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CONSIDERATION ABOUT THE ENERGY LOSS IN LONG
PIPES WITH WATER - AIR FLOW.
P. Boeriu
IHE Delft, The Netherlands, Senior lecturer
Associate Professor at Technical University of Timisoara, Romania
email: pet@ihe.ni
Abstract
Many times
in practice the water flow transports an important quantity of air. This air
can be provided by the vortices at the water intake, pumping stations,
aerators, etc. Under certain conditions air may be induced also in pressurized
intake systems. The included air may develop in some portion of the pipes air
accumulations in the form of big bubbles; this particular phenomenon will not
constitute the object of this paper. In other cases the air can be induced
continuously; in this case a two-phase flow will take place. The experiments as
well as practical experience show in
this case an increased friction loss. The immediate explanation of this
phenomenon is the increase of the flow turbulence, and consequently of the
friction coefficient that must be correctly evaluated for the design purposes.
Background
Up to
present, numerous studies on the motion of an air-water mixture were carried
out with applications in different technical field as industrial chemistry, oil
extraction and transport, etc. These researches are characterized by the fact
that they are studying a variety of motion forms with a high air content in the
flow and by the use of empirical formulae. The results obtained by using these
formulae are rather different. To calculate the friction losses in a pipe of
diameter D and length L with two phase flow the classical formula of Darcy -
Weisbach is usually used, by analogy with homogenous flow:
(1)
where: 
- the friction
coefficient of the two phase flow
V -
the average velocity of the flow
It is
obvious that the computation by using equation (1) is conventional. It is
applied to a complex flow where change of energy is taking place between the
two phases. The formula would apply for a simpler flow when a single phase is
in contact with the whole pipe perimeter. In the case of the aerated flows the
energy loss of the flow is determined by more parameters, additional loss due
to the sliding between the two phases at the contact surface taking place.
There is also variation of the pressure at the water surface due to
modification of the air pressure. For practical computation the equation (1)
still may be used if 
would be considered
as an experimental parameter with different interpretation compared to
classical hydraulics. This would be possible if experimental studies may
establish satisfactory correlations between 
and other parameters
that are influencing the phenomenon.
From
dimensional considerations, a simpler form for this kind of relations can be
accepted:
(2)
where Fr,
Re, We can be written for the mixture flow or for each phase separately. The We
number may be important if the motion will take place with bubbles. In this
case the We number can characterise the bubbles dimension. Experimental studies
have shown that the Weber number does not have a significant influence for the
range of concentrations and velocities considered in this study.
From an
extensive literature survey, the main conclusions are:
·
Most of the experiments were carried out on pipes
with small diameters. The possibility of extrapolation is doubtful.
·
The range of air concentration is larger than that
met in common water system pipes Although the different researches are
expressed in a variety of forms, these can be reduced to three particular forms
of the relation (2):
(3)
(4)
(5)
where: - Re - Reynolds number
written for the liquid phase only
- Fr - Froude number written for the two phase
flow
·
Equation (1) can be derived after some
transformations from a rather spread form proposed by Martinelli - Lockhart.
The relationship proposed by the authors gives a graphic form between a
so-called function of the stream 
and the module of the
two phases or mixture flow, expressed as:
(6)
(7)
where: -
, 
, 
are specific weight
of liquid, gas and mixture, respectively
- IL , IG are the energy gradient line of the one
phase flowing alone in the pipe, calculated with classical hydraulics
relationships
·
Writing equations (6) and (7) in other way and
rearranging the terms, the following relationship can be derived:
(8)
·
It is obvious now that equation (8) can be reduced to
a relationship of type (3). As an example considering the water air mixture at
20 oC temperature, the curve shown in figure 1 can be derived.
However, this curve is based on experiments done on small-diameter pipes (e.g.
lower than 25 mm), and the extrapolation of this curve for larger diameters is
controversial.
Figure 1.
Relationships
similar as expressed by equation (4) have been used by the Italian researchers
Mongiardini, Marchi, Piva. From their studies the following conclusion may be drawn:
·
If we will take in consideration only the experiments
performed on a certain pipe at the concentrations and values of Re number which
are common for hydraulic pipes, then one may conclude that the influence of Re
number on the ratio 
is reduced.
·
Large differences are obtained between different
relations based on the experimental studies carried out on the pipes with small
diameters, expressing the variation of the ratio 
for the same value of
Re number, figure 2.
Figure 2
Relationships
similar to type (5) have been used by Russian researchers Semenov and Kosterin.
Their results are given in figure 3 and figure 4 in graphic form. It is easy to
conclude from these figures that significant differences are resulting when
these diagrams are used. The values of the ratio 
given by Kosterin are larger than the values of the same ratio
given by Semenov. These differences cannot be explained because the
experimental data are not available. Additional experimental studies are
obviously needed.
Figure 3
Figure 4
Experimental
study
An
important aspect easy to be observed during the experiments using industrial
pipes (D > 50mm) is that the two phase flow may have a distinct separation surface
between air and water, that approximately coincides with the values from which
the increase of the friction loss coefficient becomes significant. Considering
the physical aspect of the flow, and accepting the forces of inertia and
gravity as dominant forces, the following relation may be written:
(9)
where: 
= the pressure loss of air water flow
= water, air, specific density respectively
vwater , vair
= medium velocity of the water, air phase, respectively
µ = water dynamic
viscosity coefficient
D = pipe diameter
L = pipe length
k = absolute pipe
roughness
Cm = medium
concentration of air in water
From
consideration of dimensional nature we may write:
(10)
Equation
(12) requires the following hypothesis:
a)
(11)
acceptable
when the flow in the pipe experiences a separation surface between the water
phase and the air phase
b)
(12)
This is an
incompatible criterion because it assumes the maintenance of an identical
pressure on the model and on the prototype. However, as it results from many
papers the variation of 
is small and can be
neglected for most situations.
