Two Phase Flow, Rheology and Powder Flow

Chapters 6, 9 & 10 in Fundamentals

Professor Richard Holdich

R.G.Holdich@Lboro.ac.uk Course details: Particle Technology, module code: CGB019 and CGB919, 2nd year of study.

Watch this lecture at www.vimeo.com

Also visit; http://www.midlandit.co.uk/particletechnology.htm

for further resources.

Two Phase Flow, Rheology and Powder Flow

Rheology – Section 6.7 Homogeneous systems; Newtonian and

non-Newtonian, laminar/turbulent Homogeneous but with slip

pneumatic conveying - dilute phase Heterogeneous systems

pneumatic conveying - dense phase hydraulic conveying

Powder flow

Flow of dispersions

Rheograms

Non-time dependent

r

vR

d

d)(

n

r

vkR

d

d)(

Newtonian:

Power law:

Rheograms

Time dependent

Apparent viscosity

Is the viscosity of a Newtonian fluid that flows under the same conditions of shear rate and stress as the non-Newtonian fluid.

R (Pa)

dv/dr (s-1)

Apparentviscosity

Apparent viscosity

In order to use Newtonian flow equations we really need “apparent viscosity for pipe flow” - from the “flow characteristic”, etc.

In order to predict flow rate and pressure drop use simpler approach - appropriate to power law fluids.

n

r

vkR

d

d)(

Force balance on a wall gives: L

PaRw 2

Wilkinson’s equation

Combine the power law viscosity equation with the shear stress on the wall - much like the derivation of Hagen’s equation and integrate to give:

n

Lk

Pa

n

anQ

/13

213

Laminar flow of non-Newtonian power law fluids and suspensions.

Turbulent flow

Generalised expression based on a friction factor:

)(Re,f2 2

nv

Rf

For Newtonian fluids:

3.02

Reln5.22

2/12/1

ff

Turbulent flow

Dodge and Metzner - turbulent flow power law fluid:

2.1

)2/1(75.0

2/1 4.0Re*ln

4

nf

nf n

Turbulent flow

Need a Reynolds number that reduces to Newtonian equation when n=1, and the turbulent friction expression should reduce to Wilkinson’s equation given f=16/Re* - i.e. for laminar flow.

n

nn

nnk

dv

268

*Re2

Turbulent flow

The Generalised Reynolds number - threshold value of 2000 for laminar to turbulent flow.

n

nn

nnk

dv

268

*Re2

Turbulent flow - Q from known 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, and check agreement, if doesn’t then

…………. the flow rate - iterate until it agrees.

Turbulent flow - Q from known pressure drop

Friction factor equation:

22 v

Rf w

Wall shear equation:

L

PaRw 2

Summary for suspensions

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

mean suspension density, then Treat as homogeneous fluid (i.e.

CGA001) For non-Newtonian

Wilkinson’s equation for LAMINAR Dodge & Metzner for TURBULENT

Pneumatic conveying

Distinction between homogeneous (+slip) and heterogeneous:

Pneumatic conveying

Pneumatic conveying

Positive pressure:

Pneumatic conveying

Negative pressure:

Pneumatic conveying

Mixed:

Pressure drops in pneumatic conveying

acceleration of the gas - Bernoulli

acceleration of the solids - Bernoulli friction of gas on pipe wall - friction

factor friction of solids on pipe wall - friction

factor static head of gas -

Bernoulli static head of solids - Bernoulli additional drop due to bends

See Fundamentals – Problem 9.6

Saltation velocity

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

bbbas

salt

DgMU

Dimensional constants in SI units

96.11440 xa

5.21100 xb

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

Slip velocity (solid-gas)

Solids will slip in the gas flow:

)0638.01( 5.03.0sos xUv

Dimensional constants in SI units, empirical equation relating solid velocity to superficial gas velocity.

Dense phase design

Difficult! Dense phase design:

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

Hydraulic transport

Firstly, identify occurrence of boundary between homogeneous and heterogeneous transport.

4/12/1)(9.11 xDUv tt

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

Hydraulic transport

Secondly, use homogeneous non-Newtonian (or Newtonian) transport equations - if appropriate.

If heterogeneous, correlation due to Durand (1953) but much better to empirically investigate own materials.

Powder Flow

Powder flow issues Hopper failure Explosion Powder flood

Hopper discharge Mass flow Core flow Wall and powder pressure - FRICTION

Testing

Powder Flow & Storage

Definitions: Hopper:

Conical section, bottom

Bin Cylindrical section,

top Silo

Used for both Interchangeable in

use

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 at http://picasaweb.google.com/rjleaskhttp://creativecommons.org/licenses/by/3.0/

Explosion

Powder Flow Disasters

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

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

Flow Patterns

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

Comparison of flow patternsMass flow Core flow

Flow is uniform and Erratic flow which can well 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 discharge

Tall and thin May have higher capacityfor capital cost

High stress where Arrangement may direction changes relieve wall stresses

Angle of Repose

For a FREE FLOWING powder the hopper angle needs to be greater than the angle of repose for flow to occur. This is typically 30o BUT a different approach is required for COHESIVE powders. Angle of repose is difficult to measure - best to pour powder into an upside down glass funnel and carefully remove to leave heap in place.

Bulk Density

Is the combined density of the powder and the void space. Remembering the definition of porosity:

Porosity = = void volume/total volume

Hence the bulk density will be:

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

sb )1(

Pressure transmission and powder discharge

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

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

tan2

45gDM b

Pressure transmission Janssen’s analysis

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

dkzPdkzk

gdP wvow

w

bv

zd

k

k

gdP w

w

bv

4

4

k

gdP

w

bv

4

Importance of Janssen’s work

Stress is not transmitted in a similar way to hydraulic head, and

Wall friction has a very significant influence on the internal powder stresses.

Hopper design

Mass flow discharge is based upon two factors: the hopper angle steep enough and the discharge opening wide enough to provide the flow.

The Powder Flow Function (PFF or sometimes called the Material Flow Function), characterises the ease, or otherwise, of powder transport and storage.

Stable Arch Formation

Thus the minimum hopper opening diameter needs to be

g

HfB

b

c

)(

The main stage is to identify the unconfined yield stress for a powder inside a hopper, and to know more about the functional relation H().

Mohr’s circle and principal planes

a

a

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

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

Material or Powder Flow Function

Unc

onfi

ned

yiel

d st

ress

Maximum principal stress

PFF

Obtained from a series of yield locii giving the maximum principal stress and unconfined yield stress; one data point from each yield locus.

Jenike shear cell

Two rings are used. The powder in the rings has a consolidating (normal) load applied. This load is removed and a lower load used, together with a shear stress applied via the bracket on the side of the top ring.

When the shear stress is sufficient the top ring will slide over the bottom, and the powder has sheared. This gives one value for shear and consolidating stress, that may be plotted on a Mohr circle.

Useful sitesDescription of Jenike and other techniques for yield locus determination – 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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