CHE 321 Lecture Notes

8/19/2019 CHE 321 Lecture Notes

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filtration, slurr6 flow, catal6tic reactors, settling, agglomeration, dr6ing,

floatation, surface phenomena, ion e"change, packed beds.

SOE E+85$EN& 8SE= O# $'#&5C82'&E >'N=25N%

Crushing * Screening, con7e6or, 7ibrating feeders, floatation cells,

thickeners, dr6ers, grinding mills, rotar6 kiln, classifiers, screw con7e6or,digesters, four brinier, o7en bin, storage bins, slurr6 mi"ers, dust collectors,

c6clone separator, cake washing, cr6stalli!ers, beaters.

'll these operations ha7e in common the handling and processing of

particulate solids.

MODULE 2 PARTICLE SI#ES $ SHAPES

Aefore we can discuss operations for handling and separating fluid?particle

s6stems we must understand the properties of the particles.

4.1 $'#&5C2E C>'#'C&E#5S&5CS: &he wa6 particles are characteri!eddepends largel6 on the techni9ue used to measure them. &he wa6 that we

measure a particle si!e is as important as the 7alue of the measured si!e.

easurement techni9ues such as circumference, diameter, and length of a

particle are important. &hese measurement techni9ues make the particle

si!e t6picall6 related to e9ui7alent sphere diameter b6B

a. &he sphere of the same 7olume of the particle.

b. &he sphere of the same surface area as the particle.

c. &he sphere of the same surface area per unit 7olume.

d. &he sphere of the same area when proected on a plane normal to the

direction of motion.

e. &he sphere of the same proected area as 7iewed from abo7e when l6ing in

a position of ma"imum stabilit6 (as with a microscope).

f. &he sphere which will ust pass through the same si!e of s9uare aperture

as the particle (as on a screen).

g. &he sphere with the same settling 7elocit6 as the particle in a specified

fluid. &here are two other methods that are known for si!ing particles that

are not based upon comparison to a standard (sphere) shape.

a. &he first method is to fit the particle area proected shape to a

pol6nomial t6pe of relation. &he 7alues of the pol6nomial coefficients

characteri!e the particle shape.

b. &he second method is through the use of ractals. ' fractal length

can be determined which characteri!es the si!e of the particle and its

dimensionalit6 somewhere between linear and two-dimensional.

&>E E+85D'2EN& S>$E#E

&here is onl6 one shape that can be described b6 one uni9ue number and that

is the sphere. 5f we sa6 that we ha7e a /F sphere, this describes it e"actl6.2


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@e cannot do the same e7en for a cube where /F ma6 refer to an edge or

to a diagonal. @ith a matchbo" (it has -=imensions), for e"ample, there are

a number of properties of it that can be described b6 one number. or

e"ample the weight is a single uni9ue number as is the 7olume and surface

area. So if we ha7e a techni9ue that measures the weight of the matchbo",we can then con7ert this weight into the weight of a sphere, remembering

thatG (@eight 3 Dolume of sphere " densit6 of particle)

and calculate one uni9ue number (4r ) for the diameter of the sphere of the

same weight as our matchbo". &his is the e9ui7alent sphere theor6. @e

measure some propert6 of our particle and assume that this refers to asphere, hence deri7ing our one uni9ue number (the diameter of this sphere)

to describe our particle.

&his ensures that we do not ha7e to describe our -= particles with three

or more numbers(e.g. for the matchbo" instead of sa6 1""4mm, 6ou

cannot correctl6 sa6 the matchbo" is 1mm, as this is onl6 one aspect of the

si!e, as a result it is not possible to describe the -= matchbo" with one

uni9ue number) which although more accurate is incon7enient for

management purposes. @e can see that this can produce some interesting

effects depending on the shape of the obect and this is illustrated b6 thee"ample of e9ui7alent spheres of c6lindersB

5%8#E 1: E9ui7alent Spheres of C6linders

E%&i"ale'! sherical (iame!er o) C*li'(er 1++ , 2+-m

5magine a c6linder of diameter =1 3 4/Fm (i.e. r31/Fm) and height 1//Fm.

&here is a sphere of diameter, =4 which has an e9ui7alent 7olume to the

c6linder. @e can calculate this diameter as follows:

Dolume of c6linder 3

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&he 7olume e9ui7alent spherical diameter for a c6linder of 1//Fm height and

4/Fm in diameter is around H/Fm. &he table below indicates e9ui7alent

spherical diameters of c6linders of 7arious ratios. &he last line ma6 be

t6pical of a large cla6 particle which is disc shaped. 5t would appear to be

4/Fm in diameter, but as it is onl6 /.4Fm in thicknessB normall6 we would not

consider this dimension. On an instrument which measures the 7olume of the

particle we would get an answer around Fm. >ence the possibilit6 for

disputing answers that different techni9ue gi7eI Note also that all these

c6linders will appear the same si!e to a sie7e, of sa6 4Fm where it will be

stated that Jall material is smaller than 4FmJ. @ith laser diffraction theseKc6linders’ will be seen to be different because the6 possess different7alues.

5%8#E 1: E9ui7alent Spherical diameters of C6linder of 7arious ratios.

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igure 1: Si!es of common materials

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$robabl6 among the earliest forms of particle classi)ica!io' .si/i'0) to be

de7eloped is sie"i'0. Se7eral sie7e standards e"ist which classif6 particles

according to the si!e hole through which the particles can pass.

SI#ES OF SOME GRAINS OFSAND.ha! si/e o&l( *o& choose !o (escrie

i!4

STANDARD MESH SI#ET*ler US mm I'ches

H H H./ /.1;

. /.11

; ; 4. /./


7/90

H; / /.4O=S O E'S8#5N% $'#&5C2ES S5LES * S5LE

=5SA8&5ONS

&here are a number of methods for measuring particle si!es and si!e

distributions. an6 of these techni9ues are shown in &able 4 * &able

Some of these methods depend upon calibration with known particle si!es. '

number of suppliers sell small spherical particles of nearl6 uniform si!e

distributions for calibration purposes.

Some of the more ad7anced methods of particle si!e measurement not onl6

measure the particle si!es but the6 will also pro7ide the si!e distributions ofthe particles.

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&'A2E 4: =efinitions of E9ui7alent * Statistical =iameters

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&'A2E : 2aborator6 methods of particle si!e measurements.

One of the better known instruments for this is the Coulter Counter. ' brief

description of the electronic particle counter principle is gi7en in igure 4.

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5%8#E : $articles in the aperture bend the electrical current flu" lines.

or a gi7en material, there are four t6pes of particle si!e distributions that arepossible: (1) b6 number, (4) b6 length, () b6 surface, and (H) b6 mass (or

7olume).

=istributions can be reported either in terms of fre9uenc6 (differential form)

or b6 cumulati7e (integral form) as shown below.

&o e"plain how we mathematicall6 represent the distribution data, let’s suppose

that 6ou measure the mass of particles b6 si!e b6 some unspecified process. 's

an e"ample 6our measured data ma6 be plotted as shown in igure H. Mou can

normali!e the plot b6 di7iding the masses of each si!e b6 the total mass, to

obtain the mass fractions as shown in igure .inall6, if we add the mass fractions cumulati7el6 we get the Cumulati7e ass

raction plot, shown in igure .

5%8#E H: ass 9uantities of an imaginar6 sample of particles.

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5%8#E : ass fraction from data in igure .

5%8#E : Cumulati7e mass fraction plot of data from figure .

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rom these igures we see that the cumulati7e mass fraction can be written

mathematicall6 as

...1

as a function of the nth particle si!e. urthermore, we can write the increment

in the cumulati7e mass,

...4

@here is the slope of the cur7e on the cumulati7e mass fraction plot. @e

define this slope to be the fre9uenc6 distribution of the mass fraction,

where

G

>ence, we can relate the cumulati7e mass fraction to the fre9uenc6 distribution

b6

GH

2et the fractional amount of particles of si!e " be for an6 t6pe of measurement

(b6 mass, number, area, etc.) be represented as

G13


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5f the particle si!e distribution is determined as the number fraction then the

number fre9uenc6 distribution is gi7en b6

G

@here

, is the differential range abo7e and below si!e " that the number count

represents. 5f the particle si!e distribution is determined on a microscope b6

measuring proected areas or b6 laser attenuation then the surface fraction or

fre9uenc6 distribution based on surface area is

G

, is a fractional amount, then integrating o7er all particle si!es gi7es

the whole, or

G;and if we integrate o7er onl6 the range from !ero to some si!e " we get the

cumulati7e fraction , (") ,

G<

which is the area under the f(") cur7e from / to " .

$lots of and f ha7e the general form shown in igure . @here f and are

also related b6

G1/

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5%8#E : &6pical f(") and (") cur7es.

G11

@here k1, k4, k are geometric shape factors,

Similarl6, the cumulati7e distributions can be related b6

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G14

Often, e"perimental data are reported in discrete form (such as from a sie7e

anal6sis). or these data it is easier to work with discrete forms of the integral

e9uations.

