Selection of solitary waves in vertically falling liquid films

Two-dimensional solitary waves at the surface of a film flow down a vertical plane are considered. When the system is subjected to inlet white noise, solitary waves are formed after an inception region and interact with each other. Using open-domain simulations of reduced equation models, we investigate numerically their late time process dynamics. Close to the instability threshold, the waves synchronize themselves into bound states. For higher values of the Reynolds number, the separation distance between the waves increases and the synchronization process at work is weaker. Performing statistics, we show that the mean characteristics of the waves correspond to the minimal value of the mean film thickness along the traveling-wave branch of solutions. In this regime, synchronization occurs through the waves tails which is associated with a change of scaling of the waves features. A similar behavior is observed performing simulations in periodic domains: the selected waves maximize the mean flow rate.     

The shape of a vertically falling stream of a Newtonian liquid

Summary The diameters of the axially symmetric streams of two Newtonian liquids of different viscosities falling vertically under gravity have been measured as a function of the distance from the nozzle. The measurements have been compared with two theories, (I, II) based on different approximations. Theory I leads to a second-order ordinary differential equation and theory II gives an asymptotic solution of theNavier-Stokes equations. It was found that the first approach was sufficient to explain the measurements in the high viscosity liquid (30,000 cS) but the asymptotic solution was needed to explain the low viscosity (1,000 cS) measurements.

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Zusammenfassung Die Durchmesser der achsensymmetrischen Str?mungen zweier newtonischer Flüssigkeiten verschiedener Viskosit?ten, die unter ihrem Eigengewicht senkrecht austreten, werden als Funktion des Abstandes von einer Düse gemessen. Die Messungen werden mit zwei Theorien (I, II) verglichen, die auf verschiedene N?herungen beruhen. Die Theorie I führt auf eine gew?hnliche Differentialgleichung zweiter Ordnung, w?hrend die Theorie II eine asymptotische L?sung derNavier-Stokes-Gleichung ergibt. Es zeigt sich, da? die erste N?herung ausreicht, um die Messungen der hochviskosen Flüssigkeit (30.000 cS) zu beschreiben. Zur Erl?uterung der Messungen des niedrig-viskosen Silikon?ls (1.000 cS) bedarf es der asymptotischen L?sung.

     

Linear stability in the flow of a liquid film concurrent with a gas flow

The two-phase flow of liquid films are often encountered in practice, but the number of theoretical papers devoted to this problem is limited. The problem of the linear stability of a viscous liquid film subjected to a gas flow has been formulated in [1] and, in somewhat different form, in [2]. The linear stability of plane-parallel motion in films has been studied analytically in [1–8] for some limiting cases. The range of validity of the analytic approaches remains an open question. Therefore, an exact numerical analysis of flow stability over a fairly broad range is required. In the present paper a separate solution of the problem for the gas and the liquid is shown to be possible. The Orr-Sommerfeld equation has been integrated numerically, and the results are compared to the results of analytic calculations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 143–146, January–February, 1976.The author is grateful to É. É. Markovich for directing the work and to V. Ya. Shkadov for his interest in the work and many useful comments.     

Heat transfer to non-newtonian laminar falling liquid films with smooth wave free gas—liquid interface

The present paper investigates analytically the problem of heat transfer to a non-Newtonian laminar falling liquid film flowing along an inclined wall for the thermally developing and thermally developed regions. In the developing region of the temperature profile, the Nusselt number decreases monotonically until the thermal boundary layer touches the interface. But immediately after this point, the liquid film thickness decreases as well as the temperature difference in the film. The influence of parameters such as α (i.e. Fr/Re_mod ratio), γ (i.e. modified form of ?μ), modified Prandtl number and the flow behaviour index “n’ on heat transfer results is also presented.     

Wave formation on vertical falling liquid films

The method of integral relations is used to derive a nonlinear “two-wave” structure equation for long waves on the surface of vertical falling liquid films. This equation is valid in a wide range of Reynolds numbers and reduces to the known equations for high and low Re. Theoretical data for the fastest growing waves are compared with the experimental results on velocities, wave numbers and growth rates of the waves in the inception region. The validity of theoretical assumptions is also confirmed by the direct measurements of the instantaneous velocity profiles in a wave liquid film.     

