Turbulent Pressure Loss of Yield-fluids In Pipes
Authors: Desouky S.E.D.M. , Al-Saddique M.A. 
Published: March 2010

Abstract
A friction factor equation for yield fluids (yield-pseudoplastic, yield-dilatant, Bingham) was derived from flow equation under turbulent pipe flow conditions. The rheological characteristics of the yield fluid were described by Herschel and Bulkley model.
To verify the validity of the proposed approach a comparison was made between published experimental friction factors and those calculated from the developed equation and Torrance correlation. Results show that the average absolute relative error between the experimental data and calculated data ranges from 2.84 to 5.05% for the developed equation and from 6.97 to 13.68% for Torrance correlation. A comparison was also made between the field measurements of the pressure loss in Mobil pipeline and those based on the proposed friction factor, the results showed good agreement. The evaluation emphasizes the accuracy of the developed equation in comparison to earlier work.

Introduction
All over the world, there are several pipelines transporting yield-fluids under turbulent flow conditions. Design of such pipelines requires an adequate relationship to determine the pressure loss. The yield-fluids include yield-pseudoplastic, yield-dilatant and Bingham fluids. The rheological characteristics of these fluids can be described by Herschel and Bulkley(1) model.
Equation 1 (available in full paper)
where:
n &gt; 1.0 for yield-pseudoplastic fluids
n &lt; 1.0 for yield-dilatant fluids
n = 1.0 for Bingham fluids
In view of the correlations developed for predicting the friction factor in a pipeline handling a yield fluid, there are two groups: Bingham correlations and yield-pseudoplastic and yield-dilatant correlations. For Bingham fluids, Workers(2–5) had applied the Prandtl mixing length concept to the turbulent flow in smooth tubes to derive a friction factor formula. Hanks and Dadia(6) extended the analysis of Hanks(7) for the turbulent flow of Newtonian fluids to the case of Bingham plastics. Thomas(8) presented Bingham fluid data on the turbulent pipe flow and correlated his data with Blasius equation. For yield-pseudoplastic and yield-dilatant fluids, there exists only one correlation developed by Torrance(9). He extended Clapp's treatment to the power law yield-pseudoplastic fluids and deduced the following correlation.
Equation 2 (available in full paper)
A comparison between the friction factor calculated from the above equation and the published experimental data(5, 7, 10–14) show that, Equation (2) can be used to predict the friction factor with average absolute relative error ranging from 6.97 to 13.68%. Therefore, it is necessary to develop a more accurate equation for pipeline design calculations.
In the present analysis, an equation relating the friction factor to Reynolds number and yield-fluid characteristics is developed. The validity of the equation is confirmed with published experimental data.

Theoretical Analysis
The flow equation of an incompressible fluid under steady state, isothermal and fully developed conditions is given by(15).
Equation 3–18 (available in full paper)
Equation (17) represents the velocity distribution at the turbulent core. The maximum velocity (Umax) is obtained by equating the velocities at the interface between the turbulent core and the laminar sublayer, that is
Equation 19–23 (available in full paper)

Friction Factor
To obtain an equation for friction factor, the following definitions are used,
Equation 24–28 (availabl