Coagulation and Fragmentation:  The Variation of Shear Rate and the Time Lag for Attainment of Steady State

The dynamic behavior and the attainment of steady state by a flocculating suspension in a stirred tank are evaluated using a population balance model. At long times, shear-induced coagulation and fragmentation reach a steady state, resulting in a particle size distribution (PSD) that is invariant (self-preserving) with respect to shear. The geometric standard deviations, σ g, of the self-preserving number or volume PSDs are 2.22 or 1.79, respectively, for the employed coagulation and fragmentation rates of flocculation. The time required to reach a steady-state PSD (time lag) is determined as a function of a dimensionless group comprised of the relative rates of coagulation and fragmentation. The effect of the omnipresent variable shear rate in stirred tanks during shear-induced flocculation is investigated through a sinusoidal function of the spatially averaged velocity gradient. Increasing the amplitude of the shear rate fluctuation decreases the steady-state mass mean floc size, the maximum σ g, and the time lag for attainment of steady state. The asymptotic (self-preserving) σ g is not affected by the shear rate amplitude provided that >99% of the primary particles have grown to larger sizes.