Particulate Systems Lecture Note

8/12/2019 Particulate Systems Lecture Note

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Kumasi Polytechnic

Department of Chemical Engineering

CME 309: Particulate Systems

Lecture Note

Prepared

By

Francis Attiogbe

March, 2014


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Reference Textbooks

1. Unit operations of Chemical Engineering by Warren L. McCabe, Julian C.

Smith & Peter Harrriott

2. Chemical Engineering, Volume Two by J.M. Coulson & J.F. Richardson

Size Reduction Processes

The term size reduction is applied to all the ways in which particles or solids are cut

or broken down into smaller pieces. Comminution is the general term used for size

reduction.

Purposes of Comminution

1. Preparation e.g. chunks of crude ore to workable size by crushing

2. To help chemical reaction by increasing the surface area of material3. It permits separation of unwanted ingredients by mechanical methods

4. It reduces the bulk of fibrous materials for easier handling and waste for waste

disposal.

Basic Operations of Comminution

Solids may be broken in many different ways but only four of them are commonly

used in size reduction machines: 1. Compression 2. Impact 3. Attrition or rubbing and

4. Cutting

Selection of size reduction equipment is based on 1. Feed size 2. Hardness of material

3. Ratio of size reduction

Mohr Scale of Hardness

The hardness of a mineral as measured by the Mohr scale is a criterion of its

resistance to crushing. It is a fairly good indication of the abrasive character of the

mineral, a factor that determines the wear on the grinding media. Arranged in

increasing order of hardness, the Mohr scale is as follows:

1. Talc

2. Gypsum

3. Calcite

4. Fluorspar

5. Apatite

6. Felspar

7. Quartz

8. Topaz

9. Carborundum

10.Diamond


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Materials of hardness 1-3 inclusive are classed as soft, 4-7 as intermediate and the

others as hard.

Particle Size Distribution

It is the most description of solid from size reduction operations.

Average Particle Size of a Sample

It can be defined in several ways:

_

Volume- Surface Mean Diameter (ds): It is the most used diameter and can be

calculated from the mass fraction.

_ n _

ds= 1/ xi/ di

i =1

di = average particle diameter taken as the arithmetic average of the diameter of thelargest and smallest particles in increment.

xi= mass fraction in a given increment

=

=n

ii

i

s

d

xd

1

1

Particle size Particle size

Differential Analysis Cumulative Analysis

%massfraction

Cumulativemass

fraction


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Arithmetic Mean Diameter

_ n _ n

dN= Nidi Nii =1 i =1

Ni = number of particles in each fraction.

=

=

=n

i

i

i

n

i

i

N

N

dN

d

1

1

Volume Mean Diameter

_ 1/3dv= 1

n _

xi/di3

i = 1

31

1 3

1

=

=

n

i

i

v

id

xd

_

Volume mean diameter

vd or mass mean diameter in terms of particle numbers, rather

than mass fraction is given by:

_

dv= nidi4/ nidi

3

=

3

4

ii

ii

vdn

dn

d

For samples consisting of uniform particles these average diameters are of course all

the same. For mixtures containing particles of various sizes, however, the several

average diameters may differ widely from one another.


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Particle Size Measurement

Sieve Method

Sieve analysis may be carried out for particle size (>50m) using a nest of sieves,

each lower sieve being of smaller aperture size.

Microscopic Method (1-100m)

In microscopic methods of size analysis, direct measurements are made on enlarge

images of the particles. A small is put on a glass and placed under the microscope.

Sedimentation Method (>1m)

This method depends on the fact that the terminal falling velocity of a particle in fluid

increases with size. Sedimentation methods are of two main types. In the first, the

pipette method, samples are abstracted from the settling suspension at a fixedhorizontal level at intervals of time.

t = h/v = (9h/2g)(2/d)2(/sf)

h = height of liquid in the cylinder

d = particle diameter

v = velocity of particle

= dynamic viscosity of fluid

s= density of solid particle

f = density of fluid

Other particle size measurement methods are :

Permeability methods ( 1m)

Electronic particle counters

Laser diffraction analyzers

X-ray or photo-sedimentometers

Sub-micron particle sizing

Work Required for Size Reduction

There are empirical relations which are used:

1. Kick's Law

He proposed that on the bases of stress analysis for plastic deformation within the

elastic region the work required for crushing a given quantity of material is constant

for the same reduction ratio irrespective of the original size.

m

P=

m

W= CkIn

2

1

d

d


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P = power (kW)

W= energy input

Ck= constant

d1= feed mean particle size

d2 = product mean particle size

m = feed rate kg/s

m = mass (kg)

2. Rittingers Law

It states that the work consumed in crushing is proportional to the new surface

produced.

m

P=

m

W= CR

2

1

d -

1

1

d

CR= constant

It applies to fine grinding and ball milling.

3. Bonds Law

m

P=

m

W= CB

2

1

d -

1

1

d

CBis a constant that depends on the type of machine and the material being crushed.

To use the above equation a work index Wi defined as the gross work (energy)

required to reduce a very large feed to such a size that 80% of the product passes a

100-m screen is introduced. This definition leads to a relation between CBand Wi.

CB=310100 Wi= 0.3162 Wi

m = ton/h

P = kWd = mm

For dry grinding, Wimust be multiplied by 1.34 i.e. 1.34Wi.


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Example Question

What is the power required to crush 100ton/h of limestone if 80% of the feed passes a

2-in screen and 80% of the product a 81 -in screen?

Solution

From Table, the work index Wifor limestone is 12.74.

m = 100t/h

d1= 2in = 2 25.4 = 50.8mm

d2= 81 25.4 = 3.175mm

Power required P =

m 0.3162Wi

2

1

d -

1

1

d

=1000.316212.74175.3

1 -

8.50

1

= 169.6kW

Energy Balance of Comminution Process

Win= Wuseful+ Wloss

Wuseful


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Classification of Comminution Equipment

Classification is done according to:

1. principle of operation

2. capacity

3. particle size (d1)4. wet or dry crushing

Feed size Product size

Coarse crushers 1500-40mm 50-5mm

Intermediate crushers 50-5mm 5-0.1mm

Fine crushers 5-2mm 0.1mm

Wet grinding is generally applicable only with low speed mills.

