materi 4 - flow of fluid through fluidised beds

8/17/2019 Materi 4 - Flow of Fluid Through Fluidised Beds

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Flow of Fluids throughFluidised Bed


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• 6.1.2. Efect o uid velocity onpressure gradient and pressure drop

• When a uid ows slowly upwards through abed of very ne particles the ow isstreamline and a linear relation existsbetween pressure gradient and owrate.

• If the pressure gradient ( − ∆P/l) is plottedagainst the super cial velocity (u c ) usinglogarithmic co ordinates a straight line ofunit slope is obtained! as shown in Figure

".#.


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buoyancy effect startsto work. Incipient

fluidised bed

frictional wall effect on thevessel or container isworking in transport bed

Uc,t = terminal velocity.Incipient transport ofparticles

u c > terminal velocity(transport or elutriation orentrainmen

t or carryover

∆! is due to ρ s "ρ from incipient fluidisation until incipienttransport of particles. #uring fluidisation, particles can be treated

as hindered settling. $ransport of particles starts at terminalvelocity of free settling

%indered settling


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• $s the super cial velocity approaches theminimum uidising velocity (u mf ) ! the bedstarts to expand and when the particles areno longer in physical contact with oneanother the bed is uidised . %he pressuregradient then becomes lower because of theincreased voidage and! conse&uently! theweight of particles per unit height of bed issmaller ' due to buoyancy e(ect ).


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• %his fall continues until the velocity is highenough for transport of the material to ta*eplace ! and the pressure gradient then starts toincrease again because the frictional drag ofthe uid at the walls of the tube starts tobecome signi cant.

• When the bed is composed of large particles!the ow will be laminar only at very lowvelocities and the slope of the curve may notbe constant! particularly if there is aprogressive change in ow regime as thevelocity increases.


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• If the pressure across the whole bed insteadof the pressure gradient is plotted againstvelocity! also using logarithmic coordinatesas shown in Figure ".+ ! a linear relation isagain obtained up to the point whereexpansion of the bed starts to ta*e place '$)!although the slope of the curve thengradually diminishes as the bed expands andits porosity increases.

• $s the velocity is further increased! ∆,passes through a maximum value 'B) andthen falls slightly and attains an


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frictional forces betweenparticles work to allparticles when u c isincreased until point &

frictional forces interparticles decreasemore and more untilnone at '

∆! is due to ρ s "ρ

frictional forces betweenparticles are less after bed e pansion

!articles are )ust restingon one another

*inimumfluidising vel.

frictional forces interparticles are less due toloose physical contacts,bed porosity is larger


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• I the uid velocity is reduced again ! thebed contracts until it reaches the conditionwhere the particles are just resting on oneanother (E).

• %he porosity then has the maximum stablevalue which can occur for a xed bed of theparticles.

• If the velocity is further decreased ! thestructure of the bed then remains una(ectedprovided that the bed is not sub/ected to

vibration.•


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• If the velocity is now increased again! it mightbe expected that the curve 'F0) would beretraced and that the slope would suddenlychange from # to 1 at the uidising point .

• %his condition is di2cult to reproduce!however! because the bed tends to becomeconsolidated again unless it is completelyfree from vibration.

• In the absence of channelling ! it is the shapeand si3e of the particles that determine both

the maximum porosity and the pressure dropacross a iven hei ht of uidised bed of a


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• In an ideal uidised bed ∆, corresponding to0- is e&ual to the buoyant weight ofparticles per unit area.

• In practice! it may deviate appreciably fromthis value as a result of channelling and thee(ect of particle wall friction.

•,oint B lies above - because the frictionalforces between the particles have to beovercome before bed rearrangement canta*e place.


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• %he minimum uidising velocity! u mf ! may bedetermined experimentally by measuring ∆,across the bed for both increasing anddecreasing velocities and plotting the results asshown in Figure ".+.

• %he two 4best5 straight lines are then drawnthrough the experimental points and the

velocity at their point of intersection ' E) is ta*enas the minimum uidising velocity.• 6inear rather than logarithmic plots are

generally used! although it is necessary to use

logarithmic plots if the plot of pressure gradienta ainst velocit in the xed bed is not linear.


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• %he theoretical value of the minimumuidising velocity may be calculated from the

e&uations given in -hapter 7 for the relationbetween pressure drop and velocity in a xedpac*ed bed ! with the pressure drop throughthe bed 8 the efective eight o particlesdue to s ! per unit area ! and theporosity set at the maximum value that can beattained in the xed bed.

