Theoretical Analysis of Turbulent Mass Transfer with Rotating Cylinders

Abstract

Dissipation theory provides a new approach to understanding turbulent shear flow. The universal velocity profile and the law of the wall rely on the stress at a somewhat distant wall to provide a correlating parameter, but they are not adequately universal and general, as observed with rotating cylinder flow. Instead, the dissipation theorem expresses the governing physics by means of local laws describing interactions of statistical quantities of the flow. More work is needed to refine the method, but it can predict fluid flow and mass transfer with perhaps only one or two empirical parameters. The data of Eisenberg and of Mohr provide an excellent basis for examining mass transfer in the system where the inner cylinder rotates within a larger stationary cylinder. Neither author fully exploited the data, and this work correlates the data more carefully with respect to the dependence of the Stanton number on the Reynolds number, the Schmidt number, and the geometric ratio κ of the inner cylinder radius to that of the outer cylinder. Three methods are deployed in this endeavor. Simple empiricism involves fitting the data according to St = A(κReg)nScp, where the parameters A, n, and p are functions of κ. Enlightened empiricism expands on the traditional method of fitting the eddy viscosity as a function of the distance from the wall in the form of y+ = (y/ν)(τ0/ρ)0.5, where τ0 is the shear stress at the solid wall. This method does not work well in a predictive manner for rotating systems or systems where the flow is in the direction of the curvature of the surface (as with Goertler vortices). Dissipation theory relates the local dissipation to its generation, transport, and decay, involving a parameter for the decay constant of the dissipation. The electrochemical data of Eisenberg (with 1-κ values from 0.1723 to 0.871) seem to be in the regime of wide gaps, while Mohr focuses on thin gaps (1-κ ranging from 0.0172 to 0.0839) to explore the failure of Eisenberg's data to show an expected behavior where thin gaps blend into turbulent Couette flow. The dissipation theorem should provide a new method of understanding turbulent shear flow and a way of predicting the effect of varying the radius ratio.