Laminar jets of a plane liquid sheet falling vertically in the atmosphere

The steady laminar flow of a liquid sheet falling vertically from a slit die into the atmosphere is studied theoretically under the assumptions of flat profiles of velocity and pressure over its horizontal cross-section. An equation predicting the sheet thickness is derived in a way similar to that which has been found useful for a falling round jet.The proposed equation is compared with previous results of experiments and numerical simulations as well as Clarke's exact solution to the oridanary differential equation of a one-dimensional vertical jet, and the results seem to be rather good.The present work is concerned with an integral invariant of the jet, and lower-boundary conditions on the vorticity are presented for a viscous-gravity jet as well as for an inertia-gravity jet.     

Two-phase flow characteristics in gas–liquid microreactors

Multiphase chemical microreactors require a detailed knowledge of the flow conditions inside the reaction system. This paper reports flow visualization measurements of the two-phase gas–liquid flow pattern and the liquid velocity distribution inside liquid plugs of an intermittent flow. Rectangular cross-section silicon microchannels with hydraulic diameters between 187.5 and 218 μm are fabricated. Laser Induced Fluorescence (LIF) is used to determine the flow pattern. To analyze the influence of the liquid properties and the channel diameter on the two-phase flow pattern, we present flow regime maps using different channel geometries and fluids. A universal flow pattern map based on dimensional analysis is presented. In contrast to microchannel flows, a great number of correlations for flow characteristics for multiphase flow in (round) pipes with diameters >1 mm exist. We compare our experimental results from optical flow visualizations in microreactors with common flow correlations and regime maps for macro- and microchannels. The recirculation motion in the liquid segments of an intermittent gas–liquid flow is analyzed using micron-resolution particle image velocimetry (μPIV). The velocity distribution influences the mixing and the mass transport towards the reactive phase interface dealing with two-phase chemical reactions. For straight microchannels hardly any mass transport over the center line is quantified. For enhanced mixing geometrical adaptations are suggested.     

Numerical analysis on flow dynamics and heat transfer of falling liquid films with interfacial waves

Flow dynamics and heat transfer of falling liquid films with interfacial waves flowing on a vertical plate have been studied with originally proposed numerical simulation method. To discretize basic equations a staggered grid fixed on a physical space is employed. A small amplitude disturbance generated at inflow boundary develops to a solitary wave which consists of a large amplitude roll wave and small amplitude capillary waves. Instantaneous streamwise velocity profiles at the wave crest and trough are very different from a laminar flow. A circulation flow occurs in the roll wave and it affects temperature distributions, especially the strong effect is observed for high Prandtl number liquids. The interfacial wave enhances the heat transfer by two kinds of effects which are a film thinning effect and a convection effect. The dominating effect depends on the Prandtl number. Received on 23 December 1998     

Three-dimensional surface characteristics of a falling liquid film

Optical methods are described for examining the three-dimensional character of waves on a falling liquid film. This involved monitoring the motion of the local film surface normal through the use of laser beam refraction. The wavy motion was found to be primarily of a two-dimensional nature only for Re (equal to 4Q/v) less than 1500.

Surface characteristics were examined for Reynolds numbers from 217 to 4030 and for different distances along the direction of flow.     

Characteristics of solitary waves on a falling liquid film sheared by a turbulent counter-current gas flow

