234
P. Liu
formula of hydraulic elements derived by Jukowsky is
p = −ρCV, ,h = −
C
g
V
For example, the propagation velocity of water intake shock wave C =
1435 m/s, and the velocity V 0 = 5 m/s when the pipeline is in constant flow.
Substituting the above formula, the water hammer pressure value generated
by the sudden closing of the valve is P = 7.175 × 10 6 pa.
The valve in the actual pipeline is not closed in a moment, but gradually closed in a certain period of time. The resulting water hammer pressure
is much smaller than that generated by a sudden closing of the valve. The
maximum water hammer pressure in front of the valve is the superposition of
pressure waves generated by gradually closing the valve, which is called indirect water hammer. Relatively speaking, the water hammer caused by one-off
shutdown is called direct water hammer. In addition to the water hammer
generated by closing the valve, the sudden opening of the valve will also
cause the water hammer wave in the pipeline, but the water hammer wave
is negative at this time, causing the pressure to decrease.
If the elasticity of the pipe wall is not considered, the propagation velocity
of the pressure wave caused by the compressibility of the water is the propagation velocity of the sound in the water C = 1435 m/s, but if the propagation
velocity changes after considering the elasticity of the pipe, it can be derived
using the continuity equation and momentum theorem
C =
K w
ρ
1 +
DK w
δ E
=
1435
1 +
DK w
δ E
where, K w is the bulk modulus of elasticity of water body (= 2.1 × 10 9
Pa), D is the diameter of pipe, δ is the thickness of pipe wall, and E is the
modulus of elasticity of pipe wall. Generally, the velocity of water hammer
wave in Penstock of hydropower station is between 1000–1200 m/s.
Considering the compressibility of water and the elasticity of pipe wall, it
can be assumed that the area of pipe is A = A(s, t ), ρ = ρ (s, t ), and the
relationship of the derivative with the body can be used
d A
dt
=
∂ A
∂t
+ V
∂ A
∂s
,
dρ
dt
=
∂ρ
∂t
+ V
∂ρ
∂s
P. Liu
formula of hydraulic elements derived by Jukowsky is
p = −ρCV, ,h = −
C
g
V
For example, the propagation velocity of water intake shock wave C =
1435 m/s, and the velocity V 0 = 5 m/s when the pipeline is in constant flow.
Substituting the above formula, the water hammer pressure value generated
by the sudden closing of the valve is P = 7.175 × 10 6 pa.
The valve in the actual pipeline is not closed in a moment, but gradually closed in a certain period of time. The resulting water hammer pressure
is much smaller than that generated by a sudden closing of the valve. The
maximum water hammer pressure in front of the valve is the superposition of
pressure waves generated by gradually closing the valve, which is called indirect water hammer. Relatively speaking, the water hammer caused by one-off
shutdown is called direct water hammer. In addition to the water hammer
generated by closing the valve, the sudden opening of the valve will also
cause the water hammer wave in the pipeline, but the water hammer wave
is negative at this time, causing the pressure to decrease.
If the elasticity of the pipe wall is not considered, the propagation velocity
of the pressure wave caused by the compressibility of the water is the propagation velocity of the sound in the water C = 1435 m/s, but if the propagation
velocity changes after considering the elasticity of the pipe, it can be derived
using the continuity equation and momentum theorem
C =
K w
ρ
1 +
DK w
δ E
=
1435
1 +
DK w
δ E
where, K w is the bulk modulus of elasticity of water body (= 2.1 × 10 9
Pa), D is the diameter of pipe, δ is the thickness of pipe wall, and E is the
modulus of elasticity of pipe wall. Generally, the velocity of water hammer
wave in Penstock of hydropower station is between 1000–1200 m/s.
Considering the compressibility of water and the elasticity of pipe wall, it
can be assumed that the area of pipe is A = A(s, t ), ρ = ρ (s, t ), and the
relationship of the derivative with the body can be used
d A
dt
=
∂ A
∂t
+ V
∂ A
∂s
,
dρ
dt
=
∂ρ
∂t
+ V
∂ρ
∂s