3 Hydrodynamics
231
3.6.2 Basic Equation of One-Dimensional Unsteady
Flow
The hydraulic elements of unsteady flow in pressure pipeline, such as the
average velocity V and the average pressure p, can be expressed as the function
of time t and flow path s, that is, v = v (s, t ), p = p (s, t ). The derivation of
the basic equation can be obtained from the continuity equation, momentum
equation, and energy equation. Considering the general requirements, it is
assumed that both pipe section A and liquid density ρ are also functions of
flow path s and time t. In any pipeline system, take the microelement control
volume as shown in Fig. 3.59. According to the law of conservation of mass
(continuity equation), in dt period, the mass difference of outflow and inflow
microelement control volume should be equal to the reduction in the control
volume mass in the same period, that is
∂
∂s
(ρ AV dt)ds = −
∂
∂t
(ρ Ads)dt
We can obtain
∂(ρ A)
∂t
+
∂(ρ AV )
∂s
= 0
This equation is the general form of one-dimensional unsteady flow
continuous equation.
Take any microelement along the pipe, as shown in Fig. 3.60. Let the crosssectional area of the microelement be a, perimeter P, length ds, D as the
pipe diameter, the angle between the pipe axis and the horizontal direction
be θ (the downward inclination of the pipe axis along the flow direction
is positive, sin θ = −
∂z
∂s ), the shear stress acting on the pipe wall is τ w ,
1
2
1
ds
2
t+dt
t
AVdt+ s
ρ
AVdt
ρ
AVdt)ds
(ρ
∂
∂
Fig. 3.59 Microelement control volume of unsteady flow in pressurized pipeline