Galileo Number Research Articles

Measurements of mass transfer rates in a rectangular liquid fluidised bed using LCDT

Ionic mass transfer from electrolyte to a plate cathode in liquid fluidised bed filled with spherical inert particles has been experimentally studied. The mass transfer rates were measured by using the limiting current diffusion technique (LCDT) in ferri- and ferrocyanide electrolyte systems. The effects of bed height and electrode location from the bottom of test chamber in different electrolyte concentrations and flow velocities on mass transfer coefficient were evaluated. Ranges of the variables used in the study are as follows: Reynolds number=78–300, Galileo number=282,000–298,000, density number=0.335–0.370 and Schmidt number=1831–1995. Experimental data are correlated as follows, by non-linear regression using Statistica Package Program: Sh = 0.199 Re p 3 5 Ga 1 10 Mv 7 2 Sc 1 3 ( d eq E L ) 3 10 ( d eq L Y ) − 3 4 . A comparison between the present data and related several literature correlation's has been achieved by calculating absolute ( σ A) and relative ( σ R) deviations. The deviations in present study are less than those in the literature cited.

Motion and wake structure of spherical particles

This paper presents results from a flow visualization study of the wake structures behind solid spheres rising or falling freely in liquids, under the action of gravity. These show remarkable differences to the wake structures observed behind spheres held fixed. The two parameters controlling the rise or fall velocity (i.e. the Reynolds number) are the density ratio between the sphere and liquid and the Galileo number.

Instabilities and transition of a sphere falling or ascending freely in a Newtonian fluid

The scenario of transition to chaos for a sphere falling or ascending under the action of gravity in a Newtonian fluid is investigated by numerical simulation. The mathematical formulation is parameterized using two non-dimensional parameters: the solid/fluid density ratio and the generalized Galileo number expressing the ratio between the gravity–buoyancy and viscosity effects. The study is carried out fully in this two-parameter space. The results show that for all density ratios the vertical fall or ascension becomes unstable via a regular axisymmetry breaking bifurcation. This bifurcation sets in slightly earlier for light spheres than for dense ones. A steady oblique fall or ascension follows before losing stability and giving way to an oscillating oblique movement. The secondary Hopf bifurcation is shown not to correspond to that of a fixed sphere wake for density ratios lower than 2.5, for which the oscillations have a significantly lower frequency. Trajectories of falling spheres become chaotic directly from the oblique oscillating regime. Ascending spheres present a specific behaviour before reaching a chaotic regime. The periodically oscillating oblique regime undergoes a subharmonic transition yielding a low-frequency oscillating ascension which is vertical in the mean (zigzagging regime). In all these stages of transition, the trajectories are planar with a plane selected randomly during the axisymmetry breaking. The chaotic regime appears to result from an interplay of a regular and of an additional Hopf bifurcation and the onset of the chaotic regime is accompanied by the loss of the remaining planar symmetry. The asymptotic chaotic states present an intermittent character, the relaminarization phases letting the subcritical plane and periodic trajectories reappear.

Effect of bubble deformation on the properties of bubbly flows

Direct numerical simulations of the motion of 27 three-dimensional deformable buoyant bubbles in periodic domains are presented. The full Navier–Stokes equations are solved by a parallelized finite-difference/front-tracking method that allows a deformable interface between the bubbles and the suspending fluid and the inclusion of surface tension. The Eotvos number is taken as equal to 5, so that the bubbles are ellipsoidal, and the Galileo number is 900, so that the rise Reynolds number of a single bubble in an unbounded flow is about 26. Three values of the void fraction have been investigated: 2%, 6% and 12%. At 6%, a change in the behaviour of the bubbles is observed. The bubbles are initially dispersed homogeneously throughout the flow field and their average rise Reynolds number is 23. After the bubbles have risen by about 90 bubble diameters, they form a vertical stream and accelerate. The microstructure of the bubble suspension is analysed and an explanation is proposed for the formation of these streams. The results for the ellipsoidal bubbles are compared to the results for nearly spherical bubbles, for which the Eotvos number is 1 and the Galileo number is 900. The dispersion of the bubbles and the velocity fluctuations in the liquid phase are analysed.

