Effect of evaporation and diffusion of a surface-active substance on the instability of a falling liquid film

The linear stability of a falling film of a surface-active solute is analyzed in an active evaporation zone. The phase velocities and the gain coefficients are calculated for the hydrodynamic and diffusion modes of perturbations.     

Viscous falling film instability around a vertical moving cylinder

The present work discusses both the linear and nonlinear stability conditions of a viscous falling film down the outer surface of a solid vertical cylinder which moves in the direction of its axis with a constant velocity.After studying the linear conditions,a generalized nonlinear kinematic model is then derived to present the physical system.Applying the boundary conditions,analytical solutions are obtained using the long-wave perturbation method.In the first step,the normal mode method is used to characterize the linear behaviors.In the second step,the nonlinear film flow model is solved by using the method of multiple scales,to obtain Ginzburg-Landau equation.The influence of some physical parameters is discussed in both linear and nonlinear steps of the problem,and the results are displayed in many plots showing the stability criteria in various parameter planes.     

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.     

Hall effects on hydromagnetic falling liquid film

An investigation is made of flow of an electrically conducting falling liquid film over a smooth vertical surface taking Hall effects into account, the liquid being permeated by a transverse magnetic field. Consideration of Hall current into the flow indicates a similarity between the flow of a rotating liquid and that due to the non-rotating system in presence of Hall currents. Discussion has been made for electrically conducting falling film in presence of cross-flow due to hall effect in non-rotating system.     

Heating of non-Newtonian falling liquid film on a horizontal tube

The problem of heat transfer to non-Newtonian laminar falling liquid films on horizontal tubes is investigated theoretically for constant heat flux and isothermal conditions imposed at the inner periphery of the tube. The local and average heat transfer coefficients are obtained as function of the system parameters by conjugating the convective transport of heat to the liquid film to the thermal conduction in the material of the tube in the peripheral direction. The results indicate that the average heat transfer coefficient can be described successfully by three dimensionless groups characterizing the dynamic flow characteristics of the film, modified Prandtl number and the index of the power law variation of the rate of angular shear deformation of the fluid with respect to the shear stress.Das Problem des Wärmeübergangs an laminar ablaufende Fallfilme nicht-Newtonscher Flüssigkeiten an horizontalen Rohren wird unter Voraussetzungen konstanter Temperatur bzw. konstanten Wärmeflusses bezüglich des Rohrinnenumfangs theoretisch untersucht. Die örtlichen und gemittelten Wärmeübergangskoeffizienten erhält man als Funktion der Systemparameter unter gleichzeitiger Berücksichtigung des konvektiven Wärmetransportes an dem Flüssigkeitsfilm und der Wärmeleitung im Rohrmaterial in Umfangsrichtung. Die Ergebnisse zeigen, daß sich der gemittelte Wärmeübergangskoeffizient zufriedenstellend in Abhängigkeit von drei dimensionslosen Kenngrößen darstellen läßt. Diese charakterisieren das dynamische Fließverhalten des Films bzw. entsprechen einer modifizierten Prandtl-Zahl und dem Exponenten des Potenzgesetzes, welches die Abhängigkeit der Scherdeformation der Flüssigkeit von der Schubspannung wiedergibt.     

Characteristics of critical heat flux in two phase thermosyphon – relationship between maximum falling liquid rate and instability on interface of falling liquid film

Critical heat flux in a two-phase thermosyphon is usually dealt with from two different ways in a limitation of liquid flow rate falling along a vertical tube: one is a maximum falling liquid rate due to the envelope method, and the other one due to instability of falling liquid film. The difference between the maximum and instability criteria is first made clear. The CHF in the thermosyphon is shown to be predicted well by the maximum liquid rate due to the maximum criterion better than due to the instability criterion. In addition, the comparison implies that the CHF phenomenon in the thermosyphon is considered to be caused when the falling liquid reaches the maximum value rather than when the instability of the falling liquid on the interface is brought about. Received on 1 December 1997     

Thermocapillary instability of a deformable liquid film

The instability of a plane liquid film with a uniform transverse temperature gradient under conditions of weightlessness is considered. The surface tension is assumed to depend linearly on the temperature. On the basis of an exact solution of the neutral perturbation problem for a layer with deformable boundaries, the instability domains, the dispersion curves, and the shape of the perturbations are determined. It is shown that on the interval of low Prandtl numbers both thermocapillary waves with predominantly longitudinal flow and capillary waves, supported by the thermocapillary effect, with intense transverse liquid flow can develop on the film.Perm'. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 30–36, September–October, 1996.     

