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Technical Report No. 18
ENTRANCE EFFECTS IN A DEVELOPING SLUG FLOW
by
Raphael Moissis
Peter Griffith'
For
The Office of Naval Research
Nonr 1841(39)
DSR Project No. 7-7673
June
1960
Division of Sponsored Research
Massachusetts Institute of Technology
Cambridge, Massachusetts
Assistant Professor of Mechanical Engineering, MIT
Associate Professor of Mechanical Engineering, MIT
ABSTRACT
The kinetics of a Taylor bubble, as it rises behind a series of
other bubbles in a gas-liquid slug flow, have been determined.
The
rise velocity of a bubble is expressed as a function of separation
distance from the bubble ahead of it.
Using this information, the pattern of development of bubbles
which initially enter a tube at regular intervals is determined by
means of finite difference calculations.
The density and, to a first
approximation the pressure drop, of the developing flow are then calculated from continuity considerations
The density distritution in the
entrance region is found to be a function of flow rates of the two
phases, of distance from the inlet, and of initial bubble size0
Density calculated by the present theory is compared with experimental measurements by the present and other investigators.
experiments are generally in good agreement.
Theory and
iii
ACKNOWLEDGMENTS
This work has been entirely supported by the Office of Naval
Research and has been performed in the Heat Transfer Laboratory of the
Massachusetts Institute of Technology, which is under the direction of
W. M. Rohsenowo
Part of the calculations were done at the MIT
Computation Center.
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TABLE OF CONTENTS
Page
I
II
INTRDDUCTION.
.
.
.
.
.
. .
FULLY DEVELOPED SLUG FLOW.
.
.
.
.
.
..
. .
..
.
.
. ..
.
.
.
..
.
.
..
. . . . .
.
.
.
. .
.
.
4
. . .
7
. . .
12
. . .
12
.
.
1
. .
DEVELOPING SLUG FLOW. . . . . . . . . . . . . . . .
Introduction
.
4
. .
B. Fully Developed Dlug flow Theory.......
A.
.
.....
A. Theory of Emptying a Vertical Tube..
III
.
(i) Agglomeration of bmall Bubbles to Form
G. I. Taylor Bubbles . .
. .
. . . .
(ii) Agglomeration of G. I. Taylor Bubbles.
B. Velocity Profile Behind a Taylor Bubble. . .
(i)
(ii)
Theory . . . . . .
.
.
.
.
.
.
.
.
.
12
.
.
.
.
15
.
.
.
15
15
.
Measurement of Velocity Profiles in the
Wake of a Plastic Bubble . . . . . . . . . .
(iii) Calculation of Momentum Transfer Length.
(iv)
C.
....
Discussion of Results.
. . ..
. .
Rise Velocity of Trailing Bubble . . . . . . . . .
(i)
Theoretical Considerations . . .
. . *
. . .
(ii) Measurements of Rise Velocity of a Trailing
Bubble . . . . . . . . . . . . . . . . . . .
(iii)
Discussion of Results.
. . . . .
.
. . .
D. Pressure Drop in a Developing Slug Ilow. .
.
. . .
. .
(i) Applicability of Griffith and Wallis Theory.
(ii) Calculation of Developing Flow Pattern
(iii)
Results of Computer Calculations . . . .
(iv) Measurement of Pressure Drop in a Developing
Slug Flow. . . . . . .
.
.
.
.
.
.
.
.
.
.
.
Page
(v)
IV
Comparison of Theory and Bxperiments. . . . .
49
53
SUMMARY AND CONCLUSIONS.
NOMENCLATURE
.
.
.
...
LIST OF REFERENCES
.
.
.
.
.
.
.
. .
. .
.
.
.
.
. . .
. . . . .
.
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.*
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0
-
-
0
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0
0
0
.
0
0
.
0
APPENDIX I
DATA OF PRESSURE DROP MEASUREMENT
0
.
.
0
0
.
0
0
.
APPENDIX II
SAMPLE PRESSURE DROP CALCULATION.
.
0
.
0
.
.
0
.
0
0
.
0
0
0
.
.
0
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0
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0
.
APPENDIX III
CAPTIONS TO FIGURES.
APPENDIX IV
IGURES
.
.
. . ......
.
.
.
I
INTRODUCTION
When two phases flow in a tube the distribution of the phases can
These configurations are commonly
assume one of several configurations.
referred to as two phase flow regimes.
The pattern assumed by the flowing
mixture is affected primarily by the volume flow rates, the direction of
flow of the two components, the channel orientation, and the distance from
the tube inlet.
Figure 1 illustrates the flow regimes that can exist when
a gas and a liquid flow concurrently in a vertical pipe.
The pictures are
in the order of increasing gas flow for a fixed liquid flow.
Bubble flow is characterized by bubbles which are small compared to
the tube diameter.
The bubbles are dispersed randomly in the tube and
rise with different velocities depending on the bubble size.
When the
concentration of bubbles is increased beyond some critical value the small
bubbles agglomerate to form large bubbles which are bullet-shaped with
cross-section almost equal to the cross-section of the tube.
Slug flow is thus characterized by large bubbles - which will be
referred to as G. I. Taylor bubbles - separated by slugs of liquid.
One
may or may not find small bubbles in the slug following the Taylor bubble,
as illustrated in figure 3. When the separation distance between two
Taylor bubbles is large all bubbles have smoothly rounded heads and rise
with a uniform velocity (Figure 3B).
developed slug flow.
This type of flow is termed fully
On the other hand when the separation distance
between two Taylor bubbles is smaller than some critical value, the
trailing bubble is influenced by the wake of the leading one, rises faster
and eventually the two agglomerate.
When a bubble rises in the wake, the
stable character which is seen in fully developed slug flow is lost.
The
nose of the bubble is distorted and becomes alternately eccentric on one
side or another (Figure 3A and C).
These are the characteristics of
developing slug flows.
Semi-annular flow is in effect a type of developing slug flow in
which the separation between Taylor bubbles approaches zero.
In annular
flow the liquid flows in an annulus while the gas occupies the core of
the tube.
Finally, with even higher gas flow rates, the liquid is dis-
persed to form a mist.
With the advent of processes and equipment working with two-phase
flowing mixtures, (steam generators, boiling nuclear reactors, flow of
gas and oil in pipe lines, heat exchangers and fractionators in chemical
plant to name but a few) the knowledge of quantities such as pressure
drop, density and maximum flow rate has become increasingly important.
Much research has been done on the subject in the last twenty years and
the literature on two phase flow has grown immensely (see, for example,
ref. 1).
The greater part of the work reported in the literature consists
of experimental measurements and attempts to correlate data.
There are
also some vain attempts to define a single mathematical model which
would encompass all possible two-phase flow regimes.
It is only recently
that the importance of specifying the flow regime at the outset of any
two phase flow investigation has been appreciated.
The analysis of any
two phase flow problem should begin by specifying what the flow regime is.
Figure 2 shows a flow map which allows one to determine what the flow
regime is when the pipe size and the flow rates of the two phases are
known.
It should be stressed that the divisions shown as lines in
Figure 2 should really be bands.
In addition, it should be stressed that
this map applies only in fully developed flow in a vertical pipe without
heat addition and in the absence of surface active agents.
The present investigation is concerned with the slug flow regime and
more particularly with developing slug flow.
In addition to its interest
as part of the general two phase flow problem, slug flow is important
because it is encountered in various engineering applications.
For
example, slug flow may occur in-the riser section of boiling nuclear
reactors.
The flow instability which has been observed in natural circu-
lation loops has been attributed to the variations in mean density in the
riser section (ref. 2) which in term may be- attributed to the oscillatory
nature of developing slug flows.
Also, air lift pumps which are commonly
employed in the oil industry operate in the slug flow regime.
The work reported here is an extension of Griffith and Wallis's
fully developed slug flow theory (ref. 3).
The method of approach
followed by the present investigator-is the same as that adopted by
Griffith and Wallis.
Instead of seeking a correlation of the gross
behavior of the system without examining the fundamental mechanisms
involved, a simple kinematic model has been isolated and the basic
principles of mechanics applied to the model. The results of an analysis
of this type may be of some value as far as direct application to industrial problems is concerned.
They are particularly valuable, however,
in the sense that they offer a complete description of at least one
fundamental mechanism of two phase flow, and form a solid ground on which
further work may be based.
FULLY DEVELOPED SLUG FLOW
II
This chapter contains a brief review of the existing knowledge on
the problem of slug flow.
subject:
There have been two major contributions on the
on the one hand the solution of the problem of emptying a
vertical tube by Dumitrescu (ref. 4) and by G. I. Taylor (ref. 5); on the
other, the application of Dumitrescu and Taylor's work to the analysis of
fully developed slug flow by Griffith and Wallis (ref. 3).
Both theories
will be described in some detail.
A. Theory of Emptying a Vertical Tube
The exact analysis of the motion of a large bubble in a tube filled
with stagnant water constitutes a formidable problem. The main difficulty
in the matter is to obtain a correct description of the currents in the
wake which follows the nearly flat lower surface of the bubble.
This
difficulty may be overcome, however, if one assumes the bubble to be
infinitely long.
In making this assumption one really reduces the problem
to the following question.
How rapidly will a closed tube full of liquid
empty when the bottom is suddenly opened to the atmosphere?
Since the
tube is open to the atmosphere at its lower end, it is not necessary for
the bubble to have a lower surface. All difficulties associated with the
wake are thus eliminated.
Assuming potential flow and symmetry about the axis of the tube, the
equations governing the flow may be expressed in terms of a dimensionless
velocity potential
0
and Stoke' s stream 'function
a
af
:(1
f-L-r
-
W
as follows.
5.
where r and x are dimensionless quantities defined as r/R and x/R.
The
distance x is measured from the top of the bubble. R is the pipe radius.
The general solution to these equations may be written in the
following form (see, for example, ref. 6).
~1)
i.1
'L=I
L
L
('
r)C
(2)
"L/
If one superimposes a parallel flow of dimensionless velocity V , defined
as V /g
the last two equations become
The constants V
and C and the eigen values b may be determined by
imposing the following boundary conditions:
First, at the bubble boundary the pressure must be constant. This
corresponds to the mathematical statement that on the surface
Y
the equation
must be satisfied.
Equation (4) is obtained by writing Bernoulli's
equation for the film of liquid running down the pipe wall.
0
Second, there is a stagnation point at the origin of coordinates, that
is at the nose of the bubble.
This means that at r = x = 0
The final condition that the velocity is axial at the pipe wall is already
satisfied by equations (2).
On imposing each of these conditions and truncating the infinite
series after the first term, one obtains the final result that
V*
= 0.328
(ref. 5)
Whereas on truncating the infinite series after the third term the
final result is
V*
=
0.350
(ref. 4)
Both Dumitrescu and Davies and Taylor carried out experiments and
compared their measurements with the results given above.
The agreement
is remarkably good for large tubes but the theory tends to overestimate
the velocity for smaller tubes.
A reasonable explanation for the discrepancy
in small tubes is that viscosity effects become important at low Reynolds
numbers (based on bubble velocity).
The work of Dumitrescu and Davies and Taylor served as the starting
point for the investigation of fully developed slug flow by Griffith and
Wallis.
This work will be summarized in the following section.
----------
B. Fully Developed Slug Flow Theory
Consider a vertical pipe through which a two phase mixture flows in
fully developed slug flow. As was mentioned in the introduction, slug
flow is characterized by Taylor bubbles which are separated by slugs of
liquid.
Fully developed slug flow is further characterized by the fact
that the separation distance between two bubbles is large compared with
the tube diameter so that all bubbles are uninfluenced by wake effects.
An equation is required which will predict the pressure drop for
such a system.
If one considers a typical section of pipe which contains
one slug and one bubble, and takes a control volume moving with the bubbles
at bubble velocity (see Figure 4 control volume b), then the momentum
equation for the control volume at steady state is
(PI - p2 )Ap -
Pag(Ls + Lb)Ap -TwA
=
Ap(V -Vg)
(6)
Now for steady state conditions it is clear that the input and output
momentum fluxes are identical, so that the right hand side of equation (6)
vanishes.
Also, for many applications, the shear stress term is much less
significant than the gravity term so that to a good approximation the time
average pressure gradient at a point becomes
&P =
(7)
The pressure gradient may thus be determined once the average density
is known.
