Chairmen: W. Zijl, F.E. Avila
A STUDY OF THE THERMODYNAMIC COUPLING OF MASS AND MOMENTUM TRANSPORT AT LIQUID/LIQUID-INTERFACES
Alexander Tokarz and Dieter Mewes
Institut für Verfahrenstechnik, Universität Hannover
Callinstr. 36, 30167 Hannover, Germany
The mass transfer across a liquid/liquid-interface is influenced by the molecular transport
phenomena in the immediate vicinity of the interface. Recent theoretical investigations on the
thermodynamics of transport processes at interfaces conclude that there is a thermodynamic
coupling between the stresses acting within the interfacial region and the mass transport
processes. Applying this theory to the industrially important case of the transport of a
component across an interface, it is deduced that there is an influence of the dilation of an
interface on the mass transport.
To experimentally verify this assumption, experiments are carried out in a novel flow cell
where the plane interface between two immiscible liquids - toluene and water - is dilated in a
flow field. The effect of the dilational stress on the mass transfer across the interface is
measured. The velocity field in the immediate vicinity of the interface is measured using Laser-
Doppler-Velocimetry (LDV). The mass transfer as a function of the dilation rate of the
interface is calculated from photometric concentration measurements. The concentration
profiles in the immediate vicinity of the interface are obtained using holographic interferometry.
In parallel the mass transfer is numerically simulated.
From the LDV measurements, it is shown that the toluene/water-interface is particularly
sensitive towards interfacial contaminants. These cause the interface to become rigid which
results in a reduced mass transfer. By continuously removing the contaminants from the
interface, a mobile interface free of contaminants, moving at a constant dilation rate is
maintained for a long period.
The results from the transfer of toluene into the water phase in the binary system toluene/water
indicate an increase of the mass transfer rate with increasing interfacial dilation rate. However,
this is shown to be entirely due to the enhanced convection. No additional effect is found.
Similar experiments for the transfer of acetone across the toluene-water-interface reveal that at
the moving interface being free of any contaminants, Marangoni effects can be observed even
at very low acetone concentrations. At these low concentrations, the instabilities cease almost
immediately once the flow is stopped. This is shown by means of the holographic
interferometry. If interfacial contaminants are not constantly removed, blocking of the interface
by the contaminants is observed. This greatly reduces the velocity and the mass transfer at the
interface and also reduces the intensity of the instabilities.
To validate the theoretical assumptions described above, mixtures of various concentrations in
the ternary systems toluene/water with acetone, acetic acid and propionic acid, respectively, as
the transfer components are brought into thermodynamic equilibrium. Successively, the
interface is dilated in the flow cell and the region close to the interface is observed using the
holographic interferometry. By this, any changes in the interfacial concentration can be
measured with an accuracy of better than 10-2 mol%. However, in the investigated system, no
effect of the dilation on the thermodynamic equilibrium is detected for dilation rates up to 1 s-1.
Consequently, in these systems, the dilation either does not act as a driving force for the mass
transfer across the interface or the effect is negligibly small.
MASS TRANSFER IN ANNULAR LIQUID JETS*
J. I. Ramos
Departamento de Lenguajes y Ciencias de la Computación
E.T.S. Ingenieros Industriales, Universidad de Málaga
Plaza El Ejido, s/n, 29013-Málaga, Spain
Annular liquid jets are thin, cylindrical sheets of liquid falling under gravity from an annular orifice. Surface
tension makes the sheet converge onto the symmetry axis to become a round jet some distance beneath the
nozzle exit (Figure 1). Annular liquid jets have found many applications in inertial-confinement pulsed fusion
reactors where a pellet containing deuterium and tritium is caused to undergo nuclear fusion through intense
compression and heating by lasers, electron or ion beams. Annular liquid jets have also been employed to
measure the dynamic surface tension of liquids and the diffusivities of sparingly soluble gases in liquids. Other
applications include stack emission scrubbing, halide reduction processes, gas-liquid and liquid--liquid reactions,
reaction and control of toxic wastes, scrubbing of both radioactive and non-radioactive particulates and soluble
materials, the formation of hollow spherical shells in microgravity environments, etc.
Figure 1. Schematic of a compound, annular liquid jet
Although steady, inviscid and high Reynolds number, compound jets have been analyzed in the past, previous
studies of the fluid dynamics of annular liquid jets have been concerned with steady, inviscid flows by using
the equations developed by Boussinesq for annular sheets (membranes) of liquid or with equations averaged
across the jet. Only recently, there have been some analytical and numerical studies for thick and thin, slender,
inviscid, irrotational, annular liquid jets which have been concerned with the derivation of the leading-order
equations for these flows by means of perturbation methods.