Advantages of Wet Grinding

1. Specific energy requirement is less

2. The number of auxiliary plants is less therefore less energy consumption

3. The material handling is simpler

4. The particle size of the product is more uniform

5. Better cooling because energy loss appears as heat

6. the tendency of agglomeration in the product is less

7. The fluid in which the particles are dispersed can act as a grinding aid.

8. Dust formation is eliminated

Disadvantages

1. Needs a lot of fluid (water)

2. The product has less fine particle and the more uniform particle size of the

product in some cases is not preferable

3. The cost of drying

Size Reduction Equipment

Size reduction equipment is divided into crushers, grinders, ultrafine grinders andcutting machines.

CRUSHERS

They do the heavy work of breaking large piece of solid materials into lumps. Primary

crushers accept anything that come from the mine and break it into 150 200 mm

lumps. Secondary crushers reduce these lumps into 6mm.


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GRINDERS

They reduce crushed feed to powder. The product from an intermediate grinder must

be under 0.42mm but most of the product from a fine grinder would be under

0.074mm = 74m.

ULTRAFINE GRINDERS

They accept no larger than 6mm and the product is typically 1 50 m.

CUTTERS

They give particles of definite size and shape; 2-10mm.

Crushers (Coarse and fine)

1. Jaw Crusher

2. Gyratory Crusher

3. Roll Crusher

4. Toothed roll Crusher

Grinders

1. Hammer mill

2. Rolling compression mill (ball & roller mills)

3. Attrition mills

4. Tumbling mills

a. Rod mills

b. Ball mills; pebble mills

c. Tube mills; Compartment mills

Ultrafine Grinders

1. Fluid-energy mills

2. Agitated mills

3. Special hammer mills

Cutters

1. Dicers

2. Slitters


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CRUSHERS

Jaw Crusher

Blake jaw crusher is hinged at the top. Dodge jaw crusher is hinged at the bottom. The

reduction ratio of this crusher is within 3-5 to 1 i.e. d1/d23-5 to 1.

Capacity: 10 t/h 600 t/h

d1max up to 3m.

The jaws open and close 250 400 times per minute,

Advantages

Very simple and can accept up to 3m

Disadvantages

1. Abrasion

2. Dynamic stress is high

3. Very noisy

n

Feed Pivot

Discharge

Toggle Toggle

Pitman

Flywheel

Eccentric shaft

Fixed Part

or

Stationary

Jaw Swinging Jaw


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Gyratory Crusher

Conical crushing head inside a funnel shaped casing. An eccentric drives the bottomend of the shaft. The solids within the crushing head and casing are broken down until

they are small enough to come out from the bottom.

The discharge from the crusher is continuous. The load on the bottom is nearly

uniform. Less maintenance is required than in the jaw crusher and energy requirement

per ton of material crushed is smaller than in the jay crusher.

The speed, n of the crushing head is typically 125 425 gyrations per minute.

Capacity up to 4500 ton/h. d1max ; up to 2m.

It has much higher capacity than the jaw crusher if the heat capacity is the same. The

capacity varies with the impact strength of the feed, speed of gyration (n) and the jaw

setting. Operates best on hard and brittle materials and do not work on plastic

minerals like clay.

Dischar e

n

Eccentric

Crushing head

Casing

Pivoted at the to

Feed


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Cone Crusher

It is a version of the gyratory crusher for secondary reduction. The shape of the conedetermines the minimum size d2and maximum size d1. Usually n: 400 570 gyration

/ min. Capacity up to 1000 ton/h. It is the most commonly used on ores compared to

roll and hammer mills.

Discharge

Pivot at the top

n

Gearing

Eccentric

Cone

Feed


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Smooth-Roll Crushers

Roll = 600 2000mm

n: 50 3000 r/min

d1 = 12 75 mm

d2 = 12 0.84mm

d1/d2= 3 : 1

The basic operation of the smooth roll crusher is compression. The periphery speed is

1 6 m/s. The typical rolls are between 600 2000mm. The rotating speed is between

50 3000 r/min.

Operation

The two rolls of the same rotate at different speeds. The distance between the rolls

is negligible. Roll faces usually carry corrugated breaker bars or teeth. These kinds of

roll crusher which may contain a rotating roll working against a stationary curved

Discharge

Spring

roll

Feed

d


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breaker plate. Toothed roll crushers are much more versatile than the smooth roll but

cannot handle very hard solids and they operate by compression, impact and shear.

They can reduce much larger particles. The feed size up to 500mm. Capacity up to

500 ton/h and reduction ratio is 4:1. The maximum size of particles depends on

friction between the particle and surface of the roll. It can be estimated from the

relation:

d1 max = 0.04R + d2max

2

R = radius of the roll.

Smooth roll crushers give few fines and no oversize. They are often arranged in

series. Each pair of rolls being closer than the previous one. Rolls are most widely

used for materials which are sticky usually due to high moisture content or clayey

nature. Coal, lignite, petrol coke, gypsum, limestone are materials which can be

ground in the roll crusher.


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GRINDERS

1. HAMMER MILL

d2 = 300mm

d1 = 500mm 1m

n = 50 1800 times/min

Rotor disk = 150 - 450mmPeripheral speed of the hammer tips 20 50m/s

Weight of hammer 50 100kg

Capacity = 0.1 1500 ton/h

Operation

This mill contains a high speed rotor turning inside a cylindrical casing. The feed

drops into the top of the casing. It is broken and falls through a cylindrical grating. Insmaller machines we have flying hammers swinging freely and in larger machines the

Feed

Grate bars

Screen

Hammer

Product

Product

Rotor


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hammer are fixed rigidly to the rotor. The mechanism is impact. The hammer mill

grinds almost anything tough fibrous, solids, sticky clay and hard rock. For fine

reduction they are limited to the softer materials.

