Chee2940 Lecture 7 Part A - Particle Fluid Interaction

CHEE2940: Particle Processing Lecture 7: Particle Fluid Interactions This Lecture Covers Fluid flows Fluid resistance on particles Terminal settling velocity of particles Chee 2940: Particle Fluid Interactions

IMPORTANCE OF PARTICLE-FLUID INTERACTIONS Are an integral part of many operations in particle processing. Examples: - Particle fluidisation - Particle settling by gravity - Counter-current flow of fluid - Balancing of particle settling by fluid flow causes the fluidisation of particles - Applications: drying Chee 2940: Particle Fluid Interactions

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7.1 FLUID FLOW What is a fluid? A continuous substance - whose molecules move freely past one another, and - that has the tendency to assume the shape of its container.

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Gas is a fluid having - neither independent shape nor volume, and - being able to expand indefinitely (Gases have low density and viscosity, and high expansion/compression) Liquid is a fluid having - no fixed shape but - a fixed volume (Liquids have low compressibility) Order of inter-atomic/molecular forces: Solid > liquid > gas. Chee 2940: Particle Fluid Interactions

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Types of fluid flows 1) Laminar flows: smooth motion in layers

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Laminar flow in a pipe: - parallel fluid layers - Profiling of velocity from the wall surface.

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Laminar flow around a particle: - Fluid layers are compressed at the surface but - stil slide over one another. 5

2) Turbulent flow: velocity fluctuates with time and position

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Turbulent flow in a pipe

Turbulent flow around a particle

Fluid molecules move freely and chaotically. Turbulence is decribed by statistical theories.

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Reynolds number: Re Fluid flow can be either laminar or turbulent. Laminar flow is governed by viscous force. Turbulent flow is dominated by inertia. The ratio of inertial to viscous forces determines the flow type, and is called the Reynolds number Chee 2940: Particle Fluid Interactions

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Inertial force ρ DW Re = = Viscous force µ

ρ … fluid density µ … fluid viscosity W … fluid characteristic velocity D … characteristic length (e.g. diameter of a pipe or a particle).

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Typical values of Reynolds number −2

Colloids ~ 1×10 (laminar flow) 2 Blood flow in brain ~ 1×10 (laminar flow) 3 Blood flow in vein ~ 1×10 (turbulent flow) 6 Swimmers ~ 4×10 7 Aircraft ~ 1×10 8 Blue whale ~ 3×10 9 A large ship ~ 5×10 . Chee 2940: Particle Fluid Interactions

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Flows high Re are turbulent. Flows with low Re are laminar. Onset of turbulent flows 3

Pipe flow: Re ~ 2×10 Flow around a particle: Re ~ 500 Special Re flows: Low Re flows = creeping flows Non-viscous flows = potential flows Chee 2940: Particle Fluid Interactions

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Governing Equations for Fluid Flows

- Continuity equation: describes the mass balance. G div ρW = 0

(

)

- Navier-Stokes equations: describes the momentum balances.

G G G P µ ∂W G + W ⋅ grad W = − grad   + div  grad W  +   ρ ρ ∂t   

   

  inertial force

( )

( )

pressure gradient

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viscous force

G g

ρ N

gravity

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The Bernoulli equation can be used to describe the inviscid flow.

ρV 2

2

+ ρ gh + p = constant

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When the flow velocity is zero (i.e. statics), the fluid is governed by the laws of fluid statics. - Static pressure is isotropic - Hydrostatic pressure: P = ρ gh - Atmospheric pressure (Maxwell-Boltzman law): ρ ( h ) = ρ ( 0 ) exp  − gh / ( k BT )  where h … distance from the Earth surface. kBT… thermal energy. - Buoyancy - Liquid-fluid free surface: surface tension & capillary effect (Young-Laplace equation). Chee 2940: Particle Fluid Interactions

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7.2 RESISTANCE ON SPHERES Stokes Law - Applied for low Re # (low particle-fluid relative approach velocity, small size, or high viscosity) - Re < 0.3 - Drag force Fd = 3πµ DW Drag Coefficient, Cd Experiments show that drag force is proportional to Chee 2940: Particle Fluid Interactions

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2

- Square of particle velocity, W - Density of fluid, ρ - Particle area, Ap, projected to its path Scaling law gives the definition of the drag coefficient Fd Cd = 2 Ap ρW / 2 Factor 2 is used for convenience. 2 2 For spheres: Ap = π R = π D / 4 Chee 2940: Particle Fluid Interactions

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Drag Coefficient and Stokes Law Fd 3πµ DW 24 µ Cd = = = 2 2 Ap ρW / 2 π D D ρW 2 ρW / 2 4

24 Cd = Re

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Newton Law (for high Re #) - Big particles move very fast in fluid - Flow is turbulent => independent of viscosity - Drag force is independent of viscosity 2 - Drag force is proportional to W , ρ, and Ap. - Drag coefficient is constant and ~ 0.44. Cd = 0.44

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Drag Coefficient for Intermediate Re # ( 0.3 < Re < 500) Both viscosity and inertia are important. Experimental data are shown below.

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24 0.687 Empirical correlation: Cd = 1 + 0.15Re ) ( Re Chee 2940: Particle Fluid Interactions

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7.3 SETTLING OF SINGLE SPHERES Fd Fb

Particle

Fg

- The force of gravity (particle weight), Fg, pulls the particle down. - The drag force and buoyancy resist gravity. - The particle initially accelerates, then reaches a steady velocity when a force balance is reached.

