Chapter 2: Introduction
and Applications of
Turbomachinery
Turbomachinery
The jet engines on modern
commercial airplanes are
highly complex
turbomachines that include
both pump (compressor) and
turbine sections.
Turbomachinery is an
important application of fluid
mechanics in practice.
Two categories: pumps and
turbines.
© Stockbyte/Punchstock RF
Preliminary design and overall
performance analysis and
turbomachinery scaling laws –a
practical application of
dimensional analysis
2
Learning Objectives
• Identify various types of pumps and turbines, and understand
how they work
• Apply dimensional analysis to design new pumps or turbines
that are geometrically similar to existing pumps or turbines
• Perform basic vector analysis of the flow into and out of pumps
and turbines
• Use specific speed for preliminary design and selection of
pumps and turbines
3
2–1 Classifications and Terminology
• Pumps: Energy absorbing devices
since energy is supplied to them,
and they transfer most of that
energy to the fluid, usually via a
rotating shaft. The increase in
fluid energy is felt as an increase
in the pressure of the fluid.
• Turbines: Energy producing
devices they extract energy from
the fluid and transfer most of that
energy to some form of
mechanical energy output,
typically in the form of a rotating
shaft. The fluid at the outlet of a
turbine suffers an energy loss,
typically in the form of a loss of
pressure.
(a) A pump supplies energy to a fluid,
(b) a turbine extracts energy from a fluid.
4
Pressure Changes
• The purpose of a pump is to add
energy to a fluid, resulting in an
increase in fluid pressure, not
necessarily an increase of fluid
speed across the pump.
• The purpose of a turbine is to
extract energy from a fluid,
resulting in a decrease of fluid
pressure, not necessarily a
decrease of fluid speed across the
turbine.
For the case of steady flow,
conservation of mass requires that the
mass flow rate out of a pump must
equal the mass flow rate into the
pump; for incompressible flow with
equal inlet and outlet cross- sectional
areas (Dout = Din), we conclude that Vout
= Vin, but Pout > Pin.
5
Further Classification
• Pump: machines that move liquids.
• Fan: A gas pump with relatively low
pressure rise and high flow rate, such
as ceiling/house fans, and propellers.
When used with gases, pumps are
named fans, blowers, or
compressors, depending on the
relative values of pressure rise and
volume flow rate.
• Blower: A gas pump with relatively
moderate to high pressure rise and
moderate to high flow rate, such as
centrifugal blowers and squirrel cage
blowers in automobile ventilation
systems, furnaces, and leaf blowers.
• Compressor: A gas pump designed to
deliver a high-pressure rise, at low to
moderate flow rates, such as air
compressors that run pneumatic
tools and inflate tires at automobile
service stations, and refrigerant
compressors used in heat pumps,
refrigerators, and air conditioners.
6
Turbomachines
(a) Photo by Andrew Cimbala. (b) © Bear Dancer Studios/Mark Dierker.
• Turbomachines: Pumps and turbines
in which energy is supplied or
extracted by a rotating shaft.
• The words turbomachine and
turbomachinery are often used in the
literature to refer to all types of
pumps and turbines regardless of
whether they utilize a rotating shaft
or not.
• Not all pumps have a rotating shaft;
(a) energy is supplied to this manual
tire pump by the up and down
motion of a person’s arm to pump
air; (b) a similar mechanism is used to
pump water with an old-fashioned
well pump.
7
More Definitions
• Fluid machines may also be classified as positive-displacement machines or
dynamic machines, based on how the energy transfer occurs.
•
In positive-displacement machines: Fluid is directed into a closed volume. Energy transfer
to the fluid is accomplished by movement of the boundary of the closed volume, causing
the volume to expand or contract, thereby sucking fluid in or squeezing fluid out,
respectively.
(a) The human heart is an example of a
positive- displacement pump; blood is
pumped by expansion and contraction of
heart chambers called ventricles.
(b) The common water
meter in your house is
an example of a positivedisplacement turbine;
water fills and exits a
chamber of known
volume for each
revolution of the output
shaft.
(b) Courtesy of Badger Meter, Inc. Used by permission.
8
• Dynamic machines: There is no
closed volume; rotating blades
supply or extract energy to or from
the fluid.
• For pumps, these rotating blades
are impeller blades, while for
turbines, the rotating blades are
runner blades or buckets.
• Examples of dynamic pumps include
enclosed pumps and ducted pumps
and open pumps.
• Examples of dynamic turbines
include enclosed turbines, such as a
hydro turbine that extracts energy
from water in a hydroelectric dam,
and open turbines such as the wind
turbine that extracts energy from
the wind.
The Wind Turbine Company. Used by permission.
Dynamic Machines
A wind turbine is a good example of a dynamic
machine of the open type; air turns the blades, and
the output shaft drives an electric generator.
9
2–2 Pumps
• The mass flow rate of fluid through the pump is an obvious primary
pump performance parameter
• For incompressible flow, it is common to use volume flow rate rather
than mass flow rate
• In turbomachinery industries, volume flow rate is called capacity and is
simply mass flow rate divided by fluid density:
• The performance of a pump is characterized additionally by its net head
H, defined as the change in Bernoulli head between the inlet and outlet
of the pump:
10
Brake Horsepower
• For incompressible flow through a pump:
• Special case with
• Net head is proportional to the useful power delivered to the fluid:
• This power is named water horsepower:
Water horsepower:
• Brake horsepower (bhp) is the external power supplied to the
pump:
Brake horsepower:
• Here:
• ω is the rotational speed of the shaft (rad/s)
• Tshaft is the torque supplied to the shaft
11
Pump Efficiency
• Since any pump suffers from irreversible losses due to
friction, internal leakage, flow separation on blade surfaces,
turbulent dissipation etc.
• The pump efficiency is defined as the ratio of useful power to
supplied power as:
12
Pump Performance Curves
• Pump Performance Curves: Curves of H,
, and bhp as
functions of volume flow rate are pump performance curves
(or characteristic curves)
• Operating point or duty point of the system: In a typical
application, Hrequired and Havailable match at one unique value
of flow rate — this is the operating point or duty point of
the system
• For steady conditions, a pump can operate only
along its performance curve
• The steady operating point of a piping system is established at
the volume flow rate where:
Hrequired = Havailable
13
Pump Performance Curves
• Free delivery: The maximum volume flow rate through a pump
occurs when its net head is zero, H = 0;
• This flow rate is the pump’s free delivery
• Shutoff head: The net head that occurs when the volume flow
rate is zero and is achieved when the outlet port of the pump is
blocked off
• Under these conditions, H is large, but V is zero;
• The pump’s efficiency is again zero, because the pump is
doing no useful work
• Best Efficiency Point (BEP): The pump’s efficiency reaches its
maximum value somewhere between the shutoff condition
and the free delivery condition.
• It is denoted by an asterisk (H*, bhp*, etc.)
14
Pump Performance Curves
Typical pump performance curves for a centrifugal pump with backward- inclined
blades; the curve shapes for other types of pumps may differ, and the curves
change as shaft rotation speed is changed.
15
Operating Point
The operating point of a piping system is
established as the volume flow rate
where the system curve and the pump
performance curve intersect.
Situations in which there can be
more than one operating point
should be avoided, since the
system may ‘hunt’ for an operating
point, leading to an unsteady-flow
situation. In such cases a different
pump should be used.
16
Energy Balance
• Energy balance equation for the required net head, Hrequired:
,
,
• It involves elevation change, major and minor losses, fluid
acceleration.
• Correction factors α1 and α2 are often assumed to be unity
since the flow is likely to be turbulent.
17
Energy Balance
,
,
The equation emphasizes the
role of a pump in a piping
system; namely, it increases (or
decreases) the static pressure,
dynamic pressure, and elevation
of the fluid, and it overcomes
irreversible losses.
18
Summary
,
,
The useful pump head delivered to the fluid does 4 things:
• It increases the static pressure of the fluid from point 1 to
point 2 (first term on the right).
• It increases the dynamic pressure (kinetic energy) of the
fluid from point 1 to point 2 (second term on the right).
• It raises the elevation (potential energy) of the fluid from
point 1 to point 2 (third term on the right).
• It overcomes irreversible head losses in the piping system
(last term on the right).
Operating Point:
19
Example 2.1: Operating Point of a Fan
in a Ventilation System
Manufacturer’s performance data for the fan of
3–1*
Example
2.1
Havailable,
inches
V, cfm
H2O
0
0.90
250
0.95
500
0.90
750
0.75
1000
0.40
1200
0.0
* Note that the head data are listed as inches of water,
even though air is the fluid. This is common practice in the
ventilation industry.
20
Example 2.1
A local ventilation system is used to remove air and
contaminants produced by a dry-cleaning operation. The
duct is round and is constructed of galvanized steel with
longitudinal seams and with joints every 30in (0.76 m). The
inner diameter of the duct is D = 9.06in (0.23 m), and its
total length is L = 44 ft, (13.4m). There are 5 CD3-9 elbows
along the duct. The equivalent roughness height of this duct
is 0.15 mm, and each elbow has a minor loss coefficient of
KL= C0 = 0.21.
To ensure adequate ventilation, the minimum required
volume flow rate through the duct is 𝑉̇ = 600 cfm, or 0.283
m3/s at 25 oC. Literature from the hood manufacturer lists
the hood entry loss coefficient as 1.3 based on duct velocity.
When the damper is fully open, its loss coefficient is 1.8. A
centrifugal fan with 9.0in inlet and outlet diameters is
available. Its performance data are shown in table 3.1 above.
Predict the operating point of this ventilation system and
draw a plot of required & available fan pressure rise as
functions of volume flow rate. Is the chosen fan adequate?
21
Example 2.1
• Assumptions: 1) steady, 2) concentration of the contaminants in the air
is low and the mixture properties are those of the air alone, 3) the flow at
the outlet is fully developed and turbulent pipe flow with α=1.05.
• Air properties at 25 oC : ρ=1.184 kg/m3, ν=1.562×10-5 m2/s and
patm=101.3 kPa.
• Solution: Energy balance equation for steady flow:
• Required net head:
22
Example 2.1: Solution
• Solution: Total irreversible head loss:
,
• Roughness factor ε/D = 0.15/230 = 6.52×10-4.
• Reynolds number:
• At minimum required flow rate, V = V2 = 6.81 m/s.
23
Example 2.1: Solution
• Solution: From the Moody chart (Colebrook
equation) at this Reynolds number and roughness
fact, the friction factor is f = 0.0209.
• The total minor losses are given by:
• Then the required net head of the fan at the
minimum flow rate is:
24
Example 2.1: Solution
• The head expressed in terms of
water height can be shown to be:
• The calculations at several
values of volume flow rate are
compared to the available net
head of the fan in the figure.
• The operating point is at a volume
flow rate of about 650 cfm, at which
both the required and available net
head equal about 0.83 inches of
water.
• Thus, the chosen fan is more than
adequate.
25
Example 2.1: Solution
• Discussion:
1) The purchased fan is somewhat
more powerful than required,
yielding a higher flow rate than
necessary.
2) The butterfly damper valve
could be partially closed to
cutback the flow rate to 600
cfm, if necessary.
3) For safety reasons, it is clearly
better to oversize than
undersize a fan, when used with
an air pollution control system.
26
Pump Curve Family
It is common practice in the pump
industry to offer several choices of
impeller diameter for a single pump
casing.
There are several reasons for this:
1. To save manufacturing costs
2. To enable capacity increase
by simple impeller
replacement
3. To standardize installation
mountings
4. To enable reuse of equipment
for a different application
Typical pump performance curves for a
family of centrifugal pumps of the same
casing diameter but different impeller
diameters.
27
Courtesy of Taco, Inc., Cranston, RI. Used by permission. Taco® is a registered trademark of Taco, Inc.
Example of a manufacturer’s performance plot for a family of centrifugal pumps. Each pump
has the same casing, but a different impeller diameter.
28
Example 2.2
A washing operation at a power
plant requires 370 gpm of water.
Hrequired is about 24 ft at this flow
rate. A junior engineer looks
through some catalogs and
decides to purchase the
8.25in impeller option of the Taco
Model 4013FI series centrifugal
pump. If the pump operates at
1160 rpm, as specified in the
performance plot, its
performance curve intersects
370 gpm at H=24 ft.
The chief engineer, concerned about efficiency, glances at the performance curves and notes
that the pump efficiency at this operating point is only 70%. He sees that the 12.75in impeller
efficiency is 76.5% at the same flow rate. He notes that a throttle valve can be installed
downstream of the pump to increase Hrequired so that the pump operates at this higher efficiency.
He asks the junior engineer to calculate which impeller option would need the least amount of
electricity to operate. Perform the comparison and discuss.
29
Example 2.2
Assumptions:
1. Water is at 70 oF,
2.
The flow rate and
head are constant.
Properties for water at
70 oF, ρ=62.30 lbm/ft3.
Solution:
• From the performance plot
of the figure on the right,
the junior engineer
estimates that the pump
with 8.25 in impeller
requires about 3.2 hp from
the motor. She verifies this
estimate by using:
30
Example 2.2
• Solution: the required bhp for the 8.25-in impeller option:
• The required bhp for the 12.75-in impeller option:
31
Example 2.2
Conclusions:
• The smaller-diameter impeller option is the better choice in spite
of its lower efficiency, since it uses less than half the power
Discussion:
• The larger impeller pump could operate at a higher efficiency. But
it would deliver about 72 ft of net head at this required flow rate.
This is overkill, and the throttle valve would be required to make
up the difference between this net head and the required flow
head of 24 ft of water.
• A throttle valve does nothing more than dissipate/waste
mechanical energy. However, so the gain in efficiency of the pump
is more than offset by losses through the throttle valve
• If the flow head or capacity requirements increase at some time in
the future, a larger impeller could be applied
32
Example 2.2
• In some applications, a less
efficient pump from the
same family of pumps may
require less energy to
operate
• An even better choice
would be a pump whose
best efficiency point occurs
at the required operating
point of the pump, but such
a pump is not always
available
33
Pump Cavitation and Net Positive
Suction Head
