# ME436 Applied Fluid Mechanics

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24 Οκτ 2013 (πριν από 4 χρόνια και 8 μήνες)

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ME436 Applied Fluid
Mechanics

Spring 201
2

Lecture 3:
Euler
Turbomachinery Equation

Turbomachinery type

Axial

Mixed

Velocity Components

r

a

w

inlet

outlet

At any point in the pump, velocity is in

axial (a),

tangential (w) directions.

Any change in

C
a

=> Axial force; thrust bearing
to the stationary rotor casing

C
r

C
w

=> angular motion; rotational
effect

Axial

Compact

Same inlet and outlet area

High rpm

Axial Pumps

Axial Pumps

Axial Pumps

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:

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

Technical Drawings

Open
impeller

Closed
impeller

Open impeller

Length

Thickness or width

Inlet angle

Outlet angle

Trailing edge

Tip

Pressure side

Suction side

Ideal Impeller

thickness

This way fluid follows the geometry of

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.

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

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

Value of C
w

=> Value of energy transfer

Degree of Reaction

Degree of Reaction

backward

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.

2

= 90
0
) have some
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

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.

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
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

hs = ps / γ + vs2 / 2 g

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.