Dissipation of Energy due to Solid Particles Suspended in a Viscous Liquid

Abstract

A general method is developed for calculating the additional mechanical energy dissipation incurred by the presence of solid particles suspended in an incompressible viscous fluid in laminar flow. The development is applicable to flow in any apparatus within which the steady motion of a homogeneous fluid can be described by omitting the inertial terms in the Navier-Stokes equations or in which the motion occurs in concentric circles. The validity of the results obtained is limited to situations in which the particle dimensions are small compared to those of the apparatus in which the motion is occurring. The technique, based largely upon a reciprocal theorem for the inertialess flow of viscous fluids, is employed to calculate the permanent pressure drop accompanying the passage of liquid through a bed of fluidized solids. When the particles are uniformly distributed throughout the bed, this pressure diminution is equal to the bed weight (corrected for buoyancy) per unit area of duct—in agreement with experimental data on systems in a state of particulate fluidization. Calculations are also presented for the permanent pressure arising from the passage of fluid past an immobilized spherical particle within a cylindrical duct. Einstein's formula for the apparent viscosity of a dilute suspension of spherical particles is also deduced. By means of a new definition of suspension viscosity the heretofore different points of view advanced by Einstein, Jeffery, and Burgers are seen to be consistent with one another. Einstein's result is shown to be independent of the type of viscometer employed provided that the motion of a homogeneous fluid within the device fulfills the previous criteria. In the case of nonspherical particles it is demonstrated that the constant in Einstein's viscosity equation (which has a value of 2.5 for spheres) can never be less than unity, whatever the shape of the rigid particles or their orientation with respect to the axes of the apparatus.