Rise velocity of single circular-cap bubbles in two-dimensional beds of powders and liquids

https://doi.org/10.1016/S0255-2701(99)00108-7Get rights and content

Abstract

An expression for the rise velocity of single circular-cap gas bubbles in two-dimensional (2D) beds consisting of powders or liquids is developed with the aid of experimental data and computational fluid dynamics. Experiments were performed in a two-dimensional rectangular column of width DT=0.3 m by injecting air bubbles in fluidised beds of silica (mean particle size, dp=38 μm) and polystyrene (mean particle size, dp=570 μm) and in water. The rise velocity of single gas bubbles in the size range db=0.015–0.12 m were found to decrease significantly with increasing ratio of bubble diameter to bed width, db/DT. Computational fluid dynamics simulations of single gas bubbles rising in water, carried out using the volume-of-fluid (VOF) method, showed good agreement with experiment and were used to develop a common expression for the rise velocity of single gas bubbles in gas–solid fluidised beds and bubble columns. The 2D circular-cap bubble rise velocity is found to ∼10–30% lower than that of a 3D spherical-cap bubble having the same equivalent diameter.

Introduction

Experimental work to study bubbling behaviour and hydrodynamics is often carried out using two-dimensional rectangular gas-solid fluidised beds [1], [2] and gas–liquid bubble columns [3]. In order to be able to translate the information from 2D beds to columns of cylindrical cross-section, it is important to be able to inter-relate the single bubble rise velocity in these two column configurations.

For a single gas bubble of equivalent diameter db rising in a liquid inside a cylindrical column of diameter DT, Collins [4] gives the following expression for the rise velocity (see also Clift et al. [5], Davidson et al. [6], Fan and Tsuchiya [7], Wallis [8]) wherein a scale factor SF is introduced into the classical Davies–Taylor [9] relation:Vb=0.71gdbSF

The expression derived empirically by Collins [4] for the scale factor SF isSF=1fordbDT<0.125SF=1.13expdbDTfor0.125<dbDT<0.6SF=0.496DTdbfordbDT>0.6

, are valid for spherical cap bubbles rising in inviscid flow; this condition is satisfied when the Eötvos number, Eö>40 [5]. The same expression is valid for a single bubble rising in a gas–solid fluidised bed [1], [5], [6], [7]. Bubbles in a smaller diameter column tend to rise slower than bubbles in a larger diameter column due to the restraining effects of the column walls. Such wall effects can be expected to diminish with increasing column diameter. A corresponding set of relations for bubbles, of circular-cap shape, rising in 2D columns is not available in the literature.

The experimental data of Pyle and Harrison [10] for 2D gas bubbles in rectangular gas–solid fluid beds of Ballotini, sand and iron shots lie significantly below that calculated from the Davies–Taylor–Collins relation for 3D spherical cap bubbles; see Fig. 1. Pyle and Harrison [10] correlated their experimental data with the following expressionVb=0.48gdbwhere db is the diameter of a bubble having the same area as the 2D bubble. The experimental data of Pyle and Harrison [10], however, shows considerable scatter and their developed relation in Eq. (3) is not very convincing.

There have been some attempts to develop fundamental relationships for the rise velocity of 2D circular cap bubbles. Collins [11] modelled the effect of the constraining walls in terms of the known potential flow due to a doublet in a uniform stream between two walls and applied the classical Davies–Taylor analysis to obtain the rising velocity in terms of the radius of curvature of the nose. The expression for the rise velocity is given in terms of the radius of curvature of the nose and the half-width of the channel. This work was extended by Hills [12], who replaced the doublet by a separated source and sink to give an ellipse-like closed streamline. Garabedian [13] has derived the theoretical slug flow limit, valid for narrow columns. Gera and Gautam [14] have attempted a theoretical analysis of the rise of 2D gas bubble in a gas–solid fluidised bed as a parallel to the Davies–Taylor [9] treatment but the influence of the wall is not taken into account.

The aim of the present paper is to develop an expression for the rise velocity of single circular-cap gas bubble in 2D columns, in terms of the equivalent bubble diameter, to parallel , . Both experimental data and computational fluid dynamics (CFD) are used to develop this expression.

Section snippets

Experimental

Single bubble rise velocities were measured in a rectangular column made up of two parallel glass plates of 0.3 m width and 4 m height; see Fig. 2. The distance between the glass plates was 5 mm. A sintered plate distributor (of 50 μm pore size) ensured uniform gas distribution at the bottom. Additionally, there was provision to inject gas bubbles via a central tube of 2 mm diameter. The column was filled with either water, porous silica particles (Geldart A powder with skeleton density=2100 kg

Volume-of-fluid simulations

The VOF model [15], [16], [17], [18], [19], [20], [21], [22] resolves the transient motion of the gas and liquid phases using the Navier–Stokes equations, and accounts for the topology changes of the gas–liquid interface induced by the relative motion between the dispersed gas bubble and the surrounding liquid. The finite-difference VOF model uses a donor–acceptor algorithm, originally developed by Hirt and Nichols [17], to obtain, and maintain, an accurate and sharp representation of the

Conclusions

The following conclusions can be drawn:

(1) The rise velocity of circular-cap bubbles in powders and liquids decreases with increasing ratio of bubble diameter to column width, db/DT. This wall-effect is described quantitatively by the empirical relation in Eq. (11). Eq. (11) is the 2D analogue of the Davies–Taylor–Collins relation describing the rise velocity of spherical cap bubbles.

(2) VOF simulations for air–water systems in 2D rectangular columns provide some insight into the physics

Acknowledgements

The Netherlands Organisation for Scientific Research (NWO) is gratefully acknowledged for providing financial assistance to J.M. van Baten.

References (37)

  • J.U. Brackbill et al.

    A continuum method for modelling surface tension

    J. Comput. Phys.

    (1992)
  • A. Boemer et al.

    Eulerian simulation of bubble formation at a jet in a two-dimensional fluidized bed

    Int. J. Multiphase Flow

    (1997)
  • J.A.M. Kuipers et al.

    A numerical model of gas fluidized beds

    Chem. Eng. Sci.

    (1992)
  • B.G.M. Van Wachem et al.

    Eulerian simulations of bubbling behaviour in gas–solid fluidized beds

    Comput. Chem. Eng.

    (1998)
  • B.G.M. Van Wachem et al.

    Validation of the Eulerian simulated dynamic behaviour of gas–solid fluidised beds

    Chem. Eng. Sci.

    (1999)
  • D. Geldart (Ed.), Gas Fluidization Technology, Wiley, New York,...
  • R. Collins

    The effect of a containing cylindrical boundary on the velocity of a large gas bubble in a liquid

    J. Fluid Mech.

    (1967)
  • R. Clift et al.

    Bubbles, Drops and Particles

    (1978)
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