Ball Mills

L : D = 1 : 1 - 2 : 1

d1 ~ D/20

They have a conical or cylindrical shell usually made of steel slowly rotating on a

horizontal axis and charge with grind medium. This grind medium can be metal rod (

in this case the mill is called rod mill) or it can be charged with lengths of chain or

ball. The ball can be made of metal, rubber or wood, porcelain, flint or zircon. If the

balls look like a pebble the mill is called pebble mills.

The shell is lined with high-carbon steel plates, porcelain, silica rock or rubber. The

mill can be operated wet or dry in either batch or open circuit continuously or closed

circuit with size classifiers. The load of ball normally occupy about 1/5 - of the

mill when the mill is stopped. The void space in the static bulk volume of balls is

approximately 41%. Since the medium is expanded as the mill is rotated the actual

Feed

Drive Gear

Discharge

L

Large

Balls

Small

Ball


17/75


running volume is not known. The amount of material in the mill is expressed by the

material to the volume ratio. If the solid material just fill the void this ratio is 1.

The rotation speed should be high enough to raise the ball to 2/3 of the .

Autogenous Mill

There are no grinding medium just the feed. The feed is not hard. The of these mills

are very large compared to their length and can be used for ores as mined. It is very

cheap and the reduction ratio can be very large. The feed can be up to 25cm and the

product d2 < 1mm. It is good for materials with a granular texture. Aside these

especially, its just like the Ball mill.

Autogenous grinding has two advantages:

(1)it reduces metal wear and (2)eliminates secondary and tertiary crushing

stages. Thus it offers a savings in capital and operating costs.

ULTRAFINE GRINDERS

It reduces solids to 1 to 20m but all particles are below 45m. One member of the

ultrafine grinders is the hammer mill. They can be used to make fine products. The

rotor diameter is 10 60 cm. The maximum rotating speed nmax should be 1600

3400 rev/min and up to 3 4cm. The capacity is 0.05 7 tonnes.

It can be used for non aggressive material like sugar, pharmaceuticals, plastics, dye

colour, cosmetics, coal and coke in the petroleum industries.


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Fluid Energy Mills

The solid particles are suspended in air or super heated steam and conveyed at high

velocity in a circular or elliptical path. Some reduction occurs as the particles rubbedagainst the walls of the closed chamber but most of the reduction is believed to be

caused by interparticle attrition. Internal classification keeps the larger particles in the

mill. The suspending gas admitted at about a pressure of 7bar through the energizing

nozzles. Classification of the ground particles takes place at the upper end of the loop.

As the gas stream, coarser particles are thrown outward against the outer wall while

the fines congregate at the inner wall. There is a discharge opening at the inner wall

and this lead to a cyclone separator and a bag collector for the product.

Energizing Nozzles

Air or Steam

Venture

injector

Feed

Classifier

outlet

Material and

fluid outlet

Reduction

Chamber


19/75


Remarks on the Selection of Equipment

Very important that the feed must have a suitable size and enter at a uniform rate. The

product must be removed as soon as possible after the particles are of the desired size.

Unbreakable materials must be kept out of the machine. For heat sensitive materials

the mill must be cooled. Because of this some auxiliary equipment are important such

as coolers, metal separators, blowers and constant rate feeders.

Open Circuit Operation

It is one in which we let the feed goes through the mill only once and the oversize not

return. The energy consumption is very high in this operation.

Closed Circuit Operation

It is one in which the oversize is returned to the mill after passing the feed. Usually it

is economical to remove partially ground materials from the mill and passes it through

a size separation device. The undersize then becomes the product. The oversize is

returned to the mill. In this case the energy consumption is less to achieve the same

particle size.


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Flow sheet for close circuit grinding

In a system like this the energy has to be supplied to the conveyor belts, the different

separators but still the total energy required is 25% less than the energy consumed

compared to an open circuit. The charge and the discharge rate must be equal.

FeedCoarse recycle

Rod Mill

Ball Mill

Coarse

Intermediate

Fine

Centrifugal

classifier

Repulper

Water

Screens

Gyratory

Crusher

Oversize

Product

Slurry

Oversize

Recycle


21/75


Size Separation Processes

Separation of solids from gases, liquid drops from gases, solids from solids and solids

from liquids. Two (2) general methods exist:

1. Screen or Filter It retains one component and allows other to pass.

2. Sedimentation It utilizes the difference in the rate of settling of particles or

drops as they move through a liquid or gas. Other methods are floatation

utilizes the difference in wettability. Magnetic Separation utilizes the

properties of a magnet.

All the methods are based on the physical difference between the particles.

Screening

It separates the particles according to their size. The undersize is called fines and the

oversize called tails. For only one screen we have unsized fraction. Having series of

screens will result in sized fraction. The screen can be wet or dry.

The industrial screens are made from woven wire, silk or plastic cloth, metal bars,

perforators or slotted metal plates. The most common material is steel or stainless

steel. Standard screens are ranged from 10mm - 45m and between 4mm 300m is

called ultrafine screening.


22/75


Mechanically vibrator

Gyratory in a horizontal plane

Duct

Eccentric

Gyration in vertical plane

Shaking

Gyratory at one end,

shaking at other end

Eccentric

Electricallyvibrated


23/75


The screens used to be 16 30oto the horizontal. The screens are usually rectangular

and fairly long. Their sizes lie between 50cm x 130cm to 160cm x 370cm. The rate of

gyration n is between 600 1800 r/min. Motor size for 160 x 370cm is about 1 2

kW. The speed of gyration and angle of tilt can be adjusted.

The screens angle influences the capacity of the screen. The steepest possible angle is

the best. Usually the screen surfaces are doubled. The greatest difficulty in screening

is the blinding of the screen because the particles half way can stack in the screen

openings. Near size particles can cause blindness of the screen. Sticky, soft, elongated

and flaky particles tend to blind the screen and are more difficult to handle. The

screens which are rapidly vibrated and small amplitude are less likely to blind than the

gyratory screens. The vibration screens vibrate between 1800 3600 vibrations/min.

Grizzly

It is a grid of parallel metal bars set in an inclined stationary frame. The slope and the

path of the material are parallel to the length of the bars. It is used for very coarse feed

from a primary crusher. The particle sizes are 5 20cm.

