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Chemical Engineering Journal 166 (2011) 10901094

Contents lists available at ScienceDirect

Chemical Engineering Journal

journa l homepage: www.e lsev ier .c

Enhanc de

Yacine BLaboratoire de

a r t i c l

Article history:Received 16 AReceived in reAccepted 19 N

Keywords:Unsteady stateOptimal catalyForced perturbTemperature mConcentration

ributin a

to thing talso fe per

1. Introdu

One ofreactions, tconditions,

In our prperiod of the cyclic perturbation was very large compared with thecharacteristic time of the catalytic act. In this paper intermediatemodediscussedbySilvestonet al. [3] is consideredwhere the aboveassumption [4,5], is not valid.

In this paper, the objective is the determination of the optimalcatalyst disat anymomscheme pro

2. Mathem

Let us creaction sch

Ak1B k2CFor a sym

of active cathe form [6

Cit

= De,i1x

CorresponE-mail add

=

indiand

Here the parameter represents different geometries (=0 forslab, =1 for cylindrical and =2 for sphere).

With the rates Fi given for each constituent as:

F1 = r1 + r3; F2 = r2 r1; F3 = r2; F4 = r3 (3)

1385-8947/$ doi:10.1016/j.tribution, which maximizes the catalytic effectivenessent of time for a consecutive-parallel chemical reactionceeding under created unsteady state conditions.

atical formulation of the problem

onsider the following consecutive-parallel chemicaleme

; Ak3D (1)

metric porous support with a nonuniform distributiontalyst, the nonsteady-state mass and heat balance have]

x

(xCix

) Fi(C, T) (2a)

ding author. Tel.: +213 553604453; fax: +213 36925133.ress: benguerbayacine@yahoo.fr (Y. Benguerba).

It is assumed that thekinetic rates are simplepower lawmodels:

ri = k0i (x) exp(

EiRT

)Ci (4)

The rate constant density function (x) is dened as the ratio ofthe local preexponential factor k0

i(x) to its volume averaged value

k [7].

a(x) = k0i(x)

ki(5)

The density function must satisfy the following integral:

1vp

p

a(x) dV = 1 (6)

The boundary conditions used here are given by:

dCidx

= dTdx

= 0 at x = 0 (7a)

De,idCidx

= Kc,i(Cf,i(t)CR,i); edTdx

= h(Tf(t)TR) at x = R (7b)

see front matter 2010 Elsevier B.V. All rights reserved.cej.2010.11.073ement of the catalytic performances un

enguerba , Brahim DjellouliGnie des Procds Chimiques, Universit Ferhat Abbas, Stif, Algeria

e i n f o

ugust 2010vised form 22 October 2010ovember 2010

st distributionationodulationmodulation

a b s t r a c t

The problem of optimal activity distparallel reaction scheme proceedingprevious paper [1], is now extendedreactants inside the grain vary accordas in the case of quasi-steady state, isbetter catalytic performances for larg

ction

the objectives of this work is to show, for catalytichat while working under controlled nonsteady statethe catalytic performances will be improved.ecedent works [1,2] we have consider the case that the

CpT

t

The1, 2, 3,om/ locate /ce j

r non-steady state conditions

ion in an nonisothermal catalyst pellet with a consecutive-created nonsteady state conditions already studied in our

e case where the temperature and the concentrations of theo time. The same activity distribution, i.e., a Dirac function,ound optimal in this case. The studied perturbed system givesiods compared to time characteristic of the catalytic process.

2010 Elsevier B.V. All rights reserved.

e1x

x

(xT

x

)+

3j=1

(Hj)rj(C, T) (2b)

ces (i) for the constituents A, B, C, and D are respectively4.

Y. Benguerba, B. Djellouli / Chemical Engineering Journal 166 (2011) 10901094 1091

Nomenclature

a(s) catalyst distribution functionBimBihC

Cf

CR

CpDeEr(c,T)f(u,)hkcLeRUURRGsS21(0)S21(t)

S21tTTfTRVpx

Greek letHjwie

(0)(t)

c

With thebation impis the same

Cf,i(t) = Cf,i

Tf(t) = Tf +

Cf,i andthe bulk uthe perturb

Sinus function is one of the perturbation functions used in theperiodic operations.

Cf,i(t) = Cf,i + Cf,i1 sin(t) (9a)Tf +h1e rotfollo

,1t2

;

