Quantitative predictions of phase distributions in a circular
pipe
Citation for published version (APA):
Geld, van der, C. W. M. (1985). Quantitative predictions of phase distributions in a circular pipe. (Report WOP-WET; Vol. 85.033). Technische Hogeschool Eindhoven.
Document status and date: Published: 01/01/1985
Document Version:
Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)
Please check the document version of this publication:
• A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website.
• The final author version and the galley proof are versions of the publication after peer review.
• The final published version features the final layout of the paper including the volume, issue and page numbers.
Link to publication
General rights
Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain
• You may freely distribute the URL identifying the publication in the public portal.
If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement:
www.tue.nl/taverne
Take down policy
If you believe that this document breaches copyright please contact us at: openaccess@tue.nl
providing details and we will investigate your claim.
QUANTITATIVE PREDICTIONS OF PHASE DISTRIBUTIONS IN A CIRCULAR PIPE
C.W.M. van der Geld
Eindhoven University of Technology Report nr. WOP-WET 85.033
eiaLIOTHE~K
_
8511031 .
IH.EtNDHOVEN
~
€:>
'L
'\'"Le>,\S
The financial SUpport by the Netherlands Organization for the Advancement of Pure Research (ZWO) is highly appreciated
FIGURE CAPTIONS NOMENCLATURE ABSTRACT 1 II\ITRDDUCTION CON TEN T S 1.1 General introduction
1.2 Bubbles and evaporator tubes
1.3 Bubble-plug flow pattern transition
2 CONSIDERATIONS ON PHASE DISTRIBUTIONS
2.1 Experimental observations
2.2 Physical considerations
3 qUANTITATIVE PREDICTIONS OF PHASE DISTRIBUTIONS
3.1 Minimisation of energy dissipation
3.1.1 On the principle of minimisation 3.1.2 Single object 3.1.3 Swarms of objects 2 3 5 6 7 9 11 13 16
3.2 Determination of the eccentricity function 24
3.2.1 Turbulent flow 3.2.2 Creeping objects
3.3 Calculation of phase distributions
4 BUBBLE-PLUG FLOW PATTERN TRANSITION
4.1 Two types of transition
4.2 Criterion for transition
4.3 Dependencies on physical parameters
4.4 Comparison with experiments
5 CONCLUSIONS REFERENCES 30 39 40 41 42 44 45
2
FIGURE CAPTIONS Figure 1
Shape regimes for bubbles ; summary of results of
Grace (1973), Clift et al. (1978), Shaga and Weber (1981) Figure 2
Comparison of experimental and theoretical observations in flow pattern map
Figure 3
Void fraction profiles (Serizawa, 1974)
Atmospheric air-water flow; U
=
1.03 mlss
Figure 4
Fluid flow approaching a particle at rest Figure 5
Coordinate systems Figure 6
Influence of transverse forces on bubble concentration Figure 7
Void fraction profiles in upward flows Figure 8
Void fraction profiles in downward flows Figure 9
Void fraction profiles for bubble flows The wall influence is estimated
Figure 10
Void fraction profiles for bubble flows
Liquid velocity profiles and wall influence estimated Figure 11
Droplet density profiles ; C
s
and Cs
are negativeFigure 12
Droplet density profiles in downward flow Figure 13
List of symbols Roman letters a eq c 1 c 2 c 3 c 4 c5 c 6 e e(O)trans etrans e ,x f Fbuo F drag g g(r) hex) I I rad Itot J 1 a N
o
o 0 1 O 2 P . Sl. NOMENCLATUREequivalent bubble or particle radius wall influence function (see 3.2.1) constant (see 3.2.1)
constant (see 3.2.1)
viscosity function (see 3.2.2) relative velocity (see 3.1.3) constant (see 3.1.3)
particle volume fraction NV ol
m
m/s
volume fraction at centre of the tube corresponding to transition to plug flow (section 3.2)
averaged particle volume fraction corresponding to transition to plug flow (section 3.2)
de dx
eccentricity function (see 3.1)
-1 m
buoyance force (see 3.1) N
drag force on a particle or bubble
N
gravity constant m/s2
liquid velocity function (see 3.1) eccentricity function (see 3.2.2) local energy dissipation
local energy dissipation due to radial movements
total energy dissipation total particle flux (3.1) Lagrange multiplier
number density of particles vorticity function (see 3.2.1) vorticity function (see 3.2.1) vorticity function (see 3.2.1) stream function Nm/s Nm/s N/s -1 s N -3 m -1 s -1 s -1 s
4 r
r'
R o 5 vort t U :::: U relo
U 5 U' 5 V max V o Vol x y z Greek letters llUQ llOBJ PUQ POBJ a Acronyms Eo LHS Mo OBJ Re p RHS DISTRradial coordinate defined in figure 4
distance from particle centre
(3.2.1)
tube radius vorticity
coordinate defined in figure 4
velocity component in radial direction velocity component in circumferential direction
relative velocity (see
3.1)
m m m -1 s rad m/s m/s m/s mean bubble velocity relative
liquid velocity
superficial liquid velocity superficial gas/vapor velocity maximum liquid velocity
liquid velocity
volume of bubble or particle r/R to averaged m/s m/s m/s m/s m/s ma o
coordinate defined in figure 4
axial coordinate defined in figure 2
viscosity of the liquid phase viscosity of bubble or droplet mass density of the liquid phase mass density of bubble or particle surface tension
Eotvos or Band number (see section 1) left-hand side
Morton number (see section 1)
object : bubble, drop, particle
particle Reynolds number (section 1)
right-hand side m m kg/m.s kg/m.s kg/m3 kg/m3 N/m
ABSTRACT
A model for the calculation of radial phase distributions in a vertical, circular tube is presented. Both the upflow and downflow of fully-developed bubble or droplet mixtures are considered.
Energy dissipation is minimised by selecting proper places for flow
disturbances. It is not minimised in order to determine the motion of
individual fluid particles. With this minimisation
concentration of bubbles or particles in central parts of explained.
procedure a tube can
the be The interaction of the relative velocity, induced by the buoyancy force, with the vorticity in the undisturbed liquid motion is seen to be counteracted by bubble diffusion, and to be responsible for differences in the trends of distribution profiles for upward and downward flows of bubbles or droplets.
Calculated phase distribution profiles are in agreement with experimental observations.
The model was applied to increase the accuracy of predictions of bubble plug flow pattern transitions. A small tube radius dependency has been found.
