Morton Number Research Articles

Influence of a Pair of Unequal Rotational Fluxes on Entrained Gaseous Filament

Abstract Efforts are made to elucidate a comprehensive analysis of entrainment dynamics triggered by a couple of unequal rotational fluxes within a viscous pool. Cylindrical rollers are employed to establish the rotational field. The top drum is equally submerged in both phases and also it provides a constant rotational inertia. Concomitantly, the bottom roller is completely submerged in the viscous bath, and it provides an unequal rotational strength in reference to top roller. The average rotational strength of both rollers is measured using average Capillary number (Caavg). The entrainment phenomenon is strongly influenced by both Caavg and gap between the rollers (W/D). Characterization of entrained filament is elucidated by predicting the horizontal distance (L*), radial distance (r*), temporal vertical displacement (Y*), maximum vertical displacement (Ymax*), width (H*), and location of V-shaped diversion (Øs*). Characterization of liquid tip is performed by measuring the travel rate (γ*) along periphery of drum from receding to advancing junction. Air mass ejection from filament tip is analyzed by estimating the first bubble ejection time (t1st*), volume of accumulated of ejected gaseous masses (v*), and ejection frequency (f). Furthermore, the effect of gravitational pull (specified by Archimedes number, Ar) and viscous drag (measured by Morton number, Mo) on the pattern of entrained air filament is described. Lastly, an analytical framework is established to determine the width of the V-junction by balancing the important influencing forces, which are strongly affecting the filament. Analytical observations show a satisfactory agreement with numerical findings.

Lift force on single bubbles in subcooled pool boiling systems: Experimental measurements and CFD simulations

Lift coefficient has been evaluated for a subcooled pool boiling system with water as continuous phase and vapor (steam) bubble as dispersed phase generated on vertical walls with constant heat flux for a specific Rayleigh number (Ra) range (8x1012 < Ra < 2x1013) for the first time. Experimental investigation has been done with High Speed Camera while two phase CFD simulations have been performed using Volume of Fluid (VOF) method. The Morton number (M) for this system is log10M = -10.6 and Eötvös number (Eo) range is 0.2 < Eo < 1.3 while the shear rates have been maintained in the range 2 ≤ Sr ≤ 2.8. Bubble size of the bubbles arising from the sites on vertical surface have been determined by both experiments and simulations. The CFD predictions for bubble size and lift coefficients agree well with experimental measurements with a deviation of less than 10%. Further, empirical expressions from experimental data for the estimation of bubble size and lift coefficients have also been presented. The data for lift coefficients obtained is different from the ones of bubbles generated by single sparger in air-water systems.

Numerical Study of Bubble Column Water-Air System by the VOF Method

This paper aims to present a comprehensive study of the dynamics of a bubble using the Volume of Fluid (VOF) model in Fluent software. The simulation of two-phase flows is carried out by calculating the terminal velocity, bubble flow contours at different column heights, and the evolution of bubble circularity and Reynolds number at different times. The calculation was carried out on an air bubble with a diameter equal to 10 mm and zero introduction velocity by modifying the simulation parameters, such as the surface tension, to study their influence on the deformation of the bubble. This study will present four different shape regimes, which are obtained by varying the Bo (Bond number) and Mo (Morton number) values within the corresponding ranges of 1 &lt; Bo &lt; 103 and 5×10-8 &lt; Mo &lt; 102 . In addition, simulations are performed using large density and viscosity ratios of 1000 and 100, respectively. The results are comparable with great precision to the numerical simulation and experimental data.

Optimizing Mass Transfer in Multiphase Fermentation: The Role of Drag Models and Physical Conditions

Detailed knowledge of the flow characteristics, bubble movement, and mass transfer is a prerequisite for the proper design of multiphase bioreactors. Often, mechanistic spatiotemporal models and computational fluid dynamics, which intrinsically require computationally demanding analysis of local interfacial forces, are applied. Typically, such approaches use volumetric mass-transfer coefficient (kLa) models, which have demonstrated their predictive power in water systems. However, are the related results transferrable to multiphase fermentations with different physicochemical properties? This is crucial for the proper design of biotechnological processes. Accordingly, this study investigated a given set of mass transfer data to characterize the fermentation conditions. To prevent time-consuming simulations, computational efforts were reduced using a force balance stationary 0-dimension model. Therefore, a competing set of drag models covering different mechanistic assumptions could be evaluated. The simplified approach of disregarding fluid movement provided reliable results and outlined the need to identify the liquid diffusion coefficients in fermentation media. To predict the rising bubble velocities uB, the models considering the Morton number (Mo) showed superiority. The mass transfer coefficient kL was best described using the well-known Higbie approach. Taken together, the gas hold-up, specific surface area, and integral mass transfer could be accurately predicted.

