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Chemical Engineering Journal 166 (2011) 10901094Contents lists available at ScienceDirectChemical Engineering Journaljourna l homepage: www.e lsev ier .cEnhanc deYacine BLaboratoire dea r t i c lArticle history:Received 16 AReceived in reAccepted 19 NKeywords:Unsteady stateOptimal catalyForced perturbTemperature mConcentrationributin ato thing talso fe per1. IntroduOne 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 pro2. MathemLet us creaction schAk1B k2CFor a symof active cathe form [6Cit= De,i1x CorresponE-mail add= indiandHere 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 problemonsider 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 k0i(x) to its volume averaged valuek [7].a(x) = k0i(x)ki(5)The density function must satisfy the following integral:1vppa(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 unenguerba , Brahim DjellouliGnie des Procds Chimiques, Universit Ferhat Abbas, Stif, Algeriae i n f ougust 2010vised form 22 October 2010ovember 2010st distributionationodulationmodulationa b s t r a c tThe 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 largctionthe 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 theCpTtThe1, 2, 3,om/ locate /ce jr non-steady state conditionsion in an nonisothermal catalyst pellet with a consecutive-created nonsteady state conditions already studied in oure 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.e1xx(xTx)+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 1091Nomenclaturea(s) catalyst distribution functionBimBihCCfCRCpDeEr(c,T)f(u,)hkcLeRUURRGsS21(0)S21(t)S21tTTfTRVpxGreek letHjwie(0)(t)cWith thebation impis the sameCf,i(t) = Cf,iTf(t) = Tf +Cf,i andthe bulk uthe perturbSinus 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,molm3reactant concentration in the extra-granular uidphase, molm3reactant concentration at the surface of the catalyticpellet, molm3heat capacity of the uid, J kg1 K1effective diffusion coefcient, m2 s1activation energy, calmol1rate of reaction, molm3 s1Tf(t) =Wit is thThe = DeRs = xR;dimensionless rate of reactionheat transfer coefcient, calm2 K1 s1mass transfer coefcient, ms1Lewis number, CpD/radius of catalytic pellet, mdimensionless concentrationdimensionless concentration at the surface pelletideal gas constant, Jmol1 K1dimensionless 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, m3distance from the center of pellet, mtersheat of the jth reaction, Jmol1Lobatto coefcientsdimensionless temperatureThiele modulusdimensionless Prater numberdimensionless Arrhenius numbereffective pellet thermal conductivity,cal K1 m1 s1catalyst effectiveness under stationary conditionscatalytic effectiveness under the nonstationary con-ditionsaverage effectiveness over the period, tcdimensionless timedimensionless perturbation perioddensity of the uid, kgm3constraint 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), tc0Cf,i(t) dt = 0 (8a)T f(t), tc0Tf (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 exp3 =(HUsing thdimensionlu1= 1s[uXu2= 1sLe= 1s++KWith theK31 =k3 exk1 exAnd theduids= dds=du1ds= Bdu2ds= Bdds= BihHere weThe dim(+ 1) 10Tf2 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,iRDe,i; Bih =hRe; 2i =R2ki,fDe,i; i =EiRTf;( EiRTf); j =(Hj)De,jCf,jeTfavec j = 1,2;3)De,1Cf,1eTfese parameters in Eqs. (2a) and (2b) the followingess differential equations are obtained:s(su1s) a(s)211 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(ss)+ a(s)[121u1 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,2boundary conditions:0 at s = 0 (11a)im,1(1 + 1 sin(2c) u1,R)im,2(1 + 1 sin(2c) u2,R)(1 + 2 sin(2c) 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) 10901094Catalytic parameters are given by [8]:For the catalytic effectiveness:() = (+ 1)1 + 1a(s)u1(i)[exp(1(1 1))+K3And theS21() = 10a 10The meaobtained va = c0(There isthe mean vstate obtain3. NumeriWe needwhich max(13b), and sequations (The numcollocation mNC was takmore internu1=NC+1j=1[exu2=NC+j=1+ Le=NC+j=1++KAvec i=1, NEq. (11bthe orthogu2(NC+1) aNC+1j=1A(NC +ariatischemNC +NC +resoary cungevaluwn aoptimr (13turelocatizeshm [ultsperally 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 dsa(s){21u1 exp(1(1 1/)) + K31 exp(3(1 1/))}s ds(13b)n values are calculated to compare them with thelues in the steady state case.) d; S21 = c0S21() d (14)an enhancement in the catalytic performances if onlyalues calculated by Eq. (14) are greater than the steadyed values.cal procedureto determine the optimal catalyst distribution a(s),imizes the performance index given by Eq. (13a) oratises the constraint (12) and the diffusion-reaction(10a)(10c)).erical resolution technique is based on the orthogonalethod [9,10]. The number of internal collocation pointsen equal to 7. No more accuracy was observed whenal collocation points were used.B(i, j)u1(j) a(i)21u1(i)xp(1(1 1(i)))+ K31 exp(3(1 1(i)))](15a)1B(i, j)u2(j) a(j)[22u2(i) exp(2(1 1(i)))21u1(i) exp(1(1 1(i)))](15b)1B(i, j)(j) a(i)[121u1(i) exp(1(1 1(i)))222u2(i) exp(2(1 1(i)))31321u1(i) exp(3(1 1(i)))](15c)C) is used for calculating u1,R, u2,R and R noted inonal collocation method terminology as u1(NC+1),nd (NC+1) respectively.1, j)u1(j) = Bim(1 + 1 sin(2ttc) u1(NC + 1))(16a)Fig. 1. VreactionNC+1j=1A(NC+1j=1A(Theboundorder RTheunknoof the(13a) oquadrathe colmaximalgorit4. ResThepracticters: Tempthe e Optimis timsitesmustmal pthe pFig.ters acof theThe grbut whsoidal)ons of the optimal position of the active sites for a consecutive-parallele.1, j)u2(j) = Bim(1 + 1 sin(2ttc) u2(NC + 1))(16b)1, j)(j) = Bih(1 + 2 sin(2ttc) (NC + 1))(16c)lution of differential equations ((15a)(15c)), with theonditions ((16a)(16c)), is achieved with the fourthKutta method.es of the catalyst activity at the collocation points,t this stage, were regarded as the adjustable parametersization problem. The objective function is given by Eq.b), which can be evaluated by using the GaussJacobiformulae, where the quadrature points coincide withion points [9]. The search of the catalyst activity whichEq. (13a) or (13b) is operated with the Interior-point1,2].and discussionformance enhancement of such systems is achievedY. Benguerba, B. Djellouli / Chemical Engineering Journal 166 (2011) 10901094 1093Fig. 2. Catalytic parameter variations with dimensionless time.4.1. