Scaling Pumping Requirements - Inelastic Fluids Turbulent Flow and Inelastic/Elastic Fluids In Laminar Flow
Authors: Savins J.G. , Burdyn R.F. , Wallick G.C. 
Published: December 2007

Abstract
Procedures are described for scaling up turbulent friction pressure drops of inelastic fluids and laminar friction pressure drops of inelastic/elastic fluids in circular ducts. The laminar scale-up calculation method does not require the use of arbitrarily defined flow models, and can be performed directly from laminar capillary viscometry data, or by means of a numerical solution of the integral equation for Poiseuille flow, inputing true shear rate-shearing stress data from other simple shearing experiments, e.g. Couette and plane Poiseuille flows. The turbulent scale up procedure, based on the Dodge-Metzner correlation for inelastic fluids, requires evaluating the characteristic rheological parameters at the existing wall stress under laminar flow conditions. Since the wall stress is the quantity sought in a pipe flow scale-up problem, a trial-and-error solution is indicated, and therefore a calculation method has been developed and programmed for a medium-size magnetic-drum-memory digital computer. Laminar flow data from other sources may also be adapted to this turbulent scale-up method for inelastic fluids. Comparisons between predicted and experimentally observed friction losses for generalized Reynolds numbers ranging from 3 - 32,000 indicate an error band width of less than 10 per cent in circular tubes ranging in size from 0.5 to 1.89 in, in diameter.

Introduction
Considerable effort has been devoted to the problem of establishing an accurate procedure for predicting the laminar and turbulent flow behavior of non-Newtonian materials in pipes from data obtained with small scale laboratory or pilot plant equipment, as evidenced by the variety of reviews on the subject. Although there does not appear to be a universally accepted scale-up method for laminar flow. the various methods currently employed can be divided into two principal categories depending upon whether a functional relationship between shear rate and shearing stress is assumed, or whether a generalized correlation is adopted. The methods of Hedstrom and of Weltmann are typical of the first category, while the generalized correlation of Metzner and Reed illustrates the second category. Under appropriate conditions either approach yields a reasonably satisfactory description of laminar flow behavior.
In contrast, the prediction of non-Newtonian transitional and turbulent friction losses is much more complex. For example, certain types of materials have been observed to sustain laminar flow at much higher flow rates than Newtonian liquids. When turbulence does develop, the level of agitation is damped to such an extent that the friction factor is lower than that of the original low viscosity Newtonian medium. Extremely broad transition regions and "diameter effects" have also been noted in the turbulent flow of certain chemically complex materials. Aside from the latter complications, which are attributed in part to viscoelasticity, the existence of a variety of scale-up procedures for predicting transitional and turbulent friction losses is due principally to the choice of viscosity parameter used in the definition of the Reynolds number. In contrast, Dodge and Metzner have demonstrated that the friction factor-Reynolds number relationship for inelastic fluids can be satisfactorily generalized provided the fluid property parameters are evaluated at the existing wall shearing stress. However, the quantity usually sought in a pipe flow scale-up problem is the wall stress, and hence a trial-and-error procedure is required when dealing with non-power law materials.In this report we discuss two calculation methods:for determining the wall stress, and hence the friction pressure drop, for inelastic fluids under conditions of turbulent flow by means of an iterative procedure, andfor predicting laminar friction pressure drops without recourse to arbitrarily defined flow models.
SPEJ
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