Heat transfer in vertical annular laminar flow of two immiscible liquids

The paper investigates heat transfer in annular laminar undisturbed flow of two immiscible liquids, with constant heat-flux generated at the wall of the tube. It presents an analytical solution for the fully developed temperature field. This is used to obtain a more general solution from a model, describing the temperature field as a superposition of the fully developed and the developing fields. This superposition model is solved by an orthogonal collocation method. An asymptotic model for short entry lengths is also described. Calculations for a kerosene-water system, show that the superposition solution converges to the entrance solution below 100 diameters and converges asymptotically to the solution of the fully developed temperature field beyond 5000 diameters. The effect of the wavy interface is assessed experimentally for annular kerosene-water flow, by comparing predicted and measured temperature profiles. It is found that experimental profiles are considerably flatter and measured Nusselt numbers for the kerosene phase are accordingly higher by 40–320% as compared to the undisturbed flow analyses.     

Thermocapillary instability of the interface between two immiscible liquids in the presence of volume heat sources

The problem of the stability of the interface between two infinite layers of different immiscible liquids is considered. It is assumed that within the liquid a distributed volume heat source, simulating Joule heating, is given. The stability of the rest state with respect to small unsteady disturbances is investigated. The investigation is carried out using the real boundary conditions at the interface between the two liquids rather than the model boundary conditions usually employed in such problems [5]. The problem considered is related to the practical question of the stability of electrolyzer processes. In the present case a possible threshold mechanism of development of oscillations of the electrolyte-aluminum interface is examined. A numerical example with liquid parameters that coincide with those of the electrolyte and aluminum shows that the thermocapillary instability mechanism can, in fact, be the source of surface waves at the electrolyte-aluminum interface.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 156–160, September–October, 1990.     

Weakly nonlinear theory versus theory of secondary instability for plane poiseuille flow

In [1] the theory of secondary instability for plane Poiseuille flow was proposed. In this paper the weakly nonlinear theory is used to investigate the same problem. The results are consistent with those of the theory of secondary instability. It indicates that for plane Poiseuille flow there exists a certain intrinsic relation between the weakly nonlinear theory and the theory of secondary instability. This work is supported by the Science Fund of the Chinese Academy of Sciences.     

Convective instability of a system of horizontal layers of immiscible liquids with a deformable interface

The convective stability of equilibrium is considered for a system of two immiscible fluids which differ little in density. A generalized Boussinesq approximation is developed, making it possible to take the interface deformations properly into account. The stability of the equilibrium state of two fluids in a horizontal layer with a vertical temperature gradient is investigated. Several instability mechanisms are identified: long-wave and cellular monotonic disturbances and oscillatory disturbances. Increasing the deformability is shown to cause switching between instability mechanisms.Perm. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 32–39, March–April, 1996.     

Motion of two viscous immiscible incompressible liquids in a rotating cylinder

The shape of the interface is determined using the boundary layer method and the position of the center of mass of the liquid and the value of the braking torque exerted by the liquid on the walls of the rotating cylinder are calculated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 92–99. January–February, 1990.     