In this paper, the fluid dynamics equations of slender, compound, annular liquid jets (Figure 1) at high Reynolds
numbers are first derived by means of perturbation methods for the case that the gases enclosed by the inner
jet and those surrounding the outer one are dynamically passive; this assumption is justified since the density
and dynamic viscosity of gases are, in general, much smaller than those of liquids and, therefore, the gases may
not introduce strong radial variations in the liquid although they do affect the fluid dynamics of annular liquid
jets through friction. It is shown that the leading-order equations of slender, compound, annular liquid jets
depend on the pressure of the gases enclosed by the inner jet. These gases are absorbed by the flowing liquid
so that the fluid dynamics equations are coupled with those for mass transport in the liquid. This coupling
intoduces an integro-differential dependence on the governing equations which has been studied numerically
by means of a domain-adaptive finite difference method which maps the time-dependent, curvilinear geometry
of the annular liquid jet into a unit square, in order to examine the nonlinear behaviour of slender, compound,
annular liquid jets in the presence of mass transfer under isothermal conditions. It is shown that, for slender
jets, the leading-order fluid dynamics equations only depend on the interfaces' leading-order radii and that, due
to the small mass diffusivities of gases in liquids, the steady state mass absorption rate of the gases enclosed
by the compound, annular jet is small, and that there are certain parameters for which, due to the coupling
between the fluid dynamics and mass transport equations, Hopf bifurcations occur. When these bifurcations
are periodically forced by, for example, a sinusoidal injection rate, quasiperiodic and chaotic phenomena may
take place which can substantially increase the mass transfer between the compound, annular liquid jet and
the gases that it encloses. These dynamiuc phenomena have been analyzed by means of time series, Fourier
transforms, and Poincaré diagrams.
An example of the periodically-perturbed Hopf bifurcation that occurs in annular liquid jets with mass transfer
is illustrated in Figure 2 which shows the pressure coefficient, the jet's mean radius at the axial location where
the annular jet becomes a round one, the phase diagram of the jet's radius at the axial location where the
annular jet becomes a round one, and the mass absorption rate at the jet's inner interface for a solubility ratio
of 5, and nondimensional amplitude and frequency of excitation equal to 0.01 and 0.50, respectively. This
figure clearly shows that the flow is characterized by tori with incommensurate frequencies, and an oscillatory,
non-periodic pressure coefficient and mass absorption rate.
In the absence of excitation, it has been shown that, upon increasing the solubility ratio, the flow first undergoes
a Hopf bifurcation from a steady state to a limit cycle which, upon a further increase in the solubility ratio
undergoes another Hopf bifurcation which results in a two-period limit cyele. Further increases in the solubility
ratio cause another bifurcation to a three-frequency motion, followed by strange attractors. This sequence of
bifurcation is reminiscent of the Newhouse-Ruelle-Takens route to chaos in contrast to the period doubling route
of Feigenbaum and the intermittent route of Pomeau and Manneville. The results shown in Figure 2 indicate
that a periodic forcing of the first Hopf bifurcation results in a substantial increase in the mass absorption rate
by the liquid. However, for solubility ratios smaller than that at which the Hopf bifurcation occurs, excitation
of the mass flow rate at the nozzle exit does result in a periodic flow where the amount of gas absorption during
the compression part of the oscillations is nearly identical to the gas desorption in the expansion part.
Figure 2. Pressure coefficient (top left), jet's mean radius at the convergence point (top right), phase diagram for the jet's mean radius at the convergence point (bottom left) and mass absorption rate (bottom right). ( = 5, a = 0.01, St = 0.50).
*Research supported by DGICYT Project no. PB94-1494
INVESTIGATION OF THE PROCESS OF WATER-OIL DISPLACEMENT IN LAMINATED RESERVOIRS TAPED BY HORIZONTAL HOLES
Yu.A.Volkov*, V.M.Konyukhov**, A.N.Chekalin**
* The center for improvement in methods of oil
fields development, Kazan, Russia
** Kazan State University, Kazan, Russia
Oil reservoirs have complex geological structure, and, as a rule, they are formed by layers with
essentially different physical properties. The two-phase filtration process in such reservoirs is
accompanied by mass exchange overflows between layers. The research of these processes becomes
rather untrivial with reservoir drilling in by horizontal holes (HH) which are widely used nowadays in
practice of oiloutput because horizontal section of the hole can be located in the various reservoir
layers. If injecting and extracting HH are put in row, the oil displacement is close to a plane-parallel
one. The latter means that in this case the use of plain-parallel filtration analog in the description of
the process in a reservoir is well founded.
