chee2940 lecture 7 part a - particle fluid interaction

8/14/2019 CHEE2940 Lecture 7 Part a - Particle Fluid Interaction

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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


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IMPORTANCE OF PARTICLE-FLUID INTERACTIONS

Are an integral part of many operations inparticle 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: dryingChee 2940: Particle Fluid Interactions 1


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7.1 FLUID FLOW

What is a fluid?

A continuous substance

- whose molecules move freely past oneanother, and- that has the tendency to assume the shape

of its container.

Chee 2940: Particle Fluid Interactions 2


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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 3


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Types of fluid flows

1) Laminar flows: smooth motion in layers



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Laminar flowin a pipe:- parallel fluid layers- Profiling of velocityfrom the wall surface.

Laminar flowaround a particle:- Fluid layers are

compressed at thesurface but- stil slide over one

another.Chee 2940: Particle Fluid Interactions 5


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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 Reynoldsnumber



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Inertial forceRe

Viscous force

DW

= =

fluid density

fluid viscosityW fluid characteristic velocityD characteristic length(e.g. diameter of a pipe or a particle).



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Typical values of Reynolds number

Colloids ~ 110

2 (laminar flow)Blood flow in brain ~ 110

2(laminar flow)

Blood flow in vein ~ 110

3

(turbulent flow)Swimmers ~ 4106

Aircraft ~ 1107

Blue whale ~ 3108A large ship ~ 510

9.



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Flows high Re are turbulent.

Flows with low Re are laminar.

Onset of turbulent flows

Pipe flow: Re ~ 2103Flow around a particle: Re ~ 500

Special Re flows:Low Re flows = creeping flows

Non-viscous flows = potential flowsChee 2940: Particle Fluid Interactions 12


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Governing Equations for Fluid Flows

- Continuity equation: describes the massbalance.

( ) 0div W =

- Navier-Stokes equations: describes themomentum balances.

( ) ( )

inertial forcegravityviscous forcepressure gradient

W P gW grad W grad div grad W t

+ = + +



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The Bernoulli equation can be used to describe

the inviscid flow.

2

constant2

V gh p

+ + =



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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 isotropicP =- Hydrostatic pressure: gh

- Atmospheric pressure (Maxwell-Boltzmanlaw):

( ) ( ) ( )0 exp / Bh gh k T = 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 15


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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 3d F DW =

Drag Coefficient,Cd

Chee 2940:

Experiments show that drag force isproportional to

Particle Fluid Interactions 16


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- Square of particle velocity, W2

- Density of fluid, - Particle area,Ap, projected to its path

Scaling law gives the definition of the dragcoefficient

2 / 2dd

p

FC

A W=

Factor 2 is used for convenience.2 2 / 4 A R DFor spheres: p = =



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Drag Coefficient and Stokes Law

222

3 24

/ 2 / 24

d

dp

F DW C

DW D WW

= = =

24

RedC =



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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- Drag force is proportional to W

2, , andAp.

- Drag coefficient is constant and ~ 0.44.

0.44dC =



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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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Empirical correlation:

( )0.68724 1 0.15Re

RedC = +



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7.3 SETTLING OF SINGLE SPHERES

- The force of gravity (particleweight), Fg, pulls the particle

down.

Fd

Fb

Fg

Particle

- The drag force and buoyancyresist gravity.

- The particle initiallyaccelerates, then reaches a

steady velocity when a forcebalance is reached.

Steady velocity = terminal settling velocity.Chee 2940: Particle Fluid Interactions 22


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Force balance at the steady settling:

Gravity Buoyancy Drag= +

b d F F F = +

( )2

2

f d p f T

s

C A Vmg mg

= +

VTterminal settling velocity of particlemparticle mass; gacceleration due to gravity

ffluid density; s solid densityChee 2940: Particle Fluid Interactions 23


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FOR SPHERES

22

3 3

4

6 6 2

d ffs s

s

DC V

D D

g g

= +

(2 43

s f

d T

f

DgC V

=

We need Cd to calculate the settling velocity!


