**5. Fluent® simulation**

**Figure 1.** Estimation of *kp* / *kt*

270 Modeling and Simulation in Engineering Sciences

**4. Computational fluid dynamics (CFD)**

*t* ¶

¶

1/2 from viscosity-time experimental data.

CFD is concerned with the numerical solution of the following partial differential equations that express the conservation principles of mass, energy, and momentum transport (5–7) [2].

Ñ **v** Ñ Ñ (5)

( )( ) 0 *<sup>A</sup> A AB A A A*

+× -× -+=

*<sup>x</sup> x Dx R S*

A typical simulation comprises the formulation of the problem, physical assumptions to simplify the mathematical model, the numerical solution of the conservation equations, data processing, and the discussion of results. The mathematical models, initial conditions, and other adjustments can be implemented through Fluent® software.

In **Figure 2**, there are at least two critical stages: convergence of the numerical solution and validation of the mathematical model.

**Figure 2.** Development of a Fluent® simulation.

## **6. Turbulence and RANS equations**

Turbulence is described as a random and chaotic movement of a fluid. Mathematically, a turbulence model is a nonlinear system in which a minimum modification on its boundary conditions produces severe alterations in the global behavior of the system [17]. Ranade proposes three approaches to model turbulence in fluids: statistical, deterministic, and structural [2]. Reynolds-average Navier-Stokes (RANS) equations are part of the statistical approximation in which turbulence is described as a combination of average variables (θ¯i ) and fluctuations θ<sup>f</sup> [18]:

Impact of Fluid Flow on Free Radical Polymerization in a Batch Reactor http://dx.doi.org/10.5772/64156 273

$$
\theta\_i = \overline{\theta}\_i + \theta\_f \tag{9}
$$

Semi-empirical *k*- is a turbulence model commonly used in stirred tanks. The model assumes complete turbulence and neglects molecular viscosity effects. *k*- is part of the RANS equations and it is composed by a two-equation system with two parameters to solve: *k* (turbulence kinetic energy) and (turbulence dissipation rate). Standard *k*- was the first model; RNG (renormalization group theory) and realizable models were developed from subsequent modifications [19]. In contrast to standard *k*- , RNG improves flow with eddies and it adds a term to the equation (*R* ).

$$\frac{\partial \left(\rho k\right)}{\partial t} + \mathbf{V} \cdot \mathbf{v}\left(\rho k\right) = \mathbf{V} \cdot \left[\boldsymbol{\kappa}\_k \,\,\mu\_{qt} \nabla k\right] + G\_k + G\_b - \rho \boldsymbol{\epsilon} - Y\_M + S\_k \tag{10}$$

$$\frac{\partial(\rho\epsilon)}{\partial t} + \mathbf{V} \cdot \mathbf{v}(\rho\epsilon) = \mathbf{V} \cdot \left[ \boldsymbol{\alpha}\_{i} \; \mu\_{qt} \nabla \epsilon \right] + \boldsymbol{C}\_{1i} \; \frac{\epsilon}{k} \left( \boldsymbol{G}\_{i} + \boldsymbol{C}\_{3i} \boldsymbol{G}\_{b} \right) - \boldsymbol{C}\_{2i} \rho \frac{\epsilon^{2}}{k} - \boldsymbol{R}\_{i} + \boldsymbol{S}\_{i} \tag{11}$$

Realizable *k*- has a superior performance for rotational flows and for boundary layers under adverse conditions like high-pressure gradients, separation, and recirculation [18].

$$\frac{\partial(\rho k)}{\partial t} + \mathbf{V} \cdot \mathbf{v}(\rho k) = \mathbf{V} \cdot \left[ \left( \mu + \frac{\mu\_t}{\sigma\_k} \right) \nabla k \right] + G\_k + G\_b - \rho \epsilon - Y\_M + S\_k \tag{12}$$

$$\frac{\partial(\rho\epsilon)}{\partial t} + \mathbf{V} \cdot \mathbf{v} \left(\rho\epsilon\right) = \mathbf{V} \cdot \left[ \left(\mu + \frac{\mu\_t}{\sigma\_\epsilon}\right) \mathbf{V}\epsilon \right] + \rho C\_\text{l} \mathbf{S}\_\epsilon - \rho C\_2 \frac{\epsilon^2}{k + \sqrt{\nu\epsilon}} + C\_\text{l} \frac{\epsilon}{k} C\_3 \mathbf{G}\_b + \mathbf{S}\_\epsilon \tag{13}$$