The latter may be calculated as follows:
Consider a control volume fixed in space extending from the entrance
of a vertical pipe to the middle of one slug (Figure 4 control volume a).
The velocity in the slug well ahead of the tip of the bubble is equal to
the total volume flow rate divided by the pipe area.
If the velocity of
the bubble with respect to the liquid ahead of it is V , then the
velocity of the bubble with respect to the fixed control surface is
Q ,+
A
4+V
p
The time taken by a bubble and a slug to pass a given section of
pipe is
(L + Lb) A
Q Q + V A
Consider now the time average flow rate of the gas phase past a
section.
During an interval L t one bubble will pass.
If vb is the
volume of one bubble, the average flow rate for the gas is
Qg
=
b
(9)
9 7t
The average density of the mixture in the control volume is
af 1-[
v
(10)
+
Combining equations (7), (8), (9), and (10) and using the fact that
v = (L + Lb
p the desired expression is obtained
f Qf + QgfgO
+ V Ap
g Qf + Qg + Vl
p
1
Equation (11) may be applied to the prediction of pressure drop for
fully developed slug flow when the pipe diameter, the flow rates, and
densities of the two phases are specified, provided of course that the
bubble rise velocity V, is known.
At this point the following question must be raised.
Can the results
of Dumitrescu's analysis be used for the rise velocity V, in equation (11)
above?
In order to answer this question one must examine critically the
difference between Dumitrescu's idealized model and the flow system
presently under consideration.
Two basic differences appear at once.
irst, Dumitrescu's analysis applies to a bubble of infinite length,
whereas in slug flow the bubbles have a finite length and are followed by
a wake.
Experiments by uriffith and Wallis have shown that the rise
velocity of bubbles in stagnant water are in good agreement with Dumitrescu
and Taylor's theory. Apparently, therefore, the finiteness of the bubbles
is not an important factor. The hydrodynamics at the nose of the bubble
is the sole factor determining the motion. The same conclusion was
arrived at from the experiments of Laird and Chisholm (ref. 7) with
certain reservations about very long bubbles (over 20 times the tube
diameter).
Secondly, the validity of the irrotational flow assumption must be
questioned.
F'rom the original work of Taylor it appears that the equation
V
=
0.350'VgD
is no longer valid for low bubble-velocity-Reynolds numbers, even when the
liquid is stagnant, in other words when the velocity profile of the liquid
ahead of the bubble is flat.
In addition to this low bubble.-velocity-
Reynolds number effect, Griffith and Wallis observed that V, depended on
the water velocity, or on the shape of the liquid velocity profile.
In an attempt to rationalize both these effects the constant of
Taylor and Dumitrescu was split up into two parts, C1 and C2.
Thus the
expression for the rise velocity is written as
V
=
CICa r
(12)
C, is the governing coefficient in static water. It is a function of
the Reynolds number based on bubble velocity, tube diameter and liquid
10.
viscosity. The relationship between C1 and the Reynolds number NRe is
shown in Yigure 5. This relationship is derived in reference 3 from the
experiments of Dumitrescu and Griffith and Wallis.
C2 depends both on bubble velocity and on the velocity profile of
the liquid.
Its variation has been rationalized by plotting values of C2
against a Reynolds number based on the liquid velocity and with a
Reynolds number based on the bubble velocity as a parameter.
This plot
is shown in figure 6.
Equation (12) together with Figures 5 and 6 provide the final piece
of information needed in order to use equation (11).
Eauation (11) with Ve
given by equation (12) has been applied to the
prediction of slug flow pressure drop.
These calculations are compared
with experimental observations by Schwartz (8), Griffith and Wallis (3),
Govier, et al (17) and Behringer (9) in reference (10).
The agreement
between theory and experiments is quite satisfactory in most cases.
results of Behringer are an exception.
The
all of Behringer' s measurements
show pressure gradients which are considerably larger than those predicted
by the slug flow theory. Behringer' s experimental set-up differed from
those of other investigators in the following respect.
Whereas all
other investigators used a long "calming section" and took pressure drop
data far downstream from the pipe inlet, Behringer measured pressure drops
in the entrance region.
In other words, whereas all other data apply to
fully developed slug flow, Behringer's measurements were in a developing
flow region.
The basic difference between fully developed and developing slug flow
has already been explained in the introduction.
In fully developed slug
flow all bubbles rise with approximately the same velocity.
In developing
slug fiow, on the other hand, each bubble is influenced by the waxe uf the
bubble ahead of it and rises faster.
but some larger value.
The mean bubble velocity is not V
Inspection of equation (11) indicates that an
increase in the value of V, increases the pressure gradient.
The argument given above explains, qualitatively, the discrepancy
between the fully developed slug flow theory and Behringer's experiments.
The purpose of the present investigation is to study the mechanism of
developing slug flow, and to determine how Griffith and Wallists theory
must be modified in order that it will be applicable to developing slug
flows.
12.
III
DEVELOPING SLUG FIOW
A. Introduction
Two basic processes must be distinguished in the description of
developing slug flow. first, the agglomeration of small spherical bubbles
to form G. I. Taylor bubbles.
This is the process of transition from the
bubble regime to slug flow. Second, the agglomeration of two or more
G. I. Taylor bubbles due to wake effects.
Since the mechanics of these two processes are fundamentally
different, each process must be considered as part of a separate investigation.
In the present paper the first process is described qualitatively.
A
possible explanation for the agglomeration process is suggested, and
some recommendations on the direction of future work on this subject are
put forward.
The analysis of the second process - the agglomeration of two
G. I. Taylor bubbles - constitutes the essential part of the present
investigation.
(i) Agglomeration of Small Bubbles to Form U. I. Taylor Bubbles
The agglomeration of small bubbles to form Taylor bubbles takes
place in two stages.
In the first stage bubbles which were originally
dispersed randomly, come together and form a group whose outline is the
same as that of a Taylor bubble.
This stage is seen in photograph C
Figure 7 in the location between 22 and 24 inches.
Also in photograph A
between 16 and 21 inches. The group maintains this shape with each small
bubble retaining its individual character for a finite interval of time.
The second stage is the collapse of the interfaces between the small
bubbles and the formation of a single Taylor bubble.
The collapse of the
13.
interfaces may take place instantaneously for the whole group if the group
is short, or in parts starting from the top if the group is longer.
This
stage is seen in photographs B and D in V'igure 7.
It is clear that the mechanisms of the two stages of the agglomeration
are quite different. The mechanism of the grouping together of the small
bubbles must be determined from a study of the hydrodynamics of the motion
of a bubble in a vertical tube and in the vicinity of other bubbles.
On
the other hand, in order to determine the mechanism of collapse of the
interfaces one must study the dynamics and thermodynamics of the surfaces
and of the film of liquid separating the bubbles.
Of the two mechanisms just mentioned the first one appears to be the
most difficult one to describe analytically.
Inertial and viscous forces
are equally important which means that the Navier-Stokes equations must be
written down in their most general form.
In addition, the presence of
other bubbles makes the boundary conditions extremely complicated.
The
boundary conditions are further complicated by the effects of surface
tension.
The problem is simplified somewhat if one assumes that inertia forces
are negligible compared to the viscous forces, in other words if one
assumes Stokes flow.
The slow motion of two or more spheres through a
viscous fluid has been analyzed recently by Kynch (ref. 11) and by
Hasimoto (ref. 12).
Although it is doubtful whether the Stokes flow
assumption is permissible in the present problem, it is interesting to
observe that the results of Hasimoto may, in principle, explain the
process of agglomeration.
Hasimoto obtained an analytical solution to the problem of the motion
of a cubic array of spheres through a viscous fluid.
The following inter-
esting result comes out of his calculations. The velocity of a lattice is
14.
a function of the concentration of spheres in the lattice.
Those lattices
which are more closely packed move slower than lattices of lower concentration.
The velocity-concentration, curve-is rather flat for low concentrations,
but its gradient increases exponentially beyond a critical concentration.
It is quite simple to develop an argument which explains the agglomeration process using Hasimoto's results.
in a fluid.
Consider an array of spheres rising
If, due to some disturbance, part of the array becomes more
packed than the rest, that part moves slower. At once some of the following
bubbles reach the slow moving section, and due to their inertia become part
of it.
The array becomes still more packed, moves even slower, and the
process repeats itself until the bubbles agglomerate.
It must be stressed however, that the argument given above must not
be taken as anything more than an interesting observation.
The notion of
a lattice is not strictly applicable to the motion of bubbles in a tube.
In addition, the argument ignores the presence of the walls, which undoubtedly play an important role in the agglomeration process.
The second stage of the agglomeration process, namely the collapse
of the interface between the small bubbles, appears a little easier to
describe quantitatively.
The group of bubbles is in a state of metastable
equilibrium since the total surface energy of such a system is much larger
than that of a Taylor bubble.
The collapse is delayed because a finite
interval of time is needed for the film of water separating the bubbles
to be removed.
An analysis of the dynamics of that film should provide a
rate constant for the collapse process.
cinematographic observations.
This can then be compared with
The one difficulty is in the definition of
the correct boundary conditions for the film.
It appears, however, that
the problem is solvable, and should be the obvious extention of the present
investigation.
15.
(ii) Agglomeration of u. I Taylor Bubbles
The problem which the present investigation seeks to solve may be
stated as follows:
Given a slug flow pattern at the entrance of a pipe,
establish the process of flow development along the length of the pipe.
Hence calculate the pressure gradient as a function of distance from the
inlet.
In order to solve this problem one must study the kinetics of a
bubble as it rises following a series of other bubbles.
The rise velocity
of a trailing bubble is influenced by the velocity profile of the liquid
behind a leading bubble.
The velocity profile in the wake is in turn
influenced by the velocity and the length of the leading bubble.
In order
to establish the kinetics of a trailing bubble, one must determine the
following.
a. The velocity profile of the liquid behind a leading bubble.
b. The rise velocity of the trailing bubble in a liquid with
specified velocity profile.
The problem of the velocity profile in the wake behind a bubble is
considered in part B of this chapter.
That of the rise velocity of the
trailing bubble is treated in part C.
B. Velocity Profile Behind a Taylor Bubble
(i) Theory
Consider a Taylor bubble rising in a vertical tube with velocity V.
The bubble may be made stationary by superimposing a downward velocity V
to the liquid and to the tube walls.
The problem is then equivalent to
that of a jet of water entering a circular pipe through an annular orifice
while the pipe walls are moving in the same direction as the jet.
It is clear that the problem as stated above is a steady state
problem provided that the bubble velocity V is not a function of time.
In
16.
the actual situation V is indeed constant with time provided that the
bubble is uninfluenced by wake effects; in other words, provided that the
leading bubble is not itself trailing some other bubble at short separation
distance.
The steady state assumption will be made in the present analysis.
The
complications arising when V is a function of time will be discussed at a
later point.
Even with the steady state assumption, the problem at hand is a
complicated one.
It is a combination of a turbulent jet problem and a
boundary layer problem. Of the two, the boundary layer problem is a
formidable one because of the complicated free-stream conditions.
A hint for a solution is offered by some evidence that the core mixing
process is far more rapid than the boundary layer growth.
Measurements of
velocity profiles in an annular jet ejector by Reid (ref. 13) show that
the leveling of the velocity profile in the core is almost complete before
Another piece of evidence
boundary layer effects become noticeable.
supporting the idea of neglecting boundary layer effects is offered by a
simple experiment carried out by the writer. A plastic bubble is placed
in a plexiglass tube and water is run downwards.
introduced from the bottom of the tube.
A Taylor bubble is
The water flow is varied so that
the air bubble is stopped at various distances from the plastic bubble.
A curve of water velocity against separation distance is thus obtained.
At small separation distances the water velocity decreases rapidly as the
separation distance is increased. The curve levels off at a separation
distance of about 10 tube diameters.
This experiment was carried out with cold and with hot water.
The
viscosity differed in the two experiments by a factor of 2. The results
show that the velocity-separation distance curves are identical for the
17.
two conditions in the first 10 diameters from the plastic bubble, but
differ somewhat at larger separation distances.
This seems to indicate
that boundary layer effects become important only at large separation
distances.
On the basis of these considerations, the assumption that boundary
layer effects are negligible is made in the analysis.
In the actual
situation this assumption is justified further by the fact that the
relative movement of the walls effectively controls the growth of the
boundary layer.
The problem may then be formulated as follows:
Consider a jet of
fluid entering a pipe through an annular slot adjacent to the pipe wall.