6 DISTR
1 INTRODUCTION
1.1 General introduction
The purpose of this paper is to outline the principles of a means to predicting radial distributions of the dispersed phase for fully-developed two-phase flows in a vertical, circular pipe. Phase-distribution mechanisms
are studied, and it is hoped that more understanding gained of, inter
alia, related phenomena in complex geometries such as tube bends and elbows. Differences in observed trends of distribution profiles are explained with the help of a hydrodynamic transverse force, but the local maximum of such profiles at the tube axis is explained with the aid of an energy priciple. Energy dissipation will be minimised by selecting proper places for elements of the dispersed phase.
Possible effects of these distribution mechanisms on bubble~slug flow
pattern transition are explored.
In section 1.2, application areas of interest are indicated. In section 1.3, some experimental results on bubble plug flow pattern transition are
presented. These observations seem to indicate that with constant
superficial water velocity and increasing tube radius the transition occurs at decreasing superficial air velocity.
In chapter 2, an interesting compilation of experimental data on
distributions of bubbles and particles is followed by a digression on possible explanations for the main trends of these observations. In chapter 3, the findings are used in the main part of the paper: the set-up of a means of calculating phase distributions without entering into flow details or force balances. The so-called eccentricity function is approximated in section 3.2 from theoretical considerations, but the function could also have been determined from a set of accurous experimental data.
In section 3.3, the experimental observations on void fraction or density profiles are compared with the theoretical predictions. In chapter 4, the above mentioned observations of bubble plug flow pattern transition are explained with the aid of the knowledge of void distributions.
1.2 Bubbles and evaporator tubes
In evaporator tubes, bubbles of many shapes and sizes are encountered. For
example, during test runs at the Eindhoven University of Technology, the
system pressure attained values between 10 and 250 bars, temperatures of
water and vapor varied between 100 DC and 370 DC, and the superficial mass
flow rate had values in the range from 800 to 2500 kg/m2s.
There are several physical reasons to expect a wide variety of bubbles in such differing circumstances. Wall heating probably does not affect the bubble departure radius in boiling water (Slooten, 1984), but system pressure does have influence (Van Stralen and Cole, 1979), as do contact angle, surface roughness, and mass flow rate (AI-Hayes and Winterton, 1981). At high pressure, say higher than 150 bar, the difference between liquid mass density P
LIQ and bubble mass density PoBJ is much smaller than at low
pressure.
Bubbles rising feely in unrestricted media, at rest in infinity, can be characterised in terms of the following dimensionless groups :
Rep
=
2aeq P LIQUre/ lJ LIQ (Reynolds number)Here a is defined by equating (4/3) 7T a 3 to the bubble volume;
a
denoteseq eq
surface tension, g the gravity constant, lJ
LIQ viscosity of the liquid and
U 1 the rise velocity of the bubble. Results of Grace (1973), Clift et ale re
(1978), and Bhaga and Weber (1981) are summarized in figure 1.
The range of fluid properties and bubble volumes covered by this graph is
very broad indeed. The Morton number for most practical steam-water flows is
. - 1 3 - 2
In the range 10 to 10 • When for given parameters, such as the mass flow
rate, a eq is known, the chart can be used to determine, for example, under
which practical circumstances the Reynolds number is high.
Of major concern in the analysis of sections 3.2 and bubbles with very low (less than 1) or very high (larger
3.3 are spherical than 600) Reynolds
8 DISTR
numbers. Allowances can be made for ellipsoidal bubbles, since small bubble distortion is accounted for by the use of a correction factor for the steady frictional force, derived by Moore (1965).
104 1---~---_,---,_----_r~~---, 1000~---+_--~~----~~---~----_+--~---~,
l
t
IO~---~L---~~L---~~o
spherical 0.1 10 100 _ E o 1000 Figure 1Shape regimes for bubbles summary of results of
For high-pressure water-vapor systems the analysis is therefore restricted
either to cases when is larger than 0.03 mm and less than 1 mm, or to
system pressures of order 150 bars and larger (see figure 1). In the latter -4
case Eo is less than 10 ,which corresponds to very small values of a • eq
1.3 Bubble-plug flow pattern transition
In the literature no systematic study was found of any relationship between bubble plug flow pattern transition and, for example, the tube radius. One explanation for this is probably the fact that objective criteria have to be derived before a meaningful experimental comparison can be made for tubes with only slightly different diameters.
ij s 0.1 0.5 - - - ij' (m/s) s Figure 2 Measurements Bubble - Semi Churn No plug flow observed
Ro z 5.5 em ; 4 m tube
Flow Pattern Indicator Measurements Bubble - Plug Ro ~ 1.95 em ; 8 m tube Theory Ro = 5.5 em ; a = 2 mm Theory R 1.95 em a = 2 mm o
Comparison of experimental and theoretical observations in flow pattern map
10 DI5TR
No theoretical criterium was found in the literature that would predict a dependency on the radius for bubble slug flow regime transition. In contrast, in section 4, a rather straightforward application of some results on phase distributions of section 3 will be seen to yield theoretical predictions that show a dependency on the tube radius. These predictions are also shown in figure 2.
Visual observations were used to detect bubble plug flow pattern transition
in two tubes with really different radii: 19.5 mm and 55 mm. The tube
lengths were 8 m (19.5 mm tube) and 4 m (55 mm tube). Visual observations in an air-water system at atmospheric pressure were cumbersome in the 55 mm tube. No plug flow was observed in this tube, but reasonably intermittent developing plug like phenomena were observed near the upper end of the tube. This is an entry length effect caused by the short length of the tube. The points of transition from bubble to (semi) churn flow in the flow pattern map were found scattered in a band with a certain width (see figure 2)
Visual observations in the 19.5 mm tube correspond to Flow Pattern Indicator designations (see Van der Geld, 1985).
Fully developed bubble flow was never found in the 55 mm tube. In view of this and in view of the absence of plug flow, conclusions can only have a qualitative character.
With increasing tube radius, bubble plug flow pattern transition is expected to occur at decreasing superficial air velocity.
The trend in the theoretical results is the same as the one of the experimental observations. In view of the approximate character of the latter observations, it can only be concluded that experiments seem to support the validity of the criterion for bubble plug flow regime transition (section 4.2), and to emphasize the need for a theoretical way to calculate phase distributions such as the one in section 3.
2 CONSIDERATIONS ON PHASE DISTRIBUTIONS
2.1 Experimental observations
Detailed measurements of time-averaged, local properties of vertical, co-current adiabatic air-water upflow in a circular pipe were taken by Serizawa
(1974). For low-quality flows a pronounced bubble layer occurred near the wall (see figure 3).
