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Chemical Engineering Journal 166 (2011) 1090–1094

Contents lists available at ScienceDirect

Chemical Engineering Journal

journa l homepage: www.e lsev ier .com/ locate /ce j

nhancement of the catalytic performances under non-steady state conditions

acine Benguerba ∗, Brahim Djellouliaboratoire de Génie des Procédés Chimiques, Université Ferhat Abbas, Sétif, Algeria

r t i c l e i n f o

rticle history:eceived 16 August 2010eceived in revised form 22 October 2010

a b s t r a c t

The problem of optimal activity distribution in an nonisothermal catalyst pellet with a consecutive-parallel reaction scheme proceeding in a created nonsteady state conditions already studied in ourprevious paper [1], is now extended to the case where the temperature and the concentrations of the

ccepted 19 November 2010

eywords:nsteady stateptimal catalyst distributionorced perturbation

reactants inside the grain vary according to time. The same activity distribution, i.e., a Dirac ı function,as in the case of quasi-steady state, is also found optimal in this case. The studied perturbed system givesbetter catalytic performances for large periods compared to time characteristic of the catalytic process.

© 2010 Elsevier B.V. All rights reserved.

emperature modulationoncentration modulation

. Introduction

One of the objectives of this work is to show, for catalyticeactions, that while working under controlled nonsteady stateonditions, the catalytic performances will be improved.

In our precedent works [1,2] we have consider the case that theeriod of the cyclic perturbation was very large compared with theharacteristic time of the catalytic act. In this paper intermediateode discussed by Silveston et al. [3] is considered where the above

ssumption [4,5], is not valid.In this paper, the objective is the determination of the optimal

atalyst distribution, which maximizes the catalytic effectivenesst any moment of time for a consecutive-parallel chemical reactioncheme proceeding under created unsteady state conditions.

. Mathematical formulation of the problem

Let us consider the following consecutive-parallel chemicaleaction scheme

k1−→Bk2−→C; A

k3−→D (1)

For a symmetric porous support with a nonuniform distributionf active catalyst, the nonsteady-state mass and heat balance havehe form [6] ( )

∂Ci∂t

= De,i1x˛∂

∂xx˛∂Ci∂x

− Fi(C, T) (2a)

∗ Corresponding author. Tel.: +213 553604453; fax: +213 36925133.E-mail address: benguerbayacine@yahoo.fr (Y. Benguerba).

385-8947/$ – see front matter © 2010 Elsevier B.V. All rights reserved.oi:10.1016/j.cej.2010.11.073

�Cp∂T

∂t= �e

1x˛∂

∂x

(x˛∂T

∂x

)+

3∑j=1

(−�Hj)rj(C, T) (2b)

The indices (i) for the constituents A, B, C, and D are respectively1, 2, 3, and 4.

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)

It is assumed that the kinetic rates are simple power law models:

ri = k0i (x) exp

(− EiRT

)Ci (4)

The rate constant density function ˛(x) is defined 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)

Y. Benguerba, B. Djellouli / Chemical Engine

Nomenclature

a(s) catalyst distribution functionBim Biot number for the mass transfer (kcR/De)Bih Biot number for the heat transfer (hR/�e)C reactant concentration inside the catalytic pellet,

mol m−3

Cf reactant concentration in the extra-granular fluidphase, mol m−3

CR reactant concentration at the surface of the catalyticpellet, mol m−3

Cp heat capacity of the fluid, J kg−1 K−1

De effective diffusion coefficient, m2 s−1

E activation energy, cal mol−1

r(c,T) rate of reaction, mol m−3 s−1

f(u,�) dimensionless rate of reactionh heat transfer coefficient, cal m−2 K−1 s−1

kc mass transfer coefficient, m s−1

Le Lewis number, �CpD/�R radius of catalytic pellet, mU dimensionless concentrationUR dimensionless concentration at the surface pelletRG ideal gas constant, J mol−1 K−1

s dimensionless distance from center of the pelletS21(0) catalyst selectivity factor in steady state conditionsS21(t) catalyst selectivity factor in unsteady state condi-

tionsS21 mean selectivity factor over the period, tc

t time of the process (s)T fluid temperature inside the catalytic pellet, KTf fluid temperature in the bulk fluid, KTR fluid temperature at the surface of the pellet, KVp catalytic pellet volume, m3

x distance from the center of pellet, m

Greek letters�Hj heat of the jth reaction, J mol−1

wi Lobatto coefficients� dimensionless temperature� Thiele modulusˇ dimensionless Prater number� dimensionless Arrhenius number�e effective pellet thermal conductivity,

cal K−1 m−1 s−1

(0) catalyst effectiveness under stationary conditions(t) catalytic effectiveness under the nonstationary con-

ditions average effectiveness over the period, tc

dimensionless time

bi

C

T

tt

c dimensionless perturbation period� density of the fluid, kg m−3

With the constraint that the mean value of the periodic pertur-ation imposed for extragranular temperature and concentration

s the same as used for the steady state case. And so we can write:

f,i(t) = Cf,i + Cf,i(t),

∫ tc

0

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

�

∫ tc�

f(t) = Tf + T f(t),0

Tf (t) dt = 0 (8b)

Cf,i and Tf are the reactant concentrations and temperatures ofhe bulk fluid in the steady state case; tc is the period of change ofhe perturbation functions,

�Cf,i(t) and.