Mass transfer correlation of NH 3–H 2O bubble absorption

The objectives of this paper are to study the effect of key parameters on absorption performance and to develop an experimental correlation of mass transfer coefficient for ammonia–water bubble absorption. The orifice diameter, liquid concentration and vapor velocity are considered as the key parameters. This study successfully visualized the bubble behavior and measured the volumetric diameter of bubbles during the bubble absorption process. The bubble absorption is grouped into two processes, bubble growth (process I) and bubble disappearance (process II), respectively. The following conclusions were drawn from the present study. A new experimental correlation for the volumetric bubble diameter was proposed with ±15% error band, which could be applied to calculate the mass transfer coefficient. The mass transfer coefficient increased with a decrease of the liquid concentration. In process II, the mass transfer coefficient increased with an increase of the Galileo number. Finally, experimental correlations of mass transfer coefficient were developed for processes I and II with ±18% error bands.

Improving the prediction of liquid back‐mixing in trickle‐bed reactors using a neural network approach

AbstractCurrent correlations aimed at estimating the extent of liquid back‐mixing, via an axial dispersion coefficient, in trickle‐bed reactors continue to draw doubts on their ability to conveniently represent this important macroscopic parameter. A comprehensive database containing 973 liquid axial dispersion coefficient measurements (DAX) for trickle‐bed operation reported in 22 publications between 1958 and 2001 was thus used to assess the convenience of the few available correlations. It was shown that none of the literature correlations was efficient at providing satisfactory predictions of the liquid axial dispersion coefficients. In response, artificial neural network modeling is proposed to improve the broadness and accuracy in predicting the DAX, whether the Piston–Dispersion (PD), Piston–Dispersion–Exchange (PDE) or PDE with intra‐particle diffusion model is employed to extract the DAX. A combination of six dimensionless groups and a discrimination code input representing the residence‐time distribution models are used to predict the Bodenstein number. The inputs are the liquid Reynolds, Galileo and Eötvos numbers, the gas Galileo number, a wall factor and a mixed Reynolds number involving the gas flow rate effect. The correlation yields an absolute average error (AARE) of 22% for the whole database with a standard deviation on the AARE of 24% and remains in accordance with parametric influences reported in the literature.© 2002 Society of Chemical Industry

Radioisotope tracer study in trickle bed reactors

AbstractThe residence time distribution (RTD) of liquid phase in trickle bed reactors has been measured for air‐water system using radioisotope tracer technique. Experiments were carried out in a glass column of internal diameter of 0.152 m packed with glass beads and actual catalyst particles of two different shapes. From the measured RTD curves, mean residence time of liquid was calculated and used to estimate liquid holdup. The axial dispersion model was used to simulate the experimental data and estimate mixing index, ie. Peclet number. The effect of liquid and gas flow rates on total liquid holdup and Peclet number has been investigated. Results of the study indicated that shape of the packing has significant effect on holdup and axial dispersion. Bodenstein number has been correlated to Reynolds number, Galileo number, shape and size of the packing.

Loading Capacity in Packing Towers - Database, Correlations and Analysis

Experimental results on the flooding capacity of randomly dumped packed beds were collected from the literature to generate a working database. The reported measurements were first used to review the accuracy of existing predictive tools in that field. A total of 14 correlations were extracted from the literature and cross-examined with the database. Many limitations regarding the level of accuracy and generalization came to light with this investigation. Artificial neural network modeling was then proposed to improve the broadness and accuracy in predicting the flooding capacity, which is an important design parameter for packed towers. A combination of six dimensionless groups, namely, the Lockhart−Martinelli parameter (χ); the liquid Reynolds (ReL), Galileo (GaL) and Stokes (StL) numbers; the packing sphericity (φ); and one bed number (SB) outlining the tower dimensions were used as the basis of the neural network correlation. With an initial database containing 1019 measurements, the correlation yield...

