Simple bed expansion correlations for magnetically assisted gas-fluidized tapered beds

W/.r.rtl"!

International Review of Chemical Engineering (I.RE.CH.E.), Vol. I, N. 4July 2009?ndte

/

al

ao

a b

Da

bp

Db

DL

dp

H

nb

huo

hL

Ms

Mp

oo

Q.y-t

?62-e

Simple Bed Expansion Correlations for Magnetically AssistedGas-Fluidized Tapered Beds

Jordan Hristov

Abstract - Direct scaling relationships allowing to fit bed expansion (porosity) curyesmagnetically assisted gas-fluidized beds (Magnetization FIRST mode) hove been tested.Exponential-decay, dsymptotic and power-latv relationships have been used as approximatingfunctions. The numerical experiments have demonstrated the applicability of the approximatingfunction chosen to predict either the entire expansion curve profiles or the maximum bedexpansion through simple asymptotes. The errors ofapproximations performed revealed less thanl0% in all trials and relationships established. Copyright @ 2009 Praise llorthy Prize S.r.I. - Allrights reserved

Keywotds: Fluidization, Magnetization FIRST, Tapered Bed, Porosity Correlations, ScalinRelationship

Nomenclature

Dimensionless pre-factors in (Eq. 2), o

Exponential pre-factors in Eq. (2), (tl *t )Dimensionless free term (asymptote) of theasymptotic function (Eq. 3), C)Dimensionless pre factor in Eq. 4, oDimensionless pre factor of the asymptoticfunction (Eq. 3), C)Dimensionless exponent in Eq. (a)

Rate of the Asymptotic Function,dimensionless, (-)Diameter of the flow entrance, ( rn )Top diameter of the bed, ( z )Particlediameter,(n)

Gravity acceleration, 9.81 m2 f s

Magnetic f i eld intensity, ( 4 ^)

Bed height, ( zr )Initial bed height, ( z )Bed length at the wall (see Fig. l), ( ^)

Magnetizalion at saturation, ( lf m )Mass of particles charged into the vessel,( t g )

Volumetric gas flow nte, (m3 f s )Volumetric gas flow rate at the onset ofinitial bed expansion (onset of MSB),

@'l ' )Volumetric gas flow rate at the fluidization

onset in the bottom part of the bed, (# f s )

Volumetric gas flow rate at the fluidizationonset over the entire bed in absence of a

field,(m3fs)

Volumetric gas flow rate at the fluidization

onset over the entire bed, (. ' /" )Volumetric gas flow rate at the minimum

spouting (unstable) point, (. ' /" )Volumetric gas flow rate at the minimum

spouting (stabte) points, ( nr/s )Cross section area of the gas inlet orifice,( * ' )

Superficial gas velocity (denoted also asU, see the text), (lll/s)

Superficial fluid velocity defined throughthe cone entrance cross sectional area.

@l ' )Volume occupied by the solids, ( z3 )

(nno, I n)(ol + D,Du * 4), (, ' )Independent variableDependent variableDimensionless free terms in Eqs. (2), C)

Greek LettersCone angle, (deg)Porosity, (-)Initial bed porosity, (-)

Annular bed porosity, (-)

Maximum bed porosity, (-)

Fluid density , (kcl .t )

Q^yo

Q.J-z

Q,," u

Qr" s

su=rf i fa

U

uh

a

a^

€o

t^*

P1

V,

v!r

Manuscript received and revised June 2009, accepted July 2009

316

Copyright @ 2009 Praise lVorthy Prize S.r.l. - All rights resemed

Jordan Hristov

g

,NB

t {

mt

ps

AFEDFMSBMATBPLF

ps Gas density, (ksl*t )P, Solid particle density, (kel*t )

Subscripts

of bed expansion profiles of magnetically assisted gasfluidized tepered beds with Magnetization FIRST mode.The first report in this new trend in magnetically assistedfluidization has reported by }lristov [6] with a lot ofprimary experimental data.

