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Two Phase Flow, Rheology

and Powder Flow

Chapters 6, 9 & 10 in Fundamentals

Professor Richard Holdich

[email protected] Course details:Particle Technology,

module code: CGB019 and CGB919,2 nd year of study.

Watch this lecture at www.vimeo.com

Also visit;http://www.midlandit.co.uk/particletechnology.htm for further resources.
mailto:[email protected]://www.vimeo.com/http://www.midlandit.co.uk/particletechnology.htmhttp://www.midlandit.co.uk/particletechnology.htmhttp://www.vimeo.com/mailto:[email protected]


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Two Phase Flow, Rheologyand Powder Flow

Rheology Section 6.7Homogeneous systems; Newtonian and non-

Newtonian, laminar/turbulentHomogeneous but with slip

pneumatic conveying - dilute phase

Heterogeneous systemspneumatic conveying - dense phasehydraulic conveying

Powder flow


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Flow of dispersions


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Rheograms

Non-time dependent

r v

Rdd

)(

n

r v

k R

dd

)(

Newtonian:

Power law:


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Rheograms

Time dependent


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Apparent viscosity

Is the viscosity of a Newtonian fluid thatflows under the same conditions of shear

rate and stress as the non-Newtonianfluid.

R (Pa)

dv /d r (s -1)

Apparentviscosity


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Apparent viscosity

In order to use Newtonian flow equationswe really need apparent viscosity for pipe

flow - from the flow characteristic, etc. In order to predict flow rate and pressuredrop use simpler approach - appropriate

to power law fluids.n

r v

k R

dd

)(Force balanceon a wall gives: L

Pa Rw 2


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Wilkinsons equation

Combine the power law viscosity equationwith the shear stress on the wall - much

like the derivation of Hagens equationand integrate to give:

n

Lk Pa

nan

Q

/ 13

213

Laminar flow of non-Newtonian power law fluids andsuspensions.


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Turbulent flow

Generalised expression based on a frictionfactor:

)(Re,f 2 2

nv

R f

For Newtonian fluids:

3.02

Reln5.22

2 / 12 / 1

f f


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Turbulent flow

Dodge and Metzner - turbulent flowpower law fluid:

2.1)2 / 1(75.02 / 1 4.0Re*ln4 n f n f n


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Turbulent flow

Need a Reynolds number that reduces toNewtonian equation when n =1, and the

turbulent friction expression shouldreduce to Wilkinsons equation given

f =16/Re* - i.e. for laminar flow.

n

nn

nnk

d v

26

8

*Re2


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Turbulent flow

The Generalised Reynolds number -

threshold value of 2000 for laminar toturbulent flow.

n

nn

n

nk

d v

26

8

*Re2


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Turbulent flow - Q fromknown pressure drop

Solution to turbulent equation - note that f occurs on both sides of equation:

estimate Q from laminar equation,calculate v and Re,calculate f from wall shear and friction factor

equations, then square root of f ,calculate RHS of D&M correlation, andcheck agreement, if doesnt then . theflow rate - iterate until it agrees.


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Turbulent flow - Q fromknown pressure drop

Friction factor equation:

22 v R f w

Wall shear equation:

LPa

Rw 2


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Summary for suspensions

For Newtonian:Use Krieger for viscosity f(C) and use mean

suspension density, thenTreat as homogeneous fluid (i.e. CGA001)

For non-Newtonian

Wilkinsons equation for LAMINAR Dodge & Metzner for TURBULENT


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Pneumatic conveying

Distinction between homogeneous (+slip)and heterogeneous:


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Pneumatic conveying


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Pneumatic conveying

Positive pressure:


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Pneumatic conveying

Negative pressure:


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Pneumatic conveying

Mixed:


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Pressure drops inpneumatic conveying

acceleration of the gas - Bernoulliacceleration of the solids - Bernoulli

friction of gas on pipe wall - friction factorfriction of solids on pipe wall - friction factorstatic head of gas - Bernoulli

static head of solids - Bernoulliadditional drop due to bends

See Fundamentals Problem 9.6


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Saltation velocity

Comes from Rizk correlation:)1 /(1)22 / (2 / 104

bbbas

salt

Dg M U

Dimensional constants in SI units

96.11440 xa

5.21100 xb

M s

is mass flow rate (kg/s) and D is pipe diameter (m).


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Slip velocity (solid-gas)

Solids will slip in the gas flow:

)0638.01(5.03.0

sos xU v

Dimensional constants in SI units, empiricalequation relating solid velocity to superficial

gas velocity.


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Dense phase design

Difficult!Dense phase design:

http://www.cheresources.com/pnuconvey.shtml


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Hydraulic transport

Firstly, identify occurrence of boundarybetween homogeneous and heterogeneous

transport.4 / 12 / 1)(9.11 x DU v t t

Empirical correlation due to Kim et al, 1986, Int.Chem. Eng ., p 731.


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Hydraulic transport

Secondly, use homogeneous non-Newtonian (or Newtonian) transport

equations - if appropriate.If heterogeneous, correlation due toDurand (1953) but much better to

empirically investigate own materials.


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Powder Flow

Powder flow issuesHopper failureExplosionPowder flood

Hopper dischargeMass flowCore flowWall and powder pressure - FRICTION

Testing


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Powder Flow & Storage

Definitions:Hopper:

Conical section, bottom

BinCylindrical section, top

SiloUsed for both

Interchangeable in use


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Powder Flow Disasters

Powder flood Silo failure

Images removed from copyright reasons.