**Figure 2.** Development of a Fluent® simulation.

272 Modeling and Simulation in Engineering Sciences

[18]:

fluctuations θ<sup>f</sup>

**6. Turbulence and RANS equations**

Turbulence is described as a random and chaotic movement of a fluid. Mathematically, a turbulence model is a nonlinear system in which a minimum modification on its boundary conditions produces severe alterations in the global behavior of the system [17]. Ranade proposes three approaches to model turbulence in fluids: statistical, deterministic, and structural [2]. Reynolds-average Navier-Stokes (RANS) equations are part of the statistical approximation in which turbulence is described as a combination of average variables (θ¯i ) and

$$C\_1 = \max\left(0.43, \frac{\eta}{\eta + 5}\right) \tag{14}$$

*S* = 2**S Sij ij** (15)

$$
\eta = S \frac{k}{\epsilon} \tag{16}
$$

where *Gk* is the generation of "*k*" by velocity gradients and *Gb* by buoyancy, *Sk* and *S* are source terms, *C*2 and *C*<sup>1</sup> are constants, and *σk* and *σk* are *σ* are the turbulent Prandtl numbers for *k* and , respectively. Turbulent viscosity (*μt* ) is calculated as indicated by Eq. (17):

$$
\mu\_r = \rho C\_\mu \frac{k^2}{\epsilon} \tag{17}
$$

In contrast to RNG, realizable model uses a variable *Cμ* that satisfies the "realizability" through Schwarz shear rate inequality and by making the normal stress tensor positive.

$$C\_{\mu} = \frac{1}{4.04 + \frac{\sqrt{6 \cos \phi k U}^{\prime}}{\epsilon}} \tag{18}$$

*U*\* is calculated by Eq. (19), where **Ωij** ¯ is the rotation average speed tensor on a rotating reference frame with angular velocity **ωk**:

$$\left(U\right)^{\*} = \sqrt{\mathbf{S}\_{\text{ij}}\mathbf{S}\_{\text{ij}} + \widetilde{\mathbf{D}}\_{\text{I}}\widetilde{\mathbf{D}}\_{\text{I}}}\tag{19}$$

$$
\tilde{\mathbf{Q}}\_{\rm ij} = \mathbf{Q}\_{\rm ij} - 2\epsilon\_{\rm ijk} \mathbf{o}\_{\rm k} \tag{20}
$$

$$\mathbf{Q}\_{\rm ij} = \mathbf{Q}\_{\rm LJ} - \epsilon\_{\rm ijk} \mathbf{op}\_{\rm k} \tag{21}$$

## **7. Stirring model**

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 higher.

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

$$\frac{\partial \rho \mathbf{v}\_r}{\partial t} + \left[\mathbf{V} \cdot \rho \mathbf{v}\_r \mathbf{v}\_r\right] + \rho \left[2\mathbf{w} \times \mathbf{v}\_r + \mathbf{w} \times \mathbf{w} \times \mathbf{r}\right] - \mathbf{g}\rho + \nabla P + \left[\nabla \cdot \mathbf{r}\right] = 0\tag{22}$$

$$\frac{\partial \rho \mathbf{v}}{\partial t} + \left[\mathbf{V} \cdot \rho \mathbf{v}\_r \mathbf{v}\right] + \rho \left[\mathbf{w} \times \mathbf{v}\right] - \mathbf{g}\rho + \mathbf{\nabla}P + \left[\mathbf{\nabla} \cdot \mathbf{r}\_r\right] = \mathbf{0} \tag{23}$$

The term *ρ*[**2w** × **vr** + **w** × **w** × **r**] is composed by the Coriolis acceleration (**2w** × **vr**) and the centripetal acceleration (**w** × **w** × **r**). The stress tensor **τ<sup>r</sup>** keeps its mathematical structure, but it uses relative velocities.

SM models the rotation of the grid by adding a source term as a function of time in Eq. (8) allowing a relative movement of the adjacent grids among themselves. The SM equation is formulated for a scalar (*ϕ*) as follows:

$$\frac{\text{d}}{\text{d}} \frac{\text{d}}{\text{d}t} \int\_{\text{\text{\textquotedblleft}}} \rho \phi dV + \int\_{\text{\textquotedblleft}} \rho \phi \left(\mathbf{u} - \mathbf{u}\_{\text{\textquotedblleft}}\right) \cdot \text{d}\mathbf{A} = \int\_{\text{\textquotedblleft}} D \nabla \phi \cdot \text{d}\mathbf{A} + \int\_{\text{\textquotedblleft}} S\_{\phi} dV \tag{24}$$

where *V* is the control volume, *D* is the diffusion coefficient, *Sϕ* is a source term, **u** is the flow velocity vector, **ug** is the velocity of the moving grid, and *∂V* is the control volume interface.

A third manner to model the movement of mechanical parts is through the boundaries of the walls; this approximation was named tangential rotation (ROT) and holds true only for viscous flows (non-slip condition).