Dimensionless groups provided by the "pressuretransform" approaches [6], [8], [14] were used todevelop dimensionless scaling of pressure drop [6], [8]and bed porosity [8], U4l.

Owing to the specific vessel geometry andmagnetically controlled bed structures the bed expansionprofiles expressed through the porosity evolution curvese = f (Q) exhibit plateaus [6], [13] (see the experimentalresults-Fig. 2(a)) which cannot be modelled by theexisting data correlations developed for non-magneticspouted beds [5]-[17].

Because ofthat the present article refers to developeddirect scaling relationship expressions of the bedporosity curves in the form

" = f (Q) enabling easily

practical applications and accurate estimation of themaximum bed porosity attainable.

Such direct scaling relationships are additional ones tothose developed through a more complex dimensionalanalysis [4].

Additionally, the scaling was performed with well-know functions (power-law, exponential-decay andasymptotic functions) available in most of thecommercially available.

IL Experimental

The experimental set-up is described in details elsewhere16l, l7l, [4] so we will provide some basic informationpertinent to the results reported in this work.A conical vessel (15 deg, opening angle, 30 mm ID-bottom diameter and 190 mm ID-top diameter)surrounded by saddle coils [6], [9] with 200 mm ID and400 mm in height was used.The field lines were oriented transversely to the coneaxis of symmetry and the fluid flow (see Fig. l).The field was steady and the maximum field intensity

attained in these experiments was about 27 kAlm .Magnetite sand and ammonia catalyst "Haldorf Topsoe",KM-l is narrow sieve fractions (see Table I) were usedin the experiments.

Fig. 1. Experimental set-up: schematically

International Review ofChemical Engineering, Vol. I, N.4

FluidGasMaximumMinimum fluidizationMinimum spoutingParticleSolids

AbbreviationsAsymptotic FunctionExponential Decay FunctionMagnetically Stabilized BedMagnetically Assisted Tapered BedPower -law Function

I. Introduction

Tapered beds have been originally conceived for [l]for gas-fluidization of Geldart's D particles [2] and haveencountered in many applications Il]-t181.

In contrast to the original idea in the past years suchbeds have been successfully applied to fluidization ofcohesive particles [3] and drying of pasty materials [4],[5] in two and three-dimensional tapered vessels. Theseexamples address beds with strong interparticle forceswhich cannot be controlled. The common approach tocreate fluidized beds with controlled interparticle forcesand remotely managed fluidization behaviour is byapplication of extemal magnetic or electric fileds [6]-[10]. The idea to perform fluidization in tapered vesselswas conceived recently 16], I7l involving magnetizablesolids and extemal transverse magnetic field.

The results in this new trend in magnetically assistedfluidization have reported by [6] with a lot of primaryexperimental data and ideas and trends of application ofpotential applications. Recently, dimensionless groupsprovided by the "pressure transform" approaches [6], [8],[14] were used to develop dimensionless scaling ofpressure drop [6], [8] and bed porosity t8l, tl4l.

Magnetically controlled tapered beds offer newpossibilities of magnetically assisted beds such ascontrolled formation of the central channel, controlledcirculation ofthe solids phase, and creation ofa spoutedabed and magnetically stabilized bed in a series.

All these features are feasible for creation of fluid-solid contacting devices such as dust filters, adsorber ,heat transfer (gas-solids) devices, and gas-liquid-solidmass transfer contactors.

The combination of the non-constant vessel cross-section and controlled interparticle forces offer newapplications thus extending the applications of themagnetically assisted fluidized bed commonly created incylindrical vessels t7l-tl 31.

The present article contributes to this new trend ofmagnetically assisted fluidization and addresses scaling

Copyright @ 2009 Praise Worthy Prize S.r.l. - All rights reserved

B e d F l c l dcros34.e t l6n

3r7

Jordan Hristov

Matqial Fractio Density . Ms (M / m) ,

' i^ ' (*s r ̂ ' )

Amonia 500 -

catalyst KM-I, 613H. Topsoe

6 1 3 -

800

5100 236.34

| " o II I

\average )(p^)

257.5

357.5

)) /.)