0.3
LocalVOidl
Fraction 0.2 0.1 \ I\
" ... _-""""... .... ,.--,,----r---
... - ... , ... ... _-.,., I:
Bubble Flow o+---~~---~ .~ " 1.0..
o
1.0 Figure 3Void fraction profiles (Serizawa, 1974)
Atmospheric air-water flow; U
=
1.03 m/s12 DISTR
When the void fraction in the centre of the tube exceeded a value close to 0.25, transition to plug flow was observed. The wall peaking of void fraction curves for plug flow was much less pronounced than for bubble flow. Similar findings have been reported by Kobayasi et al.(1970), Nakoryakov and Kashinsky (1981), and Galaup (1975).
For vertical co-current air-water downflow, Oshinowo and Charles (1974) have observed opposite trends. No bubble layers at the wall,but only some bubble agglomeration near the central part of the pipe occurred. No void fraction
profiles have been reported for downward flows. A model of phase
distribution should predict these observed trends.
For vertical co-current downflow of droplets and gas, Schafer (1984) has found wall peaking of particle concentration profiles. He has reported on a holographical measuring method ("00ppelimpulsholographie") that had been
applied succesfully to flows of droplets with sizes varying from 'fa-6 m to
0.1 mm.
For vertical co-current upflow of water droplets in air, Cousins and Hewitt (1968) have measured a reduction of droplet concentration near the interface of the annular liquid film that was moving up the wall. They have also reported a depression of droplet diffusion coefficients in the zone near the interface. These trends are just opposite to those for bubble flows, but should also be predicted with a model of phase distribution.
A
typical value for the bubble radius a in the above mentionedeq
measurements is 2 mm, whereby large particle Reynolds numbers (ca. 600) were involved. Low Reynolds number (less than 10) data for rigid particles have been reported by Segre and Silberberg (1962) and by Oliver (1962) for example. A high concentration of particles with a higher density than the
liquid has been observed at a radial location of about 0.6 R • Though these o
experiments have been performed with laminar (Poiseuille) flows, the
calculation principle of section 3.1.1 is applicable to various kinds of well-developed gas/liquid velocity profiles. Therefore, it is interesting to predict void distributions for relatively slow particles too, especially since energy dissipation mechanisms in this case are rather different from those in cases where boundary-layer ideas are applicable.
The entrapping of gas by transient large eddies or coherent structures in free shear layer flows is a commonly observed phenomenon in two-phase flows (see for example Foussat and Hulin, 1983). At the university of Eindhoven too it has been observed (see Van der Geld, 1985) that at higher mass flow rates, bubbles are gathered in the turbulent wake behind an obstacle. Pressure gradient forces are largely responsible for these findings.
However, pressure gradient effects in radial direction of a tube must be the same for upflow and downflow, and therefore cannot account for the different trends described earlier in this section. This argument is considered in more detail in the next section. The carrying of bubbles by large eddies is not dealt with any further in this paper.
2.2 Physical considerations
The problem of determining the density distribution of a cloud of bubbles moving relative to each other and relative to and together with a liquid flow in a circular tube is too complicated to find an exact solution to. The best approach is to solve simplified versions of the problem first, and subsequently extend these versions to match the real situation as much as possible. This approach is followed throughout section 3. As a first step the motion of a single bubble close to a wall is considered.
Attention is focussed on stationary, vertical flow situations that are invariant for translations in an axial direction. Therefore, the Kelvin
impulse and virtual mass effects become negligable (Lamb, 1963). As noted
already in section 2.1, radial pressure gradient forces can be neglected as well.
If one bubble is released in a vertical upflow of water, it has a tendency to move towards the tube wall, while a bubble released in down flow migrates
to the tube axis (see section
2.1).
Obviously, a force experienced by thebubble exists that is different from the common buoyancy force Fb and the
uo
drag force antiparallel to the bubble velocity. This transverse force accounts for the observed tendencies. Physical origins of this transverse force are now explored with the aid of figure 4.
Consider the case that one particle or bubble is moving at some radial
14 DISTR
reference in which the particle is stationary, approach velocities of the liquid correspond to one out of four typical flow situations (see figure 4). Obviously, the presence of the wall alone can not account for differences between these four cases.
The difference between co-current upward (A or C in figure 4) and co-current
downward flow (8 or 0 in figure 4) is the sign of the vorticity of the
approaching liquid. The difference between case A and case C (figure 4} is
the sign of the approach velocity U 1 at the location of the particle. The
re
same difference exists between cases 8 and O.
A WAll
c
Figure 4 COCURRENT UPWARD P bubble < ~iquid COCURRENT UPWARD Pparticle > ~IqUid U=O -,--------I I I I I I
,
I WAllo
Fluid flow approaching a particle at rest
COCURRENT DOWNWARD
P bubble < PllqUld
COCURRENT DOWNWARD
Pparticle > PUqUid
These observations can be explained by assuming that the transverse force in question is a function of both Uland the vorticity in
re
vorticity S vor t in the liquid may, in general, be generated
the liquid. The by the presence of other particles. The above observations in correlation with figure 4 can be explained further by taking the direction of the transverse force to be indicated by the vector product -vor -re S t XU 1.
There is some evidence for the validity of the above assumptions regarding a general expression of the transverse force. In some particular circumstances a vorticity induced lift force is experienced by a very small bubble (see for example Vil'khovchenko and Yakimov, 1980). In that case the force has been found to be proportional to -vor -re S tXU l' with S vor t being the vorticity of the unperturbed liquid only. The proportionality constant has been approximately calculated for weak shear fields only.
For larger bubbles the above considerations in correlation with figure 4
indicate that higher (odd) powers of S t a n d U I can become involved as
vor~ re
well, while in bubble swarms S t will also include vorticity generated at vor
other bubble surfaces.
It will be shown in this paper that, for fully-developed flows, it is possible to calculate phase distributions without any detailed knowledge of the transverse force or bubble diffusion. In such cases migration due to the transverse force balances bubble diffusion, and the extra information that can be obtained by estimating diffusion coefficients will be obtained by using a principle of minimum energy dissipation.
Though severe approximations are involved in the calculations of section 3.2, directions of the transverse forces that can be calculated from expressions given in this section do agree with the force directions that follow from the above observations.
The above considerations and figure 4 show that the vorticity in the liquid as well as the approach velocity U I have to be embodied in a phase
re
distribution model in order to account for differences in observed trends.