Interfacial Dynamics Around a Swirling Roller in the Presence of Oppositely Imposed Horizontal Crossflows

Abstract The behavior of interfacial dynamics around a swirling circular drum is reported in the presence of horizontal crossflows of gaseous phase. A constant submergence ratio of the rotary drum is considered throughout this work. Forward and reverse crossflows are defined based on the direction of rotation of drum. A clockwise rotation is imposed on the drum and the rotation direction remains fixed throughout the present investigation. The pattern of film flow and cusp entrainment is observed thoroughly by considering rotational speed (ω) and strength of crossflow as influencing parameters. Strength of crossflow of gaseous phase is calibrated using flow Reynolds number based on roller diameter (Reflow=ρgUD/μg). Subsequently, we characterized the coated film thickness (h*=h/D) and cusp width (H*=H/D) by varying ω and Reflow. Furthermore, efforts are made to observe the influence of gravitational pull (calibrated using Archimedes number, Ar) and viscous drag (determined using Morton number, Mo) on interfacial morphology. The interfacial structures respond more to the rotational inertia compared to crossflow with the continuous increase of Mo. Again, the trailing end of cusp gets widened for the case of forward crossflow, whereas the cusp growth is obstructed in the presence of reverse crossflow, which is visualized using velocity vectors. We have also predicted appropriate correlations for film thickness (h*), cusp width (H*), and steady cusp length (θs*) in the presence of forward as well as reverse crossflow of air stream. Lastly, a theoretical framework is also developed to estimate film thickness and cusp width for both cases of crossflows and this framework agrees well with the simulated predictions.

Numerical simulation of the motion and interaction of bubble pair rising in a quiescent liquid

In the present study, the motion and interaction of a bubble pair rising in a stationary liquid are simulated in two dimensions using the volume of fluid method (VOF) in the open-source CFD package OpenFOAM. For validation, the single bubble rising for different regimes was simulated. Comparing the present study results and previous numerical and experimental results showed that the results have a good agreement. In the present study, the bubble pair is arranged along a horizontal line, and results showed that in 2D domain four different interactions happen between the bubble pair rising based on the Bond (Bo) and Morton (Mo) numbers which are: coalescence, bouncing, zigzagging, and breakup. The simulations are done at three different slopes of the line connecting the center of two bubbles (θ=[0°, 45°, and 90°]), which shows the different arrangements of bubble pairs. At θ=90°, due to the attraction created by the wake of the leading bubble, the trailing bubble gets a significant acceleration in its motion, and will coalesce in the absence of surfactants. The results showed that the value of the trailing bubble velocity increases with decreasing the separation distance between the bubbles. Also, the rising velocity of the trailing bubble is always greater than the velocity of a single bubble due to the attraction effect of the leading bubble’s wake. The drag force of a trailing bubble decreases with decreasing the separation distance between the bubbles. Finally, a general criterion based on dimensionless numbers of Bo and Mo is presented to determine the type of interaction between bubble pair.

Fluctuation analysis and spatial distribution model of particle concentration in a lab-scale scrubbing-cooling chamber

The motion and distribution of ash from coal gasification during the scrubbing and cooling process are crucial for revealing the fluid flow pattern and improving the efficiency of multiphase separation. Experimental measurements of the local solid holdup (εs) in the liquid pool of a laboratory-scale gas-liquid-solid three-phase scrubbing-cooling chamber were carried out using an optical fiber probe. The fluctuations of εs in time and frequency domains under different operating conditions were discussed. The axial and radial distribution of εs was quantitatively analyzed. The results showed that inter-particle collision and particle-fluid interaction are the key factors leading to a fluctuation in particle concentration. Axial uniformity of particle concentration can be quantified by solid-phase volume fraction, particle Reynolds number, Archimedes number, and Morton number; εs in different regions is expressed according to the degree of influence of bubble wake as a correlation equation consisting only of cross-sectional average solid holdup and dimensionless radial position; a dimensionless expression for εs concerning operating conditions was established.

