user's manual for corhyd: an internal diffuser hydraulics model - IfH

elevation difference between diffuser centerline and jet centerline, plus dynamic pressure
difference between the diffuser and one single jet plus the losses occurring in all pipe
segments between these points.
p
d,i
=
p
( ) ( ) i
2
ρe
2 ρe
⎛ ⎞
α
2 iqi
− ⎜ q
2
k ⎟
C A
2Ad,i
⎝ k=
1 ⎠
a,i
+ ρeg(z
jet,i
− zd,i
) +
∑
ρeq
Losses i =
2
2
i
⎡
⎢
⎢
⎣
n
p,i
∑
j=
1
2
⎛
⎜
α
⎝ A
i
p,i, j
c,i
2
p,i
⎞ ⎛
⎟ ⎜ζ
⎠ ⎝
p,i, j
λ
p,i, jL
+
D
p,i, j
p,i, j
⎞
⎟ +
⎠
n
r ,i
∑
j=
1
⎛
⎜
⎝
1
A
r,i, j
+ Losses i
2
⎞ ⎛
⎟ ⎜ζ
⎠ ⎝
r,i, j
λ
r,i, jL
+
D
C c,i denotes the jet contraction coefficient either given by the user or calculated iteratively if
Duckbill Valves are applied C c,i,DBV = α i q i / (V DBV,i A p,i ) with V DBV,i = duckbill jet velocity
dependent on discharge. If multiple ports are applied a single jet discharge is q jet,i = α i q i with
α i = 1/(number of ports at a riser at position i).
Solving eq. (18) = (19) for an individual discharge q i gives
i−1
2
nd ,i−1
2
⎛ ⎞ 1 1
Ld,i−
1,j
( p
−
) (
−
) ∑ ⎢ ∑
⎜
⎟
d,i 1
− pa,i
+ 2g zd,i
1
− z
jet,i
+ ⎜ qk
⎟ +
ζ
−
+ λ
2
2
d,i 1,j d,i−1,j
⎥
ρ
e
⎝ k=
1 ⎠ ⎢⎣
Ad,i−
1 j= 1 A
− ⎝
D
d,i 1,j
d,i−
j ⎠⎥
q =
1, ⎦ (20)
i
2
2
2 np,i
n
α ⎛ ⎞ ⎛ λ ⎞
r ,i
⎛ ⎞ ⎛ λ ⎞
i
( )
∑⎜
αi
p,i,jLp,i,j
r,i,j r,i,j
+ ⎟ ⎜ζ
+ ⎟ + ∑⎜
1
L
⎟ ⎜ζ
+ ⎟
2
p,i,j
r,i,j
C
c,iAp,i
j=
1 A
p,i,j
Dp,i,
j j=
1 A
r,i,j
Dr,i,
j
⎝
⎠
⎝
⎡
For simple diffusers equation (20) reduces to equation (21) if no risers and no port
configurations are applied and the diffuser is just represented by simple holes in the pipe wall.
Equation (21) is the one presented in Fischer et al., 1979 which has been used for simple
diffuser calculations.
i−1
2
⎡
n −1
2
⎛ ⎞ 1
d,i
1 ⎛
L ⎞⎤
d,i−
j
q = ( − ) + ⎜∑
⎟ ⎢ + ∑
⎜ζ
+ λ
1, ⎟
i
C
c,iA
p,i
p
d,i−1
p
a,i
q
⎥ (21)
k
2
2
ρ
⎝ ⎠ ⎢
d,i−1,
j d,i−1,
j
e
k=
1
=
⎣A
d,i−1
j 1 A
d,i−1,
j ⎝
D
d,i−1,
j ⎠⎥⎦
Fischer et al. (1979) defined a loss coefficient C c,i for sharp-edged entrances:
⎠
⎝
⎠
⎝
⎛
r,i, j
⎠
r,i, j
⎞⎤
⎟⎥
⎠⎥
⎦
⎞⎤
(19)
C
c,i
⎛
0.58 ⎜⎛
= 0.63 − ⎜⎜
2g
⎝
⎝
i−1
∑
k=
1
q
k
2
⎞
⎟
⎠
⎡ 1
⎢
⎢⎣
A
d,i
2
−1
⎤⎛
⎜ 2
⎥
⎥
⎜
⎦
ρ
⎝
e
( p − p )
d,i−1
a,i
⎛
+ ⎜
⎝
i−1
∑
k=
1
q
k
2
⎞
⎟
⎠
⎡ 1
⎢
⎢⎣
A
d,i
2
−1
⎤⎞
⎥
⎟
⎥⎟
⎦⎠
−1
⎞
⎟
⎟
⎠
and for bell-mouthed ports:
C
c,i
⎛
⎜
= 0.975⎜1
−
⎝
1
2g
⎛
⎜
⎝
i−1
∑
k=
1
q
k
2
⎞
⎟
⎠
⎡ 1
⎢
⎢⎣
A
2
d,i−1
⎤⎛
⎜ 2
⎥
⎥
⎜
⎦
ρ
⎝
e
( p − p )
d,i−1
a,i
⎛
+ ⎜
⎝
i−1
∑
k=
1
q
k
2
⎞
⎟
⎠
⎡ 1
⎢
⎢⎣
A
2
d,i−1
⎤⎞
⎥⎟
⎥⎟
⎦⎠
−1
⎞
⎟
⎟
⎠
3 / 8
CorHyd furthermore allows to apply Duckbill valves also on simple diffuser systems and
therefore uses the previously defined additional local loss formulations, which are
additionally integrated in the calculations of the coefficient C c .
Institut für Hydromechanik, Universität Karlsruhe 30