**8. Numerical method**

2

(17)

(18)

W (19)

(20)

(21)

t

(22)

**0** (23)

is the rotation average speed tensor on a rotating reference

*k*

In contrast to RNG, realizable model uses a variable *Cμ* that satisfies the "realizability" through

1

*cos kU*

ò

**ij ij**

w

w

CFD simulation of moving parts, e.g. impellers and turbines, requires approximations that consider the displacement and rotation of mechanical parts on a computational grid. The most used models for stirred tanks are the moving reference frame (MRF) and the sliding mesh (SM). In contrast to MRF, SM requires more computational resources and its convergence time is

MRF is defined by a rotational and a stationary region. The equations are solved on a reference frame that rotates with the impeller and the problem is solved on a stationary grid [20]. When the momentum equation is solved, an additional acceleration term is incorporated in the

[ ][2 0 ] *<sup>P</sup>* [ ] *<sup>t</sup>*

+ × + ´- + + × = ¶ **r r <sup>v</sup>** <sup>Ñ</sup> **vv w v g** <sup>Ñ</sup> <sup>Ñ</sup>

**<sup>v</sup>** <sup>Ñ</sup> **vv w v w w r g** <sup>Ñ</sup> <sup>Ñ</sup>

[ ][ ] *<sup>P</sup>* [ ] *<sup>t</sup>*

+ × + ´ +´´- + + × =

 r r

t

velocity vector formulation as relative Eq. (22) or absolute Eq. (23).

 r

**r r r**

rr

f

*t*

Schwarz shear rate inequality and by making the normal stress tensor positive.

m

is calculated by Eq. (19), where **Ωij** ¯

frame with angular velocity **ωk**:

274 Modeling and Simulation in Engineering Sciences

**7. Stirring model**

r

¶ **r**

¶

r

r

¶

higher.

*U*\*

<sup>6</sup> 4.04 *\* <sup>C</sup>*

 <sup>=</sup> +

*<sup>U</sup>*\* ° ° = + **S Sij ij** <sup>W</sup>

°**ij** = -**ij ijk k** <sup>W</sup> <sup>W</sup> **<sup>2</sup><sup>ò</sup>**

W**ij** = - W*IJ* **òijk k**

m r = *C*m ò

> Two numerical solvers can be selected in Fluent®. The first is a pressure-based solver that was initially developed for high-speed incompressible flow. The second is a density-based that was developed for high-speed compressible flow. Regardless of the solver being used, the velocity field is calculated from the momentum equations [19]. The general solution algorithm can be divided in three stages:


The pressure-based solver is established from the pressure-correction equation obtained from the momentum and continuity equations. Convergence is reached when the estimated velocity field satisfies the continuity condition:

$$\sum\_{f}^{N\_{\text{flocc}}} J\_f A\_f = 0 \tag{25}$$

where *Jf* is the mass flux and *Af* is the surface area of face "*f*". In the pressure-based methods, there are segregated algorithms based on corrector-predictor approximations (e.g. SIMPLE, SIMPLEC and PISO). SIMPLE algorithm or semi-implicit method for pressure-linked equa‐ tions satisfies Eq. (25) by correcting the flux *Jf* through *J′<sup>f</sup>* and by the corrected pressure *p*'. The algorithm postulates that *J′<sup>f</sup>* follows Eq. (26) [19]. *J\*f* is calculated by using the pressure field *p*\*.

$$J\_f^\cdot = d\_f \left( \dot{p\_{c0}^\cdot} - \dot{p\_{c1}^\cdot} \right) \tag{26}$$

$$J\_f = J\_f^\* + J\_f^\* \tag{27}$$

$$J\_f = J\_f^\* + d\_f \left(\dot{p\_{c0}} - \dot{p\_{c1}}\right) \tag{28}$$