Oscillatory Instability and High-Frequency Wave Modes in a Marangoni-Bénard Layer with Deformable Free Surface

A linear analysis of oscillatory thermocapillary instability associated with deformation of the free surface in a finite-depth horizontal liquid layer heated from above is carried out. First, we numerically solve the problem and determine the conditions of marginal stability. Then, assuming that Galileo and Marangoni numbers are large enough while the capillary number is small enough, which corresponds to the time scales of transverse and longitudinal oscillations related to the free surface being much smaller than the characteristic viscous and thermal time scales, we construct an asymptotic solution of the problem. Two high-frequency overstable modes—transverse (capillary-gravity waves) and longitudinal—are identified and analyzed. Mode-mixing occurring at their possible resonance is also studied.

Flooding Capacity in Packed Towers: Database, Correlations, and Analysis

Experimental results on the flooding capacity of randomly dumped packed beds were collected from the literature to generate a working database. The reported measurements were first used to review the accuracy of existing predictive tools in that field. A total of 14 correlations were extracted from the literature and cross-examined with the database. Many limitations regarding the level of accuracy and generalization came to light with this investigation. Artificial neural network modeling was then proposed to improve the broadness and accuracy in predicting the flooding capacity, which is an important design parameter for packed towers. A combination of six dimensionless groups, namely, the Lockhart−Martinelli parameter (χ); the liquid Reynolds (ReL), Galileo (GaL) and Stokes (StL) numbers; the packing sphericity (φ); and one bed number (SB) outlining the tower dimensions were used as the basis of the neural network correlation. With an initial database containing 1019 measurements, the correlation yielded an absolute average relative error (AARE) of 16.1% and a standard deviation of 20.4%. Another database containing over 100 measurements on the flooding capacity was used to validate the correlation. The prediction based on these results yielded an AARE of 11.6% and a standard deviation of 13.7%. Through a sensitivity analysis, the Stokes number in the liquid phase was found to exhibit the strongest influence on the prediction, while the liquid velocity, gas density, and packing shape factor were determined to be the leading physical properties defining the flooding level. As a matter of fact, the neural correlation remains in accordance with the design recommendations and trends reported in the literature.

Linear and nonlinear Rayleigh–Bénard–Marangoni instability with surface deformations

Thermoconvective instabilities in a bilayer liquid–gas system with a deformed interface are investigated. In the first part of the work which is devoted to a linear approach, emphasis is put on the role of the upper gas layer on the instability phenomenon. The condition to be satisfied by the gas to remain purely conductive is established. The so-called Oberbeck–Boussinesq approximation is discussed and its range of validity is carefully defined. Instead of the classical Rayleigh, Marangoni, crispation, and Galileo numbers, new dimensionless groups are introduced. A critical comparison with several previous works is made. The nonlinear analysis consists in studying the different convective patterns which can appear above the threshold. Particular attention is devoted to the shape of the interface and the so-called “hybrid” relief. The amplitude of the deformation is also determined and comparison with experimental data is discussed.

Tracer vs. Pressure Wave Velocities through Unsaturated Saprolite

Saprolite is a form of weathered bedrock that is commonly used as the host material at waste disposal sites in the Southeastern Piedmont. However, estimating the unsaturated hydraulic and transport properties of saprolite is difficult due to saprolite's low permeability. We demonstrate the use of short‐duration fluid irrigation pulses for maintaining unsaturated conditions in intact saprolite columns. Concomitant Cl− tracer experiments demonstrate that irrigated waters moved through an effective volumetric porosity (0.038–0.108 cm3 cm−3) substantially less than the ambient water‐filled porosity (0.44 cm3 cm−3). We observed the unexpected result that irrigation‐induced pressure wave velocities (1983–3670 cm d−1) were ≈1000 times faster than tracer velocities (2.04–6.00 cm d−1). The relationship between pressure wave velocities and fluid velocities is described using kinematic wave theory, presented for four parametric representations (Brooks–Corey, van Genuchten–Mualem, Broadbridge–White, and the Galileo Number), that predicts fluid pressure velocities to be from approximately two to fifteen times faster than saprolite tracer velocities. None of the kinematic models was able to reproduce observed rapid pressure wave velocities. A hydraulic form of the advection–diffusion equation based on Richards' equation is presented that favorably predicts the shape of pressure response curves only when the kinematic velocity is ignored and the hydraulic diffusivity of the unsaturated saprolite is considered. Based on the advection–diffusion equation, diffusion‐dominated soil water pressure wave velocities should decrease with depth, eventually conforming with kinematic wave theory. Pressure pulse velocity monitoring may be an additional tool for estimating unsaturated hydraulic properties in low permeability media.