G1

@ith fs(") 3 k4"

4fN("), upon rearrangement

&here are se7eral e9uations that are t6picall6 fitted to the distribution. &he

most widel6 used function is called the log-normal distribution. 5t is a two-parameter function that gi7es a cur7e, which is skewed to the left compared to

the familiar bell cur7e. &his function is normall6 used because in most cases

there are man6 more measured fine particles than larger particles.

&he lognormal function is best described first b6 considering the normal

distribution of the %aussian (bell shaped) cur7e shown in igure ;a:

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5%8#E ;a: Normal %aussian cur7e

5%8#E ;b: 2og normal cur7e

G1H

where is the cumulati7e undersi!e fraction of particles, " is the particle si!e,

is the standard de7iation, and " is the mean particle si!e.

&o fit E9.1H to e"perimental data (such as from a sie7e anal6sis) first make an6

adustment necessar6 for left or right bias (that is, use the diameters

associated with the center of each bin and not the left or right edges). &he

a7erage diameter and standard de7iations are determined from

and G1

G1

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&o obtain the log-normal distribution, igure ;b, we substitute ln(") for " and

ln ( g) for . &his gi7es

G1

where " g is the geometric mean and is e9ual to the median si!e (where /0 of

the particles are greater in si!e and /0 are smaller in si!e).

&o fit e9. 1, use the following e"pressions

G1;

G1<

#earranging e9uation 1 and then appl6ing the substitution

G4/

@e get the more con7enient form gi7en b6

G41

in which "m represents the mode because it is the si!e at which d?d" has its

ma"imum (recall f(") 3 d?d", hence f is ma"imum at its mode, at "m).

S7aro7sk6 (2. S7aro7sk6, $owder &echnolog6 , 5, 1-4, 1


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to simplif6 the calculations.

EP'$2E 1

' sample of peanuts are weighed as listed in &able H below. 8sing an a7erage

densit6 of 1.4 grams per cubic centimeter, the a7erage cand6 diameter

(assuming spherical shape) is calculated. $lot the fre9uenc6 distribution and the

cumulati7e fre9uenc6 distribution of the a7erage diameter of the candies.

&able H.

8sing the formulas

in E9s.11 and 14, the fre9uenc6 and cumulati7e fre9uenc6 distributions are

calculated. &he particle si!es are added up in Q" increments of /./ cm. &he

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si!e ranges start with 1.H to 1./ cm. 'll si!es less than 1./ are counted in

the first increment, all si!es between 1. and 1. are in the second increment,

and so on.. &he 7alues for n s are determined b6 counting the number of peanuts

that fall in a gi7en si!e increment and are assigned to the a7erage si!e in theincrement.

or e"ample, there are numbers of peanuts in the si!e increment range of 1.

to 1. cm and are assigned to the a7erage si!e of 1.4 cm.

fd" is determined b6 ?413/., f is /.?/./ 3 .. is

determined b6 cumulati7e summing the 7alues fd".

&he results of the summation are plotted in figure <

re9uenc6 =istribution of the peanuts

5%8#E


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5%8#E 1/: Comparison of si!e distributions

G4

could be gi7en in an6 of the form in table below or e"ample, suppose we want the cubic mean of a set of particles for which we

know the number distribution. &he mean is defined such that,

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hence,

&able : Dalues of g(") and their mean representations

Suppose 6ou ha7e the mass distribution fre9uenc6 of a set of particles and 6ou

want the geometric mean. >ow would 6ou calculate the geometric mean from the

gi7en mass distribution fre9uenc6R

>ence

&he mean particle si!e is rarel6 9uoted in isolation. 5t is usuall6 related to some

measurement techni9ue and application and used as a single number to

represent the full si!e distribution. &he mean represents the particle si!e

distribution b6 some propert6 which is 7ital to the application or process under

stud6. 5f two si!e distributions ha7e the same mean (as measured using the

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same methods) then the beha7ior of the two materials are likel6 to beha7e in

the process in the same wa6.

5t is the alica!io' therefore which go7erns the selection of the most

appropriate mean. 8suall6 enough is known about a process to identif6 somefundamentals, which can be used as a starting point. &he fundamental relations

ma6 be o7erl6 simple to describe the process full6, but it is better than

randoml6 selecting mean definition.

EP'$2E 4: Comparison of mass 7ersus number count.

Consider measuring the si!e distribution b6 sie7ing. &he results of a sie7e

anal6sis ma6 gi7e the si!e distribution as shown in the table below, with

calculations alread6 made to generate other data.Sie7e anal6sis of a sample of particles. ass, number, and area fractions are

calculated.

&he mass fraction is found simpl6 b6 di7iding the sample masses (sie7e mass) b6

the sum of the masses. =i7iding the sample mass b6 the particle intrinsic

densit6 (assumed here to be 4. g?cm) gi7es the 7olume of the particles in the

sample. =i7iding the sample 7olume b6 the 7olume of one particle34

3 Rπ where #

is the sie7e si!e opening, gi7es the number of particles for that sample. &he

total surface area of the particles of a gi7en si!e is obtained b6 multipl6ing the

number of particles times the surface area of one particle (H π #4). &he number

and area fractions are found b6 di7iding the sample 7alues b6 the totals.

&he plot in igure 1/ shows that the modes of the three distributions 7ar6

widel6. &he number distribution and surface area distribution are skewed

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greatl6 to the small particle si!e. &his shows that a small mass of the fines

contains a large number of particles.

' propert6 such as turbidit6 is sensiti7e to the total number of particlesB hence

the large number of fines will cause the fluid to be cloud6. ' process such asfiltration is sensiti7e to the total surface area of the particles due to the drag

(to be treated later) resistance to flow across the surface.

5%8#E 11: Comparison of the fractional distributions of the particle si!e

distributions.

EP'$2E : '$$25C'&5ONB cake filtration, cake washing, dewatering, flow

through packed beds and porous media.

5f the particle si!e distribution is known, what definition of the mean should be

usedR

5n flows through a packed bed we can consider the pores to be conduits. @e

can appl6 the concept of a friction factor and a #e6nolds number. Since the

geometr6 of an arbitrar6 pore is not c6lindrical, we appl6 the h6draulic radius,

#h.

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G4Hwhere is the bed porosit6 and a is a surface area. &his surface area is related

to the specific surface area, a s , of the solids (total particle surface?7olume of

particles) b6

G4

&he specific surface area in turn is related to the mean particle diameter

(assuming the particle can be represented b6 a sphere)

G4or spheres the total 7olume of particles is gi7en b6

G4

and the total surface area of the particles is gi7en b6

>ence, we get the mean particle diameter to be

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G4;

where the latter e"pression is the anal6tical formulation.

&his latter e"pression defines the mean to be the arithmetic mean,

see e9.44 of the distribution b6 surface.

Ne"t, we must relate this to a si!e distribution b6 mass (the usual wa6 of

measurement). &he surface distributions b6 surface and mass can be related b6

G4<

where k is a constant that accounts for the geometric shape of the particles. 5t

is assumed here that k is independent of ".

Since the mean si!e is gi7en in e9. 4; then combining e9s. 4; * 4


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, of the mass distribution gi7en b6 e9. 4. >ence this shows that the

surface arithmetic mean is e9ual to the mass harmonic mean,

&herefore, for flow through packed beds, filter cakes, etc.B

the appropriate mean particle si!e definition is the arithmetic a7erage of the

surface distribution. &his is shown to be e9ui7alent to the mass distribution

harmonic mean.

MODULE 36 FORCES ON SPHERICAL $ NON7SPHERICAL PARTICLES

&he most important force acting on particles in fluid-particle medium is the

=#'% O#CE. 5t has two components (1.Skin drag, 4. orm drag). &he Skin drag

is due to e"changes with the boundar6 la6er ( a &angential stress on the bod6arising from the boundar6 la6er). &he form drag is the summation of the effect

on the bod6 of acceleration and deceleration in the streamlines as a fluid flows

pass a bod6. &heoreticall6, the drag force can be deri7ed for onl6 the simplest

of the case i. e, 2aminar flow around a sphere, all other cases are deri7ed

empiricall6. &his is due to the relati7e motion between the fluid and the

particles. Summari!ed 7ersion of this force is gi7en hereB

rom a free bod6 diagram, igure 14, we can write a balance of forces acting on

a spherical particle. &he balance of forces shows that the accelerating force

acting on the particle is gi7en b6

G4

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5%8#E 14: ree bod6 diagram on particle of diameter #

5nitiall6, when a particle falls through a fluid the particle 7elocit6 accelerates.

'fter a short distance the particle reaches its terminal 7elocit6 and its

acceleration goes to !ero. &his means that the force of acceleration, a is !ero.

>ence, at terminal 7elocit6 the kinetic force acting on the particle is gi7en b6

G

&he &E#5N'2 DE2OC5&M can be defined as TFi'al Co's!a'! See( o) a

Falli'0 O8ec!U(or &he &erminal 7elocit6 of a bod6 is the 7elocit6 attainment at

the end of its acceleration or deceleration) B that is, the constant speed that a

falling obect reaches when the downward gra7itational force e9uals the

frictional resistance of the medium through which it is falling.

mp is the mass of the particle and mf is the mass of the displaced fluid with the

same 7olume as that of the particle. &hese masses are e9ual to the 7olume of

the particle times the respecti7e particle or fluid densities. &he kinetic force

becomes

GH

@e define the Dra0 Coe))icie'!, Cd, b6 the e"pression

G

where ' is the proected area normal to the flow and VE is the characteristic

kinetic energ6. @hen we substitute in the proected area of a sphere, W# 4, and

the kinetic energ6, 1?4 Xu4, into e9.

then we can deri7e a working e9uation for determining the drag coefficient as28


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G

5n order to use this e"pression to determine 7alues for C= we must rune"periments. &he e"periments ma6 be in the laborator6 or the6 ma6 be thought

e"periments for limiting case of creeping flow around the sphere(&his operation

is discussed in some detail b6 Aird et.al. (1


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which is a constant. &his region usuall6 lies

// Z N#E Z 4//,///.