In the following sections, a method of predicting radial phase distributions in a vertical tube will be outlined.
16 DISTR
3 QUANTITATIVE PREDICTIONS OF PHASE DISTRIBUTIONS
3.1 Minimisation of energy dissipation
3.1.1 On the principle of minimisation. In this section, general expressions are derived for phase distributions in the cross-section of a tube. Only (approximately) axisymmetrical two-phase flows are considered.
-- From a physical point of view, minimisation of energy dissipation as a
criterion for dispersed flow phase distributions seems worth while
investigating. In two-phase flows, energy dissipation can be minimised by the selection of a particular distribution of bubbles/particles. In the following, no given liquid flow field in a tube with prescribed boundary conditions (on the interfaces) is studied, but rather the flow field is created by the changing position of the boundaries. That the distributions derived in this way correspond to well-developed flows as observed in experiments is demonstrated by a comparison with experimental observations. The theoretical results of this section will be applied to different kinds of undisturbed liquid velocity profiles. Only one profile, the parabolic
two-dimensional one, has been shown, by Raleigh (see Lamb, 1963, page 618),
to have the property that the dissipation in a cross-section of the tube is less than in any other motion consistent with the boundary conditions. In this section, the addition of a few, tiny bubbles is considered as a first order perturbation with one degree of freedom. It stands to reason to postUlate tentatively that in this nearby single phase flow, bubbles take up positions in the flow fields in such a way that again dissipation in a cross-section of the two-dimensional two-phase flow is less than in any other motion. Note that in this laminar liquid flow diffusion is only effectuated by thermal (Brownian) movements.
The principle of minimisation of energy dissipation by selecting a
particular distribution for the dispersed phase, is henceforth assumed to hold for turbulent liquid profiles as well.
-- From a mathematical point of view it appears that no general variational
principle can be derived for transport processes, despite the hope expressed by the mathematician Euler in 1744 (quotation adapted from Finlayson and Scriven, 1967):
being the handiwork of an all-wise Maker, nothing can be met with in the world in which some maximal or minimal property is not displayed. There is, consequently, no doubt but that all the effects of the world can be derived by the method of maxima and minima from their final causes as well as from their efficient ones.
The formal definition that well-developed flows correspond to a stationary distribution of OBJ's is adopted. That such a distribution
bubble/particle flows, described as well-developed
circumstances too, will be demonstrated in section 3.3.
corresponds to
in practical
3.1.2 Single object. Let R denote the tube radius and I' the radial distance
o
to the tube axis. Let x
=
r/R • Object OBJ may be a bubble, droplet oro
particle of any shape, volume Volar mass density P oBr However, the
equivalent radius a eq of OBJ is supposedly less than ca. 0.3 Ro' In 3.2.1
oBJ will be a long cylinder. OBJ supposedly travels at an (approximately)
uniform axial velocity at a (time-averaged) radial location roo In this
section only non-deformable OBJ's are considered.
In the frame of reference in which OBJ is at rest, the liquid velocity
vector v(r=r) has the magnitude U 1 at a far distance from Cl8J. By
- 0 re
definition U 1 is positive when v(r=r ) is parallel to the gravity vector. re - 0
Let I(r) denote the energy dissipation caused by the presence of oBJ at the
location r. Let the eccentricity function f be defined by fer)
=
I(r)/I(O)on the domain (O,R -a). Here a denotes half the maximum radial width of 08J. o
For spheres, a is the radius. Calculations of f for values of x laying
outside the range (0, 1-a/R) will be seen in section 3.3 to have no
o
physical meaning. The importance of the eccentricity function f will become evident at the end of section 3.1.3.
Let F buo denote the buoyance force on oBJ. In a liquid of density P LIQ'
F buo is equal to Vol g ( P LIq - P o8J) • Because of this definition, F buo is
positive when the liquid phase is more dense than oBJ.
Energy dissipation I t in the boundary layer of oBJ due to its rotation is
ro
presumably much less than the energy dissipation caused by the non-zero axial movement of ClBJ. Consider the case that the radial velocity of oBJ is much less than Urel " It can now be argued that, in many cases of interest, the energy dissipation due to radial movements of oBJ can be neglected as
18 DISTR
well.
Stochastic movements of OBJ in a radial direction are supposedly caused by turbulent structures in the liquid phase and hydrodynamic transverse forces
(see 1.2). Turbulent eddies with sizes up to ca. a eq support a fluctuating
diffusion force Fdiffo
A
similar force has been introduced by Langevindiffusion is
( 1908) to explain Brownian motions. Thermally induced negligible in most practical circumstances.
Eddies larger than OBJ may entrain OBJ in their flow field. Such eddies are considered as a kind of flow disturbance that mayor may not be helpful in redistributing particles; their effect is further assumed to be included in
F dUf'
In stationary flowst migration induced by the diffusion force. balances
migration induced by the hydrodynamic force F caused by the
time-hydr
averaged vorticity in the main stream. From the force balance :
(1) F (t)
=
Fd·ff(r,t) + Fh d (r) - Fdacc ~ y r rag
the mean square of radial displacements can be deduced via
differential equation (see e.g. Gardiner, 1983). Here F
ace to the acceleration of OBJ; t denotes time. Axial components be neglected.
a stochastic is proportional
of F will
hydr
In the Stokes region. the radial component of the friction force Fd rag can
be taken to be equal to 61T ~ UQaeqUrad. U
rad( t) is the radial velocity of
OBJ. Stokes law has been deduced for constant motions of translation. and in principle it can not be applied at the moments of strong acceleration that
occur in the radial movements. However, on account of the
character of the motion, departures will on average cancel out
irregular (see also
Einstein, 1926). The extra energy dissipation I due to radial movements
rad
of 08J in the Stokes region is proportional to ~ UQU:ad• In other cases
F drag can be taken to be proportional to Ura 2 d' making I d proportional to ra
U;ad'
In this section, attention will be focussed on phase-distributions for
stationary situations where the time-averaged effect of Fdiff is
counteracted by F
hydr in every region of the tube. The very small values of
U d then make it possible to neglect I d(r) with respect to I(r) at first
ra ra
an expression for I
rad, for example:
(2)
In transients, the momentary value of the RHS of (2) may have a value that can not be neglected and the diffusion coefficient must be calculated in order to determine e.g. the length of the developing zone ••
At the present level of approximation we have :
(3) I(r)
=
Fb .U 1uo re
Note that (3) can be used under all circumstances, if U
rel is considered to
be an "effective" or "apparent" relative velocity.