We report an experimental investigation of a falling water film sheared by a turbulent counter-current air flow in an inclined rectangular channel. Film thickness and wave velocity measurements associated with visual observation are conducted to study the influence of the air flow on controlled traveling waves consisting of a large wave hump preceded by capillary ripples. First, we focus on the variation of the shape, amplitude and velocity of the waves as the gas velocity is gradually increased. We demonstrate that the amplitude of the main hump grows substantially even for moderate gas velocities, whereas modification of the wave celerity becomes significant above a specific gas velocity around 4 m/s, associated with an alteration of the capillary region. The influence of the gas flow on 3D secondary instabilities of the solitary waves detected in a previous study Kofman et al. (2014), namely rugged or scallop waves, is also investigated. We show that the capillary mode is damped while the inertial mode is enhanced by the interfacial shear. Next, the gas velocity is increased until the onset of upstream-moving patterns referred to as flooding in our experiments. At moderate inclination angles (typically < 7 ^○), flooding occurs for a gas velocity around 8 m/s and is initiated at the scallop wave crests by a backward wave-breaking phenomenon preceded by the onset of ripples on the flat residual film separating two waves. At high inclination angle, a rapid development of solitons is observed as the air velocity is increased preventing the waves to turn back. Finally, at high liquid Reynolds number, sudden and intermittent events are triggered consisting of very large amplitude waves that go back upwards very fast. These “slugs” either extend over the whole width of the channel or are very localized and can thus potentially evolve towards atomization.     

The effect of liquid viscosity on mixing in falling films

The effect of liquid viscosity on mixing in falling liquid films has been investigated using a Macropore as a means of simulating conditions in a packed column.-Mass transfer coefficients and transfer unit heights have been calculated for the absorption of stagnant carbon dioxide at atmospheric pressure and 25° C into aqueous glycerol for a range of liquid rates and glycerol concentrations, and the values compared with those from Penetration theory correlations.-Deviations from expected behaviour have been observed, notably that the exponent on the overall mass transfer coefficient with respect to film Reynolds number is significantly greater than the value of one-third predicted from the Higbie model.-It has been shown that mixing at the discontinuities is more than 75% complete for pure water but that this mixing factor decreases with increasing viscosity.     

Experimental studies of a hygroscopic condenser-evaporator heat exchanger based on concentration differences of vertically falling films

A concentration-driven power cycle motivated by differences in vapor partial pressures (boiling point rise) and latent heats of brine and water is studied. The condensation of relatively low-pressure, low-temperature vapor occurs on the free interface of a relatively hot falling film of a hygroscopic salt solution due to the reduced vapor pressure of the brine. The heat released is transferred to the evaporating/cooling water film on the other side of a vertical plate separating the brine and water films. The process is maintained because the latent heat of condensation on the brine film is higher than the latent heat of evaporation of pure water. The condensation driving force is the difference between the partial pressure of condensing water vapor and that of water in the brine solution. The simultaneous mass and heat transfer mechanisms associated with this nonisothermal absorption can occur even against an opposing thermal driving force in the condensing vapor phase. Complementing earlier studies by the same authors, a vertical film-type condenser-evaporator heat exchanger is considered. The experimental study deals with the effects of the various parameters involved in this rather unique process and the mechanisms that control them. The experimental results prove the potential of operating this new heat transfer modality and provide the background for the theoretical determination of the optimal performance of this direct-contact power cycle.     

The effect of cocurrent and countercurrent gas shear on entrance region mass transfer in turbulent falling liquid films

Predictions of average film thickness and mass transfer coefficients in the entrance region of gas absorption with high cocurrent and countercurrent gas flow in a turbulent falling film are presented. The model used is a modified van Driest eddy diffusivity in the inner wall region and an interface damping eddy diffusivity in the outher region of the film modified to include the effect of interfacial shear. The calculations show that the entrance length for mass transfer decreases with increasing film Reynolds number and interfacial shear for cocurrent flow. Also the model predicts a decrease in mass transfer coefficient with an increase in Schmidt number in accordance with experimental data. On the other hand, for counterflow and at a fixedRe, an increase in upward gas shear increases the film thickness and eventually leads to rising film flow. The entrance length for mass transfer decreases with increasing Reynolds number but slightly increases with countercurrent interfacial shear. The calculations show that, in practice, the entrance region for mass transfer for gas absorption with shear-thinned film can be neglected for cocurrent shear but often cannot be neglected for countercurrent shear with a shear-thickened film.Es werden Berechnungen für die mittlere Film-dicke und die Stoffübertragungs-Koeffizienten in der Eintrittsregion der Gasabsorption mit starkem Gleich- und Gegenstrom von Gas in einem turbulent fließenden Film aufgestellt. Das benutzte Modell ist ein modifizierter van Driest-Wirbel-Diffuser für die Innenwand und ein gedämpfter Grenzflächen-Wirbel-Diffuser in der Außenregion des Filmes, der so modifiziert ist, daß die Effekte der Grenzschichtschubspannungen berücksichtigt werden. Die Berechnungen zeigen, daß die Eintrittslänge für die Stoffübertragung mit zunehmender Grenzflächenschubspannung für Gleichstrom abnimmt. Das Modell sagt ebenso ein Sinken des Stoffübertragungs-Koeffizienten mit einer Zunahme der Schmidt-Zahl in Übereinstimmung mit den experimentellen Daten voraus. Auf der anderen Seite führt im Falle von Gegenstrom und einer konstanten Reynolds-Zahl ein Ansteigen der Gasschubspannung zu einem Anwachsen der Filmdicke und eventuell zu wachsender Filmströmung. Die Eintrittslänge für die Stoffübertragung sinkt mit zunehmender Reynolds-Zahl, steigt aber langsam bei Grenzflächenschubspannungen in Gegenstromrichtung. Die Berechnungen zeigen, daß in der Praxis die Eintrittsregion für die Stoffübertragung bei der Gasabsorption mit verdünntem Film für Gleichstromschubspannung vernachlässigt werden kann, bei Gegenstromschubspannung mit verdicktem Film diese jedoch meist berücksichtigt werden muß.     

