MODELLING MULTIPHASE FLOWS OF DISCRETE PARTICLES IN VISCOELASTIC FLUIDS

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Benedict Barton
5 years ago
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Azurém 4800-058 Guimarães, Portugal 2 Department of Mechanical Eng.

1 MODELLING MULTIPHASE FLOWS OF DISCRETE PARTICLES IN VISCOELASTIC FLUIDS R. RIBEIRO 1, C. FERNANDES 1, S.A. FAROUGHI 2, G.H. McKINLEY 2 AND J.M. NÓBREGA 1 1 Institute for Polymers and Composites/i3N, University of Minho, Campus de Azurém Guimarães, Portugal 2 Department of Mechanical Eng., Massachusetts Institute of Technology, Cambridge, MA USA 28/05/ nd Iberian Meeting of OpenFOAM Technology Users Santiago de Compostela

2 Outline Motivation Objectives Computational methods Case studies Results Conclusions 2

3 Motivation Flow of particle-laden complex fluids. Numerical tools 3

4 Motivation In multiphase fluid systems (MFS) is common to follow an Eulerian- Lagrangian approach. For this, drag models are required. Contrarily to the amount of existing drag models for MFS of Newtonian fluids, the models available in the literature for viscoelastic fluids are scarce. They basically only consider single sphere sedimentation cases. 4

5 Objectives The main objective is to study numerically drag forces acting on spherical particles in inelastic and viscoelastic media. Predict the rotation of a single sphere in Newtonian and viscoelastic shear flows. Predict drag forces on clusters of rotating spheres in Newtonian and viscoelastic flows. 5

6 Computational methods Geometry Generation A Python script was created to automatically build the geometries (in STL files format). The level of surface refinement is set by the user. x1=d/2*cos(β)*.cos(α) y1=d/2*cos(β)*sin(α) z1=d/2*sin(β) x2=d/2*cos(β)*cos(α + dα) y2=d/2*cos(β)*sin(α + dα) z2=d/2*sin(β) x3=d/2*cos(β + dβ)*cos(α) y3=d/2*cos(β + dβ)*sin(α) z3=d/2*sin(β + dβ) x4=d/2*cos(β + dβ)*cos(α + dα) y4=d/2*cos(β + dβ)*sin(α + dα) z4=d/2*sin(β + dβ) 6

7 Computational methods Methodology Single sphere case was used to calculate angular velocities, while drag forces were calculated in multi sphere case studies. The spheres were created to avoid its superposition (distance between sphere centers > 1*diameter) and its interception with the domain boundaries (distance between sphere centers and domain boundaries > 0.6*diameter to the wall). 7

8 Computational methods Spheres rotation The spheres angular velocity was imposed as a boundary condition, where the angular momentum balance is given by: T is the torque I is the sphere moment of inertia Ω is the sphere angular velocity t is the time. I dω dt = T 8

9 Computational methods Drag coefficient C D = 2 n τ pi ρua2 x ds I - identity tensor τ - contribution of the extra-stress tensor n - unit normal vector to the sphere surface S x - is the unit vector parallel to the flow direction A - projected area of the spheres ρ - fluid density p - pressure U - mean flow velocity For Newtonian fluids τ = τ s, while for the viscoelastic fluid (Oldroyd-B) τ = τ P + τ s For the multi-sphere case, C D was calculated using the mean value of τ for all. 9

10 Case study Rotation of a single sphere Re = ρ U d μ = 8 De = λ U H U is the difference between Top and Bottom walls velocity H is the channel height ρ is the fluid density d is the sphere diameter µ is the fluid dynamic viscosity λ is the fluid relaxation time. De={0.15, 0.4, 1,2} The ratio (U/H) was kept constant in all cases and equal to 250, thus to obtain different values for De only the values of λ 10 were adjusted.

11 Case study Drag of a cluster of spheres in Newtonian and viscoelastic fluid Oldroyd-B fluid: De [1; 4] µ P /(µ P +µ S ) [0.25; 0.9] For both fluids: Spheres volume fraction {4%, 8%,12%} Hundreds of runs!! Creeping flow Re = ρ U d μ =

12 Results Sphere rotation in Newtonian fluids Theoretical value (*) (*) F. Snijkers et al., Effect of the viscoelasticity on the rotation of a sphere in shear flow, J. Non Newtonian Fluid Mech, 166 (2011)

13 Results Sphere rotation in viscoelastic fluids (Oldroyd B) F. Snijkers et al., Effect of the viscoelasticity on the rotation of a sphere in shear flow, J. Non Newtonian Fluid Mech, 166 (2011)

14 Results Drag force, F, versus spherical particles volume fraction, c, in Newtonian fluids F = f Τ( 6 π µ c u) F = 10 cτ 1 c 3 this work - rotation BC on the spheres surface; this work - no slip BC on the spheres surface; + results of Hill et al [1]. Brinkman s theory with corrections up to O(c) [2] Carman correlation [3]. F = ( c c ln c + 16,456c + O^(3 2 ) ൱ [1] Hill, R., Koch, D., & Ladd, A. (2001). The first effects of fluid inertia on flows in ordered and random arrays of spheres. Journal of Fluid Mechanics, 448, [2] Kim, S., & Russel, W. (1985). Modelling of porous media by renormalization of the Stokes equations. Journal of Fluid Mechanics, 154, [3] C. Carman, P. (1997). Fluid Flow Through Granular Beds. Chemical Engineering Research & Design - CHEM ENG RES DES

$25; 4% spherical particles volume fraction$

15 Results Drag forces acting on spheres in viscoelastic fluids (simulations still running) viscosity ratio=0.25; 4% spherical particles volume fraction De=1 De=2 τ xx / η 0 U H 15

16 Conclusions The boundary condition used to calculate the angular velocity of a sphere under shear flow in Newtonian and viscoelastic media was implemented and verified against theoretical and experimental results; According to the performed numerical simulations, the rotation of a sphere on those flows decreased with the fluid elasticity, quantified by the dimensionless Deborah number; The current results of the drag force calculations in viscoelastic media show that drag force acting on spheres increases with Deborah number. 16

17 Acknowledgments This work is funded by FEDER funds through the COMPETE 2020 Programme and National Funds through FCT - Portuguese Foundation for Science and Technology under the project UID/CTM/50025/2013. Minho University cluster under the project Search-ON2 Revitalization of HPC infrastructure of UMinho, (NORTE FEDER ), co-funded by the North Portugal Regional Operational Programme (ON.2-0 Novo Norte), under the National Strategic Reference Framework (NSRF), through the European Regional Development Fund (ERDF). Gompute - Thank you for your attention 17

V (r,t) = i ˆ u( x, y,z,t) + ˆ j v( x, y,z,t) + k ˆ w( x, y, z,t)

V (r,t) = i ˆ u( x, y,z,t) + ˆ j v( x, y,z,t) + k ˆ w( x, y, z,t) IV. DIFFERENTIAL RELATIONS FOR A FLUID PARTICLE This chapter presents the development and application of the basic differential equations of fluid motion. Simplifications in the general equations and common