# Impact of Particulate Size During Deep Loading on DPF Management
## 2 Numerical Models
### Fluid Dynamic Model  
Basing on the channel study provided by References [10; 18], and assuming the exhaust gas as an incompressible fluid, the volumetric flow rate, \(Q\), passing through the monolith is evaluated by summing the single channel flow rates \(q_{i}\) passing through the inlet channels:  
\[Q=\sum_{i=0}^{N_{channel,in}}q_{i}=N_{channel,in}q, \tag{21}\]\[q=U\times A_{in} \tag{22}\]  
where \(A_{in}\) is the inlet section surface of the single channel:  
\[A_{in}=\alpha^{2}. \tag{23}\]  
The monolith is characterized by a channel density \(\sigma\), defined as the total channels number per unit surface  
\[\sigma=\frac{1}{\left(\alpha+w_{s}\right)^{2}}. \tag{24}\]  
In this way, the number of inlet channels is given by:  
\[N_{channel,in}=\frac{\pi\,D_{trap}^{2}}{4}\frac{\sigma}{2}. \tag{25}\]  
The channel inlet velocity is related to the total flow rate \(Q\) and the geometric quantities according to:  
\[Q=(UA_{in})\Bigg{(}\frac{\pi\,D_{trap}^{2}}{4}\Bigg{)}\frac{\sigma}{2}, \tag{26}\]  
\[\mathrm{U}=\frac{8Q}{\pi\,D^{2}\sigma\alpha^{2}}. \tag{27}\]  
Applying [PERSON]'s law across the porous wall for a single channel and taking into account the loss terms, due to passing flow inside the square channel in laminar condition, the total pressure drop for the clean filter is expressed as:  
\[\Delta P_{tot\ 0}=\Delta P_{null}+\Delta P_{in}+\Delta P_{out}=\frac{\mu Q}{2V _{tmp}}\,\left(\alpha+w_{s}\right)^{2}\Bigg{[}\frac{w_{s}}{k_{0}\alpha}+ \frac{4 FL^{2}}{3\alpha^{2}}+\frac{4 FL^{2}}{3\alpha^{2}}\Bigg{]}. \tag{28}\]  
\(F=F(Re)\) is a loss factor for laminar flow inside the channel, \(k_{0}\) is the porous medium permeability, \(L\) is the length of the monolith and \(V_{trap}\) is its volume.  
In order to take into account the progressive reduction of filtering surface, due to soot layer growth, the [PERSON]'s law is integrated across the deposit height (\(t\))  
\[\begin{array}{l}\Delta P_{soot}(t)=\frac{\mu}{k_{soot}}\int_{0}^{w(t)}u(x)dx =\frac{\mu}{k_{soot}}\int_{0}^{w(t)}\frac{q}{A_{porous}}dx\\ =\frac{\mu\mu}{4L\kappa_{soot}}\int_{0}^{w(t)}\frac{1}{(\alpha-2w(t)+2x)}dx\\ =\frac{\mu Q}{L\pi D^{2}k_{soot}}ln\frac{\alpha}{\alpha-2w(t)}.\end{array} \tag{29}\]  
For highly loaded filter, the reduction of the channel inlet section is taken into account. The channel dimension \(\alpha^{*}(t)\) can be written in the function of layer height \(w(t)\),  
\[\alpha^{*}(t)=\alpha-2w(t). \tag{30}\]  
and channel inlet velocity increases causing a variation of the channel loss term \(\Delta Pin(t)\) compared to clean channel:  
\[U^{\prime}(t)=\frac{8Q}{\pi D^{2}\sigma(\alpha-2w(t))^{2}}, \tag{31}\]  
\[\Delta P_{in}(t)=\frac{\mu Q}{2V_{trap}}\,\left(\alpha+w_{s}\right)^{2}\Bigg{(} \frac{4 FL^{2}}{3(\alpha-2w(t)^{2})}\Bigg{)}. \tag{32}\]  
where \(m_{in,1}\) is the inlet mass in the wall and \(m_{cake}\) is trapped mass in the external layer.  
For each i-th slab, the incoming mass \(m_{in,i}\) depends on trapped mass in previous slab \(s\), corresponding at a defined loading level of unit cells,  
\[m_{in,i} = m_{in,i-1}(1-\mathrm{E}_{i-1}\left(m_{in,i-1}\,\right))=m_{out,i-1}. \tag{46}\]  
For each calculation time, the trapped mass inside porous medium \(m_{porous}\) is equal to the sum of single trapped mass in every slab  
\[m_{porous} = \sum_{i}^{N_{slab}}m_{coll,i}, \tag{47}\] \[m_{coll,i} = \mathrm{E}_{i}\ m_{in,i}. \tag{48}\]  
The total trapped mass in a single channel is given by the sum of the contributions in the porous medium and in the layer. It defines the filter total efficiency:  
\[E_{filter}=\frac{m_{in}\ \varphi+m_{in}(1-\varphi)\ E_{porous}}{m_{in}}=\varphi+(1- \varphi)\ E_{porous}, \tag{49}\]  
\[E_{porous} = \frac{m_{porous}}{(1-\varphi)m_{in}}. \tag{50}\]