CATALYTIC REACTIONS IN TRANSPORT-LINE REACTORS

PHIL&’ VARGHESE and ARVIND VARMAt Department of Chemleal Engmeermg, Umverstty of Notre Dame, Notre Dame, IN 46556, U S A

(Recerved 18 Apni 1?77. accepted 26 May 1978)

Abstract-The case of an Yreverslbk fust order cataly& reactum IS cons&red m a transport-tme reactor, wa a model wluch assumes plug flow of both the sold and gas phases, to mvesugate reactor performance as a function of malor opera- vanables Under eertam fled mechaatcai assumptions, a detaded study IS made of the effect of catalyst partacle sue on reactor convefston

tINTRomJcTloN

A transport-hne reactor IS one III which a reactant sohd or catalyst IS physIcally carrted (transported) through the reaction zone by a cocurrently upward flowmg reactant gas The gas velocity m such cases IS much wer than that reqmred for m~mrn~m flmdwtion, and the reactor bed IS then sad to be m a region of ddute phase transport [ l] The transport-he reactor has been gamfully employed m sltuat~ons of raprd catalyst deac- tivation, such as catalytic crackmg 111 the petroleum mdustry[2,3], m wlucb cases the reactor offers the attractive features of contmuous addition of fresh cata- lyst at the feed end, and sunultaneous withdraw1 of the deactivated catalyst, for subsequent regeneration, at the reactor exit

lyst de&mation, and Investigate the effect of pticle sue on reactor performance under hopefully realistic condltlons of reactor ooeraoon

Recent analyses by Pratt[4, S] have employed an ad- rmttedly sunphfied model of the transport-hne reactor to analytxcally predict reactor performance for ureverslble first order catalyttc reactions In hke fashion. we obtam an analvtlc solution for reactor conversion wth no cata-

The tirst assumption is adrmttedly crude, particularly for the sohd phase, but It IS hoped that the model predicts correct trends with respect to udiuences of the system parameters Under these sunpbfymg assump- tions, a mass balance m the gas phase yields

dC lM-&=R, 2 E (0, L) (1)

c= co, r=O (2)

where C 1s the reactant concentration UI the gas phase, and the other varmbles are defined m the Notation Smularly, for the reactant concentration, C, m the cata- lyst p&cle

ac,=o r-0 ar ‘-’

along with the nuti and boundary condltlons

C, = C(z), r = t-0, c, =O,z=O

We consider the case of an ureverstble ilrst order catalytic reactton occurrtng m an isothermal transport- hne reactor The reactor model employed herem IS the same as that of Pratt [4], wbxh mvolved the followmg

(4)

R, the volumetic reaction rate IS pven by

assumptions (a) The gas and solid phases are m cocurrent upward

ideal plug flow, thus there IS no backnuxmg of either the R = -(4~7r,,~N)D +

I ..=rQ’ (9

gas or the sold phases, although the gas and sohd travel with Merent velocities

where

(b) The gas phase IS consldered to be m steady state, while the solid catalyst phase IS m transient state

N = 3(1 - E)/(497&7 (5a)

(c) The catalyst particles, assumed spherical, are modeled yta the d&muon-reactton equauons IS the number of partxles per umt reactor volume

(d) There IS no external film resistance to mass trans- Defimng dunenslonless varmbles

fer, I e the reactant concentration on the catalyst surface IS the same as m the bulk gas around the catalyst f = CICO. fs = c*ico, Y = z/L, n = rlro.

pamcle 4’ = kro21D, A = uor,22/L.D, 9 = 3( I- l )L.LIUHO’

tTo whom correspondence should be addressed (6)

337

338 P V~GHESE and A VARMA

Equations (l)-(4) may be cast m dnnenslonless form

df __& _dL- ax ’ I Y E (0.1)

x-1 (7)

f-1, y=o (8)

Aaf.=l a 2 afs ay Tax xax ( ) - 4’f*, y E (0, 11, x E (091)

