Radial Equilribrium
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Transcript of Radial Equilribrium
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Radial Equilibrium Theory
P M V Subbarao
Professor
Mechanical Engineering Department
Long Blades are more complex in shape .
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Geometrical Details along Radial Direction
True flow through a turbo-machinery is three-dimensional.
The effect of the strong centrifugal forces are exerted by/on
blades in radial direction.
The centrifugal field distorts the flow velocity profiles
considerably.
Fluid particles tend to move outwards rather than passing
along cylindrical stream surfaces as classically assumed.
Particularly in low hub: tip ratio designs.
An approach known as the radial equilibrium method, widely
used for three-dimensional design calculations.
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Radial Variation Blade Geometry
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Radial Equilibrium Theory
Assumes that flow is in radial equilibrium before and aftera blade row.
Radial adjustment takes place through the row.
More important for Axial Flow Machines.
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Radial Equilibrium Analysis
The centrifugal force = (rrdrdq)w2r
Vq= r
The centrifugal force is
The pressure force on the element
qr q drdVF lcentrifuga2
qrdpdFpressure
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If the two forces are the only ones acting (viscous and
other effects neglected), the particle will move at
constant radius if:
lcentrifugapressure FF
rV
drdp
2
qr
rdrVdp 2q
r
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Radial Equilibrium Analysis of Compressible
Machines
An equivalent equation for compressible flow can be
developed by using the following thermodynamic relation:
0r
dpdhvdpdhTds
r
dpdh
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2222
2222
0
qVVVhVhh r
f
r
drV
dp 2q
r
0222
222
0
qVVVhddh rf
No Interactions: Conservation of Stagnation Enthalpy
r
dpdh
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0222
2222
0
qqVVV
dr
drVdh r
f
0
222
2222
0
qqVVV
dr
d
r
V
dr
dh rf
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02
0 dr
dVV
dr
dVV
dr
dVVr
V
dr
dh rr
f
fq
qq
Radial component of velocity should be constant (zero)
along radial direction for radial equilibrium of flow
02
0 dr
dVV
dr
dVVr
V
dr
dh ff
qq
q
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021
2
00
drrVd
rV
drdV
drdh
drTcd fp qq
For inert Gas
02
12
0
dr
rVd
r
V
dr
dV
dr
dT
c
f
p
qq
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0
0
0
0
1 T
dT
p
dp
For an isentropic process:
0
2
112
0
0
0
dr
rVd
r
V
dr
dV
dr
dp
p
T
c
x
p
qq
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01 0 dr
rVd
r
V
dr
dVV
dr
dp xx
qq
r
Radial Equilibrium Equation for
Incompressible Fluid Machine
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gzUVhU
hIRothalpy bladeblade
rel q0
2
,02
:
Constant in a turbo-machine along meridonial Plane
02
12
00
dr
rVd
r
V
dr
dV
dr
dh
dr
Tcd fp qq
Stagnation enthalpy is Constant in a turbo-machine
along radial direction at intake and discharge.
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Lessons from Nature
In the case of a vortex, the flow field is purely tangential.
ziiW ln2
The complex potential function:
THE VORTEX
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Free Vortex Whirl:
Forced Vortex Whirl :
General Rules for Selection of Whirl Component
r
CV q
constantfV
rCV q
2
21C rCVf
0
q
q fV
r
rV
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Twisted Blades for Large Turbines
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Radial Variation of Flow Velocity
100
120
140
160
180
200
220
240
260
280
300
0.75 0.85 0.95 1.05 1.15 1.25
Flowvelocities(m
/s)
Radius ( in m)
Intake
Discharge
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Radial Variation of Whirl Velocity
-100
0
100
200
300
400
500
0.75 0.85 0.95 1.05 1.15 1.25
whirlvelocities(m
/s)
Radius ( in m)
Intake
Discharge
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Radial Variation of Mass flow rate
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.750 0.850 0.950 1.050 1.150 1.250
massfowrate
Radius ( in m)
Intake
Discharge
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Kaplan Turbine
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DESIGN OF THE BLADE
Two different views of a blade
90% or better inefficiency
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