Dynamical adsorption in fixed bed

Ckdcnl

BnpM+Nlp

Scimce,

1977. Vol 32, pp 303-3O!l

Persamoa

DYNAMICAL PATRICK Laboratoue d’Adso@on

Prcas

Pnnted m Orcal Bntam

ADSORPTION OZIL

and LUCIEN

IN FIXED BONNETAIN

et R&action de Gaz sur Sohdes. E N S d’Electroc~e Samt Martm d’Heres, France (Recerued 15 Apnl

1976, accepted

BED

3 August

et d’ElectrometaU~e,

38401,

1976)

Abutract-The behavior of an adsorbent fixed-bed had been mvest@ed for lrreverslble eqmhbrmm when the adsorption rate 18 lmuted by mtemal dtislon Breakthrough curves equations has been estabbshed for spherical or cubic sorbent part&es constdemg axml &sperelon It has been found out a hnear law between the extrapolated breakthrough tune and the mverse of adsorbate Bowrate That allows the evahmtion of mtemal dtfEuslonand aual

tipcrslon coefficltnts The bre&tJuoughcurves of propylene on 13X molecular sieves are interpretedaccuratelywhen rt1sadm&tedthat the adsorptionrate ISInWed by the action of combmed mternal and external d8uslons and when the axml musion IS taken mto account When the ax4 dfiuslon IS neglected, the breakthrough curves can be also computed but the coefficrents of mass transfer chosen for a good fittmg do not agree w1t.h those predicted by classIcal formula

pattern assumption we denve The dynanucal behavior of an adsorbent fixed-bed 1s usually stu&ed from the breakthrough curve wbch describes the evolutron of the adsorbate concentration m the flmd phase at the outlet of the bed and pernuts the evalmtion of transfer factors For a gven adsorbent-adsorbate system, the shape of the breakthrough curve depends on the type of the adsorption eqmhbrmm, on the physlcal processes of transfer and on the phenomena related to the flow of the flmd In this paper we study the case of an ureverslble adsorption eqmhbrmm when the transfer rate 1shnuted by one of the followmg processes Internal Muslon m *cles External dtiuslon m the fled phase, Combmed external and mternal dtiuslons For these cases we take mto account the axml &spersion consldermg the following mass transfer equation We assume the followmg hypotheses (a) the flmd 1s mcompressible and the adsorption IS isothermal, (b) the sorption wave is in a quasi-steady state (“constant pattern”), (c) radml dlsperslon 1s neghgible, and (d) the coefficient of axmJ Qsperslon D IS propotional to the lmear fhud velocity (D = h,u = hi/&), tis hypothesis wdJ be JustUied later on MASS TRMBFER

%&?&_~,~=~ --I II a 87 Co

ODD

ax

y-z-x+y=o 1INTEIINALDIpRwoN Theoretwal study We are gomg to inveshgate successively the cases of spherical or cubic parkles 1 Case of spherical partrcles The rate of mtemal dtiuslon 1s expressed by

aqi_o,a a7

r2 ar

(r2&ar )

(4)

with the boundary condlfions

dr,O)=O

$yO,7)‘0 qdd,/2,7)=qX

where Q IS the adsorbate concentrahon 111sohd phase at radms r, 4 IS the parkle dmmeter and D,, LS the coefficient of lnternaJ Muslon The average concentration m the crystal is

(1) By mtegration, takmg mto account the boundary con&tions (T -**, C-G,, aCldr +O) vvlth the constant 3-E

(2)

where 4% IS the value of q m eqmhbrmm nnth Co. After mtroduction of the dtstnbutmn parameter D (= q$ps/COc) and of the reduced vanables x = C/C, and y = q/q%, the equation (2) can be wntten

EQUATION

For a bmary mutture, characterized by Its flowrate F and its adsorbate concentration Co. entermg an adsorbent fixed-bed (length H, cross-sectional area S, porosity e, bulk density pb), the dtierenti mass balance IS

