There are many applications, including time-resolved LII, in which it is crucial to model heat conduction accurately for nano-sized particles. Heat conduction from nanoparticles occurs in one of three regimes depending on the Knudsen number,
where λ(Tg) is the molecular mean free path in the gas and a is the particle radius. At large Kn, heat conduction occurs in the free molecular regime, in which gas molecules travel between the nanoparticle surface and the equilibrium gas without undergoing intermolecular collisions. The other limiting case is the continuum regime at small Kn, in which conduction is governed by the continuum equations and gas thermal conductivity.
Intermediate Knudsen numbers correspond to the transition regime, which bridges the free-molecular and continuum regimes. These conditions prevail in soot-laden aerosols at high pressures, such as in gas turbine combustors. Unlike the other two regimes, the equations governing heat conduction in the transition regime are analytically insoluble, so numerous schemes have been developed that interpolate between the results of the free-molecular and continuum equations for a particular Kn.
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1. Steady state heat conduction
One of the most widely used interpolation schemes is Fuch’s boundary sphere method1,2, in which the gas is divided into a free molecular regime (FMR) shell (Knudsen layer) surrounding the particle of thickness Δ, in turn surrounded by a continuum gas that extends to the equilibrium gas temperature, Tg. (A good biography of N. A. Fuchs is available here.) The FMR shell has a thickness approximately equal to the mean free path of the gas at the boundary, λ(TΔ). The analysis proceeds by setting the FMR heat transfer between the particle surface at Tp and the boundary sphere at TΔ equal to the continuum regime heat conduction between TΔ and Tg, and solving for the unknown TΔ.
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While many implementations simply set Δ = λ(TΔ), Filippov and Rosner3 use a simplified version of a statistical treatment proposed by Wright4,
which accounts for the curvature of the nanoparticle surface and the cosine distribution of diffusely scattered molecules. In our study5, we explore the origin of this equation and demonstrate its affect on transition-regime conduction in TiRe-LII. We also use Direct Simulation Monte Carlo (DSMC) to visualize the molecular gas dynamics underlying transition regime conduction. In this approach, thousands of simulated gas molecules are tracked through collisions with molecules and other surfaces. While other studies6 have used DSMC to calculate heat transfer rates, we use this technique to calculate probability densities of the distances incident molecules have traveled from their last intermolecular collision, δi, the distance scattered molecules travel to their next intermolecular collision, δo, and the combined distance, li+o. These results highlight the fact that the collisionless Knudsen layer is an idealization, and the boundary sphere is not, in fact, collisionless. We also show how the influence of particle curvature on Knudsen layer changes with Knudsen number.
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2. Transient heat conduction
While Fuchs's boundary sphere method provides an accurate estimation of steady state heat conduction, in TiRe-LII the nanoparticle temperature changes abruptly as it is heated with the laser pulse and then cools in the surrounding gas. There has been some speculation that transient gas dynamics phenomena could be responsible for some irregularities in observed nanoparticle cooling rates. Specifically:
To investigate these phenomena, Farzan Memarian pioneered the use of transient DSMC to model heat conduction in laser induced incandescence experiments. Transient DSMC works by carrying out a sequence of independent time-resolved DSMC simulations, and then averaging the ensemble results to reduce variance. Contrary to speculation in the literature, Farzan's simulations7 showed no shockwave formation, even at extremely high fluences. There is an enhanced cooling rate due to the nonstationary gas, however, and, a considerable amount of sublimed material will return to the nanoparticle surface due to collisions with gas molecules and other sublimed nanoclusters.