Heat Conduction Experiment

\subsection{Heat conduction}

The heat conduction mechanism represents the energy loss at the particle surface due to collisions with the surrounding gas molecules. There is no simple theory to explain the particle cooling mechanism. During the laser pulse, the particle achieves the peak temperature and starts to cool down after the laser pulse. During this time, the particles emit thermal signal. The cooling of the particles is characterized by a decay function. The decay of LII signal provides information about the primary particle size and size distribution \cite{Dankers:04, DankersSchraml,rousselle,Will:98}. The decay time of LII signal can be described by heat conduction models. The most popular heat-conduction soot particle model is

This robust method has the advantage to be used also in high-pressure situations because the method was derived based on transition regime. In order to approximate the soot particles to the real case, Liu et al extended the McCoy and Cha model to be used in aggregates \cite{liu2006}. Due to a fractal geometry, it would not be logical to think that all particles which compose the aggregate will store the same amount of energy. Liu proposed the idea to utilize an equivalent sphere for the complete aggregate which has the same effective area of the aggregate balancing the physical process. The effective heat conduction diameter ($D_{HC}$) of the equivalent sphere is defined by direct simulation Monte Carlo (DSMC) \cite{Liu_2006_lowfluence} as

The McCoy and Cha model version of Liu can be written as

Liu presented an extensive discussion about the McCoy and Cha method \cite{liu2006} and its limitations. A more sophisticated model is presented by Liu \textit{et al} \cite{liu2006}, so called Fuchs interpolation method, firstly presented by Fuchs \cite{fuchs_top} and posteriorly by Fillipov and Rosner \cite{Filippov2000127}. The Fuchs interpolation method couples the equivalent sphere with the limiting sphere which is displaced by a distance $\delta$. Figure \ref{fuchs} illustrates the Fuchs interpolation method applied by Liu and others \cite{fuchs, liu2006,Filippov2000127, Bladh}.

Both temperatures, $T_{\delta}$ and gas temperature $T_g$ should be defined as constants. This method uses the radius of the equivalent heat conduction sphere ($a_{HC}$) instead of the single particle diameter. Inside the limiting sphere, the free molecular regime is applied. Outside the limiting sphere, the continuum regime describes the heat-conduction. The heat-conduction for free molecular regime is modeled as \cite{fuchs_top, Filippov2000127, liu2006}

in which $P$ is environment pressure and the heat capacity ratio $ \gamma ^* $ is defined according