(91

af= ~=o’x=o, f.=f(yhx=l, f.=O,y=O

(10)

Usmg Laplace Transforms[6], eqns (7x10) may be solved to yield the transform of the gas phase reactant concentration

1 BP(S) =--- s w(s) (11)

where

p(s)=fi-tanh&fi*=#+As

q(s) = s tanh fi + S(B - tanh p) (12)

and s IS the transform vanable To Invert F(s), all the roots of q(s) have to be found

Rearrangmg q(s) = 0 as

sfl= 19 + ((c- @)/A}] tanh p (13)

and analyzmg the left and rrght hand sides of (13), It IS readdy estabhshed that there exists one (and only one) real value of 6 = & satisfymg (13), lymg m the range

&, E (0, /3*), 8’ = (@+ A9)“2 (14)

Then from (12), so = (&‘- 4’)/A In addltlon, q(s) also has an mfimte number of roots s,

resulbng from purely unagmary 0, B = ly, which satisfy

and

sn = -(+’ + m’)lA (16)

y,, can be shown to he m the range no < y* < (2n + l)?r/2, which ads III theu numemcal search

Using Heavlslde Expansion Theorem, reactant con- version in the gas phase, X = 1 -f, as a function of axml posItion y, 1s grven by

*

X(Y) = WBO - tanh &)[exp (soy) - ll/(saho)

+ 2 A'WY. - tan m)ll - exp H4’+ ym2)ylA31 n-f kW2 + Yn',

(17)

where

ho = (%4/Z&) + tanh PO + (so - 9)A/(2& cosh2 BO)

h, = -_(%4/2y,) + tiu~ ‘yn + G#J’+ mm*+ 5@AM2ya cos’ 7”)

(18)

The above development assumes that (1 - C/G) represents the reactant conversion, an assumption that has been shown to be sufficiently accurate except m the mlet regon of the reactor[S]

At tlus point a comparrson with the analysis of Pratt [4] IS useful The dimensionless groups M and P of Pratt141 are equivalent to l/A and A9/3, respectively As noted there, l/A IS a measure of the number of dtiusion tune constants (rO*/D) that the particle spends in the reactor, whde A913 IS proportional to the ratlo of mass of cata- lyst to mass of gas cuculated m the reactor

However, it may be noted that values of the Tbele modulus 4’ of 1, 10 and higher as employed m the computations m Ref 143 cannot be considered very real- ~shc, even under con&tlons of extremely fast reaction rates, gven the rather small particle sizes (dmmeters -50~~) that would normally be used m a transport-hne reactor Recogmtlon of ths fact also slgmficantiy alters the range of values the parameter M(l/A) may assume

Figures l-3 reproduce m part the computations per- formed earlrer[4] using somewhat more realistic values of 4’ and l/A AU qualitatively obvious features are borne out by these calculations

3.F.FFEcToFPhRTIcL&9zE

Consldenng the expression (17) denved for reactor conversion, tt may easily be seen to etiblt a complex functional dependence upon parttcle radius ro, interstitial gas velocity II, solid nse velocity uo, reactor length L, void fraction E, and the dlffuslon and reaction parameters D and k, respectively

Further, It may be seen that there must exist an interdependence between the gas velocity, II and solid velocity, u0 that could, at least m part, be represented as a function of particle radius Thus then represents an added dependence on particle radius which, taken m conJunction Hrlth the more obvious effect of diffusion lmutatlon, permits speculation about the existence, or otherwise, of optimal particle sizes

Put bnefly, an optunum would be the radius of the catalyst particle at which, for gven mass flow rates of gaseous reactant and catalyst, and grven reaction and dlffuslon parameters, the conversion IS maxumzed

3 1 Qualltatwe analysts If one unposes the condition that any vanatlon m

particle size be accomphshed m such a way that the mass flow rates of gas and catalyst are kept constant, and d one accepts the broad statement of fluid mechamcal fact that sohds nse velocity must decrease with Increasing particle radius (d gas velocity IS constant), then m fact the following effects are quahtatively feasible and may