Predicting the Splash of a Drop Impacting a Thin Liquid Film.

An experimental study is carried out to investigate droplet-film interactions when a drop impinges on a thin stagnant film of the same liquid. The impacting drop causes either liquid deposition or splash, consisting of prompt generation of secondary drops or a delayed process. By varying the drop diameter and impact velocity, measurements are made to characterize the phenomena using five different liquids that are chosen to cover a wide range of liquid properties (viscosity and surface tension). The drop impact dynamics are captured with a high-speed digital camera with real-time, high-resolution image processing. The drop-splash threshold is found to scale with inertial and viscous forces, or Reynolds number (Re), as well as capillary forces, as described by the balance of gravitational and interfacial tension forces, or Bond number (Bo); fluid properties are described by their Morton number (Mo). A correlation, functionally expressed as Re = ϕ(Bo,Mo), is devised to determine the splash/no-splash (or deposition) boundary, and the predictions for the splash/no-splash outcomes agree well with the experimental outcomes as well as those readily available in the literature.

Free‐energy lattice Boltzmann simulations of slicing of rising droplets

AbstractWe performed two‐dimensional simulations of the slicing of a rising droplet by a vertical knife and by two knives. We used a free‐energy lattice Boltzmann method. Most simulations are for an Eötvös number of 3.96 and a Morton number of 1.24 × . We carefully probed effects of the spatial resolution on the rising speed and slicing process. The effect of wettability of the knife(s) was studied by varying the equilibrium contact angle of the drop on the knife in the range of 45° to 135°. Slicing time as well as fine droplet fragments staying behind on the knife depend on the knife's wettability.

Direct numerical simulation of heat transfer on a deformable vapor bubble rising in superheated liquid

Heat transfer on a vapor bubble rising in superheated liquid is investigated by direct numerical simulation. The vapor–liquid system is described by the one-fluid formulation with the level set method capturing the interface. The proportional-integral-derivative controller is employed to keep the bubble's location fixed and evaluate interfacial forces. The heat transfer performance featured by the Nusselt number is evaluated based on the energy balance. Simulations are carried out for the bubble Reynolds number ranging from 20 to 500 and Morton number from 1.10 × 10−10 to 3.80 × 10−4. The aim of this paper is to shed some light on the effect of bubble deformation and oscillation on interfacial heat transfer. The results show that the front part of the bubble contributes to the majority of the interfacial heat transfer, while the rear part mainly affects the oscillation amplitude of the total heat transfer. The interface stretch during bubble oscillation is considered as a key mechanism in enhancing the instantaneous Nusselt number. The potential flow solution of the averaged Nusselt number is corrected by considering the influence of the aspect ratio. This research provides additional insights into the mechanism of interfacial heat transfer, and the results apply equally to interfacial mass transfer.

