Velocity of Variously Shaped Particles Settling in Non-Newtonian Fluids

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 C_D 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 d_p and to the settling velocity v_t 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 v_t 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:

$$x\left( \varnothing \right)={{e}^{\left( 4.69-5.53\varnothing \right)}}$$

$$\alpha \left( \varnothing \right)=-\frac{\left( 1.65-0.656\varnothing \right){{e}^{\left( 5.53\varnothing -4.69 \right)}}}{\left( 3.45{{\varnothing }^{2}}-5.25\varnothing +1.41 \right)}$$

$$\Omega \left( \varnothing \right)=1.65-0.656\varnothing $$

$$\theta \left( \varnothing \right)=-3.45{{\varnothing }^{2}}+5.25\varnothing -1.41$$