&his is called the Newton’s law region. >ere it is the accelerati7e forces that

are more important.= α

4 µ

#E%5ON H

&his is a region of high(E"treme) turbulence coupled with flow separation. &he

flow separation is responsible for the drop in the graph(see ig .H, Coulson *

#ichardson, Dol. 4). Some of the C= here is constant relati7e to the N#E.

C= 3 /.1/ B N#E [ 4//,/// GG..H

See the figure below for the regions.

igure 1 =rag coefficient for spheres 7ersus #e6nolds number. &hree

appro"imate cur7es are o7erla6ed onto the e"perimental cur7e. &he

appro"imate cur7es are, from left to right, C = 34H?N #E (Stoke’s 2aw range for

N#EZ1), C = 31;.?(N#E). (5ntermediate range for 1ZN#EZ1///) and C = 3/HH.

(Newton’s 2aw range 1///ZN#EZ1//,///).

or li9uid droplets mo7ing in other li9uid we can also consider it as a case of

particle in a fluid. @hen N#E is 7er6 low ( N#E Z1/ ), the particles beha7e like a

solid form. or N#E [ 1/, Oscillation comes in as well as Discosit6 and surface

tension. o7ement of gases in li9uids can also occur. 2i9uid 7iscosit6, interfacial

31


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surface tensions also come in as well as the ones we ha7e in the li9uid droplets.

8sing e9uations H, HH,and H the drag forces can be computed.

#egion =1 =8 π

4 /.: /.H 1.H 1.H4.1 = 8 πµ ρ

4 4/./ = 8 πρ

H 4 4/./4 = 8 πρ

TERMINAL :ELOCIT;

C'SE 1: 1-=imensional otion of particles through a fluid

>ere the forces acting are E"ternal forces(gra7itational, electrostatic,

magnetic). &he line of action of all forces acting on the particle are arranged inlinear order.

ig.1H

&he drag force, =, appears whene7er there is relati7e motion between the

particle * the fluid. 5t acts to oppose the motion and acts parallel with the

direction of mo7ement but in the opposite direction.

FB

FD

FE

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A, the buo6anc6 force acts parallel with the e"ternal force but opposite

direction, it also acts opposite direction to the motion of the particle. E, is the

E"ternal force.

Summing up the forces according to Newton’s lawNE& 3

du m

d θGGGG.H

83Delocit6, θ 3 timeA is defined b6 'chimede’s principle to be

( )A p E f D a ρ = aE is gra7itational., D$ Dol. of particles

E a g =

E E ma = .rom the definition of =rag force,

4

4=

$ = C '8 ρ = '$ 3 frontal area(area proected) GGGGH

$ A E fluid

$

mass a ρ

ρ

= ÷

f

A E p

ma ρ

ρ =

Substituting into e9u. H

4

4f =

p E E f p

C du ma ' u ma m d

ρ ρ ρ

− − =θ and rearranging

4

14

p = f f E

p

C ' u du a

d m

ρ ρ

ρ

= − − ÷ ÷θ

………..H;

E9u. H;, is the general e9uation for the total force on the bod6 in the force

field. @hen the force field is gra7itational it is accelerati7e. &hat is

E a g = for gra7itational force field4

E a r ω = for centrifugal force field

E a ω = for electrostatic force fieldω angular 7elocit6 of the particle (rad?sec)r radius of the path

@hen ,t du

u u d

=θ

= / , this is b6 definition of terminal 7elocit6.

&herefore b6 this definition, for gra7itational force fieldB

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( )4

4 p f t

p p = f

g m 8

C '

ρ ρ

ρ ρ

−= GGGGGGGGG..H<

or Sphere, the frontal area

4

,H p p

p

= =

m

π π

ρ = =Substituting these two 7alues into e9u. H<

( )H

p p f

t

=

g = 8

C

ρ ρ

ρ

−= GGGGGGGGG./

E9uation / is the general e9uation for the terminal 7elocit6 of a bod6 mo7ing in

a gra7itational field.

#egion 1

4H=

#E

C

N

= &herefore,( ) 4

1;

p p f

t

f

g = 8

ρ ρ

µ

−= GGGGGGGGG.1

E9uation 1 is the general stoke’s law e"pression for terminal 7elocit6 within

the laminar flow region.

#egion 4: 5ntermediate range

( ) /.1

/.1 1.1H

/.4< /.H

/.1E p p f

t

g =

8

ρ ρ

ρ µ

−

=#egion : Newton’s range

( )1.E

p p f

t

g= 8

ρ ρ

ρ

−=

'$$25C'&5ONS

1. 5n the industries, the terminal 7elocit6 e"periment can be used to know

the 7iscosit6 of unknown things (Solution and $articles)

5t can be used in the spra6 drier ( for making dr6(milk) powder) b6 the use ofnebuli!ers or atomi!ers.

4. &o find si!e distribution in a slurr6.

>ere the particles settle down a column according to the decrease in their

si!es. or e"ample because of their high terminal 7elocit6 (i.e. big particles)

settle first at the bottom of the column followed b6 the ne"t si!e.

E"ample a.

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' laborator6 Discometer consist of a steel ball. &he ball is /.//m in diameter

and the inde" mark 3 /.4m. &he 7iscosit6 of the steel oil s6rup of densit6 3

1//kg?m is to be determined. &he measured time inter7al is ;secs. @hat is

the 7iscosit6 of the s6rup. &he specific gra7it6 of the steel ball 3 .<

Solution:

Considering the flow to be laminar, using e9u. 1

( ) 4

1;

p p f

t

f

g = 8

ρ ρ

µ

−=


36/90

( )4

1

p f f

p

g V =

ρ ρ ρ

µ

− =

GGGG..

rom e9u. 4, N#E3 11;

V GGGG..H

5f the si!e of the particle is known, V can be calculated. 5f V so calculated is

less than 4., stokes law applies. (5f 6ou substitute N#E 3 1, V34.)

or Newton’s law range V is greater than ;.< but less than 4,/. or 7alues

greater than 4,/, the drag coefficient ma6 change abruptl6 with small

changes in fluid 7elocit6.

8nder these conditions, as well at the intermediate range (4.ZVZ;.


37/90

3 1m$a.seconds

5n S5 units µ 3 kg?ms or $ascal second.

5n cgs, 3g?cm.s(otherwise called poise($)).

5n general case, the direction of mo7ement of the particle relati7e to thefluid ma6 not be parallel with the direction of the E and A, as a result =makes an angle with the other two. 5n this situation, which is called !o =

(ime'sio'al mo!io'> the drag must be resol7ed into two components( 1.

>ori!ontal,4, %ra7itational)

&he E&>O= O 5NC#EEN&'2 '$$#OP5'&5ON is used in sol7ing this

approach. (&his approach will not be co7ered in our case, but students are

ad7ised to read up or be familiar with this)

&hese e"pressions for &erminal Delocit6 and =rag orce hold with thefollowing assumptionsB

a. &he settling is not affected b6 the presence of other particles in the

fluid. &his condition is known as Kfree settling’B when the interference of other

particles is appreciable, the process is known as K>indered Settling’

b. &hat the wall of the containing 7essel do not e"ert an appreciable

retarding effect.

c. &hat the fluid can be considered as a continuous medium, that is the

particle is large compared with the mean free part of the molecules of the

fluid, otherwise the particles ma6 occasionall6 TslipU( mo7e smoothl6 with sliding

motion, mo7ing awa6 from the desired position) between the molecules and thus

attain a 7elocit6 higher than the calculated one.

HINDERED SETTLING

&he settling of particles under crowded (concentrated) conditions is referred

to as Hi'(ere( Se!!li'0? &he =rag coefficient in hindered settling is greater

than in free settling. >indered settling could be caused b6B

Fx, Ux

FD, U, This is the net drag force fo !o"# $ith %e&o'it# (

F#, U#

37


38/90

a. $article $articleB $article @all collision.

b. &he particles also constitute part of the medium and impact to the

medium a higher effecti7e µ and ρ .

c. 2i9uid displaced upward is greater, flow area is less.@hat happens when particles settle in concentrated solutionsR

's each particles falls it displaces fluid which in turn must mo7e upward. 5n a

concentrated s6stem this causes an upward fluid motion which interferes with

the motion of other particles. &he net effect is a slower, hindered, settling

rate for the group of particles as compared to the free settling terminal

7elocit6 of one particle b6 itself. See figure belowB

5n this phenomenon, the particles are sufficientl6 close together to cause the

7elocit6 gradient surrounding each particle to be affected b6 the presence of

the neighbouring units. &he particles in settling therefore displace li9uidsand generate an appreciable upward 7elocit6. &he 7elocit6 of the li9uid is then

greater with respect to the particle than with respect to the 7essel.

igure 1 :>indered settling: as a particle falls its displaced fluid mo7es upward

and slows the obser7ed settling rate of neighboring particles.

