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Budget analysis of a pseudo-single-phase transport model for slurry flows

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Abstract

This study aims at understanding the influence of the various physical mechanisms involved in the transport of solid particles within slurry flows. The present model simulates such mixtures as pseudo-single-phase fluids. The approach is based on a constitutive equation, which accounts for the shear-induced particle migration. Two supplementary terms are added to this equation to account for particle buoyancy and turbulent dispersion, respectively. The present model is validated against experimental data published in the literature, and the influence of each mechanism is discussed in detail for two particle diameters \(d=125\) and \(440 \, \upmu \mathrm{m}\) and bulk Reynolds numbers \(\mathrm{Re}\) ranging between 46,000 and 109,000. The detailed budget analysis of the present transport equation reveals that the terms accounting for the particle transport due to the viscosity variation and the particle collision frequency variation get preponderant in the near-wall regions. This last term is besides at the origin of the repelling phenomenon. The turbulent agitation term counterbalances the sedimentation effect. When the convective flux term is important in the bulk region, the flow tends to be homogeneous. The present model can be used confidently to model two-phase flows under certain multiphase flow regimes while requiring less computational efforts than other more complex two-phase models.

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Abbreviations

a :

Particle radius (m)

d :

Particle diameter (m)

D :

Pipe diameter (m)

f :

Dumping function (–)

g :

Gravity acceleration (\(\mathrm{m\,s}^{-2}\))

k :

Turbulence kinetic energy (\(\mathrm{m}^2\,\mathrm{s}^{-2}\))

\(K_{c}\), \(K_{\mu }\) :

Constants (–)

L :

Pipe length (m)

p :

Pressure (Pa)

Re:

Reynolds number (\(=\rho _\mathrm{m} U_\mathrm{m} D / \mu _\mathrm{m}\)) (–)

\(T_\mathrm{L}\) :

Lagrangian integral time scale (s)

\(U_\mathrm{m}\) :

Inlet mean flow velocity (\(\mathrm{m}\,\mathrm{s}^{-1}\))

\(u_i\) :

Local mean velocity in the i-th direction (\(\mathrm{m}\,\mathrm{s}^{-1}\))

(x, y, z):

Cartesian coordinates (m)

\(x_i\) :

Coordinate in the i-th direction

\(\delta \) :

Kronecker symbol (–)

\(\varepsilon \) :

Dissipation rate of the turbulence kinetic energy (\(\mathrm{m}^2\,\mathrm{s}^{-3}\))

\({\varGamma }_\mathrm{T}\) :

Particle turbulent diffusion (\(\mathrm{m}^2\,\mathrm{s}^{-1}\))

\({\dot{\gamma }}\) :

Shear rate magnitude (\(\mathrm{s}^{-1}\))

\(\mu \) :

Dynamic viscosity (Pa s)

\(\mu _\mathrm{r}\) :

Relative dynamic viscosity (i.e. \(\mu _\mathrm{m}/\mu _\mathrm{l}\)) (–)

\(\nu \) :

Kinematic viscosity (\(\mathrm{m}^2\,\mathrm{s}^{-1}\))

\(\rho \) :

Density (\(\mathrm{kg}\,\mathrm{m}^{-3}\))

\(\phi \) :

Local particle volume fraction (\(\mathrm{m}^{+3}\,\mathrm{m}^{-3}\))

\({\varPhi }_{\mathrm{m}}\) :

Inlet mean particle volume fraction (\(\mathrm{m}^{+3}\,\mathrm{m}^{-3}\))

\(\phi _{\mathrm{max}}\) :

Maximum packing particle volume fraction (\(\mathrm{m}^{+3}\,\mathrm{m}^{-3}\))

\(\omega \) :

Specific turbulence dissipation rate (\(\mathrm{s}^{-1}\))

\(\omega _{s0}\) :

Initial particle settling velocity (\(\mathrm{m}\,\mathrm{s}^{-1}\))

\(\tau _{ij}\) :

Shear stress tensor components (Pa)

i, j :

Indexes

l:

Liquid phase, aqueous solution

m:

Mean or mixture

max:

Maximum value

s:

Solid phase, particles

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Acknowledgements

The authors would like to thank the NSERC chair on industrial energy efficiency established at Université de Sherbrooke in 2014 and supported by Hydro-Québec, Natural Resources Canada (CanmetEnergy in Varennes) and Rio Tinto Alcan. Calculations have been performed using the supercomputer Mammouth Parallèle 2 of Compute Canada’s network, which is also gratefully acknowledged.

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Authors and Affiliations

  1. Département de Génie Mécanique, Université de Sherbrooke, Sherbrooke, QC, Canada

    Aurélien Bordet, Sébastien Poncet & Nicolas Galanis

  2. Natural Resources Canada, CanmetENERGY, Varennes, QC, Canada

    Michel Poirier

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  1. Aurélien Bordet
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Correspondence to Sébastien Poncet.

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Bordet, A., Poncet, S., Poirier, M. et al. Budget analysis of a pseudo-single-phase transport model for slurry flows. Eur. Phys. J. Plus 135, 315 (2020). https://doi.org/10.1140/epjp/s13360-020-00326-7

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