ME436 Applied Fluid
Mechanics
Spring 201
2
Lecture 3:
Euler
Turbomachinery Equation
Turbomachinery type
Axial
Mixed
Radial
Velocity Components
r
a
w
inlet
outlet
At any point in the pump, velocity is in
axial (a),
radial (r) and
tangential (w) directions.
Any change in
C
a
=> Axial force; thrust bearing
to the stationary rotor casing
C
r
=> Radial force
C
w
=> angular motion; rotational
effect
Axial
Compact
Same inlet and outlet area
High rpm
Axial Pumps
Axial Pumps
Axial Pumps
Radial
Flow direction changes
Technical Drawings
Meridional Direction
In the flow direction
May be axial or radial or a combination
Definition
U: rotor speed
V: relative velocity
C: absolute velocity
Rotation
Velocity
Triangles
Euler’s Turbomachinery Equation
Assumptions:
Steady flow in a turbomachinery
Neglect turbulence effects, instabilities
etc.
Constant mass flow rate
Front View
inlet
outlet
Mass Balance:
Angular Momentum Balance:
Angular momentum balance
Torque
Power
Per unit mass:
w
r = U (rotor speed) :
Euler’s Turbomachinery
Equation
Velocity Triangles
inlet
outlet
Change of absolute kinetic
energy
Virtual pressure rise
Pump or compressor: c
2
2
may
be important
External effect + Internal diffusion effect
Change of relative kinetic
energy
Change in static pressure, if
the losses are neglected
V2>V1 : nozzle

like
V1>V2 : diffuser

like
Centrifugal effect
(energy produced by impeller)
ME436 Applied Fluid
Mechanics
Spring 201
2
Lecture
4
: Centrifugal
Turbomachinery with an
incompressible working fluid
(Pumps and Fans)
PUMP
FAN
Working fluid
Liquid (eg. Water)
Gas (eg. Air)
Material
Eg. Steel, titanium
Erosion due to
impurities
or cavitation
is a major issue
Eg. Plastik
Sealing
Important
. Leakage
may cause problems
Not important. Since
sealing increases cost,
usually avoided.
Cost
higher
Low
Pumps and fans
Hydraulic Pumps
Side and front views
tongue
Volute,
casing,
housing
Blade
Technical Drawings
Open
impeller
Closed
impeller
Open impeller
Blade
Length
Thickness or width
Breadth
Inlet angle
Outlet angle
Leading edge
Trailing edge
Tip
Blade channel
Pressure side
Suction side
Ideal Impeller
Infinitely many blades with zero
thickness
This way fluid follows the geometry of
the blades perfectly,
And also blades causes no occlusion
in the flow geometry.
Effect of the number of blades
As number of blades increases impeller can guide the fluid
better, i.e. Fluid velocity vector and blade angle will be the
same. Thus, all the work from the shaft can be transferred to
the fluid => High pressure increase.
As number of blades increases, the area that the blades
occupy due to their thickness increases. At the same flow rate
a higher fluid velocity occurs for the impeller with more blades.
Viscous losses rise with the square of the fluid velocity. =>
Large losses.
A comprimise between the two results has to be found.
Open vs. Closed Impellers
Open vs. Closed Impellers
CLOSED IMPELLER
OPEN IMPELLER
Can compensate for shaft thermal growth, but if there is too
much axial growth the vanes may not line up exactly with the
discharge nozzle.
The impeller to volute or back plate clearance must be adjusted
when the pump is at operating temperature and all axial
thermal growth has occurred
Good for volatile and explosive fluids because the close
clearance wear rings are the parts that will contact if the shaft
displaces from its centerline
You would have to use soft, non

