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The equations for each phase, i (¼1 for liquid, ¼2 forgas), are as follows:oðaiqiÞotþrðaiqi~ U Ui qiD12raiÞ¼ Si; ð1Þoðaiqi~ U UiÞotþrððaiqi~ U Ui ~ U UiÞ aiðlL;iþ lT;iÞðr~ U Ui þðr~ U UiÞTÞÞ¼ airPi þ aiðqi q0Þ~ g g þ ~ F Fi þ~ B Bi þ~ T Ti þ Si~ U Ui: ð2ÞThe turbulent viscosity lT;iin the liquid phase is calculated using the standard k e turbulencemodel, whereas lT;iis assumed zero for the gas. The term~ B Bi represents the Coriolis and centrifugalforces which apply in the rotating frame of reference only and are given by~ B Bi ¼ 2aiqi~ X X ~ U Ui aiqi~ X X ð~ X X ~ X XÞ: ð3ÞThe term Si (for i ¼ 2) is the mass source or sink of gas at the sparger and liquid surface, ~ F Fi isthe generalised interphase force and ~ T Ti is the turbulent dispersion force.There is a lack of agreement in the literature as to the exact form of the equations, with dif-ferences due to the assumed form of the instantaneous equations, due to the averaging methodapplied, and also due to differences in closure terms for the turbulent correlations which arise afteraveraging. These differences relate mostly to modelling of turbulent dispersion and interphaseforces, and thus may have a large influence on predictions of gas distribution and holdup.The approach to modelling of turbulent dispersion depends on the way in which the equationsare averaged. If time averaging is applied (so-called Reynolds averaging), then turbulent dis-persion is specified by the diffusive term in Eq. (1) and D12, the turbulent diffusivity, is usually setas a constant ratio to the liquid turbulent viscosity. However, Bakker [1] provides an alternativeequation for this diffusive term. Alternatively, if the equations are Favre-averaged, D12 in Eq. (1)is set to zero and turbulent dispersion appears as a force in the momentum equation. The tur-bulent dispersion force is derived from turbulent correlations which appear in the momentumequation after averaging, and by assuming a gradient transport hypothesis, these may be modelledin terms of gradients of volume fraction. Two such terms, as proposed by Viollet and Simonin [9],have been tested in the present modelling. These terms correspond to turbulent dispersion due tofluctuations in the pressure force and fluctuations in the drag force, given, respectively, by~ T Tð1Þ;2 ¼ ~ T Tð1Þ;1 ¼ 23b þ gr1 þ gr q1kra2; ð4Þ~ T Tð2Þ;2 ¼ ~ T Tð2Þ;1 ¼ 34a2ða1q1ÞCDd~ U Uslip D12a1a2ra2 : ð5ÞHere, the diffusivity of the gas, D12, is given byD12 ¼ lTq1b þ gr1 þ gr ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ Cbn2rq : ð6ÞThese expressions take into account the particle inertia and cross-trajectories effect due toparticle slip. Hence, the expressions are functions of the ratio of particle relaxation time to largeeddy time gr, the ratio of slip velocity to turbulent velocity, nr, and the effects of added mass givenby b, where these expressions are evaluated as follows:gr ¼ st12sp12; nr ¼~ U Uslip ffiffiffiffiffiffiffiffiffiffi2=3kp ; b ¼ 1 þ CAq2q1þ CA: ð7ÞThe most important interphase force, ~ F Fi, is the drag force acting on the bubbles. Other forcessuch as added mass and lift may also be significant under accelerating conditions, but have notbeen included thus far. Under steady conditions, a balance between drag and buoyancy forcesleads to the bubbles attaining a characteristic slip velocity relative to the liquid. As the bubblesmove through the stirred tank under the influence of liquid circulation, the bubble slip velocityessentially determines the rate of rise of gas and the proportion recirculated, and therefore slipvelocity has a large influence on the gas holdup obtained. The drag force may be written as~ F Fd ¼ 34CDda1a2q1~ U Uslip ~ U Uslip: ð8ÞFor calculation of the drag coefficient CD, the correlation of Ishii and Zuber [10] is frequentlyused for bubbles. This correlation has been applied in modelling gas flow in stirred tanks (e.g.[4,5,7]), however correlations such as this have been developed for bubbles or particles in stagnantliquids and may not be adequate where there is a high level of forced turbulence, as in a stirredtank. There is a strong interaction between bubbles and turbulent eddies, where the bubbles arecontinually being accelerated by eddies of different sizes and velocities. This may modify therelative mean velocity between gas and liquid, so that the effective drag force cannot be calculatedby standard correlations. Available data [11] indicate that the effect of turbulence may be quitelarge, and generally leads to increased drag coefficients and therefore lower slip velocities. Basedon measurements for solid particles up to 500 lm, it was found [11] that the drag coefficient, CD,under turbulent conditions could be correlated by where CD;0 is the drag coefficient in a stagnant fluid and k is the Kolmogorovmicroscale ofturbulence.Bakker [1] took into account the effect of turbulence on drag coefficient using a different ap-proach, where the standard drag curve is used, but the drag coefficient is calculated using amodified Reynolds number as a function of turbulent eddy viscosity, defined asRe¼ q1~ U UslipdlL þ 29lT;1: ð10ÞWork is in progress to test these various alternative approaches to the equations for gas–liquidflow.4. Bubble break-up and coalescenceAnother complexity is the prediction of bubble size, which is required for calculation of in-terfacial area and interphase transfer of momentum, mass and energy. The approach taken here isto calculate the bubble number density, n, given byn ¼ a2ðp=6Þd3: ð11ÞFrom the values of n at each point in the tank, the local average bubble size d can be calculated.An additional transport equation is solved for n, accounting for transport of bubbles (by con-vection and turbulent diffusion) and changes in bubble size (and therefore bubble number) bybreak-up and coalescence according toonotþrðn~ U U2 DnrnÞ¼ Sbr Sco; ð12Þwhere Sbr and Sco are the source and sink terms describing the rates of bubble break-up andcoalescence, respectively. The diffusivity coefficient,
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