Ideal and Actual Screens

Ideal screen will separate the feed mixture into two fractions.

Cut diameter dc: - Which mark the point of separation between the fractions. In ideal

screen the size of the opening is equal to the dc.

Actual Screens It does not give perfect separation as the cut diameter. The

undersize would contain particles larger than the cut diameter and the oversize

containing materials finer than the cut diameter.

The commercial screens give poorer separation than testing sieve on the same

mixture. In testing sieve the time alone can be as long as necessary. In industrial

screening each particle may be presented to the screen surface max 10 12 times

before being removed as oversize. The capacity of the screen is the mass of the

material that can be fed per unit per unit area. Factors which reduce capacity and

lower efficiency or effectiveness are:

1. Interference of the bed of particles with the motion of the other particles.


24/75


2. The blinding

3. Cohesion of particles to each other

4. Adhesion of particles to the screen surface

5. The large particles tend to segregate in a layer next to the screen and so

prevent the smaller particles from reaching the screen.

6. The particles usually do not approach perpendicular to the surface of the

screen.

Material Balances over screen

Let F, D, and B be the mass flow rates of the feed, overflow, and underflow,

respectively, and xF, xD, and xB be the mass fraction of oversize materials in these

three streams. The mass fractions of the undersize materials in the feed, overflow, and

underflow are 1- xF, 1- xD, and 1- xB.

Since the total material fed to the screen must leave it either as underflow or as

overflow,

F = D + B

The material balance on the oversize materials in the streams

BDF BxDxFx +=

Eliminating B from the above equations give:

BD

BF

xx

xx

F

D

=

Elimination of D gives:

BD

FD

xx

xx

F

B

=

Screen Efficiency

The efficiency of a screen is a measure of the success of a screen in closely separating

materials. This is commonly done by taking the ratio of the oversize material that is

actually in the overflow to the amount entering with the feed. Thus the screen

efficiency based on the oversize is given by:

F

D

DFx

DxE =

Similarly, an efficiency EBbased on the undersize materials is given by:


25/75


( )( )F

BB

xF

xBE

=

1

1

Screen Capacity

The capacity of a screen is measured by the mass of material that can be fed per unit

time to a unit area of the screen.

Classification

The term classification is usually restricted to processes in which particles are

distinguished by their different rate of travel through a fluid media.

Sedimentation

Law of Settling

Three forces act on a particle moving through a fluid. These forces are the external

forces Fe (gravitational or centrifugal), buoyant force Fb and finally the drag force.

Drag force appears whenever there is a relative motion between the particles and the

fluid.

V = Velocity of the particle relative to the fluid = density of the fluid

p = density of particle

a = acceleration

Fe = m ae

Fb = mae

p

FD = CDV2AP

2

Fb

FD

Fe


26/75


CD = drag coefficient (dimensionless)

Ap = projected area of the particle measured in a plane perpendicular to the

direction of motion of the particle.

The resultant force on the particle is given by:

2

2

2

2

PD

p

p

PD

p

e

e

Dbe

AVCaem

dt

dvm

AVCamma

dt

dvm

dt

dvmFFF

=

=

=

If the external force is gravity, ae= g, the acceleration due to gravity and the above

equation

Becomes

m

AVCg

dt

dv PD

p

p

2

2

=

The drag force increases with velocity; so the particle quickly reaches free settling

velocity or terminal velocity Vt. p

The terminal velocity Vt, for gravitational settling is found by taking 0=dt

dv, then

gmAC

VPDp

p

t 21

=

Centrifugal field; ae= r2

r = radius of path of particle

= angular velocity, radian/s

rmAC

VPDp

pt 21

=

For spherical particles the mass;

46

23d

Aandd

m pp

==

Substituting m and Apinto Vtfor gravity settling, gives the equation for gravity

settling of sphere;

( )

D

p

tC

dgV3

4 =


27/75


1. Low Re, Re< 1, laminar flow

In laminar flow tconsCD tanRe

24==

tVd

=Re ; dynamic viscosity of fluid

( )lawStokes

18

2

=

pgd

2. For turbulent flow 1000 < Rep< 200,000 CD= 0.44

lawsNewtondgV

dgV

p

t

p

t

'75.1

44.03

4

=

=

=

3. 1 < Rep< 1000 intermediate region

6.0Re

5.18=

DC

CRITERION FOR SETTLING REGIME

tVd=Re

Substituting( )

18

2gd

V p

t

= for Stokess law range

( )

( )

( ) 31

2

3

2

3

3

2

3

18Re

18

18Re

=

=

=

=

p

p

p

gdk

K

gdK

gd

If Re 1; k 2.6

If k 2.6 Stokess law.

For the Newtons law range substitute the Newtons terminal velocity into the Re.This gives:


28/75


( )

Re75.1

75.1

75.1

75.1Re

5.1

5.1

3

2

3

==

=

=

K

K

K

gd p

Re > 1000 K > 68.9

Re < 200,000 K < 2360

If 68.9 < K < 2360 Newtons law applies.

Critical Particle Diameter (dcrit)

If Re = 1

( )

( )3

2

2

18

18

1Re

gd

gdV

Vd

p

crit

p

t

tcrit

=

=

==

dcrit

is the maximum of the particle size by which Stokess law is applied.

E.g.

Estimate the terminal velocity of 80 to 100 mesh particles of limestone (p=

2800kg/m3) falling in water at 30oC. (b) How much higher would the velocity be in a

centrifugal separator where the acceleration is 50g?

d for 100 mesh = 0.147mm

d for 80 mesh = 0.175mm

= 0.801 cp; = 995.7kg/m3.

Solution

161.0

2

175.0147.0

=

+=avgd

To find which settling law applies, calculate K


29/75


( )

( )

( )86.4

10801.0

801.97.99528007.99510161.0

31

23

3

32

=

=

=

gdk

p

Slightly above Stokess law range, Assume Re = 5 CD= 7.2

( )

( )( )

( )

( )

1.82Re

41.0

3

504

15.190Re

9.17

5086.4

50

50)(

86.4

10801.0

023.07.99510161.0

Re

Re

023.0

2.77.9953

807.910161.07.99528004

3

4

3

31

2

3

3

3

=

=

=

==

=

=

=

=

=

=

=

=

=

=

Check

sm

C

gdV

CEstimate

gdk

gaeb

Vd

Check

C

gdV

D

p

t

D

p

t

D

p

t

E.g. A discrete spherical particle has a diameter of 0.15mm and a specific gravity S.G

of 1.1 calculates the settling velocity in water of 20oC.