sparking materials for the
impeller and that is not very practical.
The impeller is initially very efficient, but looses its efficiency as
the wear ring clearance increases
Efficiency can be maintained through impeller clearance
adjustment.
No impeller adjustment is possible. Once the wear ring
clearances doubles they have to be replaced. This means the
pump had to be disassembled just to check the status of the
wear rings.
The impeller can be adjusted to compensate for wear and stay
close to its best efficiency. No pump disassembly is necessary.
The impeller can clog if you pump solids or "stringy material".
It's difficult to clean out these solids from between the shrouds
and vanes.
The open impeller is less likely to clog with solids, but if it does,
it is easy to clean.
The impeller is difficult to cast because the internal parts are
hidden and hard to inspect for flaws
The open impeller has all the parts visible, so it's easy to
inspect for wear or damage
The closed impeller is a more complicated and expensive
design not only because of the impeller, but the additional wear
rings are needed.
The pump is less costly to build with a simple open impeller
design.
The impeller is difficult to modify to improve its performance.
The vanes can easily be cut or filed to increase the capacity.
The specific speed choices (the shape of the impeller) are
limited
You have a greater range of specific speed choices.
Examples
Determine the work required for a
pump with no pre

whirl at the inlet?
For the best efficiency: Cw1=0
Example
at the inlet and outlet
Radial, Backward, Forward
Impeller Blade Shape
Value of C
w
=> Value of energy transfer
Degree of Reaction
For a radial turbomachinery:
Degree of Reaction
backward
radial
forward
for a given impeller tip speed, forward

curved vanes
have a highvalue of energy transfer.
Therefore, it is desirable to design for high values of
b
2
(over 90
0
),
but the velocity diagrams show that this also leads
to a very high value of C
2
.
High kinetic energy is seldom required, and its
reduction to static pressure by diffusion in a fixed
casing is difficult to perform in a reasonable sized
casing.
However, radial vanes (b
2
= 90
0
) have some
particular advantages for very highspeed
compressors where the highest possible pressure is
required.
Radial vanes are relatively easy to manufacture and
introduce no complex bending stresses.
Characteristic Curve
W=C
w2
U
2
Forward:
Cr2
Cw2
W
Radial:
Cr2
Cw2
W
Backward:
Cr2
Cw2
W
The effect on fan performance is shown in the different performace curves.
The forward curved type
run slower than the other types,
are the quietest in operation.
use higher horsepower at low resistance, and
the least amount of horsepower at the higher pressures and low flows.
The backward curved type of centrifugal fan performance characteristic curve
shows that
for increasing delivery volume, they startout at a lower horsepower,
rise to a peak on the horsepower curve near the point of highest efficiency on the
fan performance curve and
then drop off again.
Effect of blade thickness
t
Obstruction of the fow area
Slip
Stodola
Stanitz
c.a. 0.9;
evenif the
fluid is ideal!
Slip Factor
Losses
Slip
Viscous Losses
Losses in pipes
Mechanical Losses
Volumetric Losses (Leakage)
Efficiency
Efficiency
example
Volute
Vaneless Diffuser
A simple annular passage
Suited for a wide range of operations
Assuming
m
=0 ; angular momentum is constant:
Usually C
w
>> C
r
; thus
If
r
=const. => rC
r
=const.
Thus, the flow maintains a
constant inclination to radial lines,
the flow path traces a logarithmic
spiral.
for an incremental radius dr, the
fluid moves through angle d
q
Vaned Diffuser
Smaller size
KE transferred to P at a higher rate
More efficient
More friction
Any deviation from the design point =>
changes in velocity triangles =>
decrease in efficiency
Vaned Diffuser
Inlet vane of a radial turbine
Stator
Rotor
Cavitation
Local drops in pressure => cavitation
Suction Head:
hs = ps / γ + vs2 / 2 g
Liquids vapor head:
hv = pv / γ
NPSH:
NPSH = ps / γ + vs2 / 2 g

pv /
γ
Available NPSH:
NPSH
a
= patm / γ

he

hl

pv / γ
Required NPSH:
Thermodynamics of Cavitation
P
v
Bubble Collapse
In bulk flow
Near the wall
Cavitation Damages
Local pitting of the impeller and erosion of
the metal surface
Serious damage can occur from this
prolonged cavitation erosion.
Vibration of machine and noise is also
generated in the form of sharp cracking
sounds when cavitation takes place.
A drop in efficiency due to vapor formation,
which reduces the effective flow areas.
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