When the drag force on OBJ is proportional to U
rel, i.e. Fdrag= e.Urel, then
I(r)=F b
2
/e
and f(r)=C(O)/C(r). Since Fb is independent of r, the entireuo uo
radial dependency of the energy dissipation is hidden in C, which is a
friction force parameter. In this respect, eccentricity is entirely governed by friction forces.
let the undisturbed, single-phase liquid velocity in the frame of
in which the tube is at rest be given by V (x) V g(x.R). Note
o max 0
is positive when liquid flows upward, and that U 1 is positive re
rises with respect to the undisturbed liquid velocity. In the turbulent pipe flow Nikuradse (see Hinze, 1959) found:
(4) g(r)
=
(1 _ r/R )1/n o reference that V max when OBJ case ofwhere n=n (Re), Re being Reynolds number 2Ro V gem P LIQ/ ]..I LIQ. For example
n(4000) = 6.0 ; n(110000)
=
7.0 ; n(3200000) 10.0. By partial integrationone simply derives :
(5) V max
=
V gem ( 1 + 1/ ( 2n) )( 1 + 1/ n)The velocity distribution in the convenient form given by (4) is adapted
without loss of generality. Any other expression for g(x), e.g. a
20 DISTR
3.1.3 Swarms of objects. Let us now presume that more than one OBJ is carried by the fluid. Let N(x) denote the (time-averaged) number of identical OBJ's per unit of volume around x. Swarms of unidentical particles are not considered for reasons of simplicity. It is assumed that, on the average, the dispersed phase occupies less than ca. 30
%
of a volume of space_Let e(x) be equal to the product of N and Vol' For bubbles, e denotes the void fraction. The function e(x) can be determined by minimising energy dissipation (see 3.1.1).
In a mixture, the buoyancy force has to be corrected with a factor (1-e). This yields, from
(3):
(6) I(r) Fb uo re .U 1.(1-e)
The total energy dissipated by the relative motion of the OBJ's per unit of time per unit of length of the tube is given by :
(7) I tot ;:: b of 1 dx x e(x) (1-e(x)) Fbuo urel(x)
in which b stands for 2 7T R 2
Iv
1-o 0In the frame of reference in which the tube is at rest, the total particle flux is given by :
(8) J ;:: b
f
1 dx x e ( x ) (V ( x) + U 1 ( x) )o 0 re
The flux J is kept constant while minimising the energy dissipation. For this purpose, a Lagrange multiplier will be introduced.
quantity yet undetermined, first suitable expressions ,for found.
Since U 1 re is a U 1 re have to be
Many relative velocity data have been correlated with expressions of the general form (see, for example, Richardson and Zaki, 1954, Smissaert, 1963, and Wisman, 1979):
where c
6 denotes a constant and c5 is the relative velocity of an object in the absence of other objects. The definition of the eccentricity function (see 3.1.2) yields:
At present it suffices to know that c
5(O) has a certain value that depends on some physical parameters given. The parameter c
5(O) is discussed in some detail in section 3.3.
Wisman (1979) correlated bubble data to find C
s
=
4 (and c5
=
0.36), while experiments performed with bubble swarms by Smissaert (1963) show that the value of c6 increases with increasing superficial water velocity, and that
c
S
=
2.5 when V gem=
0.244m/s.
The values 2.5 and 4.0 for bubble flows are considered as limiting values for c6' and both values will be examined in later sections.
As for swarms of droplets having a higher density than the surrounding liquid, data published by Richardson and Zaki (1954) has shown that the power c6 depends on Rep and has values in the range (-2.4, -4.7). The values -2.5 and -4 are close to the limiting one for droplet swarms and are precisely the negative of the limiting values of bubble swarms. For this reason, the values -2.5 and -4 will be examined in later sections.
The above equations lead to a variational problem with free boundary conditions (see Courant and Hilbert, 1968, page 179), for which the general Euler equation reads :
( 11)
and for which the integrand G
1 of Itot as well as the integrand G2 of J do de
not depend on e,x = dx • Using the expressions for G
1 and G2 given by (7) and (8), one derives from (11) the following expression in the phase-distribution e(x):
(12 ) (1-e)(c6-2) -1 c 1 c
s
6 +(e- -1)(1-1 -1 V g(x)(1-e)
Ie)
= 0a a a max 5
22
Lagrangian multiplier 1 :
a
(13)
OISTR
From a given value for e(O), one calculates I from (13). Subsequently e(x) a
is determined numerically from (10) and (12), for example with the aid of Newton's algorithm.
Equation (10) shows the importance of the eccentricity function f in this calculation. Expressions for f will be derived in the next section. The
velocity V max appears implicitly in f(x), but also has a latent presence in
g(x), as shown by (4) and (5).
Alternatively, the Lagrangian multiplier can be
mean volume fraction
e.
Integration of (2x/R )e(x)o
expressed in terms of the over the range (0,1-afRo) for x with the aid of (12) does not lead to a relatively simple
formula like (13). It is easier to calculate
e
by integration of (2x/R )eo for a given value of e(O) with the aid of (13) and (12). Repeating this calculation for many values of e(O), one obtains the correspondence between
e(O) and
e
which determines the functional relationship laCe) with the aidof (13). The mean volume fraction occupied by the particulate phase will be considered further in 4.2.
A preliminary check on the correctness of (12) and (13) is obtained by
putting f(x)=1 for every value of x. This means the abandonment of
hydrodynamic interaction with the vorticity in the main stream as well as
the abandonment of wall influences. Resulting curves are named (f=1)~curves.
The result of this check is that 08J's show some tendency to agglomerate near central parts of the tube. Transport velocities attain maximum values near the centre of the tube. Therefore, for a certain value of the particle flux the total number of 08J's per unit length of the tube is smaller when
concentration occurs near the centre than when concentration occurs
elsewhere. Energy dissipation in this case being proportional to N, the
minimisation of energy dissipation corresponds to conglomeration near
central parts of the tube when f is constant. This explains the result of
the check on (12).
Note that, on account of (9), conglomeration of bubbles implies a higher relative velocity. In downward flows, conglomeration therefore causes lower
gas transport velocities at each given radial location, while in upward flows the above mentioned tendency to agglomorate near the centre part is reinforced by the increasing relative velocity.
