Based on the importance of thermophoretic drift in transporting small particles across a turbulent thermal boundary layer, and the relatively small Brownian diffusivity of such particles, we present a simple asymptotic theory of particulate transport to aerodynamically smooth, solid surfaces cooled below, Te, the mainstream gas temperature. Numerical calculations based on a law-of-the-wall equilibrium velocity profile, and the assumption that the effective eddy diffusivities for mass, energy, and momentum diffusion are equal, are well-represented by
$−m˙p″≈ρeueωp,e•Sth•(αTLe)w$

$[(Te−Tw)/Tw]{1+[(Te−Tw)/Tw]•[0.07+0.93(αTLew]}$
where Sth is the local heat-transfer coefficient (Stanton number) and (αTLe)w is the ratio of the particle thermophoretic diffusivity to the gas mixture heat diffusivity. While currently being extended to cover particle size ranges for which (i) the Brownian diffusion sublayer is not negligible in thickness compared to the viscous sublayer, or (ii) eddy impaction sets in, the present theory provides a rational improvement over previous estimates, and explains several important features of the recent data of Nomura et al [1] on the fouling rate of internally air-cooled, gas turbine blades exposed to the products of combustion of Vanadium-containing residual fuel oil.
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