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```Design of stay vanes and spiral casing
Guri-2, VENEZUELA
Aguila, ARGENTINA
Sauchelle-Huebra, SPAIN
Sauchelle-Huebra, SPAIN
Three Gorges Turbine, GE Hydro
The spiral casing will
distribute the water
equally around the stay
vanes
In order to achieve a
uniform flow in to the
runner, the flow has to
be uniform in to the stay
vanes.
Flow in a curved channel
Streamline
The pressure normal to the streamline can
be derived as:
p
 p


dFn  p  ds  db   p 
 dn  ds  db 
 dn  ds  db
n
n


S tream line
Newton 2. Law gives:
m
 p
n
 dn  ds  db    dn  ds  db  a n

an 
c
2
R
S tream line
1 p
c


 n
R
2
1
The Bernoulli equation gives:
p


c
2
 const .
2
Derivation of the Bernoulli equation
gives:
1 p
c

 c
0
2
 n
n
Equation 1 and 2 combined gives:
c
n

c
R

R  dc  c  dR  0
1 p
c


 n
R
1
1 p
c

 c
0
 n
n
2
2

d (c  R )  0

R  c  const .
Free Vortex
Inlet angle to the stay vanes
 cm
a i  a tan 
 cu




cm
ai
cu
Plate turbine
Find the meridonial velocity from
R0
continuity:
Q  A cm
B

cm 
Q
A

Q
2R0 B
Find the tangential velocity:
R0
R
Q

B
y
 c u  dr
R
R0

R
Q

 By 
R0
const .
By
 dr
r

R
Q

B y  const . 

R0
dr
r

cu

Q
 R 

B y  R 0  ln 

R0 

 R 

B y  const .  ln 

R
 0
Example
C
L4
L1
q
L3
R0
R
L2
By
=
=
=
=
1,0
10
0,2
0,8
m3/s
m/s
m
m
Flow Rate
Velocity
Height
Q
C
By
R0
Find:
L1, L2, L3 and L4
Example
C
L4
L1
q
Flow Rate
Velocity
Height
L3
R0
R
L1 
L2
Cu
By

Q
C
By
R0
Q
C  By
=
=
=
=
1,0
10
0,2
0,8
m3/s
m/s
m
m
 0 ,5 m
Q
 R 

B y  R 0  ln 

R
 0
 12 ,9 m
s
Example
C
L4
L1
q
L3
R0
R
Flow Rate
Velocity
Height
Q
C
By
R0
=
=
=
=
1,0
10
0,2
0,8
m3/s
m/s
m
m
We assume Cu to be constant
along R0.
L2
At q=90o, Q is reduced by 25%
By
Example
C
L4
L1
q
L3
R0
R
Flow Rate
Velocity
Height
Q
L2

Q
Cu
By
R0
=
=
=
=
0,75
12,9
0,2
0,8
m3/s
m/s
m
m
 R 

B y  const .  ln 

 R0 

Q

 R 

B y  C u  R 0  ln 

R
 0

By
Q
R

R0  e
B y C u  R 0
Example
C
L4
L1
q
L3
R0
R
Flow Rate
Velocity
Height
Q
Cu
By
R0
=
=
=
=
0,75
12,9
0,2
0,8
m3/s
m/s
m
m
Q
R
L2
By

R0  e
L2 = 0,35 m
L3 = 0,22 m
L4 = 0,10 m
B y C u  R 0
Find the meridonial velocity from
continuity:
By
2
Q  A  cm
B
R0

cm 
Q
A

Q
2    R 0  B  k1
k1 is a factor that reduce the inlet area due to the stay vanes
Find the tangential velocity:
c u  R  c T  const .
By
2
cu 
cT
B
R0
R
Rt  r
Q 
B
y
 c u  dR
R0
B y  2  r  sin 
R  R T  r  cos 

2


sin



d
2

Q  2  r  c T    R t  r  cos  

dR  r  sin   d 
cT 
Q
2


sin 
2
d 
2  r  
  R T  r  cos  


By
2
B
cu 
Q
2


sin 
2
d 
R  2  r  
  R T  r  cos  


R0
Spiral casing design procedure
1. We know the flow rate, Q.
2. Choose a velocity at the upstream section of the spiral
casing, C
3. Calculate the cross section at the inlet of the spiral casing:
r 
Q
C 
4. Calculate the velocity Cu at the radius Ro by using the
equation:
cu 
Q
2


sin

2
d 
R  2  r  
  R T  r  cos  


Spiral casing design procedure
5. Move 20o downstream the spiral casing and calculate the
flow rate:
o
Q new 
20
360
o
 Q total
6. Calculate the new spiral casing radius, r by iteration with
the equation:
2


sin



d
2

Q  2  r  c T    R t  r  cos  

Outlet angle from the stay vanes
cm 
Q
A

Q
2R Bk
cm
c u  R  const .
 cm
a  a tan 
 cu
a cu




Weight of the spiral casing
Stay Vanes
Number of stay vanes
30
N u m b e r o f S ta y V a n e s
28
26
24
22
20
18
16
0 ,0
0 ,2
0 ,4
0 ,6
0 ,8
Speed N um ber
1 ,0
1 ,2
1 ,4
1 ,6
Design of the stay vanes
• The stay vanes have the main purpose of
keeping the spiral casing together
• Dimensions have to be given due to the
stresses in the stay vane
• The vanes are designed so that the flow is
not disturbed by them
Flow induced pressure oscillation
f  1 .9  B 
c
t  0 . 56
Where
f = frequency [Hz]
B = relative frequency to the Von Karman oscillation
c = velocity of the water [m/s]
t = thickness of the stay vane [m]
Where
A = relative amplitude to the Von Karman oscillation
B = relative frequency to the Von Karman oscillation
```
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