707.5

TABLEIMATETATS USED IN Trc EXPERIMETnS

approximations have to fit directly the basic variables

€= f(Q) and any effects ofthe field applied and the

solids inventory are accounted for through theestabl i shed coefficients.

Numerical experiments done with help of Origin(version 6.0) employed experimental data publishedelsewhere [6], [7], [4].

Three successful functions were selected, with y = 5'

as dependent variable and x = Q Ets independent

variable, namely:c Exponenlial decayfunction (EDF):

€ = ! r - Arup(-qQ) Q)

Asymptotic function (AF) with a rate co = 0.5

(standard value of the function provided by Origin6.0- The shape of (3) and the asymptotes are shownin the Inset of Fig. 5:

€ = ao _boe

o Power-lav (alometiQ function (PLF):

e = aoQoo (4)

These special functions were especially chosen not onlydue to functional simplicities but also as a result of theiralmost linear behaviour in logarithmic coordinates, as itis illustrated by the example in Fig. 2(c).In logarithmic co-ordinates we have:

o hponential decayfunction (EDF):

l n (y -e)= lnArarQ (s )

with ln(y, - e\ = ln(yr - €,^) - see the asymptotic

analysis in the next point.c Asymptoticfunction (AF):

ln(a" - e) =

= lnbo + Qlnco > ln(ao - e) = lnbo + Qtn0-5 (6)

1n0.5 * -0.693

o Power-law (alometric) function (PLF):

l ne= lnao+bo lnQ Q)

The first two expressions (EDF and AF) give linearfunction of Q, while the expression (7) exactly

demonstrates the applicability of the power-law functionto the data at issue.

Magnetite 200 - 5140 477.36

(Fe,Q) ' - 315

d315 -400

o gc [9] ** - avqagc sieve dimeter used in calculations

The bed porosity to o known as annular porosity was

calculated through [6], [7]:

e - = l - v P =" Vo"d

- l -

lt+"to+DLDb.";)]

( u, \ ( r )\.tl

Where D. = Dt*2hutan(al2\

ilI. Results

ill.I. Bed Porosity Cuwes andApproximating Functions

Sample of experimental results concerning porosity

evolution with increase in the volumetric gas flow rateare shown in Fig. 2(a). Many such curyes were publishedelsewhere [6], t7l, U4l with relevant physicalexplanations. In general, the porosity curves of gas

fluidized magnetically assisted beds exhibit plateauxwith levels dependent on the field intensity applied.

Such plateaux result from the vessel geometry and bedexpansion to different extents along the vessel axis [6],[7], [4] as well as the regime performed in accordancewith the Magnetization FIRST mode [6].

The regimes exhibited by the magnetically assistedtapered beds are presented in a common manner througha phase diagram in H-Q co-ordinates t9l and

illustrated in Fig. 2(b) (more phase diagrams areavailable elsewhere [6]). The next steps of this workaddress reliable approximations of curves with typicalshapes (see Fig. 2(c)) and asymptotes approaching the

extremes in the bed porosity: the initial porosity eo and

the maximum attainable porosity e,,^ .

These functions have to fit the data points in generalavoiding the "steps" (see Fig. 2(c)) [6], [14] due tospouting in the centre of the vessel accompanied by amagnetically stabilized section the bed top. The

Copy'ight @ 2009 Praise Wortlry Prize S.r.l. 'All igh* resened

(3)

3 1 8

International Review ofChemical Engineering, Vol. I, N.1

0.5

o.5

clo . 0 5 0 _ 5 1 2 3

h Q

Figs- 2. Bed porosity cuwes and a phase diagram of: (a) Tapered bedporosity as a function of the gas flow rate. Ammonia catalysts KM-l

( 613 - 800 pn ); (b) Phase diagram in 11 - Q co-ordinates. The small

schematic pictures illustrate the bed intemal structures. Ammoniacatalysts KM-1 ( 613 - 800 pz ); (c) Bed expansion curves

dernonstrating the almost linear behaviour in logarithmic co-ordinates.