�T f(t)

ering Journal 166 (2011) 1090–1094 1091

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

Cf,i(t) = Cf,i + Cf,i�1 sin(ωt) (9a)

Tf(t) = Tf + Tf�2 sin(ωt) (9b)

With�1 and�2 are the amplitudes of the perturbation functions;ω is the rotation frequency (s−1).

The following dimensionless parameters are defined:

= De,1t

R2; c

De,1tcR2

; Le = �CpDe,1

�; ui =

CiCf,i

; � = T

Tf;

s = x

R; Bim,i =

kc,iR

De,i; Bih = hR

�e; �2

i = R2ki,fDe,i

; �i =EiRTf

;

ki,f = ki exp

(− EiRTf

); ˇj = (−�Hj)De,jCf,j

�eTfavec j = 1,2;

ˇ3 = (−�H3)De,1Cf,1

�eTf

Using these parameters in Eqs. (2a) and (2b) the followingdimensionless differential equations are obtained:

∂u1

∂= 1s˛∂

∂s

(s˛∂u1

∂s

)− a(s)�2

1

×[u1 exp

(y1

(1 − 1

�

))+ K31 exp

(y3

(1 − 1

�

))](10a)

X∂u2

∂= 1s˛∂

∂s

(s˛∂u2

∂s

)− a(s)

×[�2

2u2 exp(y2

(1 − 1

�

))− �2

1u1 exp(y1

(1 − 1

�

))]

(10b)

Le∂�

∂= 1s˛∂

∂s

(s˛∂�

∂s

)+ a(s)

[ˇ1�

21u1 exp

(y1

(1 − 1

�

))

+ˇ2�22u2 exp

(y2

(1 − 1

�

))

+K31ˇ3�21u1 exp

(y2

(1 − 1

�

))](10c)

With the constants � and K31 given by:

K31 = k3 exp(−E3/RTf)

k1 exp(−E1/RTf); = De,1cf,1

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

De,2

And the boundary conditions:

duids

= d�ds

= 0 at s = 0 (11a)⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

du1

ds= Bim,1

(1 + �1 sin

(2�

c

)− u1,R

)du2

ds= Bim,2

(1 + �1 sin

(2�

c

)− u2,R

)d�ds

= Bih

(1 + �2 sin

(2�

c

)− �R

)at s = 1 (11b)

Here we suppose Bim,1 = Bim,2 = Bim.The dimensionless form of Eq. (6) is given by:

(˛+ 1)

∫ 1

0

a(s)s˛ ds = 1 (12)

1 ngineering Journal 166 (2011) 1090–1094

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092 Y. Benguerba, B. Djellouli / Chemical E

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

() = (˛+ 1)1 + K31

∫ 1

0

a(s)u1(i)[

exp(�1

(1 − 1

�

))

+K31 exp(�3

(1 − 1

�

))]s˛ ds (13a)

And the selectivity:

21() =∫ 1

0a(s){�2

2u2 exp(�2(1 − 1/�)) + �21u1 exp(�1(1 − 1/�))}s˛ ds∫ 1

0a(s){�2

1u1 exp(�1(1 − 1/�)) + K31 exp(�3(1 − 1/�))}s˛ ds(13b)

The mean values are calculated to compare them with thebtained values in the steady state case.

¯ =∫ c

0

() d; S21 =∫ c

0

S21() d (14)

There is an enhancement in the catalytic performances if onlyhe mean values calculated by Eq. (14) are greater than the steadytate obtained values.

. Numerical procedure

We need to determine the optimal catalyst distribution a(s),hich maximizes the performance index given by Eq. (13a) or

13b), and satisfies the constraint (12) and the diffusion-reactionquations ((10a)–(10c)).

The numerical resolution technique is based on the orthogonalollocation method [9,10]. The number of internal collocation pointsC was taken equal to 7. No more accuracy was observed whenore internal collocation points were used.

∂u1

∂=

NC+1∑j=1

B(i, j)u1(j) − a(i)�21u1(i)

×[

exp(�1

(1 − 1

�(i)

))+ K31 exp

(�3

(1 − 1

�(i)

))](15a)

∂u2

∂=

NC+1∑j=1

B(i, j)u2(j) − a(j)[�2

2u2(i) exp(�2

(1 − 1

�(i)

))

+ �21u1(i) exp

(�1

(1 − 1

�(i)

))](15b)

e∂�

∂=NC+1∑j=1

B(i, j)�(j) − a(i)[ˇ1�

21u1(i) exp

(�1

(1 − 1

�(i)

))

+ˇ2�22u2(i) exp

(�2

(1 − 1

�(i)

))

+K31ˇ3�21u1(i) exp

(�3

(1 − 1

�(i)

))](15c)

vec i = 1, NCEq. (11b) is used for calculating u1,R, u2,R and �R noted in

he orthogonal collocation method terminology as u1(NC + 1),(NC + 1) and �(NC + 1) respectively.

