Problem 3.21. Microscopic particles that are suspended in gas are driven from high temperature to low

temperature regions. This process is called thermophoresis. In the absence of other particle diffusive

transport mechanisms, the slip velocity (velocity between gas and particle) caused by thermophoresis can

be found from [Talbot et al., 1980; Friedlander, 2000]:

UTP

=

mol

k

(17+ C,(2)

k₁

P

-2C,V

= V

k

(1+6C Kn) 1+2 +4C, Kn

where d is the particle diameter, kp is the thermal conductivity of the particle, all properties without a

subscript represent the gas, and

C, 1.17; C = 2.18; C = 1.14.

Kn₁ = 2/d

πΜ

2R T

u

VT

T

+C, (2Kn) C-

1/2

P

(Knudsen number)

(1.5.10) (Gas molecular mean free path)

C=1+2Kn [1.257 +0.4cxp(-0.55/ Kn)] (Cunningham correction factor)

Consider a flat and horizontal surface that is at a temperature of 398 K, and is cooled by a parallel air

flow. The air has a pressure of 0.1 bar and a temperature of 253 K, and flows with a far-field velocity of

20 m/s with respect to the surface. At a distance of 0.5 m downstream from the leading edge of the

surface, calculate the thermophoretic velocity in the vertical (y) direction of a metallic spherical particle

that is 0.5 µm in diameter and has the thermophysical properties of cupper, when it is 1 mm away from

the surface.

How does this velocity compare with the fluid velocity in the y direction?