38


39/90

Coe and Cle7enger (&rans. 'm. 5nst. in. et. Eng. @@, , 1


40/90

igure 1. $orosit6 as a function of sphericit6 and packing structure. 2oose

packed materials are ones that ha7e not had much time to settle due to

7ibrations. ' normal packed material is one that ma6 ha7e sat for se7eral da6s

or weeks. =ense packed materials are ones that ha7e sat for a 6ear or more and

ha7e had ample time to settle and 7ibrate into its most densel6 packed

structure under force of gra7it6.

aude * @hitmore (Ar. \. 'ppl. $h6. , H-H;4, 1ere 8t is calculated as beforefor a single particle falling through a clear fluid and accounts for the hindered

settling effects. &he parameter n (is an Emperical factor) is determined

e"perimentall6. 8nfortunatel6 n is not a constant but 7aries as a function of the

particle geometr6 and the #e6nolds number. $err6’s >andbook (th edition, pg -

;) shows that n 7aries from 4. to H.(See also cCabe * Smith fig. .) for

spherical particles and has a dramatic effect on the calculated 7alues for the

hindered settling 7elocit6.

40


41/90

n 3 H. for low N#E,B and n 3 4. for high N#E' rational approach to hindered settling is described in which the particle

settles through the slurr6 instead of the clear fluid. &his approach is a

preferred alternati7e to the aude * @hitmore approach.5f particles of a gi7en si!e are falling through a suspension of much finer

solids, the terminal 7elocit6 of the larger particles should be calculated using

the densit6 and 7iscosit6 of the fine suspension. E9u. ma6 then be used to

estimate the settling 7elocit6 with the porosit6 taken as the 7olume fraction of

the fine suspension, not the total 7oid fraction. Suspensions of 7er6 fine sand in

water are used in separating coal from hea76 minerals, and the densit6 of the

suspension is adusted to a 7alue slightl6 greater than that of coal to make the

coal particles rise to the surface, while the mineral particles sink to the

bottom.

Dra0 Force o' No'7Sherical Par!icles

&he shape and orientation of the particle has an important effect on the flow

profiles around the particle. cCabe and Smith (8nit Operations of Chemical

Engineering, th ed, c%raw->ill, N.M., 4//1), igure ., and $err6’s Chemical

Engineer’s >andbook, (th ed., c%raw->ill, N.M., 1


42/90

#aschig rings /.

Coal dust, natural (up to ?; inch) /.

%lass, crushed /.

ica flakes /.4;Sand

'7erage for 7arious t6pes

lint sand, agged

Sand, rounded

@ilco" sand, agged

/.

/.

/.;

/.

ost crushed materials /. to /.;

Vunii and 2e7enspiel (luidi!ation Engineering, \ohn @ile6, N.M. 1


43/90

igure 1; =rag coefficient #e6nolds number relationship for non-spherical

particles. &he particle diameter is the 7olume e9ui7alent diameter, " 7, of the

sphere with the same 7olume as the particle.

GGGGGG.

&he last e9uation is known as %alileo Number.

@ith this chart and the correlation in E9. (line 4) the terminal 7elocit6 can

be calculated from the material properties and the sphericit6.

43


44/90

>aider and 2e7enspeil ($owder &echnolog6 , @9, , 1


45/90


46/90

BUL PROPERTIES OF PODERS AND SLURRIES

5n the process industries economics usuall6 re9uires us to handle and process

man6 particles at a time rather than indi7iduall6. Aecause of this we must ha7e a

working knowledge of the collecti7e or TbulkU properties of these materials.&he Tcollecti7eU properties are the measurable properties of groups of

particles. Some of these properties are analogous to properties that are

measured on indi7idual particles while other properties ma6 not be defined for

indi7idual properties.

Some important e"amples are discussed here, but the list is not

complete.

Brie) O"er"ie o) Some B&l Proer!ies

'N%2E O #E$OSE

&he angle of repose is a characteristic of solids which characteri!es the pilingor stacking nature of the particles. &he wa6 that particles stack when poured

into a pile is a function of the si!e?shape, particle intrinsic densit6, surface

forces (stickiness, electrostatic), and roughness of the particles. an6 factors

can influence the wa6 particles stack hence it is difficult to predictB normall6 a

simple measurement can be made to determine the angle of repose.

igure 4/. 'ngle of repose, α , of (a) a pile of powder, (b) powder in a container,

and (c) powder in a rolling drum.

&he angle of repose is considered to be mostl6 a measure of the internal

friction between the particles as a whole, but not between indi7idual particles.

5t is a rapid method of assessing the beha7iour of a particulate mass. 5t is used

in a number of correlations and estimates for the beha7ior properties of thebulk solids. One e"ample gi7en in Coulson and #ichardson’s te"t relates the angle

of repose to the height of the longest mo7able plug in a piston. &he angle of

repose ma6 is often incorrectl6 be used to estimate the angle re9uired for the

bottom of a hopper to ensure proper discharge.

46


47/90

'ngle of repose 7ar6 from about 4/o with free flowing solids, to about / o with

solids with poor flow characteristics. 5n e"treme cases of highl6 agglomerated

solids, angles of repose up to nearl6


48/90

2oose packed materials are ones that ha7e not had much time to settle due to

7ibrations. ' normal packed material is one that ma6 ha7e sat for se7eral da6s

or weeks. =ense packed materials are ones that ha7e sat for a 6ear or more and

ha7e had ample time to settle and 7ibrate into its most densel6 packedstructure under force of gra7it6.

Since porosit6 is defined as a fraction it must ha7e a 7alue between / and 1

inclusi7e. is the fluid phase 7olume fraction. (1-) is the solid phase 7olume

fraction. &heir sum is 1 (See figure 44 ).

BUL DENSIT;

Aulk densit6 is the effecti7e densit6 of a powder or particulate solid taking into

account the 7olume occupied b6 both the solid and fluid phases. &he bulk

densit6 is calculated from the porosit6 and the intrinsic densities of the fluidand solid phases:

GGGGG.

EP'$2E: 'n e"ample of appl6ing bulk densit6 is determining the weight of

sand in a bucket. &he intrinsic densit6 of one sand particle is about the same as

that of glass, 4. g?cm. 5f sand packs with a porosit6 of /.H, how much will a

twent6 fi7e liter bucket filled le7el to the top with dr6 sand weighR

&he mass of sand in the bucket is gi7en b6

GGGGGG.'ppl6ing E9.(H-4), neglecting mass of the air (air densit6 is about 1?4// that

of the sand), we get

H.4 omentum &ransport $roperties&here are se7eral properties of dispersed multiphase mi"tures that are

important to predicting handling and transporting properties. &hese include bulk

7iscosit6, coefficient of friction, \anssen’s Coefficient, and permeabilit6.

48


49/90

BUL .SLURR;4 :ISCOSIT;

Slurries, which are mi"tures of fluids and solids, displa6 a number of

interesting properties including Aingham $lastic, $ower 2aw, =ilatent, and time

dependent beha7iors.Aingham $lastic (Mield Stress) flow occurs when particles in the slurr6 resist

motion between each other and with the pipe or container wall. &he shear

stress must e"ceed a certain 7alue (the 6ield 7alue) before the fluid will flow.

(see $atel, #.=., JNon-Newtonian low,J in >andbook of luids in otion , N.$.

Cheremisinoff and #. %upta eds., Chapter B 'nn 'rbor Science, 1-1;, 1


50/90

&o estimate the 7alues of c one can use the porosit6 of a loosel6 packed bed.

oust, 'ppendi" A, gi7es a correlation between sphericit6 and porosit6 and

2oose, Normal, and =ense packing,figure 44 ('.S. oust, 2.'. @en!el, C.@.

Clump, 2. aus, and 2.A. 'ndersen, $rinciples of 8nit Operations , @ile6, NewMork, 1


51/90

igure 4. &6pical $acked Aed.

$ermeabilities of t6pical materials

&he =arc6’s law e"pression pro7ides a means of estimating the flow rate for a

gi7en pressure drop of fluid. &he permeabilit6 coefficient must be determined

from e"periment.

51


52/90

' few correlations are a7ailable for predicting the permeabilit6. One of the

more common correlations is Ergun’s e9uation (Aird et.al., &ransport $henomena ,

@ile6, New Mork, 1


53/90

GGGGG

&he ne"t e"pression is useful for estimating permeabilit6 for a powder of a

particular si!e, or estimating particle si!e from pressure drop flow rate data.

GGGGG.

5f the particles are appro"imatel6 spherical the sphericit6 is 1./, and assuming

normal packing the porosit6 is about /.; (from igure H-). or small #e6nolds

number the 1;/ term dominates the denominator this e9uation reduces to

GGGGG.;

EP'$2E H-4

's an e"ample, 7er6 slow flow of water (1 liters?minute per s9uare meter)

through a 1/ cm thick packed bed of spherical 2ucite particles produces a

pressure drop of 1/k$a. @hat is the appro"imate si!e of the 2ucite particlesR

SO28&5ON:

Sol7ing =arc6’s 2awB

Substitute this 7alue for permeabilit6 into

and sol7ing for dp gi7es

or the a7erage particle si!e is about 1 microns.