Magnetrte (3 1 5 - 400 pm). The solid line shows the linear trends,

*til,* tlre dashed one marks the data scattering. Particular equations ofPLF are available in Table III

III.2. Numerical Results

The data fitting with the exponential decay functionp)was quite successful as it is shown in Figs.3(a) and 3(b! Examples of equations derived are summarized inTlrbfe tr. The asymptote of EDF at x-+ 0(fixed bed) is

h- 4 * €0 and when at high flow rates, i.e. r -) co the

as;rmptote is y, -+ e^* . The independent variable r

denotes the independent variable Q in the asymptoticarralysis- The parity plots in Figs. 4 reveal that the EDFfedicls satisfactory a- whilst the initial porosity

li -.4 = ao is underestimated. The initial bed porosity is

nd at issue in the approximating procedure since s0 can

be calculated from (1). The asymptotic function (3)

ryroximates adequately the bed expansion curves (see

Cqyglt O 2009 Praise lltorthy Prize S.r.l. - All rights resemed

Figs. 5) with minor variations in the coefficienta" (seeTable II).

y= 0.682- 2.53exp ( -2.75Q )

H = 4 . 7 7 k A t m

Fe3O4 ( 315 - rllXt )G = 2(g, hbo = 250 mm

1.5 2.O 2.s s.0

Q ( m 3 / s ) x t o n

y = 0.684 - 2791 exp ( -2.07 Q )

H = 9 . 5 5 k A r mK M - r ( 5 0 0 - 6 1 3 )G=2f tg ,hbo=250mm

1.5 23 L5 3.0 3.5 4.0 4.5

Q ( m 3 r s l x l o - 3

Figs. 3. Direct scaling ofthe porosity evolution " = f (Q) fV

Exponential-decay function approximations

Utsrb( rtf -,'e t

1 0,r*,F

B6 -#. 's

0 4

o-? s.{

* ^,{erperiine*tuIi

0.80

^ 0.75

F o.to

E o.rt

@ 0.60

0.56

e;

9

KM-1 ( 613 - m0 lG = 2 k S

hu - 250 mm

3.0 4.0 5-0 6-0

Q ( m 3 r s ) x l o - 3

H ( u / m )

Jordan Hristov

H ( k A r m )

+ 0

+ 2-384 4-77

+ 9.55-4-- 14.32

+19,1

+23.a7

0.70

0.68

^ 0.56

J 0.6/r

3 0.82

E 0.60Tg o.ss

f, 0.56

o-54

o.52

0.50

Al

0.72

0.70

0-68

o.55

3 o..l

3 0.62E 0.60tsg 0.58

3 0.55

0-54

0-52

0.50

B)

0

6

7

6

I

1

0

B)

E \sH E *'- o-.t jE9.-.'--'--6l};; \1H'-:l'c'-i-"*'\'-ig,*.;F./#-'f--:- E )a.-.-'- Et-3-.-,-€-Hili-- F

e 1 ( 6 1 3 - e ) ; G = 2 k g ; h m = 2 $ n m

E i y H Et 'tT H L---3-^*

10 t5 20

H ( m t m l

+ 0+ + n+ 9.55+ 11.32

+ 23.A7

G'I r,tgEeg{

i O.t

a.€

t''

??,e

t* o,7

ff i4 lr l r .s l

it* tr

/6I';4"

*.6 3,' or D,!

s*" {gxtE ime{trl}

Figs. 4. Parity plot of the maximum attainable bed porosity t* with

the asymptote \(x + -) The i.rt"g"..rrr-bers near the labels

indicate the cnrrent strength f (;) *rougn tne coils ( 11 = 2387.2 x I )be clarity of the pictures. Data: M€netite : G = 2kg; Km-l : G = 2kg

