Civil Engineering Reference
In-Depth Information
By linear combination of Euler and continuity equations in characteristic solution
Method we have:,
∂
v
1
∂
p
dz
f
∂
V
1
∂
p
2
+
−
(4.15)
λ
+
+
g
+
VV
+
C
+
= λ=
0,
c
&
λ=
c
∂
s
ρ ∂
s
ds
2
D
∂
s
ρ ∂
t
dV
1
dp
dz
f
+
+
g
+
VV
=
0
(4.16)
dt
ρ
c
dt
ds
2
D
dV
1
dp
dz
f
−
+
g
VV
=
0
(4.17)
dt
ρ
c
dt
ds
2
D
Method of characteristics drawing in (s-t) coordination:
dV
g
dH
−
=
0
(4.18)
dt
c
dt
=
c
dH
gx
(Joukowski Formula),
(4.19)
g
By Finite Difference method:
(
)
(
)
fV
V
VV
−
H H
−
g
Le
P
Le
P
Le
Le
c
+
:
+
+
=
0
,
(4.20)
(
)
(
T
−
0)
c
t
−
0
2
D
p
P
fV
V
VV
−
g
H H
−
Ri
Ri
P
Ri
P
Ri
c
−
:
+
+
=
0
,
(4.21)
t
−
0
c
t
−
0
2
D
P
p
fV
V
g
+ − +
(
)
(
)
Le
Le
,
(4.22)
c
:
V V
H
− +∆
H
ff
=
0
P
Le
P
Le
c
2
D
VV
g
− − +
(
)
(
)
Ri
Ri
c
:
V V
H
− +∆
H
ff
=
0
,
(4.23)
P
Ri
P
Ri
c
2
D
The MOC approach transforms the water hammer partial differential equations
into the ordinary differential equations along the characteristic lines. Theses lines de-
fined as the continuity equation and the momentum equation are needed to determine
V and P in a one-dimensional flow system. Solving these two equations produces
a theoretical result that usually corresponds quite closely to actual system measure-
ments. This is happened when the data and assumptions used to build the numerical
model are valid. Transient analysis results that are not comparable with actual system
measurements are generally caused by inappropriate system data (especially boundary
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