Check the #e6nolds number, to make sure the low #e6nolds number assumption

holds:

53


54/90

Since # ep is much smaller than 1 then the assumption in deri7ing

holds.

FLUIDI#ATION• Pace( e(s> )l&i(i/e( e(s> ress&re (ro i' )l&i(i/e( e(s> hea! $

mass !ra's)er i' )l&i(i/e( e(s?

#egions of luidi!ation

@hen a fluid is passed upwards through a bed of particles the pressure loss in

the fluid due to frictional resistance increases with increasing fluid flow. '

point is reached when the upward drag force e"erted b6 the fluid on the

particles is e9ual to the apparent weight of particles in the bed. 't this point

the particles are lifted b6 the fluid, the separation of the particles increases,and the bed becomes fluidi!ed. &he force balance across the fluidi!ed bed

dictates that the fluid pressure loss across the bed of particles is e9ual to the

apparent weight of the particles per unit area of the bed. &hus:

weight of particles - upthrust on particles

$ressure drop 3 -------------------------------------------

bed cross sectional area

or a bed of particles of densit6 p, fluidi!ed b6 a fluid of densit6 f to form abed of depth > and 7oidage in a 7essel of cross sectional area ':

' plot of fluid pressure loss across the bed 7ersus superficial fluid 7elocit6

through the bed would ha7e the appearance of.

&he straight line region O' is the packed bed region. >ere the solid particlesdo not mo7e relati7e to one another and their separation is constant. &he

pressure loss 7ersus fluid 7elocit6 relationship in this region is described in

general b6 the Er0&' e%&a!io', E9uation . (&hat is this is the e9uation for

pressure loss in a packed bed

54


55/90

igure 1: $ressure drop 7ersus fluid 7elocit6 for packed and fluidi!ed beds

&he region AC is the fluidi!ed bed region where E9uation 1 applies. 't point ' itwill be noticed that the pressure loss rises abo7e the 7alue predicted b6

E9uation 1. &his rise is more marked in powders which ha7e been compacted to

some e"tent before the test and is associated with the e"tra force re9uired to

o7ercome interparticle attracti7e forces. &he superficial gas 7elocit6 at which

the packed bed becomes a fluidi!ed bed is known as the mi'im&m )l&i(i/a!io'

"eloci!*> Um)? &his is also sometimes referred to as the "eloci!* a! i'ciie'!

)l&i(i/a!io' .i'ciie'! mea's ao&! !o e0i'4 . 8mf increases with particle

si!e and particle densit6 and is affected b6 fluid properties. 5t is possible to

deri7e an e"pression for 8mf b6 e9uating the e"pression for pressure loss in afluidi!ed bed (E9uation 4) with the e"pression for pressure loss across a packed

bed. &hus substituting the e"pression for (- p) for a fluidi!ed bed from

E9uation 4 into the e"pression for (- p) for a packed bed from E9uation :

55


56/90

'nd

where 'r is the dimensionless number known as the 'rchimedes number and #emfis the #e6nolds number at incipient fluidi!ation,

5n order to obtain a 7alue of 8mf from E9uation we need to know the 7oidage

of the bed at incipient fluidi!ation, 3 mf. &aking mf as the 7oidage of the

packed bed, we can obtain a crude 8mf. >owe7er, in practice 7oidage at the

onset of fluidi!ation ma6 be considerabl6 greater than the packed bed 7oidage.

' t6pical often used 7alue of mf is /.H. 8sing this 7alue, E9uation becomes:

@en and Mu (1


57/90

&he @en and Mu correlation is 7alid for spheres in the range

/./1 Z #emf Z 1///.

or gas fluidi!ation the @en and Mu correlation is often taken as being most

suitable for particles larger than 1// m, whereas the correlation of Aae6ens(1


58/90

%eldart (1


59/90

where "i is the arithmetic mean of adacent sie7es between which a mass

fraction mi is collected. &his is the harmonic mean of the mass distribution,

which is e9ui7alent to arithmetic mean of a surface distribution.

B&li'0 a'( 'o'7&li'0 )l&i(i/a!io'

Ae6ond the minimum fluidi!ation 7elocit6 bubbles or particle-free 7oids ma6

appear in the fluidi!ed bed.

'n e9uipment T&wo - dimensional fluidi!ed bedU ,a fa7ourite for researchers

looking at bubble beha7iour consists of a 7essel with a rectangular cross

section, whose shortest dimension ( in the direction being 7iewed is usuall6 onl6

1cm or so. &he e9uipment can re7eal the effect of gas flow on bubbles in powder

either at (1) lower gas 7elocit6 (4) higher gas 7elocit6

't superficial 7elocities abo7e the minimum fluidi!ation 7elocit6, fluidi!ation

ma6 in general be either bubbling or non-bubbling. Some combinations of fluid

and particles gi7e rise to onl6 bubbling fluidi!ation and some combinations gi7e

onl6 non-bubbling fluidi!ation. ost li%&i(7)l&i(i/e( s*s!ems, e"cept those

in7ol7ing 7er6 dense particles, do not gi7e rise to bubbling. ' bed of glass

spheres fluidi!ed b6 water can e"hibit non-bubbling fluidi!ed bed beha7iour.

igure H: E"pansion of a li9uid-fluidi!ed bed. (i) ust abo7e 8mf, (ii) li9uid

7elocit6 se7eral times 8mf. Note the uniform increase in 7oid fraction

59


60/90

Gas7)l&i(i/e( s*s!ems, howe7er, gi7e either onl6 bubbling fluidi!ation or non-

bubbling fluidi!ation beginning at 8mf , followed b6 bubbling fluidi!ation as

fluidi!ing 7elocit6 increases. Non-bubbling fluidi!ation is also known as

ar!ic&la!e or homo0e'eo&s fluidi!ation and bubbling fluidi!ation is oftenreferred to as a00re0a!i"e or he!ero0e'eo&s fluidi!ation.

Classi)ica!io' o) Po(ers

%eldart (1


61/90

igure : %eldartYs classification of powders

Since the range of gas 7elocities o7er which non-bubbling fluidi!ation occurs in

%roup ' powders is small, bubbling fluidi!ation is the t6pe most commonl6

encountered in gas-fluidi!ed s6stems in commercial use. &he superficial gas

7elocit6 at which bubbles first appear is known as the minimum bubbling 7elocit6

8mb. $remature bubbling can be caused b6 poor distributor design or

protuberances inside the bed. 'brahamsen and %eldart (1


62/90

referred to as e( e,a'sio' (see igure on E"pansion of a li9uid fluidi!ed

bed).

&he relationship between fluid 7elocit6 and bed 7oidage ma6 be determined b6

recalling the anal6sis of multiple particle s6stems (see #hodes, 1


63/90

of a fluidi!ed bed the time-a7eraged actual 7ertical particle 7elocit6 is !ero (8 p3 /) and so:

where 8fs is the downward 7olumetric fluid flu". 5n common with fluidi!ation

practice, we will use the term superficial 7elocit6 (8) rather than 7olumetric

fluid flu". Since the upward superficial fluid 7elocit6 (8) is e9ual to the upward

7olumetric fluid flu" (-8fs), and 8fs 3 8f , then:

#ichardson and Laki (1


64/90

&hus E9uations, 41, 44 and 4 in conunction with the correlation of Vhan and

#ichardson abo7e, permit calculation of the 7ariation in bed 7oidage with fluid

7elocit6 be6ond 8mf. Vnowledge of the bed 7oidage allows calculation of the

fluidi!ed bed height as illustrated below:

mass of particles in the bed 3 A 3 (1 - ) p ' > . . . . . (E9.4H)

5f packed bed depth (>1) and 7oidage ( 1) are known then if the mass remains

constant the bed depth at an6 7oidage can be determined:

B&li'0 Fl&i(i/a!io'

&he simplest description of the e"pansion of a bubbling fluidi!ed bed is deri7ed

from the &wo-$hase &heor6 of fluidi!ation of &oome6 and \ohnstone (1


65/90

+ is the actual gas flow rate to the fluid bed and +mf is the gas flow rate at

incipient fluidi!ation, then

gas passing through the bed as bubbles = + - +mf 3 (8 - 8mf) ' . . . (E9. 4)

gas passing through the emulsion phase

3 +mf 3 8mf ' . . . . . (E9.4)

E"pressing the bed e"pansion in terms of the fraction of the bed occupied b6

bubbles, A:

where > is the bed height at 8 and >mf is the bed height at 8mf and 8A is the

mean rise 7elocit6 of a bubble in the bed (obtained from correlations -- see

below). &he 7oidage of the emulsion phase is taken to be that at minimum

fluidi!ation mf. &he mean bed 7oidage is then gi7en b6:

5n practice, the elegant &wo-$hase &heor6 o7erestimates the 7olume of gas

passing through the bed as bubbles (the 7isible bubble flow rate) and better

estimates of bed e"pansion ma6 be obtained b6 replacing (+ - +mf) in E9.4; with

7isible bubble flow rate, +A 3 M ' (8 - 8mf) . . . . . (E9./)

where for %roup ' powders /.; Z M Z 1./

for %roup A powders /. Z M Z /.;

for %roup = powders /.4 Z M Z /.