• When pumping liquids, it is possible
for the local pressure inside the
pump to fall below the vapor
pressure of the liquid, Pv.
• When P < Pv, vapor-filled bubbles
called cavitation bubbles appear.
• The liquid boils locally, typically on
the suction side of the rotating
impeller blades, where the pressure
is lowest.
• Net positive suction head (NPSH): The
difference between the pump’s inlet
stagnation pressure head and the
vapor pressure head.
Cavitation bubbles forming and collapsing on
the suction side of an impeller blade.
•
34
Required Net Positive Suction Head
(NPSHR) The minimum NPSH necessary to avoid cavitation in the pump.
Typical pump performance curve in
which net head and required net
positive suction head NPSHR are
plotted versus volume flow rate.
The volume flow rate at which the available
NPSHA and the required NPSHR intersect
represents the maximum flow rate that can
be delivered by the pump without the
occurrence of cavitation.
35
Example 2.3
• The 11.25 in impeller option of the Taco
Model 4013 FI series centrifugal pump
(see performance figure above) is used to
pump water at 25 oC from a reservoir
whose surface is 4.0 ft above the
centreline of the pump inlet. The piping
system from the reservoir to the pump
consists of 10.5 ft of cast iron pipe with an
ID of 4.0-in and an average inner
roughness height of 0.02 in. There are
several minor losses: a sharp- edge inlet
(KL=0.5), 3 flanged smooth 90o regular
elbows (KL=0.3 each), and a fully open
flanged globe valve (KL=6.0).
Inlet piping system from the reservoir (1) to
the pump inlet (2) for Example 2.3.
• Estimate the maximum volume flow rate
(gpm) that can be pumped without
cavitation. If the water were warmer,
would this maximum flow rate increase
or decrease? Why? Discuss how you
might increase the maximum flow rate
without cavitation.
36
Example 2.3
• Assumptions: 1) steady and incompressible, 2) the flow at the pump inlet
is turbulent and fully developed with α=1.05. 4) no turbines involved, 5)
water speed in the reservoir is 0, i.e. V1≈0.
• Properties: for water at T= 25 oC, ρ=997.0 kg/m3, μ=8.91 ×10-4kg/ms
and Pv=3.169 kPa. Patm=101.3 kPa.(Table A-3, page 950)
• Solutions:
Energy balance equation:
• Pump inlet pressure head:
37
Example 2.3: Solution
• Available NPSHA:
• Irreversible head loss:
• For a given volume flow rate, speed V can be determined and
Reynolds number.
• From Re and the known pipe roughness, apply Moody chart
to obtain friction factor f.
• The minor loss coefficient is:
38
Example 2.3: Solution
• For demonstration, when
• The average speed of pipe water is
• Reynolds number is
• With this Re and roughness factor ε/D = 0.005, applying the Colebrook
equation yields f = 0.0306. Then the available net positive suction head is:
39
Example 2.3: Maximum Flow Rate to
Avoid Pump Cavitation
• Applying EES to calculate NPSH as a
function of volume flow rate .
• Cavitation occurs at flow rates above
approximately 600 gpm.
• As water is getting warmer, viscosity and
density are decreased. However, vapor
pressure is increased. A similar calculation
can be repeated for ρ=983.3 kg/m3,
μ=4.67×10-4kg/m.s and Pv=19.94 kPa
@T=60 oC. The maximum volume flow
rate is about 550 gpm. It decreases with
increased temperature.
• Cavitation is predicted to occur at flow
rates greater than the point where the
available and required values of NPSH
intersect.
Net positive suction head as a function of
volume flow rate for the pump at two
temperatures.
40
Example 2.3: Maximum Flow Rate to
Avoid Pump Cavitation
• Warmer water is closer to its
boiling point.
To increase the maximum flow rate:
• Increase the height of the reservoir
surface,
• Rerouting the pipe so that only 1
elbow is needed and replacing the
globe valve with a ball valve to
decrease minor losses.
• Increasing the diameter of the pipe
and decrease the surface
roughness (to decrease major
losses).
• In practice, decreasing both minor
and major losses could be needed.
However, increasing pipe diameter
is more effective. That is why many
centrifugal pumps have a larger
inlet diameter than outlet diameter.
41
Pumps in Series and Parallel
• When faced with the need to
increase volume flow rate or
pressure rise by a small amount,
you might consider adding an
additional smaller pump in series
or in parallel with the original
pump.
• Arranging two very dissimilar
pumps in (a) series or (b) parallel
can sometimes lead to problems.
• A better solution is to increase
the original pump’s speed and/or
input power, replace the impeller
with a larger one, or replace the
entire pump with a larger one.
42
Pumps in Series and Parallel
• For series-operated pumps,
the smaller pump may be
forced to operate beyond its
free delivery flow rate.
• For parallel-connected
pumps, the smaller pump
may not be able to handle
large head imposed on it,
and the flow in its branch
could be reversed.
• In either case, the power
supplied to the smaller
pump would be wasted.
43
Pumps in Series and Parallel
• There are many applications where 2 or 3 more similar (identical) pumps
are operated in series or in parallel.
• When operated in series, the combined net head is the sum of the net
heads of each pump as:
• Combined net head for n pumps in series:
• When operated in parallel, the combined volume flow rate is the sum of
each individual volume rate as:
• Combined capacity for n pumps in parallel:
44
3 Pumps in Series
• Pump performance curve (dark blue)
for three dissimilar pumps in series.
• At low values of volume flow rate, the
combined net head is equal to the
sum of the net head of each pump by
itself.
• To avoid pump damage and loss of
combined net head, any individual
pump should be shut off and
bypassed at flow rates larger than
that pump’s free delivery, as
indicated by the vertical dashed red
lines.
• If the three pumps were identical, it
would not be necessary to turn off
any of the pumps, since the free
delivery of each pump would occur at
the same volume flow rate.
45
3 Pumps in Parallel
• Pump performance curve (dark blue)
for three pumps in parallel. At a low
value of net head, the combined
capacity is equal to the sum of the
capacity of each pump by itself. To
avoid pump damage and loss of
combined capacity, any individual
pump should be shut off at net heads
larger than that pump’s shutoff head,
as indicated by the horizontal dashed
gray lines.
• That pump’s branch should also be
blocked with a valve to avoid reverse
flow. If the three pumps were
identical, it would not be necessary
to turn off any of the pumps, since
the shutoff head of each pump
would occur at
46
Several identical pumps are
often run in a parallel
configuration so that a large
volume flow rate can be
achieved when necessary.
Three parallel pumps are
shown.
47
Positive-Displacement Pumps (Nonexaminable)
Adapted from F. M. White, Fluid Mechanics 4/e. Copyright © 1999. The
McGraw-Hill Companies, Inc.
• Fluid is sucked into an
expanding volume and then
pushed along as that volume
contracts, but the mechanism
that causes this change in
volume differs greatly among
the various designs.
• Positive-displacement pumps
are ideal for high-pressure
applications like pumping
viscous liquids or thick slurries,
and for applications where
precise amounts of liquid are
to be dispensed or metered, as Positive-displacement pumps: (a) flexible-tube
peristaltic pump, (b) three-lobe rotary pump, (c)
in medical applications.
gear pump, and (d) double screw pump.
48
Positive-Displacement Pumps (Nonexaminable)
Four phases (one-eighth of a turn apart) in the operation of a two-lobe rotary pump, a type
of positive-displacement pump. The blue region represents a chunk of fluid pushed
through the top rotor, while the red region represents a chunk of fluid pushed through the
bottom rotor, which rotates in the opposite direction. Flow is from left to right.
49
Positive-Displacement Pumps (Nonexaminable)
• Comparison of the pump
performance curves of a
rotary pump operating at
the same speed, but with
fluids of various viscosities.
To avoid motor overload the
pump should not be
operated in the shaded
region.
50
Positive-Displacement Pumps (Nonexaminable)
• Positive-displacement pumps
have many advantages over
dynamic pumps.
• For example, a positivedisplacement pump is better
able to handle shear sensitive
liquids since the induced shear
is much less than that of a
dynamic pump operating at
similar pressure and flow rate.
A pump that can lift a liquid even when the
pump itself is “empty” is called a selfpriming pump.
• Blood is a shear sensitive liquid,
and this is one reason why
positive-displacement pumps are
used for artificial hearts.
51
Positive-Displacement Pumps (Nonexaminable)
• Positive-displacement pumps have some disadvantages:
– The volume flow rate can not be changed unless the rotation
rate is changed.
– Very high pressure at the outlet side may be created, if the
outlet is blocked.
– Overpressure protection is needed.
– Sometimes, PD pumps deliver a pulsating flow, which may
not be acceptable for some applications.
• The closed volume (
) that is filled for every n rotations of the
shaft. Then volume flow rate is equal to rotation rate time
divided by as
• Volume flow rate, positive-displacement pump:
52
Positive-Displacement Pumps (Nonexaminable)
Example-2.4
• A two-lobe rotary positivedisplacement pump, moves
0.45 cm3 of SAE30 motor oil
in each lobe volume Vlobe.
• Calculate the volume flow
rate of oil for the case
where =900 rpm.
53
Example 2.4 (Non-examinable)
• Assumption: 1) steady and no leakages, 2) incompressible oil.
• Solution: For half of a rotation (180o for n=0.5 rotations) of
the two counter-rotating shafts, the total volume of oil
pumped is:
• The volume flow rate is then calculated as:
• The higher the fluid density, the higher the required shaft
torque and brake horsepower.
54
Dynamic Pumps
•
•
•
There are three main types of dynamic
pumps that involve rotating blades
called impeller blades or rotor blades,
which impart momentum to the fluid.
They are sometimes called
rotodynamic pumps or simply
rotary pumps.
Rotary pumps are classified by the way
flow exits the pump:
1. centrifugal flow,
2. axial flow, and
3. mixed flow
The impeller (rotating portion) of the three main categories
of dynamic pumps: (a) centrifugal flow, (b) mixed flow, and
(c) axial flow.
55
Pump Flow Directions
• Centrifugal-flow Pump: Fluid enters axially (in the same
direction as the axis of the rotating shaft) in the center of
the pump but is discharged radially (or tangentially) along
the outer radius of the pump casing.
• For this reason, centrifugal pumps are also called radial-flow
pumps.
• Axial-flow Pump: Fluid enters and leaves axially, typically
along the outer portion of the pump because of blockage by
the shaft, motor, hub, etc.
• Mixed-flow Pump: Intermediate between centrifugal and
axial, with the flow entering axially, not necessarily in the
center, but leaving at some angle between radially and
axially.
56
Centrifugal Pumps
• Centrifugal pumps and blowers can
be easily identified by their snailshaped casing, called the scroll.
• They are found all around your home;
in dishwashers, hot tubs, clothes
washers and dryers, hairdryers,
vacuum cleaners, kitchen exhaust
hoods, bathroom exhaust fans, leaf
blowers, furnaces, etc.
• They are used in cars; the water
pump in the engine, the air blower in
the heater/air conditioner unit, etc.
• Centrifugal pumps are ubiquitous in
industry; they are used in building
ventilation systems, washing
operations, cooling ponds and
cooling towers.
Courtesy of the New York Blower Company, Willowbrook, IL. Used
by permission
A typical centrifugal blower with its
characteristic snail-shaped scroll.
57
Centrifugal Pump Components
•
Impeller or Rotor: In pump terminology, the rotating assembly that consists of the shaft,
the hub, the impeller blades, and the impeller shroud.
•
A shroud often surrounds the impeller blades to increase blade stiffness.
Side & frontal view of a centrifugal pump. Fluid enters axially in the middle of the pump (the
eye), is flung around to the outside by the rotating blade assembly (impeller), is diffused in the
expanding diffuser (scroll), and is discharged out the side of the pump. r1 and r2 are the radial
locations of the impeller blade inlet and outlet; b1 and b2 are the axial blade widths at the
impeller blade inlet and outlet, respectively.
58
Pump Blade Orientation
There are three types of centrifugal pump based on impeller blade geometry: (a)
Backward-inclined blades, (b) radial blades, and (c) forward-inclined blades.
• Centrifugal pumps with backward-inclined blades are the most common. These
yield the highest efficiency of the three because fluid flows into and out of the
blade passages with the least amount of turning.
• Centrifugal pumps with radial blades (or straight blades) have the simplest geometry
and produce the largest pressure rise of the three.
• Centrifugal pumps with forward-inclined blades produce a pressure rise that is
nearly constant.
59
Centrifugal Pump Types
The three main types of centrifugal pumps are
those with (a) backward-inclined blades, (b)
radial blades, and (c) forward- inclined blades;
(d) comparison of net head and brake
horsepower performance curves for the three
types of centrifugal pumps.
60
Mass Conservation
Mass conservation holds, since
the volume flow rate entering the
pump eye (defined with width b1
at radius r1)
Pass through the circumferential
cross- sectional area defined by
width b2 at radius r2.
Using the average normal velocity
components , and , , Volume
flow rate is given by:
Close-up side view of the simplified
centrifugal flow pump used for
elementary analysis of the velocity
vectors; V1, n and V2, n are defined as
the average normal (radial)
components of velocity at radii r1
and r2, respectively.
,
,
,
,
61
Velocity Triangles
Close-up frontal view of the simplified centrifugal flow
pump used for elementary analysis of the velocity
vectors. Absolute velocity vectors of the fluid are shown
as bold arrows. It is assumed that the flow is
everywhere tangent to the blade surface when viewed
from a reference frame rotating with the blade, as
indicated by the relative velocity vectors.
•
The inlet of the blade (@r1) moves at
tangential velocity ωr1. The outlet of the
blade moves at ωr2.
•
The 2 tangential velocities differ not
only in magnitude, but in direction,
(inclination of the blade).
•
Leading edge angle β1 as the blade angle
relative to the reverse tangential
direction at r1.
•
Trailing edge angle β2 as the blade angle
relative to the reverse tangential
direction at r2
•
It is assumed that the flow is everywhere
tangent to the blade surface (no flow
separation) when viewed from a reference
frame rotating with the blade.
•
In other words, the flow impinges on the
blade parallel to the blade’s leading edge
and exits the blade parallel to the blades’
trailing edge.
62
Turbomachine Equation
The flow is assumed to be everywhere tangent to the blade surface when viewed from a
reference frame rotating with the blade.
•
Shaft torque is equal to the change in moment of
momentum from inlet to outlet, as given by the
Euler turbomachine equation (or Euler’s turbine
formula)
Euler turbomachine equation:
Tshaft  Vr2V2, t  r1V1, t 
Alternative form:
Tshaft  Vr2V2 sin 2  r1V1 sin 1 

