A model for pressure loss, film thickness, and entrained fraction for gas-liquid annular flow

A two-component (air-water) annular flow model is presented requiring only flow rates, absolute pressure, temperature, and tube diameter. Film thicknesses (base film and wave height) are calculated from a critical film thickness model. Modeled pressure gradient is weighted by wave intermittency to compute average pressure gradient. Film flow rate and wave velocity are estimated using the universal velocity profile in the waves and a piecewise linear profile in the base film. For vertical flow, mean absolute errors for film thickness, wave velocity, and pressure gradient are 9%, 9%, and 19%, respectively. In horizontal flow, mean absolute errors for pressure gradient, base film thickness, and disturbance wave velocity are 17%, 10%, and 14%, respectively, on par with those from single-behavior models that require additional film thickness or other data as inputs.     

Pure steam condensation model with laminar film in a vertical tube

A new physical model for calculating the liquid film thickness and condensation heat transfer coefficient in a vertical condenser tube is proposed by considering the effects of gravity, liquid viscosity, and vapor flow in the core region of the flow. To estimate the velocity profile in the liquid film, the liquid film was assumed to be in Couette flow forced by the interfacial velocity at the liquid–vapor interface. For simplifying the calculation procedures, the interfacial velocity was estimated by introducing an empirical power-law velocity profile. The resulting film thickness and heat transfer coefficient from the model were compared with the experimental data and the results obtained from the other condensation models. The results demonstrated that the proposed model described the liquid film thinning effect by the vapor shear flow and predicted the condensation heat transfer coefficient from experiments reasonably well.     

An accurate model for the thin film flow

This paper deals with the linear stability of a liquid film flowing down an inclined plane. The Navier-Stokes equations were reduced into four evolution equations that describe the development of the film depth, the flow rate, the free surface velocity, and the wall shear stress, using the Karman-Polhausen boundary layer integral method. Thus, we were able to determine the stability threshold and approach well the critical wave number for long waves. The obtained results were found to be in good agreement with the experiments of Liu et al.     

Long waves on a viscoelastic film flow down a wavy incline

Long waves on a viscoelastic film flow down a wavy inclined plane is investigated. The analysis is performed to see how long non-linear waves on viscoelastic film down an uneven inclined wall are deformed due to the non-uniformity of the basic flow. The results are then compared with those corresponding to Newtonian film down a wavy inclined wall as well as viscoelastic film down a plane inclined wall.     

An extended study of the hydrodynamics of gravity-driven film flow of power-law fluids

A new similarity transformation has been devised for extensive studies of accelerating non-Newtonian film flow. The partial differential equations governing the hydrodynamics of the flow of a power-law fluid down along an inclined plane surface are transformed into a set of two ordinary differential equations by means of the dimensionless velocity component approach. Although the analysis is applicable for any angle of inclination (0<π/2), the resulting one-parameter problem involves only the power-law index n. Nevertheless, physically essential quantities, like the velocity components and the skin-friction coefficient, do depend on and relevant relationships are deduced between the vertical and inclined cases. Accurate numerical similarity solutions are provided for n in the range from 0.1 to 2.0. The present method enables solutions to be obtained also for highly pseudo-plastic films, i.e. for n below 0.5. The mass flow rate entrained into the momentum boundary layer from the inviscid freestream is expressed in terms of a dimensionless mass flux parameter Φ, which depends on the dimensionless boundary layer thickness and the velocity components at the edge of the viscous boundary layer. Φ, which is thus an integral part of the similarity solution, turns out to decrease monotonically with n. This parameter is of particular relevance in the determination of the streamwise position at which the entire freestream has been entrained and viscous stresses prevail all the way to the free surface of the film. A short-cut method to facilitate rapid and yet accurate estimates of the mass flux parameter is developed to this end.     

A rheological theory for liquid crystal thin films

A macroscopic rheological theory for compressible isothermal nematic liquid crystal films is developed and used to characterize the interfacial elastic, viscous, and viscoelastic material properties. The derived expression for the film stress tensor includes elastic and viscous components. The asymmetric film viscous stress tensor takes into account the nematic ordering and is given in terms of the film rate of deformation and the surface Jaumann derivative. The material function that describes the anisotropic viscoelasticity is the dynamic film tension, which includes the film tension and dilational viscosities. Viscous dissipation due to film compressibility is described by the anisotropic dilational viscosity. Three characteristic film shear viscosities are defined according to whether the nematic orientation is along the velocity direction, the velocity gradient, or the unit normal. In addition the dependence of the rheological functions on curvature and film thickness has been identified. The rheological theory provides a theoretical framework to future studies of thin liquid crystal film stability and hydrodynamics, and liquid crystal foam rheology. Received: 9 October 2000 Accepted: 6 April 2001     

Planar laser-induced fluorescence (PLIF) measurements of liquid film thickness in annular flow. Part I: Methods and data

Most approaches to the modeling of annular flow require information regarding the thin liquid film surrounding the central gas core. This film is hypothesized to present a rough surface to the gas core, enhancing interfacial shear and pressure loss, with the roughness closely linked to the height of the film. This height is typically obtained from conductance probe measurements. The present work used planar laser-induced fluorescence to provide direct visualization of the liquid film in upward vertical air–water annular flow. Images were processed to produce the distribution of film heights. The standard deviation and average film thickness are found to be an increasing function of liquid flow and a decreasing function of gas flow, with the standard deviation approaching 0.4 times the average at sufficient liquid flow.     

