Technical note: The Graetz problem at small Peclet number and its application to sizing condensation particle counters

Abstract

We extend to small Peclet number Pe the familiar high-Pe solution to the Graetz problem of heat/mass transfer in a tube with parabolic flow with a wall temperature discontinuity at an axial position z = 0. Retaining diffusion in the axial direction results only in slight shifts in the eigenfunctions describing the problem. This enables the characterization of the saturation field S by separation of variables, which we implement for Pe = 0, 0.1, 0.25. Calculations based on discretization of the transport partial differential equations are also used to analyze the problem at Pe up to 5. As Pe tends to 0, the heat and mass transfer problems are both described by the same dimensionless function θ(z,r) of the axial and radial coordinates z and r. S then depends on the single quantity θ, and its maximum value Smax is exactly the same along all streamlines. This would result in sharp activation curves in a hypothetical low-Pe condensation particle counter, similarly as in the special circumstance when the ratio Le between heat and mass diffusivities is unity. Calculations are run to determine how the radial uniformity of Smax is degraded at increasing Pe. When accounting for axial diffusion, Smax at the wall is independent of Pe or Le. The commonly considered high Pe limit fails to satisfy this requirement, and therefore predicts an incorrect Smax at distances from the wall discontinuity small compared to Pe−1/2.