bhp  Tshaft  V r2V2, t  r1V1, t
Control volume (shaded) used for angular
momentum analysis of a centrifugal pump;
absolute tangential velocity components V1,t and
V2, t are labeled.

 Wwater horsepower  gVH
Net head:
1
H  r2V2, t  r1V1, t 
g
63
Example 2.5
A centrifugal blower rotates at
rpm
(183.3 rad/s). Air enters the impeller normal
to the blades (α1=0o) and exits at an angle of
40o from radial (α2=40o) as shown in the
figure.
The inlet radius is r1=4.0 cm, and the inlet
blade width b1=5.2 cm. The outlet radius is
r2=8.0 cm, and the outlet blade width b2=2.3
cm. The volume flow rate is 0.13 m3/s.
For the idealized case, 100% efficiency,
calculate the net head produced by this
blower in equivalent millimetres of water
column height. Also calculate the required
brake horsepower in Watts.
64
Example 2.5
• Assumptions: 1) steady mean flow and no leakages, 2) air
flow is incompressible, 3) efficiency is 100% (no irreversible
losses).
• Air properties: ρair=1.2 kg/m3.
65
Example 2.5: Solution
Solution:
The normal velocity components at the inlet is
calculated as:
Similarly:
66
Example 2.5: Solution
Solution: V1=V1,n and V1,t=0, since α1=0
The net head:
67
Example 2.5: Solution
Required brake horsepower:
• Note that the actual net head delivered
to the air is lower than that predicted
above due to inefficiencies.
• The actual brake horsepower is higher
than that predicted above due to
inefficiencies in the blower, friction on
the shaft etc.
68
Blade Design
To design the blade shapes, trigonometry
expression for V1,t and V2,t are applied in terms
of β1 and β2. Applying the cosine law, we have
V 2 V 2
2
2 2