[Kinematic viscosity of water at 20oC is 1.1 x 10-6m2/s]

Assume that the flow is laminar.


30/75


Vt = gd2(p- )

18

It is necessary to check that this calculated settling velocity produces a value of Re

within the laminar range which is assumed in using the Stokess law/equation.

Re = 0.002 x 1.5 x 10-4

1.01 x 10-6

If the calculated value of R was > 1 it would have been necessary to recalculate v t

using the appropriate equations.

Hindered Settling

In hindered settling, the velocity gradients around each particle are affected by the

presence of nearby particles, so the normal drag correlations do not apply.

Particles in settling displace liquid which flows to the fluid greater than the absolute

settling velocity. The settling velocity also depends on the shape and size distribution.

Discrete particles do not change in size, shape or mass during settling. The theory of

sedimentation assumes the presence of discrete particles.

The drag coefficient values usually 2-5 times greater than the theoretical. Because of

this, for accurate design we need an experiment. For a uniform suspension, the

settling velocity Vscan be estimated from the terminal velocity using this formula

Vs = Vt . n

= Total void fraction

= Vliquid = Vliquid

Vliquid + Vsolid Vsuspension

n = Function of the Reynoldss number

Re 0.1 1 10 102 103

n 4.6 4.3 3.7 3.0 2.5

For very small particles Vs/Vt= 0.62 for = 0.09; 0.095 for = 0.6.

If particles of a given size are falling through a suspension of much finer solids, the

terminal velocity of the larger particles should be calculated using the density and

viscosity of the fine suspension. The Vscan be calculated in this case from Vs = Vt .

n


31/75


= volume fraction of the finer suspension

( )4

15.01

+=s

Applies only when > 0.6 and is most accurate when > 0.9.s = effective viscosity of the suspension

= viscosity of the liquid.

E.g. Particles of sphalerite are settling under the force of gravity in carbon

tetrachloride (CCl4) at 20oC. The diameter of the sphalerite particles is 0.10mm. The

volume fraction of sphalerite in CCl4 is 0.20. What is the settling velocity of the

sphalerite?

Solution

( )

( )29.3

1003.1

81.9159440001594101.0

/1003.103.1

1594

4000

1.0

31

23

3

23

3

3

=

=

==

=

=

=

mNscp

mkg

mkg

mmd

p

The settling is almost in the Stokess law range


32/75


( )

( ) ( )

s

mV

n

VV

V

VV

dV

s

m

s

m

s

m

gdV

s

sphccl

ccl

nn

ts

t

p

t

052.08.0013.0

056.4

8.02.08.0

8.0

013.0

01.2

1003.1

101.0013.01594

Re

013.0

13

101273

1003.118

81.9101.015944000

18

05.4

4

4

3

3

5

3

23

2

==

=

=+

=+

=

==

=

=

=

=

=

=

When the particles are very small, the diameter of the particles (d = 0.1m) settling

does not occur only Brownian movement.

Clarifier

A settler which virtually removed all the particles from a liquid.

Classifier

A settler which separates the solids into two fractions

Sedimentation Process (Gravity Settling Process)

1. B = A Suspension of uniformly distributed in the liquid.

BZ


33/75


2. A = Clear liquid

D = Solids resting lightly on one another

C = transition layer in which the solid concentration vary

3. As the settling continuous, the layers A and D increases. Layer C remains the same

and B decreases.

4 Eventually zone B disappears and all solids are in zones C & D.

5 In compression some of the liquid which accompanied the particles to zone D

splits out the thickness of the zone and D decreases.

The critical point is the moment when compression first occurs.

A

B

A

B

A

D

A

D


34/75


The settling process stops when the weight of the solid reaches mechanical

equilibrium with the compressive strength of the particles. C is the point where the

sludge reaches the ultimate height. In industry, these processes are carried out in a

thickener.

Differential Settling Methods

Most processes which depend on differences in behaviour of particles in a stream of

fluid separate materials according to their terminal falling velocities, which in turn

depend primarily on the density and size and to a lesser extent on shape. Thus, inmany cases it is possible to use the method to separate a mixture of two materials into

its constituents, or to separate a mixture of particles of the same material into a

number of size fractions.

Suppose that it is desired to separate particles of a relatively dense material A (density

A) from particles of a less dense material B. If the size range is large, the terminal

falling velocities of the largest particles of B (density B) may be greater than those of

the smallest particles of A, and therefore a complete separation will not be possible.

The maximum range of sizes that can be separated is calculated from the ratio of the

Ultimate Hei ht

Height of

Sludge ConstantRate

C

Settling time


35/75


sizes of the particles of the materials which have the same terminal falling velocities.

This condition is given by the equation:

s

A

B

d

d=

B

A , where s = 0.5 when Stokes law applies and s = 1 when Newtons

law applies.

It is seen that the size range becomes wider with increase in the density of the

separating fluid and, when the fluid has the same density as the less dense material,

complete separation is possible whatever the relative sizes.

E.g.

It is desired to separate into two pure fractions a mixture of quartz and galena of a size

range from 0.015mm to 0.065mm by the use of hindered settling process. What is the

minimum apparent density of the fluid that will give this separation? Density of

galena is 7500kg/m3and that of quartz is 2650kg/m3.

Solution

Assume shapes of galena and quartz particles are similar.

Required density of fluid when Stokes law applies

0.5

015.0

065.0=

2650

7500 = 2377 kg/m3

Required density of fluid when Newtons law applies

015.0

065.0=

2650

7500 = 1196 kg/m3

Thus the required density of the fluid is between 1196 and 2377 kg/m3.