319

International Review ofChemical Engineering, Vol. I, N. 4

Jordan Hristov

TABLEIISAMPLES OF EeuATroNs DEvELopED TIs.oucH

DrREcr AppRoxMATIoNSOF E)eERIMEI.ff AL DATA

Exoerimental conditions Approximating function ExperimentalData

H(kA/m)

Code

O M3-EDF-O

4.77 M3-EDF-2

9.55 M3-EDF-4

14.t2 M3-EDF-6

23.87 M3-EDF-10

Exponential decay function (EDF)

! = \ - A r e x p ( - a r x )

I - + € - ' a t t - ) o

e = 0.512 - 0.46t exp(-o.oseg)

z' =8.8265x10t , R' = 0.967

s = 0.814 - 0.48t exp (-o.szog)

,x' = 3.l88lx1o-, R' = 0.943

e = 0.7 60 - 0.44+ exp (-o.ed+q)

12 =5.6937x104, R ' =0 .903

e = 0.7 46 - 0.48* exp (-O.t eZg)

z' =2.7926x10', R'? =0.956

e -- 0.728 - 0.40+ exp(-o.tttg)

I ' =5 .2788x10 ' , R '? =0 .903

Asymptotic function (AF)

Y = a, - b"c: with rate

c. = 0.5 ao -+ e. at

.x -) co

e = 0.7 -0.149x0.54

12 = 0.00017, R2 = 0.926

e = 0.767 -0.486x0.54

, f ' =0 .00028 , R2 =O942

e = 0.754-0.446x0.54

12 = 0.00Q5, R'? = 0.903

e = 0.755 - 0.469 x0.54

X2 = 0.00026, R'] = 0.956

e = 0.73't -0.371x0.54

12 = 0.0005, R' = 0.902

. Data"rc

flrolntSexp.

0 636 t2

0.721 10

0.703 l0

0.721 13

0.753 19

Code

M3-AF-O

M3-AF-2

M3.AF-4

M3-AF-6

M3.AF-l0

Magretite

(3 1 s-400)

G:3 kg

y = 0 . 7 5 0 - 0 . 2 0 { ( o . q A

tlrnlfft

*'r'-- { H r i e , l K r nt

* a o o , * - " r o ,

0 0.5 1 1.5 2 2-5 3 3.5 4 4.5

Q ( m 3 r 6 ) x 1 0 {

y=o .TBz -0 .3s2 (o . s )a

.-r'"'1;1_ a

A ) o ( m 3 r s ) ! 1 0 {

Figs. 5. Porosity curves approximation by "aqrmptotic function " .Inset: General function shape and asymptotes

The asymptote ! ) ao a!. x -+ @ approximates

e** with errors close to those provided by EDF (seeFigs. 6). The power-law approximation) was quitesuccessful (see Figs. 7(a) and 7(b)).

A specific feature ofthis equation [18] is that fittingsuccessfully experimental points both the pre-factorapand the exponent 6rvaries in different directions,

This implies, that when decreasing curve slope meanslow value of bp, successful approximation yields

increasing ap, and vice versa.

CoWiSht @ 2009 Praise l4lorthy Prize S.r.l. - All rights resemed

0.8

o.7

0.65

0.6

0.$

0.5

c l

y=0.764-0 .206(0 .5)A

H r 2tiC kA, h

1 2 3 4 5 4

Q ( m 3 r s l x ' l o 3

I

tI

o

o.75

o.7

0.65

0.6

0.5

B )

o

o-7

0.65

0.55

0.5

If

I

H . 9 . 9 U , h

8r 0.tP

Ei.7

*.qmlF. | 2!tr - 315 I

,c* f1,.11-,

o,, o.Erdr {spdlrftdrD

Figs. 6(a)

320

International Review ofChemical Engineering, Vol. I, N.4

Jordan Hristov

KS"t t 6rt " st l

E.tt6 ./ot' t/

,/ , ./ , 'o

, ki'i.4"

fact is PLF) and magnetically assisted fluidized bedswere discussed. Numerical and developed equations aresummarized in Table III.

