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Dn, is assumed to have the same value as thediffusivity coefficient D12.In modelling coalescence [1,12,13], it is generally considered that coalescence occurs due tobinary collisions between bubbles, and expressions for collision rate are derived by assumingrandom collisions induced by turbulent eddies, analogous to the model for molecular collisions inthe kinetic theory of an ideal gas. Hence the coalescence rate term has the following form:Sco / d2utn2: ð13ÞHere, ut is taken to be the velocity of eddies in the inertial subrange of the turbulent eddyspectrum, which may be written as An additional factor should be introduced [13] to account for the reduced mean free path ofbubbles with increasing gas volume fraction, a2, so that the expression for coalescence rate be-comesSco ¼ Ccogcod2ðedÞ1=3n21ð1 a1=32 Þ; ð15Þwhere gco is the coalescence efficiency, which is set to unity for the time being, until a suitableexpression can be incorporated.The tendency for bubbles to break up or remain stable may be defined in terms of aWeber number, the ratio of the disruptive forces to the restoring surface tension force being givenbyWe ¼ qu2tdr: ð16ÞBubble break-up occurs only when the Weber number exceeds a critical value, Wecrit, which isapproximately 1.2 in turbulent flows [14]. However, the Weber number is an equilibrium condi-tion which defines the maximum stable size after exposure to a given level of turbulence for a longtime. An expression is also needed to define the time rate of bubble break-up. Following otherauthors [12,13], the bubble break-up is considered to depend on the frequency of collisions be-tween bubbles and eddies of a similar size, somewhat analogous to the frequency of bubblecollisions in the coalescence model. Furthermore, only a certain fraction of the eddies may havesufficient energy to break up a bubble, and this leads to an efficiency factor in the form of anexponential function. The final expression follows the equation developed by Wu et al. [13], wherethe break-up rate is given bySbr ¼ CbrnðedÞ1=3d1 WecritWe 1=2exp WecritWe ; We > Wecrit: ð17Þ5. Simulation methodThe CFD model was set up to simulate a gas-sparged tank for which published experimentaldata [15,16] are available for the distribution of bubble sizes and local gas volume fraction.Specifically, the gas is air and the liquid is water, and properties are set in the model accordingly.The tank is baffled and has a 1.0 m diameter with a standard Rushton turbine 0.333 m diameterlocated at a clearance of 0.25 m. Impeller speed is set to 180 rpm and the gas flow rate is 0.00164m3s 1. The tank is modelled by a finite volume grid in cylindrical coordinates (see Fig. 1), with61, 36 and 20 cells in the axial, radial and azimuthal directions, respectively. To reduce theproblem size, one-sixth of the tank is modelled, including just one blade and one baffle withperiodic boundary conditions assumed in the azimuthal direction.
All walls are treated as no-slipboundaries applying wall functions to calculate the velocity profiles near the wall, except that theliquid surface is treated as a zero stress boundary. Gas is added at the sparger and removed at theliquid surface using source terms in the equations. The bubbles at the inlet are assumed to have amean size of 2 mm. The set of equations are solved numerically using the commercial code CFX4.2. The code hasbeen augmented by several user-supplied routines, to implement the multiple frames of referencemethod; to add and remove the gas from the tank; and to specify equations for turbulent dis-persion, interphase forces, and bubble number density. Satisfactory completion of each simulationis based on several criteria, including sufficient reduction of the mass residuals, an accurate bal-ance between the rates of gas entering and leaving the tank, and a constant final value of gasholdup.6. Results and discussionSimulations of the gas-sparged tank have been run using several possible versions of the two-phase equations, with different terms for turbulent dispersion and drag force (as described inSection 3), in each case comparing results with the published experimental data [15,16]. Turbulentdispersion was modelled firstly by using a diffusive term in Eq. (1), with the ratio qD12 to lT;1 set to0.9. Modelling of dispersion was also tested using the turbulent force terms of Eqs. (4) and (5) inEq. (2). The second method was found to give a slightly higher gas holdup, but both modellingmethods give a similar pattern of gas distribution. It was found that the results were most sensitiveto the approach taken to modelling the drag force, Eq. (8), and in particular the method ofspecification of the drag coefficient. The results presented here (see Figs. 2–5) were obtained using
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