CEs-Vd32. No

qx

3 -q - (4r2)3

I

l ,n o r2q, dr

Takmg mto account the boundary cond&ons 303

(6) (5) the

PA~CK

304

Ozn.

and

resolution of eqns (4) and (6) IS known[l] and leads to

BONNET~IN

bK!IEN

with boundary conditions

qicx Y,Z,O)=O

(7)

q&a,

Y, Z ~1= 4(X.

fa, Z, 7) = q,(X Y, fa, T) = qi

With

4n2T2DE

8.=

d2 P

The average concentration m the crystal is

where 7. IS the breakthrough ume The mtroducfion of the expresslon of y mto the mass balance (3) mves us after calculations[2] 1 - x = exp

[-a,(7 - TV)]+ -q, m2

n2

(13)

The solution of eqns (13) and (11) Hrlth boundary condltlons (12) can be wntten[l]

’

(

Oq, dXdYdZ

1-k

1

1-p l-43. (7 - TON-exp [-ad7 - 7011)

(9) with

wlth

al=m

u

(‘0)

am, = +

[(m + l12)2+ (n + l/2)’ + (p + l/WI

Bmnp= (m + 1j2)2(n + 1/2)2(p + 112)2 Fve 1 penrmts to apprecmte the strong mlluence of ax& dlspersum on the breakthrough curves A small mcrease of the axtal dlsperslon coefficient leads to an important broademng of the adsorption front 2 Case of cubic parhcles The dlffuslon equation for cubic parttcles of side 2a IS

(15)

(‘6)

The mtroductlon of eqn (14) mto the mass balance allows us to obtam the followmg equation of the breakthrough curves 1 -x = exp [-a,(7 - TV)]

(1’)

I

bxp

-

[-amp(7 - ~011- exp [-ad7

- &I)

(17)

The mfluence of axml dlsperslon on the breakthrough curves 1s nearly the same as for spherical pa.r&les 3 Lmeansatuw The compkxlty of eqns (7) and (17) makes them hard to be exploited If approxunate values of coefficients of Internal drffuslon and axial &spersmn are unknown This fact leads us to look for a convemcnt hneansatlon allowmg the evaluation of parameters hD and

x

05

DP The mass balance between the beqnmng and the end of the breakthrough 1s

i-

Hence the evaluation of the Integral mves the value of T,, For spherical parWles we denve

10

5 T-70,

Fhg 1

m-l

Inlluenceof the axial &sperslon on the curves of internal

ddTuslon m spherical particles (0, = 3 75 x lo-” cm’ s-‘, F = 1490 cm’ nm-‘. C, = 10%) 1, Axial &sperslon neglected, 2, hD=06cm,3,hD=11cm,4,hD=21cm

and for cubic particles

Dynanucaladsorpuonm tlxedbed Wlth

(zAnP= (m + l/2)2 + (?I+ l/2)2 + (p + l/2)*

(21)

&he-e m a former paper 131we have used the lmear tivmg force approxunation of rate provided by Glueckauf and Coates[4]

305

adnutted that the value of r0 1s the one defined by the mt§ion of the tangent at the &non pomt with the tune axis. Furthermore tis choice allows to ehmmate the mfluence of other processes occurrmg at begmmng of breakthrough such as external dtiston which renders the begmnmg of curves slumsh As shown on Fig 2, the “break&o& trme” vaes hnearly vvlth the inverse of adsorbate flowrate accordmg to the proposed theory Hence we can find hD = 0 8 cm and, for the Merent patterns, the followmg values for internal dtiuslon coefficients (Table 2)

that had led us to

Table 2 D,, (cm’ s-l)