Heat transfer solutions in laminar co-current flow of immiscible liquids

Thermal entry region solutions are analytically determined for horizontal, co-current laminar flow of immiscible liquids in direct contact, inside circular tubes and parallel plate channels. The related eigenvalue problem for such a composite media is readily solved by extending the ideas in the recently advanced sign-count method. It is assumed a core-annular flow configuration for circular tubes and sheat-core flow for the parallel plates channel, without consideration of interface instabilities and stratified flow. First, the velocity problem is solved for fully developed flow and pumping power expressions established for different operating conditions. Then, the temperature problem is analytically handled to yield expressions for quantities of practical interest such as total heat exchange rates, along the duct length and, again, for different flow rates and pressure drop requirements. The analysis is illustrated through consideration of an application dealing with pumping of a very viscous oil with the addition of an external thin layer of a less viscous fluid (water). Pumping power and total heat exchange are then evaluated for both geometries and critically compared to the single fluid flow problem.Hier sind Lösungen für die thermische Eintritts-strecke von horizontalen, laminaren Gleichströmungen in unvermischbaren Flüssigkeiten, die in direktem Kontakt untereinander sind, analytisch bestimmt worden. Die Lösungen gelten für Gleichströmungen in Röhren und in parallelen flachen Kanälen. Das betreffende Eigenwertproblem für solch ein zusammengesetztes Medium ist vollkommen mit dem Gedanken des kürzlich weiterentwickelten Zeichenzählverfahrens gelöst worden. Für die Rohre sind kreisring- und kernförmige Strömungen und für die parallelen Plattenkanäle Schichtkernströmungen angenommen worden. Hierbei ist die Grenzflächeninstabilität nicht in Betracht gezogen worden. Als erstes ist das Geschwindigkeitsproblem für eine vollkommen entwickelte Strömung gelöst und es sind Ausdrücke für die Pumpleistung für verschiedene Betriebsbedingungen ermittelt worden. Dann ist das Temperaturproblem analytisch behandelt worden, um Ausdrücke für die Größen von praktischem Interesse zu erzielen, wie die gesamte Wärmeaustauschrate über die Kanallänge für verschiedene Strömungsgeschwindigkeiten und Druckverlustanforderungen. Die Berechnung ist durch die Betrachtung eines Anwendungsbeispiels veranschaulicht worden, bei dem sehr zähflüssiges Öl mit einer zusätzlichen äußeren, dünnen Schicht, die weniger zähflüssig ist, gepumpt wurde. Die Pumpleistung und der gesamte Wärmeaustausch sind für beide Geometrien ausgewertet und kritisch mit dem einfachen Strömungsproblem von Fluiden verglichen worden.     

Weakly nonlinear stability analysis of a falling film with countercurrent gas flow

Weakly nonlinear stability analysis of a falling film with countercurrent gas–liquid flow has been investigated. A normal mode approach and the method of multiple scales are employed to carry out the linear and nonlinear stability solutions for the film flow system. The results show that both supercritical stability and subcritical instability are possible for a film flow system when the gas flows in the countercurrent direction. The stability characteristics of the film flow system are strongly influenced by the effects of interfacial shear stress when the gas flows in the countercurrent direction. The effect of countercurrent gas flow in a falling film is to stabilize the film flow system.     

Mixing of mutually soluble liquids in turbulent flow in a pipe

The problem concerned with mixing of mutually soluble liquids in turbulent flow in a pipe [1–11] is considered. To describe the distribution of concentration in the region of mixture, taken average across the section of the pipe, we use a model based on a one-dimensional model of the type of heat-conduction equation with an effective coefficient which, as tests show, is different from the coefficients of molecular and turbulent transfer. The dimensionless value of this coefficient depends on a number of parameters, such as the Reynolds number calculated for one of the liquids, roughness, ratio of the densities and viscosities of the liquids, as well as on the concentration, gradients of concentration, etc. These relationships can be established either by means of tests or on the basis of theoretical consideration of the mixing phenomenon. In this paper we theoretically derive a dispersion model with an effective diffusion coefficient which depends on Reynolds and Schmidt numbers, as well as on roughness.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 96–102, September–October, 1971.     

Direct contact heat transfer between two immiscible liquids flowing in a horizontal concentric annulus

Direct contact heat transfer between water and a heat transfer oil was investigated under non-boiling conditions in co-current turbulent flow through a horizontal concentric annulus. The ratio of the inner pipe diameter to the outer pipe diameter (aspect ratio) κ = 0.730−0.816; total liquid velocity (mixture velocity) V_T = 0.42−1.1 m/s; inlet oil temperature T_oi = 38−94°C; oil volume fraction in the flowing mixture φ_o = 0.25−0.75 were varied and their effects on the overall volumetric heat transfer coefficient U_v were determined at constant interfacial tension of 48 dynes/cm.