Figure 6, to be discussed in section 3.3, shows some (f=1)-curves. These
curves are seen to depend on the sign of V • This dependency is entirely
max
governed by the correction on bubble interaction, such as the one given by
(9),
in the way described above. However, when the correction given by(6.9)
is expanded and only the lowest powers of e(x) are retained, yielding Urel=
c5(1+c6e), it can be shown that resulting (f=1)-curves become independent of
the sign of V •
max
The assumptions underlying (12) are summarized below:
- axisymmetrical, fully-developed steady state flows
- e(x) piecewise continuous on
(0,
1-a/R)
eq 0
- only identical, non-deformable bubbles/particles occur; no phase-change
- radial velocities much smaller than U 1
re
- I ro t(r) much smaller than I(r)
- negligible influence of any transverse pressure gradient
- interactions between bubbles/particles are expressed by correlations of the general form :
-c6
Urel
=
c5(1-e)In the following, the mean, local liquid velocity profile will be
approximated by the undisturbed profile given by equation (4). Despite of all the approximations involved, the trends of the observations as discussed in section 2.2, can be explained by the theory, as is demonstrated in the following sections.
24 DISTR
3.2 Determination of the eccentricity function
3.2.1 Turbulent flow. In this section, the vorticity in the flow around
bubbles or particles is assumed to be confined to a thin boundary layer and a narrow wake. Outside this region, perturbations to the rotational pipe flow are described as an irrotational flow. An estimate for the eccentricity function is obtained by considering the case of infinitely long cylinders moving between two parallel plane walls. This obvious simplification still
gives results applicable to the three-dimensional case, since the
eccentricity function represents proportionalities rather than magnitudes of physical quantities. Also, the radius a is assumed to be small as compared to the tube radius R • The free stream velocity is expanded in a Taylor
o
sequence of which only three terms are considered. For ease of reference, bubbles are considered first. Particles with higher density than the liquid phase are treated seperately, at the end of this section.
Since only relative dissipation rates are of importance in the context, dissipation in the boundary layer of an interface has accounted for only when the vorticity in the liquid affects the
present to be boundary layer. The occasional presence of this interplay between boundary layer and free stream vorticity has been demonstrated by Taylor (1936).
The eccentricity function depends on the number N of bubbles and on the distance r to the tube axis. The former dependence is supposedly dealt with
by a correction of the form given by (9). Furthermore, in the first part
of the current section the latter dependence is assumed to be apparent only
through the "free stream" function g (see (4». This means that the
presence of the tube wall is neglected in first approximation. This can be done at locations at some distance from the wall for highly turbulent flows, for example. In the second approximation, the presence of the wall will be accounted for in the last part of this section.
Consider the flow around a circular cylinder bubble with radius a in the reference
stationary (see figure
frame in which the cylinder is at rest. Let the centre cylinder be the origin of co-ordinate systems (y,z) and
5).
The liquid approaches at large distances from the body at velocity
of the
(14 )
The coefficients 00 and 0
1 depend on the position of the cylinder, and hence
on r (or x). I
k/r.
" , Z I I Figure 5 Coordinate systemsI
i
~
II
.
The disturbance motion due to the presence of the bubble is represented by potential flow, the acyclic part of which is determined uniquely by the condition of zero flux at the boundary surface. The potential flow solution satisfies the condition of vanishing velocity difference in directions normal to the surface, but not of that of zero tangential stress. The
rotational part of the flow is confined to a thin (boundary) layer of
thickness O(Re -0.5), and viscous dissipation per unit of volume in these p
regions is of the same order as in the much larger region of irrotational
flow (Moore, 1965). Hence, in first order approximation, one may consider
dissipation in the irrotational flow alone.
By adapting the constants of a truncated Laurent series to the boundary condition, one finds the stream function P . for the whole motion to be:
Sl
(15 )
26 DISTR
applied to three-dimensional bubble flow, and there is no counterpart to the cyclic constant in a three-dimensional field. Velocity components u
r and ut are determined locally with the aid of:
( 16) u r = r -1 P.
e
S~ , ut = - p . s~ ,r
The rate of dissipation of mechanical energy by viscosity, per infinitesimal
volume dV, can be evaluated from (see Batchelor, 1967):
( 17) I
dV
=
dV IJ LIQ(u . . ~,j + U • • J , l )(u . . 1,J + u . . J , l )/2where the Einstein summation convention has been used. After combining (15)
through to (17) and integrating over the region outside the cylinder one
obtains per unit length of cylinder :
in which c
1 equals
the wall effect to
Velocity U in (14)
1 but has been introduced to facilitate the discussion of
come.
and (18) equals the relative velocity U l' and is also re
determined by
(3).
The buoyancy force Fb per unit length of cylinder is• 2 uo .
gwen by TIa g( PUQ- PoBJ). Let c
3 denote g( PUQ- POBJ)/ IJUQ• Solvlng (3)
and (18) with the trivial constraint on U that in case 0
0=01=0 U is unequal
to zero, one obtains :
(19 )
The eccentricity function f now follows from U(r)/U(O).
By expanding Vo(x) in a Taylor series around the point x one can easily show from (14) that when (4) is valid:
The "apparent" first order dipole strength in P . is the factor multiplying Sl -1 . ( ) .
r Sln t 1n
P .•
If we were to useSl 2 3
Vo
= -
U - 00 - 01 Y - 02Yinstead of (14), then the "apparent" first order dipole strength would amount to:
2 4 6
a U + 01a /4 + 320
2a
Using (20) and a similar expression for 03, one can show that for x=0.9 the
expansion given by (14) is insufficient when a is about 1 mm, n=7 and
R
=19.5 mm, but that, under the same conditions, the expansion suffices wheno
a equals 0.1 mm. It can be demonstrated that every even power of y in an expansion like (14) adds to the magnitude of the "apparent" first order dipole.
In second order approximation, wall effects are included. As will be demonstrated, c
1 now becomes an increasing function of x with values larger
than 1. The "apparent" first order dipole strength, which already changed
value from a 2U to a 2(U + 01a2/4) (see equation (15», will be shown to
increase with decreasing distance to a wall. Therefore, the value of c1
increases with increasing x. Note that, in third order approximation, the
other terms on the RHS of (18) are affected by the presence of walls as
well.