• In a uidised bed! the total frictional force on

the particles under condition! which is similarto hindered settlin 8 the e(ective wei ht


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• %hus! in a bed of unit cross sectional area! depthl! and porosity e ! the additional pressure dropacross the bed attributable to the layout weight

of the particles is given by9 (general orce"alance or uidised "ed)• .• where9 g is the acceleration due to gravity and

ρ s and ρ are the densities of the particles andthe uid respectively.• ∆, is applied to overcome friction on the particle

surfaces in hindered settling until terminal

velocity 'drag force at u : u 1 8 gravity force ;bouyancy force).


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• %here may be some discrepancy between thecalculated and measured minimum velocitiesfor uidisation.

• %his may be attributable to channelling ! as aresult of which the drag force acting on the bedis reduced ! or to the action of electrostatic

forces in case of gaseous uidisation particularly important in the case of sands! or toagglomeration which is often considerable withsmall particles! or to friction between the uid

and the walls of the containing vessel


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• If ow conditions within the bed arestreamline ! the relation between uid velocityu c! pressure drop '− ∆P) and voidage e is given!for a xed bed of spherical particles ofdiameter d ! by the -arman


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• 6.1.&. 'inimum uidising velocity• $s the upward velocity of ow of uid through

a pac*ed bed of uniform spheres is increased !the point o incipient fuidisation isreached when the particles are justsupported in the uid .

• %he corresponding value of the minimumuidising velocity (u mf ) is then obtained bysubstituting e mf into e&uation ".> to give9

+ . -. using u mf is derived from + . -./ using uc, so it is

based on cross"sectional area of the column


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• =ince e&uation ".7 is based on the -arman;


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• When the ow regime at the point ofincipient uidisation is outside the rangeover hich the arman! o*eny e+uationis applica"le ! it is necessary to use one ofthe more general e&uations for the pressuregradient in the bed such as the Ergune+uation as9

• where d is the diameter of the sphere• 0rgun e&uation is to correlate between ∆P

and u c ' #$ed "ed % minimum velocity

+rgun e . forring packing


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0ubstituting e = e mf at the incipient fluidisationpoint and for 1 ∆P from e uation -.2 , e uation -.- isthen applicable at the minimum fluidisation velocityu

mf , and gives3

4orks on fi ed bed =minimum velocityfluidised bed

4orks on fluidisedbed (general forcee

&y definition, Re mf is isbased on bed cross-

sectional area (u

mf =u

c


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• where Re mf is the ?eynolds number at the

minimum uidising velocity and e&uation".@ then becomes9


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• E$pansion o udised "ed• Beyond u mf the particle separation increases with

increasing uid super cial velocity whilst the pressureloss across the bed remains constant.

• %his increase in bed voidage with uidi3ing velocity isreferred to as bed expansion.

• ?ichardson and Aa*i '# C7) found u c is the function f' e )which applied to hindered settling . In general u c 8 u cT εn where u cT is the terminal velocity of super cial velocity

' u mf : u c : u cT ).• For ?e5:1.> '=to*e5s law)! u c 8 u cT ε7."C

• For ?e5DC11 'Eewton5s law)! u c 8 u cT ε+.7


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• ρ f ' ρ p ρ f )g Gµ+) and d p is the particlediameter and D is the vessel diameter.


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• 0stimation of the bed voidage can bedetermined by following calculation9


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• E$ample 6.2• Hil! of density 11 *gGm > and viscosity >

mEsGm +! is passed vertically upwardsthrough a bed of catalyst consisting ofapproximately spherical particles of diameter1.# mm and density +"11 *gGm >.

•$t approximately what mass rate of ow perunit area of bed will 'a) uidisation! and 'b)transport of particles start to occur


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• ,olution 'assumption of laminar ow'=to*es5 law))

• In this problem! ρ s 8 +"11 *gGm >! ρ 8 11

*gGm>

! μ 8 > 1 J #1−>

EsGm+

and d 8 1 # mm 8# 1 J #1 −7 m.• $s no value of the voidage is available or can

be estimated! e will be estimated by

considering eight closely pac*ed spheres of


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5ssumption at minimumfluidisation for unknownbed si6e and mass of

the bed


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78/9:alileonumer

$rial anderror

#ensity of oil

!articles in bed are treated as independent particles, then Ga can be applied

materi 4 - flow of fluid through fluidised beds

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