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3 Similarity Considerations Valid when: –Geometric similarity –All velocity components are equally scaled –Same velocity directions –Velocity triangles are kept the same –Similar force distributions –Incompressible flow
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1
Fundamental similarity considerations
• Similarity Considerations• Reduced parameters• Dimensionless terms• Classification of turbines • Performance characteristics
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Similarity Considerations
Similarity considerations on hydrodynamic machines are an attempt to describe the performance of a given machine by comparison with the experimentally known performance of another machine under modified operating conditions, such as a change of speed.
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Similarity Considerations
• Valid when:– Geometric similarity– All velocity components are
equally scaled – Same velocity directions– Velocity triangles are kept the
same– Similar force distributions– Incompressible flow
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These three dynamic relations together are the basis of all fundamental similarity relations for the flow in turbo machinery.
1
2
3 .Constu
Hg2.Const
cHg2
.ConstcpAF
.Constuc
22
2
5
Velocity triangles
ru
wc
.c Constu
1
6
Under the assumption that the only forces acting on the fluid are the inertia forces, it is possible to establish a definite relation between the forces and the velocity under similar flow conditions
tcmF
dtdcmF
cQFQt
m
In connection with turbo machinery, Newton’s 2. law is used in the form of the impulse or momentum law:
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For similar flow conditions the velocity change c is proportional to the velocity c of the flow through a cross section A.
It follows that all mass or inertia forces in a fluid are proportional to the square of the fluid velocities.
2
2
F p Const cA
p ch Constg g
2
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By applying the total head H under which the machine is operating, it is possible to obtain the following relations between the head and either a characteristic fluid velocity c in the machine, or the peripheral velocity of the runner. (Because of the
kinematic relation in equation 1)
2 .g H Constc
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2 .g H Constu
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For pumps and turbines, the capacity Q is a significant operating characteristic.
2
3 .Q
QD Constn D n D
.c Constu
c is proportional to Q/D2 and u is proportional to n·D.
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22 2
2
.. .g H H H D ConstConst Constc Q gQ
D
22 2 2
.. .g H H H ConstConst Constu n D gn D
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Affinity Laws
3
31 1 1
32 2 2
1 1
2 2
.Q Constn D
Q n DQ n D
Q nQ n
This relation assumes that there are no change of the diameter D.
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Affinity Laws
2 2
2 21 1 1
2 22 2 2
21 1
22 2
.H Constn D
H n DH n D
H nH n
This relation assumes that there are no change of the diameter D.
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Affinity Laws
3 2 2
3 2 2 3 51 1 1 11 1 1 1 1 1 1
3 53 2 22 2 2 2 2 2 22 2 2 2
31 1
32 2
. .Q HConst Const P g H Qn D n D
n D n DP g H Q H Q n DP g H Q H Q n Dn D n D
P nP n
This relation assumes that there are no change of the diameter D.
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Affinity Laws
32
31
2
1
nn
PP
22
21
2
1
nn
HH
This relations assumes that there are no change of the diameter D.
2
1
2
1
nn
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Affinity Laws Example
Change of speed
n1 = 600 rpm Q1 = 1,0 m3/sn2 = 650 rpm Q2 = ?
smQ
nnQ
nn
3
11
22
2
1
2
1
08,10,1600650
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Reduced parameters used for turbines
The reduced parameters are values relative to the highest velocity that can be obtained if all energy is converted to kinetic energy
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Hgc
Hgzzgc
zgchz
gch
2
2
22
2
21
22
2
22
21
21
1
Bernoulli from 1 to 2 without friction gives:
Reference line
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Reduced values used for turbines
Hg2cc
Hg2uu
Hg2ww
22u11uh ucuc2
Hg2QQ
Hg2
Hhh
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Dimensionless terms
• Speed– Speed number – Specific speed NQE
– Speed factor nED, n11
– Specific speed nq, ns
• Flow– Flow factor QED, Q11
• Torque– Torque factor TED, T11
• Power– Power factor PED, P11
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Fluid machinery that is geometric similar to each other, will at same relative flow rate have the same velocity triangle.For the reduced peripheral velocity:
For the reduced absolute meridonial velocity:
.u D Const ~
2 .m
Qc Const
D~
We multiply these expressions with each other:
2 .Q
D Q ConstD
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21
Speed number
Q***
Geometric similar, but different sized turbines have the same speed number
D
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Speed number
2
2
12
21 2
1D Const
Q ConstConst
Q Const Const
cm
cmD
122
2
4
m
Q Qc Const
DD
u D D Const
1
2
ru
wccm
cuFrom equation 1:
Inserted in equation 2:
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Speed Factorunit speed, n11
11
260 2nu D Const D
g H
n D Const nH
ru
wccm
cu
If we have a turbine with the following characteristics:
• Head H = 1 m• Diameter D = 1 m
we have what we call a unit turbine.
HDnn11
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Speed FactornED
260 2
ED
nu D Const Dg H
n D Const ng H
ru
wccm
cu
If we have a turbine with the following characteristics:
• Energy E = 1 J/kg• Diameter D = 1 m
EDn DnE
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Energy
Reference line
z1
ztw
h1c1
abs
221
1 1
2 21
1 1
twabs atm tw n
twn abs atm tw
ccg h g z g h g z g Hg g
c cE g H g h g h g z g zg
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Specific speed that is used to classify turbines
75,0q HQ
nn
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Specific speed that is used to classify pumps
nq is the specific speed for a unit machine that is geometric similar to a machine with the head Hq = 1 m and flow rate Q = 1 m3/s
43q HQ
nn
43s PQ
n333n
ns is the specific speed for a unit machine that is geometric similar to a machine with the head Hq = 1 m and uses the power P = 1 hp
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29
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Flow Factorunit flow, Q11
11 2
QQD H
ru
wccm
cu
If we have a turbine with the following characteristics:
• Head H = 1 m• Diameter D = 1 m
we have what we call a unit turbine.
22
112
4
m
Q Qc Const
DD
Q Const QD H
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Flow FactorQED
2EDQQ
D E
ru
wccm
cu
If we have a turbine with the following characteristics:
• Energy E = 1 J/kg• Diameter D = 1 m
22
2
4
m
ED
Q Qc Const
DD
Q Const QD g H
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Exercise• Find the speed number and
specific speed for the Francis turbine at Svartisen Powerplant
• Given data:P = 350 MWH = 543 mQ* = 71,5 m3/sD0 = 4,86 mD1 = 4,31mD2 = 2,35 mB0 = 0,28 mn = 333 rpm
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27,069,033,0Q***
Speed number:sm10354382,92Hg2
srad9,34
602333
602n
1m33,0s
m103s
rad9,34
Hg2*
2
3
m69,0s
m103s
m5,71
Hg2QQ*
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Specific speed:
43q HQ
nn
03,25543
5,71333n 43q
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Performance characteristics
200.00 400.00 600.00 800.00Turta ll [rpm ]
0.50
0.60
0.70
0.80
0.90
1.00
Virk
ning
sgra
d
Speed [rpm]
Effic
ienc
y [-
]
NB:H=constant
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Kaplan
37
EDn Dng H
0
1.3
1.0
0.7
0.3
0.60.8
0.9
0.7
2ED
gHD
