Geoscience Reference

In-Depth Information

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.