36/75


Batch Settling Tank

Vo = volumetric flow rate of the feedV1 = volumetric flow rate of the clear liquid

V1 = volumetric flow rate of the under flow or sludge

V0, V1, V2: m3/s

Xo, X1, X2are concentrations (kg/kg)

A = Cross sectional area (m2)

V = Velocity (m/s)

Settling height h (m)

Settling time t (s)

V1 = Vo- V2

For dry matter

V1, X1

Clear fluid

Feed Vo, Xo

V

V2, X2

Sludge


37/75


38/75


Mechanically Agitated Thickener (Dorr-Thickener)

= 10 - 100m

2.5 3.5 deep

n = 0.015 - 0.5 r/min

X2 = max 35 - 55%

Dorr thickener is a large fairly shallow tank with a slow moving rake. The liquormoves radially at a constantly decreasing velocity allowing the solids to settle at the

bottom of the tank. Clear liquor spills over the edge of the tank into a launder. The

rake gently agitates the sludge and move into the centre of the tank. Then it moves

through a large opening to the inlet of a sludge pump.

The volume of clear liquor produced in a unit time depends on the cross-sectional area

available for settling and is independent of the liquid depth. To remove very fine

particles, sometime we use flocculation. Flocculation removes particles of a few

micrometer of diameter and the settling velocity is too low.

Feed

well

VerticalShaft

BladesArm

Discharge cone

Overflow launder

Feed pulp

Mechanism

supportn


39/75


The particles are agglomerated (are made to come together in a group) or flocculated

before they can be removed. Usually the particles carry electric charges. Because of

these charges they tend to remain dispersed. If some electrolyte is added, the ion form

in solution neutralizes the charges on the particles. These neutralize contains many

particles. The greater the value of valence of the ion the more effective the ion as

flocculation agent. The amount used is between 0.4 2% of the weight of the solute

particles.

Flocculation agent can be active SiO2or poly acryl amine. Other flocculation methods

used are surface active agent and the addition of materials such as glue, lime, alumina,

sodium silicate. For the design of a thickener, the smallest particle required to settle so

the retention time (t) should be greater or equal to the settling time of the smallest

particles.


40/75


Centrifugal Settling Processes

Cyclones are very good representative of centrifugal settling equipment

For liquids. For Gas

A cyclone consists of a vertical cylinder with a conical bottom. A tangential inlet near

the top and the outlet for the dust or sludge. The outlet pipe is extended into the

cylinder to prevent short circuiting of fluid from inlet to outlet. The incoming fluid

with the particles travel in a spiral path around and down the cylindrical body of the

cyclone. The centrifugal force moves the particles radially towards the wall and the

particles that reach the wall slide down into the cone and are collected. The clear air

or liquid in the centre of the cyclone moves upward in a small spiral path in the low

pressure.

( )

18

22 =

p

t

rdV

X1, C1clear liquid

R = radi

of cylin

gas

ro

r

R

V

VoSettlingVelocity

Vo

Xo

Co

H

Dust

&

Gas

dust

Discharge (sludge)


41/75


( )r

vdV

p

o

18

22 = Stokess law

For gases pp

o

r

vdV


42/75


= collection efficiency = Co - C1

Co

[g] = g/m3

Co= incoming concentration of the particlesC1= outgoing concentration of the particles

= 70 - 80%

Temp T: up to 100oC Operation temperature &

pressure

Pressure P: up to 5oo x 105Pa

Cyclones are used to remove particles from gas stream. It is a cheapest way of

collecting dust. It is cheapest in operation and design.

Sedimentation Centrifuges

In sedimentation centrifuge a particle of a given size is removed from the liquid if

sufficient time is available for the particles to reach the wall of the separator.

Operation

Surface of

liquid

h

Bowl

wall

ro

R

n

Liquid

Discharge

Trajectory

Feed


43/75


The feed point is at the bottom and the liquid discharge at the top. The heavy larger

solid particles are thrown out of the liquid, finer light particles may not settle in the

time available and be carried out with the liquid.

They also have the separation factor.

9009003600.

4 2222222 rn

g

rn

rg

rn

rg

v

F

Ff

g

c =====

If the separation factor 3000 (i.e. a normal industrial centrifuge)

: 3000 10,000 Super Centrifuge

> 10,000 Ultra Centrifuge

Principle of Centrifugal Separation

We assume that the particle is at all time moving radially with its terminal velocity.

( ) op

r

R

dt ln

182

= in the Stokess law range t = resident time.

dcut = A cut point is the of that particle which just reaches the distance between roand R. A particle of dcutmoves a distance R - ro during the settling time alone.

2

If a particle of dcutis to be removed it must reach the bowl wall in the available time.

The resident timerateflowvolumetric

bowlliquidofvol

V

Vt

o

.==

[V] = m3

[Vo] = m3/s

V = (R

2

- ro2

) h

( )

o

ocutp

ocut

rR

R

rRdh

t

VV

+

==

2ln

18

2222

Vocut = Volumetric flow rate corresponding to dcut. If the thickener of liquid layer is

small compared to radius of bowl ro R. In this case the equation above is

indeterminant.


44/75


( )

18

22rd

VRr p

to

=

Let the thickener of liquid layer be and the settling distance for particle of dcutbe /2

( )

( )

=

=

=

=

=

tgravocyt

tgrav

pcut

pcut

ocut

t

VV

Vgrd

g

v

grd

g

vV

tV

2

18;

18

2

2

.

22

22

Thickener

This is a large, fairly shallow tank with slow moving radial rakes driven from a

central shaft. Its bottom may be flat or a shallow cone. Dilute feed slurry flows from

an inclined trough or launder into the centre of the thickener. The feed slurry, being

more dense than water, tends to flow downward until it reaches a zone of equal

density. Then it moves radially outward at a constantly decreasing velocity, and the

flow gradually divides between the downward moving suspension and the upward

moving flow that is nearly free of solids. Liquor moves radially at a constantly

decreasing velocity, allowing the solids to settle to the bottom of the tank. Clear liquor

spills over the edge of the tank into a launder. The rake arms gently agitate the sludge

and move it to the centre of the tank, where it flows through a large opening to the

inlet of a sludge pump. The volume of clear liquor produced in a unit time by a

continuous thickener depends primarily on the cross-sectional area available for

settling and in industrial separators is almost independent of the liquid depth.

