DEM Model for Soot Restructuring


Developing a discrete element model for restructuring of soot aggregates. Aggregates are represented as a collection of rigid spherical particles. Various contact models are added to create a realistic system.

Binary interactions

Elastic contact force between two rigid spherical particles can be modeled with Hertz theory:

\[\mathbf{F}_{cn}=-\frac{2}{3}\frac{E}{\left(1-\nu^2\right)}\sqrt{\frac{R}{2}}\delta_n^{3/2}\mathbf{n}\]

In order to simulate kinetic energy loss during a collision, a damping component is added to the normal force:

\[\mathbf{F}_{dn}=c_n\left(\mathbf{v}_{ij}\cdot\mathbf{n}\right)\mathbf{n}\]

Van der Waals attraction between small particles can be modeled with the Hamaker equation:

\[\begin{multline*}
\mathbf{F}_{vw}=\frac{Ad}{(\lvert \mathbf{r}_i-\mathbf{r}_j\rvert-d)^2}\mathbf{n}\\
\mathrm{if}\ \lvert \mathbf{r}_i-\mathbf{r}_j\rvert>r_0
\end{multline*}\] \[\begin{multline*}
\mathbf{F}_{vw}=\frac{Ad}{(r_0-d)^2}\mathbf{n}\\
\mathrm{if}\ \lvert \mathbf{r}_i-\mathbf{r}_j\rvert\leq r_0
\end{multline*}\]

Tangential contact force resulting from an oblique collision can be calculated with the following equation:

\[\mathbf{F}_{ct}=-\mu_s\lvert\mathbf{F}_{cn}\rvert\left[1-\left(1-\frac{\min{\{\delta_t,\delta_{t,max}\}}}{\delta_{t,max}}\right)^{3/2}\right]\mathbf{t}\]

Tangential damping force can be modeled in a manner similar to normal damping:

\[\mathbf{F}_{dt}=c_t\left(\mathbf{v}_{ij}\times\mathbf{n}\right)\times\mathbf{n}\]

The torque arising from the tangential forces can then be expressed as:

\[\mathbf{T}=\mathbf{R}\times\left(\mathbf{F}_{ct}+\mathbf{F}_{dt}\right)\]

Finally, the torque generated by rolling friction can be modeled with the equation:

\[\mathbf{M}=-\mu_r\lvert\mathbf{F}_{cn}\rvert\boldsymbol{\hat{\omega}}\]

Stable aggregates


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