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.
StreamlineStreamline
The pressure normal to the streamline can be derived as:
dbdsdnnpdbdsdn
nppdbdspdFn ⋅⋅⋅
∂∂−
=⋅
⋅
∂∂
+−⋅⋅=
Newton 2. Law gives:
StreamlineStreamline
Rc
np
adbdsdndbdsdnnp
n
21=
∂∂⋅
⇓
⋅⋅⋅⋅=⋅⋅⋅∂∂−
ρ
ρ Rca
2
n =
1
m
The Bernoulli equation gives:
.const2cp 2
=+ρ
Derivation of the Bernoulli equation gives:
0ncc
np1
=∂∂⋅+
∂∂⋅
ρ2
Equation 1 and 2 combined gives:
.constcR
0)Rc(d
0dRcdcR
Rc
nc
=⋅⇓
=⋅⇓
=⋅+⋅⇓
−=∂∂
Free Vortex
Rc
np 21=
∂∂⋅
ρ
0ncc
np1
=∂∂⋅+
∂∂⋅
ρ 2
1
Find the tangential velocity:
⋅⋅
=
⇓
⋅⋅=⋅⋅=
⇓
⋅⋅=
⇓
⋅⋅=
∫
∫
∫
00y
u
0y
R
Ry
R
Ry
u
R
Ry
RRlnRB
Qc
RRln.constB
rdr.constBQ
drr
.constBQ
drcBQ
0
0
0
By
R0 R
Example
By
Flow Rate Q = 1,0 m3/sVelocity C = 10 m/sHeight By = 0,2 mRadius R0 = 0,8 m
Find: L1, L2, L3 and L4
L1
θ
C
L3
L2
L4
R0
R
Example
By
Flow Rate Q = 1,0 m3/sVelocity C = 10 m/sHeight By = 0,2 mRadius R0 = 0,8 mL1
θ
C
L3
L2
L4
R0
R
sm
RRRB
QC
y
u 9,12ln
00
=
⋅⋅
=
mBCQL
y
5,01 =⋅
=
Example
By
Flow Rate Q = 1,0 m3/sVelocity C = 10 m/sHeight By = 0,2 mRadius R0 = 0,8 m
We assume Cu to be constant along R0.
At θ=90o, Q is reduced by 25%
L1
θ
C
L3
L2
L4
R0
R
Example
By
Flow Rate Q = 0,75 m3/sVelocity Cu = 12,9 m/sHeight By = 0,2 mRadius R0 = 0,8 mL1
θ
C
L3
L2
L4
R0
R
00
00
0
ln
ln.
RCBQ
uy
y
uyeRR
RRRCBQ
RRconstBQ
⋅⋅⋅=
⇓
⋅⋅⋅=
⇓
⋅⋅=
Example
By
Flow Rate Q = 0,75 m3/sVelocity Cu = 12,9 m/sHeight By = 0,2 mRadius R0 = 0,8 mL1
θ
C
L3
L2
L4
R0
R0
0RCB
Q
uyeRR ⋅⋅⋅=
L2 = 0,35 mL3 = 0,22 mL4 = 0,10 m
Find the meridonial velocity from continuity:
10m
m
kBR2Q
AQc
cAQ
⋅⋅⋅π⋅==
⇓
⋅=R0B
k1 is a factor that reduce the inlet area due to the stay vanes
2yB
Find the tangential velocity:
Rcc
.constcRc
Tu
Tu
=
==⋅
∫
∫
⋅−⋅⋅⋅=
⇓
⋅⋅=+
π
ϕ
ϕϕ
ϕ drRcrQ
dRcBQ
tT
u
rR
Ry
t
cossin
2
2
2
0
R0B R0B
ϕϕϕ
ϕ
drdRrRR
rB
T
y
⋅⋅=⋅−=
⋅⋅=
sincos
sin2
2yB
Spiral casing design procedure1. We know the flow rate, Q. 2. Choose a velocity at the upstream section of the spiral
casing, C3. Calculate the cross section at the inlet of the spiral casing:
4. Calculate the velocity Cu at the radius Ro by using the equation:
π⋅=CQr
ϕ
ϕ⋅−
ϕ⋅⋅=
∫π
ϕ
dcosrR
sinr2R
Qc
T
22
u
Spiral casing design procedure5. Move 20o downstream the spiral casing and calculate the
flow rate:
6. Calculate the new spiral casing radius, r by iteration with the equation:
totalo
o
new QQ ⋅=36020
∫
⋅−⋅⋅⋅=
π
ϕ
ϕϕ
ϕ drRcrQ tT cos
sin2
2
2
Number of stay vanes
16
18
20
22
24
26
28
30
0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6
Speed Number
Num
ber o
f Sta
y Va
nes
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
56.09.1
+⋅⋅=tcBf
Where f = frequency [Hz]B = relative frequency to the Von Karman oscillationc = velocity of the water [m/s]t = thickness of the stay vane [m]