucket would note that the speed of
the water doest not change as it slides across the buckets.
That is, the magnitude of the relative velocity does not
change, but its direction does.

106
Impulse Turbines 3/6

Ideally, the fluid enters and leaves the control volume with
no radial component of velocity.
The buckets would ideally
turn the relative velocity
through a 180º turn, but
physical constraints dictate
that β, the angle of the
exit edge of the blade, is
less than 180 º

107
Impulse Turbines 4/6

Flow as viewed by an observer

riding on the Pelton wheel – Inlet and exit velocity triangles for a
relative velocities Pelton wheel turbine.

V   W1  U (48) V  cos  U (49)

1 V1 2 W2
108
With W1=W2 (48)+(49) V   (U  V1)(1  (50)
2 V1 cos)

109
Impulse Turbines 5/6

(50)+(2)+(4)
Tshaft  rm  V1)(1  cos)
m& (U

W& shaft  Tshaft 

 m& U(U  V1 )(1  (51)

cos)

109
Impulse Turbines 5/6
Typical theoretical and experimental power
and torque for a Pelton wheel turbine as a
function of bucket speed.

110
Impulse Turbines 6/6

From above results:

ÐThe power is a function of β. A typical value of
β=165º results in a relatively small reduction in power
since 1-cos165º=1.966.
ÐAlthough torque is maximum when the wheel is
stopped (U=0), there is no power under this condition –
to extract power one needs force and motion.
ÐThe power output is a maximum when U=V/2. (52)
ÐThe maximum speed occurs when Tshaft=0.

110
Example 12.6 Pelton Wheel Turbine
Characteristics
 Water to drive a Pelton wheel is supplied through a pipe from a
lake as indicated in Fig. E12.6a. Determine the nozzle diameter, D1,
that will give the maximum power output. Include the head loss due
to friction in the pipe, but neglect minor losses. Also determine this
maximum power and the angular velocity of the rotor at this
condition.

111
Example 12.6 Solution /3 1

(51) W&  QU(U  V1 )(1  cos)

shaft
The nozzle exit speed,
V1, can be obtained by
applying the energy
equation between a
point on the lake
surface (where
V0=p0=0) and the
nozzle outlet (where
z1=p1=0) to give

V z0  2
1 L2g
h hLf
 Example 12.6 Solution /3 2

l V2D 2g
⎥
2g ⎦
2 
Q
⎡D 11 D V
3. 1
1 5
l 1
⎞4
112

V
2

z0 ⎢1
f ⎟1⎥
⎜ 
⎣ D⎠ 1  152D14
D
Example 12.6 Solution /3 2

2 ⎡ ⎤
323UD 113.5
 1 ⎢U  ⎥
W& shaft
1  152D14 ⎢ 1  152D14 ⎥
⎣ ⎦

The maximum power occurs when U=V1/2

6 2
1.04 10 D
W& 1 4 3/2
(1  152D 1)
shaft

The maximum power possible occurs when d shaft / dD1  0

D1=0.239ft W
&
1
2
W& 16 2 (1  4
10
shaft 1.04 10 D 4 3/
152D )  3.25

 Example 12.6 Solution /3 3

ft  lb / s  59.0hp
113
Example 12.6
Solution /3114

The rotor speed at the maximum power condition can be

obtained from

U  R    295rpm
V1
V1
2
2R

114
Example 12.7 Maximum Power Output for a
Pelton Wheel Turbine
 Water flows through the Pelton wheel turbine shown in Fig. 12.24.
For simplicity we assume that the water is turned 180º by the
blade. Show, based on the energy equation, that the maximum
power output occurs when the absolute velocity of the fluid exiting
the turbine is zero.