The simulation of water-oil displacement in a vertical section of a thin (in comparison with the
distance between holes) laminated reservoir was carried out within the large-scale approximation
framework with no regard for a capillary pressure and a force of gravity. The ternary two-phase
filtration model was used in the description of the polymeric flooding. It was considered that the
adsorption is equilibrium and irreversible. In modeling a horizontal hole by a gallery, it is possible to
suppose that the gallery uncovers a reservoir from roof to bottom or it only some layer.
Let us note only, not dwelling on a numerical method of the system solution, that the elaborated
efficient algorithms for the examined tasks solution are based on the methods described in [1,2].
These algorithms are used in the program complex "Armaris" allowing to keep track of a filtration
process on the display screen and to determine basic characteristics of a reservoir development. All
necessary computing experiments were carried out using this complex.
The computing experiments have shown that:
- oiloutput of the reservoir and each layer significantly depends on the ratio of their absolute permeabilities;
- the overflows between layers caused by the difference of their absolute permeabilities have a pronounced effect on the water-oil displacement process and decrease nonuniformity of the layers development;
- when the thin low-permeable layer is positioned between two more permeable layers, its oil output may be higher than the oiloutput of one of them due to the overflows;
- with the availability of bottom water, injecting HH should be spaced in the water-containing layer, otherwise a rational HH spacing along the reservoir depth depends on a choice of optimization criterion;
- the thickening agent receives to less permeable layer largely because of the overflow from more permeable layers;
- the laminated of a reservoir heterogeneity can cause the break of the polymeric margin at the expense of great difference movement rates of margin back front in layers;
- high-permeable thin layer renders a great negative influence on the reservoir oiloutput and water-oil ratio than a low-permeable layer;
- the polymeric flooding increases a reservoir oiloutput and decreases water-oil ratio, but does not lead to a leveling of the oiloutput for reservoirs with the different absolute permeabilities, at small margin size.
- Chekalin A.N., Mikhailov V.V., A numerical modeling of two-phase three-component filtration in multiconnected domains. Mathematical Modeling and Applied Mathematics Samarskii A.A. and Sapagovas M.P. (Editors) Elsevier Science Publishers B.V. (North-Holland) 1992 IMACS, 1992, pp 78-89.
- Chekalin A.N., Kudryavtsev Y.V., Mikhailov V.V., Investigation of two- and three-component filtration in oil reservoirs. Publishing house of Kazan University, Kazan ,1990 (in Russian)
INVESTIGATION OF EFFECT OF A SPATIAL INHOMOGENEITY ON THE VAPOUR-LIQUID FLOW DYNAMICS IN HEAT-GENERATION POROUS MEDIUM
Oleg V. Khoruzhii, Vladimir V. Likhanskii, Alexander I. Loboiko
National Research Center of Russian Federation "TRINITI", Troitsk Institute for Innovation and
Thermonuclear Researches, Troitsk, Moscow Reg., Russia
ABSTRACT. Heat and mass transfer in heat-generating porous medium are important issues in both
scientific studies and practical applications. Analysis of the Three Mile Island (TMI-2) accident
revealed significance of these factors for the problem of safety of nuclear reactors. Postaccident
analysis has shown that fragments of the reactor core produced two different porous beds: the first
one was formed within the core itself, the another at the bottom of the reactor vessel. Only high
efficiency of heat removal due to water boiling in the lower porous region prevented the reactor
vessel from a damage and helped to localise the accident. Presently, interaction of porous heat-
generating debris bed with reactor vessel is recognised as an important stage during the severe
accident of light-water reactors caused by decreasing core cooling capacity. The vessel may be
damaged if it is exposed to heat attack of melted material, which appears due to drying up and
overheating of certain regions of the porous layer. To evaluate this risk, it is important to know
critical value of heat loading that corresponds to threshold of formation of drained areas in the debris
Though there is a number of experimental works, theoretical studies, and computer simulations
addressed to this matter, the present knowledge of the problem is quite limited. For instance, it is still
a question why heat removal from the lower porous debris bed in TMI-2 accident was so efficient
that the reactor vessel was not overheated. Analysis shows that if take into account the size of the
debris bed in TMI-2 accident the rate of heat generation should be high enough to cause draining and
overheating of the debris layer. As a result, thermal stress applied to the reactor vessel should be
much higher than it was in reality. The above conclusion is mostly based on correlations, that
represent one-dimensional approach to the problem and well-agree with corresponding experimental
data. Therefore, to eliminate the obvious contradiction between predictions and real scenario of
TMI-2, it is necessary to take into consideration multidimensional effects. Spatial non-homogeneity
of such parameters as heat generation rate, porosity of the debris bed, size of particles in bed, etc.
could play a crucial role in this respect.