(23) Hence for the three consldered cases we can wrote formally

when defimng for cubic pmcles

an eqmvalent &ameter by

The only Merence accordmg to these patterns IS the value of K (Table 1) The graph of the function r0 = KlIFCJ 1s a &a&t hne Its intercept on abscissa axis allows the calculation of the coefficient 0, of mternal dlffuslon and Its slope sves the coefficient of axial dlsperslon (D = hDu) Expenmental study We have tested our theoretical results for adsorption of propylene of 13X pelletied molecular sieve (16 mm dm 5 mm length) m a range of volunuc concentrations 2-IO!% and flowrates from 1 to 41 mn-‘, m a temperature regulated column at 30°C under atmospherrc pressure (20 cm length, 2 8 cm dla ) These cylmdr~cal type pellets of molecular sieve are formed with quasi cubic crystals of zeohte agglomerated by an morgamc non adsorbent binder The mtemal Muslon occurs m these crystals for which the size &stibution law 1s a log normal one[5,6] and the average side 1s about 0 8pm The determmation of a true breakthrough tune 1s expenmentally unposslble because it depends on the detection senslbdity of the apparatus Therefore we have

Sphericalpar&les lmear approxunatin exact solution

3 75 x loI= 2 92 x 10-12

Cubw partwles

These values of internal dtiuslon coefficients are m agreement unth those obtamed by other authers for smular adsorbent-adsorbate systemsfl-lo] The assumption of propotionahty between the axial dlsperslon coefficient and the fled hnear velocity IS These Justdied by several correlations Pe = f(Re) correlations [ll-131 show an horrzontal plateau for Reynolds number between 2 and 10, which 1s the case of our expenmental work The value that we have found, hD = 0 8 cm, 1s srrmlar to the one proposed by Convers [9] for adsorption of heptane on zeohte with Reynolds numbers between 5 and 16 However it 1s four to eet tunes greater than those predicted by other authors{12,14] Yet it IS necessary to notice that those last stuches are concernmg beds of spherical particles Such packmgs are obviously much more regular than the one used durmg our experunents Then we have compared our expernnental breakthrough curves with the theorekal curves of dtiuslon mto spherical and cubic particles (Fig 3) It 1s clear that for mtml adsorbate concentrations which are relatively h@b (about 10%) the agreement between theory and experrment 1s very satisfactory Nevertheless, the dls-

Table 1 Value of K

Sphericalparucles Lmear approxuntion Exact solution Cubic pmcles

A=00167 @$,=&=00’67 I/%.

m-lcm3

Hg. 2 Test of Imeanaataon

PATRICK OZIL and LUCIEN B~NNETAIN

derived by Drew, Spooner and Douglas[f6], when axial dlsperslon 1s bemg neglected, I e ho tends to 0

equation

I

x

Expenmental study We have apphed these theoretical results to the propylene-13X molecular sieve system However before undertakmg thus study, it 1s mterestmg to examme the theory neglectmg the mtluence of ax& dispersion 1 Interpretation neglectmg amal drspersron The breakthrough curves equation provided by Drew, Spooner and Douglas can be wntten

05

& [log x + 11= 5

0

r. mn

3 Representation of breakthrough curves d&won and axml tiperslon -, expementai curve of mtemal dtffuston m cub= parttcles Rg

by Internal curve, ---, (0, = 2 92 x

lo-‘* cm’s-l hD=08cm), - -, curve of mternal dlffuslon m spherical p&cles (0, = 3 75 X lo-“cmz s-‘, ho =08cm) 1, F = 2390cm3mn-‘, C,= 10 1%. 2, F=2170cm’mn-‘, C,= 963%. 3. F=2230cm3mn-‘. C,=7%%, 4. F=236Qcm3mn-‘, G-533%

agreement ISmcreasmg when the adsorbate concentration decreases We shall show that these Merences are due to the action of external ddfuslon wluch lnAuences mamly the breakthrough beglnmng 2..AL-ON

ik

D

(26)