Type Bowl

Diameter

(cm)

n (1/min) value

(m2)

f = Fc/Fg Through put liquid m3/h

Tubular 10 15000 2500 12500 0.03 0.3

Disk

(liq liq)

24

35

6500

4650

5600

4200 1.4 14


45/75


= Cross sectional area of a gravity settling tank of the same separation capacity as

the centrifuge.

Vgrav. = terminal velocity under gravity settling conditions.

Typical Values

Scaling up from lab test sedimentation performance should be the same if the value of

Vo/ is the same for the lab and industrial machine. This is a dependable criteria for

the comparison of centrifuges of similar geometries and proportion developing the

same centrifugal force.

If we consider centrifuges of different configuration

Vo = Vo2 = . . . . . .

1n1 2n2

The efficiency factor is about 90% for tubular bowl and about 55% for disk bowl. But

performance of any type of centrifuge may deviate from the theoretical because of

factors like:

1. Particle size distribution

2. Particle shape

3. The agglomeration of flocs in the feed system

4. Re-agglomeration of flocs within the centrifuge

5. non-uniform distribution of flow within the centrifuge.

6. Hindered settling effect

Critical speed for the centrifuge It is at which the frequency of rotation matches the

natural frequency of the rotating part. At this speed any vibration induces by slight

unbalance in the rotor is strongly re-enforced resulting in large deflection, high stress

and even failure of the equipment. Nearly all the centrifuges operate at speed well

above the critical speed and therefore must pass through this speed during

acceleration and deceleration to let them to do so safely, some degree of damping in

their mounting is provided.

50 4240 5000

Helical

conveyor

Liq - solid

35

64

4000

3000

1200

5700

3100

3200

To 20 solid(t/h) 0.5 1.5

To 70 solid (t/h) 2.5 - 12


46/75


Type of centrifuges

1. Batch lab centrifuges

2. Tubular bowl centrifuges (liq liq)

3. Disk centrifuges (liq liq)

4. Nozzle discharge (liq solid)

5. Cylindrical conical centrifuges (liq solid)

The viscosity of the liquid has an important effect on centrifuge capacity.

Heating of feed will frequently give great improvement in performance. Good

performance of high speed disk and tubular-bowl centrifuges has been obtained by the

difference in specific gravity. Commercial centrifuges separate particle 1 m P 0.1

E.g.

Calculate the diameter of the smallest particle that settle in a rectangular flue gas duct

if the length of the duct is 16m, the height is 2m, linear gas velocity is 0.5 m/s ; =

0.03 x 10-3Pas

of gas = 0.8kg/m3, p = 4000kg/m3

(1) Calculate the time which the gas passes through the duct

Solution


47/75


( )

( )( )

( )

m

gd

m

gd

s

mV

V

VV

s

mmV

m

V

Lt

p

p

crit

t

t

tt

t

g

3.4181.9104

124.0103.018

18

37.0

10037.0

81.98.08.04000

1003.018

18

1Re

124.0

062.02

5.0

062.0325

2

3255.0

16

3

3

2

2

23

3

2

1

max

=

=

=

=

=

=

=

=

=

==

==

===

Hydrodynamic of Packed Beds

As a fluid flows through beds of solid particles there is some resistant due to the flow.

This resistant is the resultant of the total drag of all the particles in the bed. Depending

on the Re; the fluid flow can be laminar, turbulent, form drag, separation and wake

formation occur.

The fluid flows through channels in the bed and these channels are irregular in shape,

have a variable cross section and they are highly unconnected. The main aim is to

calculate the pressure drop through the bed. For calculation, it is assumed that:

1. The actual channels may be effectively replace by a set of identical parallel

conduit of cross-section.

2. The total drag per unit area of channel wall is the sum of 2 kinds of force (Fv=

viscous drag force and Fi= internal forces)

3. The particles are packed randomly


48/75


4. All the particles are of the same size and shape

5. The end and wall effects are negligible. It means, number of particles adjacent

to the wall and end are very few compared to the total number of particles.

The last assumption is valid when the diameter and depth of the bed are large

compared to the particles.

As = Area of channel boundaries (m2)

= Viscosity of fluid (Ns/m2= Pa.s = kg/ms)

V = Average velocity of the bed

= Porosity of the bed

L = Depth of the bed

So = Cross-sectional area of the empty tower of the bed

P = Pressure drop through the bed (N/m2)

d = Diameter of the particle (m)

Vo = Superficial or empty tower velocity

= Density of the fluid (kg/m3)

rH = hydraulic radius of channel

w = Shear stress at channel boundaries (N/m2)

Vp = Volume of single particle (m3)

Sp = Surface area of single particle (m2)

rw = radius of channel

D

L

r

L

A

FP

o

v

42

===

2

32

D

LVP

= Hagen Poiseuille equation


49/75


( )

( )

( )

( )2

1

2

2

2

2

1

2

2

2

2

1

2

2

1

2

2

1

2

2

22

6136

1

11

sec.

.1

1

2

...

66

.1

.

1

..

sec

2

2

2

2

8

kdV

k

V

d

L

P

VkV

SVk

V

S

L

P

ChanneloftionalCrossSPF

Vk

S

V

Vk

V

SLS

ASVkr

VkA

A

F

A

FF

Vkr

Vk

A

F

A

F

A

F

Assumption

VV

VAVA

dV

S

dS

V

solidofvolV

LSSNA

S

V

A

LSr

solidtheofareatotal

voidtheofvol

L

L

ChannelofPerimeter

ChanneloftionalCrossr

LV

PDf

Vf

VVffactorfriction

D

V

OsO

s

O

P

PO

P

P

OD

O

P

P

O

P

PO

H

S

s

I

S

V

D

Hs

i

s

v

s

s

o

o

sp

p

p

p

s

p

o

pps

p

p

s

o

H

H

+

=

+

=

=

+

=

+=

+=

+=+=

=

=

==

==

==

=

=

=

=

==

=

Ergun had the value of the constant 36k1= 150 and 6k2= 1.75 from experiment.