&he abo7e anal6sis re9uires a knowledge of the bubble rise 7elocit6 8A, which

depends on the bubble si!e dA7 and bed diameter =. &he bubble diameter at a

gi7en height abo7e the distributor depends on the orifice densit6 in the

distributor N, the distance abo7e the distributor 2 and the e"cess gas 7elocit6

(8 - 8mf).

65


66/90

or %roup A powders:

or %roup ' powders: bubbles reach a ma"imum stable si!e which ma6 be

estimated from:

E'!rai'me'!

&he term entrainment will be used here to describe the eection of particles

from the surface of a bubbling bed and their remo7al from the 7essel in the

fluidi!ing gas. 5n the literature on the subect other terms such as carr*o"er,and el&!ria!io' are often used to describe the same process. >ere we are going

to stud6 the factors affecting the ra!e o) e'!rai'me'! of solids from a

fluidi!ed bed and de7elop a simple approach to the estimation of the

entrainment rate and the si!e distribution of entrained solids.

Consider a single particle falling under gra7it6 in a static gas in the absence of

an6 solids boundaries. @e know that this particle will reach a terminal 7elocit6

when the forces of gra7it6, buo6anc6 and drag are balanced (see #hodes, 1


67/90

• (b) in turbulent flow: the particle ma6 mo7e up or down depending on its

radial position. 5n addition the random 7elocit6 fluctuations superimposed

on the time-a7eraged 7elocit6 profile make the actual particle motion less

predictable.

5f we now introduce into the mo7ing gas stream a number of particles with a

range of particle si!e, some particles ma6 fall and some ma6 rise depending on

their si!e and their radial position. &hus the entrainment of particles in an

upward-flowing gas stream is a comple" process. @e can see that the rate of

entrainment and the si!e distribution of entrained particles will in general

depend on particle si!e and densit6, gas properties, gas 7elocit6, gas flow

regime - radial 7elocit6 profile and fluctuations and 7essel diameter. 5n addition

• (i) the mechanisms b6 which the particles are eected into the gas stream

from the fluidi!ed bed are dependent on the characteristics of the bed -

in particular bubble si!e and 7elocit6 at the surface,

• (ii) the gas 7elocit6 profile immediatel6 abo7e the bed surface is

distorted b6 the bursting bubbles. 5t is not surprising then that

prediction of entrainment from first principles is not possible and in

practice an empirical approach must be adopted.

&his empirical approach defines coarse particles as particles whose terminal

7elocit6 is greater than the superficial gas 7elocit6 (8& [ 8) and fine particles as

those for which 8& Z 8 and considers the region abo7e the fluidi!ed bed surfaceto be composed of se7eral !ones shown in igure below:

67


68/90

ig. $article distribution image and rotated plot of particle densit6 7s height

showing the !ones in the freeboard of a fluidi!ed bed.

• Freeoar(: the entire region between the bed surface and the gas outlet.

• Slash /o'e: the region ust abo7e the bed surface, in which coarse

particles fall back down.

• Dise'0a0eme'! /o'e: the region abo7e the splash !one, in which both the

upward flu" and the suspension concentration of fine particles decreases

with increasing height.• Dil&!e7hase !ra'sor! /o'e: region abo7e the disengagement !one, in

which all particles are carried upwardsB particle flu" and suspension

concentration are constant with height.

Note that, although in general fine particles will be entrained and lea7e the

s6stem and coarse particles will remain, in practice fine particles ma6 sta6 in

the s6stem at 7elocities se7eral times their terminal 7elocit6 and coarse

68


69/90

particles ma6 be entrained. &he height from the bed surface to the top of the

disengagement !one is known as the !ra'sor! (ise'0a0eme'! hei0h! (&=>).

'bo7e &=> the entrainment flu" and concentration of particles is constant.

&hus, from the design point of 7iew, in order to gain ma"imum benefit from theeffect of gra7it6 in the freeboard, the gas e"it should be placed abo7e the

&=>. an6 empirical correlations for &=> are a7ailable in the literature (e.g.

Len!, 1orio 1orio, which is presented in

E9uation .

(dA7s 3 e9ui7alent 7olume diameter of a bubble at the surface).

&he empirical estimation of entrainment rates from fluidi!ed beds is based on

the following rather intuiti7e e9uation:

where elutriation rate constant (the entrainment flu" at height h abo7e the bedsurface for the solids of si!e "i, when mAi 3 1./).

A 3 total mass of solids in the bed

' 3 area of bed surface

mAi 3 fraction of the bed mass with si!e " i at time t.

or continuous operation, mAi and A are constant and so:

and total rate of entrainment,

&he solids loading of particles of si!e " i in the off-gases is i 3 #i ? (8 ') and

the total solids loading lea7ing the freeboard is & 3 i.

69


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or batch operation, the rates of entrainment of each si!e range, the total

entrainment rate and the particle si!e distribution of the bed change with time.

&he problem can best be sol7ed b6 writing E9uation E9.; in finite increment

form:

where (mAi A) is the mass of solids in si!e range i entrained in time increment

t.

&hen total mass entrained in time

(for k si!e ranges) and mass of solids remaining in the bed at time

(where subscript t refers to the 7alue at time t.) Aed composition at time

Solution to a batch entrainment problem is b6 se9uential application of

E9uations H1 to HH for the re9uired time period.

&he elutriation rate constant V` ih cannot be predicted from first principles and

so it is necessar6 to rel6 on the a7ailable correlations which differ significantl6

in their predictions. Correlations are usuall6 in terms of the carr6o7er rate

abo7e &=>, . &wo of the more reliable correlations are gi7en below:

or particles [ 1// m and 8 [ 1.4 m?s %eldart et al. (1


71/90

or particles Z 1// m and 8 Z 1.4 m?s Len! and @eil (1


72/90

Consider an element of the bed of height 2 at a distance 2 abo7e the

distributor (igure below).

ig 'nal6sis of gas-particle heat transfer in an element of a fluidi!ed bed

2et the temperature of the gas entering this element be & g and the change in

gas temperature across the element be & g. &he particle temperature in the

element is & s.

&he energ6 balance across the element gi7es:

where

a 3 surface area of solids per unit 7olume of bed

Cg 3 specific heat capacit6 of the gas

g 3 gas densit6

hgp 3 particle-to-gas heat transfer coefficient8 3 superficial gas 7elocit6

5ntegrating with the boundar6 condition & g 3 & g/ at 2 3 /,

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&he distance o7er which the temperature distance is reduced to half its initial

7alue, 2/. is then:

or a bed of spherical particles of diameter ", the surface area per unit 7olume

of bed, a 3 (1 - ) ? " where is the bed 7oidage. 8sing the correlation for hgp

in E9. H, then

's an e"ample we will take a bed of particles of mean si!e 1// m, particle

densit6 4// kg?m fluidi!ed b6 air of densit6 1.4 kg?m, 7iscosit6 1.;H " 1/-

$as, conducti7it6 /./44 @?mV and specific heat capacit6 1// \? (kg V).

8sing the Aae6ens e9uation for 8mf (E9uation 11), 8mf 3 ence, assuming a fluidi!ed bed 7oidage of /.H, 8rel 3 /./4 m?s.

Substituting these 7alues in E9uation 1, we find 2/. 3 /.


74/90

hpc is the particle con7ecti7e heat transfer coefficient and describes the heat

transfer due to the motion of packets of solids carr6ing heat to and from the

surface. hgc is the gas con7ecti7e heat transfer coefficient describing the

transfer of heat b6 motion of the gas between the particles. hr is the radiantheat transfer coefficient. igure below, after Aotterill (1


75/90

igure 1: >eat transfer from bed particles to an immersed surface

&he particle-to-surface contact area is too small to allow significant heat

transfer. actors affecting the gas film thickness or the gas conducti7it6 will

therefore influence the heat transfer under particle con7ecti7e conditions.

=ecreasing particle si!e, for e"ample, decreases the mean gas film thickness

and so impro7es hpc. >owe7er, reducing particle si!e into the %roup C range will

reduce particle mobilit6 and so reduce particle con7ecti7e heat transfer.

5ncreasing gas temperature increases gas conducti7it6 and so impro7es hpc.

$article con7ecti7e heat transfer is dominant in %roup ' and A powders.5ncreasing gas 7elocit6 be6ond minimum fluidi!ation impro7es particle

circulation and so increases particle con7ecti7e heat transfer. &he heat

transfer coefficient increases with fluidi!ing 7elocit6 up to a broad ma"imum

hma" and then declines as the heat transfer surface becomes blanketed b6

bubbles. &his is shown in igure below for powders in %roups ', A and =.

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igure: Effect of fluidi!ing gas 7elocit6 on bed-surface heat transfer

coefficient in a fluidi!ed bed.

