Water Hammer and Hydrodynamics’ Instability - Advances in Control and Automation of Water Systems - page 66

figure 4.5 Characteristic lines network for assumed pipe.

4.2.1 Modeling and numerical simulations of water hammer
Two fundamental laws apply to steady state, or transient models:

• Conservation of mass—also expressed as the continuity equation, which states
that matter cannot be created or destroyed.
• Conservation of energy—also expressed as the momentum equation, which
states that energy cannot be created or destroyed. The best way to arrive at
sound, physically meaningful conclusions and recommendations is to keep
these principles in mind whenever research interprets the results of a hydraulic
model. Present research has made this easy by tracking the mass inflow or out-
flow of air or water at any location and by plotting or animating the resulting
total energy at any point and time in the system. Friction factor at steady state
was the same as unsteady state [6-10]:

ρ fv

2

1 ∂ z f V

2

1 dV

(4.3)

τ= − , − = ,

0

8 γ ∂ s D 2 g g dt

For flow direction changes:

V

2

= VV , (4.4)

∂

+

dV 1 p dz f

(Euler equation), (4.5)

+ g + VV = 0

dt ρ ∂ s ds 2 D

Continuity equation and for fluid fine element:

ρρ

1 d d dA ∂ V 1 d

− + ( ds ) = 0,

(4.6)

ρ d + 1/ A dt ∂ s ds dt