1. 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

2. Absolute specific stagnation energy
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]
The absolute specific stagnation energy is
constant along a streamline in a frictionless
system.
Ref. Grunnkurs i Hydrauliske Strømningsmaskiner

3. Relative specific stagnation energy Rotalpy
Relative acceleration in a rotating channel can be derived as:
Where
I = Rotalpy [J/kg]
c = Velocity [m/s]
f = Friction force [N/kg]
The Rotalpy is constant along a streamline

4. Energy conversion w c

5. w cu cm c

6. Euler’s pump and turbine equation

7. Velocity Triangles for a Radial Turbinew1 u1 c2 w2 u2

8. Velocity triangles for an axial turbine
Guidevanes
Runnerblades

9. c c1 w1 u1 w c2 w2 u2

10. Velocity Triangles for a Radial Pumpw1 u1 c1 w2 u2 w

11. SVARTISEN c1 w1 hh = 96 % cm1 = 13,9 m/s cu1 = 68 m/s b1 = 63o
P = 350 MW
H = ? m
Q* = 71,5 m3/S
D0 = 4,86 m
D1 = 4,31m
D2 = 2,35 m
B0 = 0,28 m
n = 333 rpm c1 w1
b1 = 63o
hh = 96 %
cm1 = 13,9 m/s
cu1 = 68 m/s