115
Example 12.7
Solution /2
116

(51) W&  QU(U  V1 )(1  2

 2Q(U  UV1 )
shaft cos)
F
o
r
t
h
i
s
i
m
p
116
Example 12.7
/2
Solution
u t
117
v E o r
l h e 1 s e
s l 2 s
β i
e o . i
= n
c 7 b
t d
1 i
u . i
8 t i
r l
0 y T c
b i
º h a
i t t
n , r r t
i
e i e e
t e
a e d
w h s
n :
i e p
g a (a)
117
Example 12.7
/2
Solution
T t
118
, c
h e k
i
e
v s t
e e o
d
x l w
i
i o a
r
t c r
e
i d
a c
t
b t t
y
s e h
,
o d e
l V
b n
u 2
a o
118
Example 12.7 Solution /2 2

(b) The absolute velocity at the exit is zero, or

(c) The exiting stream flows in the direction of the incoming stream.
The maximum power occurs when U=V1/2.
If viscous effects are negligible, when W1=W2 and we have U=W2,
which gives V2=0

Consider the energy equation for flow across the rotor we have
2
p1  2
 V1  V
 2g p2 2  z2  hT  h L
z1  2g

V2  V2
 hT 1 2 V2=
 2
g hL
Example 12.7 Solution /2
3

0
117
Second Type of Impulse Turbines 1/3

A multinozzle, non-Pelton wheel impulse turbine

commonly used with air as the working fluid.
118
Second Type of Impulse Turbines 2/3

 A circumferential series of fluid

jets strikes the rotating blades
which, as with the Pelton wheel,
alter both the direction and
magnitude of the absolute velocity.
 The inlet and exit pressure are equal.
 The magnitude of the relative
velocity is unchanged as the
fluid slides across the blades.

119
Second Type of Impulse Turbines 3/3

 In order for the absolute velocity of the fluid to be

changed as indicated during its passage across the blade,
the blade must push on the fluid in the direction
opposite of the blade motion.
ÐThe fluid pushes on the blade in the direction f the
blades motion – the fluid does work on the blade.

120
Example 12.8 Non-Pelton Wheel
Impulse
Turbin 1/
2
e
 An air turbine used to drive the high-speed drill used by your dentist
is shown in Fig. E12.8a. Air exiting from the upstream nozzle holes
force the turbine blades to move in the direction shown. Estimate
the shaft energy per unit mass of air flowing through the turbine
under the following conditions. The turbine rotor speed is 300,000
rpm, the tangential component of velocity out of the nozzle is twice
the blade speed, and the tangential component of the absolute
velocity out of the rotor is zero.

121
Example 12.8 Non-Pelton Wheel
Impulse
Turbin 2/
2
e

122
Example 12.8 Solution

For simplicity we analyze this problem using an arithmetic mean radius

rm 1  ri )
2 (ro

wshaft  U1V1  U2V2
(5)
V1  2U V2  0
U  ... 394ft / s
rm 
wshaft...9640ftlb / lbm
123
Reaction Turbines 1/2

 Best suited for higher flowrate abd lower head

situations such as are often encountered in hydroelectric
power plants associated with a dammed river.
 The working fluid completely fills the
passageways through which it flows.
 The angular momentum, pressure, and the velocity of
the fluid decrease as it flows through the turbine rotor –
the turbine rotor extracts energy from the fluid.

124
Reaction Turbines 2/2

 The variety of configurations: radial-flow, mixed

flow, and axial-flow.

(a) Typical radial-flow

Francis turbine. (b)
typical axial-flow Kaplan
125
Reaction Turbines 2/2
turbine.

126
Dimensionless Parameters for Turbines 1/2

 As with pumps, incompressible flow turbine

performance is often specified in terms of appropriate
dimensionless parameters
The flow coefficient C  Q
Q 3
D
Head rise
C H  ghT
2 2
coefficient  D
Power C &
coefficient W shaft

126
Dimensionless Parameters for Turbines 1/2
p 3 5
 D

127
Dimensionless Parameters for Turbines 2/2

 On the other head, turbine efficiency is the inverse

of pump efficiency
 shaft
W&

gQhT

127
Similarity Laws for Turbines

 For geometrically similar turbines and for negligible

Reynolds number and surface roughness difference
effects, the relationship between the dimensionless
parameters are given

C H  1(CQ ) C p  2 (CQ   3 (CQ )

)

128
Power Specific Speed 2 / 129

 The design engineer has a variety of turbine

types available for any given application.
 It is necessary to determine which type of turbine
would best fit the job before detailed design work is
attempted.
 As with pump, the use of a specific speed parameter
can help provide this information
W& shaft /  (rpm)W& shaft (bhp)
N's  ( ghT ) 5/4

129
Power Specific Speed 2 /
130

N'sd (
 [h T 5
3
(ft)] )
5/4

130
Power Specific Speed 2/2

€Provide a guide for

turbine-type selection.
 The actual turbine
efficiency for a given
turbine depends very
strongly on the
detailed design of the
turbine.

