Euler’s pump and turbine equation - Wieser · • Derivation of Euler’s pump and turbine...
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Transcript of Euler’s pump and turbine equation - Wieser · • Derivation of Euler’s pump and turbine...
• 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