2

C+1∑j=1

A(NC + 1, j)u1(j) = Bim

(1 + �1 sin

(2�t

tc

)− u1(NC + 1)

)

(16a)

Fig. 1. Variations of the optimal position of the active sites for a consecutive-parallelreaction scheme.

NC+1∑j=1

A(NC + 1, j)u2(j) = Bim

(1 + �1 sin

(2�t

tc

)− u2(NC + 1)

)

(16b)

NC+1∑j=1

A(NC + 1, j)�(j) = Bih

(1 + �2 sin

(2�t

tc

)− �(NC + 1)

)(16c)

The resolution of differential equations ((15a)–(15c)), with theboundary conditions ((16a)–(16c)), is achieved with the fourthorder Runge–Kutta method.

The values of the catalyst activity at the collocation points,unknown at this stage, were regarded as the adjustable parametersof the optimization problem. The objective function is given by Eq.(13a) or (13b), which can be evaluated by using the Gauss–Jacobiquadrature formulae, where the quadrature points coincide withthe collocation points [9]. The search of the catalyst activity whichmaximizes Eq. (13a) or (13b) is operated with the Interior-pointalgorithm [1,2].

4. Results and discussion

The performance enhancement of such systems is achievedpractically with difficulty by the control of two important parame-ters:

• Temperature variation: any large thermal inertia tends to defeatthe effect of sudden changes in this variable.

• Optimal catalyst position: the location of the catalyst distributionis time dependent (Fig. 1). To control the position of the activesites which must be changed in a periodic way, technically onemust make migrate collectively the active sites from the old opti-mal position to the new one periodically according to the time ofthe process [2].

Fig. 2 shows us the mode of variation of the catalytic parame-ters according to the sinusoidal perturbations of the concentrations

of the reactants and the temperature of the extra-granular fluid.The graphs obtained show the periodic character of the answersbut which is not sinusoidal (the perturbation of the entries is sinu-soidal).

Y. Benguerba, B. Djellouli / Chemical Engineering Journal 166 (2011) 1090–1094 1093

4

tTttmvtt

4p

iambb

Fig. 4. Effects of the concentration modulation on the catalytic parameters.

4.3. Effects of the Lewis number on the catalytic parameters

Fig. 2. Catalytic parameter variations with dimensionless time.

.1. Effects of perturbation period on the catalytic parameters

Fig. 3 shows that the increase in the period of the perturba-ions gives an significant improvement of the catalytic parameters.his gives us an idea on the order of magnitude of the period ofhe perturbations which must be very important compared withime characteristic of the system R2/De,1. Thus if one wants to opti-

ize the catalytic output it is necessary to work with periods ofery large cyclic perturbations compared to time characteristic ofhe process being held in a catalytic pellet of unspecified geome-ry.

.2. Effects of the amplitudes of the perturbations on the catalyticarameters

Figs. 4 and 5 show that only the modulation of the temperaturen the case of a consecutive-parallel chemical reaction scheme hasn effect on the catalytic performances. In the case of temperatureodulation, a good enhancement is obtained for the effectiveness

ut the selectivity decreases slightly if the amplitude of the pertur-ation is increased.

Fig. 3. Effects of the period of the perturbations on the catalytic parameters.

Fig. 5. Effects of the temperature modulation on the catalytic parameters.

The results represented in Fig. 6 enabled us to determine theoptimal values for the Lewis number (relationship between mass

Fig. 6. Effects of the Lewis number on the catalytic parameters.

1 ngine

diiwbndtFc

5

ta

•

•

•pellets. VIII. General nonisothermal reacting systems with arbitrary kinetics,

094 Y. Benguerba, B. Djellouli / Chemical E

iffusivity and thermal diffusivity). These values are given in thenterval Le ∈ [1, 10]. This can be explained by the fact that in thisnterval the heat transfer and the mass diffusion rather favored

ith rather ideal proportions and are balanced what created theest conditions for an optimal catalytic yield. For too large Lewisumber, there is a very great mass diffusion with a weak thermaliffusion process what created an imbalance between the mass andhe quantity of energy available for the conversion of this mass.or weak Lewis number there is too much energy which diffusesompared to the mass available for conversion.

. Conclusion

This second part of our study concerning the case where theime of the disturbances (period) is of the same order of magnitudes time characteristic of the catalytic process (c ≈ 1) showed that:

The studied disturbed systems give better catalytic performancesfor periods of rather large perturbation compared to time char-acteristic of the catalytic process.Optimal distribution of the active sites, in any moment of time ,

is given by the function ı of Dirac.The creation of the nonstationary conditions must be accompa-nied by a control of the catalytic activity profile. The location ofthe optimal distribution of the active sites, given by the functionı of Dirac, for a given moment, is variable with time.

[

ering Journal 166 (2011) 1090–1094

• The answer of the catalytic process in this case is not done withthe same function of variation of the cyclic disturbances, becausethe system cannot follow the sinusoidal variations of the inlet toofast compared to the catalytic characteristic time. In addition, thisanswer remains always periodic.

References

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