r2  2r2V 2, relative cos 2
2, relative
It can also be shown that:
V2, relative cos 2  r2 V2, t
Substituting it into the above equation yields:
1
2
2 2
2
V

V

r2 
rV


2 2, t
2
2, relative
2
A similar equation holds for the blade inlet
(change subscript 2 to 1)
The law of cosines is used.
69
Rotating Reference Frame
• Substitute
and
into net head:
• This leads to:
• Net head is proportional to the change in absolute kinetic
energy plus the rotor-tip kinetic energy change minus the
change in relative kinetic energy from inlet to outlet.
70
Rotational Bernoulli Equation
• As shown previously:
• Equating gives:
• This is the Bernoulli Equation in a rotating reference frame:
Relative Frame of Reference
V1, t  r1 
V1, n
tan 1
When V 1,t=0, V 1,n=V 1 and r1 
V1, n
tan 1
For the approximation of flow through an impeller
with no irreversible losses, it is often more
convenient to work with a relative frame of
reference rotating with the impeller; in that case,
the Bernoulli equation gets an additional term, as
indicated in Bernoulli equation in rotating
reference frame.
Close-up frontal view of the
velocity vectors at the impeller
blade inlet. The absolute
velocity vector is shown as a
bold arrow.
72
Preliminary Design of Impeller Blades
V1, n
r1 
tan 1
,
Volume flow rate is expressed as:
,
Setting V1,t=0 can be used for
preliminary design of the
impeller blade shape.
Close-up frontal view of the velocity vectors
at the impeller blade inlet. The absolute
velocity vector is shown as a bold arrow.
73
Axial Pumps
• Axial pumps do not utilize
centrifugal forces. Instead, the
impeller blades behave more like
the wing of an airplane, producing
lift by changing the momentum of
the fluid as they rotate.
• The lift force on the blade is caused The blades of an axial-flow pump behave
by pressure differences between
like the wing of an airplane. The air is turned
the top and bottom surfaces of the downward by the wing as it generates lift
blade, and the change in flow
force FL.
direction leads to downwash (a
column of descending air) through
the rotor plane.
• From a time-averaged
perspective, there is a pressure
jump across the rotor plane that
Downwash and pressure rise across the rotor
induces a downward airflow.
plane of a helicopter, which is a type of axialflow pump.
74
Propeller
•
Imagine turning the rotor plane vertically; we now have a propeller.
•
Both the helicopter rotor and the airplane propeller are examples of open axial-flow fans,
since there is no duct or casing around the tips of the blades.
•
The casing around the house fan also acts as a short duct, which helps to direct the flow and
eliminate some losses at the blade tips.
•
The small cooling fan inside your computer is typically an axial-flow fan; it looks like a
miniature window fan and is an example of a ducted axial-flow fan.
u   r

 
Vrelative Vin Vblade
Axial-flow fans may
be open or ducted:
(a) a propeller is an
open fan, and (b) a
computer cooling
fan is a ducted fan.
Photos by John M. Cimbala.
75
Propeller Performance
Typical fan performance curves
for a propeller (axial-flow) fan.
BEP
• Unlike centrifugal fans, brake
horsepower tends to decrease with
flow rate.
• The efficiency curve leans more to
the right compared to that of
centrifugal fans.
• The result is that efficiency drops
off rapidly for volume flow rates
higher than that at the best
efficiency point.
• The net head curve also decreases
continuously with flow rate
(although there are some wiggles),
and its shape is much different than
that of a centrifugal flow fan.
76
Photo courtesy of United Technologies Corporation/Pratt & Whitney. Used
by permission. All rights reserved.
Turbofan engine (both multistage
axial-flow compressors and turbines)
Pratt & Whitney PW4000 turbofan engine; an example of a multistage axial-flow
turbomachine.
77
2–3 Pump Scaling Laws
Dimensional Analysis
 V
bhp
 function of  3 ,
D
3 D5
 D2  
, 
D

gH
 V
 function of  3 ,
2 D2
D
 D2  
, 
D

D2
Re 

Dimensional analysis of a pump.
78
2–3 Pump Scaling Laws
Dimensionless pump
parameters:
C H  Head coefficient 
gH
 2 D2
V
C Q  Capacity coefficient 
D3
bhp
C P  Power coefficient 
3 D5
• Cavitation is of concern and the required net
positive suction head is applied to create
another dimensionless parameter CNPSH as:
Dimensional analysis of a pump.
C NPSH  Suction head coefficient 
gNPSHrequired
2 D2
79
Dimensional Analysis
Dimensional analysis is useful
for scaling two geometrically
similar pumps.
• If all the dimensionless pump
parameters of pump A are
equivalent to those of pump B,
the two pumps are dynamically
similar.
• When dynamic similarity is
achieved, the operating point
on the pump performance
curve of pump A and the
corresponding point on the
pump performance curve of
pump B are ‘homologous’.
80
Pump Efficiency
•
For many engineering problems, the effects of
both Re and ε/D can be neglected. Thus:
CH  function of CQ
•
The pump efficiency can be transformed by using
dimensionless parameters as:
pump 
•
CP  function of CQ
 VgH 
bhp
 D 3C Q 2 D2C H  C QC H
 function of C Q