The effect of total reflection in PLIF imaging of annular thin films

Planar laser-induced fluorescence (PLIF) imaging techniques are increasingly used to study wavy annular flows in vertical and horizontal pipes. We present a ray-tracing study that analyzes the imaging properties of the PLIF technique for uniform and wavy annular films. Part of the emitted fluorescence will hit the water–air interface at angles greater than the critical angle of total reflection. As a consequence, PLIF imaging probes the film twice: From bottom to top and then again from top to bottom. Deflected fluorescence widens PLIF images of uniform films by about 30%. Surface waves further increase the contribution of deflected rays to the overall signal. At the steep slopes of large amplitude axial waves, total reflection might widen PLIF signals by up to a factor of two.     

Planar laser-induced fluorescence (PLIF) measurements of liquid film thickness in annular flow. Part II: Analysis and comparison to models

Film thickness distributions in upward vertical air–water annular flow have been determined using planar laser-induced fluorescence (PLIF). Film thickness data are frequently used to estimate interfacial shear and pressure loss. This film roughness concept has been used in a number of models for annular flow of varying complexity. The PLIF data are presently applied to the single-zone interfacial shear correlation of Wallis; the more detailed model of Owen and Hewitt; and the two-zone (base film and waves) model of Hurlburt, Fore, and Bauer. For the present data, these models all under-predict the importance of increasing liquid flow on pressure loss and interfacial shear. Since high liquid flow rates in annular flow induce disturbance wave and entrainment activity, further modeling in these areas is advised.     

A two-phase, two-component model for vertical upward gas-liquid annular flow

In this paper, a new two-fluid two-component computational fluid dynamics (CFD) model is developed to simulate vertical upward two-phase annular flow. The two-phase VOF scheme is utilized to model the roll wave flow, and the gas core is described by a two-component phase consisting of liquid droplets and gas phase. The entrainment and deposition processes are taken into account by source terms of the governing equations. Unlike the previous models, the newly developed model includes the effect of liquid roll waves directly determined from the CFD code, which is able to provide more detailed and, the most important, more self-standing information for both the gas core flow and the film flow as well as their interactions. Predicted results are compared with experimental data, and a good agreement is achieved.     

Measurement of the liquid film thickness in micro tube slug flow

Slug flow is one of the representative flow regimes of two-phase flow in micro tubes. It is well known that the thin liquid film formed between the tube wall and the vapor bubble plays an important role in micro tube heat transfer. In the present study, experiments are carried out to clarify the effects of parameters that affect the formation of the thin liquid film in micro tube two-phase flow. Laser focus displacement meter is used to measure the thickness of the thin liquid film. Air, ethanol, water and FC-40 are used as working fluids. Circular tubes with five different diameters, D = 0.3, 0.5, 0.7, 1.0 and 1.3 mm, are used. It is confirmed that the liquid film thickness is determined only by capillary number and the effect of inertia force is negligible at small capillary numbers. However, the effect of inertia force cannot be neglected as capillary number increases. At relatively high capillary numbers, liquid film thickness takes a minimum value against Reynolds number. The effects of bubble length, liquid slug length and gravity on the liquid film thickness are also investigated. Experimental correlation for the initial liquid film thickness based on capillary number, Reynolds number and Weber number is proposed.     

Critical friction factor modeling of horizontal annular base film thickness

Measurements of liquid base film thickness distribution have been obtained for 206 horizontal annular two-phase (air–water) flow conditions in 8.8 mm, 15.1 mm, and 26.3 mm ID tubes. It is found that the trends in base film thickness measurement do not match trends in the literature for average film thickness, which considers waves and base film together. An iterative critical friction factor model is used to model circumferentially-averaged base film thickness; an explicit, empirical correlation is also provided. Asymmetry is well-correlated by a modified Froude number based on the correlated base film thickness and the gas mass flux. The iterative model is also extended to estimate the critical film flow rate.     

A new solution for the rotation-driven spreading of a thin fluid film

Lie groups are used to solve the equation governing the flow of a thin liquid film subject to centrifugal spreading and viscous resistance. A new implicit solution is found. It is shown how this relates to the previous known solutions for the spreading of an initially flat film, the steady state and a separable solution. New permissible forms for the film evolution are also studied, including solutions exhibiting finite time blow-up. Near the contact line, where the film height tends to zero, an approximate explicit solution is obtained which may be used to describe a film with any size contact angle.     