Natural convection of a viscoelastic fluid with deformable free surface

In this paper, linear overstable convection of a viscoelastic fluid layer with upper deformable free surface is investigated. For the sake of comparison, the corresponding case for nondeformable free surface is also investigated. The Oldroyd's viscoelastic model constitutive equation with relaxation and retardation times is made use of. Calculations are made with the Maxwell's model as a particular case. It is supposed that the lower flat surface is at fixed temperature and that the upper deformable free surface is at fixed heat flux. Two cases are investigated which differ due to the mechanical boundary conditions imposed on the lower flat wall, that is, it may be solid or stress free. It is found, for different values of the Galileo number, that for relaxation times that are large enough the curves of the critical Rayleigh numbers are lower than those of stationary convection and those of overstability of the Newtonian fluid with deformable free surface. When the lower surface is rigid maxima in the curves of criticality against the relaxation time are found. When the lower surface is stress free, jumps and maxima are found in the curves of the critical wavenumber and frequency against the relaxation time. To explain these jumps and maxima, based on results previously reported in the literature, it is shown that gravity acts as an extra body force through the surface deformation, playing a dual role on the stability of the convective viscoelastic fluid layer.

Two-layer Bénard-Marangoni instability and the limit of transverse and longitudinal waves

The oscillatory instability of a B\'enard-Marangoni system composed of two liquid layers, of finite depth, separated by a deformable interface is considered. We show that in the limit of large Galileo and Marangoni numbers and small capillary number (boundary layer approximation), two modes of high-frequency interfacial excitation can be distinguished: transverse (capillary-gravity) waves and longitudinal waves.

Nonlinear analysis of the coupling between interface deflection and hexagonal patterns in Rayleigh-Bénard-Marangoni convection.

An analytical study is conducted to examine the coupling between buoyancy and surface tension forces in a shallow pool of a Boussinesq fluid. The liquid layer is bounded below by a rigid and poorly conducting plate, and is open to the air at its upper deformable free surface. At the onset of convection, the coupling between buoyancy and surface-tension-induced instabilities is represented by a single parameter m. A pattern selection study predicts the existence of a critical value for the parameter m, ${\mathit{m}}_{\mathit{c}}$, that separates subcritical hexagons with upflow at their centers from subcritical hexagons with downflow at their centers. Several experimental situations are identified in terms of the Prandtl, Galileo, and capillary numbers, and the dependence of ${\mathit{m}}_{\mathit{c}}$ on these parameters is analyzed. Further, expressions for the interface deflection that accompanies the onset of Rayleigh-B\'enard-Marangoni convection are derived. These expressions show explicitly the dependence of the interface morphology on the coupling parameter m and on the Prandtl, Galileo, and capillary numbers. It is also noted that, for a specific set of experimental parameters, spatial resonance will occur which leads to a situation in which the interfacial deformations do not conform to the flow pattern. \textcopyright{} 1996 The American Physical Society.

Steady-state particle distributions in shear flows of colloids and finely dispersed suspensions

On the basis of an analysis of the pseudoturbulent motion of both the suspended particles and the carrier fluid, the normal stress components in the dispersed phase are obtained for the problem of inclined confined flows of finely dispersed suspensions and colloids. These hydrodynamic pulsations are due to the shear and the work done by the average relative flow of the fluid phase on random concentration fluctuations of the disperse system because of the substantial slip of the phases of the suspension under gravity. The momentum conservation equations for the particles are obtained with allowance for the angle of inclination of the flow to the vertical and on the basis of these equations the suspension capacity of the flow as a function of the angle of inclination, particle size, Galileo number and other parameters is illustrated.

