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Fundamental similarity considerations
• Reduced parameters• Speed number and Specific
speed• Classification of turbines• Similarity Considerations• Performance characteristics
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
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
Reduced values used for turbines
Hg2cc
⋅⋅=
Hg2uu
⋅⋅=
Hg2ww
⋅⋅=
( )22u11uh ucuc2 ⋅−⋅⋅=η
Hg2QQ
⋅⋅=
Hg2 ⋅⋅ω
=ω
Hhh =
Speed number
Q*** ⋅ω=Ω
Geometric similar, but different sized turbines have the same speed number
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
Specific speed that is used to classify turbines
75,0q HQnn ⋅=
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
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
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* ==
⋅⋅=
Specific speed:
43q HQnn ⋅=
03,25543
5,71333n 43q =⋅=
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.
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
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
Velocity triangles
ru ⋅ω=
wc
.constuc
= 1
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:
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
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 =
⋅
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 =⋅
=⋅
=⋅
⇒=⋅⋅
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.
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.
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.
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
=
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
=⋅
=⋅
=
⇓
=
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
Kaplan