On comparison of exact and series solutions for thin film flow of a third‐grade fluid

The present investigation derives the exact and series solutions for steady thin film flow of a third‐grade fluid. The series solution is constructed by a homotopy analysis method. The obtained solutions are compared and an excellent agreement between these is achieved. Copyright © 2009 John Wiley & Sons, Ltd.     

On study of horizontal thin film flow of Sisko fluid due to surface tension gradient

The present paper is concerned with the steady thin film flow of the Sisko fluid on a horizontal moving plate, where the surface tension gradient is a driving mechanism. The analytic solution for the resulting nonlinear ordinary differential equation is obtained by the Adomian decomposition method (ADM). The physical quantities are derived including the pressure profile, the velocity profile, the maximum residue time, the stationary points, the volume flow rate, the average film velocity, the uniform film thickness, the shear stress, the surface tension profile, and the vorticity vector. It is found that the velocity of the Sisko fluid film decreases when the fluid behavior index and the Sisko fluid parameter increase, whereas it increases with an increase in the inverse capillary number. An increase in the inverse capillary number results in an increase in the surface tension which in turn results in an increase in the surface tension gradient on the Sisko fluid film. The locations of the stationary points are shifted towards the moving plate with the increase in the inverse capillary number, and vice versa locations for the stationary points are found with the increasing Sisko fluid parameter. Furthermore, shear thinning and shear thickening characteristics of the Sisko fluid are discussed. A comparison is made between the Sisko fluid film and the Newtonian fluid film.     

Liquid film flow due to an unsteady stretching sheet

We have studied two-dimensional flow of a thin liquid film over an impulsively stretching sheet under assumption of uniform initial film thickness. Using singular perturbation technique both momentum and film evolution equations are solved analytically for small Reynolds number and these solutions are verified numerically. Numerical computation for large Reynolds number shows an anomalous behaviour of film thinning rate in different time zone. These results are explained physically and the crucial role-played by viscosity in this case is highlighted. It is found that faster rate of thinning can be obtained if the sheet is stretched impulsively with continuously increasing stretching speed.     

A note on the flow of a non-newtonian fluid film

Film flow of liquids simulated by the third-grade model down a vertical longitudinallyoscillating wall is investigated. The non-linear partial differential equation resulting from themomentum equation is solved by the method of Galerkin and flow enhancement is predicted fordifferent material constants. It is also found that an increase in either the amplitude or frequency ofthe vibration always leads to an increase in the magnitude of the flow enhancement.     

A multiphase,micro-scale PIV measurement technique for liquid film velocity measurements in annular two-phase flow

Prediction methods for two-phase annular flow require accurate knowledge of the velocity profile within the liquid film flowing at its perimeter as the gradients within this film influence to a large extent the overall transport processes within the entire channel. This film, however, is quite thin and variable and traditional velocimetry methods have met with only very limited success in providing velocity data. The present work describes the application of Particle Image Velocimetry (PIV) to the measurement of velocity fields in the annular liquid flow. Because the liquid is constrained to distances on the order of a millimeter or less, the technique employed here borrows strategies from micro-PIV, but micro-PIV studies do not typically encounter the challenges presented by annular flow, including very large velocity gradients, a free surface that varies in position from moment to moment, the presence of droplet impacts and the passage of waves that can be 10 times the average thickness of the base film. This technique combines the seeding and imaging typical to micro-PIV with a unique lighting and image processing approach to deal with the challenges of a continuously varying liquid film thickness and interface. Mean velocity data are presented for air–water in two-phase co-current upward flow in a rectangular duct, which are the first detailed velocity profiles obtained within the liquid film of upward vertical annular flow to the authors’ knowledge. The velocity data presented here do not distinguish between data from waves and data from the base film. The resulting velocity profiles are compared with the classical Law of the Wall turbulent boundary layer model and found to require a decreased turbulent diffusivity for the model to predict well. These results agree with hypotheses previously presented in the literature.     

A model for droplet entrainment in heated annular flow

The ability to accurately predict droplet entrainment in annular two-phase flow is required to effectively calculate the interfacial mass, momentum, and energy transfer, which characterizes nuclear reactor safety, system design, analysis, and performance. Most annular flow entrainment models in the open literature are formulated in terms of dimensionless groups, which do not directly account for interfacial instabilities. However, many researchers agree that there is a clear presence of interfacial instability phenomena having a direct impact on droplet entrainment. The present study proposes a model for droplet entrainment, based on the underlying physics of droplet entrainment from upward co-current annular film flow that is characteristic to light water reactor safety analysis. The model is developed based on a force balance and stability analysis that can be implemented into a transient three-field (continuous liquid, droplet, and vapor) two-phase heat transfer and fluid flow systems analysis computer code.