Modeling One and Twophase Water Hammer Flows - Advances in Control and Automation of Water Systems - page 46

head. Because changes in velocity of several feet or meters per second when a pump
shuts off or a hydrant or valve is closed, it is easy to see how large transients can occur
readily in water systems. The mass of fluid that enters the part of the system located
upstream of the valve immediately after its sudden closure is accommodated through
the expansion of the pipeline due to its elasticity and through slight changes in fluid
density due to its compressibility. This equation does not strictly apply to the drop in
pressure downstream of the valve, if the valve discharges flow to the atmosphere.
Celerity and Pipe Elasticity-- The elasticity of any medium is characterized by
the deformation of the medium due to the application of a force. If the medium is a
liquid, this force is a pressure force. The elasticity coefficient (also called the elasticity
index, constant, or modulus) is a physical property of the medium that describes the
relationship between force and deformation. Thus, if a given liquid mass in a given
volume ( V ) is subjected to a static pressure rise (d p ) a corresponding reduction (d V <
0) in the fluid volume occurs. The relationship between cause (pressure increase) and
effect (volume reduction) is expressed as the bulk modulus of elasticity ( E n) of the
fluid, as given by:

( )

E

= - dp / dV / V = dp / d r / p

(3.6)

A relationship between a liquid's modulus of elasticity and density yields its char-
acteristic wave celerity:

a = Er / r = dp / d r ,

(3.7)

The characteristic wave celerity ( a ) is the speed with which a disturbance moves
through a fluid. Its value is approximately 4,716 (ft.Sec - ¹) or 1,438 (m. Sec - ¹) for water
and approximately 1,115 (ft.sec - ¹) or 340 (m. Sec - ¹) for air. Injecting a small percent-
age of small air bubbles can lower the effective wave speed of the fluid/air mixture,
provided it remains well mixed. This is difficult to achieve in practice, because diffus-
ers may malfunction and air bubbles may come out of suspension and coalesce or even
buoy to the top of pipes and accumulate at elbows, for example. In 1848, Helmholtz
demonstrated that wave celerity in a pipeline varies with the elasticity of the pipeline
walls. Thirty years later, Korteweg developed an equation to determine wave celerity
as a function of pipeline elasticity and liquid compressibility. Research uses an elastic
model formulation that requires the wave celerity to be corrected to account for pipe-
line elasticity [5, 6].

= +Y

(3.8)

( )

a E / /1 ED

E

O r

e

This is valid for thin walled pipelines ( D / e > 40). The factor Y depends on pipeline
support characteristics and Poisson's ratio. Y depends on the following:

• Pipe is anchored throughout against axial movement: Y = 1 - µ 2 , where µ is
Poisson's ratio
• Pipe is equipped with functioning expansion joints throughout: Y = 1 - µ/2