Take the origin of coordinates x = 0 r = 0 at the pipe inlet and at the
center of the pipe.
Let D be the pipe diameter and d-the diameter of the
core.
For an incompressible fluid, assuming that boundary layer effects
are negligible, that the flow is symmetric about the x-axis, and that
steady state conditions exist, the governing equations of motion are
a
F
+
=0
x
(15)
Equations (13) and (14) are the momentum equations in the x and r directions
respectively.
Equation (15) is the continuity equation.
u and v are the
18.
components of velocity in the x and r direction whereas p stands for pressure
and F for body forces.
The exact solution of these equations for the case of turbulent flow
is not possible, at least at the present time.
One must resort, therefore,
to one of the semi-empirical theories of turbulence.
The singular success of Reichardt's inductive theory of turbulence
when applied to jets and other problems where there is a principal flow
direction, suggests that his theory may be applicable to the present problem.
This theory has been applied successfully to problems of free jets, jets
discharging parallel to a wall, and Jets discharging at the center of a
duct, by a group of investigators at the Engineering Experimental Station
of the University of Illinois (ref. 15).
The approach to the present
problem is the same as that adopted in reference 15.
The theoretical reasoning of Reichardt's approach is described by
Schlichting (ref. 14).
After a critical examination of experimental data
on turbulent. flows, Reichardt discovered that the velocity profiles under
consideration could be approximated very successfully by Gauss' error
function.
Starting with this premise he attempted to cover all cases of
free turbulent flow with the aid of a simple set of formulae instead of
endeavoring to solve the differential equations such as (13), (14), and
(15) above.
His theory is based on the momentum equation for the time-
averages of velocity in the principal direction of flow, that is of
equation (13) above.
LL.4 J_
r
31
--
l---(16
-r.1
In equation (16) a function w denotes
I
19*
w = Lim
{
t0 + t
pdt
t
0
Body forces have been omitted, but can always be superimposed to the final
solution, if that should be necessary.
It must be stressed that the symbols in equation (16) represent total
quantities.
Vor example,
U=
U + U
where u is the mean component of velocity and u' the turbulent fluctuation
component.
The second basic equation in Reichardt's theory is of an empirical
nature and has the form
uv
The parameter
A
=
is a "momentum transfer length."
(17)
It has the
dimension of length, is assumed to be a function of x only, and must be
determined empirically.
stress.
The product uv represents the turbulent shearing
Thus equation (17) states that the flux of x-component of momentum
which is transferred in a transverse direction is proportional to the
transverse gradient of momentum. One may observe the similarity of
equation (17) to Fourier's law of heat conduction, which relates the heat
flux to the temperature gradient.
Combining equations (16) and (17) and rearranging:
20.
Assuming that the pressure gradient in the transverse direction is of
small order compared to 3 /(:
, one may add
)
/ar
to the above
equation without disturbing its validity
Setting m =
+
u
Then
_ /A
<o
31
ram!
0
3rL 3rJ
In order to simplify the boundary conditions it is convenient to define a
function M as follows
m - MO
m -m a
c
where subscripts c and a refer to the core and annulus at x
m
0 respectively.
is the value of m as x approaches infinity.
Defining further a dimensionless radial distance
r
2r
*
=
D
the final problem formulation is reduced to the equation
-a-
---
-(g
r
D13r*
01
3c
Together with the following boundary conditions:
= 0
at
r* =
0
(a)
21.
This is the symmetry condition
= 0
(b)
at r* =1
3r
Since in the absence of friction no momentum is transferred to the wall
M =
0
oo
x -..
at
(c)
This condition results from the definition of M.
Finally at x = 0
m - M
M = me
Mc
d/D
for 0 4 r* (
ao
ma
(d)
M =
M -m
c
a
Mc
a
for d/D<
1
r*
It is assumed here that the velocity profile is flat at the inlet to the
tube.
The problem described by equation (16) together with boundary
conditions a, b, c, and d is analogous to a transient heat conduction
problem. The analogous problem is that of the transient temperature
distribution in an insulated, infinitely long cylinder which at time
x = 0 has a specified temperature distribution.
The conductivity
A
of
this cylinder is a function of time.
The product solution to equation (18) may be written as
x
M
A
A
((K
r
B
2.I
d-
Kr')
The constants A and B and the eigen values Kn are determined by application
of the boundary conditions
22.
(a)
B = 0
gives
AKJI(K) = 0 hence Kn are determined as the zeroes of J,
(b) gives
(c) is satisfied automatically
(d)
may be satisfied by the orthogonality of the Bessel series.
In this
manner An is obtained
2
An
Ji(K
d)
KnJ2 (Kn
no
n
so that the desired solution is as follows
x
The function
empirically.
A (x)
which
appears in the exponent must be determined
What is required is a set of experimental values for the
momentum function M. The function
A(x)
should then be defined in such
a manner that experimental and theoretical values for M will be identical.
Experiments were carried out in order to measure M. These are
described in the following section.
(ii)
Measurement of Velocity Profiles in the Wake of a Plastic Bubble
The apparatus which is used in order to investigate the velocity
profile in the wake of a Taylor bubble is shown in schematic form in
Figure 8.
Since it is virtually impossible to measure velocity profiles behind
an air bubble which is rising in a liquid, the situation is simulated by
placing a plastic bubble inside a 2 inch tube in which water (at 700F)
23.
flows downwards.
This model represents the actual situation quite closely. The one
significant difference is the boundary condition at the wall-liquid interface. However, it is expected that this difference will not alter the
character of the velocity profile in the core, at least at small distances
from the bubble.
The second apparent difference, namely the fact that the bubble has
solid walls instead of a gaseous surface, is quite insignificant,
according to some preliminary measurements.
(An air bubble was allowed
to become attached at the flat surface of the plastic bubble.
The
velocity profile behind the real bubble remained identical to the profile
behind the plastic.)
Velocity profiles are obtained by means of a total head probe which
is traversed radially across the tube.
measure static pressure.
A wall-tap is used in order to
A wall-tap is preferred to a pitostatic tube
because it is suspected that secondary flows may make the reading of
static pressure in the stream inaccurate.
For ease of construction the plastic bubble, rather than the pitot
tube, is moved axially along the tube.
In this manner variations in the
hydrostatic component of pressure are eliminated.
The water flow is induced by means of a pump driven by a 1 h.p.
motor.
Variations in the flow are minimized by using a constant head
tank at the pump supply.
Pressure measurements are taken on two inclined tubes, with water as
the manometer fluid. The manometer tubes are connected through their
upper ends to a bottle which is pressurized by raising a column of mercury
against it.
In this manner, indicated pressures are kept within the
reading scale.
24.
Mean flow rates are measured to a good accuracy by means of a weigh
bucket.
Continuity of the measurements is established by comparing the
weighed water flow to the flow rate obtained by integrating the velocity
profiles.
The fact that continuity is satisfied from measurement of the
total pressure (p + 1 Du2 ) in the stream and of the static pressure (p)
at the wall, indicates that the variation of the static pressure in the
radial direction is not significant.
Reproductibility of the measurements is checked by taking readings
along a diameter at both sides from the center of the tube.
Measurements of static pressure and total pressure were taken at
distances between bubble and probe of 0, 0.5, 1, 2 . . . 10 tube diameters.
The results for the velocity profiles are shown in Figure 9. It should be
noted that no points are shown for the axial position of zero.
The reason
for this is that measurements at that location do not satisfy continuity.
This is not surprising, since the static pressure at that location must
vary appreciably along the radius.
The dotted line shown in figure 9 is
calculated from continuity considerations alone.
All other profiles shown in Aigure 9 satisfy the continuity condition
of 100 lbs of water per minute within an accuracy of 5%.
In order to calculate the function M of equation (19) a knowledge of
the variation of the static pressure with x is needed. This variation is
shown in figure 10.
The results of Figures 9 and 10 supply the information needed in
order to calculate the momentum transfer length. The calculation of
is described in the next section.
A
I
25.
(iii) Calculation of Momentum Transfer Length
There is no systematic method of determining the momentum transfer
length.
Point by point application of equation (19) is an extremely
The problem would be simplified greatly if equation
tedious procedure.
(19) were one-dimensional.
The problem may be reduced to a one dimensional one by defining a
function which is integral in r*.
The integral of M itself is not suitable
since, by continuity, it is always zero. The integral of M2 however is
not zero.
Hence, it may be used to define a one-dimensional function I as
follows
=2
21T r*dr*
(20)
0
Combining equations (19) and (20)
Now all cross-product terms of the squared summation are zero because of
the orthogonality of the Bessel series.
That is, all terms of the form
5
J (K r )J(Kir )dr* = 0
0
when j i i, whereas
J J(Knr*)dr*
0
=
$
J(K) + Jj(Kn
26.
Using these equations, together with the fact that Ji(Kn) = 0 by the
boundary conditions of the problem, the mixing parameter I is obtained
I
(21)
____5;7_
At this point it is convenient to write A
(x)in
some explicit form.
Again following the example of reference 15, the function may be written
as follows
A
=
(22)
c2
2 (1 +)
The reasons for selecting equation (22) for the momentum transfer
length in reference 15, for the case of jet centered at the inlet of a
duct, may be summarized as follows:
In the case of free jets, the transfer length has been found to be
of the form.A = c2 x. In the case of a ducted jet, one expects the
turbulence 'pattern to be similar to that in free jets in the immediate
vicinity of the nozzle. On the other hand, in a duct, A
cannot grow
indefinitely, as in a free jet, because of the presence of the wall.
However, with wall friction effects ignored, as in the present analysis
as well as in reference 15, it
seems reasonable that the presence of the
wall should have no other effect on
its growth.
The function
A
A
than to place an upper limit on
given in equation (22) is a purely arbitrary
function that reduces to the free jet relation as x approaches zero, but
which imposes on
diameter.
A
an upper limit which is a function of the duct
27.
Substituting
A
from equation (22) into equation (21) and carrying
out the integration in the exponent, the function I becomes
00
4K(
DPL-IP?
(23)
For any set of values b and c, equation (23) yields the variation of
I as a function of x only. The task of comparing this analytic function
with experimental results is now a relatively simple one.
kigures 9 and 10 give all the information needed in order to calculate the distribution of measured values of m and M at any cross section
of the duct.
The value of I corresponding to any position x may be found
by performing graphically the integration
1
J
M2 - 21 r*dr
0
The results of such graphical integrations are shown in Figure 11.
It must be noted that values of I shown in iigure 11 are not obtained by
direct use of the raw data of Figures 9 and 10.
for compatibility with
the assumptions of the theory, the raw data have been normalized as
follows.
First, the small boundary layer effects seen in Figure 9 have been
eliminated by assuming the velocity profile to approach the wall along a
tangent drawn to the point of maximum velocity. The normalized velocity
profiles are better suited to the rising bubble than the profiles of
Figure 9.
As has been mentioned earlier the relative motion between wall
28.
and bubble effectively controls the boundary layer growth.
Secondly, the profiles of Figure 9 do not satisfy continuity exactly
but are in error by about 5%.
the measuring instrument.
This error is partly due to inaccuracies in
The more likely reason for the error is the
variable scale of turbulence that exists in the tube.
A total pressure
probe placed in a turbulent flow records the following effects
Ptot
p
=
p
static
2
+u
++
2
p (7+2)
In the interpretation of the data in the present experiment, the
second term on the right hand side has been neglected.
Available data
for free jets show that u'2 is of the order of 0.04 ~UT (ref. 16).
indicates that the assumption that u
2
This
= 0 alone -may well explain the
reasons for the discrepancy in the velocity data. Since it is essential
that the data used to calculate the length
A
should satisfy continuity,
all velocity readings were normalized so that all profiles- yield a total
The normalizing was carried out by multi-
flux of 100 lbs per minute.
plying all velocity readings at a section by a constant factor.
The problem of determining the momentum transfer length
A
is now
reduced to selecting values of the constants b and c such that equation
(23) will yield the same curve as the one shown in Figure 11.
Figure 12
shows the curve obtained from equation (23) with the following values for
the constants
b =
c
=
100
1.07
Since comparison of the theoretical curve with experimental points
is excellent, it may be concluded that the expression for the momentum
transfer length in an annular ducted jet is
29.
A
l072
=
(24)
2(1 + 1O)
(iv) Discussion of Results
Equations (19) and (24) constitute the solution to the first of the
two problems which must be solved in order to establish the kinetics of a
trailing bubble.
