SESSION 10

INTERFACIAL PHENOMENA

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
dms@c36.uni-hannover.de

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:

1. oiloutput of the reservoir and each layer significantly depends on the ratio of their absolute permeabilities;
2. 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;
3. 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;
4. 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;
5. the thickening agent receives to less permeable layer largely because of the overflow from more permeable layers;
6. 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;
7. high-permeable thin layer renders a great negative influence on the reservoir oiloutput and water-oil ratio than a low-permeable layer;
8. 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.

REFERENCES

1. 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.
2. 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 bed.

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.