BimBiot number for the mass transfer (kcR/De)Biot number for the heat transfer (hR/e)reactant concentration inside the catalytic pellet,molm3

reactant concentration in the extra-granular uidphase, molm3

reactant concentration at the surface of the catalyticpellet, molm3

heat capacity of the uid, J kg1 K1

effective diffusion coefcient, m2 s1

activation energy, calmol1

rate of reaction, molm3 s1

Tf(t) =Wit

is thThe

= DeR

s = xR;dimensionless rate of reactionheat transfer coefcient, calm2 K1 s1

mass transfer coefcient, ms1

Lewis number, CpD/radius of catalytic pellet, mdimensionless concentrationdimensionless concentration at the surface pelletideal gas constant, Jmol1 K1

dimensionless distance from center of the pelletcatalyst selectivity factor in steady state conditionscatalyst selectivity factor in unsteady state condi-tionsmean selectivity factor over the period, tctime of the process (s)uid temperature inside the catalytic pellet, Kuid temperature in the bulk uid, Kuid temperature at the surface of the pellet, Kcatalytic pellet volume, m3

distance from the center of pellet, m

tersheat of the jth reaction, Jmol1

Lobatto coefcientsdimensionless temperatureThiele modulusdimensionless Prater numberdimensionless Arrhenius numbereffective pellet thermal conductivity,cal K1 m1 s1

catalyst effectiveness under stationary conditionscatalytic effectiveness under the nonstationary con-ditionsaverage effectiveness over the period, tcdimensionless timedimensionless perturbation perioddensity of the uid, kgm3

constraint that the mean value of the periodic pertur-osed for extragranular temperature and concentrationas used for the steady state case. And so we can write:

+ Cf,i(t), tc

0

Cf,i(t) dt = 0 (8a)

T f(t),

tc0

Tf (t) dt = 0 (8b)

Tf are the reactant concentrations and temperatures ofid in the steady state case; tc is the period of change ofation functions,

Cf,i(t) and.

T f(t)

ki,f = ki exp

3 =(H

Using thdimensionl

u1

= 1s

[u

Xu2

= 1s

Le

= 1s

+

+K

With the

K31 =k3 ex

k1 ex

And the

duids

= dds

=

du1ds

= B

du2ds

= B

dds

= Bih

Here weThe dim

(+ 1) 1

0Tf2 sin(t) (9b)

and2 are theamplitudesof theperturbation functions;ation frequency (s1).wing dimensionless parameters are dened:

cDe,1tcR2

; Le =CpDe,1

; ui =CiCf,i

; = TTf

;

,i =kc,iR

De,i; Bih =

hR

e; 2i =

R2ki,fDe,i

; i =EiRTf

;

( EiRTf

); j =

(Hj)De,jCf,jeTf

avec j = 1,2;

3)De,1Cf,1eTf

ese parameters in Eqs. (2a) and (2b) the followingess differential equations are obtained:

s

(su1s

) a(s)21

1 exp(y1

(1 1

))+ K31 exp

(y3

(1 1

))](10a)

s

(su2s

) a(s)

[22u2 exp

(y2

(1 1

)) 21u1 exp

(y1

(1 1

))]

(10b)

s

(s

s

)+ a(s)

[1

21u1 exp

(y1

(1 1

))

222u2 exp

(y2

(1 1

))

31321u1 exp

(y2

(1 1

))](10c)

constants and K31 given by:

p(E3/RTf)p(E1/RTf)

; = De,1cf,1De,2cf,2

; x = De,1De,2

boundary conditions:

0 at s = 0 (11a)

im,1

(1 + 1 sin

(2

c

) u1,R

)

im,2

(1 + 1 sin

(2

c

) u2,R

)(1 + 2 sin

(2

c

) R

)at s = 1 (11b)

suppose Bim,1 =Bim,2 =Bim.ensionless form of Eq. (6) is given by:

a(s)s ds = 1 (12)

1092 Y. Benguerba, B. Djellouli / Chemical Engineering Journal 166 (2011) 10901094

Catalytic parameters are given by [8]:For the catalytic effectiveness:

() = (+ 1)1 +

1a(s)u1(i)

[exp

(1

(1 1

))

+K3

And the

S21() = 1

0a 10

The meaobtained va

= c

0

(

There isthe mean vstate obtain

3. Numeri

We needwhich max(13b), and sequations (

The numcollocation mNC was takmore intern

u1

=NC+1j=1

[e

xu2

=NC+j=1

+

Le

=NC+j=1

+

+K

Avec i=1, NEq. (11b

the orthogu2(NC+1) a

NC+1j=1A(NC +

ariatischem

NC +

NC +

resoary cungevalu

wn aoptimr (13turelocatizeshm [

ults

perally with difculty by the control of two important parame-

erature variation: any large thermal inertia tends to defeatffect of sudden changes in this variable.al catalyst position: the location of the catalyst distributione dependent (Fig. 1). To control the position of the activewhich must be changed in a periodic way, technically onemakemigrate collectively the active sites from the old opti-osition to the new one periodically according to the time ofrocess [2].

2 shows us the mode of variation of the catalytic parame-cording to the sinusoidal perturbations of the concentrationsreactants and the temperature of the extra-granular uid.aphs obtained show the periodic character of the answersich is not sinusoidal (the perturbation of the entries is sinu-.K31 0

1 exp(3

(1 1

))]s ds (13a)

selectivity:

(s){22u2 exp(2(1 1/)) + 21u1 exp(1(1 1/))}s ds

a(s){21u1 exp(1(1 1/)) + K31 exp(3(1 1/))}s ds(13