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Fundamental similarity considerations
• Reduced parameters• Speed number and Specific
speed• Classification of turbines• Similarity Considerations• Performance characteristics
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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
Hgzzc
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 ⋅−⋅⋅=η
Hg2QQ
⋅⋅=
Hg2 ⋅⋅ω
=ω
Hhh =
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Speed number
Q*** ⋅ω=Ω
Geometric similar, but different sized turbines have the same speed number
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Fluid machinery that is geometric similar to each other, will at same relative flowrate have the same velocity triangle.For the reduced peripheral velocity:
For the reduced absolute meridonial velocity:
.constdu =⋅ω~
.2 constdQ
cm =~
We multiply these expressions with each other:
.2 constQdQ
d =⋅=⋅⋅ ωω
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Specific speed that is used to classify turbines
75,0q HQnn ⋅=
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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 HQnn ⋅=
43s PQn333n ⋅⋅=
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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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*** =⋅=⋅ω=Ω
Speed number:sm10354382,92Hg2 =⋅⋅=⋅⋅
srad9,34
602333
602n
=Π⋅⋅
=Π⋅⋅
=ω
1m33,0s
m103s
rad9,34
Hg2* −==
⋅⋅ω
=ω
2
3
m69,0s
m103s
m5,71
Hg2QQ* ==
⋅⋅=
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Specific speed:
43q HQnn ⋅=
03,25543
5,71333n 43q =⋅=
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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 turbomachinery.
.constu
Hg2.constc
Hg2
.constcpAF
.constuc
22
2
=⋅⋅
=⋅⋅
⋅⋅ρ==
= 1
2
3
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Velocity triangles
ru ⋅ω=
wc
.constuc
= 1
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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∆⋅⋅ρ=⇒⋅ρ=
∆
In connection with turbomachinery, 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.
gcconsth
gp
cconstpAF
2
2
⋅==⋅ρ
⇓
⋅ρ⋅== 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)
.constc
Hg22 =⋅⋅
3
.constu
Hg22 =⋅⋅
.constg2
cH
2 =
⋅
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For pumps and turbines, the capacity Q is a significant operating characteristic.
.constDn
QDn
DQ
3
2=
⋅=
⋅.const
uc
= ⇒
c is proportional to Q/D2 and u is proportional to n·D.
.constg2
.constQ
DH
DQ
H.constc
Hg22
4
2
2
2 =⋅
=⋅
=
⇒=⋅⋅
( ).const
g2.const
DnH
DnH.const
uHg2
2222 =⋅
=⋅
=⋅
⇒=⋅⋅
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Affinity Laws
2
1
2
1
322
311
2
1
3
nn
DnDn
.constDn
Q
=
⇓
⋅⋅
=
⇓
=⋅
This relation assumes that there are no change of the diameter D.
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Affinity Laws
22
21
2
1
22
22
21
21
2
1
22
nn
HH
DnDn
HH
.constDn
H
=
⇓
⋅⋅
=
⇓
=⋅
This relation assumes that there are no change of the diameter D.
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Affinity Laws
( ) ( )( ) ( )
32
31
2
1
52
32
51
31
22
22
322
21
21
311
22
11
22
11
2
1
223
nn
PP
DnDn
DnDnDnDn
QHQH
QHgQHg
PP
QHgP.constDn
H.constDn
Q
=
⇓
⋅⋅
=⋅⋅⋅⋅⋅⋅
=⋅⋅
=⋅⋅⋅ρ⋅⋅⋅ρ
=
⇓
⋅⋅⋅ρ==⋅
=⋅
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 LawsExample
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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Performance characteristics
200.00 400.00 600.00 800.00Turtall [rpm]
0.50
0.60
0.70
0.80
0.90
1.00
Virk
ning
sgra
d
α = 5
α = 10
α = 15
α = 20
α = 25
Speed [rpm]
Effic
ienc
y [-
]
NB:H=constant
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Kaplan