Thermophoresis of a Spherical Particle: Reassessment, Clarification, and New Analysis

Abstract

The thermophoretic force acting on a spherical particle depends on the Knudsen number and the particle-to-gas thermal conductivity ratio, and it can be estimated using various analytical and numerical methods for solving the Boltzmann equation. A substantial body of experimental data also exists. Nevertheless, the situation is not as clear as it might be and this article assesses the current predictive capabilities. First, some issues of nondimensionalization and data presentation are discussed. Then, the Grad 13-moment (G13) method of solution is examined in detail, and it is shown how the method generates a hierarchy of expressions for the thermophoretic force at low Knudsen number including all the well-known results. The non-Navier-Stokes-Fourier thermal stress and pressure-driven heat flux and their relation to the phenomenon of reversed thermophoresis are discussed. Theories of thermophoresis at arbitrary Knudsen number are then examined and it emerges that there are essentially only two theories extant. The available experimental measurements of the thermophoretic force and velocity are compared with these theories. Finally, it is shown that the G13 solution can be adapted to provide an interpolation formula for the transition regime, which gives a good approximation for practical calculations and is quantitatively very different from the commonly used prescription.

Effects of thermal stress slip on thermophoresis and photophoresis

“…1, where e z is the unit vector in the z direction, E N is taken to be positive, and (r,y,f) are the spherical coordinates originating from the particle center. The dimensionless group E 1 a=T 0 , where T 0 is the prescribed temperature at the particle center or the mean gas temperature in the vicinity of the particle, has been named as the Epstein number and found to be small in practice (Young, 2011), so all the physical properties of the particle and fluid are taken to be constant. It is assumed that the Knudsen number is moderately small so that the fluid motion is in the slip-flow regime and the Knudsen layer at the particle surface is thin in comparison with the particle radius.…”

Fundamental solutions to moment equations for the simulation of microscale gas flows

“…The analytical solution derived by Young (2011) is for the general case of a conducting sphere within both a uniform flow and a temperature gradient. In this section we consider the specific solution pertaining to a monatomic gas, purely diffusive surfaces, an infinitely conducting sphere (producing a uniform temperature) and a uniform far-field temperature.…”

Thermophoresis of an Aerosol Sphere with Chemical Reactions

“…It is assumed that the Knudsen number l/a is small (of the order 0.1) so that the fluid flow is in the slip-flow regime and the Knudsen layer at the particle surface is thin in comparison with the particle radius. The dimensionless group E ∞ a/T 0 has been named as the Epstein number and found to be small (and thus T p − T 0 )/T 0 << 1) in practice (Young 2011), so all physical properties are assumed constant. Gravitational effects, which can be considered separately and added directly due to the linearity of the problem, are ignored here.…”