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Modelling the structure of deposits formed by polydisperse suspensions

Modelling the structure of deposits formed by polydisperse suspensions. Andrew Hogg. Robert Dorrell, Peter Talling, Esther Sumner, Tom Harris, Herbert Huppert. Outline. Settling through otherwise quiescent fluid: hindered settling Particle-driven gravity currents : turbidity currents

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Modelling the structure of deposits formed by polydisperse suspensions

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  1. Modelling the structure of deposits formed by polydisperse suspensions Andrew Hogg Robert Dorrell, Peter Talling, Esther Sumner, Tom Harris, Herbert Huppert

  2. Outline • Settling through otherwise quiescent fluid:hindered settling • Particle-driven gravity currents: turbidity currents • How are the dynamics affected by polydisperse suspensions? u f h ws Initial suspension Deposit Gravitationally driven flows of suspensions over rigid boundaries

  3. Settling of an individual particle • Stokes settling velocity for an individual spherical particle settling through an unbounded fluid domain gives • The velocity field around the particle decays as d/r • Particle settling is influenced by its neighbours: Hindered Settling. • ‘Return’ flow of fluid • Enhanced viscosity of suspension Streamlines in a frame moving with the settling particle aaa

  4. Hindered settling of a suspension • Mass conservation for a suspension of concentration, f(z,t) • The settling flux, q, depends on the volume fraction, expressing the hindered settling • For example: • Discontinuous solutions are possible q(z,t)=Settling flux (Kynch,1952) q z=s(t) • ‘Shock’ speed 1 0 f/fm f(s-) f(s+)

  5. Monodisperse settling (i) Settling from an initially well mixed state: f=f0 at t=0 for 0<z<h • f0determines whether there is a shock in the profiles of f(z,t) • f0 =0.1 Discontinuous profiles; f0 =0.5 Continuous profiles • The final deposit with f=fm is only formed as t f0=0.5 f0=0.1

  6. Revised model of hindered settling • Mass balance between the settling flux of particles and the return flow of suspending fluid (wp: particle velcoity; wf: fluid velocity) • Force balance: (fm: Maximum packing)If f < fmSubmerged weight = Drag If f > fm Submerged weight = Force from particle contacts • Settling flux q f 0 fm 1

  7. Monodisperse settling (ii) Settling from an initially well mixed state: f=f0 at t=0 for 0<z<h f0=0.1 f0=0.2 Discontinuous jump to fm Discontinuous jump to f<fm; discontinuous jump to fm f0=0.35 f0=0.5 Continuous evolution and discontinuous jump to fm Discontinuous jump to fm

  8. Bidisperse settling (i) • Hindered settling for two classes of particles:Mass conservation:Force balance: Submerged weight = Drag (if f1+f2<fm) • Fluxes of particles q1=wp1f1 , q2=wp2f2 (if f1+f2<fm)and q1=q2=0 (if f1+f2>fm) • Coupled evolution equations for f1(z,t) and f2(z,t) • Solutions with discontinuous volume fractions may arise.Across the shock, both volume fraction change discontinuously. wpi: settling velocity of ith particle class Note: q1(f1,f2) and q2(f1,f2)

  9. Bidisperse settling (ii) Settling from an initially well mixed state: f1=f2=f0 at t=0 for 0<z<hRatio of particle sizes (d1/d2)2=1/2 • Dilute suspension f0=0.025F0=f1(0)+f2(0)=0.05 • Discontinuous jump from well mixed suspension into stationary deposit with total volume fraction Ff1+f2=fm • Deposit composed of ungraded mixture of fine/coarse particles overlain by an interval of pure fines The total volume fraction F=f1+f2 Volume fraction of larger particles Volume fraction of smaller particles

  10. Bidisperse settling (iii) Settling from an initially well mixed state: f1=f2=f0 at t=0 for 0<z<hRatio of particle sizes (d1/d2)2=1/2 F0=0.05 F0=0.2 Discontinuous jump to fm Discontinuous jump to f<fm; discontinuous jump to fm F0=0.3 F0=0.5 Continuous evolution and discontinuous jump to fm Discontinuous jump to fm

  11. Deposit from bidisperse suspension z • The deposit comprises an ungraded region, overlain by a region consisting purely of fine particles • The relative proportion of larger to smaller particles in the ungraded portion is determined by • Size ratio (d1/d2) • Initial ratio (f1(0)/f2(0)) • Combined initial volume fraction (F(0)) fm Ratio of volume fractions in ungraded deposit R= f1/f2 (d1/d2)2=0.5 f1(0)/f2(0)=1 F(0)