Eo"!-i

^ 0-t

I

* -6 -

: *

a.

F"3O4{200-3 te l ;G=2t (g

FBOa{3r5- {oO} ;G. tkg

Fa3g{ { !15-40o1;6=t tg

Klld-r {S{!- 6lt}; €-aq

Xr&. t {513-dx t } ; G-2kg0.r 0.E 0,9{ftr {S$drlnrrtrl,

(b)

Figs. 6. Parity plot of the maximum attainable porosity e,^ and the

qmptote a, provided by Eq. (3) as y(r -+ -) . Similarly to Fig. 4,

rhe integer numbers near the labels indicate the current strength

/ (/) ttnough the coils ( 11 = 238'7.2 x I ) for clarity of the pictures.

Data: Magnetite : G=2kg; Km-1 : G: 2kg

FEa

o

t-J&

0,6 l{0.8

K*tt - ' l { t15- 8d{l l i A - 2 kg

cl ; * i-, : ," ' : a

Fig. 7(b)

Fgs- 7. Direct powerlaw correlations r" = aoQb' - examples (a) , (b)

Good data fitting in specific cases comparablewith those performed by EDF

This specific behaviour is clearly demonstrated bythe variations of the alometric function parameters withincreasing field intensity in Fig. 8. Because the generalindependent variable is represented by the volumetricflow rate Q,then the exponent D, is responsible for the

material properties, vessel conditions and the magneticfietd intensity. We have to mention that interpretationsslch as: ao varies from 0.45 p to 0.75 , whilst

Dodecreases from 0.46svyn 1s0.15 with the field

inrensity, are purely formalistic and do not refer to basicfedures of the power-law function mentioned above. Inrhis context we refer to [8] where these problems withrespect to the Richardson-Zaki (RZ) equation (which in

Cryyight @ 2009 Praise l{orthy Prize S.r.l. - All rights reserved

c.t

1 0 1 6 mH { k A l m l

Fig. 8. Field intensity effect on the coefficient a, and the

exponent 6o of the powerJow function

III.3. Evaluation of the Scaling Performance of theAppr oxi mati n g Func ti ons

The performed direct correlations have goodperformances over the entire porosity curves withsatisfactory errors of approximation (see Figs. 3-8).This impression comes from their graphicalperformances and the ability to fit points either from therising sections or the plateaux of the porosity curyes.The Exponential-Decay Function (EDF) (2a) manifestsitself as the best approximation of the entire porositycurves. To some extent, similar results were developedby the approximating function (3), with less accuracy ofapproximation ofthe plateaux but keeping the tendencyin the curve shape. The direct power-law scaling (4) fitswell the rising curve sections but over-predicts the pointofthe plateau. The pre-factor and the exponents oftheseapproximations vary as the parameters of gas-solidsystem and the field intensity are varied that nopredictions of the maximum bed porosity is possibleunlike those provided by EDF and AF. Albeit the lackof generalization these commonly used functions andthe results developed drew the hend for theirdimensionless implementations. These empirical directcorrelations provide satisfactory results but manyfactors such as particle size, solid density, field intensityeffectso etc. remain implicit. The latter addresses morecomplex approaches such a dimensional analysis [8],[9], [14], a topic beyond the scope ofthe present work.

However, omitting the generalization task, EDF, AFand PLF are practical tools which might be successfullyused in practical situations with magnetically assistedtapered fluidized beds.

p

&

= o.?a

.er o"6f

a $"so€

o-63

o.50

o.itn

c-76

c.70

c s 6

o,6{'

o"6g

o.5(,

o.8!

o.7g

o t ? 3 4 5 G

e ( m * r B l x t O - 3

Fig. 7(a)

}* = 14.32 k4 ' m

\*- y*o"carQc'tg

F€5O4t31S. - 4Od ! ; $ = ? ks

321

Internationol Review ofChemical Engineering, Vol. I, N.4

Jordan Hristov

TABLEIII

PowER-LAw CORRELATIONS 6 = aoQo'

