# Heat Conduction Theory

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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

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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