## 2 Materials and Methods
### Methods
#### 2.2.2 Experimental Characterization of Model Input Parameters  
(a) Monolith pressure loss characterization. The classical approach that links pressure drop and velocity in flows through porous media is the Forchheimer equation (Equation (1)), that can be derived from the Navier-Stokes equation for one-dimensional, incompressible and steady laminar flow of a Newtonian fluid in a rigid porous medium [29]:  
\[-\frac{dp}{dx}=\frac{1}{\kappa}(\mu\cdot v)+\beta\left(\rho\cdot v^{2}\right) \tag{1}\]  
The second term in the right can be interpreted as a second-order correction to account for the contribution of inertial forces, but, at sufficiently low velocities, this effect is negligible and Equation (1) can be reduced to [PERSON]'s law (Equation (2)) [30]:  
\[-\frac{dp}{dx}=\frac{1}{\kappa}(\mu\cdot v) \tag{2}\]  
[PERSON]'s law can be rewritten in the form of Equation (3), relating the average fluid velocity \(v\) through the pores with pressure drop \(\Delta p\) along a segment of length \(L\). \[-\frac{\Delta p}{\mu L}=\frac{1}{\kappa}\,v \tag{3}\]  
If the equation from the linear fit of the scatter plot of \(-\frac{\Delta p}{\mu L}\) vs. \(v\) is (Equation (4)):  
\[Y=BX \tag{4}\]  
then [PERSON]'s constant \(\kappa\) can be derived as follows (Equation (5)):  
\[\kappa=\frac{1}{B} \tag{5}\]  
(b) Heat transfer to surroundings. Natural convection to pipe surroundings in the test-bench needs to be included. Wall temperatures along the exhaust pipe were measured using thermography. Figure 4 shows, as an example, views of the external surface of the pipe and the DOC. Figure 3: Test matrix. Circles represent selected engine operating points. An average skin temperature was used to calculate an effective convection heat transfer coefficient. This skin temperature simplification is considered as reasonable, since calculation of heat transfer rates is not very sensitive in the values of pipe wall temperatures [5]. Total transferred heat from pipe wall to surroundings is \(\dot{Q}\) (W) (Equation (6)):  
\[\dot{Q}=\overline{h}_{\infty}\pi d(T_{wall}-T_{\infty})+\varepsilon\pi dL \sigma\Big{(}T_{wall}^{4}-T_{amb}^{4}\Big{)} \tag{6}\]  
Here, all terms in the right side of Equation (6) are grouped in one single equivalent convection term (Equation (7)):  
\[\dot{Q}=\overline{h}_{eff}\pi dL(T_{wall}-T_{\infty}) \tag{7}\]  
The effective convection coefficient accounts for convection and radiation heat losses, although radiation heat transfer to the environment is expected to be low, since wall temperature is below 400 \({}^{\circ}\)C [4]. A similar approach is explained in [PERSON] et al. [8]. Based on the measured exhaust gas and pipe wall temperatures, an energy balance for the exhaust gas is employed for the calculation of the heat flux. The resulting heat fluxes are employed in the estimation of a mean gas-to-wall convection coefficient. Since temperature at both ends of the exhaust pipe are measured, total heat losses \(\dot{Q}\) (W) can be obtained as Equation (8):  
\[\dot{Q}=\dot{m}_{g}c_{p,g}\left(T_{g,in}-T_{g,\;out}\right) \tag{8}\]  
Equaling Equations (7) and (8), \(\overline{h}_{eff}\) can be derived as Equation (9):  
\[\overline{h}_{eff}=\frac{\dot{m}_{g}c_{p,g}\left(T_{g,in}-T_{g,\;out}\right)}{ \pi d(T_{wall}-T_{\infty})} \tag{9}\]  
(c) Estimation of kinetic parameters. Under light-off temperatures (about 200 \({}^{\circ}\)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 [PERSON] et al.