The resolution of system (3), (26) with the boundary condlhons

tlOLl

Independently of the breakthrough curves, kf can be calculated from the ddfus~v~ty D, of propylene m hydrogen, its tamer gas The formula of Huschfelder, Bird and Spotz[17] moddied by W&e and Lee{181 ytelds a value for the ddTus1wty (D, = 0 488 cm/s-‘) m agreement wrth the work of Welssman[19] Then the mod&d formula of Thoenes[20] b=O38u

dgu --Oso 6( 1 - e)D, I

leads to values of the transfer coefficient varymg m our expenmental domame between 5 and 6 cm s-’ Hence the apparent fled-phase mass transfer coefficient deduced from the breakthrough curves 1s about five tuues lower than the one forecast by theory In other words the theoretical coefficient leads to adsorption waves too much steep (Fig 5)

to leads to 0+

(27) 7 *

with

-lO-

3 (27 1)

(27 2) Thts solution IS formally identical vvlth the one suggested by Acxwos when the saturation of the adsorbent bed ISnot reached [15] It pernuts to find out the

(28)

Thus form allows a smgle hnemsation for all the breakthrough curves If we assume that 4 is constant The graph l/CJlog x + 11 = f[~ - rat] fits a strzught hne up to x = 0 6 (Fe 4) From its slope we deduce kf = 1 1 cm/s-’ As shown m Fu 5 for one experunent, the use of tlus apparent transfer coefficient mves a satisfactory representation of the breakthrough begmmng (same curvature) Nevertheless, It IS necessary to compare Gus transfer coefficient mth the one commg from theoretical calcula-

Theoretical study For an lrreverslble adsorption equihbrmm the transfer equation IS

ay 45!ex -=

17 - T,tl

9 -

-20

_

P h .

387 499 533

1710 2683 2360

n

796

2230

-3c-

1

I

-5

0 T-T*,.

inn

E+g4 Test of lmcar~sat~on

Dynanucal adsorptin

1

307

m 6xed bed study

The0ret1cal

In so far as external dfiuslon controls the breakthmugh the rate equation is the eqn (26) The resolutmn of equations system (3) (26), after mtroductmn of the transition pomt coordmates T=and xn leads to x =

xT

exp A ,( r - T.,)

(30)

Beyond tis transition pomt, internal rllffuslon predormnates and the hnear dnvmg-force approxunation of transfer rate can be wWten[Zl]

2 = ba;(l-

1

0 50

56 T.

60

I

(31)

where a L IS the effective mass-transfer area between flurd and crystals, R, 1s the mass transfer coefkxnt m sohd phase connected to the mtemal dSusion coefficient Q by

65

k.a:=y

mn

of break&rough curve by external expemnental ddfus~on (F = 2680 cm’ mn-‘, C. = 4 89%) -, curve, tbeoretrcrtl curves, ---, k, = 1 1 cm s-‘, - -, rC,= 6 53 cm s-l, wthout axial duqerslon, - - - - - - - -, k, = 6 53 cm s-‘, h, = 0 8 cm, with axml hsperslon PM

y)

(32)

5 Representation

The solution of eqns (3) and (30) 1s 1 -_x

dl-x+2--~x~exp

[-al(T

-T

St

al-a2

Then it 1s necessary to look for an explanation of the slumsh begmmng of the front wbch reveals a lower efficacIty of the adsorbent bed Therefore we have taken into account tbe axliil dispersion 2 hterpretatron rncludmg axrai daspersaon. #en takmg for kf the values obtamed by calculation and for ho the value 0 8 cm found, the theoretical curves plotted from eqn (26) gve a satisfactory representation of the slope of expenmental adsorption fronts at the breakthrough begmmng (Fu 5) So the slugg&mess of breakthrough curves at begmmng can be explamed by the action of axml dispersion which mduces a decrease of the column efficaclty As it has been seen, tius fact causes a dunmution of the mass transfer coefficient m fled phase For adsorption of propylene of 13X molecular sieve, the breakthrough begmmng seems to be controlled by the external ddfuslon and the breakthrough end by the mtemal Muslon as shown previously This fact leads us to study the action of combmed mtemal and external cGfEuslonswith ax& dlsperston 3.

