Catalytic reactions in transport-line reactors

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 operavanables 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 deactivation, such as catalytic crackmg 111 the petroleum mdustry[2,3], m wlucb cases the reactor offers the attractive features of contmuous addition of fresh catalyst at the feed end, and sunultaneous withdraw1 of the deactivated catalyst, for subsequent regeneration, at the reactor exit Recent analyses by Pratt[4, S] have employed an adrmttedly 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 catalyst de&mation, and Investigate the effect of pticle sue on reactor performance under hopefully realistic condltlons of reactor ooeraoon

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 assumptions, a mass balance m the gas phase yields dC lM-&=R,

c= co,

(1)

(2)

r=O

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 catalyst p&cle

along with the nuti

ac,=o ar ‘-’

and boundary

condltlons

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

r-0

We consider the case of an ureverstble ilrst order catalytic reactton occurrtng m an isothermal transporthne reactor The reactor model employed herem IS the same as that of Pratt [4], wbxh mvolved the followmg assumptions (a) The gas and solid phases are m cocurrent upward ideal plug flow, thus there IS no backnuxmg of either the gas or the sold phases, although the gas and sohd travel with Merent velocities (b) The gas phase IS consldered to be m steady state, while the solid catalyst phase IS m transient state (c) The catalyst particles, assumed spherical, are modeled yta the d&muon-reactton equauons (d) There IS no external film resistance to mass transfer, I e the reactant concentration on the catalyst surface IS the same as m the bulk gas around the catalyst pamcle

2 E (0, L)

c, =O,z=O

(4) R, the volumetic

reaction

rate IS pven

R = -(4~7r,,~N)D +

by

I ..=rQ’

(9

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

(5a)

IS the number of partxles per umt reactor Defimng dunenslonless varmbles

f = CICO.fs = c*ico,

volume

Y = z/L, n = rlro.

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

l)L.LIUHO’ (6)

tTo whom correspondence should be addressed 337

P

338

Equations

(l)-(4)

may be cast m dnnenslonless

df __& _dLax

I

Aaf.=l ay

a

Tax

2

y E (0, 11,x E (091) (91

f.=O,y=O

f.=f(yhx=l,

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

=---1

BP(S) w(s)

s

(11)

where p(s)=fi-tanh&fi*=#+As q(s) = s tanh fi + S(B - tanh p)

and s IS the transform vanable roots of q(s) have to be found Rearrangmg q(s) = 0 as

sfl=

19 +

((c-

(12)

To Invert F(s),

all the

@)/A}] tanh p

&, E (0, /3*), 8’ = (@+

(13)

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 conversion in the gas phase, X = 1 -f, as a function of axml posItion y, 1s grven by * = WBO -

+ 2 n-f

tanh &)[exp

A'WY. -

(soy) - ll/(saho)

tanm)ll

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 catalyst 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 143cannot 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 performed 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

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

X(Y)

VARMA

(8)

afs (xax )- 4’f*,

af= x=o, ~=o’

(7)

y=o

f-1,

and A

where

form

Y E (0.1)

’

x-1

V~GHESE

- exp H4’ + kW2 + Yn',

ym2)ylA31 (17)

Consldenng the expression (17) denved for reactor a complex conversion, tt may easily be seen to etiblt 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-

OIO4

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

P

340

VARGHESE

make then contnbutlons felt when parucle radius mcreases (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 residence 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 conversion 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

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 sunultaneously 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’, VScosity 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)

(19)

Uo=U-U*

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

u=-

Uh&

(23)

E

u, = 4g@;i/)ro*,

Re, < 0 4

@a)

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

l

ute2- KE + ubeb = 0,

and 113 rO, 04
(500

(2Ob)

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

(24)

c = {K - (K* - 4uru,+&“3 2u*

(25)

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, respectively 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 thus

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 dependences 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 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

loo

341

1

I

1

1

+:-IO' 90-

ub = 100 cm/set

80 -

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

c,.,-098,

, L = 800

D= 0

cm

I cm2/sec

90

4:s lo-*. E-098

80

L=800cm

I D=Ol cm%ec

/’

0

I

0

1

IO

I

20

-

30

Partrcle

+‘a* Ki* I

I

I

40

50

60

Rodus

(microns

70

1

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

“I +f

:f

Ib

,

,

- Id5,

D = 0

I cmf/sec

L=8OOcm

:-yq

1 20 -

30 Particle

40 Radws

50

60

FIN 4 Reactor conversion

as a function of catalyst partde for various values of base gas velocity

4,’ = IQ5.

30

t 2 z g

‘b’O98

ub=

70

(mfcrons)

100cm /set I cm2/sec

, D=O

size,

A

25205 t

0

Rg

-0

IO

20 +

Fu

1

_:/

IO

5

30 Particle

40 Radius

50

60

70

(microns)

Reactor conversion as a function of catalyst partde for various values of reactor length

we,

7

, I

I

0

IO __c

I

20

I

30 Partlcle

I

40 Radws

1

I

50

60

( microns)

Reactor conversion as a function of catalyst partxle for vartous catalyst mass flow rates

computations that were decrease was ever noted as m particle radius Instead, sharply wth partrcle radius

I

70

we,

carned out, no converslon a consequence of an increase converston increases rather for some base gas velocltles

P VARCHESE and A

342 Table

1

Effect

of w-tlcle

SW Ub

vmatlon =

100

on gas velocity,

sohd veioclty and void fracuon at lower base gas velocity "b

cmfsec

Gas Velocity u(cm/sec)

Radius ro(microns)

V-A

=

0

98 Void

Solid Velocity "o(cm/sec)

Fraction F

99

897

98

223

0

9810

20

100

000

93

301

0

9800

30

100

196

85

125

0

9780

831

0

9766

3.0

35 37

5

40

100

345

79

101

939

48

287

0

9613

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 = Radius ro(nncrons)

800

Eb

cm/set Gas Velocity u(cm/sec)

SolId Velocity uo(cm/sec)

=

0

98 Void

Frac~xon 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

801

005

747

352

0

9787

40

801

086

743

857

0

9786

50

801

421

729

884

0

9782 9778 9773

37

5

60

801

768

715

924

0

70

802

129

701

978

0

This would mdlcate that dIffusIona hmltatlons are not That dlffusional considerations were minor m the exmajor and that the last of the four quahtatlvely antltreme, was confirmed by computations carried out at clpated results, namely that of mcreasmg solids content , widely varymg values of dlffuslvlty, keeping all other m the reactor, predommates parameters constant Conversion was not found to vary This conclusion IS further supported by an inspection m a noteworthy manner with such vanatlon of Tables 1 and 2 which show that the computation at Figure 7 vanes the base void fraction wMe keepmg lower base gas velocity shows a slgmficant lowenng of the mass flow of gas constant, thus essentially varying void fraction when particle radius increases, as a the catalyst supply to the reactor Proportionately mconsequence of the mass balance, whereas computations creased conversion 1s not obtamed smce the increased at a higher base gas velocity do not gas velocity necessitated alters the residence times Figure 5 compares conversion vs particle radms at avsulable within the reactor for the transient system different reactor lengths The entuely expected result 1s Fuures 4-7 show that d m fact the velocity behavior demonstrated by the curves of solids m a tUgh voidage reactor (2-3% sohds) can be Figure 6 Isolates the effect of varying the mtrmslc approxunated to the behavior of a single particle, with reaction rate constant on the conversion vs particle the associated transition from one flow regme to anoradius curve It 1s seen that at very low and very high ther, then under most combmatlons of reaction and values of the rate constant, the curves are vu-tually flat reactor parameters, there 1s slgmficant advantage to be and display no advantage to be gamed by employmg gamed by operating in the range of higher radu, smce larger par&les and lower void fractions However for there IS often a sharply drscontmuous mcrease m conintermediate ranges of #,,’ such a course of action shows version associated with the transltlon itself to be of no small merit In Fig 6, this 1s m part due At any rate, for a reactlon system of known to the choice of a low base gas velocity for all the curves parameters, plottmg graphs such as the above can help but nevertheless this effect may be expected to show up, confirm the existence or otherwise of such an advantage even If m a more moderate form, for other velocities This then consldered m conJunctIon with other external

Catalytic reactlons In transport-lme reactors factors can lead to a more studied choice of reactlon system parameters Some of these factors surely mclude and conveying power separation solids atmtlon. 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 NOTATION

A c CO CS D !3 I! ho. f ; K L M N P(S) P q(s) r r. R s t

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

u,, uI x X y Z

343

interstituil gas velocity solids nse velocity terminal velocity of sohds rfrO fractional conversion ZIL 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

[I] Carberry J J , Chemical and Catalytrc Reactton Engmeemng McGraw-Ha, New York 1976 121 Brother C W , Vennfflion W L and Conner A J , Hydroc Proc 197289 [31 Weekman V W , Jr, Id Engng Chem , Proc Bs DEV 1968 I41 Erz K C , Chem Engng Scr 1974 29 747 151 Robertson A D and Pratt K C , Chem Engrtg Six

1975 30 1185 [6] Crank J , The Mathematrcs of L%fluswn, 2nd Edn , Oxford Umverslty Press, Oxford 1975 [7] Kunu D and Levensplel 0, Flrcrdrrahon Englneenng Wdey, New York 1969