Closed-form solutions for batch settling height from model settling flux functions

Abstract

A novel approach for relating suspension batch settling data (settling height vs time) to model settling flux functions is presented. The approach exploits a new closed-form solution relating settling height and time for a very simple functional form of the settling flux. The simple form in question employs a so-called hindered settling factor (a key material property in suspension rheology) that is taken to be a straight-line function of the solids fraction. The closed-form solution for settling height and time has a parametric dependence on the slope and intercept of the above-mentioned (straight line) hindered settling property: thus a functional relationship between batch settling height and suspension material property parameters is established. Moreover by adjusting the slope and intercept parameters, the closed-form solutions for settling height vs time can be matched to batch settling experimental data, and thereby the settling flux can be directly obtained. Unlike classical approaches for determining settling flux functions from batch settling data (i.e. Kynch theory), nowhere does the new approach require data for settling velocity: this gives it an in-built robustness to experimental noise compared to any approach that obtains experimental settling velocities via finite differences of settling heights, since such velocities tend to be far more noisy than the settling heights themselves. In a typical physical system, a straight-line relationship between hindered settling factor and solids volume fraction will only be a reasonable approximation over a very restricted domain of solids fraction. However, over a wider domain, it is possible to approximate the hindered settling factor vs solids fraction dependence via a sequence of straight-line relations, each taken over a narrow interval of solids fraction. A functional form for settling flux is thereby obtained, useful for suspension dewatering calculations and engineering equipment design for suspension/sludge processing. The novel approach for determining settling flux has been applied both to experimental and synthetic batch settling data, and has performed robustly.