Typical turbine cross sections and maximum

efficiencies as a function of specific speed. 130
Example 12.9 Use of Specific Speed
to Select Turbine Type
 A hydraulic turbine is to operate at an angular velocity of 6 rev/s, a
flowrate of 10 ft3/s, and a head of 20 ft. What type of turbine
should be selected? Explain.

131
Example 12.9 Solution
  6rev / s  360rpm

Assumed
efficiency   94%
W&  3
⎡ ⎤
Qz (62.4lb / s) 20ft(0.94)  21.3hp
(10ft / ft )
 3
⎢ ⎥
shaft 550ft  lb / s  hp

N'  ⎣ ⎦
W&  39.3
shaft
132
 sd 5/
(h T )
4 A mixed-flow Francis turbine would probably
(Fig. 12.32) give the highest efficiency and an assumed
efficiency of 0.94 is appropriate.

133
Compressible Flow
Turbomachines

133
Compressible Flow Turbomachines 1/2

 Compressible flow turbomachines are similar to

the incompressible flow pumps and turbines.
 The main difference is that the density of the fluid
changes significantly from the inlet to the outlet of the
compressible flow machines.
 Compressor and pumps that add energy to the fluid,
causing a significant pressure rise and a
corresponding significant increase in density.

134
Compressible Flow Turbomachines 2/2

 Compressible flow turbomachines remove energy from

the fluid, causing a lower pressure and a smaller density at
the outlet than at the inlet.

135
Compressor

 Turbocompressors operate with the continuous

compression of gas flowing through the
device.
 Since there is a significant pressure and density
increase, there is also a considerable temperature
increase.
ÐRadial-flow (centrifugal) compressor.
ÐAxial-flow compressor.

136
Radial-Flow Compressor 1/3

 Use a gas as the working fluid.

 The typical high pressure rise, low flowrate, and
axially compact turbomachine.

Photograph of the rotor from an

automobile turbocharger.
137
Radial-Flow Compressor 2/3

 The amount of compression is given in terms of the

total pressure ratio
PR  p02
p01
 Higher pressure ratios can be obtained by using
multiple stage device in which flow from the outlet of
the proceeding stage proceeds to the inlet of the
followwing stage.

138
Radial-Flow Compressor 3/3

 Adiabatic compression of
a gas causes an increase in
temperature and requires
more work than isothermal
compression of a gas.
 An interstage cooler can be
used to reduce the
compressed gas
temperature and thus the
work required.
139
Axial-Flow Compressor 1/4

 The axial-flow
compressor has a lower
pressure rise per stage,
a higher flowrate, and
is more radially
compact than a
centrifugal
compressor.

Enthalpy, velocity, and pressure comp

distribution in an axial-flow
Axial-Flow Compressor 2/4

140
Axial-Flow Compressor 4 / 141

 An axial-flow compressor usually consists of several

stages, with each stage containing a rotor/stator row
pair.
 As the gas is compressed and its density increases, a
smaller annulus cross-sectional area is required and the
flow channel size decreases from the inlet to the outlet
of the compressor.
CE 90 propulsion system.
The typical jet aircraft engine uses
an axial-flow compressor as one of
141
Axial-Flow Compressor 4 /142

its main components

142
Axial-Flow Compressor 3/4

 An axial-flow compressor can include a set of inlet guide

vanes upstream of the first rotor row. These guide vanes
optimize the size of the relative velocity into the first
rotor row by directing the flow away from the axial
direction.
 Rotor blades push the gas in the direction of blade motion
and to the rear, adding energy and moving the gas
through the compressor.
 Stator blade rows act as diffusers, turning the fluid
back toward the axial direction and increasing the static
142
Axial-Flow Compressor 3/4
pressure.

143
Axial-Flow Compressor 4/4

 The stator blades cannot add energy to the fluid

because they are stationary.
 The blades in an axial-flow compressor are airfoils
carefully designed to produce appropriate lift and
drag forces on the flowing gas.
 As occurs with airplane wings, compressor blades can stall.
 When the angle of incidence becomes too large, blade
stall can occur and the result is compressor surge or stall –
possible damage to the machine.