H

(At n) Magnetite(200-315) ;2kg

Material

Magnetit€(315-400);2kg

Magnetite(315-400) ;3kg

KM-I(s00-6ls);2kg

0 €o = 0.5g7Qo'*

12 =000017;R2 =0.9327

7 data points

1771.1 s" = 0.599Qo220

/' = 0.00013; R'? = 0.95t

l0 data points

9548.8 e" = 0'6l3Qo't 'e

X2 = 0.00029: R2 = 0.926

l3 data points

t4323.2 e" = 0'635Qotso

X2 = Q.00073; R'z = 0.912

l2 data points

^ - - . ^0 .09519097.6 E, = v.otta

,X'z = 0'00045,'R'? = 0'837

l9 data points

23872 e. = 0'66lQo8n

1 2 = 0 0 0 0 1 7 : R ' z = 0 . 9 3 2'

20 &ta ooints

€" = 0.527Qil'3

z2 = o.oool4; R2 -- 0.936

8 data points

e . = o.566Qo'n"

Z' = 0.00082; R' = 0.7t5

I I data points

e" = o.565Qo"t

t2 = 0.00083; R' = 0.E12

9 data points

e" = o.599Qot'6

/ ' = 0 0 0 0 8 7 ; R ' ? = 0 . 7 6 318 data points

e. = o 576Qot6'

Z'! = 0 001 12; R'? = 0.806

20 data points

e "

= o.597Qut$

Z' = 0.0008: R' = 0.835

e . = 0.52'tQo"1

/'? = 0.00014; R'? = 0.936

8 data points

s" = 0.528Q020

12 =0.00026:R2 =0.946

9 data points

e" = 0.533Qo'2@

Z' = 0.00053; R'? = 0.898

l0 data points

e, = o.531Qo:ot

Z' = 0.00035; R'z = 0.940

13 data points

e" = o 545Qo'223

Z2 = o.00078; R' = 0.923

l9 data points

t , = o 5 6 l Q o t "

,z' = 0.0005?; R'? = 0.888

e "

= o.4|lQo2n8

Z'z = 0.00018; R'z = 0.915

l0 data points

e" = o.529Qo''2t

X2 = 0.00048; R' = 0.822

13 data points

s" = o.524Qo-2q

Z' = 0.00057: R' = 0.E51

15 data points

e. = 0.557Qu72

,ft = 0.0005; R'? = 0.E31

l8 data points

€, = o.6ooQo t'o

/'? = 0.00051; R'? = 0.795

24 data points

€. = o-58190.r'o

,2 = 0.00055; R'? = 0.814

25 data pointsl8 data points 18 data points

IV. Conclusion

Numerical experiments and dimensional performed inthis work revealed that approximating functions providedby common computer codes work satisfactory in fittingbed expansion profiles of gas-fluidized magneticallyassisted tapered beds with magnetization FIRST mode.The results briefly could be outlined in several generalpoints, among them:o All functions tested with direct scaling (EDF, PLF

and AF) work successfully and could be used in dataprocessing of particular porosity evolution curveswith magnetization FIRST mode.

. These approximations allow easily predicting themaximum bed expansion through simple asymptotes.The errors of approximations performed reveal less

than l0% in all trails and relationships established.

Acknowledgments

This study was partially supported by grant No10476-2008 of UCTM and the help is highlyacknowledeed.

References

tll K. B. Mathur, N. Epstein, Spouted Ded. (Academic Press, NewYork,1974).

