Modeling One and Twophase Water Hammer Flows - Advances in Control and Automation of Water Systems - page 43

Governing Equations for Unsteady (or Transient) Flow— Hydraulic transient
flow is also known as unsteady fluid flow. During a transient analysis, the fluid and
system boundaries can be either elastic or inelastic:
• Elastic theory describes unsteady flow of a compressible liquid in an elastic
system (e.g., where pipes can expand and contract). Research uses the Method
of Characteristics (MOC) to solve virtually any hydraulic transient problems.
• Rigid-column theory describes unsteady flow of an incompressible liquid in a
rigid system. It is only applicable to slower transient phenomena.
Both branches of transient theory stem from the same governing equations. 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 theoreti-
cal result that usually corresponds quite closely to actual system measurements if 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 conditions) and inappropriate as-
sumptions.
Governing Equations for Steady-state Flow-- Steady-state models, such as Wa-
ter CAD or Water GEMS, are capable of two modes of analysis: steady-state and ex-
tended period simulation (EPS). The EPS solves a series of consecutive steady states
using a gradient algorithm and accounting for mass in reservoirs and tanks (e.g., net
inflows and storage). Both methods assume the system contains an incompressible
fluid, so the total volumetric or mass inflows at any node must equal the outflows,
less the change in storage. In addition to pressure head, elevation head, and velocity
head, there may also be head added to the system, for instance, by a pump, and head
removed from the system by friction. These changes in head are referred to as head
gains and head losses, respectively. Balancing the energy across two points in the
system yields the energy or Bernoulli equation for steady-state flow: the components
of the energy equation can be combined to express two useful quantities, the hydraulic
grade and the energy grade:

( ) ( )

( ) ( )

2 2

P / γ + += ++ + ZV gh P /2 / γ Z V gh /2 (3.1)

1 11 p 2 2 2

• Hydraulic grade —the hydraulic grade is the sum of the pressure head (p/γ) and
elevation head (z). The hydraulic head represents the height to which a water
column would rise in a piezometer. The plot of the hydraulic grade in a profile
is often referred to as the hydraulic grade line or HGL.
• Energy grade —the energy grade is the sum of the hydraulic grade and the ve-
locity head (V2/2g). This is the height to which a column of water would rise in
a pitot tube. The plot of the hydraulic grade in a profile is often referred to as the
energy grade line or EGL. At a lake or reservoir, where the velocity is essentially
zero, the EGL is equal to the HGL [2, 3].