Civil Engineering Reference
In-Depth Information
-Fluid elasticity property,
ρρ
1
d
=
0
ρ
d
-Pipe elasticity property
1
dA
∂
v
1
dp
−
2
c
+
=
0
(Continuity equation),
(4.7)
A
dt
∂
s
ρ
dt
dV
∂
1
p
dz
f
+
+
g
+
VV
=
0
(Euler equation),
(4.8)
dt
ρ ∂
s
ds
2
D
The Method of Characteristic (MOC), to solve governing equations for unsteady
pipe flow. Using the MOC, the two partial differential equations can be transformed to
the following two pairs of equations:
fV V
g
dH
dV
ds
+ +
=⇒ =
0
c
+
(4.9)
c
dt
dt
2
d
dt
fV V
g
dH
dV
ds
− + +
(4.10)
=⇒ =
0
c
−
c
dt
dt
2
D
dt
Method of characteristic solution for partial differential equation:
The method of characteristics is a finite difference technique where pressures are
computed along the pipe for each time step. Calculation automatically sub-divides the
pipe into sections (i.e. reaches or intervals) and selects a time interval for computa-
tions.
=
dp
∂
p
∂
p
ds
=
(4.11)
dt
∂
t
∂
s
dt
dv
∂
v
∂
v
ds
=
+
(4.12)
dt
∂
t
∂
s
dt
P and V changes due to time are high and due to coordination are low then we can
neglect coordination differentiation, we have:
∂
v
1
∂
p
dz
f
+
(Euler equation),
(4.13)
+
g
+
VV
=
0
∂
t
ρ ∂
s
ds
2
D
+
∂
v
1
∂
p
(Continuity equation),
(4.14)
C
2
=
0
∂
t
ρ ∂
t
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