Evaporation of laminar,falling liquid film on a horizontal cylinder

An analytical study of falling film evaporation on a horizontal tube is accomplished for the case of constant heat flux condition imposed at the wall. The process of evaporation from the liquid film interface is conjugated to thermal conduction in the wall of the tube around its periphery. In addition, the flow of the film is considered to be thermally developing throughout its travel around the tube. Influence of various governing parameters on local and average heat transfer coefficient is presented. The theoretical investigation is compared with the available experimental data. The agreement is found to be satisfactory.     

Experimental study of the falling film of liquid around a Taylor bubble

The present work reports an experimental study of the falling liquid film around single Taylor bubbles rising in vertical tubes filled with stagnant liquids by using a pulse-echo ultrasonic technique. The experiments were carried out in acrylic tubes 2.0  m long, with inner diameters of 0.019, 0.024 and 0.034  m, with five water-glycerin mixtures, corresponding to inverse viscosity number ranging from 15 to 22422. The rising bubble and the falling liquid film were measured by using ultrasonic transducers located at the one side of the tube. The velocity and profile of the Taylor bubble, and the development length and equilibrium thickness of the falling liquid film around the bubble were obtained by the ultrasonic signals processing. Based on the experimental results of the present study, several correlations available to estimate the equilibrium thicknesses of liquid films falling around Taylor bubbles were evaluated and new correlations were proposed to estimate the dimensionless equilibrium film thickness and the film development length respectively.     

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.     

Heat transfer in non-Newtonian falling liquid film on a horizontal circular cylinder

 This study aims to investigate numerically the laminar flow and heat transfer in a pseudoplastic non-Newtonian falling liquid film on a horizontal cylinder for the constant heat flux and isothermal boundary conditions. The inertia terms are taken into account. An implicit finite difference method is carried out to solve the governing boundary layer equations. The effects of operational parameters on the hydrodynamic and heat transfer characteristics are examined and discussed in detail. The results presented show that the local and average Nusselt numbers varies significantly as a function of the concentration of aqueous carboxymethylcellulose (CMC) solutions and the cylinder diameter. Higher concentration of aqueous CMC solutions generate larger heat transfer coefficients. Finally, a comparison with the experimental and numerical results available in the literature for Newtonian fluids shows clearly that the present analysis is reasonably accurate. Received on 29 March 2001 / Published online: 29 November 2001     

Experimental study on surface wave and film breakdown of falling liquid film flow

In order to evaluate characteristics of the liquid film flow and their influences on heat and mass transfer, measurements of the instantaneous film thickness using a capacitance method and observation of film breakdown are performed. Experimental results are reported in the paper. Experiments are carried out at Re = 250–10000, T _in = 20–50°C and three axial positions of vertically falling liquid films for film thickness measurements. Instantaneous surface waveshapes are given by the interpretation of the test data using the cubic spline method. The correlation of the mean film thickness versus the film Reynolds number is also given by fitting the test data. It is revealed that the surface wave has nonlinear behavior. Observation of film breakdown is performed at Re = 1.40 × 10³–1.75 × 10⁴ and T _in = 85–95°C. From experimental results, the correlation of the film breakdown criterion can be obtained as follows: Bd = 1.567 × 10⁻⁶ Re ^1.183     

Model of a wavy flow in a falling film of a viscous liquid

A new model is developed for describing long-wave perturbations in a falling film of a viscous liquid. The model is based on an integral approach and an expansion of the velocity profile into a series in linearly independent basis functions of a boundary-value problem. A linear analysis of film flow stability is performed, and dispersion dependences are obtained. Results predicted by the new model are demonstrated to be in good agreement with available experimental data on the film flow over a gently sloping surface.     

A numerical study of the non-absorbable effects on the falling liquid film absorption

A numerical study for the simultaneous heat and mass transfer in a falling liquid film absorption process with the presence of non-absorbable gases is presented. Water vapor mixed with air as the non-absorbables being absorbed into a falling smooth aqueous lithium chloride film flow was chosen as the model problem for the study. The finite difference numerical calculation was proceeded by marching downward from the top end, owing to the parabolic type energy and concentration equations for both liquid and gas phases. The results indicate that the local non-absorbable gas concentration is much higher at the gas-liquid interface than that in the ambient, hence the local vapor pressure is lowered there such that the absorption driving potential of the vapor pressure difference is reduced. The resulting reduction of the absorption rate due to the presence of the non-absorbables suggests that its effect must be carefully considered in the application of absorption heat pump design. The present study can provide some useful information for this purpose.

