chapter8 particle techn imp

8/4/2019 Chapter8 Particle Techn IMP

1/16

1

Lecture Particle Technology, Prof. Heinrich summer term 2011 1

8. Basics of Fluid-Solid-Flow8.1 Stationary motion of single particles in fluid

Question: Which types of forces act on a freely movingsingle particle?


Overview of forces acting on particles

following situation:

centre of massw

Particle described by

position of centre of mass and orientation of the particle (relative to flow)

translation velocity

rotation velocity (not taken into account)

Flow profile described by

fluid velocity

Relative velocity acts on the particle

rw

rv

= r r r

relv v w

w

v

vrel

rrelv

rw

rv

S

v



2/16

2


Different types of stress can act on particles:

1. Field forces

1.1 Gravity:

1.2 Electrostatic forces (e.g. electrostatic filters)

1.3 Magnetic forces (e.g. magnetic separator)

2. Interaction forces with the surrounding fluid

The relative velocity induces a torsional moment and a force .

The force is divided in two components

in the direction of : Resistance force

and perpendicular to : dynamic buoyancy Ad(in contrast to static buoyancy)

= r

sG V g

rrelv

rM

rF

r

Fr

relvr

Wr

relv



3. Pressure

can be exerted from the medium, if a pressure gradient is present inthe flow profile (also if no relative velocity between particle and fluidexist).

static buoyancy

4. Apparent forces

(Terms from mechanic) :

4.1 DAlambertsche inertia force for accelerated motion:

4.2 Coriolis-forcefor particle motion in rotating systems, e.g. centrifuges:

= rr&

sT V W (mass acceleration)

{

=

r r r

svektro product

C V 2 W Speed of rotation in the direction fothe axis (centrifuge)



3/16

3


5. Forces between solid surfaces (contact forces)

5.1 Impact

5.2 Friction

5.3 Adhesion

For continuous transport of a particle collective, e.g. pneumatic transport, theimpulse flux is changed by these forces.

6. Diffusion force

6.1 Interaction of molecules of liquid with molecules of the particle

6.2 Transient states resulting from temperature of concentration gradients6.3 Light pressure



Flow resistance correlation for the flow around a particle

The term of a physical parameter

generally:

physical parameter = {value} [unit]

e.g.

= = =2

3 3 3

3Nm kg m1kW 10 W 10 10s s

Unit e.g.

[ ]

2

3

Nm kg mkW , , ,

s s

this means different units can have the same physical meaning. The here consideredunit examples can all be reduced to the basic structure

[ ] [ ]

[ ]

[ ][ ]

[ ]=

2

3

Mass Length Force Length,

TimeTime



4/16

4


Static buoyancy

Considering the fluid:

Force balance:

On a Fluidelement (dA dx) pressure and gravity is acting

p(x) dA

pp(x) dx dA

x

+

dAdx g(pressure forces from the side compensateeach other)

pp(x)dA gdAdx p(x) dx dA

x

+ = +

pg :

x

=

because g is independent of x and for liquids-

also it follows:



Shape of solid particles in process engineering normally angular. For flowaround spheres detailed experimental and theoretical investigations for fluiddynamics exist.

The resistance of flow around a sphere

a) Sphere in nonviscous flow

Nonviscous flow is a model for many technical flows. Well developed mathematicaltool Theory of potential flow.

Nonviscous means, that in each point of the liquid only pressure but no shearforces exist.

Integration of the motion equation for a fluid element neglecting the weight yieldsfor stationary flow of an incompressible, nonviscous liquid

+ = =2 0v p const p total pressure (Bernoulli)

2



5/16

5


applying potential theory the flow (and pressure) profile can be calculated

from pressure profile the pressure distribution on the sphere surface results

from the symmetric couture results a symmetric pressure distribution

no pressure resultant is acting on the sphere

it can be shown generally:

in incompressible, nonviscous and stationary flow no resistance acts on thesphere ( or any differently shaped bodie) (DAlembert-Paradoxon)

Bild(Tafel)



b) Viscous flow

Real flows are always viscous bodies have finiteresistance

Boundary layer theory of Prandtl (1904):

for nonviscous flow only the component of the velocity perpendicular to the wall of

the body is zero at the bodys surface, but still a finite velocity of the fluid elementfor real, viscous flow no relative velocity of the fluid at the wall, because ofinteractions of liquid molecules with the wall (Adhesion). The reduction of thevelocity from a high value at a high distance to the bodies surface to the zero valueat the surface occurs in a relatively thin boundary layer.

