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Velocity of Variously Shaped Particles Settling in Non-Newtonian Fluids

  • Moacyr Laruccia
  • Cesar Santana
  • Eric Maidla

Abstract

This research [1] concerns the development of a drag coefficient correlation for nonspherical particles settling in purely viscous non-Newtonian fluids. The dynamic interaction term between fluids and particles was studied using both the dimensional analysis and a large number of experimental data covering the laminar, transitional and turbulent flow regime to obtain a generalized correlation for the determination of the settling velocity valid for particles on a sphericity (ø) range from 0.5 to 1.

Unlike the previous published research in this area, this generalized correlation does not depend on a particular rheological model.

The developed correlation for the drag coefficient CD assumes the form
$${{C}_{D}}={{\left\{ {{\left[ \frac{24\Omega \left( \varnothing \right)}{{{\operatorname{Re}}_{gen}}} \right]}^{m}}+{{\left[ x\left( \varnothing \right) \right]}^{m}} \right\}}^{1/m}}$$
(1)
being the Reynolds number Re defined here as
$$\operatorname{Re}=\frac{\rho V_{t}^{2}\theta \left( \varnothing \right)}{\tau \left( \overset{\cdot }{\mathop{\gamma }}\, \right)}$$
(2)
In equation (2), θ(ø) is a known form factor and τ(\(\overset{\cdot }{\mathop{\gamma }}\,\) ) is the shear stress correspondent to a shear rate \(\overset{\cdot }{\mathop{\gamma }}\,\) related to the particle diameter dp and to the settling velocity vt by the following equation:
$$\overset{\cdot }{\mathop{\gamma }}\,=\frac{{{V}_{t}}}{{{d}_{p}}}\theta \left( \varnothing \right)$$
(3)

In equation (1) the functions Ω(ø) and X(ø) known from experiments considering the limit cases of laminar fully turbulent flow and the exponent m is determined from the data reduction using the Churchill's asymptotic method and an extensive data file from the literature.

A form for vt can be obtained by combination of the above dimension-less numbers resulting
$${{V}_{t}}={{\left\{ {{\left[ \frac{4}{3}g\frac{{{d}_{p}}\left( {{p}_{s}}-p \right)}{x\left( \varnothing \right)p} \right]}^{m}}-{{\left[ \frac{24\tau \left( \overset{\cdot }{\mathop{\gamma }}\, \right)}{\rho }\alpha \left( \varnothing \right) \right]}^{m}} \right\}}^{\frac{1}{2m}}}$$
(4)
The match of experimental data led to the following sphericity (ø) dependent parameters:
$$x\left( \varnothing \right)={{e}^{\left( 4.69-5.53\varnothing \right)}}$$
$$\alpha \left( \varnothing \right)=-\frac{\left( 1.65-0.656\varnothing \right){{e}^{\left( 5.53\varnothing -4.69 \right)}}}{\left( 3.45{{\varnothing }^{2}}-5.25\varnothing +1.41 \right)}$$
$$\Omega \left( \varnothing \right)=1.65-0.656\varnothing $$
$$\theta \left( \varnothing \right)=-3.45{{\varnothing }^{2}}+5.25\varnothing -1.41$$

Keywords

Shear Rate Settling Velocity Rheological Model Creeping Flow Nonspherical Particle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Elsevier Science Publishing Co., Inc. 1990

Authors and Affiliations

  • Moacyr Laruccia
    • 1
  • Cesar Santana
    • 1
  • Eric Maidla
    • 2
  1. 1.Chemical Engineering DepartmentState University of Campinas (UNICAMP)CampinasBrazil
  2. 2.Petroleum Engineering DepartmentState University of Campinas (UNICAMP)CampinasBrazil

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