**3. Rheokinetics and inverse problem**

kinetics, the molecular weight, and the composition of polymers can be altered due to local

In this study, a batch chemical reactor is analyzed. This type of reactor is defined as a closed and spatially uniform system where the chemical species are transformed only as a function of time. The transformation of chemical species can be quantified by following any physico‐ chemical property associated with either reagents or products. During free radical polymeri‐ zations, the viscosity of the medium increases dramatically while products are formed [4]. The

Rapid computational development has made the numerical analysis of phenomena associated with stirred tanks easier [5]. For example, through CFD, Patel [6] studied the mixing process on a continuous stirred tank reactor and how the thermal polymerization of styrene is affected.

Computational analysis in stirred reactors has to consider at least two models: one for turbulence and the other for stirring. The turbulence model describes the random and chaotic movement of a fluid [7], while the stirring model describes the displacement of the fluid as a

The study of batch reactors with a tracer is the basis for understanding flow behavior [8]. Tracer evolution curves allow to identify regions with turbulence, dead zones, recirculation cycles,

In this work, a tracer test was used to validate a mathematical model. The mixing process was analyzed by using both the experimental and simulated behavior of the tracer. The experi‐ mental kinetics of polymerization was obtained by a multiparametric nonlinear regression of

The uses of polyacrylamide have been extended to different applications in the oil industry,

In EOR applications, acrylamide (AAm) polymers are dispersed in water to increase their viscosity. However, at high temperatures the viscosity of AAm polymer decreases due to hydrolysis [11]. This can be mitigated by using co-monomers such as sodium 2-acrylamido-2 methylpropane sulfonate (AMPSNa) [12]. This sodium sulfonate monomer is well known because it confers stability to the polymer against divalent cations and high temperatures (above 90°C). In view of the benefits, it is necessary to develop a process for the synthesis of the AAm-AMPSNa copolymer that guarantees product quality and synthesis reproducibility in order to properly design the polymer. The next sections will be focused on studying the relation between the mixing time and the mixing process during the synthesis of copolymer

such as water conformance, fracking, and enhanced oil recovery (EOR) processes.

concentration gradients as a consequence of bad mixing [3].

268 Modeling and Simulation in Engineering Sciences

consequence of the local movement of mechanical parts.

viscosity-time data.

in a batch reactor.

**2. Problem definition**

kinetics of polymerization can be followed from the change of viscosity.

closed circuits, or even to determine the mixing time of the reactor [9, 10].

The kinetics of polymerization was analyzed through the evolution of rheological behavior of the reactive system. Rheology has entered into science fields, such as biology and polymer science [13]. Historically, polymer science and rheology converge in what is known as rheokinetics. This field was created more than 30 years ago to have a better understanding of the phenomenological nature of polymerizations.

There are two main problems related to rheokinetics: the direct and inverse problems. The inverse problem, which is the principal focus of this work, deals with the determination of kinetic parameters given by the experimental data of viscosity-time curves. Equation (4) reproduces the viscosity-time curve behavior (rheokinetic model) and it was obtained from Eqs. (1–3). This equation assumes a linear free radical polymerization and it does not consider mass and energy transport effects. Equations (1) through (4) are deeply analyzed by Malkin, see [14].

$$
\eta = Kx^b \overline{N}^a \tag{1}
$$

$$\mathbf{x} = 1 - \exp\left(-\frac{k\_{\rho}k\_d^{1/2}f^{1/2}}{k\_\iota^{1/2}}[I]\_o^{1/2}t\right) \tag{2}$$

$$\overline{N} = \frac{\left[M\right]\_o}{\left[I\right]\_o} \ge \left(1 - \exp\left(-k\_d t\right)\right) \tag{3}$$

$$\eta = K \left[ M \right]\_o^a \left[ I \right]\_o^{-a} \left[ 1 - \exp \left( -\frac{2f^{1/2}}{k\_d^{1/2}} \frac{k\_p}{k\_l^{1/2}} \left[ I \right]\_o^{1/2} \left[ 1 - \exp \left( -k\_d t \right) \right] \right) \right]^{a \star b} \tag{4}$$

where *<sup>η</sup>* is the viscosity, *<sup>t</sup>* is the reaction time, *N*¯is the polymerization degree, *<sup>x</sup>* is the conver‐ sion, [*M*]*<sup>o</sup>* is the initial monomer concentration, [*I*]*<sup>o</sup>* is the initial initiator concentration, *K*, *a*, *b*, and *f* are system parameters and *kp*, *kt* , *kd* are the rate constants of propagation, termination, and initiation, respectively.

The ratio *kp* / *kt* 1/2 is estimated as proposed by **Figure 1**. To guarantee the quality of the adjust‐ ment, *kd* must be a number between 0.01 and 1 [4]. To know which process dominates the polymerization, the magnitude of *kp* / *kt* 1/2 is used an indicator. If the ratio is *kp* / *kt* 1/2≫1, the propagation of live chains dominates; on the contrary, when *kp* / *kt* 1/2≪1, the termination of macro radicals dominates.

**Figure 1.** Estimation of *kp* / *kt* 1/2 from viscosity-time experimental data.

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

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].

$$\frac{\partial \mathbf{x}\_A}{\partial t} + \left(\mathbf{V} \cdot \mathbf{v} \mathbf{x}\_A\right) - \left(\mathbf{V} \cdot \mathbf{V} D\_{AB} \mathbf{x}\_A\right) - R\_A + S\_A = 0\tag{5}$$

$$\frac{\partial \rho C\_{\rho} T}{\partial t} + \left(\nabla \cdot \mathbf{v} \rho C\_{\rho} T\right) - \left(\nabla \cdot \nabla k \,\rho C\_{\rho} T\right) - q\_{\kappa} + q\_{\iota} = 0\tag{6}$$

$$\frac{\partial \rho \mathbf{v}}{\partial t} + \left[\mathbf{V} \cdot \rho \mathbf{v} \mathbf{v}\right] - \mathbf{g}\rho + \nabla P + \left[\mathbf{V} \cdot \mathbf{r}\right] = \mathbf{0} \tag{7}$$

where *xA* is the concentration of "*A*" species, *T* is the temperature, **v** is the velocity, *ρ* is the density, *Cp* is the heat capacity, *DAB* is the diffusion coefficient, *k* is the thermal conductivity, **g** is the gravity, *P* is the pressure, **τ** is the stress tensor, *RA y qR* are terms associated with the chemical reaction, *SA y qI* are source terms and **∇**<sup>=</sup> <sup>∂</sup> ∂ *xi* .

Equations (5–7) are supported in two assumptions. The first one is the conservation principle which states that mass, energy, and momentum are transformed without creating or destroy‐ ing themselves and the second one is the continuum hypothesis which considers continuity of its physical properties [15].

When simplifying Eq. (7) by considering a fluid of constant density and viscosity and a linear relation between the shear rate and the shear stress, the Navier-Stokes equations can be obtained Eq. (8):

$$
\rho \frac{\text{D} \mathbf{v}}{\text{Dt}} - \mathbf{g} \rho + \mathbf{\nabla} P - \mu \left[ \mathbf{\nabla} \cdot \mathbf{\nabla} \mathbf{v} \right] = \mathbf{0} \tag{8}
$$

Navier-Stokes equations are the basis of CFD and its numerical solution is fundamental to understand and describe the phenomena of fluid flow.

A CFD simulation is limited by the data processing rates and storing capacity. Nevertheless, improvements of computers capacity have stimulated the growth and diversification of CFD applications [16]. Nowadays, there are several CFD software tools as COMSOL® and Fluent®.