121 D, Geldart, Types of Gas Fluidization, Powder Technologt, 7(r973) 28s-292.

t3] V. R. Deiva, J. Chaouki, D. Klvana, Fluidization of cryogels in aconical column, Powder Technologt, 89 (1996) 179-186

Copy'ight @ 2009 Praise lilorthy Prize S.r.l. - All rights resemed

t4l M. S. Bacelos, M. L. Passos, J. T. Freire, Effect of interparticleforces on the conical spouted bed behaviour of wet particles withsize distribution, Powder Technologlt, 171 (2007) 114-126.

t5] M. S. Bacelos, J. T Freire, Flow regimes in wet conical spoutedbeds using glass bead Mixtures, Particuology, 6 (2008) '72-80.

t61 J. Hristov, Magnetically Assisted Gas-Solid Fluidization in aTapered Vessel: First report with observations and dimensionalanalysis. The Canadian Journal of Chemical Engineering, 86 (3)(2008) 470-492.

l7l J. Hristov, Magnetically assisted gas-solid fluidization in ataperedvessel:Part l.Magnetization-LAST mode, Particuologt, 7(l\(2009\26-34.

t81 J. Hristov, Magnetic field assisted fluidization-A unifiedapproach. Part 5. A Hydrodynamic Treatise on liquid-solidFluidized Bed, Reviews in Chemical Engineering, 22 (5) (2006)195-377.

t91 J. Hristov, Magnetic field assisted fluidizatiorpA unifiedapproach. Part 1. Fundamentals and relevant hydrodynamics,Reviews in Chemical Engineering, I 8 (4-5) (2002) 295-509.

[0] J. Hristov, Magretic field assisted fluidization-A unifiedapproach. Part 2. Solids batch gas-fluidized beds: Versions andRheology. Reviews in Chemical Engineering, I 9 (l) (2003) I - I 32.

[l] J. Hristov, Magnetic field assisted fluidization-A unifiedapproach. Part 6.Topics of Gas-Liquid-solid Fluidized bedHydrodynamics, Reviews in Chemical Engineering, 22 (6) (2007)195-377.

[2] J. Hristov, Magretic field assisted fluidization-A unifiedapproach. Part 3. Heat fransfer- a critical re-evaluation of theresults, Reviens in Chemical Engineering, 19 (3)(2003) 229-355.

[3] J. Hristov, Magnetic field assisted fluidization-A unifiedapproach. Part 7. Mass Transfer: Chemical reactors, basic studiesand practical implementations thereof, Reviews in ChemicalEngineering.25 (2009)- in press.

[4] J. Hristov, Magnetically Assisted Gas-Fluidized Tapered Beds.Part 2. Bed xpansion Scaling Studies with a Dimensionalanaly sis, P ar tic uo I o gSt, 7 (3) (2009) I 83- I 92.

[15] A. Kmiec, Expansion of Solid-liquid Spouted Beds, The ChemicalEngineering Jownal 1 0 (197 5) 219-223.

322

International Review ofChemical Engineering, Vol. I, N.4

l l6l

t17l

tl 8l

A. Kmiec, Expansion of Solid-gas Spouted Beds, The ChemicalEngineering Journal, I 3 (1977) 143-147.A. Kmiec, Hydrodynamics of Flows and Hpat Transfer in SpoutedBeds, The Chemical Engineering Jounal, 19 (1980) I 89-200.J. F. White, S. J. Gould, (1965) Interpretation of the coefficientsin the alometric equation, The American Naturalist,99 (No104)(r96s) s-18.

Authorst information

Jordan Hristov is associate professor ofChemical Engineering at the University ofChemical Technology and Metallurgy, Sofi4Bulgaria. He was graduated in 1979 as ElectricalEngineer (MS equivalent) at the TechnicalUniversity, Sofi4 Bulgaria. His PhD thesis onthe magnetically assisted fluidization wasawarded by the University of Chemical

Technology and Metallurgy in 1995. A./Prof Hristov's researchinterests cover the areas of patticulate solids mechanics, fluidisation,heat and mass transfer with special emphasis on scaling andapproximate solution. The main branch of his research is devoted tomagnetic field effects of fluidisation. Additionally, specific heattransfer topics are at issue, especially to thermal effects in accidents(fire). Relevant information is available at htto:iAristov.com/jordan.

Jordan Hristov

Copyright @ 2009 Praise Worthy Prize S.r.l. - All rights resemed International Review ofChemical Engineering, Vol. I, N.4