With the velocity field ahead of the- trailing bubble
specified, one can at least formulate the problem of the- bubble motion.
As will be seen in a later section the analytical solution to this problem
is extremely difficult. Nevertheless, inspection of equations (19) and
(24) can reveal the important parameters which influence the motion and
indicate which quantities should be controlled in experimental
investigations.
Inspection of equations (19) and (24) reveals the following points.
The momentum flux M depends only on the length to diameter ratio
x/D and on the ratio of the core-to-tube diameters d/D
The fact that the momentum flux M depends on x/D and not on D
directly, suggests that the rise velocity of a trailing bubble is a
function of the separation distance x/D.
The tube diameter D should not
be an independent parameter, at least in the range of applicability of
the present theory.
The dependence of M on d/D suggests a dependence of the rise velocity
of the trailing bubble on the length of the leading bubble. It is shown
in reference 4 that the ratio d/D varies as the square root of the bubble
length. The magnitude of the dependence of the velocity of the trailing
bubble on the length of the leading one is not revealed from equations
(19) and (24) alone. The direction is certainly indicated.
leading bubble corresponds to a large d/D.
A long
This in turn corresponds to
§*MMMN6WM1WAOMWW
30.
a large M and consequently to a higher rise velocity.
In the range of applicability of the theory, the magnitude of the
total throughput velocity should not affect the rise velocity of the
trailing bubble.
This conclusion follows from-the fact that in a friction-
less system, the throughput velocity, (Qf + Q )/A, is simply superimposed
on the entire system; that is, on the leading and trailing bubbles as well
as on the liquid. This argument breaks down at large separation distances,
where boundary layer effects become important.
In that region the rise
velocity is a function of the total throughput velocity.
(See Chapter II
and Figure 6.)
The analysis presented in the previous sections is based on the
assumption that the rise velocity of-the leading bubble is a constant.
At this point it is important to examine the implications of that
assumption.
If the leading bubble is not itself influenced by wake effects, its
In that case the conclusions
velocity will be constant and equal to V
drawn from the analysis are valid.
The rise velocity of the trailing
bubble at short separation distances will be a function of x/D, and d/D
only. It can be expressed in a plot of Vb/V, against x/D, possibly
with length of leading bubble as a parameter.
In the case of a number of bubbles rising at short separation
distances from each other, as may be the case in developing slug flow,
Consequently its
a "leading" bubble is itself trailing another bubble.
velocity is not constant with time.
The rationalization of the variation
of Vb is not simple as in the case of constant V .
Since an analysis of
the transient state problem is not feasible, one must resort to some
idealizing assuimtions.
31.
Two possible assumptions may be made.
in V
The first is that the variation
is negligible and that the rise velocity of a trailing bubble can
be expressed in a plot of Vb/V, against x/D, with V considered as
T
constant. The second is that the variation in the velocity of the leading
bubble (VL) is immediately felt by the trailing bubble.
In that case one
would expect the rise velocity of a trailing bubble to be expressible in
a plot of (Vb ~ L)/V
,
against x/D, with VL considered variable.
It is clear that neither of these assumptions is exactly true in the
case of developing slug flows.
The first one is obviously incorrect when
the "leading" bubble is about to agglomerate with a bubble ahead of it.
The second assumption neglects the important factor of time lag.
If both
leading and trailing bubbles are accelerating, the velocity profile "seen"
by the trailing bubble at a certain instant corresponds to the velocity of
the leading bubble not at that instant, but at-a-small fraction of time
earlier.
Consequently, the rise velocity of the trailing-bubble at the
certain instant is not characteristic of the separation distance at that
instant, but of the separation distance which existed a small fraction of
time earlier.
This "time lag" effect is observed most clearly in the following simple
experiment.
When a Taylor bubble rises in a vertical tube which is open
at the top, it normally rises with a uniform velocity up to the point where
it reaches the free surface.
If two bubbles rise following one another,
the second one accelerates as it approaches the first.
This acceleration
of the trailing bubble and the characteristic distortion of the same,
continues even after the leading bubble has left the tube.
As a result of
the time lag, the trailing bubble appears to be influenced by the wake of
a bubble which no longer existal
32.
Of the two assumptions discussed here, the first one seems to be more
suitable if the theory is to be applied to a developing slug flow.
The
results of the analysis indicate that the rise velocity of a trailing
bubble is not too different from V
less than 1 diameter.
unless the separation distance is
It is only rarely that two consecutive bubbles will
both have slugs of liquid less than 1 diameter long ahead of them, at the
same instant. Furthermore, the second assumption implies that in order to
find the rise velocity of a bubble Vb one must add the value of (Vb
-
VL
which is a function of separation distance to the value of VL at that
instant.
In the case of a developing slug flow, this superposition of all
VL's on all VbIs means that if two bubbles agglomerate somewhere in the
tube, so that (Vb
-
L) increases rapidly at that point, the entire column
of bubbles is also accelerated, no matter how long the column is. This
is clearly not consistent with actual observations.
The conclusions of the analysis may now be summarized as follows:
1. When two bubbles rise in a tube, the rise velocity of the
trailing bubble is expected to depend on the separation
distance and on the length of the first bubble only.
2. In the case of a series of bubbles rising in a tube, the
rise velocity of a particular bubble depends on the separation and length of the bubble ahead of it, but also, in some
way, on the velocity of the leading bubble.
However, in
most cases, it is reasonable to assume that the velocity of
the leading bubble is constant.
The analysis which is presented in this section is useful to the
general investigation in two respects.
First, the conclusions just sum-
marized are valuable in the planning of experiments to determine the rise
velocity of a trailing bubble.
Second, the knowledge of the velocity
330
distribution in the wake is important in studies of the process of
agglomeration of small bubbles which follow a Taylor bubble.
C. Rise Velocity of Trailing Bubble
(i) Theoretical Considerations
The results of section B define the velocity field behind a leading
bubble.
This information is essential to the complete formulation of
the problem of the rise velocity of the trailing bubble.
Although the problem is in some respects similar to the one solved
by Dumitrescu (see Chapter II), it is a far more complicated one.
The
solution of Dumitrescu's problem was made possible after two basic
assumptions were made:
that the bubble had infinite length, and that
the flow ahead of the bubble was irrotational. Whereas the first of
these idealizations could well be made in the present problem, the second
one is clearly not acceptable.
The fact that the flow is not irrotational makes an analytical
approach to the problem well neigh hopeless.
In order to determine the
rise velocity of the trailing bubble one must therefore resort to
experiments.
The analysis of section B has revealed the important parameters that
determine the bubble motion.
One can therefore plan some simple experi-
ments, which will establish the relationship between rise velocity and all
the pertinent quantities.
These experiments are described next.
(ii) Measurements of Rise Velocity of a Trailing Bubble
The apparatus used in order to determine the rise velocity of a
trailing bubble as a function of separation distance, is shown schematically
in Figure 13.
It consists of an 18 ft. long lucite glass tube mounted on
a vertical board.
The board is painted in black with while-line graduations
I -
34.
at 0.1 inch intervals.
Air and water inlets are at the bottom of the
tube. The air inlet tap is brought into the test section in an inclined
downward direction.
As the air is blown downwards, a larger bubble is
formed before it begins to rise.
In this manner one obtains slug flow
almost at the tube entrance.
A movie camera is mounted on a vertical track facing the test section.
The camera may be moved up and down the track by means of the crank, wire
and pulleys arrangement shown in Figure 13.
and test section may be varied.
The distance between camera
The section of the tube just opposite
the camera is lit by means of four flood lamps which are connected to the
camera platform.
The range of view of the camera is marked on the tube
by two markers which are also attached to the camera platform.
As the two phases flow upwards in the test section, a film is taken
of two consecutive Taylor bubbles as they rise in the tube and agglomerThe rise velocity of the trailing bubble as a function of separation
ate.
distance is found by reading the film.
The position of the top and bottom
of the leading bubble and of the top of the trailing bubble at each frame
of the film are noted.
Since the camera speed is calibrated, a curve of
bubble position against frame number is equivalent to a distance-time
curve.
Velocity is obtained by differentiation of the latter curve.
Air flow is measured by means of a sharp-edge orifice which is
calibrated by the soap-bubble technique.
The camera speed is calibrated
by taking a film of a Taylor bubble as it rises in stagnant water with a
known, uniform velocity.
Water flow is measured by means of a weigh
bucket.
There are two limitations in the set-up.
The task of keeping the
camera in line with the bubbles by turning the hand-crank is a difficult
one at high bubble speeds. The maximum bubble speed is limited to
Ism.
'35.
2 ft/sec for this reason.
The second limitation concerns the maximum ratio of Qg/Wf. As this
ratio is increased, the number of small bubbles in the slugs increases.
The presence of small bubbles makes the film reading difficult,
particularly in identifying the bubble' s tail.
In the experiments, the first tube tested was a 1 inch diameter one.
Measurements were taken for nine different combinations of air flow Qg,
and water flow Qf. a 1 plot of Vb/ V
against dimensionless separation
distance L D, for the 1 inch diameter tube, is
shown in Figure 14.
The form of the average velocity separation distance curve is just
like that predicted by the results of section III B. The resemblance in
shape of Figures 12 and 14 is quite marked.
Two points must be discussed in connection with the results of
Figure 14:
first, the reasons for the nather large scatter in the data;
second, the effect of length of leading bubble which is predicted in
section III B.
Part of the scatter in the data may be due to experimental inaccuracies.
Variation of the air flow rate is one possible reason.
Inability to define
exactly the bottom of the leading bubble in the film is another.
It should
be remembered that the velocity Vb is, in many cases, a relatively small
number which is obtained by subtracting one large number from another
(Vb = velocity with respect to the ground minus total throughput velocity.)
Any percentage error in Q
error in Vb'
will therefore result in a far larger percentage
The inability to define the bottom of the bubble is another
effect which is magnified by the presentation of Figure 14.
What may be
nothing more than a 2 to 3% error in T/D may appear as a 40 to 50% error
in Vb/V 0
Both of these errors can possibly be eliminated by taking great pains
36.
to keep the air flow absolutely constant and by refining the photographic
techniques. No such steps were taken, however.
The reason for this is
that it became evident in the course of the experiments that the main
reason for the scatter did not lie in experimental inaccuracies, but in a
physical instability of the bubble motion.
It has been mentioned earlier that a bubble rising in the wake of
another, loses the stable profile which is characteristic of Taylor
bubbles.
Instead of assuming a smooth steady shape, the nose of the
bubble distorts, becomes alternately eccentric on one side or another, and
leans over to one side of the tube.
The rise velocity changes quite
randomly as the shape changes, being higher the greater the distortion of
the nose.
As a result of this instability the rise velocity at a given
separation distance shows considerable variation from one experiment to
the next.
The distance-time curve is not uniform but "wavy" in any one
experiment.
No distinct effect of leading-bubble length was observed in the
experiments.
The length of leading bubbles varied from 2 to 15 diameters.
Some of the high points shown in Figure 14 correspond to the longer
leading bubbles.
The trend was not sufficiently clear to call for a
separate correlation.
It is probable that the leading bubble length
effect is "buried" in the scatter of the experimental data.
The dotted curve in Figure 14 may be considered as a good representation of the mean velocity-separation distance relationship.
The scatter
is statistical. As expected from the theory of section B, Q and Q do
not affect the rise velocity.
After the experiments with the 1 inch tube, measurements were taken
with tubes of sizes 3/4 inch, 2 inches, and 1/2 inch internal diameter.
The results for these tubes are shown in Figures 15, 16, and 17 respectively.
37.
Figures 15 and 16 are essentially similar to figure 14.
The trends
are the same and the scatter in the data is of the same order of magnitude.
Points for the 1, 3/4, and 2 inch tubes are collected in Figure 18.
Inspection of Figure 18 shows no apparent difference between the 1 inch
and 2 inch tubes.
Values of V /V
for the 3/4 inch tube tend to be lower,
but not appreciably so.
The results for the 1/2 inch tube, on the other hand, fall quite
apart from the other three tubes.
tube than in the other three.
The wake effects are shorter in this
The obvious reason for this is that whereas
bubble-velocity Reynolds numbers for the three larger tubes are well in
the turbulent zone, in the case of the 1/2 inch tube the Reynolds number
is of the order of 2000.