It was found that, in each concentric pipe set, the overall volumetric heat transfer coefficient increased with increasing dispersed phase volume fraction at each constant mixture velocity and reached a maximum at around φ_o = φ_w ≈ 0.5. The maximum U_v values increased with increasing total liquid velocity and decreasing aspect ratio of the annulus. The volumetric heat transfer coefficient was also found to increase with increasing inlet oil temperature and increasing total liquid velocity but to decrease with length along the test section keeping all other parameters constant. Empirical expressions for the volumetric heat transfer coefficient were obtained within the ranges of the experimental parameters.     

Oscillating flows in capillaries filled with immiscible liquids

Hydrodynamic flows generated by mechanical vibrations of a capillary filled with immiscible liquids are investigated. Air bubbles are contained at the hermetically sealed ends of the capillary. Equations for the change in the volumes of the air bubbles as functions of time and velocity distribution in the liquids are obtained for the case when the radius of the capillary is much less than the lengths of the liquid columns. Results of numerical calculations are given for a capillary filled with two liquids: water and mercury. Amplitude-frequency dependences of the change in volumes of the air bubbles are constructed which have a resonance nature. Graphs of the dependence of the velocity of the water and the mercury on the radial coordinate at different times are given.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 13–18, September–October, 1987.     

A note on co-current laminar flows of two immiscible elastico-viscous liquids

Summary An analysis is presented of steady (isothermal) co-current laminar flows of two immiscible elasticoviscous liquids in cylindrical channels to include (i) unidirectional stratified flow with ripple-free, plane liquid interface, and (ii) concentric-layered swirling flow with ripple-free cylindrical liquid interface. The general conditions are derived for such two-phase channel flows to be physically realizable. It is shown that, whereas (under certain circumstances)single-phase laminar flows are physically possible,two-phase flows, on the other hand, of liquids of the same class may not be. But liquids of theRoberts type (Roberts 1953), with a normal stress difference equivalent to an extra simple tension along the streamlines in simple shearing, are capable of steady unidirectional flowin all circumstances (whether in single or two-phase flow), though they are not in a privileged position so far astwo-phase swirling flows are concerned.     

Calculation of heat transfer in laminar pipe flow of liquids with structural viscosity

Expressions have been obtained for the dimensionless heat-transfer criteria of liquids with structural viscosity for the conditions t_w=const and q_w=const in the case of laminar quasi-isothermal flow.     

Magnetohydrodynamics instability of interfacial waves between two immiscible incompressible cylindrical fluids

The problem of nonlinear instability of interfacial waves between two immiscible conducting cylindrical fluids of a weak Oldroyd 3-constant kind is studied. The system is assumed to be influenced by an axial magnetic field, where the effect of surface tension is taken into account. The analysis, based on the method of multiple scale in both space and time, includes the linear as well as the nonlinear effects. This scheme leads to imposing of two levels of the solvability conditions, which are used to construct like-nonlinear Schr6dinger equations （1-NLS） with complex coefficients. These equations generally describe the competition between nonlinearity and dispersion. The stability criteria are theoret- ically discussed and thereby stability diagrams are obtained for different sets of physical parameters. Proceeding to the nonlinear step of the problem, the results show the appearance of dual role of some physical parameters. Moreover, these effects depend on the wave kind, short or long, except for the ordinary viscosity parameter. The effect of the field on the system stability depends on the existence of viscosity and differs in the linear case of the problem from the nonlinear one. There is an obvious difference between the effect of the three Oldroyd constants on the system stability. New instability regions in the parameter space, which appear due to nonlinear effects, are shown.     

