contents
• Description of
the waterhammer phenomenon
• Protective systems
• Method of air chamber determination
• Experimental validation
• The future of air chambers
The protection of the pipes in a pumping station
against waterhammer has proved to be one of the most important
problems in systems today, and owing to the economic aspects of
these installations the test pressures are generally minimised
compared with the normal operating conditions. The optimisation
of systems providing protection against the waterhammer is therefore
essential :
• to limit pressure variations caused
by shutting down or starting up pumping stations,
• to minimise the investments costs of
providing waterhammer protection.
the shutting down of pumping sets generates
the well-known pressure variations, but we shall nevertheless
describe this phenomenon, as it conditions the selection of the
waterhammer protective system.
Figure 1.
The sudden deceleration of the fluid circulating
in the conduits (figures 1 and 2) causes a pressure drop in the
whole installation, and then an oscillation of a pressure wave
(pressure rises and falls in reducing amplitude owing to the dissipation
effects).
The initial pressure drop is the most critical
one, as its extent determines the pressure which follows. Hence
the necessity to limit in accordance with the maximum pressure
permissible in the installation.
Moreover, this pressure drop must be limited
to avoid the cavitation phenomena. This cavitation will materialise
as a local transformation of the fluid into vapor, thus forming
a "vacuum" pocket, which will close up, causing very
high pressure rises, entailing very serious damage.
Figure 2.
This phenomenon can appear:
- either at the pump level,
- or at any point of the conduit depending on its longitudinal
section.
The waterhammer protective system must therefore
be optimised to:
- limit the pressure drop to avoid any risk
of cavitation at all points of the installation.
- limit the pressure rise to the value of the test pressure of
the installation.
Numerous systems have been developed to provide
protection of the installations against waterhammers:
- the inertia flywheel installed on the pumping
set to increase the shutdown time and thus reduce the amplitudes
of the pressure drops and rises. This solution usually proves
impractical for economic reasons.
- the bypass on the pump to eliminate the risk
of cavitation only at the pump, but which clearly does not allow
the control of the pressure rise. Its utilisation is therefore
very restricted, even though it is possible to limit the pressurisation
by judiciously locating a number of non-return valves on the conduits
to spread the waterhammer energy over all of them (see figure3).
Figure 3.
There are various types of air chamber:
- chamber with gas-liquid separator,
- chamber without separator,
- surge chamber.
This system provides perfect control of pressure
drops and rises in the installation. Moreover, it offers complete
operating safety, as none of the mechanisms is moving.
The sizing and performance appraisal of this
system may be obtained by using mathematical methods (2) based
on wave propagation equations and which can be solved using a
graph on the basis of the method proposed by Bergeron (1). with
today's powerful computers these equations can be solved digitally
in a minimum of time, thus providing sophisticated and accurate
means of determination.
1/ Analysis of the phenomenon of wave
propagation
When a valve is closed very quickly (example
figure 4) the velocity of the fluid particles which strike the
valve is annihilated, causing a local pressure increase, and their
kinetic energy is transformed to cause a deformation of the pipe
and of the liquid, whose compressibility must be taken into account.
Figure 4.
As the particles strike against the preceding
layer, the pressure rise zone propagates and extends fro the valve
towards the chamber (1). Section xx', thus propagates from A towards
R at velocity A, pressure rise wave propagation velocity or speed.
When the valve reaches the end of the line (in
this case the chamber), the high pressure extends throughout the
pipe and all the 'compressed' fluid particles are stopped. the
fluid speed is therefore zero (2).
As the pressure in the pipe is higher than that
in the chamber, a flow is established between pipe an chamber
at velocity Vo, in such a way that the pressures are equalised
(Po pipe = Po chamber).
Section xx', returns towards valve at velocity
A. The deformation effect is transformed into kinetic energy and
the liquid in the pipe regains velocity Vo in the opposite direction
(4).
This displacement of fluid particles takes place
from the valve, causing a pressure drop (negative pressure rise
- dp), entails a tightening of the pipe and an expansion of the
liquid. The kinetic energy is again transformed into deformation
(with the opposite sign) (5).
At the moment the negative shock wave reaches
the chamber, the pressure throughout the pipe is lower than that
in the chamber and a flow from R to A is again established (6
and 7).
The initial conditions are re-established at
the end of this phase.
2/ Interest of the protective system
the source of the water hammer is a sudden variation
of low or of velocity, giving a high velocity gradient. The resultant
pressure variations are propagated at the speed of sound in the
pipes at a high frequency.
This results in wave oscillation.
Figure 5.
Accordingly, waterhammer protective devices
aim at transforming wave oscillation into mass oscillation. In
the case od a hydropneumatic chamber, the energy supplied by the
pressure rise will be transformed into kinetic energy of the fluid
(see figure5). Thanks to its construction, the chamber provides
a gas-liquid boundary of lower resistance than that of the conduit.