For a suitable example please see

http://www.jenike.com/Solutions/silofail.html

Image created by R J Leask found athttp://picasaweb.google.com/rjleask http://creativecommons.org/licenses/by/3.0/
http://www.jenike.com/Solutions/silofail.htmlhttp://www.jenike.com/Solutions/silofail.htmlhttp://picasaweb.google.com/rjleaskhttp://picasaweb.google.com/rjleaskhttp://www.jenike.com/Solutions/silofail.htmlhttp://www.jenike.com/Solutions/silofail.html


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Explosion

Powder Flow Disasters

Image removed from copyright reasons.For a suitable example please see

http://www.teachersdomain.org/asset/lsps07_int_expldust/
http://www.teachersdomain.org/asset/lsps07_int_expldust/http://www.teachersdomain.org/asset/lsps07_int_expldust/


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Flow Patterns

MASS FLOW: first in first outCORE FLOW: first in last out


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Comparison of flowpatterns

Mass flow Core flow

Flow is uniform and Erratic flow which canwell controlled cause powder to aerate

and flood (avalanche)

No dead (static) regions Static zones at sides- no perishable spoilage - may empty at the end

Channelling and bridging Piping may occur should be absent

Less segregation Particles roll in dischargeTall and thin May have higher capacity

for capital cost

High stress where Arrangement may

direction changes relieve wall stresses


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Angle of Repose

For a FREE FLOWING powder the hopper angleneeds to be greater than the angle of repose forflow to occur. This is typically 30 o BUT a differentapproach is required for COHESIVE powders.Angle of repose is difficult to measure - best topour powder into an upside down glass funneland carefully remove to leave heap in place.


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Bulk Density

Is the combined density of the powder andthe void space. Remembering the definitionof porosity:

Porosity = = void volume/total volumeHence the bulk density will be:

the above densities are, in order: bulk, solid& fluid. If the fluid is air the furthest rightterm can be ignored.

sb )1(


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Pressure transmission andpowder discharge

Unlike fluids there isn't a linear increase inpressure with height - for all heights. In fact,the pressure stabilises after a few metres andthe rate of discharge from a hopper will,therefore, be remarkably constant. For freeflowing powders the empirical equation:

where D is the opening diameter. Note that thisequation does not include powder height.

tan2

4

5 g D M b


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Pressure transmissionJanssens analysis

where P vo is the pressure at z=0, called the'surcharge' or uniform stress applied at the top of thepowder. For P

vo=0 and at small values of z:

as exp(- Az) 1 - Az for low z

Thus, - a similar result to that of liquids BUT only for

small values of z. At large values of z:as the exponential term disappears.

i.e. pressure asymptotes to the above uniform value.

) / 4exp() / 4exp(14

d kzPd kzk

gd P wvow

w

bv

zd

k k

gd P w

w

bv

44

k gd

Pw

bv

4

f


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Importance of Janssenswork

Stress is not transmitted ina similar way to hydraulic

head, andWall friction has a verysignificant influence on theinternal powder stresses.


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Hopper design

Mass flow discharge is basedupon two factors:the hopper angle steep enoughand the discharge opening wideenough to provide the flow.

The Powder Flow Function (PFF orsometimes called the Material FlowFunction), characterises the ease, orotherwise, of powder transport andstorage.


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Stable Arch Formation

Thus the minimum hopperopening diameter needs tobe

g H f

Bb

c

)(

The main stage is to identify theunconfined yield stress for a powderinside a hopper, and to know moreabout the functional relation H( ) .

M h i l d i i l


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Mohrs circle and principalplanes

a

a

The maximum principal plane stress for the circle formed byconditions of a and a is given by the Mohr's circle drawnthrough those points and is read off at the =0 axis.

The unconfined yield stress is the stress (Pa) given by theMohr's circle that goes through the origin AND is a tangent tothe yield locus. It is the maximum principal plane stress for

this circle.

M i l P d Fl


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Material or Powder FlowFunction

U n c o n

f i n e d y

i e l d s t r e s s

Maximum principal stress

PFF

Obtained from a series of yield locii giving themaximum principalstress and unconfinedyield stress; one datapoint from each yieldlocus.


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Jenike shear cell

Two rings are used. The powder inthe rings has a consolidating(normal) load applied. This load isremoved and a lower load used,together with a shear stressapplied via the bracket on theside of the top ring.

When the shear stress is sufficient the top ring willslide over the bottom, and the powder hassheared. This gives one value for shear andconsolidating stress, that may be plotted on a

Mohr circle.


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Useful sitesDescription of Jenike and other techniques for yield locusdetermination then how to use the data for hopper design.

http://members.aol.com/SchulzeDie/grdle1.html

Also, try the freeware program spannung.exe

A well known name and company with many useful resources:

http://www.jenike.com

On-line magazine for powder and bulk handling:

http://www.powderandbulk.com/

Highly recommended article on different flow types:

http://www.erpt.org/992Q/bate-00.htm

and more generally on this subject:

http://www.erpt.org/technoar/powdmech.htm

http://www.erpt.org/technoar/powddyna.htm


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This resource was created by Loughborough University and released as an open educational resource through

the Open Engineering Resources project of the HE Academy Engineering Subject Centre. The Open EngineeringResources project was funded by HEFCE and part of the JISC/HE Academy UKOER programme.

2009 Loughborough University

This work is licensed under a Creative Commons Attribution 2.0 License .

The name of Loughborough University, and the Loughborough University logo are the name and registered marks of Loughborough University. To the fullest extent permitted by law Loughborough University reserves all its rights in its nameand marks which may not be used except with its written permission.

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