  12. Polydisperse suspensions: Experiments • Settling tube experiments (Amy et al. 2006) • Well mixed, polydisperse sand suspensions at different initial volume fractions (F(0)=0.05-0.6) • Settle until no particles remain in suspension • Deposit frozen, sliced into horizontal strips and composition measured. z • Key observations: • The deposit has ungraded at base overlain by finer material • The average grain size in the ungraded portion decreases with increasing F(0) • The relative depth of the ungraded portion increases with increasing F(0) Proportion with diameter d d50

  13. Polydisperse suspensions: models • Extend the hindered settling framework to an arbitrary number of particle classes • Represent the suspension with 28 classes • Suspension evolution represented by 28 coupled equations • Integrate until deposit fully formed from given F(0) Evolution of the total volume fraction,F(z,t)=i fi(z,t) F(z,0)=0.3

  14. Polydisperse suspensions: results Composition of the deposit: d50, d10, d90 as functions of relative depth in deposit F(0)=0.05 F(0)=0.1 F(0)=0.2 F(0)=0.3 F(0)=0.4 F(0)=0.5

  15. Polydisperse suspensions: interpretation Key observations: • The deposit has ungraded at base overlain by finer material • From a well-mixed suspension, the flux of each class of particles into the deposit remains constant until all of the largest class of particles has deposited. Thus the composition of the deposit does not initially vary, forming an initially ungraded deposit. • The average grain size in the ungraded region decreases and its relative depth increases with increasing F(0) • Increasing F(0) reduces the settling velocity of the particles and thus the time during which the deposit is ungraded and of constant composition is increased. This means that a greater proportion of the particle load is deposited in the ungraded region.

  16. Turbidity currents • Motion driven by gravitational force associated with density difference between the cloud of particles and surrounding fluid: r=ra+ f (rs-ra) • Sedimentation reduces the concentration of the suspension and thus reduces the density difference • Passage of the current leaves a deposit on the underlying boundary Flow over an impermeable, horizontal boundary u f h ws

  17. Models h(t) • ‘Box’ models capture the essential dynamics Conservation of fluid mass Evolution of concentration Front (dynamic condition) • Flow stops when the volume fraction of particulate falls from its initial value f=f0tof=0. • This occurs at L(t) f(t) ws (monodisperse)

  18. Comparison with experiments Laboratory flume experiments in which varying initial volumes of monodisperse suspension (A), of varying particle sizes and settling velocities (ws) and differing initial concentrations (f0) were instantaneously released. Scaled data and theory Raw data L/L t/T

  19. Bidisperse gravity currents • For bidisperse suspensions: • Separate evolution of the volume fraction of each species because the particle settle with different velocities • The total volume fraction F=f1+f2 determines the excess density and drives the motion. • How well is the runout represented by treating the suspension as effectively monodisperse with settling velocity given by the initial average (ws1f+ws2(1-f)) and with the combined initial volume fraction (F0)? At t=0 f1=fF0 f2=(1-f)F0

  20. Bidisperse currents: runout lengths The runout length: L=L when f1=f2=0 and ws2/ws1=1/4 Relative to current with average settling velocity Relative to current of only coarse particles Enhancement of Runout length L /Lc Enhancement of Runout length L /La 100% coarse 100% fine 100% fine 100% coarse Fraction of coarse particles Fraction of coarse particles

  21. Bidisperse currents: Temporal evolution • Length of current L(t) for ws2/ws1=1/4 L(t)/La Final runout 100% coarse 100% fine Fraction of coarse particles

  22. Bidisperse currents: deposit h(t) The depth of the deposit and its composition for ws2/ws1=1/7 L(t) f(t) 100% fine ws Surface of deposit Vertical and longitudinal grading of deposit Depth within deposit z/[F0A/La] 100% coarse Distance from release x/La

  23. Polydisperse gravity currents • Suppose the suspension was made up of N classes of particles, with initial concentrations fi(0)(i=1..N) and settling velocities wsi (i=1..N) • Is it reasonable to estimate runout by treating the suspension as monodisperse with initial volume fraction and average settling velocity ? • If it is reasonable, then runout • Models for the runout for the polydisperse current show

  24. Summary and conclusions • Hindered settling • Formulation for polydisperse suspensions • Deposit formation and composition • ‘Shocks’ in the volume fraction play an important role. • Quantitative agreement with laboratory experiments • Turbidity currents • Simple ‘box’ model to predict the runout of dilute suspensions • Generalise to polydisperse systems • Mixture of particle sizes always leads to propagation further than had it been monodisperse with the average settling velocity • Deposit progressively thins and exhibit vertical and streamwise normal grading

  25. Worldwide University Network SymposiumEarth Surface Sedimentary Flow Processes 11-13 April 2011, Bristol Registration now open : http://www.wun.ac.uk/events/earth-surface-sedimentary-flows-symposium Full funding available for registration and accommodation costs Organisers: Dr Andrew Hogg (Bristol), Dr Peter Talling (Southampton), Prof. Nico Gray (Manchester), Dr Chris Keylock(Sheffield)

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