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Research Papers: Fundamental Issues and Canonical Flows

# Friction Coefficients for Bingham and Power-Law Fluids in Abrupt Contractions and Expansions

[+] Author and Article Information
Sergio L. D. Kfuri

LFTC-PPGEM,
Department of Mechanical Engineering,
Vitória,
Espírito Santo 29075-910, Brazil
e-mail: skfuri@hotmail.com

Edson J. Soares

Professor
LFTC-PPGEM,
Department of Mechanical Engineering,
Vitória,
Espírito Santo 29075-910, Brazil
e-mail: edson.soares@ufes.br

Roney L. Thompson

Professor
Department of Mechanical Engineering,
COPPE,
Universidade Federal do Rio de Janeiro,
Centro de Tecnologia,
Ilha do Fundão,
Rio de Janeiro 21941-450, Brazil
e-mail: rthompson@mecanica.coppe.ufrj.br

Renato N. Siqueira

Professor
LPMF-Department Mechanical Engineering,
Instituto Federal de Educação,
Ciência e Tecnologia do Espírito Santo,
São Mateus,
Espírito Santo 29932-540, Brazil
e-mail: renatons@ifes.edu.br

1Corresponding author.

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received January 12, 2016; final manuscript received July 19, 2016; published online November 3, 2016. Assoc. Editor: Mhamed Boutaous.

J. Fluids Eng 139(2), 021203 (Nov 03, 2016) (8 pages) Paper No: FE-16-1035; doi: 10.1115/1.4034521 History: Received January 12, 2016; Revised July 19, 2016

## Abstract

Industrial processes with non-Newtonian fluids are common in many segments such as petroleum, cosmetic, and food industries. Slurries, emulsions, and gas–liquid dispersions are some examples with industrial relevance. When a fluid flows in a pipe system, pressure losses are always present. For Newtonian fluids, a quite reasonable understanding of this phenomenon was already achieved and is available in the literature. The same cannot be stated for non-Newtonian fluids owing to their complex characteristics, such as pseudoplasticity, viscoplasticity, elasticity, and thixotropy. The understanding of the influence of these characteristics on flow behavior is very important in order to design efficient pipeline systems. The design of such systems requires the estimation of the pressure drop due to friction effects. However, there are few works regarding friction losses for non-Newtonian fluids in pipeline systems, making this task a difficult one. In this study, two classes of fluids are investigated and compared with the Newtonian results. The first category of fluids are the ones that exhibits pseudoplastic behavior and can be modeled as a power-law fluid, and the second category are the ones that possesses a yield stress and can be modeled as a Bingham fluid. Polyflow was used to compute the friction losses in both abrupt contractions and expansions laminar flow conditions. It shows that for the expansion cases, the aspect ratio affects more the local friction coefficients than for the contraction cases. The influence of the power index n on local friction losses is similar for both cases, abrupt contractions and abrupt expansions. At low Reynolds numbers, dilatant fluids present the lowest values of the friction coefficient, K, independent of geometry. At high Reynolds numbers, a reversal of the curves occurs, and the dilatant fluid presents larger values of K coefficient. For the cases investigated, there is also a Reynolds number in which all the curves exhibit the same value of K for any value of the power-law index. The effect of $τy′$ shows a different behavior between contractions and expansions. In the case of contractions, the material with the highest dimensionless yield stress has the highest K value. In the case of the expansions, the behavior is the opposite, i.e., the higher the yield stress, the lower is the values of the K coefficient. Equations for each accessory as a function of the rheological parameters of the fluid and the Reynolds number of the flow are also proposed. The data were adjusted according to two main equations: the two Ks method proposed by Hooper (1981, “The Two-K Method Predicts Head Losses in Pipe Fittings,” Chem. Eng., 81, pp. 96–100.) is used for all the contractions cases, and the equation proposed by Oliveira et al. (1997, “A General Correlation for the Local Coefficient in Newtonian Axisymmetric Sudden Expansions,” Int. J. Heat Fluid Flow, 19(6), pp. 655–660.) is used for all the expansions cases. The equations found were compared with the numerical results and showed satisfactory precision and thus can be used for engineering applications.

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## Figures

Fig. 1

Abrupt contraction geometry

Fig. 2

Abrupt expansion geometry

Fig. 3

Mesh architecture used in this study

Fig. 4

Reynolds number influence on friction coefficients of power-law fluids for a contract ratio CR = 2.6:1. Points (numerical data) and curves (master equation).

Fig. 5

Reynolds number influence on friction coefficients of power-law fluids for a contract ratio CR = 4:1. Points (numerical data) and curves (master equation).

Fig. 6

Reynolds number influence on friction coefficients of Bingham fluids for a contract ratio CR = 2.6:1

Fig. 7

Reynolds number influence on friction coefficients of Bingham fluids for a contract ratio CR = 4:1

Fig. 8

Reynolds number influence on friction coefficients of power-law fluids for a expansion ratio ER = 1:2.6. Dots (numerical data) and lines (model of Oliveira et al. [11], according to Eq.(3)).

Fig. 9

Reynolds number influence on friction coefficients of power-law fluids for a expansion ratio ER = 1:4. Dots (numerical data) and lines (model of Oliveira et al. [11], according to Eq.(3)).

Fig. 10

Reynolds number influence on friction coefficients of Bingham fluids for a expansion ratio ER = 1:2.6. Dots (numerical data) and lines (model of Oliveira et al. [11], according to Eq.(3)).

Fig. 11

Reynolds number influence on friction coefficients of Bingham fluids for a expansion ratio ER = 1:4. Dots (numerical data) and lines (model of Oliveira et al. [11], according to Eq.(3)).

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