Heat removal rate in the system of boiling coolant and porous medium is controlled by the rates of
water supply and steam removal. In one-dimensional case both processes take place over the same
regions of the medium. That is, increase in steam production decreases the water supply, and vice
versa. As a result, the highest rate of heat removal is achieved when fraction of liquid in the system is
of some optimum value. At the same time, heat removal under these conditions can be significantly
lower than that in the case of over-steamed medium when the fraction of water at its upper boundary
equals to zero. Situation changes when the medium is no longer treated as one-dimensional, since it
is quite possible that the coolant can be delivered to the boiling region from the neighbour areas. This
effect may become even more prominent when heat generation within the system is not uniform. In
this case, the region with maximal heat removal and minimal quantity of liquid is surrounded by
regions, where boiling is less intense and fraction of liquid is higher. Thus, coolant supply to the
region of intense boiling is simplified. Hence, local critical values of heat loading in non-uniform
porous layer can substantially exceed these in one-dimensional case.
It is worth to point out at least two effects, that should be taken into account when heat and mass
transfer in non-uniform medium are considered. Firstly, if fractions of steam in neighbour regions are
different, then steam pressures are different and a natural convection should appear in the system. In
the regions with higher pressures up-streaming flow will appear, and vice versa. Closing of flow lines
assumes that liquid will flow from the regions with water excess towards the regions with excess of
steam. This will provide extra liquid to the regions of intense boiling and thus will increase critical
value of heat removal. Secondly, due to variation of water supply non-uniform distribution of
capillary pressure will appear. The origin of this effect may be easily understood if to take into
consideration the fact that local distribution of steam and water is governed by capillary forces: water
trends to occupy narrower channels, leaving wider ones to steam. Consequently, characteristic size
of capillaries, filled-up with water, should depend on water fraction the higher the latter, the wider
the channel. Thus, variation of water fraction should result in variation of meniscus curvature and,
correspondingly, in variation of difference between steam and water pressures. The above
consideration shows that within the region where fraction of steam is higher vapours' pressure should
exceed that in the region with lower fraction of steam. Therefore, capillary effects may encourage
increase of steam removal rate and may provide extra water supply to the boiling region. As a result,
due to convection and capillary effects within the layer with non-uniform heat generation, critical
value of thermal loading should increase.
In this work, we applied developed numerical code to model two-phase convection and heat transfer
in porous medium with non-uniform heat generation. A model based on the concepts of relative
permeabilities and Leverett function has been used.
To study qualitative effects that appear due to non-uniform heating, we considered two-phase
convection of water and steam in porous medium at external atmospheric pressure. It was assumed
the debris bed to be constructed by particles of the same size, which are placed into a two
dimensional layer, which has impermeable adiabatic bottom and walls, and totally permeable
isothermal upper boundary. Particle size was varied. Heat generation was assumed to alter with the
distance from the central vertical plane of the layer as cosine function. The main parameter that
determines the process of boiling when heat generation occurs in the whole bulk of medium, is the
heat flux integrated over the depth of debris bed. Therefore, representation of convection in the
debris bed with constant depth, but with non-uniform distribution of heat generation is qualitatively
similar to the case of debris bed, which depth is varied, but generation of heat over the volume of the
medium is uniform. For example, cosine function, that describes distribution of heat generation,
corresponds to the case of semi-cylindrical shape of the debris bed. Simplified geometry was chosen
only in order to make qualitative analysis easier. In our following works we are planning to consider
more realistic representation of debris bed geometry.
Special emphasis has been made on the arising multidimensional effects. It is shown that under
certain conditions dynamics of steam-liquid flows is significantly affected by effect of "capillary
pump" which supplies the regions of intense boiling with extra liquid. As a result maximum heat flux,
which can be removed without draining and overheating of the medium, can significantly exceed that
estimated in 1-D approach. For example, in case of cavity with depth 10 cm, which is filled up with
particles with diameters 1.5 mm, an average rate of heat generation as high as 20 W/cm3 , which
corresponds to the heat flux density at the upper boundary of heat-generating layer 200 W/cm2, can
be succesfully removed. At the same time, without taking into account of the multidimentional
effects, drained region appears at average rate of heat generation about 12.5 W/cm3.