Catidytlc reactions In transport-lme reactors 339

FIN 1 Reactor converslon as a function of the parameter M (dunenslonless catalyst residence time) for varrous values of P (catalyst/gas cuculatlon ratio). & = lo-’

80 -

60 -

40-

ZO-

O- IO4 10s IO6 IO7 108

-M

FQ 2 Reactor converston as a ftmctlon of the parameter Ad (dlmenslonless catalyst residence tune) for various values of P (catalyst/gas cuculatlon ratlo), s$* = lo+

-P

Fii 3 Reactor converslon as a function of the parameter P (catalyst/gas clrculatlon ratlo) for varfous values of A4 (dImensIontess catalyst residence ttme) &* = low5

340 P VARGHESE

make then contnbutlons felt when parucle radius m- creases

(a) Decrease in conversion. because of mcreasmg Qffuslonal hmuatlons and thus decreasing catalyst effectiveness

(b) Decrease m conversion, because of decreased residence time of gas wuhm the reactor, whmh follows duectly from decreased sohds nse velocity leading to lower void fracuons, under the fixed mass flow rates constraint

(c) Increase in conversion, as a result of greater resi- dence time of the sohd leading to greater dtiuslonal penetration of the solid and hence higher solid phase utlllzatlon

(d) Increase m conversion, because of the lower mass ratio of gas to solid wlthm the reactor, due to lower void fracuons

The relative magmtudes of the above effects should determine the conversion vs radius curve Predommance of one or more effects cannot be ruled out either, smce some effects, e g dlffuslonal lmutatlons, may turn out to be minor gven the range of particle s=es customanly employed m transport-he reactors

3 2 Flu& mechanrcal assumptrons Already unphclt in the denved expression for con-

version are rather drastic assumptions, the most notable bemg the assumption of plug flow m the solid phase In addition now, It becomes necessary to make further assumpuons regardmg the relation between parucle radius, r0 and sohds nse velocity, u,,

Fust, It 1s assumed that the difference between the mterstmal gas velocity and the termmal velocity of the particle represents the sohds nse velocity

Uo=U-U* (19)

Second, that the terminal velocuy 1s that of a single particle under the prevalent umform gas flow ([7], p 76)

u, = 4g@;i/)ro*, Re, < 0 4 @a)

and ute2 - KE + ubeb = 0,

113 rO, 04<Re, (500 (2Ob)

where

Rep =%&I!5 CL

(21)

and g 1s acceleration due to gravity, p 1s vlscoslty of the gas, and pS and pB are densmes of solid and gas, respec- tively

The above, whtle constltutmg a gross sunplrficatlon of physical reality, yet enable one to examme the effects attendmg the choice of dtierent pticle sues

3 3 Computational aspects For the purposes of computauon, It becomes neces-

and A VARMA

sary to assume certam base values for the parameters associated with the problem The procedure employed after the choice of these base values IS to vary the particle radius wtthm the reqmred range, whde sunul- taneously makmg the necessary corrections in the values of the parameters affected by such vanauon

Parameters and theu base values whnzh were common to all computations were Density of catalyst, pS = 2 gmlcm”, density of gas, pB = 7 185 x lO+ gm/cm’, VS- cosity of gas, CL = 0 026 cp, radius of catalyst parucle, rO= 20pm

Compuatlons were camed out for a range of values of the other vmables

Eb, 0 95-O 99

Ub, l&2000 cmlsec

0, 0 001-O 5 cm*/sec

L, 20&2OOOcm

&,z,lod-lo-’

where the subscnpt b refers to base values The last, the Thlele modulus represents, Hrlth D = 0 1 cm*/sec and r. = 20 pm at base values, a vartauon of mtrmslc rate constant k in the range 0 025-25 set-‘, and thus covers fully the range of reaction rates that are hkely to be encountered