&he ma"imum in hpc occurs relati7el6 closer to 8mf for %roup A and = powders

since these powders gi7e rise to bubbles at 8mf and the si!e of these bubbles

increase with increasing gas 7elocit6. %roup ' powders e"hibit a non-bubblingfluidi!ation between 8mf and 8mb and achie7e a ma"imum stable bubble si!e.

Aotterill (1


77/90

Aotterill (1


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igure: ' fluidi!ed bed solids cooler

luidi!ed beds are used for coating particles in the pharmaceutical andagricultural industries. etal components ma6 be plastic coated b6 dipping them

hot into an air-fluidi!ed bed of powdered thermosetting plastic.

Chemical Processes

&he gas fluidi!ed bed is a good medium in which to carr6 out a chemical reaction

in7ol7ing a gas and a solid. 'd7antages of the fluidi!ed bed for chemical

reaction include:

• &he gas-solids contacting is generall6 good.

• &he e"cellent solids circulation with the bed promotes good heattransfer between bed particles and the fluidi!ing gas and between the

bed and heat transfer surfaces immersed in the bed.

• &his gi7es rise to near isothermal conditions e7en when reactions are

strongl6 e"othermic or endothermic.

• &he good heat transfer also gi7es rise to ease of control of the reaction.

• &he fluidit6 of the bed makes for ease of remo7al of solids from the

reactor.

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>owe7er, it is far from idealB the main problems arise from the two phase

(bubbles and fluidi!ed solids) nature of such s6stems. &his problem is

particularl6 acute where the bed solids are the catal6st for a gas-phase

reaction. 5n such a case the ideal fluidi!ed bed chemical reactor would ha7ee"cellent gas-solid contacting, no gas b6-passing and no backmi"ing of the gas

against the main direction of flow. 5n a bubbling fluidi!ed bed the gas b6-passes

the solids b6 passing through the bed as bubbles. &his means that unreacted

reactants appear in the product. 'lso, gas circulation patterns within a bubbling

fluidi!ed bed are such that products are back-mi"ed and ma6 undergo

undesirable secondar6 reactions.

&hese problems lead to serious practical difficulties particularl6 in the scaling-

up of a new fluidi!ed bed process from pilot plant to full industrial scale. &his

subect is dealt with in more detail in references: Vunii and 2e7enspiel (1


80/90

igure 1: VelloggYs odel ' Orthoflow CC 8nit

&his figure is a schematic diagram of one t6pe of fluid catal6tic cracking (CC)

unit - a celebrated e"ample of fluidi!ed bed technolog6 -- for breaking down

large molecules in crude oil to small molecules suitable for gasoline etc. Other

e"amples of the application of fluidi!ed bed technolog6 to different kinds of

chemical reaction are shown in &able 4.

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8N5& O$E#'&5ONS 1

&he fi7e ke6 components thumb rules of chemical engineering are : material

balance, energ6 balance, momentum balance, e9uilibria relations or ratee9uations, and cost e9uations or econom6 e9uations. 'll chemical engineering

problems and calculations fall under categories. Chemical engineering

operations used to be classified as 1? U'i! oera!io's, 2? U'i! rocesses. 5n

unit operations, there is no chemical reaction but onl6 ph6sical change. E"amples

include filtration, dr6ing, e7aporation etc. 5n unit process, there is chemical

reaction. E"amples includeB o"idation, h6drogenation and halogenation etc. 5n

chemical engineering and related fields, a &'i! oera!io' is a basic step in a

process . or e"ample in milk processing, homogeni!ation, pasteuri!ation , chilling ,

and packaging are each unit operations which are connected to create the

o7erall process. ' process ma6 ha7e man6 unit operations to obtain the desired

product. >istoricall6, the different chemical industries were regarded as

different industrial processes and with different principles. 5n 1.

@alker, @arren V. 2ewis and @illiam >. c'dams wrote the book &he

$rinciples of Chemical Engineering and e"plained the 7ariet6 of chemical

industries ha7e processes which follow the same ph6sical laws. &he6 summed-up

these similar processes into unit operations. Each unit operation follows the

same ph6sical laws and ma6 be used in all chemical industries. &he unit

operations form the fundamental principles of chemical engineering.

Chemical engineering unit operations consist of fi7e classes:

1. luid flow processes, including fluids transportation, filtration, solids

fluidi!ation etc.

4. >eat transfer processes, including e7aporation, condensation etc.

. ass transfer processes, including gas absorption, distillation,

e"traction, adsorption, dr6ing etc.

H. &hermod6namic processes, including gas li9uefaction, refrigeration etc.

. echanical processes, including solids transportation, crushing andpul7eri!ation, screening and sie7ing etc.

Chemical engineering &'i! oera!io's and chemical engineering &'i! rocessi'0

form the main principles of all kinds of chemical industries and are the

foundation of designs of chemical plants, factories, and e9uipment used.

81

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82/90

FILTRATION

iltration is the remo7al of solid particles from a fluid b6 passing the fluid

through a filtering medium (' 52&E#), or septum , on which the solids aredeposited. =epending on the application, the solid, the fluid, or both ma6 be

isolated. E"amples of filtration include a coffee filter which separates the

coffee grounds from the brewed coffeeB the use of consumer water filters to

impro7e the taste or appearance of municipal waterB and the use of >E$'

filters in air conditioning to remo7e particles from air. &o separate a mi"ture of

chemical compounds, a sol7ent is chosen which dissol7es one component, while

not dissol7ing the other. A6 dissol7ing the mi"ture in the chosen sol7ent, one

component will go into the solution and pass through the filter, while the other

will be retained. &his is one of the most important techni9ues used b6 chemists

to purif6 compounds.

luid flows through a filter medium b6 7irtue of a pressure differential across

the medium. &he6 are also classified into those that operate with a pressure

abo7e atmospheric on the upstream side of the filter medium * those that

operate with atmospheric pressure on the upstream side and a 7accum on the

downstream side. &he simplest method of filtration is to pass a solution of a

solid and fluid through a porous interface so that the solid is trapped, while the

fluid passes through. &his principle relies upon the si!e difference between the

particles making up the fluid, and the particles making up the solid. 5n thelaborator6, a Achner funnel is often used, with a filter paper ser7ing as the

porous barrier.

igure: showing a simple filtration process

ost industrial filters are ress&re )il!ers, "acc&m )il!ers, and ce'!ri)&0al

seara!ors. &he6 are either continuous (discharge of both solids * fluid is

interrupted as long as the e9uipment is in operation) or discontinuous(&he flow82

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of fluid through the de7ise is continuous, but it must be interrupted periodicall6

to permit discharging the accumulated solids), depending on whether the

discharge of filtered solids is stead6 or intermittent.

Fil!ers are (i"i(e( i'!o !hree mai' 0ro&s: cake filters, clarif6ing filters, and

cross flow filters. Cae )il!ers separate relati7el6 large amounts of solids as a

cake of cr6stal or sludge. &he6 often include pro7isions for washing the cake

and for remo7ing some of the li9uid from the solids before discharge. Clari)*i'0

)il!ers remo7e small amounts of solids to produce a clean gas or sparkling clear

li9uids such as be7erages. &he solid particles are trapped inside the filter

medium or on its e"ternal surfaces. Clarif6ing filters differ from screens in

that the pores of the filter medium are much larger in diameter than the

particles to be remo7ed. 5n Cross )lo Fil!er the feed suspension flows under

pressure at a fairl6 high 7elocit6 across the filter medium. ' thin la6er of solids

ma6 form on the surface of the medium, but the high li9uid 7elocit6 keeps the

la6er from building up. &he medium could be ceramic, metal, or pol6mer

membrane with pores small enough to e"clude most of the suspended particles.

Some of the li9uid passes through the medium as clear filtrate, lea7ing a more

concentrated suspension behind. 8ltra filtration is a cross flow t6pe containing

a membrane with e"tremel6 small openings, used for the separation and

concentration of colloidal particles and large molecules.

EAMPLES OF FILTERS

DISCONTINUOUS PRESSURE FILTERS( $ressure filters appl6 a large

pressure differential across the septum to gi7e economicall6 rapid filtration

with 7iscous li9uids or fine solids, most pressure filters are discontinuous):

Fil!er ress (pg /H, Coulson * #ichardson, Dol 4), Shell = a'( = lea) )il!ers,

A&!oma!ic el! )il!er

:ACUUM FILTERS. &he6 are usuall6 continuous, though a discontinuous

7acuum filter is a useful tool. &he pressure difference across the septum in acontinuous 7acuum filter is not high). E"ample of Continuous t6pe is the Ro!ar*

= (r&m )il!er( e.g. in a Linc leaching operation). @here 7acuum filtration cannot

be used, other means of separation, such as continuous centrifugal filters,

should be considered.

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CENTRIFUGAL FILTERS< &he main t6pes of filtering centrifuges are

s&se'(e( a!ch machi'es, which are discontinuous in their operations:

a&!oma!ic shor! = c*cle a!ch machi'esB and co'!i'&o&s co'"e*or ce'!ri)&0es.

&he most important factors on which the rate of filtration depends will be

1. &he drop in pressure from the feed to the far side of the filter

medium.

4. &he area of the filtering surface.

. &he 7iscosit6 of the filtrate.