Bild(Tafel) in the boundary layer velocity increase fromzero to outer fluid velocity

= thickness of the boundary layer



6/16

6


Resulting from the high velocity also in liquids with low velocity significant shearstress is created:

Bild(Tafel)

in the outer flow velocity gradients can be neglected, hence it is justified to assumenonviscous liquidde for calculation of this region.

The boundary theory yields the explanation for the experimentally observedlaminar separation and turbulent effects of flow around edgeless bodies (e.g.sphere, cylinder):

Bild(Tafel)

Its distinguished between: outer flow and boundary region (red)

v

y

=



Initial consideration of a fluid element in the outer flow on a flow line D over E to F:

in the outer flow the model of nonviscous liquid is valid thus also Beroulli.

From D to E the velocity increases pressure decreases

From E to F the velocity decreases pressure increases

Because no pressure gradient perpendicular to the flow line can exist (it would

mean liquid would flow, which disagree to the definition of a flow line), it follows:

Pressure profile of the outer flow is impressed on the boundary layer.



7/16

7


Consequence:

In the boundary layer flowing fluid element must follow the impressed pressureprofile. Due to strong friction effects the fluid element already lost so much energy

(r/2v2) on the way from D to E, that it can over come the pressure increase on theway from E to F.

The fluid element stops moving at A; and is by pressure distribution of the outerflow moved in the backward direction.

Increase of boundary layer thickness, laminar separation, vortexcreation



Viscous flow

The equation of motion (Navier-Stokes-equation) for nonviscous, incompressible,stationary flow in the direction x is (applied on the volume of the fluid element):

+ + = + + + 1424314444244443 14444244443

Pr essureInertiaterm Fricti

2 2 2

2 2 2

on

u u u u u uu v w u v w

x y z x x yV

zdV d dV

x

z

y

u

w

v

With the characteristic gas velocity U and the characteristic length l thedimensionless length and gas velocity is defined:

= =

= =

= =

u' u /U x ' x / l

v ' v / V y ' y / l

w ' w /U z ' z / l



8/16

8


= = =

2 2

2 2 2

x ' 1 1;

x x ' x l x ' x l x '

Substitution yields:

2 2 2 2

2 2 2 2

U u' u' u ' 1 U u' u' u 'u' v ' w '

l x ' y ' z ' l x ' l x ' y ' z '

+ + = + + +

Multiplying with yields:

2l

U

+ + = + + + 1424314444244443 144424443

Pr essureInertia Fricti

2 2

2

on

2

2 2

Ul u' u ' u ' l u ' u' u 'u ' v ' w '

x ' y ' z ' U x ' x ' y ' z '

Inertia can be neglected, if the pre-factor is sufficiently small. Neglectingthe inertia it follows:

U l

= + +

2 2 2

2 2 2

u u u

x x y z

it holds for Re


9/16

9


With the help of (a) equation of continuity and the (b) boundary conditionadhesion at sphere surface a solution for the flow and pressure profile can bededuced. Moreover from the flow profile the shear stress can be calculated.

= W 3 dv

d

(STOKES, 1845)

Important: W ~ v (linear resistance law)

This relationship holds for Re 0,25 (approximated to )Re 1

dynamic viscosity of the fluidd diameter of the spherev velocity

The integration of wall shear stress and normal pressure over the sphere surfaceyields the resulting resistibility



Resistance force results from pressure and shear forces:

The total resistance consist for STOKES flow of:

1/3 pressure force 2/3 of shear force (wall shear stress)

Bild(Tafel)

Important: for Stokes flow no laminar separation!

(Rechnungsgang z.B.: I. Szab, Hhere Technische Mechanik, Springer-Verlag, S. 458)

Definition of the value of resistance cw

= 2WW c v A

2



10/16

10


compared with the previous result for W, it yields

= 2 2W3 dv c v d

2 4

= =

W

24c 24

vd Refor STOKES flow (Re < 0,25)

The profile cW(Re) for all Reynolds numbers

The measured cW(Re) are commonly plotted as logcW versus logRe:



Resistance law for spherical particles after Schlichting(H. Rumpf, Mechanische Verfahrenstechnik, Carl Hanser Verlag, Mnchen, 1975),

Re = v d / mit Anstrmgeschwindigkeit v und Kugeldurchmesser d



11/16

11


Resistance law for spheres and thin cylindersin dependency of the Reynolds-number; Curve (1): Theory after STOKES,

Re = v d / with flow velocity v and sphere or cylinder diameter d



5 regions are distinguished:

1. STOKES-region: Re 0,25

=W24

cRe

with

= Wlogc const. logRe(straight line with slop -1)

2. Transient region: 0,25 < Re < 103

(Designation, because of transition from linear W ~ v to quadratic resistance lawW ~ v2, cW = const.)

