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2/3, corresponds to the BGK theory. D= v T D1 means infinite mean free path
(free molecule regime). One clearly sees that the agreement of the theoretical and
the experimental data is fairly good.
3.7
Heat Transfer
A new flux-matching theory is formulated and applied to the study of heat transfer
from (or to) aerosol particles whose size is comparable to the molecular mean
free path in the carrier gas. The collisionless kinetic equation is solved, and the
temperature profile in the free molecule zone (at distances less than the molecular
mean free path) is found. This profile is then matched to that derived from the
solution of the thermoconductivity equation, which describes heat transport outside
the free molecule zone. The final output of this section is the expression for particle
heat transfer efficiency, which contains its free molecule value (also derived here),
the thermoconductivity of the carrier gas, and the thermal velocity of the carrier gas
molecules.
3.7.1
Introduction
In this section we outline the derivation of the formula for the particle heat transfer
efficiency .a/ defined as the proportionality coefficient between the temperature
difference of the particle and the carrier gas and the energy flux from (to) the
particle. This formula is valid throughout the whole interval of particle sizes and
contains no free parameters . This problem received attention at the very beginning
of this century by Filippov and Rosner ( 2000 ), who applied a Monte Carlo method
for studying heat transfer from the particle to the surrounding gas. Some earlier
references can be found in this paper. This problem was studied experimentally: the
authors (Winkler et al. 2004 , 2006 ) compared their results to a modification of the
Fuchs-Sutugin formula and found satisfactory agreement between theoretical and
experimental data (Lee et al. 1985 ).
We again introduce a limiting sphere outside of which the temperature profile
can be described by the thermoconductivity equation. Inside the limiting sphere we
solve the collisionless Boltzmann equation subject to a given boundary condition
at the particle surface and put additional conditions: (1) total heat flux does not
change in crossing the border of the limiting sphere, (2) carrier gas density is
a continuous function of the radial coordinate, and (3) its radial derivative at
the surface of the limiting sphere coincides with that found from the solution of
the thermoconductivity equation. These three conditions determine three matching
parameters: heat flux, matching distance, and temperature at the matching distance.
Below we demonstrate the details of the derivation of the expression for the heat
transfer efficiency.
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