Gas velocities at other than the base radius may be calculated from void fracuon l , which itself may be obtained by the followmg mass balances

Solrd balance

uo,(l-~~)=uO(l-P)=(U-uIlt)(l-e) (22)

Gas balance

Uh& u=-

E (23)

(22) and (23) lead to the quadratic in l

where

K=u~e~+uU,+~o~(l-~b) (24)

and thus

c = {K - (K* - 4uru,+&“3 2u*

(25)

The other root of the quadratic 1s spu~lous smce It gives an Incorrect result m the limit cb = 1

It may be noted that the dfierent functlonal depen- dences of terminal velocity on radius, when in lammar and turbulent Bow, will result m drscontrnurty in the converslon vs particle radms curve The locatlon of this dtscontmulty IS at a particle radius of 35 pm, which IS

Catalytic reactions in transport-lme reactors 341

also the mid-pomt of the range of radn for which computations have been camed out This range, It IS thought, 1s sufficient to cover the range of sizes used m normal practice

3 4 Results and drscasslon Figures 4-7 represent an Illustrative cross-sectron of

the computational results Figure 4 illustrates the effect of varymg gas velocltles

and hence mass flow rate of reactant, all other parameters bemg kept constant In thn, as in all other

90

4:s lo-*. L=800cm 80 E-098 I D=Ol cm%ec

:f Ib , , :-yq

20 30 40 50 60 70 - Particle Radws (mfcrons)

FIN 4 Reactor conversion as a function of catalyst partde size, for various values of base gas velocity

30 t

4,’ = IQ5. ub= 100 cm /set

‘b’O98 , D=O I cm2/sec A

2 25-

z g 20-

loo 1 I 1 1

+:-IO' 90-

ub = 100 cm/set , L = 800 cm c,.,-098, D= 0 I cm2/sec

80 -

/’ +‘a* Ki*

0 I 1 I I I I

0 IO 20 30 40 50 60 70 - Partrcle Rodus (microns 1

Rg 6 Reactor converSIon as a function of catalyst partlcle size, for various values of reaction rate constant

“I

+f - Id5, D = 0 I cmf/sec L=8OOcm

1

IO

5 t

_:/ 1 ,

I

0 I I I I 1 I I 0 IO 20 30 40 50 60 70

__c Partlcle Radws ( microns)

Rg 7 Reactor conversion as a function of catalyst partxle we, for vartous catalyst mass flow rates

-0 IO 20 30 40 50 60 70 + Particle Radius (microns)

Fu 5 Reactor conversion as a function of catalyst partde we, for various values of reactor length

computations that were carned out, no converslon decrease was ever noted as a consequence of an increase m particle radius Instead, converston increases rather sharply wth partrcle radius for some base gas velocltles

342 P VARCHESE and A V-A

Table 1 Effect of w-tlcle SW vmatlon on gas velocity, sohd veioclty and void fracuon at lower base gas velocity Ub = 100 cmfsec "b = 0 98

Radius Gas Velocity Solid Velocity Void Fraction ro(microns) u(cm/sec) "o(cm/sec) F

3.0 99 897 98 223 0 9810

20 100 000 93 301 0 9800

30 100 196 85 125 0 9780

35 100 345 79 831 0 9766

37 5 101 939 48 287 0 9613

40 102 238 45 009 0 9585

50 103 980 32 444 0 9424

60 107 323 21 479 0 9131

70 113 679 13 528 0 8620

Table 2 Effect of partlcie sue varlatlon on gas velocity, sohd velocity and void fraction at hgher base gas velocity

“b = 800 cm/set Eb = 0 98

Radius Gas Velocity SolId Velocity Void Frac~xon ro(nncrons) u(cm/sec) uo(cm/sec) B

