(c) Estimation of kinetic parameters.

Under light-off temperatures (about 200 ◦C) catalytic processes remain basically inactive. Until activation temperature is reached, reactions are chemically controlled. On the other hand, post-light-off reaction rates are limited mainly by mass transfer and, consequently, conversion efficiency depends on the residence time within the monolith, the surface to volume ratio of the monolith and the mass transfer [31]. In operating conditions from test matrix, catalytic reactions are normally within the light-off band. Thus, estimation of kinetic parameters is needed.

Two main reactions occurring in DOCs, as in Voltz et al. [17], are modeled (Equations (10) and (11)):

$$\text{C}\_3\text{H}\_6 + \frac{3}{2}\text{O}\_2 \rightarrow 3\text{ CO}\_2 + 3\text{ H}\_2\text{O} \tag{10}$$

*Appl. Sci.* **2017**, *7*, 590

$$\text{CO} + \frac{1}{2}\text{O}\_2 \rightarrow \text{CO}\_2\tag{11}$$

Propylene is representative of the easily oxidized hydrocarbons, which constitute about 80% of the total hydrocarbons found in a typical exhaust gas. Other saturated hydrocarbons (typically represented as methane or propane) that are resistant to oxidation usually make up the remaining 20% [17]. Since only total hydrocarbon (THC) data is available and fast oxidation hydrocarbons represent the majority of the THC, propylene alone was used as representative of the HC (as is common [32]).

Usually adopted forms of the rates of *CO* and *HC* oxidations are as follows (Equations (12) and (13)):

$$R\_{CO} = \frac{k\_{CO}c\_{CO}c\_{O\_2}}{G} \tag{12}$$

$$R\_{C\_3H\_6} = \frac{k\_{C\_3H\_6}c\_{C\_3H\_6}c\_{O\_2}}{G} \tag{13}$$

where G is a term accounting for *NO*, *CO* and *HC* inhibition effects on oxidation (Equation (14)):

$$\mathcal{G} = T\_{\mathfrak{s}} \left( 1 + K\_1 \mathfrak{c}\_{\complement \complement \complement \complement \mathcal{C}} + K\_2 \mathfrak{c}\_{\complement \amspace} H\_6 \right)^2 \left( 1 + K\_1 \mathfrak{c}\_{\complement \complement \mathcal{C}} + K\_2 \mathfrak{c}\_{\complement \complement \mathcal{C}} H\_6 \right)^2 \left( 1 + K\_3 \mathfrak{c}\_{\complement \complement \mathcal{C}}^2 \mathfrak{c}\_{\complement \amspace} H\_6 \right) \left( 1 + K\_4 \mathfrak{c}\_{\complement \amspace}^{0,7} \right) \tag{14}$$

Since the intended target is an empirical model to fit experimental data and power-law reactions showed good performance, an inhibition term was not included (Equations (15) and (16)):

$$R\_{\text{CO}} = k\_{\text{CO}} c\_{\text{CO}} c\_{\text{O}\_2} \tag{15}$$

$$R\_{C\_3H\_6} = k\_{C\_3H\_6} c\_{C\_3H\_6} c\_{O\_2} \tag{16}$$

Arrhenius kinetic constants *km* are defined as (Equation (17)):

$$k\_m = A\_m e^{-\frac{E\_{g,m}}{RT\_s}} \tag{17}$$

The problem is reduced to obtain the pre-exponential factors *Am* and activation energies *Ea*,*m*. For both species, an iterative process varying the pre-exponential factor in CFD simulations is conducted until the deviation of outlet concentration value from the experimental is as much as 0.1%. Once this is achieved, values of *km* are obtained. This tuning process is done for the different operating conditions, and then *km* of each reaction is obtained as a function of monolith temperature (provided by CFD results). This iterative process needs to be followed just once (to obtain the above-mentioned constants and implement them in the model) and not for every prospective 3D simulation.

The simulations were performed on a 2D axisymmetric DOC model. The 2D model maintained the same characteristic lengths but with a simplified geometry (see Figure 5), allowing fast iterative simulations. Other aspects about set-up of these 2D simulations are the same than those presented below for the 3D calculations.

**Figure 5.** Geometry of the 2D axisymmetric DOC model for the tuning process of kinetic parameters. Darker shade of grey represents the monolith.

From results, the kinetic exponential law (Equation (17)) can be obtained, since (Equation (18)):

$$
\ln k\_m = C\_m - \frac{D\_m}{T\_s} \tag{18}
$$

where *Ci* and *Di* are constants from the linear fit. Sought kinetic constants are (Equations (19) and (20)):

$$A\_{\mathcal{W}} = \mathcal{e}^{\mathbb{C}\_{\mathcal{W}}} \tag{19}$$

$$E\_{\mathfrak{a,m}} = D\_{\mathfrak{m}} \mathbb{R} \tag{20}$$