S f S


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Stokes Law for Fine Spheres

24 24

Red

f T

CDV

= =

(2 2 4243

s f

d T T

f T f

DgC V V

DV

= =

(2

18

s f

T

D g

V

= forRe 0.3<


N t L f Bi S h


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Newton Law for Big Spheres

0.44dC =

2 2 40.44

3

s f

d T T

f

DgC V V

= =

1.74s f

T

f

V Dg

= forRe 500>


S h ith I t di t R #


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Spheres with Intermediate Re #

( )0.68724

1 0.15ReRe

dC = + ; Ref TDV

=

( )2 20.687424

1 0.15Re3

s f

d T T

f T f

DgC V V

DV

= + =

(

( )

2

0.68718 1 0.15Re

s f

T

D gV

=

+

Iteration is needed to find VT!(Iteration can be done in Excel with VBA)


N


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NON-SPHERICAL PARTICLES

( )2

2

f d p f T

s

C A Vmg mg

= +

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 VTsince Cdand Re are dependent on VT.



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Cubes

Spheres

One method of avoiding trial and error is to


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One method of avoiding trial and error is touse a modified form of the drag chart.

Using volume-equivalent diameter, Dv, we obtain

Ref v TD V

= and ( )2 4

3

v s f

d T

f

D gC V

=

Eliminating velocity gives

(

( )

32

2

4

Re 3

s f f

d v

g

C D

= Chee 2940: Particle Fluid Interactions 30


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The modified drag chart gives correlation forCdRe

2 as a function of Re.

Knowing the physical parameters of the particlewe can calculate CdRe

2and then determine Re

from the modified chart and the velocity from theRe #.


Cd/Re


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Cd/ReCdRe2

Re


CdRe2

The principal can be applied for determining the


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The principal can be applied for determining theparticle size if terminal velocity is known.

- Eliminating the diameter gives

2 24/ Re3

s f

d

f T

gCV

=

- The modified drag chart gives correlation forCd/Re as a function of Re.

- Knowing the particle velocity and otherparameters we can calculate Cd/Re and thendetermine Re from the modified chart and the

diameter from the Re #.Chee 2940: Particle Fluid Interactions 33

WALL EFFECT


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WALL EFFECT

Settling velocity in confined space, such as asmall pipe, is smaller than the velocity in infinite

fluid, due to the wall effect.

The wall effect is accounted for by the correctionfactor, fw.

,T confined w T V f V=

fw

depends on Re and distance from the wall.



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Faxen: 1 2.1wD

f x

= for Re 0.3 and D/x 0.1

Munroe:

2

1wD

f x

= for 10

3

Re 104

and

0.1 D/x 0.8.

x distance between the particle and the pipesurface.


ACCELERATION OF PARTICLES


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ACCELERATION OF PARTICLES

At the beginning, particles do not reach thesteady settling.

The inertial effect arises due to the unbalance ofgravity, buoyancy, and drag force.

Simple equation for unsteady settling of spheres:

Inertia Gravity Buoyancy Drag=



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2

2 f d p f

s

C A VdVm mg mg dt

=

For fine spheres:

33 3d3

6 d 6 6

fs s D g D V D g DVt

=

Scaling and re-arranging gives


dV


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Stokes

d

d

VV V

t

+ =

VStokes terminal settling velocity by Stokes law

2

18D

= particle relaxation time

Particle transient velocity

Stokes( ) 1 exp tV t V

=


Particle acceleration decays exponentially.

is a measure for the tendency of particles reach


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is a measure for the tendency of particles reachequilibrium.

- At t = , 2 and 3 the transient velocity iswithin 63, 87, and 95% of the settling velocity,

respectively.- ForD = 1mm, = 2500 kg/m

3, we obtain =

0.14s in water ( = 0.001) and = 13.9s in air (= 0.00001).

Terminal velocity in water will be attained almostinstantaneously. Terminal velocity in air requiresa longer time, depending on the particle size and

density.Chee 2940: Particle Fluid Interactions 39

SOME APPLICATIONS OF PARTICLE


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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


fluid. The more spheres, the more drag and


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fluid. The more spheres, the more drag andhigher suspension viscosity.

Einstein prediction (1906):

( )1 2.5 suspension fluid = +

volume fraction of particles

Volume of particlesTotal volume of particles & fluid

=


chee2940 lecture 7 part a - particle fluid interaction

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