• Derivation of Euler’s pump and turbine equation

• Velocity triangles for a radial turbine

• Velocity triangles for an axial turbine

• Velocity triangles for a radial pump

Absolute specific stagnation energy

( ) fEcc −⋅∇=×∇×

Starting with Newton 2. law, the absolute acceleration for stationary flow can be derived as:

Where:E = Specific stagnation Energy [J/kg]c = Velocity [m/s]f = Friction [N/kg]

zgcpE ⋅++=2

2

ρ

The absolute specific stagnation energy is constant along a streamline in a frictionless system.

Ref. Grunnkurs i Hydrauliske Strømningsmaskiner

Relative specific stagnation energy

Rotalpy

( ) fIcw −⋅∇=×∇×Where

I = Rotalpy [J/kg]c = Velocity [m/s]f = Friction force [N/kg]

Relative acceleration in a rotating channel can be derived as:

zg2R

2wpI

222

⋅+⋅ω

−+ρ

=

The Rotalpy is constant along a streamline

Energy conversion

( ) ( ) hHgIEIE η⋅⋅=−−− 2211

222222

222

2222222

2222

uwcRwcIE

zgRwpzgcpIE

+−=⋅

+−=−

⇓

⋅+

⋅−+−⋅++=−

ω

ωρρ

ru ⋅= ω

wc

c

ru ⋅ω=

w

cu

cm

( )

2222

22222

2222

22222

muu

mumu

cccuuw

ccuwww

++⋅−=

⇓

+−

=+=

222

222mu ccc

+=

u

muu

mu

cuIE

ucccuuccuwcIE

⋅=−⇓

+−−⋅+−+=+−=−222222222

222222222

( ) ( ) 22112211 uuh cucuHgIEIE ⋅−⋅=⋅⋅=−−− η

Euler’s pump and turbine equation

Hgcucu 2u21u1

h ⋅⋅−⋅

=η

Velocity Triangles for a Radial Turbine

ω

c1 w1

u1

c2w2

u2

Guidevanes

Runnerblades

Velocity triangles for an axial turbine

c

c1w1

u1ω

c2 w2

u2

Velocity Triangles for a Radial Pump

ωc1w2

u2

c1w1

u1

SVARTISEN

u1=75 m/s

w1c1

P = 350 MWH = ? mQ* = 71,5 m3/SD0 = 4,86 mD1 = 4,31mD2 = 2,35 mB0 = 0,28 mn = 333 rpm

β1 = 63o

ηh = 96 %cm1 = 13,9 m/scu1 = 68 m/s