H. the resistance of the filter cake.

. &he resistance of the filter medium and initial la6ers of cake

&o increase flow rate( Consider =arc6’s law), it is fa7oured b6B 5ncrease

pressure, 5ncrease area, Change media( more permeable), change fluid(less

7iscous), thinner media. The Se!&m m&s! also B 1. #etain the solids to be

filtered, with a reasonabl6 clear filtrate. 2? ust not plug or blind. 3? ust be

resistant chemicall6 and strong enough ph6sicall6 to withstand the process

conditions. ? ust permit the cake formed to discharge cleanl6 * completel6.

@? ust not be prohibiti7el6 e"pensi7e.

Fil!er Ai(s : Der6 fine solids mo7ing across the medium often plug an6 medium

that is fine to retain them. So, it is re9uired that the porosit6 of such material

be increased to permit passage of the li9uor at a reasonable rate. &his is done

b6 adding a filter aid. E"amples areB diatomaceous silica, perlite, purified wood

cellulose to the slurr6 before filtration. &he aid subse9uentl6 ma6 be separated

from the filter cake b6 dissol7ing awa6 the solids or b6 burning out the filter

aid. 'nother wa6 of using a filter aid is b6 recoa!i'0> that is, depositing a

la6er of it on the filter medium before filtration. 5n batch filter it is thin, in a

continuous precoat filter, the la6er of precoat is thick. $recoats pre7entgelatinous solids from plugging the filter medium and gi7e a clearer filtrate. &he

precoat is part of the medium not that of the cake.

ilter cakes can be di7ided into two classes, incompressible cakes and

compressible cakes. or incompressible cake, the resistance to flow of a gi7en

7olume of cake is not appreciabl6 affected either b6 the pressure difference

across the cake or b6 the rate of deposition of material. On the other hand,

84


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with a compressible cake, increase of the pressure difference or of the rate of

flow causes the formation of a denser cake with a higher resistance. 5n

filtration, the complete c6cle is made up ofB a? &he filtration process, ?

@ashingB c? =ismantling?reassembl6. 5n all, the filtration time is e7aluated b6considering the incompressible cake and the compressible cake.

FILTRATION THEOR;

iltration is a special e"ample of flow through porous media. 's the flow

increase with time the medium becomes clogged or a filter cake builds up. &he

9uantities of interest are the )lo ra!e through the filter and the ress&re

(ro across the unit. 's time goes on in the operation, either the flow rate

diminishes or the pressure drop rises. 5n Constant-pressure filtration the

pressure drop is held constant and the flow rate allowed to fall with timeB less

commonl6, the pressure drop is graduall6 increased to gi7e a constant-rate

filtration. 5n cake filtration, two resistances are a7ailable, the cake and that of

the filter medium. &he o7erall pressure drop at an6 time is the sum of the

pressure drops o7er medium and cake.

PRINCIPLES OF CAE FILTRATION

O7erall material balance based on a unit area in a filtration operation is gi7en as

ass of slurr6 3 ass of Cake ass of iltratec c

c

w w 7

s s ρ = + GGGG.1

wc 3 total mass of dr6 cake solids per unit aream

' ÷

7 3 iltrate 7olume per unit area

s 3 mass fraction of solids in the slurr6

sc3 a7erage mass fractions of solids in the cake ρ 3 densit6 of filtrate.

rom e9u. 1 1c

c

s w 7 s

s

ρ

= − c7 = GGG..4

@here 1c

s c

s

s

ρ =

−

C 3 concentration e"pressed b6 mass of dr6 cake per unit 7olume of filtrate.

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rom 4, c w

c 7

= , if the mass of solids in the cake can be obtained.

>owe7er, draining the slurr6 can often lead to difficulties in accurate

determination of cake mass. 'n alternati7e approach is to consider the cakethickness. &he cake thickness is gi7en as 2, this can be related to the cake

mass wc b6( )

( )

( )

1

1

1

c s a7

c s a7

s a7

w 2

m w 2

'm '2

ρ ε

ρ ε

ρ ε

= −

= = −

= −

GGG

s ρ 3 true densit6 of the solidB m 3 mass of solids in la6er( )1 a7 '2ε − = Dolume of solids in the la6erB a7 ε 3 a7erage porosit6 of the cake.

Combining 4 *

( )

( )

1

1

B

c s a7

s a7

2 2

w c7 2c

2 7

22 c 7 c

7

ρ ε

ρ ε

= = −=

−

= =

2c is the ratio of cake thickness 2 to the per unit area filtrate 7olume, 7. &he

thickness 2 is in fact the primar6 parameter related to filter design.

$#ESS8#E =#O$ &>#O8%> 52&E# C'VE

' t6pical diagram showing a section through a filter cake and the filter medium

at a definite time t, from the start of the flow of filtrate re7eals that in thebed the 7elocit6 is sufficientl6 low to ensure laminar flow. 'ccordingl6, as a

starting point for treating the pressure drop through the cake, an e9uation of

the form

( )4

4 4

1/ 1o

s p

7 p

2 =

µ ε

ε

−∆=

ΦGGGG..H

E9uation H is known as Vo!eng Carman e9uation and is applicable for flow

through beds at particle #e6nolds numbers up to about 1./

5f we designate the superficial 7elocit6 of the filtrate as u , e9uation H

becomes( )

( )

4

4

1/ 1

s p

u dp

d2 =

µ ε

ε

−=

ΦGGGG

E"pressing as a function of surface 7olume ratio instead of particle si!e

( )

4

4

4

H.1 1 p

p

s u

7 dp

d2

µ ε

ε

− ÷ ÷

=GGG.

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Sp 3 surface area of single particle B 7p 3 7ol. of single particle µ 3 7iscosit6 of filtrate B u 3 linear 7elocit6 of filtrate, based on filter areadp

d2

= pressure gradient at thickness 2

&he linear 7elocit6 u ?d7 dt

'= GG..

D 3 7olume of filtrate collected from the start of filtration to time t( )1s dm 'd2 ρ ε = − GG.;

Eliminating d2 from * ; gi7es us

( ) ( )4

1

? 1p p

s

k u s 7 dp dm

'

µ ε

ρ ε

−= GGG<

V1 replaces the coefficient H.1 5f $S units are used, Newton law

proportionalit6 factor gc must be included in the denominator of e9uation


88/90

52&E# E=58 #ES5S&'NCE

' filter medium resistance /,α α

can be defined b6 analog6 with the cakeresistanceα .

m m

p #

u µ

∆= m p ∆ = $ressure drop o7er medium 3 1 b p p −

&he filter medium resistance m # ma6 7ar6 with the pressure drop, since the

higher li9uid 7elocit6 caused b6 a large pressure drop ma6 force additional

particles of solid into the filter medium. m # also 7aries with the age and

cleanliness of the filter mediumB

c m p p p ∆ = ∆ + ∆

c m um

# u

'

αµ µ = +

?Bc m

m d7 dt + $ u # u

' ' '

α µ

∆ = + = = ÷

GGG..1

c m

m + $ #

' '

α µ

∆ = + ÷

, + 3 Dolumetric flow rate

c c

m w c7

' = =

c m 'CD ∴ =

1 becomes

4.

.

m

m

m

d7 'c7 p #

dt ' 'd7 c7

# dt ' '

dt c7 #

d7 ' p '

α µ

α µ

µ α

∆ = + ÷

= + ÷

= + ÷∆

m

dt c7 #

d7 ' p '

µ α = + ÷∆ GGG.1H

Or . m d7 c7

p # dt ' '

µ α ∆ = + ÷

CONS&'N& $#ESS8#E 52'&5ON

't constant pressure, the 7ariables that 7ar6 are 7,and t.

@hen t3/, 73/ and m p p ∆ = ∆ , hence

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//

1m # dt

' p d7 9

µ = = ÷∆ GGG1

1H becomes /

1 1

c

dt k 7

d7 9 9 = = +

GGG.1

@here 4c c

k ' p

µ α =

∆ GGG..1

5ntegrating e9uation 1 between limits (/,/) * (t,7), results as

// / /

1t 7 7

c dt k 7d7 d7 9 = +∫ ∫ ∫

/

4

4c

7 7 t k

9 = +

/

14

c 7 t 7 k 9

= + ÷ ÷

/

1

4c k t 7

7 9

6 m" c

= + ÷

= +

So the plot of t?7 7ersus 7 will be linear, with a slope e9ual to k c?4 and the

intercept of 1?9/. from this graph and using e9u. 1 * 1 , m # * α ma6 be

e7aluated.

' centrifugal pump will deli7er a constant head($ressure)

E$E#5C'2 E+8'&5ONS O# C'VE #ES5S&'NCE

Conducting constant pressure e"periments at 7ar6ing Y ,p s α ∆ 7ariation withp ∆ can be found. 5f α is independent of p ∆ , the cake is incompressible.

Ordinaril6 α is e"pected to increase with p ∆ ,at least to some e"tent most

t?7

7

Slope3kc?4

5ntercept31?9/

89


90/90

cakes are incompressible. or highl6 compressible cakes α increases rapidl6

with p ∆ .&he empirical e9uation is gi7en as

( )/s

p α α = ∆ GGGGG.1;/,α α are empirical constants, s 3 compressibilit6 coefficient of the cake. s3/

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CHE 321 Lecture Notes