Influence of inertia strongly increases, flow starts laminar separation at .

With increasing Re-number the laminar separation point moves to the front.

Re 24

v



12/16

12


Characteristics cW(Re) for the transient region have been determinedexperimentally

More recently numeric solutions of the Navier-Stokes-equation under considerationof the inertia by BRAUER und Mitarbeiter:Ihme,Schmidt-Traub, Brauer, Chemie-Ingenieur-Technik 44 (1972), 306.

Haas, Schmidt-Traub, Brauer, Chemie-Ingenieur-Technik 44 (1972), 1060.

v


separation point

dead water zone


In contrast to Stokes-region where the flow profile is the same in frontand on the back of the sphere, in the region 24 < Re < 130 a laminarflowing vortex is formed on the back of the sphere.

a) laminar flowing vortex, 24 < Re < 1000

b) Vortex relieve (relieve point A), unsteady wake turbulence, 130 < Re < 1000

At 130 < Re < 1000 the vortex system becomes unsteady. Singlevortex delaminate and for a wake turbulence on the back of the sphere.

v

v

A



13/16

13


Viscosity dominates the flow

Stationary vortex pair with negative

pressure, p controlls cWIncreasing vortex formation

First vortex

Flow profiles



3. Newton-region (quadratic region): 103 < Re < 105

cW = const. = 0,40,5 W ~ v2

Approximately

Laminar separation point before the Equator. Resistance is determined bypressure. The size of the laminar separation region does not change.

=Wc 0,44

v

A

4. Critical region: = 5cRe Re 3 10

At the point A the boundary layer flow changes from laminar to turbulent (in thetransition region with critical Reynolds-numbers Rec), before laminar separationoccurs.



14/16

14


The turbulence of the boundary layer causes amuch more intensive momentum exchangebetween boundary layer and free flow. Hence

energy is transferred from the free flow to theboundary layer and the backflow leading to laminarseparation is avoided.

5. Supercritical region: >

5

Re 3 10

Wc 0,09

With increasing REthe transition U occurs earlier.

bigger region of turbulent boundarylayer (with significant frictionresistance)

Bild(Tafel)

Bild(Tafel)


Consequence: The laminar separation point is moved further downstreamsmaller dead water zone cW decreases significantly


t = => wf > wfwf = stationary terminal velocity

Force balance:

G = gravity A = bouncy W = resistance

G

A

W

Force balance for stationary sinking of a particle in resting fluid:

( ) ( )2 2 3w f S

W G A

c Re d w d g4 2 6

=

=

For a spherical particle the terminal velocity is:

( )

( ) =

S2f

w

4 1w gd

3 c Re

= =

f fw d w dRe(1)



15/16

15


This equation can not be calculated directly, because Re depends on wf.

A simple solution only exist for the STOKES-region:

=W24

cRe

( ) =

S2 ff

4 w dw gd

3 24

( ) =

S 2f

1w d g

18For Re < 0,25

In general Re must be estimated then cW is determined, afterwards wf is calculated

by Equ. (1) and Re is corrected.



Special case: terminal velocity in centrifugal field

Assumption:

No relative motion between fluid and cylinder

Constant angular velocity [rad/s]

Centrifugalacceleration

Gravity



16/16


centrifugal acceleration b :

Centrifugal number, ...

Force balance for a particle (all forces act in radial direction):

centrifugal force

buoyancy

resistance force

Small forces, like inertia and Coriolis force are neglected

Commonly: z >> 1 => gravity negligible

=

ur r3 2

SZ d r6

=

ur r3 2A d r

6

=

uur ur2

f zf z wW w w d c (Re)2 4


Acceleration ratio z :

2b rz

g g

= =

2b r=


Settling velocity wfz in radial direction to outside, if particle density > fluid density (sand in water) to centre, if particle density < fluid density (gas bubble in water)

Force balance:

Applied for spherical particles compared to Equ. (1)

Centrifugation: separation of small particles:

small Reynolds-number (< 0,25)

STOKES-region (viscous flow)

cw = 24/Re

Z

AW

=Z A W

( ) ( )

= 3 2 2 2S f z wd r w d c Re6 2 4

( )( ) =

S2f

w

4 1w gd3 c Re

zg( )

( ) =

S2 2f

w

4 1w d r3 c Re

(1)


chapter8 particle techn imp

Documents