On the Flow of Bingham Plastic Slurries in Pipes and Between Parallel Plates
Authors: Hanks Richard W. 
Published: December 2007

Abstract
The method of Caldwell and Babbitt for determining Bingham plastic rheological constants from engineering pipe flow data has been erroneously used in many previous applications. A reanalysis of extensive pipe flow data from the literature is performed. Critical Reynolds numbers corresponding to the laminar-turbulent transition calculated from these data are found to agree with theoretically calculated values for the entire flow range studied. Similar agreement is found for the authors' own data for flow of 10 slurries between parallel plates. The Bingham plastic model is shown to be a reliable representation of the flow of non-Newtonian slurries provided it is properly applied. If the Hedstrom number He= pD ro/n greater than 10, then the linear approximation method of Caldwell and Babbitt is invalid for pipe flow data obtained in systems of ordinary engineering interest, and the complete nonlinear form of the pipe flow Bingham plastic equation must be used in determining rheological parameters.

Introduction
Many materials of engineering interest must be handled and transported as slurries or suspensions of insoluble particulate matter in a Newtonian liquid. These suspensions frequently exhibit non-Newtonian rheological behavior which is reasonably well described by the simple Bingham plastic rheological equation.
(1)
Using this particular equation to describe the laminar flow characteristics of slurries was discussed at length by Caldwell and Babbitt who considered the flow of various clay slurries and sewage sludges. The model has since been used by many others. More recently the problem of predicting the laminar turbulent transition Reynolds number for Bingham plastic fluids has been treated 8 for the case of flow in pipes. This paper points out errors which have existed in the analysis of Bingham plastic flow since the work of Caldwell and Babbitt and presents a reanalysis of the laminar turbulent transition calculation for Bingham plastic flow in pipes. In addition, new data obtained for flow of Bingham plastic slurries between parallel plates, both in laminar flow and in the laminar turbulent transition region, will be presented and compared with the theoretical analysis of laminar turbulent transition for flow between parallel plates.

THEORETICAL ANALYSIS: PIPE FLOW
The mathematical analysis of laminar flow of a Bingham plastic fluid leads to the following equation.
(2)
where q less than v greater than /R is a pseudo shear rate, rw is the wall shear stress xio=To/Tw, ro is the yield stress and n is the coefficient of rigidity or plastic viscosity from Eq. 1. Caldwell and Babbitt recognized that the quartic term in Eq. 2 was small compared to the other terms whenever xio less than less than 1, and that for large values of rw a plot of Eq. 2 becomes linear. The slope of such a plot with rw as ordinate is 4n and the intercept is (4/3)ro. This approximation is illustrated schematically in Fig. 1. The shaded region between the true curve and the straight-line approximation near the origin represents the contribution of the quartic term.