10 799 884 798 224 0 9801

20 800 000 793 301 0 9800

30 800 170 785 099 0 9797

35 800 283 779 769 0 9796

37 5 801 005 747 352 0 9787

40 801 086 743 857 0 9786

50 801 421 729 884 0 9782

60 801 768 715 924 0 9778

70 802 129 701 978 0 9773

This would mdlcate that dIffusIona hmltatlons are not major and that the last of the four quahtatlvely antl- clpated results, namely that of mcreasmg solids content , m the reactor, predommates

This conclusion IS further supported by an inspection of Tables 1 and 2 which show that the computation at lower base gas velocity shows a slgmficant lowenng of void fraction when particle radius increases, as a consequence of the mass balance, whereas computations at a higher base gas velocity do not

Figure 5 compares conversion vs particle radms at different reactor lengths The entuely expected result 1s demonstrated by the curves

Figure 6 Isolates the effect of varying the mtrmslc reaction rate constant on the conversion vs particle radius curve It 1s seen that at very low and very high values of the rate constant, the curves are vu-tually flat and display no advantage to be gamed by employmg larger par&les and lower void fractions However for intermediate ranges of #,,’ such a course of action shows itself to be of no small merit In Fig 6, this 1s m part due to the choice of a low base gas velocity for all the curves but nevertheless this effect may be expected to show up, even If m a more moderate form, for other velocities

That dlffusional considerations were minor m the ex- treme, was confirmed by computations carried out at widely varymg values of dlffuslvlty, keeping all other parameters constant Conversion was not found to vary m a noteworthy manner with such vanatlon

Figure 7 vanes the base void fraction wMe keepmg the mass flow of gas constant, thus essentially varying the catalyst supply to the reactor Proportionately m- creased conversion 1s not obtamed smce the increased gas velocity necessitated alters the residence times avsulable within the reactor for the transient system

Fuures 4-7 show that d m fact the velocity behavior of solids m a tUgh voidage reactor (2-3% sohds) can be approxunated to the behavior of a single particle, with the associated transition from one flow regme to ano- ther, then under most combmatlons of reaction and reactor parameters, there 1s slgmficant advantage to be gamed by operating in the range of higher radu, smce there IS often a sharply drscontmuous mcrease m con- version associated with the transltlon

At any rate, for a reactlon system of known parameters, plottmg graphs such as the above can help confirm the existence or otherwise of such an advantage This then consldered m conJunctIon with other external

Catalytic reactlons In transport-lme reactors 343

factors can lead to a more studied choice of reactlon system parameters Some of these factors surely mclude solids atmtlon. separation and conveying power requirements

Acknowledgements-P V was reclplent of a Peter C Redly Fellowship dunng the coarse of this study We are abo indebted to the Umverslty of Notre Dame Computmg Center for a grant of free time for the calculauons reported here

A c

CO CS D !3

I!

ho. f

; K L

M N

P(S) P

q(s) r

r. R s t

NOTATION Qmenslonless group, defined by eqn (6) reactant concentration m the gas phase reactant concentration m feed reactant concentration m the sohd phase effective dlffuslvlty m the solid phase dlmenslonless group, defined by eqn (6) C/CO C*l CO acceleration due to gravity defined by eqn (18) d-1 reaction rate constant defined by eqn (24) reactor length dimensionless catalyst residence time, = IIA number of particles per umt reactor volume defined by eqn (12) AL%?/3 defined by eqn (12) radral posltlon wtthn the spherxal catalyst radrus of spherical catalyst partxle volumemc reaction rate Laplace transform variable time

U interstituil gas velocity u,, solids nse velocity uI terminal velocity of sohds x rfrO

X fractional conversion y ZIL Z axial distance along reactor

Greek symbols /3 defined by eqn (12) Y -8

void fraction m reactor i Threle modulus, rO(WD)“2 pS density of gas pS density of sohd W viscosity of gas

Subscripts b base values for computation n integer values

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I41 Erz K C , Chem Engng Scr 1974 29 747 151 Robertson A D and Pratt K C , Chem Engrtg Six 1975 30

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New York 1969