The boundary conditions are:

• at the vessel wall and bottom V = ωR ;

• on the impeller V  = 0.

A series of shear thinning fluids obeying the power-law model are considered: the consistency index is 8 Pasn, and the power law indices (n) range from 0.4 to 1. In all the computations, the fluid density (ρ) is 1394 kg/m3 . The above equations are solved using the finite volume method.

For a shear thinning fluid (Ostwald model), the Reynolds

number is given by:

All variables describing the hydrodynamic state are writing

in dimensionless form, the dimensionless velocities are obtained as follow: V ∗ =V/πND; R∗ = 2R/D and Z∗  = Z/D.

4. Numerical issues

To perform calculations, the commercially available CFX

12.0 computer code developed by AEA Technology, UK, was used. A pre-processor (ICEM CFD 12.0) was used to

Figure 2. Numerical grid (tetrahedral mesh).

Table 1.  Details on mesh tests.

discretize the flow domain with a tetrahedral mesh (Fig-

ure 2). In general, the density of cells in a computational grid needs to be fine enough to capture the flow details, but not so fine, since problems described by large numbers of cells require more time to solve. In order to capture the boundary layer flow detail, an increased mesh density was used near the tank wall and the rotating impeller. In order to have a very refined mesh in the vicinity of the blades, the sufficient amount of nodes that properly define the curvature of the blades was created on the impeller edges and a size function was used to control the mesh growth. This feature allows the mesh elements to grow slowly as a function of the distance from the impeller blades. The number of cells used for discretization was determined by conducting a grid independence study. Mesh tests were performed (Table 1) by verifying that additional cells did not change the velocity magnitude in the regions of high velocity gradients around the impeller blades by more than 0.025.

To verify the grid independency, the number of cells was increased by a factor of about 2 used by other researchers in CFD modeling of the mixing processes [22, 23]. The original 3D mesh of the model had 75,251 computational cells. To verify the grid independency, the number of cells was increased from 75,251 cells to 150,502 cells. The additional cells changed the velocity magnitude in the regions of high velocity gradients and the impeller power number by more than 0.03. Thus, the number of cells was changed from 150,502 cells to 301,004 cells. The additional cells did not change the velocity magnitude in the regions of high velocity gradients and impeller power number by more than 0.025. Therefore, 150,502 cells were employed in this study. Simulations were considered converged when the scale residuals for each transport

equation were below 10−7 .

This study is restricted to the laminar and transition regimes, the Reynolds number is varying from 0.1 to 200.

 Number of cells 75251   150502  01004

max 0.79105 0.8011 0.8066

 Time required [second] 8415 15897 25586

The transition regime begins for Reynolds number between 20 and 30 for this kind of fluids (Figure 14) [24]. Even if the flow is not fully into the turbulent regime, we should simulate the flow as being turbulent. Fortunately, the turbulent properties vanish when in the laminar flow, so the correct way of simulating in the transitional regime is using a turbulence model. In our study, we have used the SST model. This model has the option of specifying how we model the transition turbulence: it is the fully turbulent, specified intermittency (requiring acquired knowledge), the gamma model or the gamma theta model.