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

QQ

DnDn

QQ

.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

QQ

=

Affinity LawsExample

Change of speed

n1 = 600 rpm Q1 = 1,0 m3/sn2 = 650 rpm Q2 = ?

smQ

nnQ

nn

QQ

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