On Reynolds Number and Transition Point of Flow of True Plastic Fluids in Circular Pipe

Abstract

In the previous paper, the author presented a correlation between a newly definad Reynolds number and friction factor. Using this experimental correlation it was confirmed that the pressure drop of flow of Bingham fluid could be treated with Fanning equation as well as that of Newtonian fluid could.In this paper, the meaning of Reynolds number is predicated by theoretical approaches.As Reynolds number is defined as the ratio of shearing stress to inertia force, We put the thickness of boundary layer as the representative length, (ua2/D) as the measure of the inertia, and the shearing stress at the inner wall of the pipe as the measure of the shearing stress.Then, for Newtonian fluid, we have(1)This Reynolds number (Re') gives 1/16 times smaller value compared with Reynolds numher commonly used (Re).For Bingham fluid, (2)where μa=η/4aα which is an apparent viscosity.The Rèynolds number show by equation (2) may be called the general form o Reynolds number available for both Bingham and Newtonian fluid flows.The pressure drop of laminar flow of Bingham fluid where the apparent viscosity in μa, the average velocity ua, and the diameter of pipe D, will be regarded identical with that of Newtonian fluid flow where the viscosity is (1-a)μa, the average velocity (1-a)ua, and the diameter of pipe (1-a)D respectively. Such being the case, the friction factor of Newtonian flowmay be regarded as the practical expression of that of Bingham fluid flow.The propriety and utility of these considerations are confirmed by experimental data obtained through the test of flow of clay suspension in 1/2″ iron pipe. It is illustrated in Fig. 2.The value of the critical Reynolds number Rec is given as a function of the relative plug radius (ac=2rp/D) at the very transition point where the state of flow changes from laminar to turbulence (Fig. 2). When the relative plug radius becomes larger or the concentration of suspension heavier, Rec tends to decrease till it reaches a definite value. This definite value will be given by the point in Fig. 2 where the extension of the curve in turblence region intersects the curve in laminer flow region.