Viscosity effects, which were apparently small
in the larger tubes, are no longer negligible in the case of the 1/2
inch one.
The relationship
V/V
=
1 + 8 exp (-1.06 L/D
shown in Figure 18, represents the mean velocity-separation distance
relation for the larger tubes.
(iii)
Discussion of Results
The analytical results of section B and the experimental measurements
of the present section are quite compatible.
The decay of the dependence
of rise velocity on separation distance is of the same character as the
decay of the wake effects.
The bubble rise velocity, corresponding to
various shapes of the liquid velocity profile, is shown in 1igure 19.
The fact that results for the three larger tubes are essentially the
same, whereas those for the 1/2 inch tube differ, shows that Reynolds
number effects in the wake are small, provided that the Reynolds number
38.
is not too low.
The choice of Vb/V,
and L.yDas coordinates, with V
constant, appears to be a good one.
This does not answer in any conclusive
manner the question raised in Section B on whether V /V
is a more suitable coordinate.
situation in which (Vb ~ L/
considered a
or (Vb
-
L
O
It must be remembered that the only
is different from Vb/\o
- 1 is when the
leading bubble is itself about to agglomerate with the bubble ahead of it.
This is a rather rare situation, and it never appeared in the 3500 feet
of film taken.
The results of sections B and C are the main pieces of information
needed in order to solve the problem stated in section A. Calculations
of the flow pattern and the pressure drop in a developing slug flow are
given in the following section.
39.
D.
Pressure Drop in a Develoing Slug Flow
(i) Applicability of Uriffith and Wallis Theory
The pressure drop for fully developed slug flow may be predicted by
means of equation (11).
The question rises how that equation must be
modified in order that it will be applicable to a developing slug flow.
If one reconsiders the control surface used in Griffith and Wallis'
analysis (Figure 4 control volume b), the momentum equation for the surface
is equation (6) regardless of whether the flow is fully developed or
*
developing
wAw = fA (V22
w
pag(Ls + L )Ap
(Pi - p 2 )A - pb
p
-
(6)
V2)
For the case of developing slug flow the statement that the input
and output momentum fluxes are identical is not strictly true.
Never-
theless, experimental observations of Laird and Chisholml(ref. 7) have
shown that the liquid momentum term
A(Vj
-
V2) is negligibly small in
comparison with the hydrostatic term. In addition, for not too high
velocities, the wall friction term may be neglected.
The pressure drop equation is
again
(7)
=
A
or, following the argument of Chapter II
Q + VAQ
g
*
g f
Q
Q
bp
+ g
+
(25)
This, of course, is only true as long as the control surface retains
its identity. As soon as two bubbles agglomerate, the control surface
must be redefined.
40.
where Vb is the bubble veloeity at the section under consideration.
Combining equations (11) and (25) the following relation is obtained
-fQ
f
+ Q
~
+ VbAR
+ Q + V A)
x
=
R
(26)
x
This relates the density (which is the same as the pressure drop
within the limitations of equation (7)), at any position x to the density
as x approaches infinity, in other words as the flow becomes fully
developed.
Equation (26) may be applied to the prediction of pressure drop in
a developing slug flow, provided that the variation of Vb in the entrance
region is known.
The information which is needed in order to calculate
the development of an initially specified slug flow pattern, is contained
in sections B and C of this chapter.
The next section describes the method of calculating a developing
slug flow pattern.
(ii) Calculation of Developing Flow Pattern
Consider a series of Taylor bubbles rising in a tube.
Suppose that
the position and length of each bubble are specified at some initial
time t = 0. The distribution of the bubbles at t = 0 may be quite random.
Now, provided that the bubble-velocity Reynolds number is sufficiently
high, the instantaneous rise velocity of any bubble is given by the
relationship
=
where L
1 + 8 exp
-1.06 L /D
(27)
is the length of the slug, ahead of the bubble under consideration.
The constant V
is determined by the methods of Chapter II
41.
With the aid of equation (27) the trajectory of each bubble at t may
be determined. A finite difference procedure is required.
A bubble with
smaller L will travel faster and eventually will reach the bubble ahead
of it.
When two bubbles agglomerate a new bubble is formed whose volume is
approximately equal to the sum of the volumes of the two.
The length of
the new bubble may then be obtained by using one of equations 28a or b.
= 1.82 vV(ApD ) for
L/D
Lb/D =
1.095 vb/(A D,)
P
vb/(APDP)
+ 0.57
0.550
vb/(APD ) >
for
0.550
(28a)
(28b)
Equations 28 are formulae which have been fitted to the bubble shape
calculated by Dumitrescu.
Strictly speaking, they are valid only in
potential flow.
As already mentioned, the method described above may be applied to
a perfectly general initial configuration.
As
part of the present
investigation, the method has been applied to the particular initial
situation described below.
Bubbles of a specified, uniform volume are assumed to enter a tube
at uniform time intervals, A to.
The first bubble enters at time t = 0.
At that instant the tube is filled with a liquid, which may be stagnant
or may be flowing in an upward direction.
Since the first bubble has no wake before it, it will rise with a
uniform velocity. The magnitude of V
for any particular set of
conditions is calculated by means of equation (12).
top of the first bubble at time t, =
kinematic equation
At
The position of the
is found by application of the
42.
X
=
(Qf + Qg)/A + V
The double subscript in X
is used in order to specify the number
of the bubble n and the time interval referred to.
The length of the bubble L
is found from equation (28).
The
bottom of the bubble is at
X1 - Ll,
Y11
If Y
is greater than zero, the second bubble appears at the inlet.
Its position is given by
1
0
The velocity of bubble 2 is found from equation (27) using the fact
that
L s21
11 - X21
If, on the other hand, Y11 is smaller than zero, the second bubble
is assumed to agglomerate with the first.
The new value of L
is found,
as usual, by equation (28).
It is easy to see that when this procedure is extended to a large
number of bubbles, the developing flow pattern is obtained.
The only
point which needs some explanation is the process of "starting" the
calculation when Y11 , Y12 ' Y1 3 *
.
all come out as smaller than zero.
The calculation can still be carried out. Equation (28b) is such that
if t is increased sufficiently Yit will eventually become positive.
In principle the calculation is trivially simple.
however, it is extremely tedious.
In practice,
kfortunately, this type of problem is
particularly suitable for programming on an electronic computer.
In order to obtain results of a sufficiently general nature, the
pertinent variables must be put in a dimensionless form.
The following
43.
dimensionless groups have been selected.
X
= X/D
= dimensionless vertical distance
V
=V /V
=
V* =
Q/(AV
V,=
Q/(A V)
p
Sg
f
p
Lvt
)
=
gas flow rate
=
liquid flow rate
oD
p
time interval
Dp
"
D=
bubble volume
Lb* =
Lb
v
=
oD
rise velocity
"
=
Lb/DP
ecpa
=
bubble length.
"
These are compatible with all thea
*
*
*
g
Vb
VV* =
g
ppropriate relations
+
Vf)t
v/Att
*1*
and so on.
In terms of the dimensionless notation, there are only three
*
*
independent variables.
The absolute velocity (V + Vf + 1); the initial
bubble period At*; and the initial bubble length L .
Calculations may
be carried out for several combinations of these three variables.
Before presenting the results of the computer calculation it is
worth while to discuss some simple results that may be obtained from
continuity considerations.
*
Assume that the initial volume of bubbles is v
frequency N
.
and the initial
Letting v5 and N denote the bubble volume and frequency
when the flow is fully developed, and assuming that N
is uniform, con-
tinuity requires that
*
V=
g
**
v N
0 0
=
*
v
oo
*
N
oC
(29)
44.
But
N
(Vf + V + Vb)/(LB + Lb)
1/t* =
=
Hence
Va-
N0
(V + V + V )(L
V0
N
(V + V + 1)(L
If one writes L*
g0
f
in terms of v
so
+ L
)
(30)
+L )
b
from (28b),
and combines
equations (29) and (30) the following relation is derived
=
(0.913 L*M + 0.526)
0
q/(l - q)
(31)
where
q = 1.095 V* V*,+ Vg + 1)
Equation (31) gives the volume of the bubbles in fully developed
flow in terms of the slug length and quantities which are .specified.
If
fully developed flow is defined arbitrarily by the condition that
VJV
= 1.05, then Figure 18 gives the corresponding value of L
Equation (31) with L
= 6.
= 6 becomes
=
6q/(1 - q)
(32)
Inspection of equation (32) shows that v. approaches infinity as q
approaches 1. Consequently, if q > 1, fully developed slug flow is not
possible.
The ultimate flow regime for a developing slug flow with
q ') 1 must be annular flow.
45.
(iii) Results of Computer Calculations
The calculations reported in this section were carried out on an IBM
704 computer at the MIT Computation Center,
out for a total of 34 sets of conditions.
Calculations were carried
The conditions were selected
in such a manner that the dependence of the flow pattern on each of the
three independent variables (absolute bubble velocity, initial bubble
length, and initial frequency) would be established.
The computer output listed the position, the length, the velocity,
and the ratio R of equation (26) corresponding to each bubble at
several instants of timeo
The development of the flow follows the same pattern in all the
calculated conditions.
Due to its constant velocity, the very first
bubble is reached by a large number of others0
The first bubble may
consist of 20 to 150 initial bubbles, depending on the flow rates,
before the flow becomes fully developed.
All other bubbles reaching the
fully developed region are considerably smaller than the first one.
This is an elightening result since, in experimental work, the first
bubble formed after the gas flow is turned on, is far longer than all
subsequent bubbleso
After the first bubble moves far downstream, the flow pattern in
the entrance region assumes an oscillatory - but not exactly periodic character. The velocity of the bubbles passing a fixed section of the
tube, varies with time.
As a result, the mean density of the mixture in
the entrance region oscillates.
The result that developing slug flows are oscillatory in character
is indeed remarkable.
It explains, in principle, the flow instability
in natural circulation loopso
These loops are important because they
form the emergency cooling system of nuclear reactors.
In an attempt to
46.
explain the flow instability Wissler, Isbin and Amundson (ref. 2) have
carried out an extensive study of the dynamics of two-phase naturalcirculation loops. Their results show that the reasons for the instability
lie in the riser section of the loop.
It is clear that slug flows develop-
ing in the riser in the oscillatory manner described above, will give rise
to an oscillation of the mean density difference-between riser and downcomer and, consequently, to an oscillation of the flow-rate.
The amplitude of the oscillation -of the density At a-point, given
by the calculations described above, is of the order of 4% to 6% of the
mean density.
In a calculation carried out on a- hand computer, in which
the rise velocity of a bubble was computed by adding the value of V V
of equation (27),
to the velocity of the leading bubble -V , the oscillation
of the density at a point was of the order of 40% to 60%.
As discussed
in an earlier section, the exact situation in a developing slug flow lies
between these two extremes.
In order to calculate the mean density at a section, the following
averaging procedure has been adopted.
The computed values of the
density ratio R , for bubbles contained in a given section (between
X = 5 and X = 10, for example),
were added up and averaged.
The value
obtained in this manner is considered to be the mean R in the- middle of
the section (at X = 7.5 in the example given above).
A convenient method of presentation of the results is that adopted
in Figures 20 through 24. The calculated points are shown in Figure 20
but are omitted in Figures 21 through 24 in order to keep the graphs clear.
The main advantage of this presentation is that interpolation between the
curves is relatively easy. For any ratio Qf/Q
the intersection of the
appropriate curve with the horizontal axis may be found by computing the
47.
value of 1/ [ 1 + (Qp/Q ) + (V, A/Q )]
as Q approaches infinity
(1/1.5 when Q/Q = 0.5 for example).
With the two extreme points known,
the process of interpolation is facilitated greatly.
Provided that the initial bubble length and flow rates are specified,
one may compute the pressure gradient as a function of distancefrom
Figures 20 through 24.
The flow rates of the two phases are quantities
which are normally specified in engineering applications.
bubble length, however, is not,
significance of L
The initial
Consequently a discussion of the
is necessary.
The results of Figures 20 to 24 indicate that the pressure drop in
a developing slug flow is quite strongly dependent on the initial bubble
lengthe
This fact suggests a method of reducing the pressure drop in a
slug flow system. The pressure drop in an air lift pump,, for example,
may be reduced by designing an air valve which introduces large air bubbles
intermittently at the inlet to the shaft.
If, on the other hand, no special arrangement is made which gives
long air bubbles, the assumption that L
one.
