 # Heat Conduction Theory

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Heat conduction mechanism represents the energy loss at the particle surface due to the collisions with the surrounding gas molecules. There is no simple theory to explain the cooling mechanism of the particles. During the laser pulse, the particle achieves the peak temperature and emit thermal signal. After the laser pulse duration, the particle starts to cool down. 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. A heat-conduction model for soot particle conduction cooling is known as McCoy and Cha …show more content…
The Knudsen number, $K_n$, correlates the ratio of the mean free path ($\lambda_{mfp}$) of the molecular gas and a characteristic particle size dimension ($L_c$), for instance the particle radius. This dimensionless number serves as an evaluation criterion and defines three distinct gas regimes: free molecular regime $Kn > 10$, continuum regime $Kn << 1$ and transition regime.
The Knudsen number is defined
The Planck's radiation law is written as a function of the temperature and wavelength

According to Kirchhoff's law, a black body in thermal equilibrium has an emissivity of $\epsilon = 1.0$ and the energy is radiated isotropically, independent of radiation direction. In other words, the absorption is equal to the emissivity \cite{Huffman}. Soot has a lower emissivity independent of the frequency and often it is referred to as a gray body. The emissivity used in this work is defined as the Rayleigh limit according to

The integration of Eq. \ref{planck_eq} does not have an analytical solution, but has only numerical solution. In this work, $\gamma$ and $\zeta$ functions were used to solve the radiation equation \cite{arfken, erwin}. Bladh solves the radiation equation by applying $\gamma$ and $\zeta$ function according to \cite{Bladh}

and from numerical evaluation $\zeta (5)= 1.0369$ and $\Gamma (5) = 24$. Reimann \cite{reimann} chose the same way to solve the radiation equation using the Rayleigh condition. This elegant solution can be seen in more detail in \cite{Liu2005301}. The Reimann simplification is presented

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