By
accepting these hypothesis, the equation (12) may be written as:
(13)
where:
(14)
and:
(15)
By using more
than 240 experiments performed in different laboratories and different pipe
diameters following a relationship of type (15), the ratio 
and the corresponding
Froude number have been calculated. The results obtained for pipes of three
different diameters were represented on restricted ranges of Froude numbers.
The concentration range was restricted to an upper limit of Cm = 45
%. The main findings of this work can be summarised as follows:
-
For the same domain of the Froude
number the ratio of 
= f (Cm )
determined for pipes of different diameters are closed one to another, the
dispersion from a pipe to another being of the same order of magnitude as those
corresponding to the same pipe.
-
The variation of the ratio 
with Cm is
different as Froude number is larger or smaller than one. For the flows with Fr
< 1, the data offered in literature are not significant enough, more
experimental measurements being needed.
-
For flows with Fr > 1, the ratio 
increases with the
air concentration. The increase inclines to be more accentuated when Fr number
increases, that corresponding to the tendency showed by Semenov's diagram.
-
The value of the ratio 
for the domain of
Froude number limited by 1.5 and 10 experimentally determined are larger than
those showed by the diagrams presented by Semenov.
-
The flow of the water air mixture
may present an irregular free water surface with variation of cross sectional
area. In this kind of phenomena the gravity forces are playing an important
role that may justify the use of Froude number as a main parameter within
relationships characterizing qualitatively and quantitatively the studied
phenomenon.
Most of
the experiments described above were done on small diameter pipes and with air
concentration that are unusual for water supplies or water system pipes but
common for chemical or other technologies. Our research as well as others show
that the maximum aeration coefficient for usual water supply pipes is:
(16)
the medium
transport concentration corresponding to the air water flow is:
(17)
Because
other flows except those with maximum aeration coefficient may enter in
conduits the most possible concentrations are larger than the overside limit.
This is why we are using on our experiments a limiting value of Cm =
0.50
In order
to validate the considerations exposed above a laboratory study was carried
out. The experimental installation has been composed by a long pipe with
constant water level supply and flow measurement devices provided at the
entrance and at the end of the pipe. The air was introduced in the water flow
by an air compressor provided as well with flow rate and pressure measurement
devices. A series of measurements has been made on steel pipes of 100 mm, 150
mm and 400 mm as well as in nature on a vertical well of a secondary intake
with a diameter of 1400 mm. The experimental results have been processed
according to equation (15). The results of the processing are shown in Figure
5, in a dimensionless parameters diagram where the axis are indicating the
medium concentration of air in a water flow and the expression 
, for specific fields of Froude numbers
Figure 5
By using
the interpolation method the following equation may be obtained:
(18)
valid in a
range of Froude numbers between 0 and 4.
Conclusions
Most of
the studies related to the calculation of the two phase flows losses in long
pipes agree that:
·
The two phase flow may be represented in different
forms that evolve one from another without apparent discontinuities
·
Despite all efforts, the general laws of these
complex phenomena have not yet been determined. It may only be observed that
the present research trends are divided in two directions: one considering the
mixture of two different fluids water and air, and other considering a
homogenous fluid with certain physical characteristics.
·
Several explanations of the air entrainment
phenomenon based on the increase of water flow turbulence are presented in
literature. It is admitted that this phenomenon appears when the degree of
turbulence is high enough so that the normal components on the free surface of
the pulsating rate may transfer to water drops the potential energy necessary
to overcome the action of the surface tension forces and to detach it from
water bulk. All authors generally agree that
air appearance in pressurized pipes induces a complex biphasic flow with
an increase friction head loss as a result. This trend is explained based on
the reduced water flow cross section, turbulence increase, irregular flow
contour at the separation surface between the two phases.
·
Large differences between the experimental results
presented by various researchers and no extrapolation possibilities for larger
diameters are given. From this viewpoint the relation (5) is more favourable
but some disagreements between the studies presented by Semenov and Kosterin
make no possible the use of their results without additional explanations.
The
considerations presented in this paper are recommending the use of a relation
with a structure defined by equation (15). Based on a laboratory study, the
author of this paper is recommending an equation to calculate the value of
friction coefficient of a two phase flow. Equation (18) is a valid formula for
a range of medium air concentrations of 0 to 60% and in a restricted domain of
Froude numbers ranging from 1 to 4.
To verify
the equation (18) we have used the measurements made by Mongiardini /3/ and
A.M. Piva /4/ performed for pipes with different diameters and in different
laboratories. The results of their experimental studies have been worked out on
restricted ranges of Froude numbers (Fr = 1 - 2 and Fr = 2 - 3), by using
equation (18), and presented in figures 6 and figure 7. The differences between
results are of the same order of magnitude as the dispersion band of errors, confirming
the validity of the hypothesis considered during our own experimental studies.
Figure 6
Figure 7
Equation
(18) represents a valid computational relationship to determine the friction
head losses in steel pipes with diameters larger than 80 mm and average air
concentration lower than 50%
References
1.
Boeriu, P. Contribution to the long
pipes two phase flow hydraulics, Doctoral Thesis
2.
Boeriu, P. Modelling of the two
phase flow, Hydrotechnica, no.9, 1986
3.
Mongiardini, V. Moto dei miscugli di
area e acqua in correnti in pressione, L' Acqua no.2, 1966
4.
Piva, A.M Moto bifase in tubazioni
orizzontali, IX Convegno di Idraulica, Genova, 1968
5. Preuss, K. Fliessverhaltnisse von Luft und Wasser in einem teilgefüllten
Rohr, Technische Hochschule München, 1970