EFFECTS OF SURFACTANT ON LIFT COEFFICIENT OF ELLIPSOIDAL BUBBLES IN THE VISCOUS-FORCE DOMINANT REGIME

Effects of surfactants on lift coefficients, &lt;i&gt;C&lt;sub&gt;L&lt;/sub&gt;&lt;/i&gt;, of single ellipsoidal bubbles rising through linear shear flows were investigated. Two types of surface-active agents, Triton X-100 and 1-octanol, were used. The liquid properties except for the surface tension were identical to those in a clean system of logM &amp;#61; -5.5, where M is the Morton number. The range of the bubble Reynolds number was 0.1 &amp;#60; Re &amp;#60; 70. Bubble shapes were either spherical or ellipsoidal. Comparing with clean bubbles, less deformation of contaminated bubbles was confirmed. A shape correlation without taking the dimensionless shear rate into account gave good evaluations of the bubble aspect ratio, which means that the shear rate is not a dominant factor causing the change of shape deformation. However, drag coefficients were affected by the shear rate. A new correlation of drag coefficients was deduced, which agreed well with the experimental data. Both clean and contaminated &lt;i&gt;C&lt;sub&gt;L&lt;/sub&gt;&lt;/i&gt; data showed a similar tendency: after a drastic decrease in &lt;i&gt;C&lt;sub&gt;L&lt;/sub&gt;&lt;/i&gt;, &lt;i&gt;C&lt;sub&gt;L&lt;/sub&gt;&lt;/i&gt; slightly increases and then decreases to a negative value while increasing the bubble Reynolds number, Re. A difference in concentration of Trion X-100 resulted in only a slight change in &lt;i&gt;C&lt;sub&gt;L&lt;/sub&gt;&lt;/i&gt; at high Re regime. Different types of surfactant resulted in noticeably different values of &lt;i&gt;C&lt;sub&gt;L&lt;/sub&gt;&lt;/i&gt; especially at low Re. The &lt;i&gt;C&lt;sub&gt;L&lt;/sub&gt;&lt;/i&gt; of small bubbles in contaminated systems could be reproduced by a correlation for solid particles, supporting that fully contaminated spherical bubbles behave like solid spheres. For deformed bubbles, &lt;i&gt;C&lt;sub&gt;L&lt;/sub&gt;&lt;/i&gt; can be expressed by relating the negative lift force due to shape deformation with the drag force.

A study of bubble size/shape evolution in microbubble countercurrent contacting flotation column

AbstractCationic collectors are widely used in the flotation of iron ore, silicate, oxides, and so forth. At this stage, so much research has been carried out on flotation column cells, but more attention needs to be paid to the hydrodynamic properties in the presence of cationic collectors in cells. In order to study foaming performance and hydrodynamic characteristics in the presence of GE609, the bubble characteristics are studied in an experimental scale flotation column. A high‐speed camera recorder is used to take bubbles' images, and the bubble size and shape under different conditions is measured by image analysis. This paper aims to explore the effect of dispersed gas and continuous liquid velocities on the bubble size and aspect ratio at different axial heights in a countercurrent flotation column in the presence and absence of GE609. In the systems, the bubble size distributions and shapes at different axial positions were obtained by using high‐speed camera technology in the superficial gas velocity (Ug) ranges of 0.0021–0.0106 m/s and superficial liquid velocity (UL) ranges of 0.00123–0.00209 m/s. Through the comparison with relevant papers and studying the experiment data, the relationship between the Sauter mean diameter and Ug, UL and axial heights have been analyzed by using multiple linear regression and dimensional analysis. The experimental data have been adopted to establish the correlation between the bubble diameter and the dimensionless array, which have been compared with other literature. And the predicted values/results of Weber and Morton number correlation in this work are relatively close to the experimental values.

Gaseous Entrainment Dynamics in a Viscous Pool Due to Combined Influence of Asymmetric Rotational Field and Crossflow of Air

Abstract Efforts are made to perform simulations to describe the gaseous entrainment dynamics in a viscous liquid pool due to the combined influence of asymmetric converging rotational field and continuous freestream flow of air. A pair of counter-rotating and equal sized rollers is placed inside the pool along a horizontal line. Gerris is an open-source solver, which is employed to carry out the present computational study. Complex interfacial configurations are illustrated with the influence of relevant input parameters, such as rotation of rollers 1 and 2 (measured by Capillary number, Ca1=Rω1μl/σ and Ca2=Rω2μl/σ, where R=D/2 is roller radius), submersion depth (b*), the gap between the rollers (2a*), and strength of air stream flow (measured by Reynolds number, Reflow=ρgUD/μg). It has been observed that the depth of steady entrainment is reduced at Reflow≠0 compared to Reflow=0 because the hydrodynamic force acts as an opposing force to viscous pumping and rotating inertia. A complete understanding of disintegration of and subsequent accumulation of gaseous bubbles from the cusp tip is characterized in detail. In addition, the influence of viscous drag (specified by Morton number, Mo=gμl4(ρl−ρg)/(ρl2σ3)) and gravitational pull (estimated by Archimedes number, Ar=gD3ρl2/μ2) on the phase contours are also reported. Finally, an analytical formulation is proposed to analyze the structure of entrainment, and this model reports an excellent match with the numerical findings.