= 0.5 is the most reasonable
This is approximately the smallest size bubble which rises with a
G. I. Taylor velocity.
Observations in tubes with air inlets of the
order of 1/4 and 3/8 of the tube size, support the same assumption. Of
course, with low gas flow rates, bubbles which are considerably shorter
than Lb = 0*5 are introduced through the same inlets.
Since, however,
these bubbles do not rise with Taylor velocities, the application of the
*
present theory with Lb smaller than 045, is clearly unreasonable.
bo
The results of the computer calculations have been applied to the
prediction of pressure drop for comparison with experiments of the present
and other investigations.
section0
The comparison is described in a subsequent
A sample calculation which illustrates the use of Figures 20 to
24 is given in an appendix
48.
(iv) measurement of Pressure Drop in a Developing Slug Flow
The test section used for pressure drop measurements is the same
as that used for the velocity-separation data (Figure 13).
A number of
static pressure taps were placed on the 1" tube at one foot intervals,
starting from a distance of 4 inches from the inlet.
Tygon tubing rising
vertically from each tap to the top of the 18 foot board is used as a
manometer.
The pressure indicated by each manometer tube varies considerably.
The reading rises while a slug passes the tap and falls when a bubble
passes.
In order to obtain the mean reading at a point -the oscillations
in the manometer were damped by restricting the flow through the manometer
by means of rubber clips.
A total of 11 runs were taken.
Of these the 8 were on a water-air
system. The remaining three were taken on a pure n-pentane-air system.
The main reason for the pentane-air experiments is,to examine
whether or not the ratio of viscosities of the two phases is an important
parameter. It must be remembered that the present theory claims that
Reynolds number effects are negligible (within the limitations expressed
earlier) in a developing slug flow. Now, all variations in the Reynolds
number in the experiments described thus far, have been achieved by varying
the characteristic velocity and length. The viscosity was not varied
appreciably. Since a dimensional analysis of the problem of the rise
velocity of a bubble in general shows the ratio of viscosities/jv/Jg
to be a parameter in addition to the Reynolds number, it is necessary to
perform at least one set of experiments in which this ratio varies.
The ratio of the viscosities of water and pentane is approximately
5. The rise velocity of a single bubble in pentane in a 1 inch diameter
tube was found to be the same as that of a bubble rising in water
49.
(o.577 ft/sec).
No measurements of velocity as a function of separation
distance were taken.
To the naked eye there is no apparent difference
between the air-water and the air-pentane systems.
The raw pressure drop data for all 11 runs are given in an appendix.
(v)
Comparison of Theory and Experiments-
Pressure drop for conditions corresponding to three of the eight
water-air runs have been calculated by means of the present theory.
The
calculation for one of these conditions is included in an appendix as a
sample calculation.
Theory and experiments are compared in iigure 25.
i!or all three
runs, comparison between theory and experiments is excellent.
Inspection
of the results for runs 2, 4, 5, 6, and 7 shows that comparison between
theory and those experiments is equally good.
Aigure 26 compares the theory with the pentane-air experiments.
Once again, the agreement between theory and experiments is excellent.
Apparently, the influence of the viscosity ratio on the rise velocity of
a trailing Taylor bubble is not appreciable.
iigure 27 compares the present theory with water-steam experiments
by Behringer (ref. 9).
The three conditions listed have been selected at
random from the much larger number of conditions reported in ref. 9. All
other sets of experiments reported by Behringer give very similar results.
In this case, the agreement between theory and experiments is
excellent again for the first 20 diameters of pipe length.
and experiments deviate at longer distances from the inlet.
deviation in pressure
rop is of the order of 25%.
But theory
The maximum
For over 70% of the
experimental points the error is smaller than 10%.
Although this type of agreement would be considered as satisfactory
I_.-
-- Ollihii
50.
when compared with most of the existing two phase flow correlations, the
striking difference between the agreement in kigures 25 and 26 on the one
hand and rigure 27 on the other, calls for a careful study of Behringer' s
experiments.
Behringer' s experimental set up was as follows.
Water at saturation
temperature was introduced at the bottom of the vertical test section.
An electric coil placed in the section was used to boil the water.
The
water vapor rose to the top of the tube, was then passed through a
condenser and the condensate was returned to the water supply for recirculation.
Heat losses were minimized by means of suitable jacketing
arrangements.
Static pressure was measured by a probe which could move
axially along the tube.
Comparison of the test conditions of Behringer's experiments and
those of the present investigation reveals several possible reasons for
the different results. Each of these reasons will be examined critically.
The first important difference between Behringer's experiments and
those reported here, is that fluids of different properties were used by
the two investigators.
The viscosity of water at 70 OF, which was used
by the writer, differs by the viscosity of water at 212 IF and 365 "' by
factors of about 4 and 7 respectively. Nevertheless, this is not believed
to be the cause of the discrepancy. The viscosity difference between the
water-air and pentane-air experiments of the writer is of the same order
as the differences given above.
Zet, no appreciable difference was
found in comparing figures 25 and 26.
Furthermore, Behringer's results
alone seem to dismiss the suggestion that viscosity effects are so
important.
His results at low pressure are in no clear way different
from his results at higher pressures, although the water viscosity
changes almost by a factor of 2 between the two conditions.
51.
The second important difference between the two experimental set-ups
concerns the inlet conditions.
In the present investigation the air-inlet
tap was especially designed so that slug flow is obtained almost at the
tube inlet.
Behringer, on the other hand, used an electric coil and
produced the gas flow by boiling in the test section.
Nucleation,
undoubtedly, produced small bubbles which eventually agglomerated to
form Taylor bubbles.
The different inlet conditions seem to offer a good reason for the
different results. A more careful examination of Figure 27, however,
soon dismisses that as a sufficient explanation.
If the present theory assumes inlet conditions which do not represent
satisfactorily Behringer's experiments, one should expect the discrepancy
between theory and experiments to be most pronounced at the immediate
vicinity of the inlet.
This, of course, is not shown by Figure 27.
On
the contrary, the first 4 or 5 points fall exactly on the theoretical
curve.
The deviation is noticeable after 20 or 25 diameters.
One might
say that in Behringer's experiments the entrance effects are more prolonged
than the theory predicts.
This last consideration leads to what the_ writer considers to be the
most likely cause of the discrepancy.
As small bubbles form at the electric coil, some agglomerate almost
at once to form Taylor bubbles.
The slug following each bubble,
undoubtedly contains a large number of small bubbles (as illustrated in
Figures 3 and 7).
As the development process continues, some of these
small bubbles agglomerate to form a new Taylor bubble.
It is easy to see
that the formation of these new Taylor bubbles prolonges the entrance
effects. The formation of a new bubble in a 6 diameter long slug induces
prolonged "entrance effects" in what would otherwise be a fully developed
52.
flow.
The results of section D may now be summarized as follows:
A
method is presented for calculating the density in a developing slug
flow. The density is a function of the flow rates of the two phases, of
the distance from the pipe inlet, and of the initial bubble size.
The
theory applies to any fluid as long as the Reynolds number, based on
bubble velocity, is greater than about 2000.
The method gives results
which agree well with experiments in a developing slug flow, provided
that slug flow already exists at the tube inlet.- If,
on the other hand,
a transition from bubble to slug flow occurs in the tube, pressure drops
calculated by the theory may be in error by factors of the order of 1.25.
For most calculations, an initial bubble size of LVDp = 0.5 is
recommended (this is the smallest bubble that will rise with a G. I.
Taylor velocity.)
Calculations based on the assumption of L DP = 0.5
are expected to be applicable to systems with most of the entrance
geometries that are commonly used in industrial components.
This is
supported by experiments by Chamberlain (ref. 18) and by Kearsley
(ref. 19) which show that inlet conditions do not affect appreciably
the performance of air lift pumps.
53.
IV
SUMMARY AND CONCLUSIONS
Two stages have been distinguished in the description of developing
slug flow. First, the agglomeration of small spherical bubbles to form
G. I. Taylor bubbles.
This is the process of transition from the bubble
regime to the slug flow regime.
Second, the agglomeration of two or more
Taylor bubbles due to wake effects.
The present investigation is concerned in particular with the
second stage. More specifically, the process of development of an
initially specified slug flow pattern, has been established.
In order to solve the problem, the kinetics of a bubble, as it
rises behind a number of other bubbles, had to be known.
This required
the knowledge of
a. The velocity profile of the liquid behind a leading bubble
b. The rise velocity of a trailing bubble in a liquid with
specified velocity profile.
An analytical expression for the velocity profile behind a leading
bubble has been obtained. The most important assumptions made in the
analysis were the following:
that the velocity of the leading bubble is
uniform; that wall-boundary layer effects are negligible in the immediate
vicinity of the bubble; and that the governing equations for turbulent
flow may be expressed in the form suggested by Reichardt in his Inductive
Theory of Turbulence. The momentum transfer length which appears in
Reichardt' s method has been determined experimentally by measuring the
velocity profile behind a plastic bubble.
Because of the complicated rotational flow, a theoretical solution
to the problem of the rise velocity of a trailing bubble has not been
possible. Nevertheless, the knowledge of the velocity profile, served
54.
in the planning of simple experiments which determined the rise velocity
as a function of separation distance.
V
=
The equation
1 + 8 exp(-1.06 L,/D
(27)
gives the best continuous curve which represents the data.
Once the kinetics of a trailing bubble are known, the developing
pattern of bubbles which initially enter a tube at regular intervals may
be determined by means of a finite difference calculation.
The density
of the developing flow may then be calculated from continuity considerations.
F'or the convenience of the reader, the procedure for determining
the density
is summarized below:
Given Qf, Qg, the pipe diameter and the fluid properties, the rise
velocity in fully developed flow is given by
V
=
cCCa2
gD
(12)
p
where C, and C2 may be read from Figures 5 and 6 respectively.
The density in fully developed flow may then be computed using
Griffith and Wallis' equation
L
LR RIg
g
gJOf (Qf + V A )(Qf+ +Q
gcfQ/(
+ +Q ++ V A
(Qf
g
p
Q ++V A)+
+Op
(11)
In the entrance region the ratio
(=f
-
)
=
(26)
R
corresponding to the particular flow conditions may be read from Figures
20 through 24. For most applications, the assumption L
recommended.
= 0.5
is
550
This method for calculation of density applies to any combinations
of fluids, provided that the bubble-velocity Reynolds number is larger
than 2000.
Pressure drops calculated by the method outlined above have been
compared with experiments by the writer and by Behringer.
The writer' s
experiments were controlled so that only the process of agglomeration of
Taylor bubbles was taking place.
In Behringer's experiments the complete
process of transition from bubble to slug flow was taking place in the
test section. Comparison between the theory and the writer's experiments
with water and pentane is excellent. But the pressure drop in Behringer's
experiments was larger than the theory predicts by as much as 25%.
The conclusions of the investigation may be summarized as follows:
1. The velocity profile in the wake behing a bubble is determined
by the bubble velocity and the bubble length. Provided that the
bubble velocity Reynolds number is not too small, the viscosity
of the liquid does not affect the profile appreciable.
2. The mean rise velocity of a trailing bubble in a developing slug
flow may be expressed as a function of the separation distance
between the leading and trailing bubbles.
Statistical deviations
from that mean occur because of the unstable character of the
bubble motion.
3.
The process of agglomeration of Taylor bubbles which are
introduced at the inlet of a tube at uniform time intervals can
persist for lengths equal to 25 to 30 tube diameters.
4.
The mean density in the entrance region may be as much as three
times greater than in fully developed slug flow.
56.
5. The density at a given section of pipe in the entrance region,
varies with time. As a result, the mean density in the entrance
length oscillates.
This result explains in principle the flow
oscillations in natural circulation loops.
6. A method for predicting density and pressure drop in a developing
slug flow is presented.
The method of calculation applies to
any fluid provided that the bubble velocity-liquid viscosity
Reynolds number is over 2000.
7. For systems in which the flow development consists of Taylorbubble agglomerations only, the density and pressure drop can be
calculated with very good accuracy.
8.
If the same method of calculation is used to predict densities
in systems in which the entire process of transition from bubble
to slug flow is taking place, the results may be in error by
factors of the order of 1.25.
9. In the case of systems with heat addition, the process of
transition from bubble to slug flow may be taking place in the
entire length of the system. In such cases the high entrance
region densities, which the present analysis predicts, will
persist for much greater lengths.