3 5
CP
 D C P
Since CH, CP and ηpump are approximated as a function
of CQ , it is often to plot these 3 parameters as a
function of CQ on the same plot, generating a set of
non-dimensional pump performance curves (see figure
below).
81
Nondimensional Pump Curves
When plotted in terms of dimensionless pump parameters, the performance curves of
all pumps in a family of geometrically similar pumps collapse onto one set of
nondimensional pump performance curves.
Values at the best efficiency point are indicated by asterisks.
82
When a small-scale model is tested to
predict the performance of a fullscale
prototype pump, the measured
efficiency of the model is typically
somewhat lower than that of the
prototype.
Empirical correction equations such as
Eq. 2–34 have been developed to
account for the improvement of pump
efficiency with pump size.
Moody efficiency correction equation for pumps:
 D model 
pump, prototype  1 1  pump, model 

D
 prototype 


1/5
14 - 34
(2-34
)
83
Pump Specific Speed NSp
• NSp is a useful dimensionless parameter as formed by CQ and
CH
Pump specific speed:
N Sp 
C1/2
Q
C3/H 4
V/D 


3 1/2
gH / D 
2
2
3/4

V1/2
gH 3/4
Pump specific speed is used to characterize the operation of a pump at its
optimum conditions (best efficiency point) and is useful for preliminary pump
selection and/or design
n, rpmV, gpm
NSp, US 
H, ft  3/4
1/2
Pump specific speed, customary U.S. units:
84
Pump Specific Speed
• Pump specific speed, customary European units:
•
NSp, Eur 


1/2
3

n, Hz V , m /s
gH, m /s 
2
2
3/4
Even though pump specific speed is a dimensionless
parameter, it is common practice to write it as a
dimensional quantity using an inconsistent set of units.
Practicing engineers have been used to using
inconsistent units in the equation that defines NSp.
85
Conversions between the
dimensionless, conventional
U.S., and conventional
European definitions of
pump specific speed NSp.
Numerical values are given
to four significant digits. The
conversions for NSp,US
assume standard earth
gravity.
86
Example 2.6
A pump is being designed to deliver 320 gpm of gasoline at room temperature.
The required net head is 22.5 ft. The pump shaft is determined to rotate at
1170 rpm. Calculate the pump specific speed in both non-dimensional form
and customary US form. Based on your results, decide which kind of dynamic
pump would be most suitable.
plication.
Assumptions: 1) the
pump operates near its
best efficiency point, 2)
the maximum efficiency
versus pump specific
speed curve follows the
figure well.
Max efficiency as a function of pump specific speed for the 3
main types of dynamic pump
87
Maximum efficiency as a function of pump specific speed for the three main types
of dynamic pump. The horizontal scales show nondimensional pump specific speed
(NSp), pump specific speed in customary U.S. Units (NSp, US), and pump specific speed
in customary European units (NSp, Eur).
89
Affinity Laws (similarity rules) for
homologous states A & B
3
VB  B  D B 


VA A  DA 
14- 38a
(2-38a)
2
Affinity laws:
H B  B   D B 

  
HA  A   DA 
3
2
bhpB B   B   D B 

   
bhpA A  A   DA 
(2-38b)
14- 38b
5
14- 38c
(2-38c)
• Eqs. (2–38) are applicable to both pumps and turbines.
• States A and B can be any two homologous states between any two
geometrically similar turbomachines, or even between two homologous
states of the same machine.
• Examples include changing rotational speed or pumping a different fluid
with the same pump.
90
Affinity Laws
For a given pump in which ω is varied, but DA=DB and ρA and ρB, affinity
laws reduce to:
When the affinity laws are applied to a single pump in which the only thing that is varied is shaft
rotational speed 𝜔, or shaft rpm, n, Eqs. (3–38) reduce to those shown above, for which a
mnemonic can be used to help us remember the exponent on 𝜔 (or on n):
Very Hard Problems are as easy as 1, 2, 3.
91
Example 2.7
An engineering uses a small closed-loop water tunnel to
perform flow visualization research. He would like to double
the water speed in the test section of the tunnel and realizes
that the least expensive way to do this is to double the
rotational speed of the flow pump. What he doesn’t realize is
how much more powerful the new electric motor will need to
be? If the engineers doubles the flow speed, by approximately
what factor will the motor power need to be increased?
• Assumptions: 1) the water remains the same temperature, 2)
after doubling the pump speed, the pump runs at a
conditions homologous to the original conditions.
• Solution: since neither diameter nor density has changed,
the affinity laws show that:
92
Example 2.7: Solution
Solution (Cont): the ratio of required shaft
power:
Since ωB=2ωA, thus bhpB=8bhpA. The power
to the pump motor is increased by a factor of
8.
As shown in the figure, both net head and
power increase rapidly as pump speed is
increased.
When the speed of a pump is increased, net head increases rapidly; brake
horsepower increases even more rapidly.
93
Example 2.7
• Discussion: No piping system is
considered.
• The friction factor results in
the required head of the
system is not increased by a
factor of 4. The assumption 2
is not necessarily correct,
when the flow speed is
doubled.
• The system will adjust to an
operating point at which
required and available heads
match. However, this point is
not necessarily be
homologous with the original
operating point.
94
Example 2.8
Manufacturer’s performance
data for a water pump
operating at 1725 rpm and
room temperature (Example 3–
11)*
V,
cm3/s
%
100
H, cm
ηpump,
180
32
200
185
54
300
175
70
400
170
79
500
150
81
600
95
66
700
54
38
* Net head is in centimeters of water.
In a pump manufacturing company, the best-selling
product is a water pump-pump A. Its impeller diameter is
DA=6.0 cm, and its performance data when operating at
nA=1725 rpm (ωA=180.6 rad/s) are shown in the Table. As
market demands, the company need to design a new
product: pump B (a larger pump) to pump liquid
refrigerant R-134a at room temperature. The new pump is
to be designed such that its best efficiency point occurs as
close as possible to a volume flow rate of VB=2400 cm3/s
and a net head of HB=450 cm (of R-134a). The chief
engineer tells you to perform some preliminary analyses
using pump scale laws to determine if a geometrically
scaled-up pump could be designed and built to meet the
given requirements.
(a) plot the performance curves of pump A in both
dimensional and dimensionless form and identify the best
efficiency point.
(b) calculate the required pump diameter DB, rotational
speed nB, and brake horsepower bhpB for the new
product.
95
Example 2.8
• Assumptions: 1) new pump B is
geometrically similar to Pump-A, 2)
both liquids are incompressible, 3)
both pumps operate under steady
conditions.
• Properties: at T=20oC, ρwater=998.0
kg/m3, ρR-134a=1226 kg/m3.
• Solution: (1) curve fitting done to the
data in the table by using a 2nd order
least-square polynomial function (see
figure). A curve for brake horsepower
is also plotted by using Eq. below
Pump efficiency:
pump 
Wwater horsepower Wwater horsepower gVH