For small and intermediate values of x, the function f is again determined approximately by studying the case of a cylinder. Let the centre of a cylinder be moving at distances band c from two parallel walls. By applying the method of reflections, one obtains infinitely many first order dipoles to ensure a zero velocity normal to the walls. These dipoles induce second order
third
with v
dipoles near each other. Again an infinitely long array is necessary; and higher order dipoles are induced in the same way_ Define:
E1(Y) == tg( rry/(2b+2c) )/(1+v.tg( rry/(2b+2c) )
E (y) == tg( rr (y+a2/(2b+2c»/(2b+2c) )
o c
E
2(y) == E (y)/(1 + v.E (y) )
2 0 0
=
cotgh (rrz/(2b+2c) ) (see figure 5). When only the first terms ofthe stream function are retained, one obtains :
Psi
=
a2U (1+v)(2b+2c)-1 (-E1 (y)+E1(-y+2b) +
+ (a/(2b+2c»2(-E
28 DISTR
(to include first, second and third order dipoles and their mutual interaction at distances as large as 2N(b+c) a total number of 2+24N different terms need to be added up). Neglecting distances between dipoles
near the centre of the cylinder, one obtains an estimate for the "apparentlf
or Ifeffectivelf dipole strength by the summation of the strengths of all
these (two-dimensional) dipoles. It is not difficult to see that, for fixed z, the total dipole strength increases when the absolute value of (b-c) increases.
For large values of x it suffices to study the case of a spherical bubble
moving parallel to a plane wall. Again the method of reflections is applied. Recently, Jeffrey (1973) used for two-sphere potential problems a method which he calls the method of twin spherical expansions. For the present
level of approximation, it sufficient to use Butler's sphere theorem (see
for example Streeter, 1961) to evaluate the flow due to a source and a sink near a sphere of radius a. In the limit where the source and the sink become a (three-dimensional) dipole of strength m, they induce in the sphere a
dipole of strength ma3/(2h)3, and a uniformly distributed dipole of total
strength -D.5(ma 3/(2h)3). Here 2h represents the distance between the sphere and the dipole of strength m. Neglecting all higher order reflections, one finds an estimate for the "apparent" dipole strength by summation, as noted before. The result is:
( 21 ) c 3 -3/
1
=
1 + a h 1 6where h = (1 - x)R denotes the distance to the wall. Equation (21) is valid
o
for values of x close to 1, which justifies the modelling with a plane wall,
but not too close to 1 to allow for the neglection of the higher order
dipoles.
objects with a higher density than the continuous phase, the group c3
(proportional to the density difference) is negative. In this case equations
(3) and (18), in combination with the constraint imposed by the case that
both 00 and 01 equal zero, yield :
This equation differs from (19) only because of a minus sign. All the other equations remain the same.
3.2.2 Creeping flow. For a sphere moving in a circular cylindrical tube at a
small relative Reynolds number, Happel and Brenner (1965) derived the drag
force on the sphere by the method of "reflections". They only considered the case, that g(x)=1_x2 (see equation 4), and rotations of the sphere were neglected. In the notation of this chapter and including a correction due to Kaplun et ala (see Happel and Brenner, 1965):
(22) Fdrag ::: C.U rel, with: C(x)::: (2 1TJ.l UQac4)(1 - c4a.h(x)/(3Ro»-1(\+ 3Rep/2 + + (9/160)Re In(Re /2) + ••• ) p p c4 ::: (2 + 3 J.loBi J.l UO )/(1 + J.loBi J.l UO )
Here Rep denotes the particle relative Reynolds number 2aeqUrel P UO/ ]J UQ.
For a bubble c4 usually is close to 2. Because of a lack of knowledge of
higher order reflections, (22) is less accurate close to the wall. Values
for the eccentricity function hex) have been obtained by numerical
integration (see Happel and Brenner, 1965). For example:
(23) h(O) ::: 2.10444 ± 0.00002 h(0.2) ::: 2.07942 ± 0.00002 h(0.4) ::: 2.04401 ± 0.00002 h(0.6)
=
2.16965 ± 0.00002 h(O.S) ::: 3.2316 ± 0.0003 2Since 1(1')
=
Fbuo/e,
one derives from (3) and (22) that f is approximately30 DISTR
3.3 Calculations of void distributions
In this section the results of sections
3.1
and 3.2 are used to calculatevoid distributions for some typical cases. Results are only given for a
tube radius R of equal to o 19.5 mm, but quite similar results were obtained
for other values of the tube radius. For each V the parameter n is
max interpolated from values given in section 3.1.2.
In the case of turbulent pipe flow with high bubble Reynolds numbers Rep
(larger than
600),
equations (4),(5), (9), (12), (13), (19)
and(20)
apply.When the influence of the tube wall is neglected, c
1 can be taken equal to 1
and results like those of figures 6, 7 and 8, which are typical for air-water systems at room temperature and atmospheric pressure can be obtained. Either the void fraction at x=O or the average void fraction can serve as the input parameter.
For adiabatic flow of non-deforming bubbles, the
decreases from some value at the location x=1-a/R o
void fraction monotonely to zero at the wall. However, in evaporator tubes the void fraction at the wall is different from zero. To avoid loss of generality, curves drawn correspond to a range of
values for x varying from 0 up to 1-a/R only. Moreover, calculations of f
o
for values of x closer to 1 are not allowed since 00 and 0
1 can not be
defined in this region.
The bubble radius a cannot be chosen too large since the truncated Taylor sequence of (14) falls short close to the wall (see 3.2.1). In usual air-water experiments, bubble radii are encountered of the order of 2 mm. Therefore, one has to be careful in comparing theoretical results to experimental observations. Also,
because of the assumption of high As for bubbles, observation shows
a can not be chosen arbitrarily small
Re (see also sections 1 and 2).
p
that due to the action of surface tension, distortion is small for bubbles of radii up to about 0.5 mm in pure water. The application of Moore's correction factor (see section 1) would allow for ellipsoidal-shaped bubbles in the theory.
hydrodynamic interaction with the vorticity is taken gradients in the unperturbed liquid profile are small in of the tube, differences between (f;1) and other curves
along. Since the the central regions are also small in those regions. Figure 6 clearly shows, how the interaction with the vorticity reinforces the tendency to agglomorate near the centre of the tube, but counteracts this tendency in upward flow.
0.5,---. 0.3
r--_
C6 • 2.5 v max - 2 mls0.2,--_____ _
C 6 - 2.5 0.1 Vmax = -1.27 rols v • -1.27 rols max f(x) • I O+-~--~--r_~--~~--_r--r_~--~ o 0.5 .. x Figure SInfluence of transverse forces on bubble concentration It can clearly be seen in figures 7 and 8 how void peaking occurs near the
tube wall for upward flows (positive V ), but not for downward flows
max
(negative V max ). Downward means parallel to the gravity vector.