WAMMONVANOMIA"
57.
NOMENTCLATURE
English Symbols
b
= Area; A for pipe cross-section; A for pipe walls
w
p
= Constant defined in equation (22) page 26
c
= Constant defined in equation (22) page 26
C11 C2
=Constants
d
= Diameter at tail of Taylor bubble
D
p
=
Pipe diameter
F
=
Body force
g
= Gravity constant
I
= Integrated momentum function, defined on page 25
L
= Length; L, = length of slug; Lb = length of bubble
L*
=
m
= Momentum function p +
M
= Momentum function defined on page 20
NRe
= Reynolds number; NReb =
A
defined in equation (12) page 9
LfDp
NRef
f + Q
, bubble Reynolds number;
)
/(A
liquid Reynolds number
N*
= Dimensionless bubble frequency = D /V A t
p
= Pressure; p = mean; p' = turbulent fluctuation component
q
= Defined in equation (31) page 44
Q
=
r
= Radial position in cylindrical tube; r = dimensionless
Flow rate; Qf = of liquid; Q = of gas
*
R=
radial distance = 2r/D
p
Ratio (fp-f)/(f -
t
Time
u
)
= Component of liquid velocity in x-direction; u = mean;
u' = turbulent component
58.
v
=
Component of liquid velocity in r-direction
vb
=
Volume of bubble; v = dimensionless volume of bubble;
*
*
*
'T = initial value of v
= Velocity; V = of bubble in fully developed slug flow;
V
VL = of leading bubble in a developing slug flow;
L*
V = of a Taylor bubble in general; V, = V // gD
b
p
V*Vb/
g = Q(Ap
); Vf = Qf/(A V )
;
x
=
Distance along axis of cylindrical tube
X
=
Distance of top of bubble n at time t measured from tube
X /D
inlet;
= Distance of bottom of bubble n at time t measured from tube
Y
inlet;e
nt
Y . D
nVp
Greek Symbols
Denotes incremental change; Lt = bubble period; At 0 = period
A
at tube inlet; Lt* =
6 tV /Dp
= Momentum transfer length, defined in equation (17) page 19
=
Viscosity, /f
Density;
f)
= of liquid
= of liquid;
g = of gas; Pa or
density in fully developed slug flow;
= mean density at
distance x from pipe inlet
wc
=
Denotes summation
=
Wall shear stress
=
Dimensionless velocity potential (
=
Dimensionless stream function
= mean
gD)
59.
REFERENCES
1. Maung Maung-Myint, "A Literature Survey on Two-Phase klow of Gas
and Liquid," MIT B.S. Thesis, 1959.
2. Wissler, E. H., H. S. Isbin, and N. R. Amundson, "Oscillatory
Behavior of a Two-Phase Natural-Circulation Loop," A.I.Ch.E. Journal,
Vol. 2, No. 2, p. 157, 1956.
3.
Griffith, P. and U. B. Wallis, "Slug Plow," Technical Report No. 15,
Division of Sponsored Research, MIT.
4. Dumitrescu, D. T., "Stromung an einer Luftblase in senkrechten Rohr,"
ZAMM, 1943, Vol. 23, No. 3, pp. 139-149.
5. Davies, R. M. and G. I. Taylor, "The Mechanics of Large Bubbles
Rising Through Extended Liquids and Through Liquids in Tubes,"
Proc. _Roy. Soc, London 1950, Vol. 200, Series A, pp 375-390.
6. Lamb, H., Hydrodynamics, Dover Publications, New York, 1932.
7. Laird, A. D. K. and Chisholm, "Pressure and Forces along Cylindrical
Chem., Vol. 48, (8),
Bubbles in a Vertical Tube," Indus. and E
1956, pp. 1316-18.
8. Schwartz, K.,
1954.
VDI-F'orschungsheft, Issue B, 20, No. 445, pp 1-44,
9. Behringer, P. "Steiggeschwindigkeit von Dampfblasen in Kesselrohren,"
VDI iorschungsheft, 365.
10.
Griffith, P., "Two Phase Flow in Pipes," MIT 6ummer session Notes,
1960.
11.
Kynch, U, J., "The Slow Motion of Two or More Spheres Through a
Viscous Fluid," J. Fluid Mechanics, April 1959.
12.
Hasimoto, H., "On the Periodic Fundamental Solutions of the Stokes
Equation and Their Application to Viscous illow Past a Cubic Array
of Spheres," J. fluid Mechanics, April 1959.
13.
Reid, E. G.,
14.
Schlichting, H.,
15.
AlexanderL. G., et al, "Transfer of Momentum in a Jet of Air Issuing
into a Tube," Univ. of Illinois Tech. Report No. 11, 1952.
16.
Corrsin, S. and M. S. Uberoi, NACA TN 1865 (1949).
17.
Govier, Radford, and iDunn, "The Upwards Vertical Flow of air-Water
Mixtures," Canada J. of Ch. Eng., August 1957, p. 58.
"Annular Jet Ejectors," NACA TN 1949, (1949).
Boundary Layer Theory, Pergamon Press, N.Y., 1955.
59a.
18.
Chamberlain, H.V., "Factors Affecting Capacity of Air Lifts,"
AEC Research and Development Report, 1)0 - 14398.
19.
Kearsley, G. W. T., "Use of an Air Lift as a Metering Pump for
Radioactive Solutions," Oak Ridge National Laboratory, ORNL - 2175w
60.
APPENDIX I
PRESSURE DROP MEASUREMENTS IN ONE INCH DIAMETER TUBE
(Values for Qf, Q are in cu. ft. per minute)
(A = Manometer reading in inches)
(B = Pressure in feet of water)
Run 1
(Air-Water)
= 0
Qg
Q9= 0.048
Run 2
(Air-Water)
Qf
= 0
Run 3
(Air-Water)
Qf = 0.057
g = 0.390
0.196
Run 4
(Air-Water)
Qf = 0.117
Q = 0.672
Position
inches
-A
B
A
49.9
55.7
61.8
B
A
B
A
B
6.25
5.73
96.5
98.1
9.72
8.85
6.06
5.24
102.9
8.25
7.67
7.08
49.6
52.8
59.5
67.2
5.33
4.89
75.8
4.24
83.5
3.88
91.4
99.4
107.2
3.54
2.58
3.81
115.9
123.8
inlet
4
16
141.7
143.5
13.84
12.99
28
147.8
12.35
40
149.4
151.2
11.48
10.63
68.0
4.75
107.9
75.5
4.38
153.0
9.78
82.8
3.99
112.9
118.2
154.9
157.0
159.3
8.94
90.8
3.65
8.12
99.0
3034
7.31
6.48
107.0
3.00
1150.
2.67
123.1
2.34
131.2
2.02
52
64
76
88
100
112
124
161.4
163.5
123.9
129.2
134.8
6.53
6.00
5.44
4.91
4053
3.21
2086
140.1
145.6
150.8
4.35
3.24
131.9
2.70
140.2
1.92
1.61
2.16
148.1
1.27
0.95
0.64
136
165.8
5.66
4.85
148
160
168.0
4.03
139.0
1.67
170.3
3022
147.0
1033
156.3
161.8
172
172.6
2.42
1.02
167.2
1.61
156.3
184
196
174.8
1.60
155.2
163.0
0.67
172.9
1.08
177.2
0.80
171.2
0.35
208
179.6
0
179.0
0
178.4
183.9
0.54
0
164.6
172.8
180.9
2.24
0032
0
610
Run 5
(Air-Water)
Qf=0
Q = 0.727
Run 6
(Air-Water)
Qf=0
Q = 0.053
Position
inches
from
inlet
A
B
4
16
28
18.0
23.5
30.1
3.45
2.90
2.45
137.3 13.12
138.8 12.24
140.6 11.39
40
40.8
2.35
142.1
10.52
52
64
49.9
59.5
2.10
1.90
76
70.1
1.78
143.9
145.9
147.8
88
79.5
100
88.8
1.56
1.34
112
98.0
124
136
148
160
172
184
--
196
-
200
A
B
Run 7
Run 8
(Air-Water)
(Air-Water)
Qf=0
Qf=0
Q = 0.029
A
B
Q
= 0.410
A
B
120.6 14.94
121.7 14.04
122.8 13.13
54.3
59.2
6.67
64.9
12.22
71.2
5.56
5.08
9.67
8.83
7.99
125.3 11.34
126.6 10.44
127.9
9.55
77.8
4.63
85.1
91.8
4.24
3.80
1.10
150.1
152.3
154.7
7.18
6.37
5.57
129.3
130.7
132.1
8.67
7.79
6.90
98.9
106.5
114.7
3.39
3.02
2.70
107.1
0.86
156.8
4.74
133.8
6.04
122.6
2.36
116.5
126.0
135.0
0.64
0.43
0.18
159.2
161.6
164.2
3.94
3.14
2.36
135.2
136.9
138.6
5.16
4.30
3.44
130.2
137.9
145.7
1.99
1.64
1.29
144.8
0
166.8
1.36
140.2
2.58
153.4
0.93
--
169.4
0.79
141.7
1.70
161.5
0.60
--
171.9
0
143.4
0.84
169.7
0.29
0
178.3
0
-
-145.3
123.9
6.08
62.
Run 9
(Air-Pentane)
Qg
= 0
Q9= 0.040
Run 10
(Air-Pentane)
= 0
Qg
Q9= 0.090
Run 11
(Air-Pentane)
Q
=
0
Q = 0.580
Position
inches
from
inlet
A
B
A
B
A
B
4
84.6
9.29
2.96
86.2
8*39
7,70
6.96
7.3
16
69.3
72.2
13.0
28
87.9
89.5
7.56
6.22
20.5
6.70
75.3
79.0
2.42
2.02
5.53
29.6
1.82
52
64
91.3
1.60
4.11
39.0
48.8
95.0
97.0
82.5
86.0
89.3
4.82
76
5.85
5.01
4.16
93.2
97.0
57.8
67.1
99.0
101.0
3.32
2.49
3.39
2.71
2.02
1.66
103.1
0.83
105.1
0
100.5
104.5
108.3
75.8
84.8
40
88
100
112
124
136
93.2
1.29
0.64
0
93.8
104.0
1.33
1.17
0.95
0.67
0.41
0.16
0
63.
APPENDIX II
SAMPLE PRESSURE DROP CALCULATION
The pressure drop corresponding to conditions of run 3 of the writer's
experiments will be calculated.
1" diameter pipe; Air-Water
Qf = 0.057, Q = 0.196 cfm
Since a calculation of NRe requires knowledge of %, which is itself
a function of NRe, a quick trial and error procedure is required.
Guess V, = 0.710 ft/sec
NReb
=
(
fD,1f = 5450;
+ Qf)D/(A1f)
(g
NRef
From Figures 5 and 6 C1 = 0.35; C2 = 1.22
V
Q:)
=ClCa
fD
pI
= 0.710 ft/sec
Qg/(Qf + Q + V A) =
From equation (11)
0.427;
J0 /jF f
Qf/Qg
= 0.290
= 0 . 595
Y'or interpolation in figures 20 to 24, intersection of
Q/Q g = 0.290 with horizontal axis is at 1.29 = 0.774
From figures 20 to 24
1.22
=
2.40
R5
R5
=
2.18
R20
114
=
1.36
R25
= 1.06
R
(1f-fo
Hence
=
RO
(A p/AL)
(t p/LL)5
(L p/LL)10
)
X)R
= 0.831
1-
0 .595)
_
0.405
x
R
x
(dp/& L)1 5 = 0.668
0.814
(L p/tLL)20
= 0.700
(L p/A L)2 5
=
0.645
0.618
= 5950
64.
Starting with the pressure of 10.2 ft at L/D = 0 which has been
measured in the experiments:
= 10.2
= 10.2 - 0
pO
5
=
10.2 - 0.831 x
P7 . 5
=
10.0
p2
1
0.814
-
.x
10.0
1
=
12
=
9.66
1
=
9.37
p1 2
5 =
9.66 - 0.700
x
p1 7
5
=
9.37 - 0.668
x
=
9.09
=
9.09
0.645 x
=
8.82
=
8.82 - 0.618 x 1
=
8.56 - 0.595 x 72
p2 7
p100
5
-
8.56
=
4.96
65.
