Friction Factors For Yielldless Fluids In. Turbulent Pipe Flow
Authors: Hemeida Adel M. 
Published: March 2010

Abstract
A new equation relating the friction factor to the generalized Reynolds number and rheological characteristics of non-Newtonian yieldless fluids was developed under turbulent pipe flow conditions. The rheological behaviour of yieldless fluids was described by the generalized approach developed by Metzner and Reed. Statistical error analysis were used to determine the accuracy of the developed equation.
To verify the validity of tire present analysis, a comparison was made between published experimental friction factors and those cal culated from the developed equation, Dodge and Metzner, and Desouky and El-Emam correlations. A comparison was also made between field measurements and calculated ones, The results showed that the developed equation provided a more accurate estimation of friction factor.

Introduction
Transportation of non-Newtonian yieldless fluids in a pipeline under turbulent conditions requires a precise method to determine the pressure loss. The yield less fluids include pseudoplastic and dilatant fluids. In view of several correlations developed for calculating pressure loss in a pipeline handling pseudoplastic or dilatant fluids, there are two approaches: power-law and generalized correlations. Shaver and Merrill(1), Tomita(2), Ciapp(2), Torrance(2l, Bogue and Metzner(3), Hall (5), Murthy and Zandi(5), and Szilas, Bobok and Navratil(6) had developed correlations applied for power-law fluids. The power-law model is one of rhe twenty blown rheological models of pseudoplastic and dilatant fluids(7). As for the second approach, odge and Metzner(H) developed a generalized correlation for time-independent fluids. The data used to determine the constants of their correlation were from power-law and viscoelastic fluids(2). Lord, Hulsey and Melron(9) developed a generalized approach for ime-independent viscous or viscoelastic fluids. Hall(5) drew attention to the actual variation in the exponent of the correlation proposed by Lord, Hulsey and Melton and cautioned against scale-up over too extensive a range. A recent generalized correlation and short method for all types of pseudoplastic were developed by Desouky and El-Emam(7), and Desouky(10). These authors indicated excellent agreement between calculated and experimental friction factors over values of 5000 to 60 000.
In the present analysis, a generalized equation for friction factors of pseudoplastic and dilatant fluids were derived under turbulent pipe flow conditions. The twenty, known rheological models of pseudoplastic and dilatent fluids were expressed through the use of the generalized approach developed by Metzner and Reed(11).

Equation Development
An equation relating the friction factor to the generalized Reynolds number and characteristics of pseudoplastic and dilatant fluids was derived. The maximum velocity was determined by equating the point velocities on both sides of the bounding surface of the turbulent core and the laminar sublayer.
For an incompressible fluid under steady stare, isothermal and fully developed conditions, the flow equation in Z-direction is given by:
Equation (1) Available In Full Paper.
Integrating equation (1) with respect to r, the following equation is obtained
Equation (2) Available In Full Paper.
The relationship between laminar shear stress and shear rate for pseudoplastic and dilatant fluids(2) is given by:
Equation (3) Available In Full Paper.
The apparent viscosity (μapp) can b