50/75


( )equationErgun

dVV

d

L

P

OsO

s +

=

75.1

1150

12

2

The LHS of the Ergun equation gives us the friction factor fp

( )

( )

( )

=

=

=

=

NB: Laminar term is eliminated. In practice this is fluidization of combustion of coal.

Type of Fluidization

1. Particulate Fluidization

Beyond the Vom the appearance of beds flooded with liquid or gas is often quite

different. When fluidizing with water the particle move farther apart and their motion

become more vigorous as velocity is increased but the average density at a given

velocity is the same in all sections of the bed and this type of fluidization is called

particulate. This particulate fluidization is characterized by a large but uniform

expansion of the bed at high velocities.

2. Bubbling Fluidization (Aggregative)

When fluidized with gases at superficial velocities Vo much greater than the Von

most of the gas passes through the bed as bubbles, which are almost free of solids, and

only a small fraction of the gas flows in the channel between the particles. In the

space between the bubbles the void fraction is about the same as an incipient

fluidization. This kind of fluidization is also called boiling. It is characterized by a

non-uniform nature of the bed. The behavior of a bubbling fluidized bed depends very

strongly on the number and size of gas bubbles. The average bubble size depends on:

1.The nature and size distribution of particles

2.Type of distributor plate

3.The superficial velocity Vo and the

4.Depth of the bed.

4. Slugging Bed (Fluidization)This type can be considered as a special type of bubbling bed. A small diameter

column is used with a deep bed of solids. The bubbles may grow until they fill the


70/75


entire cross-section. This is usually undesirable because of pressure fluctuation in bed

and increase entrainment.

NB. The generalization that liquids give particulate fluidization of solids while gases

give bubbling fluidization is not completely valid. The density different is an

important parameter and very heavy solids may exhibit bubbling fluidization with

water while gases at high pressure may give particulate fluidization of fine solids.

Expansion of Fluidized Bed

With both types the bed expands as the superficial velocity increases because

P = constant P/L decreases as increases

(a)Particulate Fluidization

( )

( )gdV

d

V

Pms

om

ms

om

=+

32

2

3

2

22

175.11150

Turbulent Flow

( ) 223 150

1 dg

V

SP

o

=

Here Vo = m-----Empirical equation proposed by Lewis,

Gas


71/75


omoo VVforV >

1

3

=

1

1m

mLL

Lm= bed height at incipient fluidization

L = height of the expanded fluidized bed

Gilliand Bauer

The equation can be applied directly or in a ratio from

m

mom

o

V

V

=

The expansion of the bed is greater than what can be calculated from the empirical

equation. Experiment shows that the value of m varies from 4.5 in laminar region to

2.5 at high Re.

Application of Fluidization

1. It becomes important in the petroleum industry in which they used it for catalytic

cracking.

2. It can be used for heat transfer eg. Drying fine solids.

0.1 1 10 102 103 Re

m

5

4

3

2


72/75


3. Can be used for solid mixing.

4. In chemical reactions either in catalytic chemical reaction

Advantages of Fluidization

1. The solid is vigorously agitated by fluid passing through the bed.

2. The mixing of the solid ensures that there are practically no temperature gradients

in the bed even with quite endothermic or exothermic reactions.

3. Because of fluidity of the solid it is easy to pass solids from one vessel to another.

Example

A bed of iron exchange bead 8ft deep = L is to be backwash with water to remove

dirts. The particles has a density of 1.2 g/cm3 and the average size is d = 1.1 mm.

What is the minimum fluidization velocity using water at 20o and what velocity is

required to expand the bed by 25%. The beads are assumed to be spherical. Sphericity

= 1, minimum fluidization porosity; = 0.40; = 1cp = 10-3Pas


73/75


74/75


Hydraulic Transport

When particles size d < 50m settle very slowly and are readily suspended in moving

liquid. Particles greater than d 0.25mm in this case, a fairly large liquid velocity is

needed to keep them (particles) moving especially in horizontal pipes.

The critical velocity Vcbelow which particles will settle out, is usually between 1 5

m/s. Vcdepends on several factors such as :

1. The density difference between solids and liquid

2. The particle size

3. The slurry concentration

4. Diameter of the pipe

Vcis large in big pipes than smaller pipes. Pressure drop P in slurries of non-settling

particles may be found from the equations for a homogenous liquid.

For settling slurries there is no simple satisfactory correlation. Usually the velocity in

a long slurry pipeline is typically 1.5 to 2 times Vc.

Pneumatic Conveying

The suspending fluid in a pneumatic conveyor is a gas, usually air, flowing at

velocities between 15 and 30m/s in pipes ranging from 50 to 400 ( 2 to 16 in) in

diameter. There are two principal types of systems: negative pressure (vacuum)

systems, useful for transferring solids from multiple intake point to a single delivery

point and positive-pressure systems, which are best with a single input station and one

or more delivery points.

In vacuum systems the mass ratio of solids to gas is usually less than 5; for such

suspensions the critical velocity in metres per second, may be estimated from the

empirical relation

Put equation at page 86

Where Dp is the diameter of the largest particle to be conveyed. Mot pneumatic

conveyor operate under positive pressure with a blower or compressor. Feeding air (

or occasionally nitrogen) at 1 to 5 atm. gauge pressure into the system. The ratio of

solids to gas is usually higher than in vacuum systems.


75/75

The pressure drop P required to pass air alone through a pneumatic conveying

system is small, but greatly increased when additional energy must be supplied to lift

and move the solids. This additional energy is given by

Put latter equation on page 86Where

r = mass ratio of solids to gas

Vsa = Velocity of solids at inlet

Vsb = Velocity of solids at outlet

s = Density of solid

Pb = Pressure at stations a and b.

The energy Esis supplied by the air. It is transmitted to the solid particles through the

action of drag forces between the air and the solid.

Mixing

It is a random distribution into and through one another of two or more initially

separate phases. The term mixing is applied to variety of operations differing widely

to the degree of homogeneity of the mixed material.

Particulate Systems Lecture Note

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