Developed turbulent transport in ducts for large prandtl or schmidt numbers



The problem of developed turbulent heat or mass transfer in a duct is considered for the limit of large σ (Prandtl or Schmidt number). The limiting results depend on the behavior of the eddy diffusivity near the solid surface. Since there is a question about whether this variation begins with ϵ ∝ y+3 + … or ϵ ∝ y+4 + … for y+ near zero, both possibilities are considered. In each case the first three terms of the asymptotic expansion for σ → ∞ are obtained. The first term of the asymptotic expansion agrees with limiting results derived earlier, while the correction terms indicate the errors associated with earlier simplifying assumptions.

By proper scalling, it is demonstrated that in the limit of σ → ∞ the results are independent of geometry and boundary conditions for situations involving parallel plates, circular tubes and concentric annuli with either constant surface heat flux or temperature. The correction terms to the σ → ∞ asymptote can be significant, although the effect of Reynolds number on the correction terms is very small.

A comparison between a typical numerical integration and the asymptotic formula shows excellent agreement. The asymptotic formulae are used to correlate large Schmidt number mass transfer data.