Wshaft
bhp
T shaft
Data points and smoothed dimensional pump
performance curves for the water pump.
96
Example 2.8: Discussion
• The desired value of ωB is determined. However, it is difficult
to find an electric motor that rotates at exactly the desired
rpm.
• The desired rotation rate is achievable, if the pump is beltdriven or if there is a gear box or a frequency controller.
• Standard single-phase 60-Hz, 120-v AC electric motors
typically run at 1725 or 3450 rpm. The new pump B can be
driven at standard speed. However, it would not operate at a
point exactly at the BEP.
101
Non-Dimensional Performance
Smoothed nondimensional pump performance
curves for the pumps of Example 2–8; BEP is
estimated as the operating point where 𝜂pump is a
maximum.
The affinity laws are manipulated to
obtain an expression for the new pump
diameter DB. 𝜔B and bhpB can be
obtained in similar fashion (not shown).
103
•
•
•
•
•
•
•
Turbines have been used for centuries
to convert freely available mechanical
energy from rivers and wind into
useful mechanical work (through a
rotating shaft).
The rotating part of a hydro turbine is
the runner.
When the working fluid is water, the
turbomachines are hydraulic turbines
or hydro turbines.
When the working fluid is air, and
energy is extracted from the wind,
the machine is a wind turbine.
windmill is used to describe any
wind turbine, whether used to
grind grain, pump water, or
generate electricity.
Turbomachines that convert energy
from the steam into mechanical energy
of a rotating shaft are steam turbines.
Turbines that employ a
compressible gas as the working
fluid is gas turbine.
© OJPhotos/Alamy RF
3–4 Turbines
An old windmill in Klostermolle, Vestervig,
Denmark that was used in the 1800s to grind
grain. (Note that the blades must be covered to
function.) Modern “windmills” that generate
electricity are more properly called wind turbines.
104
Efficiency
In general, energy-producing turbines have somewhat higher
overall efficiencies than do energy-absorbing pumps.
• Large hydroturbines achieve overall efficiencies above 95 percent, while the
best efficiency of large pumps is a little more than 90 percent.
Several reasons for this:
1. pumps normally operate at higher rotational speeds than do
turbines; therefore, shear stresses and frictional losses are higher.
2. conversion of kinetic energy into flow energy (pumps) has inherently
higher losses than does the reverse (turbines).
3. turbines (especially hydroturbines) are often much larger than pumps,
and viscous losses become less important as size increases.
4. while pumps often operate over a wide range of flow rates, most
electricity-generating turbines run within a narrower operating range
and at a controlled constant speed; they can therefore be designed to
operate most efficiently at those conditions.
105
Turbine Types
• As with pumps, we classify turbines into two broad
categories:
• positive displacement and dynamic.
• Positive-displacement turbines are small devices used for
volume flow rate measurement, while dynamic turbines
range from tiny to huge and are used for both flow
measurement and power production.
• Positive-displacement turbines are generally not used for
power production, but rather for flow rate or flow volume
measurement.
• Dynamic turbines are used both as flow measuring devices
and as power generators
106
Positive-Displacement Turbines
• A positive-displacement turbine may be thought
of as a positive-displacement pump running
backward—as fluid pushes into a closed volume,
it turns a shaft or displaces a reciprocating rod.
• The closed volume of fluid is then pushed out
as more fluid enters the device.
• There is a net head loss through the
positive-displacement turbine; energy is
extracted from the flowing fluid and is
turned into mechanical energy.
The nutating disc fluid flowmeter is a type of
positive-displacement turbine used to measure
volume flow rate: (a) cutaway view and (b) diagram
showing motion of the nutating disc.
This type of flowmeter is commonly used as a
water meter in homes.
(a) Courtesy of Niagara Meters, Spartanburg, SC.
107
Dynamic Turbines (flow measuring
and power generation)
• Hydroturbines utilize the large elevation change across a dam to generate
electricity, and wind turbines generate electricity from blades rotated by the
wind. There are two basic types of dynamic turbine—impulse and reaction.
• Impulse turbines require a higher head but can operate with a smaller volume
flow rate. Reaction turbines can operate with much less head but require a
higher volume flow rate.
(a) © Matthias Engelien/Alamy RF. (b) NASA Langley Research Center.
Examples of dynamic turbines: (a) a typical three-cup anemometer used to measure wind speed, and
108
Impulse Turbines
• In an impulse turbine, the fluid is sent through
a nozzle so that most of its available
mechanical energy is converted into kinetic
energy.
• The high-speed jet then impinges on bucketshaped vanes that transfer energy to the
turbine shaft.
• The modern and most efficient type of
impulse turbine is Pelton turbine, and the
rotating wheel is now called a Pelton wheel.
Schematic diagram of a Pelton-type impulse
turbine; the turbine shaft is turned when high-speed
fluid from one or more jets impinges on buckets
mounted to the turbine shaft. (a) Side view, absolute
reference frame, and (b) bottom view of a
cross section of bucket n, rotating reference frame.
109
A close-up view of a Pelton wheel
showing the detailed design of the
buckets; the electrical generator is on the
right. This Pelton wheel is on display at
the Waddamana Power Station Museum
near Bothwell, Tasmania.
Courtesy of Hydro Tasmania, www.hydro.com.au. Used by permission.
A view from the bottom of an operating
Pelton wheel illustrating the splitting and
turning of the water jet in the bucket. The
water jet enters from the left, and the
Pelton wheel is turning to the right.
© ANDRITZ HYDRO GmbH
110
Reaction Turbines
• The other main type of energyproducing hydroturbine is the reaction
turbine, which consists of fixed guide
vanes called stay vanes, adjustable
guide vanes called wicket gates, and
rotating blades called runner blades.
• Flow enters tangentially at high
pressure, is turned toward the runner
by the stay vanes as it moves along the
spiral casing or volute, and then passes
through the wicket gates with a large
tangential velocity component.
• A reaction turbine differs significantly
from an impulse turbine; instead of
using water jets, a volute is filled with
swirling water that drives the runner.
For hydroturbine applications, the axis
is typically vertical. Top and side views
are shown, including the fixed stay
vanes and adjustable wicket gates.
111
Reaction Turbines
• There are two main types of reaction turbine—Francis and
Kaplan
• The Francis turbine is somewhat similar in geometry to a
centrifugal or mixed-flow pump, but with the flow in the
opposite direction
• The Kaplan turbine is somewhat like an axial-flow fan
running backward
• We classify reaction turbines according to the angle that
the flow enters the runner. If the flow enters the runner
radially, the turbine is called a Francis radial-flow turbine
• If the flow enters the runner at some angle between radial
and axial, the turbine is called a Francis mixed-flow turbine
• The latter design is more common
112
Francis Turbines
• Some hydroturbine engineers use the term “Francis turbine”
only when there is a band on the runner.
• Francis turbines are most suited for heads that lie between the high heads of
Pelton wheel turbines and the low heads of Kaplan turbines.
• A typical large Francis turbine may have 16 or more runner
blades and can achieve a turbine efficiency of 90% to 95%
• If the runner has no band, and flow enters the runner
partially turned, it is called a propeller mixed-flow turbine or
simply a mixed-flow turbine
• If the flow is turned completely axially before entering the
runner, the turbine is called an axial-flow turbine
113
(a) Francis radial flow,
(b) Francis mixed flow,
(c) propeller mixed flow, and
(d ) propeller axial flow.
The main difference between
(b) and (c) is that Francis
mixed-flow runners have a
band that rotates with the
runner, while propeller mixedflow runners do not. There are
two types of propeller mixedflow turbines: Kaplan turbines
have adjustable pitch blades,
while propeller turbines do not.
•
•
Kaplan turbines are called double regulated because the flow rate is controlled in two
ways—by turning the wicket gates and by adjusting the pitch on the runner blades.
Propeller turbines are nearly identical to Kaplan turbines except that the blades are fixed
(pitch is not adjustable), and the flow rate is regulated only by the wicket gates (single
regulated).
114
(a) Francis radial flow,
(b) Francis mixed flow,
(c) propeller mixed flow, and
(d ) propeller axial flow.
The main difference between
(b) and (c) is that Francis
mixed-flow runners have a
band that rotates with the
runner, while propeller mixedflow runners do not. There are
two types of propeller mixedflow turbines: Kaplan turbines
have adjustable pitch blades,
while propeller turbines do not.
Compared to the Pelton and Francis turbines, Kaplan turbines and propeller turbines are
most suited for low head, high volume flow rate conditions.
Their efficiencies compared with Francis turbines may be as high as 94%.
115
Ideal power production:
Wideal  gVHgross
H gross  zA  zE
Net head for a hydraulic turbine:
H  EGLin  EGLout
Typical setup and terminology for a hydroelectric plant
that utilizes a Francis turbine to generate electricity.
Turbine efficiency:
turbin 
e
Wshaft
Wwater

bhp
gHV
The net head of a turbine is
defined as the difference
between the energy grade line
(EGLin) just upstream of the
turbine and the energy grade line
(EGLout) at the exit of the draft
tube.
horsepower
116
Turbine Efficiency
• Real efficiency must always
be less than 100%.
• The efficiency of a turbine is
the reciprocal of the
efficiency of a pump
117
Hydropower Turbines
(a) © Corbis RF (b) © Brand X Pictures RF
(a) An aerial view of Hoover Dam and (b) the top (visible) portion of several of the
parallel electric generators driven by hydraulic turbines at Hoover Dam.
118
Relative and absolute velocity vectors and
geometry for the inner radius of the
runner of a Francis turbine.
Absolute velocity vectors are bold.
Relative and absolute velocity vectors and
geometry for the outer radius of the
runner of a Francis turbine.
Absolute velocity vectors are bold.
Runner leading edge:
Runner trailing edge:
V2, t  r2 
V2, n
tan 2
V1, n
V1, t  r1 
tan