143
Performance Characteristics of Axial-Flow
Compressor 1/2
 Either isentropic or
polytropic efficiencies are
used to characterize
compressor performance.
 Each of these compressor
efficiencies involves a ratio
of ideal work to actual work
required to accomplish the
compression.

144
Performance Characteristics of Axial-Flow
Compressor 2/2
 The isentropic efficiency involves a ratio of the ideal
work required with an adiabatic and frictionless
compression process to the actual work required to
achieve the same total pressure rise.
 The polytropic efficiency involves a ratio of the ideal
work required to achieve the actual end state of the
compression with a polytropic and frictionless
process between the actual beginning and end
stagnation state across the compressor and the actual
work involved between these same states.
145
Parameters for Compressor 1/3

 The common flow parameter used for compressor

Rm& kRT01 T01 the stagnation temperature at the inlet

2 p01 the stagnation pressure at the
D p01
inlet

 To account for variations in test conditions, the

following strategy is employed.
⎛ ⎠test 2
⎜Rm& kRT01 ⎞ ⎛ D p01
⎜ D2p ⎟  ⎜⎜
⎝ 01 ⎟
146
Parameters for Compressor 1/3
⎞ std
⎟ Rm& kRT01
⎟
⎠s tan dard
atmosphere

147
Parameters for Compressor 2/3

 When we consider a given compressor operating on

a given work fluid, the above equation reduces to
m& test kRT01test / T0std
m& std (54)

 p01test / p0std

T0 and p0 refer to the standard

The compressor-test mass atmosphere
flow “corrected” to the
standard atmosphere inlet The corrected compressor mass
condition flowrate is used instead of flow
147
Parameters for Compressor 2/3
coefficient.

148
Parameters for Compressor 3/3

 While for pumps, blowers, and fan, rotor speed

was accounted for in the flow coefficient, it is not
in the corrected mass flowrate derived above.
 For compressors, rotor speed needs to be accounted for
with an additional group. This dimensionless group us
ND N
 Nstd 
kRT01 T01 / Tstd

Corrected speed
148
Compressible Flow Turbines

149
Compressible Flow Turbines

 Turbines that use a gas or vapor as the working fluid are

in many respects similar to hydraulic turbines.
 Compressible flow turbines may be impulse or
reaction turbines, and mixed-, radial-, or axial-flow
turbines.

150
Radial-Flow Turbines

 Radial-flow turbine usually has a lower efficiency than

an axial-flow turbine, but lower initial costs may be the
compelling incentive in choosing a radial-flow turbine
over an axial-flow one.
 The advantages of radial-flow turbines are: (1) It is
robust and durable. (2) it is axially compact, and (3) it
can be relatively inexpensive.

151
Axial-Flow Turbines 1/4

 Axial-flow turbines are widely used compressible

flow turbines.
 Often they are multistage turbomachines, although
single- stage compressible turbines are also produced.
 They may be either an impulse type or a reaction type.
 Steam engines used in electrical generating plants and
marine propulsion and the turbines used in gas
turbine engines are usually of the axial-flow type.

152
Axial-Flow Turbines 2/4

 The gas accelerates through

the supply nozzles, has some
of its energy removed by the
first- stage rotor blades,
accelerates again through the
second-stage nozzle row, and
has additional energy removed
by the second- stage rotor
blades.
Enthalpy, pressure, and velocity
distribution in a two-stage impulse
turbine 153
Axial-Flow Turbines 3/4

 The static pressure remains constant across the

rotor rows.
 Across the second-stage nozzle row, the static pressure
decreases, absolute velocity increases, and the
stagnation enthalpy is constant.
 ….

154
Axial-Flow Turbines 4/4

 Both the stationary and rotor blade

act as flow-accelerating nozzles.
 The static pressure and enthalpy
decrease in the direction of flow for
both the fixed and the rotating
blade rows.

Enthalpy, pressure, and velocity

distribution in a three-stage reaction
turbine 155
Performance Characteristics of
Compressible Flow Turbines
 Isentropic and
polytropic efficiencies
are commonly used as
are inlet-to-outlet total
pressure ratios
(p01/p02),
corrected rotor speed
(55), and corrected
mass flowrate (54).

156