So the volume increase accompanying the pressure wave is absorbed
by the chamber and not by the conduit.
As soon as the pressure in the conduit becomes
lower than that in the chamber, the latter returns the liquid
stored in the conduit (figure 6).
Figure 6.
3/ Sizing of an air chamber
Pressure rise calculation :
Let dH be the maximum pressure variation arising
when the phenomenon occurs. dH may be expressed as function of
:
• wave propagation velocity : a (m/s)
• gravity : g (m/s2)
• fluid velocity in conduit : V (m/s)
For a shutdown or a flow which will last less
than the critical time 2L/a, an initial approximation of this
pressure variation in the conduit can be estimated using the Allievi
formula :
Vo = initial flow of velocity
For a slower closure, whose time T of closure
or of pump shutdown exceeds 2L/a, this pressure rise or fall can
be estimated using the Michaud formula, assuming that the flow
rate variation is linear during the shutdown of the flow.
4/ Effectiveness of the air chamber
The method of determining an air chamber is
based on the solution of the movement quantity equations. This,
by differentiation, leads to the wave propagation equations an
to the pressure balance equations. The result is :
Upstream
Downstream
the wave speed in the pipe depends on the nature
of the fluid and of the pipe. it may be obtained from the flow
equations of the fluid by taking into account the elasticities
of the fluid an of the conduit.
Where:
p : fluid density kg/m3
D : pipe diameter in m
c : pipe wall thickness
EF : Modulus of elasticity of fluid in N/m2
Ec : modulus od elasticity of conduit
These elements allow the simulation of the behavior
of the chamber on the line. this simulation is based on a method
of iteration starting from the initial pressure and flow rate
conditions.
The chamber therefore constitutes a transformation
of energy which absorbs the energy of a wave oscillation and restores
kinetic energy, in the form a mass oscillation.
It therefore allows the pressure variations
in the circuit to be brought back to acceptable levels.
5/ Solution method
The wave propagation equations are solved digitally
using sophisticated computers now available.
The simulation of the shutdown of a pumping
station, whether or not equipped with an air
chamber, can be effected very quickly. Our software offers the
opportunity of automatic optimisation to determine the most appropriate
chamber volume in terms of the maximum permissible pressure rise
and fall stresses.
The actual structure of the equations to be
solved, which is hyperbolic, requires the definition of the initial
conditions, and the upstream and downstream limiting conditions
which determine how the waves are reflected to the pump and to
the chamber at any time. moreover, it is possible to determine
the pressure values at different points of the conduit in accordance
with a number of solution nodes selected along the conduit.
the diagram (figure 7) shows how we calculate
the values of the pressure P and of the flow Q at 5 points of
the conduit for each moment, dt starting from the pump shutdown
at To, then To + dt, To +2dt, etc.
Figure 7.
This diagram shows (as an abscissa) the various
nodes from 1 to 5, located along the conduit, assumed to be horizontal,
or of its abscissa if this conduit has any other longitudinal
section.
| at pump |
(x/L = 0) |
| at quarter point |
(x/L = 0.25) |
| at midpoint |
(x/L = 0.5) |
| at three-quarter point |
(x/L = 0.75) |
| at chamber |
(x/L = 1) |
Time is shown as ordinate, the first time step
being T = To + dt, with dt equal to L/4a (N = 4 number of sections).
By retaining the relations (1) and (2) along
a characteristic line + a/gs and - a/gs depending on whether the
wave comes from upstream or downstream, it is possible to determine
the values of the pressures and the flows at each node for each
calculation step.
For example, at point 4 at To + dt, the pressure
and flow values are obtained from the values calculated at To
at points 3 and 5.
6/ Characteristics specific to waterhammer
protective device
Waterhammer protection air chambers, with or
without separator, have specific characteristics which have to
be taken into account for the stimulations, as they from an integral
part of the conditions at the calculation limits. The most important
of these concern the filling and emptying head loss coefficients.
In fact, the energy dissipated in these irregularities can modify
all the results obtained in the simulation.
The characteristic of the polytropic behavior
of the gas, given by the polytropic coefficient n, such as P.Vn
= constant, is also important and its value has been specially
studied to determine it as function of the conditions of utilisation
of the waterhammer protective device.
7/ Characteristics specific to the pump
The limiting conditions found upstream also
concern the pump characteristics :
• its rotational inertia,
• its pressure-flow characteristics in terms of the speed
of rotation.
The works of E. Mendiluce (3) have permitted
the empirical evaluation of the shutdown time of a pump from the
geometric and hydraulic parameters of the installation only (figures
8 and 9).
Figure 8.
MENDILUCE - ROSICH FORMULA
g : gravity
L : Conduit Length
V : Flow Velocity
HMT : Pressure Head
C.K. : Empirical Constants
Figure 9.