Rotational flux influenced cusp entrainment in a viscous pool

Evolution of entrained gaseous cusp in a viscous liquid pool caused by a fully immersed horizontal revolving circular drum is elucidated thoroughly. Finite volume-based open source Gerris is employed to carry out the numerical simulations. Length θs* and width (H*) of the cusp are characterized thoroughly by varying the strength of rotational flux (measured by the capillary number, Ca) and depth of immersion (submersion height to roller diameter ratio, h/D) of the spinning roller. Locus and travel rate of cusp tip are also elaborated from origination to attainment of steady state with the progress of time for different combinations of Ca and h/D. We have also reported the bubble ejection mechanism from the advancing end of the cusp after achieving macroscopic steady length. Bubbling frequency and volume accumulation of detached bubbles show strong dependency on Ca or h/D. Entrainment patterns are also observed by investigating the effect of gravitational pull (measured by employing the Archimedes number, Ar) and viscous drag (specified by using the Morton number, Mo). Correlations are developed to estimate the cusp width (H*) at different angular locations and steady length of gaseous cusp θs*, which show satisfactory agreement within ±6%. Finally, an analytical model is proposed to determine the cusp width by using the relevant influencing forces acting on the cusp. A satisfactory agreement is obtained between the analytical solutions and computational results.

Three-dimensional phase-field lattice-Boltzmann simulations of a rising bubble interacting with obstacles: Shape quantification and parameter dependence

Exploring the bubble dynamics in the presence of obstacles can improve understanding of mechanism, design, and operation of multiphase flow. However, large deformation and complex hydrodynamics during the bubble–obstacle interaction pose challenges for shape quantification and parameter dependence. In this work, the bubble–obstacle interaction is investigated by employing a conservative phase-field lattice-Boltzmann model implemented on a parallel platform. Two shape parameters, which are the combination of the bubble geometrical parameters, are defined to characterize the bubble deformation during rising, impacting, and sliding process. The effects of the flow parameters (Reynolds, Eötvös, and Morton numbers) and the obstacle geometry settings (size and layout) are discussed, and a multilinear correlation is established to obtain a thorough evaluation. The difference induced by spatial dimension is further discussed to illustrate the necessity of simulating three-dimensional liquid–gas flow. The results can help decode the intricate bubble dynamics and lay a foundation for developing physically informed models for predicting the bubble–obstacle interaction.

Toward the Ultimate Goal on Simulating Multi-Phase Flows, Simulation of Film Boiling at High-Density Ratios

Abstract Flow boiling is a common phenomenon in modern technology and industries. A finite difference Front Tracking method is used to study film boiling on a flat plate subject to a constant heat flux. Large density ratios are examined (1000, 2000). The flow structure including the circulation zones, the vortex developing areas, and vortex pair generation are studied in detail. The numerical method is validated by comparing the result with the analytic solution for a simple problem. Large bubbles comparable to the computational domain are formed at large density ratios. The bubble either breaks up and departs from the rest of the gas, or a jet is formed depending on the heat flux imposed. It is found that the Nusselt number depends on the heat flux on the wall. Specifically, the Nusselt number decreases as the heat flux is raised. The Morton number also affects the Nusselt number in agreement with experimental correlations. The Nusselt number increases when the Morton number is raised. The results obtained agree with experimental correlations at relatively low heat fluxes. Agreement gets worse at large heat fluxes. Experimental correlations estimate larger Nusselt numbers compared to the present study. The flow enhances along the direction of the gravitational acceleration at large density ratios (2000). A wide neck is formed that stabilizes at a specific position inside the flow. The microstructure of the flow shows a vortex pair that develops beneath the root of the bubble at a large density ratio (2000).

Numerical simulation of single bubble motion along inclined walls: A comprehensive map of outcomes

The bubble interactions with a wall affect many industrial application's performance. The single bubble rising in a stagnant fluid is studied next to an inclined wall. The two phases are incompressible, and the interface between phases is captured with the volume of fluid (VoF) method. The influence of various parameters such as wall slope (θ), contact angle (φ), and dimensionless numbers (Morton and Bond) on the bubble-wall regime is investigated. The numerical results (bubble shape and terminal velocity) are validated with experiments.After comprehensive numerical simulation over a wide range of Bond (1≤Bo≤40) and Morton (2.54×10−11≤Mo≤1×100) for inclined walls (0∘<θ<90∘), three bubble-wall interaction regimes: sliding, intermittent and non-contact are proposed and a regime map is plotted. We recognized two sliding regimes based on Bond numbers at small inclined walls (θ≤30∘). In the sticking sliding regime (Bo≤5), bubbles remain attached to the inclined wall. They make the wall dry along their path. But at higher Bond in film sliding, bubbles slip over a thin liquid film. Thus, the wall surface is always wet. The film thickness is linearly proportional to bubble diameter. Non-contact regimes occur at steep walls (θ≥70∘), where the bubble never touches the wall. In moderate wall slopes (30∘<θ<70∘), a bubble repeatedly impacts on and rebounds from the wall. The intermittent regime is an unstable regime, and it alters to a sliding or a non-contact regime based on Bond and Morton numbers.