119
In some Francis mixed-flow turbines, high-power, high-volume flow rate conditions
sometimes lead to reverse swirl, in which the flow exiting the runner swirls in the
direction opposite to that of the runner itself, as sketched here.
120
© American Hydro Corporation. Used by permission.
Visualisation
• Visualization from CFD analysis of a
hydroturbine including the wicket gates,
runner, and draft tube. Five of the twenty
wicket gates are hidden to better show the
runner blades. The flow pathlines, which
can be thought of as the paths that single
molecules of water take as they flow
through the turbine, are shown relative to
the angular velocity of each stage and help
visualize how the water enters and exits
each component of the turbine.
• The surface pressure contours are coloured
by static pressure and show areas of high
and low pressure on the wicket gates and
runner blades. While numerical data of
head, flow, and power are extracted from
the CFD model during design iterations, this
type of visualization is very useful to
engineers as a qualitative analysis to
identify areas that are susceptible to
cavitation damage and unwanted vortices.
121
Example 2.9
• A hydroelectric power plant is being designed. The gross head from the
reservoir to the tailrace is 1065 ft, and the volume flow rate of water
through each turbine is 203,000 gpm at 70 oF. There are 12 identical
parallel turbines, each with an efficiency of 95.2%, and all other
mechanical energy losses (through the penstock etc.) are estimated to
reduce the input by 3.5%. The generator itself has an efficiency of 94.5%.
Estimate the electric power production from the plant in MW.
• Assumptions: 1) steady and incompressible, 2) parallel configuration
and identical efficiencies.
• Properties: the density of water at T=70 oF is 62.30 lbm/ft3.
• Solution:
The ideal power produced by one hydro-turbine is:
122
Example 2.10
A retrofit Francis radial-flow hydroturbine is designed to
replace an old turbine in a hydroelectric dam. The new
turbine meet the following design restrictions to properly
couple with the existing setup: the runner inlet radius is
r2=8.2 ft (2.5m) and its outlet radius is r1=5.8 ft (1.77m).
The runner blade widths are b2=3.0 ft(0.914 m) and
b1=8.6 ft (2.62m) at the inlet and outlet respectively. The
runner rotate at n=120 rpm (ω=12.57 rad/s) to turn the 60
Hz electric generator. The wicket gates turn the flow by
angle α2=33o from radial at the runner inlet, and the flow
at the runner outlet is to have angle α1 between -10o and
10o from radial for proper flow through the draft tube.
The volume flow rate at design conditions is 9.50×106 gpm
(599 m3/s), and the gross head provided by the dam is
Hgross=303 ft (92.4 m).
Calculate (a) inlet and outlet runner blade angles β2 and β1 and predict the power output and
required net head if irreversible losses are neglected for the case with α1=10o from radial
(rotation-swirl), (b) repeat the calculations for the case with α1=0o from radial (no swirl), c) repeat
the calculations for the case with α1=-10o from radial (reverse swirl).
124
Example 2.10
Assumptions: 1) steady and incompressible, 2) water flow at 20 oC, 3) the blades is
infinitesimally thin, 4) the flow is tangent to the runner blades, 5) irreversible losses
through the turbines are neglected.
• Properties: water at 20 oC, ρ=998.0 kg/m3.
• Solution: (a) the normal component of velocity at the inlet is:
• The tangential velocity component at the inlet is:
125
Example 2.10: Solution
• (a): The required net head (assuming
ηturbine=100%) is:
• Solution (b): the foregoing calculations
with no swirl at the runner outlet α1=0 o,
the runner blade trailing edge angle
reduces to 42.8o, and the output power
increases to 509 MW (6.83×105 hp).
Hrequired increases to 86.8 m
• Solution c) for α1=-10 o, the runner
blade trailing edge angle reduces to
38.5o, and the output power increases
to 557 MW (7.47×105 hp)
• Hrequired increases to 95.0 m. A plot of
power and net head as a function of
runner outlet flow angle is α1 shown in
the figure
127
Example 2.10: Discussion
• The output power is theoretically increased by 10% by
eliminating swirl from the runner outlet and by nearly another
10% when there is 10 deg of reverse swirl
• The gross head available from the dam is only 92.4 m. Thus, the
reverse swirl case of part c) is clearly impossible, since the
predicted net head is required to be greater than Hgross
• Here we neglect irreversibilities
• In practice, the actual output power will be lower and the
actual required net head will be higher than the values
predicted here
128
•
In a coal or nuclear power plant,
high-pressure steam is produced by
a boiler and then sent to a steam
turbine to produce electricity.
•
Because of reheat, regeneration, and
other efforts to increase overall
efficiency, these steam turbines
typically have two stages (high
pressure and low pressure).
•
Most power plant steam turbines
are multistage axial-flow devices.
•
There are also stator vanes (called
nozzles) that direct the flow
between each set of turbine blades
(called buckets).
© Miss Kanithar Aiumla-OR/Shutterstock RF
Gas and Steam Turbines
Turbine blades (buckets) in the rotating portion of
the low-pressure stage of a steam turbine used in
power plants. High pressure superheated steam
enters at the middle and is split evenly left and
right to accommodate the large volume flow rate
without requiring an enormous casing. The
curved shapes of the buckets are clearly seen in
the stages on the left side. Pressure decreases
across each stage, requiring progressively larger
buckets in the direction of flow
129
Gas Turbines
• A gas turbine generator is like a jet
engine except that instead of providing
thrust, the turbomachine is designed to
transfer as much of the fuel’s energy as
possible into the rotating shaft, which is
connected to an electric generator.
• Gas turbines used for power
generation are typically much larger
than jet engines.
• A significant gain in efficiency is
realized as overall turbine size
increases.
Courtesy of GE Energy
The rotor assembly of the MS7001F gas turbine being lowered into the bottom half of the gas turbine casing.
Flow is from right to left, with the upstream set of rotor blades (called blades) comprising the multistage
compressor and the downstream set of rotor blades (called buckets) comprising the multistage turbine.
Compressor stator blades (called vanes) and turbine stator blades (called nozzles) can be seen in the bottom
half of the gas turbine casing. This gas turbine spins at 3600 rpm and produces over 135 MW of power.
130
Wind Turbines (Non-Examinable)
• As global demand for energy increases, the supply of fossil fuels diminishes, and
the price of energy continues to rise.
• To keep up with global energy demand, renewable sources of energy such as
solar, wind, wave, tidal, hydroelectric, and geothermal must be tapped more
extensively.
• In this section we concentrate on wind turbines used to generate electricity.
• We note the distinction between the terms windmill used for mechanical
power generation (grinding grain, pumping water, etc.) and wind turbine used
for electrical power generation.
• Although the wind is “free” and renewable, modern wind turbines are expensive
and suffer from one obvious disadvantage compared to most other power
generation devices – they produce power only when the wind is blowing, and the
power output of a wind turbine is thus inherently unsteady.
• Wind turbines need to be located where the wind blows, which is often far
from traditional power grids, requiring construction of new high-voltage
power lines.
131
Wind Turbines (Non-Examinable)
• We generally categorize wind turbines by the orientation of
their axis of rotation:
• Horizontal axis wind turbines (HAWTs)
• Vertical axis wind turbines (VAWTs).
• An alternative way to categorize them is by the mechanism
that provides torque to the rotating shaft: lift or drag.
• So far, none of the VAWT designs or drag-type designs has
achieved the efficiency or success of the lift-type HAWT.
• This is why most wind turbines being built around the world
are of this type, often in clusters affectionately called wind
farms.
• For this reason, the lift-type HAWT is the only type of wind
turbine discussed in any detail in this section.
132
Wind Turbine Designs
133
Wind turbine Basic Concepts
• Cut-in speed: The minimum wind speed at which useful power can be
generated.
• Rated speed: The wind speed that delivers the rated power, usually the
maximum power.
• Cut-out speed: The maximum wind speed at which the wind turbine is designed
to produce power. At wind speeds greater than the cut-out speed, the turbine
blades are stopped by some type of braking mechanism to avoid damage and for
safety issues. The short section of dashed blue line indicates the power that
would be produced if cut-out were not implemented.
Typical qualitative wind-turbine
power performance curve with
definitions of cut-in, rated, and
cut-out speeds.
134
© Doug Sherman/GeoFile RF
© Construction Photography/Corbis RF
The disk area of a wind turbine is defined as the swept area
or frontal area of the turbine as “seen” by the oncoming
wind, as sketched here in red. The disk area is (a) circular for
a horizontal axis turbine and (b) rectangular for a vertical
axis turbine.
135
Disk area A:
• It is the circular area (normal to wind direction) swept out by
the turbine blades as they rotate.
• The available wind power
in the disk area is
calculated as the rate of change of kinetic energy of the
wind
Wavailable 

d 12 mV 2
dt
  1 V dm  1 V m  1 V VA  1 V A Available wind power:
2
2
2
dt
2
2
2
3
2
For comparison, wind power density is typically applied. It is defined as the aviable
wind power per unit area W/m2:
Wind power density:
Wavailable 1 3
 V
A
2
136
Average Wind Power Density
Wavailable 1 3
 V
A
2
• The wind power density is directly proportional to air density—cold air has a larger wind
power density than warm air blowing at the same speed, although this effect is not as
significant as wind speed.
• The wind power density is proportional to the cube of the wind speed—doubling the wind
speed increases the wind power density by a factor of 8. It should be obvious then why wind
farms are located where the wind speed is high!
• Average wind power density in terms of annual average wind speed as
Average wind power density:
Wavailable 1
 avg V 3 K e
A
2
100 W/m2, poor
400 W/m2, good
700 W/m2, great
Ke is a correction factor called energy pattern factor: it is defined as
Energy pattern factor
1 N 3
V
Ke 
3  i
NV
N=8760 (hours/year)
137
Aerodynamic Efficiency
Aerodynamic efficiency: The fraction of available wind power that is extracted by the
turbine blades. his efficiency is commonly called the power coefficient, CP
Power coefficient:
Wrotor shaft output Wrotor shaft