MENDILUCE FORMULA COEFFICIENTS
Thus our limiting conditions are established
from this shutdown time which allows the operation to be simulated
without the pump rotational inertia being known. Similarly, we
evaluate the pressure-flow characteristic curves from 'typical'
centrifugal pumps.
8/ Presentation of the results
The results are given in the form of a general
table which indicates (see figures 10 and 11)
• geometrical data,
• hydraulic data,
• results of the maximum and minimum pressure values at
different points of the conduit,
• the chamber selected and its initial gas charge.
Figure 10.
Figure 11.
Moreover, different curves show :
• the pressure time change in the conduit
at pump level,
• the pump and chamber flows (inlet and outlet)
• the sections along the conduit of minimum and maximum
pressures so as to detect the points liable to be in the "cavitation"
zone where pressure falls occur.
The follow-up of various pumping installation
start-ups has allowed the recording of a considerable amount of
experimental data.
The comparison which has been made with our
results obtained from the simulation calculation shows that the
relative error does not exceed 4% on the amplitudes of the pressure
rises and falls.
Moreover, in nearly 80% of the cases, the pressures
measured are more favorable than the calculated pressures which
makes it possible to guarantee the pressure rise and drop values
with a safety margin in the majority of the configurations. (figure12).
Figure 12.
Surge chambers are not covered here, as their
applications remain very limited and our present calculation method
does not allow their simulation.
Air chambers with or without separators vary
considerably in design and can present very original technological
solutions. in any case, with our calculation method the operation
of the pumping set can be simulated with both types of equipment.
1/ Chambers without separator
The design of this type of equipment is generally
reserved exclusively for large volumes (several tons of cubic
meters). It must therefore deal with two major problems :
• the first of these is the solubility
of gas in water. It is therefore necessary to add an air compressor
to provide for the periodic filling of the chamber with air,
• the second is to avoid vortices when emptying the chamber
at the start of the pump shutdown phase, as this would involve
a considerable volume of air in the conduit and so reduce the
effectiveness of the waterhammer protective devices.
It is possible:
• either increase the total volume of
the chamber from 50 to 100% compared with the minimum volume required,so
that the volume of liquid present in the chamber is always sufficient
to avoid vortex formation.
• or to provide a special suction system in the lower part
which avoids vortex formation without, however, increasing the
minimum volume required for the waterhammer protective device
by a significant amount.
These systems, developed by Dennis and Young
(4) are designed in accordance with certain principles which allow
the limiting of vortex formation in suction systems. One example
is shown in figure 13 where all geometrical parameters are defined
from operating conditions.
Figure 13.
2/ Chamber with separator
We can concentrate on two design types :
• liquid in bladder (ANN)
• gas in bladder (IBV)
The ANN type equipment has the special feature
of a metal envelope which is not in direct contact with the liquid
and so gives better corrosion protection.
The design of the separator in thermoplastic
material means that it can be made in large quantities by rotational
moulding, so that waterhammer protective devices with separators
of volumes of some thousands of litres are feasible.
As for IBV type equipment, it is generally limited
to volumes less than 200 litres.
The means of calculation which have been created
allow the characteristics of the chambers to be taken into account
and optimised.
Moreover, it is well known that the waterhammer
protective device is greatly improved by having an asymmetrical
head loss system at its outlet (figure 14).
Figure 14.
This system allows fast emptying with a low
P and filling with a high P, which assures considerable dissipation
of the kinetic energy entering the chamber. This system can be
optimised and allows the very marked reduction of the minimum
volume required for protection without affecting its performances.
The pressure rise values are maintained, while the pressure drops
are smaller, although the cavitation risks remain - see example
in Table 15.
The asymmetrical head loss system
Objective : to reduce the air chamber volume
by providing an identical maximum pressure.
Example : Figure 15.
AAN 500 |
300 |
200 |
Pmax 89 mcl |
Pmax 89 mcl |
Pmax 89 mcl |
Pmin 36 mcl |
Pmin 30 ml |
Pmin 27 mcl |
The technology of this system is such that :
• there are no moving metal parts,
• emptying head loss is very small,
• filling head loss is adjustable in accordance with the
installation data.
With the adoption of such a system it is possible
to envisage the installation of chambers with a capacity of 1000
litres, where before air chambers, without a separator, of nearly
6000 litres (chamber without special vortex protection), had to
be provided. This leaves the future wide open for chambers involving
technology which is fundamentally based on the principle of a
passive component as a guarantee of maximum safety in the protection
of pumping units against the waterhammer.
a = sonic speed in liquid
D = conduit diameter
c = pipe wall thickness
EF = modulus of elasticity of liquid
EC = modulus of elasticity of conduit
g = acceleration of gravity
HMT = pressure head
HG = geometric height of chamber
D
= head loss on line
L = conduit length
S = pipe cross-section
U = flow velocity
V = chamber gas volume
aR = chamber filling head loss coefficient
aV = chamber emptying head loss coefficient
p = density of fluid
Francois Brault, OLAER engineer.