Novel shear modulus equations for concentrated emulsions of two immiscible elastic liquids with interfacial tension

Starting from the shear modulus equation for a dilute emulsion system of two immiscible liquids with interfacial tension, four new equations have been developed for the shear modulus of concentrated emulsions using a differential scheme. The continuous phase and the dispersed droplets are treated as elastic liquids in the derivation. Out of the four models developed in the paper, two models predict the relative shear modulus (ratio of emulsion modulus to continuous phase modulus) to be a function of three variables: elastocapillary number, modulus ratio (dispersed phase modulus to continuous phase modulus) and volume fraction of dispersed phase. The remaining two models include an additional parameter, i.e. the maximum packing volume fraction of droplets. The proposed models are evaluated using three sets of experimental data on high frequency shear modulus of concentrated polymer-thickened oil-in-water emulsions.     

Blends of two immiscible and rheologically different fluids

The Doi and Ohta theory of the rheology of immiscible blends is extended by considering the field of the velocity with which the interface moves as an independent state variable. This type of generalization is needed in order to be able to take into account the difference in the rheological properties of the fluids involved. Expression for the extra stress tensor as well as equations governing the time evolution of the interfacial area, orientation of the interface and its velocity are derived.     

Motion and heat transfer in tubes of liquids with temperature-dependent viscosities

The problem of the velocity and temperature distribution of a liquid flowing in a cylindrical tube in the case where the viscosity varies exponentially with temperature is considered.     

One-dimensional filtration of immiscible liquids in a nonuniform porous medium

As is known, the differential equation for two-phase filtration with account for capillarity was obtained in [1], and later integrated numerically for the case of a uniform stratum of finite length in [2]. Other versions of the solution of the Rapoport-Leas equation or the system which is equivalent to it are known [3, 4]. This article presents the results of a numerical solution of an analogous problem with account for nonuniform permeability of the stratum.The authors wish to thank T. V. Startsev and L. Kh. Aminov for assistance in performing the calculations on the Ural-3 computer.     

Dynamics of two-phase bubble-droplets in immiscible liquids

This paper describes an experimental study of two-phase bubble-droplets of butane in distilled water. In the process of direct-contact heat transfer between two immiscible liquids, and with change of phase, evaporation occurs within a liquid droplet to form an interior bubble which ultimately may expand to absorb the entire droplet. The so-called stopped-evaporation two-phase bubble-droplets were formed by such evaporating droplets of butane rising in a column of water by application of pressure on the surface of the water to stop the evaporation. The configurations of such bubble-droplets, which depend on the ratio of mass of the vapor to mass of the liquid, are discussed. Results are also given for the rise velocity, and comparisons are made between the rise velocity of stopped-evaporation droplets and those of evaporating droplets.

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Dynamik zweiphasiger Blasen-Tropfen in unmischbaren Flüssigkeiten

Zusammenfassung Die experimentelle Untersuchung von zweiphasigen Blasen-Tropfen aus Butan in destilliertem Wasser wird beschrieben. Bei direktem Wärmeaustausch mit Phasenänderung zwischen zwei unmischbaren Flüssigkeiten beginnt die Verdampfung innerhalb eines Flüssigkeitstropfens und bildet eine innere Blase, die sich schließlich ausdehnt und den Tropfen absorbiert. Ein Ende der Verdampfung konnte beim Aufsteigen von Butantropfen in eine Wassersäule durch Anlegen von Druck an der Wasseroberfläche erreicht werden. Die Struktur dieser Blasen-Tropfen, die abhängt vom Verhältnis der Massen des Dampfes und der Flüssigkeit, wird diskutiert. Außerdem werden Ergebnisse für die Aufstiegsgeschwindigkeit gegeben und die Aufstiegsgeschwindigkeiten der verdampfenden Tropfen mit denen bei beendeter Verdampfung verglichen.

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Nomenclature d Equivalent spherical diameter of initial droplet (mm) - D Equivalent spherical diameter of bubble-droplet (mm) - m₀₀ Initial mass of droplet (kg) - m _v Mass of vapour (kg) - Re _c Reynolds number based on water properties (Re_c=UD/) - t Time (s) - T _c Temperature of water (°C) - U Rise velocity (m/s) - Z Position of bubble-droplet (mm) - t ₀ Overall temperature difference (C deg) - Density (kg/m³) - Viscosity (kg/m s)