3。 Numerical Simulation

3。1。 Gas–liquid mixing modelling

Gas–liquid mixing in the stirred vessel was simulated using the Eulerian–Eulerian multiphase model。 Mass conservation equation for each phase is given as follows:

Liquid phase (l) and gas phase (g) are assumed to share space in proportion to their volume such that their volume fractions sums to unity in the cells domain:

μg = 1。789 × 10−5 Pa·s) was introduced through a ring sparger located below the lower impeller with the flow rate of Q = 0。4 and 0。6 m3·h−1。

There are 94 upward-facing holes (dia。 0。8 mm) on the sparger pipe with the ring diameter of 0。43 T。 The other dimensions of the stirring system are given in Table 1, where B is the baffle width, Cs and C1 are the clearances from sparger and the lower impeller respectively to the vessel bottom, C2 is the spacing between the two impellers。 Five impel- ler rotational speeds, i。e。, N = 300, 400, 500, 600, 700 r·min−1 were se- lected。 The corresponding Reynolds number is in the range of Re = ρlND2/μl = 3。18–7。42 × 105。 Power consumptions were measured with the AKC-215 type torque transducer (China Academy of Aerospace

Aerodynamics, Beijing, China)。 Gas dispersion images were captured with a CCD camera (Nikon AF NIKKOR, 1280 × 1024 pixels)。

Momentum conservation equation for phase i  is

∂t ðαiρi uiÞ þ ∇ · ðαi ρi uiuiÞ ¼ −αi ∇p þ ∇ · τi þ αi ρi g þ F ð3Þ

Dimensions of the stirred vessel

1748 F。 Yang et al。 / Chinese Journal of Chemical Engineering 23 (2015) 1746–1754

where the pressure is shared by the liquid and gas phase, τi is the liquid phase stress tensor, and F is the drag force between phases。 Effects of virtual mass force and lift force are negligibly small and hence can   be

neglected [27,28]。 Drag force largely predominates in aerated stirred vessels and is given below:

3ρl αl αgCD。ug−ul 。。ug−ul 。

phase and air as the secondary phase。 Gravity acceleration was defined as 9。81 m·s−2, in the negative z direction。 The coupling of pressure and velocity was performed with the SIMPLE algorithm。 All the governing equations were discretised using the first order upwind scheme。 Time

step was 5 × 10− 4  s。 Given the time step and the mesh used in  this

work, the courant number is less than 2。 The solutions were considered to be converged when the normalized residuals of all the variables were

where d is the bubble diameter。 Here, the coalescence and breakup of gas bubbles were neglected due to the fact that the  computation based on this assumption gave satisfactory predictions on the gas hold-up and flow field [28–30]。 A single bubble diameter of 4 mm was used according to the experimental measurement of Alves et al。 [31]。 The drag coefficient was calculated with the Schiller–Naumann   model:

8 24。1 þ 0:15Re0:687。

4。 Results and Discussion

4。1。 Flow field

During computation, time revolutions of the fluid velocities at sever- al selected points were monitored so as to achieve the quasi-steady flow field。 In Fig。 2, the liquid phase velocity vectors at N = 300 r·min−1  and

N = 600 r·min−1 in the vertical plane containing the baffles at t = 30 s