APPENDIX III
CAPTIONS TO FIGURES
Figure 1
Two phase flow regimes in a vertical pipe
Figure 2
Flow map for vertical pipes
= bubble regime
I
II = slug regime
= mist below dotted line; annular above
III
Figure 3
Qf = 0 in 1 inch pipe
Q = 0.073 cfm
Typical low speed flow. A and C show stages in the agglomeration
The lower bubble is caught in the wake of the upper,
process.
distorts, and accelerates rapidly.
Figure 5
Dimensionless constant C, against bubble Reynolds number
x
Results of Dumitrescu
0
Results of Griffith and Wallis
Dumitrescu's theory gives C, = 0.350 for potential flow
figure 6
Coefficient C2 against liquid Reynolds number for various
values of bubble Reynolds number
"Laminar"
- - - - - - ..
Figure 7
e
Q = 0,208
..0
NReb = 0
-
3000
"Transition"
NReb
"Transition"
NReb = 5000
"Transition"
NReb
"Turbulent"
NReb = 7000
"Turbulent"
NReb = 8000
Q= 0.288 cfm
6000
1 inch diameter pipe
Agglomeration of small spherical bubbles to form Taylor
Bubbles
66.
Figure 9
Velocity distribution behind bubble. Mean water velocity
= 1.45 ft/sec. Pipe diameter 2 inches,
Bubble diameter
1.75 inches, 6" long
Figures 20 to 24
The density ratio (%
-/)/(f
-
) is
approximately the same as the pressure drop ratio
, p
//_ \p
negligible.
in those cases where friction effects are
67.
APPENDIX IV
IGURES
I.
e
l
9
oa@
I
S.
bis
slug
Annlar
Semi-Anzmal
Mint
Figure
og
Qf +Q,
0
0.2
0.4
0.6
(ALL POINTS ARE FOR SLUG FLOW)
X GROVIER,et al
o BRANCART
v GRIFFITH AND WALLIS
o KOZLOV
6 SCHWARTZ
0.8
1.0
NFrm
2
4
A
6
10
20
)/gDp
FIGURE 2 FLOW MAP FOR VERTICAL
PIPES
40
60
100
.
MPPK
.
.
Jill
MIIU
II
&..
.
.
.
.
&
....
...
%i
co
6
I1 1 1 1
I 1I 1
IiI
I
I
I
I
I
OA
(C
0.3
27
0.2
L,+ Lb
Lb
x
0.1
2
4
6
8
10
12
16
Bubble Reynolds' Number N . x 10
FIG.
5
FIG.4
0
1000
2000
3000
N8 ,
FIG 6
4000
5000
6000
Is
20
"lol11 N
r to
,twm
I
1
t0 1
kIL
A*
(9
LUCITE TUBE 2" I. D.
-RUNNING THREAD
1/2" D. 20 / INCH
MERCURY
PLASTIC BUBBLE
|3/4D. 6" LONG
6 FT
WATER SUPPLY
INCLINED
MANOMETER
(WATER)
STATIC PRESSURE TAP
TOTAL PRESSURE TAP
PUMP
CONSTANT HEAD TANK
FIGURE 8
TO WEIGH-BUCKET AND SINK
APPARATUS FOR WAKE MEASUREMENTS.
El
6
5
4
Li
L)
4
0
3
H
>2
1.00
I
.75
.50
.25
RADIAL POSITION-r/R
FIGURE9
____v
0
0
VELOCITY DISTRIBUTION IN THE WAKE BEHIND A PLASTIC BUBBLE
I
I1
I
I
I
I
I
I
0.4
-4
--
0.3
N
-6
s
-8
0.1
-101-
-/
I
I
I
I
I
I
I
I
I
1
2
3
4
5
6
7
8
9
AXIAL DISTANCE FROM
FIGURE
0.2
10
STATIC
PRESSURE
BEHIND PLASTIC
BUBBLE,
BUBBLE
01
0
10
I
2
3
AXIAL DISTANCE
x/D
FIGURE
||
MEASURED
VALUES
5
6
4
FROM TAIL OF BUBBLE,
OF INTEGRATED MIXING
7
x/D
PARAMETER I
8
0.6
0.5
0.4
I
CAMERA.
0.3
0.2
0.1
0
x/D
F IGU RE
12
CA LC UL A T ION
OF
A (W)
FIGURE 13
APPARATUS FOR VELOCITY- SEPARATION MEASUREMENTS.
80+1
Q
Qg
= 0
cfm
0
a
oQ
= 0
cfm
L Qf = .0
Q = .068
cfm
0
Of
= 0
QO = 0
QO = .101
Qg = .123
cfm
cfm
Of = 0
f = 095
Qg = .256
cfm
Of = .151
+ Qf = 0
>
6
v
0
0
5
*
0
0
g
Of
.173
Of
4
cfm
0
Qa = 0
7 Of = .045
o
= .256
cfm
.259
cfm
6
-
Of = 0
v= 0
49
Of = .I 13
cfm
Qg = .073
cfm
Q = .163
cfm
-
>
5
0
4
wLiL
3 oo
Dn
ina
0
000
0
2
2
3
4
5
6
SEPARATION DISTANCE
FIGURE
14
VELOCITY
vs
SEPARATION
7
8
9
10
I
2
3
Ls/Dp
DISTANCE-I" DIAMETER
TUBE
FIGURE
15
VELOCITY
vs
4
5
6
7
SEPARATION DISTANCE
Ls/Dp
SEPARATION
DISTANCE -3/4
"DIAMETER
8
TUBE
9
10
8
o
ofO
a
V
Of
00,
Of
O
.200
.1
9 8
.085
" O
7
cfm
Og
a .434
cfm
0,
=
0
cfm
0,
-
.300
cfm
of,=
0
0,
a 0
cfm
Af
= .042
0,
a 0
cfm
6
-
-
'5
0
-J
9w
> 4
I
w
o
\
9
1
i\
~
03
00oS\
2
I
3
2
5
4
SEPARATION
FIGURE
16
VELOCITY
vs
SEPARATION
DISTANCE-
7
6
DISTANCE
8
I
9
2
2"DIANWETER TUBE
3
4
5
6
SEPARATION DISTANCE
Ls /Dp
FIGURE
17
VELOCITY
7
8
LI/Dp
vs SEPARATION DISTANCE - I/2
DIAMETER
TUBE
9
10
I
0
INUM
UIAvltilN
Iutl
3.0 -
3/4 INCH DIAMETER TUBE
7.0 02
INCH DIAMETER TUBE
Vb
*
0
6.0
Vb
= 3.E
V
I +8EXP(-I.O6 Ls/Dp)
Vb/VO
-
--
2.5 -
Vb
-
2.1
2.0- -
0>
5.0 -
0~Vb
-J
0
QJ
IJ
w
>4.03 .
V
--
0
=
V
0
aQ
0
o
In
> 1.5 -
3.0
0
2 .0 2.0 -
00
a
a0
00 0
0
03
0_
ca
o0.%
_
0
e.00
02
_
_
_
_
_
_
_
_
_
_
_
_
00
3
4
5
6
7
0 <xn|8
8
9
0.2
O0
|C
0
0.4
0.6
1.0
0.8
RADIAL POSITION,
r/ R
SEPARATION DISTANCE Ls/Dp
FIGURE
FIGURE 18
VELOCITY - SEPARATION
DISTANCE DATA
19
DEPENDENCE OF BUBBLE RISE
LIQUID VELOCITY PROFILE
VELOCITY ON
IN
3.8
I B
L
INITIAL BUBBLE
--
=
LENGTH= =.5x TUBE
INITIAL BUBBLE LENGTH= TUBE
INITIAL BUBBLE
3.8
DIAMETER
3.4
DIAMETER
LENGTH= 2x TUBE
INITIAL BUBBLE LENGTH =O.5xTUBE
DIAMETER
INITIAL BUBBLE LENGTH =TUBE
3.0
INITIAL BUBBLE LENGTH
DIAMETER
DIAMETER
2x TUBE
DIAMETER
-
-3.4--
2-.6
2.6
<-
22
2.2
---
09
1.8-
/
1.4
91.4
Q
0
0
0 2
0.3
0.4
PRESSURE
DROP
AT PIPE INLET
0.6
0.5
Og/(Qg+Q
20
--
1.D0
--
.0
FIGURE
-
OVER
+V
0.7
0.8
0.9
-..
0.1
1.0
0.2
0.3
A)
PRESSURE
DROP AT INFINITY
FIGURE
21
PRESSURE
DROP AT
5 L/D
0.4
0.5
0.6
Qg /(Qg+Q
+V, A)
FROM PIPE
0.7
0.8
0.9
1.0
INLET OVER PRESSURE DROP AT INFINITY
4w
I1
3.8
1
INITIAL BUBBLE LENGTH
3.4
-INITIAL
BUBBLE
--
-
C
LENGTH
INITIAL BUBBLE LENGTH
3.0
2.6
'
2.2
18
-=2
1.4
10
4
05
06
-g
0
0.1
0.2
03
0.4
Q9 /(09
Qg / (Qg+ Qf+ VDA)
FIGURE
22
PRESSURE
DROP AT 10 L/D FROM
PIPE INLET OVER PRESSURE DROP AT INFINITY
FIGURE
23
PRESSURE
DROP AT 15 L/D
FROM
05
+ Q
0.6
07
08
09
10
+ V.A)
PIPE INLET OVER PRESSURE
DROP AT INFINITY
I
I
I
I
I
|
|
I
I
I
|
.%...
3.4
I
I
I
I
I
-
0
121
INITIAL BUBBLE
LENGTH
0.5xTUBE DIAMETER
INITIAL BUBBLE LENGTH= TUBE
RUN
I
DIAMETER
10
3.0
F
-42-
%
--
-~
~Q.
2.6
8
2.2
60
6
-
~-
-
RUN 3
RUN 8
-
0-
1.8
4 -
PRESENT THEORY WITH
---0
RUN I:
2
1.4
0.1
0.2
03
0.4
0.5
0,/(O,+Oq+
FIGURE
24
PRESSURE
DROP AT 20 L/D FROM
0.6
0.7
0.8
0.9
1.0
-
0
0,
/Qg
= 0
Qg /(Qf
RUN 3: Og /Q
= .3
Og /(Q
RUN8: Q0 /Q9
= 0
0,
10
20
Vu3A)
PIPE INLET OVER PRESSURE DROP AT INFINITY
POINTS
EXPERIMENTAL
30
.5
Lb
OF
+Qg
+ VDA)
=.203
+Q9
+ Vc A)
= .427
/(Q0
+ 0g + VmA)
= .630
40
50
LENGTH OF PIPE
FIGURE
25
COMPARISON
-
OF THEORY
AND
.
0
PRESENT INVESTIGATION
60
70
80
L/D
EXPERIMENTS
(AIR -WATER)
90
100
I
|
12-
I
I
1
I
----
PRESENT THEORY WITH
0
RUN
EXPERIMENTAL POINTS OF PRESENT INVESTIGATION-
RUN IO:0 /Q
RUN I :Of /Q,
Z 10-
I
I
= 0
=0
3
LI = 0.5
Q, /(Q,+Q, + V.A) =. 320
Q, /(Q, +Q, +Vw A) =.760
3
9:0O /O,2.01
RU
9Q~O~=0 Q0 (o+0+V)
2.5
-
z
1|
1
PRESENT THEORY WITH Lb =0.5
DATAOF BEHRINGER Of /Q9= 0, 0g /Q9 + Of+ V.A)= .820
INCH
.100 DATA OF BENRINGER Of /Qg 0, QO/(Og + Of +V.A)= .740
D 3.25 INCH
p
I ATM.ABS.
0 DATAOF BEHRINGER
Of /00=0,
0g/(0g + Of+VmA)
.530
3
w
u.
1
8
-
n
w-'%
w
LL
wo
0
R URN9
4r -s
2r
-RUN
10
61 6 -
2.
tI,
1.5
-f'.
0'
o
-r
0
N%
w
0
4
2
0
0
1.0
0
0
E3
'-C6
2
0
-
0
10
RUN 11
20
30
40
50
0
0.5
60
70
80
90
100
0
l
5
LENGTH OF PIPE L/D
FIGURE
I
I
I
10
15
|1
20
25
30
3
0
5
5
LENGTH OF PIPE L/D
26 COMPARISON
OF THEORY
ANDEXPERIMENTS
(AIR- PENTANE)
FIGURE27 COMPARISON
OFTHEORY
WITHEXPERIMENTS
OF BEHRINGER
(STEAM-WATE