C p output

Wavailable
1
V 3 A
2



 F   mV   mV
out
in
FR  mV2  mV1  mV2 V1 
FR  P3A  P4 A  0

FR  P4  P3  A
P3 V3 2
P1 V12

 z1 

 z3
 g 2g
 g 2g
The large and small control volumes for analysis of ideal wind
turbine performance bounded by an axisymmetric diverging
stream tube.
P4 V42
P2 V2 2

 z4 

 z2
 g 2g
 g 2g
138
Average Velocity through Turbine
Qualitative sketch of
average streamwise
velocity and pressure
profiles through a wind
turbine.
V12 V 22 P3  P 4

2


Substituting m =ρV3A and combining the results with
FR  P3A  P4 A  0
yields
V V
V3  1 2
2
 FR  P4  P3  A
The average velocity of the air through an ideal wind turbine is the arithmetic
average of the far upstream and far downstream velocities.
139
Defining a as
a
V1 V3
V1
ThenV3  V1 1  a and m  AV3   AV1(1a)
V3 
V1 V2
2
V2  V1 1 2a
For an ideal wind turbine without irreversible losses such as friction, the power
generated by the turbine is simply the difference between the incoming and outgoing
kinetic energies
2
2
2
2
2
V
V
V
1
2a
V


2
1
1
2
Wideal  m 1
  AV 1 1 a 
 2 AV 13a 1 a 
2
2
The power coefficient (i.e. efficiency of the turbine)
CP 
Wrotor shaft output
1
2
V13 A
2
Wideal
2  AV 13a 1 a
2
 1


4a
1
a


3
3
1
2 V1 A
2 V1 A
2
16
1 1
C P, max  4 1  
 0.5926
3 3
27
Betz limit
a  1/3
setting dC P /da  0
or a=1
140
Performance (power coefficient) of various types of wind turbines as a function of the ratio of
turbine blade tip speed to wind speed. So far, no design has achieved better performance
than the horizontal axis wind turbine (HAWT).
141
Example 2.11
The average wind speed at a proposed HAWT wind
farm site is 12.5 m/s (see figure). The power efficient
of each wind turbine is predicted to be 41%, and the
combined efficiency of the gearbox and generator is
92%. Each wind turbine must produce 2.5 MW of
electrical power when the wind blows at 12.5 m/s.
(a) Calculate the required diameter of each turbine
disk. Take the average air density to be ρ=1.2
kg/m3.
(b) If 20 such turbines are built on the site and an
average home in the area consumes
approximately 1.5 Kw of electrical power,
estimate how many homes can be powered by
this wind farm, assuming an additional efficiency
of 96% to account for powerline
losses.
142
2–5 Turbine Scaling Laws
The main variables used for dimensional
analysis of a turbine. The characteristic
turbine diameter D is typically either the
runner diameter Drunner or the discharge
diameter Ddischarge.
Dimensionless turbine parameters:
gH
C H  Head coefficient  2 2
D
bhp
CP  Power coefficient 
3 D5
turbine 
V
C Q  Capacity coefficient 
D3
(14
– 61)
(2-61)
bhp
turbine  Turbine efficiency 
gHV
CP
 function of C P
C QC H
145
Efficiency Correction
Moody efficiency correction equation for turbines:
turbine,
prototype
 D model 
 1 1  turbine, model 

 Dprototype



1/5
In practice, hydroturbine engineers
generally find that the actual increase in
efficiency from model to prototype is only
about two-thirds of the increase given by
this equation.
Dimensional analysis is useful for scaling
two geometrically similar turbines. If all
the dimensionless turbine parameters of
turbine A are equivalent to those of
turbine B, the two turbines are
dynamically similar.
146
Example 2.12
A Francis turbine is being designed for a hydroelectric dam.
Instead of starting from scratch, the engineers decide to
geometrically scale up a previously designed hydro turbine
that has an excellent performance history.
• The existing turbine (Turbine A) has a diameter DA=2.05 m
and spins nA at =120 rpm (ωA=12.57 rad/s). At its best
efficiency point, VA=350 m3/s and HA=75.0 m of water, and
bhpA=242 MW
• The new turbine (Turbine B) is for a larger facility. Its
generator will spin at the same speed (120 rpm), but its net
head will be higher (HB=104 m)
• Calculate the diameter of the new turbine such that it
operates most efficiently, and calculate VB, bhpB and ηB
147
Example 2.12: Assumptions
• Assumptions:
1. Turbine B geometrically similar to turbine A,
2. Reynolds number and roughness effects are
neglected
3. The new penstock is geometrically similar to the
existing penstock so the incoming flow profile,
turbulence intensity etc is similar
• Properties: Water at 20oC, ρ=998.0 kg/m3
148
Turbine Specific Speed
Turbine specific speed:




 bhp
bhp/3 D5
C P 1/2
N St 
 1/2
5/ 4 
5/4
2 2 5/4
CH
gH



gH / D
Relationship between NSt and NSp:
1/2
NSt  NSp
1/2
turbine
A pump–turbine is used by some power plants for energy storage: (a) water is pumped by the
pump–turbine during periods of low demand for power, and (b) electricity is generated by the
pump–turbine during periods of high demand for power.
150
© American Hydro Corporation. Used by permission.
The runner of a pump–turbine used at the Yards Creek pumped storage station in Blairstown,
NJ. There are seven runner blades of outer diameter 5.27 m. The turbine rotates at 240 rpm
and produces 112 MW of power at a volume flow rate of 56.6 m3/s from a net head of 221 m.
151
Capacity Specific Speed
Pump – turbine specific speed relationship at same flow rate and rpm:
3/ 4
N St  N Sp
 H pump 
turbine 

 H turbine 
 N Sp turbine 
5/4
 
3/4
pump
3/ 4
 bhp pump 


bhp
turbine 

Turbine specific speed, customary U.S. units:
n, rpm  bhp, hp1/2
NSt, US 
H, ft  5/ 4
Conversions between the dimensionless and
the conventional U.S. definitions of turbine
specific speed. Numerical values are given to
four significant digits. The conversions
assume earth gravity and water as the
working fluid.
NSt, SI 



1/2
n,rpm V, m3 /s
H, m
3/4
Capacity specific speed
Turbine specific speed is used to characterize the operation of a turbine at its optimum
conditions (best efficiency point) and is useful for preliminary turbine selection.
152
Maximum efficiency as a function of turbine specific speed for the three main types of
dynamic turbine. Horizontal scales show nondimensional turbine specific speed (NSt) and
turbine specific speed in customary U.S. units (NSt, US). Sketches of the blade types are also
provided on the plot for reference.
153
Example 2.13
• Calculate and compare the turbine specific speed for both the small (A)
and large (B) turbines of Example-2.15.
• Assumptions:
1. Turbine B is geometrically similar to turbine A,
2. Reynolds number and roughness effects are neglected,
3. The new penstock is geometrically similar to the existing
penstock so the incoming flow profile, turbulence intensity etc is
similar.
• Properties: water at 20oC, ρ=998.0 kg/m3.
• Solution: The dimensionless turbine specific speed for turbine A is:
154
Summary: Classifications and
Terminology (Pumps)
• Pump Performance Curves and Matching a Pump to a Piping System
• Pump Cavitation and Net Positive Suction Head
• Pumps in Series and Parallel
• Positive-Displacement Pumps, Dynamic Pumps, Centrifugal Pumps, Axial Pumps
• Pump scaling laws
• Dimensional Analysis
• Pump Specific Speed
• Affinity Laws
• Turbines
•
•
•
•
•
•
Positive-Displacement Turbines
Dynamic Turbines
Impulse Turbines
Reaction Turbines
Gas and Steam Turbines
Wind Turbines
• Turbine scaling laws
• Dimensionless Turbine Parameters
• Turbine Specific Speed
157