Natural convection in a rotating fluid layer with deformable free surface

Abstract In this paper we investigate the stationary thermoconvective instability of a rotating fluid layer with a deformable free surface. In order to understand phenomena which could appear in planetary and stellar systems, a variety of mechanical and thermal boundary conditions is considered, which give upper and lower stability bounds to all possible physical situations. It is shown that rotation is stabilizing when surface deformation is present, for large values of G[sgrave], the product of the Galileo and Prandtl numbers, which appears due to surface deflection. Moreover, it is shown that for certain boundary conditions the interaction between rotation and gravity through surface deformation leads to an increase in convection cell size (decrease in wavenumber) and destabilization of the system when the Taylor number Ta increases and G[sgrave] is relatively small. The Boussinesq approximation is violated when infinite size convection cells appear in our analysis. Therefore, graphs of the critical values of G[sgrave] and Ta corresponding to the limits of this approximation are given along with analytical expressions for the corresponding Rayleigh numbers.

Pattern formation in large-scale Marangoni convection with deformable interface

We derive a nonlinear evolution equation describing the evolution of large-scale patterns in Marangoni convection in thermally insulated two-layer liquid-gas system with deformable interface, and generalizing equations obtained previously by Knobloch and Shtilman and Sivashinsky. Both surface deformation and inertial effects contribute to the diversity of long-scale Marangoni convective patterns. In the space of parameters — Galileo and capillary numbers — different regions are found where not only hexagonal, but also roll and square patterns are subcritical. Stability regions for various patterns are found, as well as regions of multistability. It is shown that competition between squares and hexagons leads to formation of a stable quasicrystalline dodecagonal convective structure.

Mass transfer between fluid and immersed surfaces in liquid fluidized beds of coarse spherical inert particles

Mass transfer coefficients between fluid and immersed surfaces in liquid fluidized beds of spherical inert particles have been studied experimentally using fluidization columns 40 mm and 70 mm in diameter. Mass transfer data were obtained in three experimental systems: i) transfer of methylene blue dye from very dilute aqueous methylene blue solution as the fluidizing fluid to plane solid surfaces using the adsorption method [Končar-Djurdjević, Nature, 172 (1953) 858]; ii) transfer of benzoic acid from a large spherical particle to flowing water using the dissolution method, and iii) transfer of benzoic acid from a plane solid surface to flowing water also using the dissolution method. In all runs mass transfer rates were determined in the presence of inert fluidized particles 1.20, 1.94 and 2.98 mm in diameter. Measurements covered a particle range having Reynolds number from 15 to 400, and two Schmidt numbers (20 °C), 1361 and 1932. Experimental data are correlated by the widely used formula j D =0.261 Ga 0.324Re p −0.97 ( j D = mass transfer number; Ga = Galileo number; Re p = Reynolds number). The data show that an analogy exists between the mass transfer factor and fluid–particle interphase drag coefficient, since both quantities vary with voidage in a similar way. The normalized mass transfer factor in liquid–phase fluidized beds of active or inert particles and the dimensionless drag coefficient are shown to be the same. Experimental data agree reasonably well with the proposed model.

Collision efficiency for fine mineral particles with single bubble in a countercurrent flow regime

The collision of fine mineral particles with a bubble in a countercurrent flow regime has been modeled to predict the collision efficiency between fine mineral particles and a single bubble in a flotation column. Surface forces (DLVO forces as well as non-DLVO forces) are ignored in the collision process. The complete Navier-Stokes equations are resolved to achieve a finer angular grid for the numerical solution of a collision angle which is a function of the density of the particles represented by the Galileo number ( Ga) and of the Reynolds number of bubbles in a countercurrent flow regime ( Re). The collision efficiency given is a function of the diameters of both the particles and the bubble, and to both Ga and Re, and to the bubble slip velocity and terminal settling velocity of the particles. The value of the collision efficiency has been determined experimentally: the agreement between the theoretical predictions and the measured values is good.