Investigation of bubble dynamics in a micro-channel with obstacles using a conservative phase-field lattice Boltzmann method

Simulating bubble dynamics impacting on obstacles is challenging because of large liquid-to-gas density ratio and complex interface deformation. In this study, a conservative phase-field model, based on a modified Allen–Cahn equation, is employed to accurately capture the bubble interface, and the lattice Boltzmann model is applied to solve the flow field. The bubble rises under the influence of buoyancy force and surface tension force, and complex topology changes, such as rotation, breakup, and squeeze deformation, are predicted in the presence of obstacles. Three dimensionless numbers, including Reynolds, Eötvös, and Morton numbers, are used to characterize bubble dynamics, and two shape indicators, including the revised Blaschke coefficient and the oblateness degree, are introduced to obtain a more systematic assessment of the bubble shape. Effects of flow parameters and obstacle geometries on bubble dynamics impacting on obstacles are investigated to render a quantitative investigation with physical insights. Model extension to the 3D case, the low-viscosity flow and non-pure fluid is further remarked, which can shed light onto future development of physically informed models for predicting the bubble behavior in more real scenarios.

Development of a novel artificial neural network model for closed pulsating heat pipe with water and aqueous solutions

AbstractAqueous solutions are increasingly introduced as the working fluids of pulsating heat pipe (PHP), to balance and optimise certain thermo‐physical properties of pure water. In this paper, a first attempt is conducted to predict the heat transfer performance of closed PHP with pure water and available aqueous solutions based on a fully connected feed‐forward artificial neural network (ANN). The notable dimensionless numbers, Kutateladze number (Ku), Bond number (Bo), Morton number (Mo), Jackob number (Ja), Prandtl number (Pr), Laplace number (La), d/Le and number of turns (N) were selected as the inputs of the ANN model. It was found that the prediction of proposed ANN model posed a good agreement with experimental data with the MSE and correlation coefficient of 0.026 and 0.979, respectively. For 81.02% of the collected data, the absolute deviation of prediction for thermal resistance was within 25%. For 95% of collected data, the prediction of thermal resistance and temperature difference between the evaporation and condensation section fell within ±0.37 K/W, and ±17.1 K, respectively. The evaluation of evaporation and condensation section temperatures was also presented based on the proposed ANN model.

Effect of Oil Properties on the Generation of Nano-Aerosols During Bubble Bursting Through Crude Oil-Dispersant Slicks.

This study examines the effect of dispersant and oil properties on the aerosolization of fresh and weathered surface crude oil slicks by bursting of a plume of ∼0.7 mm bubbles. A scanning mobility particle sizer measures the size distribution of aerosols in the 20-400 nm range in a clean air chamber. The 500-μm-thick slicks contain oils with varying origin, viscosity, interfacial tension, and weathering state. Test are performed with and without premixed dispersant (Corexit 9500A), which reduces the oil-seawater interfacial tension by 2 orders of magnitude at a dispersant-to-oil ratio (DOR) of 1:25. When compared to aerosolization in clean seawater, the nano-aerosol concentration decreases for slicks without dispersant but increases by 27%-351% upon introduction of dispersant. For most cases, the airborne nanodroplet concentration increases with decreasing Capillary or Morton numbers as well as the ratio of the so-called inner to thermal length scales. To explain the airborne nanodroplet generation in an oil-dispersant mixture, we show that prior to bubble injection, even minimal agitation of the interface causes generation of a subsurface cloud of nanodroplets that diffuses away from the interface. This process appears to be caused by thermal capillary instability when the interfacial tension is low enough to increase the thermal length scale to a few nanometers.