Numerical Prediction of the Pipeline Flow Characteristics of Thixotropic Liquids
Authors: Ritter R.A. , Batycky J.P. 
Published: December 2007

Abstract
A numerical technique has been developed to permit establishing the pressure gradient associated with laminar flow of thixotropic liquids through long pipelines. For this purpose the pipeline is divided into a number of radial and longitudinal increments within which rheological properties of the fluid may be considered as constant at any time. Then, provided only that the fluid flow curve is defined at every duration of shear, it is possible to predict the instantaneous pressure gradient at any cross-section along the pipeline for each desired flow rate and pipe size. The technique consists of an iterative integration of shear rate to establish the appropriate value of the wall shear stress at each location. Consistency of fluid in the increment is determined by the flow history of that increment, while the radial flow) associated with variations in velocity profile is accounted for by adjusting the width and radial position of the increment. A number of pressure profiles, computed at each of several flow rates, provide a convenient basis for pipeline design and pump selection.

Introduction
In recent years, considerable attention has been given to predicting pressure drop associated with the isothermal laminar flow of time-independent non-Newtonian fluids in pipes and annuli. The approach generally has been m develop analytical relationships between flow rate and pressure drop based on simple constitutive models which hopefully provide an approximate description of the rheological properties of the fluid. Analytical solutions are highly desirable since the influence of all pertinent parameters can be readily determined. Unfortunately, however, this approach is restricted to simple flow geometries and frequently leads to erroneous results due to inadequacies in the model. In certain cases a solution may be obtained through applying appropriate numerical techniques For example, a digital computer program is available for predicting the velocity profile and pressure drop encountered by any Newtonian or time-independent non-Newtonian fluid flowing under laminar conditions in a cylindrical pipe or annulus. In this paper the consistency behavior of the fluid need only be described in terms of basic rheological data. Analyzing flow systems involving fluids with time-dependent rheological characteristics is considerably more complicated since substantial changes in consistency may occur because of sustained shear action. This sensitivity to shear frequently persists for several hours. Consequently, variations in pressure drop and/or flow rate resulting from the aging process and addition of unsheared or partially sheared fluid to the system must be considered for purposes of pipeline design. This paper outlines a numerical method for predicting the transient and steady-state laminar flow behavior of a thixotropic liquid in a pipeline of arbitrary length (i.e., at a specified constant flow rate, the instantaneous pressure gradient may be determined at any time after start up and at any location along the pipeline). Several such pressure gradient profiles computed at several flow rates, may be combined to produce a complete portrait of the system response. This flow portrait provides a reasonable basis for pipeline design and for selecting a suitable pump characteristic.

TIME-DEPENDENT RHEOLOGICAL BEHAVIOR
The most familiar time-dependent rheological properties are those exhibited by thixotropic liquids. Many of these materials, particularly thixotropic crude oils, generally display an apparent yield stress in that a finite pressure gradient is required to initiate flow. Then, under the influence of sustained shear at a constant shear rate, the consistency systematically decreases to some final limiting value.