Steady velocity = terminal settling velocity. Chee 2940: Particle Fluid Interactions

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Force balance at the steady settling:

Gravity = Buoyancy + Drag Fg = Fb + Fd

ρ f Cd Ap ρ f (VT ) mg = mg + 2 ρs

2

VT…terminal settling velocity of particle m…particle mass; g…acceleration due to gravity ρf …fluid density; ρs …solid density Chee 2940: Particle Fluid Interactions

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FOR SPHERES

πD

2

ρ C V d f ρ π D ρs π D ρs f 4 + g= g ρs 6 6 2 3

3

2

CdVT =

2

4 Dg ( ρ s − ρ f ) 3ρ f

We need Cd to calculate the settling velocity! Chee 2940: Particle Fluid Interactions

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Stokes Law for Fine Spheres 24 24 µ Cd = = Re ρ f DVT

4 Dg ( ρ s − ρ f ) 24 µ 2 CdVT = VT = ρ f DVT 3ρ f 2

D g ( ρs − ρ f ) 2

VT =

18µ

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for Re < 0.3

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Newton Law for Big Spheres

Cd = 0.44 2

2

CdVT = 0.44VT =

4 Dg ( ρ s − ρ f ) 3ρ f

ρs − ρ f VT = 1.74 Dg for Re > 500 ρf

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Spheres with Intermediate Re #

ρ f DVT 24 0.687 Cd = 1 + 0.15Re ); Re = ( µ Re 4 Dg ( ρ s − ρ f ) 24 µ 2 2 0.687 CdVT = 1 + 0.15Re )VT = ( ρ f DVT 3ρ f 2 D g ( ρs − ρ f ) VT = 0.687 18µ (1 + 0.15Re ) Iteration is needed to find VT! (Iteration can be done in Excel with VBA) Chee 2940: Particle Fluid Interactions

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NON-SPHERICAL PARTICLES

ρ f Cd Ap ρ f (VT ) mg = mg + ρs 2

2

We can measure m, densities, volume, and surface area. Cd is given as a function of Re # and (volume) sphericity. Trial-and-error approach is used to calculate VT since Cd and Re are dependent on VT. Chee 2940: Particle Fluid Interactions

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Cubes Spheres

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One method of avoiding trial and error is to use a modified form of the drag chart. Using volume-equivalent diameter, Dv, we obtain 4 Dv g ( ρ s − ρ f ) ρ f DvVT 2 Re = and Cd (VT ) = 3ρ f µ

Eliminating velocity gives Cd Re = 2

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4 g ( ρs − ρ f ) ρ f 3µ

2

( Dv )

3

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The modified drag chart gives correlation for 2 CdRe as a function of Re. Knowing the physical parameters of the particle 2 we can calculate CdRe and then determine Re from the modified chart and the velocity from the Re #.

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Cd/Re Cd/Re

CdRe2

Re

CdRe2 Chee 2940: Particle Fluid Interactions

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The principal can be applied for determining the particle size if terminal velocity is known. - Eliminating the diameter gives 4  ρs − ρ f Cd / Re =  2  3 ρf

 gµ  2 V  T - The modified drag chart gives correlation for Cd/Re as a function of Re. - Knowing the particle velocity and other parameters we can calculate Cd/Re and then determine Re from the modified chart and the diameter from the Re #. Chee 2940: Particle Fluid Interactions

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WALL EFFECT Settling velocity in confined space, such as a small pipe, is smaller than the velocity in infinite fluid, due to the wall effect. The wall effect is accounted for by the correction factor, fw.

VT ,confined = f wVT fw depends on Re and distance from the wall. Chee 2940: Particle Fluid Interactions

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D Faxen: f w = 1 − 2.1  for Re ≤ 0.3 and D/x ≤ 0.1 x 2 D 3 4 Munroe: f w = 1 −   for 10 ≤ Re ≤ 10 and x 0.1 ≤ D/x ≤ 0.8.

x … distance between the particle and the pipe surface.

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ACCELERATION OF PARTICLES At the beginning, particles do not reach the steady settling. The inertial effect arises due to the unbalance of gravity, buoyancy, and drag force. Simple equation for unsteady settling of spheres:

Inertia = Gravity − Buoyancy − Drag Chee 2940: Particle Fluid Interactions

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ρ f Cd Ap ρ f V dV m = mg − mg − dt 2 ρs

2

For fine spheres: 3

π D ρ s dV 6

dt

=

3

π D ρs g 6

3

−

πD ρf g 6

− 3πµ DV

Scaling and re-arranging gives Chee 2940: Particle Fluid Interactions

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dV τ + V = VStokes dt VStokes… terminal settling velocity by Stokes law 2

D ρ τ= … particle relaxation time 18µ

Particle transient velocity   t V (t ) = VStokes 1 − exp  −  τ 

  

Particle acceleration decays exponentially. Chee 2940: Particle Fluid Interactions

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τ is a measure for the tendency of particles reach equilibrium. - At t = τ, 2τ and 3τ the transient velocity is within 63, 87, and 95% of the settling velocity, respectively. 3 - For D = 1mm, ρ = 2500 kg/m , we obtain τ = 0.14s in water (µ = 0.001) and τ = 13.9s in air (µ = 0.00001). Terminal velocity in water will be attained almost instantaneously. Terminal velocity in air requires a longer time, depending on the particle size and density. Chee 2940: Particle Fluid Interactions

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SOME APPLICATIONS OF PARTICLE HYDRODYNAMICS 1) Particle settling: - Coal and mineral processing industry (gravity separation) - Dewatering industry (hindered settling) - Water treatment (particle separation) 2) Viscosity of suspension: Drag around spheres increases viscosity of a Chee 2940: Particle Fluid Interactions

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fluid. The more spheres, the more drag and higher suspension viscosity. Einstein prediction (1906):

µ suspension = µ fluid (1 + 2.5ε ) ε … volume fraction of particles Volume of particles ε= Total volume of particles & fluid Chee 2940: Particle Fluid Interactions

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