Time-average local thickness measurement in falling liquid film flow

A method for monitoring time-varying local film thickness variation through the detection of laser scattering from suspended latex particles is briefly described. This method was used in conjunction with the Jeffreys theory of drainage from a flat plate to determine time-average local film thickness.Measurements were made at Reynolds numbers (equal to ($\text{4Q/ν}$)) from 145 to 4030 at varying distances along the direction of flow. At the bottom of the flow, 134 cm from the top, average film thickness is given by the expression: where $\text{a}{}_{\mathrm{i}}$ and $\text{n}{}_{\mathrm{i}}$ are constants unique to each of the three Reynolds number regions, wavy laminar, transitional and turbulent.     

Experimental study of heat transfer and flow characteristics of liquid falling film on a horizontal fluted tube

The aim of the present study is to investigate experimentally the effect of the fluted surface tube on the heat transfer and flow characteristics of liquid falling film. Experiments have indicated that, when a liquid falling film falls on a horizontal fluted surface tube, the transition starts at low Reynolds number than that of the plain tube. The value of the film thickness has been slightly decreased by decreasing the fluted pitch. A reduction of the film thickness was observed at about 9% for tube number 4, which has lower pitch, at Reynolds number of 485. A clear reduction of the dimensionless wavelength, λ*, has occurred at low fluted pitch tube. The use of enhanced surfaces can provide heat transfer coefficients higher values than those obtained from plain tube. Heat transfer enhancement was noticed due to the use of fluted tube surface. An improvement of the Nusselt number reached about 45% for tube 4. However, the low values of the fluted pitch increased the heat transfer enhancement than that of the high values.     

Entropy generation in gas absorption into a falling liquid film

The study of mass transfer into falling films constitutes a significant aspect for numerous applications in the chemical technology and is considered the subject of many theoretical and experimental researches. Evaluating the second law of thermodynamics is one of the contemporarily used methods to determine the performances of an industrial process and to study various sources of irreversibility. Expressions of the liquid velocity, the gas concentration, the entropy generation rate as well as the main sources of irreversibility in the case of gas absorption (carbon dioxide) into a laminar falling viscous incompressible liquid film (water) without chemical reaction, are analytically derived and graphically presented and discussed.     

Entropy generation in gas absorption into a falling liquid film

The study of mass transfer into falling films constitutes a significant aspect for numerous applications in the chemical technology and is considered the subject of many theoretical and experimental researches. Evaluating the second law of thermodynamics is one of the contemporarily used methods to determine the performances of an industrial process and to study various sources of irreversibility. Expressions of the liquid velocity, the gas concentration, the entropy generation rate as well as the main sources of irreversibility in the case of gas absorption (carbon dioxide) into a laminar falling viscous incompressible liquid film (water) without chemical reaction, are analytically derived and graphically presented and discussed.