Turbulent Flow of Bingham Plastic Fluids in Pipes and Annuli
The frictional pressure loss associated with the turbulent flow of a Bingham plastic fluid is affected primarily by density and plastic viscosity. Whilst the yield point of the fluid affects the frictional pressure loss in laminar flow, in fully turbulent flow the yield point is no longer a highly significant parameter. It has been found empirically that the frictional pressure loss associated with the turbulent flow of a Bingham plastic fluid can be predicted using the equations developed for Newtonian fluids. The plastic viscosity is simply substituted for the Newtonian viscosity. This substitution can also be made in the Reynolds number used in the Colebrook function defined by Equation 14 or in the simplified turbulent flow equation given by Equation 18.
These equations are however only appropriate when the flow is in turbulence. There must therefore be an equation which can be used to determine the point at which the flow enters turbulence. The obvious solution is to use a modified form of the Reynolds number. There are two problems associated with using the Reynolds number criterion. The first is that this criterion was designed for pipe flow and an equivalent diameter must be used if the fluid is flowing in an annulus. The second problem is that non-Newtonian fluids such as Bingham Plastic fluids do not have a single parameter representation of viscosity. In the case of Bingham Plastic fluids a representative apparent viscosity is developed. The apparent viscosity most often used is obtained by comparing the laminar flow equations for Newtonian and Bingham plastic fluids. For example, combining the pipe flow equation for the Newtonian and Bingham plastic model yields an equation for ^ the apparent Newtonian viscosity:
6.66xYd
Це= Цр +
Equation 23 Apparent Newtonian Viscosity for Bingham Fluid in Pipes
A similar comparison of the laminar flow equations for Newtonian and Bingham fluids in an annulus yields:
5X(d2-di)
^e= ^ + V———-
Equation 24 Apparent Newtonian Viscosity for Bingham Fluid in Annuli
These apparent viscosities can be used in place of the Newtonian viscosity in the Reynolds number formula. As in the case of Newtonian fluids, a Reynolds number greater than 2,100 is taken as an indication that the flow pattern is turbulent.