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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
2
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.
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
4
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:
7
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
8
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
3
2 .g H Constu
9
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.
4
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.
11
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.
12
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.
13
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
14
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
15
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
16
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
17
Reduced values used for turbines
Hg2cc
Hg2uu
Hg2ww
22u11uh ucuc2
Hg2QQ
Hg2
Hhh
18
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
19
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
20
21
Speed number
Q***
Geometric similar, but different sized turbines have the same speed number
D
22
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
24
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
25
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
26
Specific speed that is used to classify turbines
75,0q HQ
nn
27
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
28
29
30
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
31
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
32
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
33
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*
34
Specific speed:
43q HQ
nn
03,25543
5,71333n 43q
35
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
36
Kaplan
37
EDn Dng H
0
1.3
1.0
0.7
0.3
0.60.8
0.9
0.7
2ED
gHD