A comparison of the results for c
6
=
2.5 and cS=
4 shows that in order toobtain reliable quantitative results the value of C
s
is important. However,the results also clearly show that in order to obtain semi-quantitative results or a good understanding of radial distribution mechanisms, the parameter cS' representing bubble interaction, is not essential.
32 DISTR 0.5
e
a - 0.4 mm Vmax )0r
0.4 c6. - 4 Vtnax• 2 tn/s 0.3 0.2 j--=::::::::;;:::::-_ _ _ 0.1 o+---~~--~--~--~~--~--~--~~ o 0.5 --004 ... X Figure 7Void fraction profiles in upward flows
0.5,---,
e
a = 0.4 mm Vmax <01
0.0 C 6 - 2.5 Vtnax• -2 tnlsO.3r--__
""",,=:b::=:::::::::::::::-+-_0.2,-__
-,======::::::::::&-__
C 6 • 2.5 0.1 V - -\ .27 tnls tnax C6 • 4 V - -1.27 tnls max O~--~~--~--~--~~--~--~--~~ o 0.5 ---004 ... X Figure 8When the influence of the tube wall is partly accounted for and c
1 is given
values larger than 1 with the aid of (21), the trends for upward and downward flow become even more different (see figure 9). Agreement with experimental data will become even better when the eccentricity function f for turbulent pipe flow is determined more accurately. This is already indicated by the fact that the "apparent" first order dipole strength is
increased when higher even powers of yare included in (14) (see 3.2.1).
e
r
0.5 a = 0.4 mm C, )1 c 6 = 2.5 0.4 0.3 0.2 , -_ _ _ _ -.--.._ O. I V max = -I • 27 ml s o O.S - - - t .. ~ X Figure 9Void fraction profiles for bubble flows The wall influence is estimated
In real circumstances, bubbles may experience some adherence to the wall. This might strongly influence the eccentricity function and promote further wall peaking.
A further correction is obtained by taking into account the perturbation of g(x) due to the presence of a bubble layer near the wall. This can be done in the following way_
34 DISTR
Let us assume that a bubble layer extends from average void fraction e
1 and average (liquid or
r:::R
1 to
bubble) the mean
V max,o and V gem,o denote the maximum and
unperturbed velocity profile respectively.
0.5
e
a '" 0.4 nun Vo corrected1
C6 = 2.5 C, )1 0.4 ... I I I I 0.3 , ' I I , I I I I I 0.2 m/s 0.1 o 0.5 - -... X Figure 10Void fraction profiles for bubble flows
r:::, that it has
o
velocity v. Let velocity of the
Liquid velocity profile and wall influence estimated Assume that the relatively slow bubble layer near the wall acts as a continuous interface. The liquid flow inside the bubble annulus then has to accomodate to this boundary condition. The perturbed profile supposedly can be represented by an expression of the form:
(25) V
1(r) = V max, 1 (1-r/R2)1/n
with R2 determined by the condition
(26) V :::
V
(1-R /R
)1/nContinuity of mass requires that the average value of V
1 on the interval (0,
R
1) is equal towhich on account of equation (5) furnishes the relationship between V
max,o from
and V max, 1. The velocity v can be conveniently estimated
v=V
«R +R1)/2)+ U l«R +R1)/2).
- 0 0 re 0
Estimated values from figure 9 yield e
1=0.35 and R2= 0.019. With e(0)=0.3
and V = 2.07 m/s the corrections were calculated for the case already
max
shown in figure 9. The results are shown in figure 10, where the broken line
represents the assumed bubble layer.
o 0.5
---t.~ X
Figure 11
Droplet density profiles ; C
s
and c6 are negativeThough the correction on g(x), preformed in this way, is not particularly
36 OISTR
shown in figure 3, surely if one takes into account that these experiments have been performed with larger bubbles (see section 2.1). The author believes, that a rigorous determination of e(x) would require a more precise knowledge, not only of the eccentricity function, but also of the dependencies of g on e and maybe even e • The variational procedure used in
,x
section 2.1.3 can easily accomodate such improved knowledge.
If the dispersed phase has a higher mass density than the liquid phase, equation (19a) has to be used instead of (19). A high concentration of particles near the wall is now found for downward flows, and not for upward flows (see figures 11 and 12). These trends are just the opposite of those of bubble flows, which is in perfect agreement with the experimental observations described in section 2.1.
e 0.5~---~ a • 0.4 mm V • -2 m/s max Droplets Vmax <0 0.3
r----.L-.-!---_---J
0.2t---~r_---~i
0.1 a • 0.5 mm V"-1.7'm/s max O+---~~--~--~--~--r_~~~--~~ o 0.5 - -... X Figure 12Droplet density profiles in downward flow
A peaking phenomenon near the wall is also calculated for the case when Re p is much less than 1 (see figure 13). For Stokes flow, equations (12), (13),
(23) and (24) apply. The relative velocity cS(O) in this situation is calculated from the equation
To the present level of approximation, results of the computations show no difference between upward and downward flow. Only a wall effect has been accounted for in the analysis leading to (22), (23) and (24), and no
interaction with the vorticity in the unperturbed liquid flow. Hence, the
similarity between upward and downward Stokes flow confirms to some extend the observations made in section 2.2.
•
0.5~---~ 0.3 0.2 O. I a • 0.01 mm V • 2 m/s max a '" 0.05 mm Vmax'" 1.6 m/s O+-~~~--~--~ ... ~~--~--r-... ~" o 0.5 - - -.... ~ X Figure 13Void fraction profiles for Stokes bubble flows
Without the wall effect, spherical particles with low Reynolds number would travel along the streamlines of the undisturbed flow. The peaking phenomenon (see figure 13) can only be observed when Brownian motion and external causes, such as small flow disturbances, can effectuate a redistribution of particles. A long observation time and a long entrance region probably are
38 DISTR
required to establish a stationary flow condition in that case. No such experimental observations are known to the author.
The experimental results of Segre and Silberberg (1962) show a high
concentration of particles at a fractional distance of about 0.6 of a tube radius. The responsible radial migration of effectively neutrally buoyant particles has been explained as an inertial effect (Cox and Brenner, 1968).
Comparison shows agreement between the predictions of void distributions in bubble and droplet flow, and the corresponding experimental observations
described in section 2.1. The trends of four typical two-phase flows (see
section 2.1) are correctly calculated.
Therefore, it can be concluded that the minimising of energy dissipation by selecting proper places for bubbles is a good procedure for predicting phase distributions.