c-

iNTERNALAND-AL DIFFUSIONS WlTB AXIAL DISPEUSION

When external dauslon and mtcmal dtiuslon control successively the adsorption, external ddfuslon predonunates at the breakthrough begmnmg as long as the concentration gradient between flmd phase and interface 1s large enough Then, when approachmg saturation, the mternal Musmn becomes prddommant unti the dent between mterface and sold phase reaches zero At the transltmn pomt between these two mechamsms statmg the equahty of both rates can be solved

-

(1- %)a1- AIXTexp

[-a2(T

_

)]

),

T

IL

al-a2

(33)

wth

a, = kpa;

(33 1)

a2=&

(33 2)

At the transihon pomt, we obtam from the equahty of dtiuslon rates, usmg eqns (3) and (30)

%n

xT=s!!s+ !$:+1 D

(

>

ba;

(34)

In short, the breakthrough curves are represented by eqn (30) untd the transition pomt and by the eqn (33) beyond Expenmentai study Before testmg these equahons for the system propylene-13X molecular sieve, we have tied the mterpretation of our expenmental breakWough curves when axial dlsperslon 1s neglected neglectmg axual drsprrslon The 1 Intetpretatum breakthrough curves equations are denved accordmg to the above exposed method but neglectmg axml bpersmn(211 The line&rfsa~on of our expenmental results mth the equat&m of Glue&auf and Coatesf41 for duration greater than the stoicluometrrc tune yields a value of &a ;

PA-lRlCKOZlL. alId LUCIBNBONNBTAIN

308

equal to 0 011 s-’ Assuming thus value of $a: we have plotted a theoretical curves for our expemnents usmg successively the theoretical and the apparent value of 4 From the Fs 6 we can see that the agreement is good when usmg the value of b obtamed from the breakthrough curves, and the calculated fllud phase transfer coefficients lead to adsorption fronts which are too steep That conforts Section 2 1 of our study and the fact that a satisfactory formal representation can be gven neglectmg the mfluence of axml dlsperslon rnclu&ng axtal drsperslon Keepmg 2 Interpretatwn the theoretical values of k, and takmg for &a; and ho the values deduced from former Section 14 &,a: = 0 0225 s-’ and hD = 0 8 cm, the theoretical curves correspondmg to the eqns (30) and (33) gve a good fittmg of our experunental fronts (Fe 7) Hence the decrease of the column efficaclty, which 1s revealed by the slwshuess of breakthrough curves, LS caused by the axml phenomena Its tiuence is appreclable all along the breakthrough Furthermore when takmg mto account the axml lsperslon the theoretically forecast coefficients can be experunentally found CONCLWIONS The ftrst part of thus study allows us, m the case of an lrreverslble adsorption hmlted by mternal dtiuslon m spherical or cubic partrcles, to estabhsh the breakthrough equattons and propose convenient formula for the determmation of the coefficients of mtemal ddfuslon and axml &spersron The second part allows us to explam the sorption kmettc of propylene on 13X molecular sieve by combmed mtemaI and external dtiuslons It IS clear that classic theones neglectmg the axial dlsperslon can lead to a good flttmg of the expenmental adsorption fronts but permits

Ftg 7 Represent&on of breakthrough curves by combmed mternal and external drtfustons w&b axml dtspersron -, Expenmental curve, - - -, tbeoreucsl curve 1, F = 2390 cm3 mn-‘, C,=lOl%, 2, F=2170cm’mn-‘. C,=963%. 3, F= 2230 cm3mn-‘. C. = 7 %%,4. F = 2360 cm’ mu-‘, C,, = 5 33% only the deternunatton of apparent thud phase transfer coefficients The rntroductron of axial dtsperston effect IS then capttal rf one wants to use the theoretically computed

