## 2 Modelling Tools
### DOC Model  
A lumped modular DOC model is applied to compute the main physical and chemical processes affecting the flow properties at the DPF inlet [26]. The model is based on the solution of the pressure drop, heat transfer and the chemical reactivity processes [36]. Since this work is focused on the analysis of the DPF response during the regeneration, the heat transfer and chemical processes are modelled directly imposing the turbine outlet flow properties, i.e., mass flow, composition, pressure and temperature, as boundary conditions. \begin{table}
\begin{tabular}{l c c} \hline \hline  & **DOC** & **DPF** \\ \hline Diameter [mm] & 172 & 172 \\ Monolith length [mm] & 82 & 100 \\ Channel cross-section & Square & Square \\ Cell density [cpsi] & 400 & 200 \\ Cell size [mm] & 1.17 & 1.39 \\ Wall thickness [mm] & 0.101 & 0.4 \\ Catalytic area [m\({}^{2}\)] & 5.5 & \\ Filtration area [m\({}^{2}\)] & & 2.11 \\ Porosity [-] & & 0.4 \\ Mean pore diameter [\(\upmu\)m] & & 21.4 \\ Permeability [\(\times 10^{-13}\)m\({}^{2}\)] & & 7.69 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Geometrical parameters of the diesel oxidation catalyst (DOC) and the diesel particulate filter (DPF). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Magnitude** & **Sensor** & **Range** & **Error** \\ \hline Temperature & Thermocouple type K & [\(-\)200–1200 \({}^{\circ}\)C] & \(\pm\)1.1 \({}^{\circ}\)C or 0.4\% (actual value) \\ Pressure & Piezoresistive PMA P40 & [0–6 bar] & \(\pm\)0.3\% (full scale) \\ Fuel mass flow & Gravimetric balance AVL733S & [0–27 kg/h] & \(\pm\)0.12\% (full scale) \\ Air mass flow & Sensylflow ABB & [0–720 kg/h] & \(\pm\)1\% (actual value) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Characteristics of the instrumentation. The pressure drop across the DOC is modelled based on a phenomenological approach assuming incompressible flow as [36]  
\[\Delta p_{DOC}=\frac{1}{2}\varsigma_{DOC}\rho_{in}u_{in}^{2}, \tag{1}\]  
where \(\rho_{in}\) is the DOC inlet density, \(u_{in}\) is the DOC inlet velocity and \(\varsigma_{DOC}\) is the pressure drop coefficient of the device. This coefficient is specific to the system geometry and obtained experimentally as a function of the Reynolds number. The heat transfer phenomena are simulated applying a lumped nodal scheme adapted from a 1D modelling approach [37]. The model accounts for the gas to wall heat exchange and the heat losses towards the environment. In addition, since the active regeneration is a transient process, the thermal inertia of the monolith substrate and the external canning is also considered to provide the temperature evolution at the DPF inlet. The nodal schemes corresponding to the monolith and the DOC canning are represented in Figure 2[26]. The model definition is based on the discretization of the general heat transfer equation by centred explicit finite differences. If axial and radial heat transfer is assumed, the wall temperature at time-step \(p+1\) and node \((i,j)\) is determined considering the gas temperature and the substrate-canning state at the previous time-step \(p\). Taking into account the definition of every control volume, i.e., thermal properties and heat transfer area, the substrate and canning temperatures are finally computed as:  
\[T_{w}^{p+1}=\frac{\Delta t}{C_{w}^{p}}\left(\sum_{k}\frac{T_{k}-T_{w}^{p}}{R_{ k}}+q_{r}^{p}\right)+T_{w}^{p} \tag{2}\]  
\[T_{sur,int}^{p+1}=\frac{\Delta t}{C_{sur,int}^{p}}\left(\frac{T_{w}^{p}-T_{sur,int}^{p}}{R_{rad}^{p}}+\frac{T_{sur,ext}^{p}-T_{sur,int}^{p}}{R_{sur}^{p}} \right)+T_{sur,int}^{p} \tag{3}\]  
\[ \begin{split} T_{sur,ext}^{p+1}=&\frac{\Delta t}{C_ {sur,ext}^{p}}\left(\frac{T_{ext}-T_{sur,ext}^{p}}{R_{ext}^{p}}+\frac{T_{sur, int}^{p}-T_{sur,ext}^{p}}{R_{sur}^{p}}+\right.\\ &\left.\frac{T_{sur,pre}^{p}-T_{sur,ext}^{p}}{R_{ax,pre}^{p}}+ \frac{T_{sur,post}^{p}-T_{sur,ext}^{p}}{R_{ax,post}^{p}}\right)+T_{sur,ext}^{p}.\end{split}