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Numerische Studie über die Einwirkung von nicht absorbierbaren Stoffen auf die fallende Flüssigkeitsfilmabsorption

Zusammenfassung Hier wurde eine numerische Studie der Wärmeund Stoffübertragung in einem Absorptionsprozeß eines fallenden Flüssigkeitsfilms in Anwesenheit von nicht absorbierbaren Gasen dargestellt. Ein Wasserdampf-Luft-Gemisch, das in Anwesenheit von nicht absorbierbaren Gasen von einer fallenden glatten flüssigen Lithium-Chlorid-Filmströmung absorbiert wird, wurde als das Modellproblem für diese Studie gewählt. Die numerische Berechnung mit dem Finite Differenzenverfahren wurde schrittweise vom obersten Ende nach unten durchgeführt. Die Berechnung bezieht sich auf den parabolischen Typ der Energie- und Konzentrationsgleichungen für die Flüssigkeits- und Gasphasen. Die Ergebnisse weisen darauf hin, daß die lokale nicht absorbierbare Gaskonzentration bei der Gasflüssigkeitsphase sehr viel höher ist als in der Umgebung. Weiter ist der lokale Dampfdruck so erniedrigt worden, daß sich das Absorptionsbewegungspotential des Dampfdruckunterschiedes reduziert. Die resultierende Reduzierung der Absorptionsrate, die auf die Anwesenheit der nicht absorbierbaren Stoffe zurückzuführen ist, verlangt eine sorgfältige Einbeziehung ihere Einflüsse auf die Gestaltung der Absorptionswärmepumpen. Diese Arbeit kann einige nützliche Informationen für diesen Zweck geben.

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Nomenclature C absorbent concentration in weight fraction of salt - C _a nonabsorbables concentration in molar fraction - C _a at inlet and infinity - C _in C at film inlet - c _p specific heat of liquid - c _{p g} specific heat of gas - D species diffusivity for LiCl-H₂O - D _g species diffusivity for air-water vapor - g gravity - h _o film thickness - H _a heat of absorption - k thermal conductivity of liquid - k _g thermal conductivity of gas - L transformation constant - water vapor mass absorption rate - P _v water vapor pressure - Re film Reynolds number=V ₀ h ₀/ - T film temperature - T _in film temperature at inlet - T _g gas temperature - T _w wall temperature - T gas temperature at inlet and infinity - u velocity inx-direction - U =1.5V ₀ - V ₀ mean film velocity=g h ₀ ² /3µ - x coordinate parallel to the wall - y coordinate normal to the wall in the film region - y _g coordinate normal to the wall in the gas region - transformedy _g Greek letters dynamic viscosity - liquid density - _g gas density     

Direct contact condensation of vapor on turbulent, falling liquid film

The problem of condensation of pure vapor on turbulent falling liquid film of the same species is analytically solved. The gradual change in enthalpy of the coolant liquid film in the flow direction is considered to take place in three successive stages. The study brings out the influence of inlet Reynolds number, Prandtl number and degree of subcooling of the coolant on condensation heat transfer coefficients. The heat transfer coefficients predicted from the theoretical analysis are in reasonable agreement with the experimental data available in literature.     

Developed steady wave motions of a liquid film falling along a vertical plane

The falling of a thin viscous fluid layer (film) along a vertical plane under the effect of gravity is accompanied by wave motions in which capillary forces play an essential part. An equation for the film thickness h(x, t) is used extensively in analyses of these motions. This equation, obtained from the Navier—Stokes equations and the boundary conditions under different assumptions, reduces to an ordinary third-order nonlinear differential equation [1–7] for steady plane motions. Periodic solutions of this equation were sought by the methods of asymptotic expansions in the amplitude or by Fourier series expansions [1–7], which assumes a sequential accounting of the nonlinearity as a small perturbation. This limits the validity of the results obtained to the domain of small amplitudes. The case of arbitrary amplitudes is considered in this paper. A solution of the problem, based on an asymptotic expansion in the parameter ε is constructed. In this expansion the equation for the first approximation remains nonlinear but admits of integration, which discloses the class of bounded periodic solutions. Moreover, strict integral relations (for any ε) are obtained, and a variational problem about seeking the lower bound of values of the mean film thickness and other characteristics of the ultimately developed optimal motions is formulated and solved on their basis. The results obtained agree with experiments.     

Stability of a falling conducting liquid film in an alternating electric field

The flow of a thin viscous conducting liquid film falling along one of the plates of a vertically positioned plane capacitor is studied. The capacitor is connected to an alternating current power supply. It is shown that the presence of the electric field leads to flow destabilization; moreover, the parametric resonance of capillary waves is observed.     

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.     

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.