transfer coefficients NOTATION

a

4

c

es 4 4 D Df D,

D F

H hr, kf k, Pe 4 qi q5 Frg 6 Representauon of breaktbrougb curves by combmed mternal and external d#us~ons w&out axial dlsperslon -, Expenmental curve, - - -, tbeereucsl curve for apparent t, - -, tbeoreucal curve for tbecreucaf t I, F = 2390 cm3 mn-‘. C, = 10 10%. 2. F = 2170 cm’mn-‘, C~=963%,3,F=2230cmSmn-‘. Co=7%%,4,F=2360cm’mn-‘,C,=533%

half side of parWe supposed to be cubic, cm area between Amd and pellets per umt volume of bed, cm2 cm-” effective mass-transfer area between fled and parkles per umt volume of bed, cm’ cm-” adsorbate concentration m fled phase nut& adsorbate concentration m fhud phase eqmvalent dmmeter of pellets, cm

a, effective mass-transfer

Re S I( x

equrvaient drameter of partrcles, cm axrrd dtsperston coefficient, cm* s-’ drffustvrty of the absorbatc m the carrter gas, ctn* S-’

internal dlffuslon coefficient 111crystals, cm’s_’ &stnbution parameter (= 4th /Cd) volumrc flowrate of the fled phase, cm3 s-’ length of the adsorbent bed, cm D/u, cm mass-transfer coefficient of the fhud phase, cm s-’ mass-transfer coefficient of the sohd phase, cm s-l PecIet number (= dgu/D) average adsorbate concentration m sohd phase over

a cross section of the bed local adsorbate concentration m a partrcle adsorbate concentration tn soltd phase m equdibrmm urlth C, Reynolds number (= d#u/v) cross sectmn area of column, cm* hnear velocity of the fimd phase m the bed (= F/ES), cm s-’ ratio of fled phase concentration to miet conccntration (= C/C,)

Dynauucaladsorpt~oon III fixed bed y

ratio of sold phase concentration to concentration at saturation (= q/q$

Greek symbols Pb bulk density of the bed, g crnm3 l

r 70

7‘1 Y

void fraction m the bed

tie, s breakthrough tune, s

stolchlometnc tune (= qXp,SflFc,), s kmemahc wscoslty of the fled phase, cm’s_’ -cEs

Cl1 Crank I , The hfathematrcsof h&slon

Oxford Uruverslty Press, Oxford 1956 [2] Ozd P , Th&e 3&mecycle Grenoble 1974 [31 OZIIP , Gmoux I L and BonaetamL , CR Acad Scl Pans 1972274 752 143Glue&auf E and Coates J I, J Chem Sot 1947 1315 153Loughhn K F , Derrah R I and Ruthven D M , Can J Chem Engng 197149 66 [6] Ozd P , Unpubhshedworks

309

[7] LaurentA and BonnetamL , CR Acad SCI Pans 1972275 [8] :uthvenD M,LoughhnK F andDerrahR 1,Adu Chem Senes 1973121 330 [9] Convers A, Tbtse Doctcur-In&ueur Grenoble 1964 1101 _. New York - _ Breck D W , Z&tie Molecular S~cuesWdev. 1973 1111GunnD J andEnglandR,Chem EngngScr 1970261413 1121Gunn D J , Tmns Instn Chem Engrs 197149 109 1131De Llgny C L , Chem Engng SCJ 197025 1177 [14] Levensplel 0 , Chemrcal Reachon Etzgmeenng Wiley. New York 1%2 [15] Acnvos A , cited m Perry’s Chemical Engmeers’ Handbook, 5th Edn McGraw W. New York 1973 [16] Drew T B , Spooner F M and Douglas J , cited by Klotz I M , Chem Revs 194639 241 [17l Hnschfelder J 0, Bud R B and Spotz E L , Tmns Am Sot Mech &QTS 194971 921 [183 Wdke C R and-b C Y .Ind Engng Chem 1955471253 1193Welssman S , I Chem Phys 196440 3397 [20] Thoenes D , Thesis Delft 1957 1211Vermeulen T , Adu Chem Engng 195811 147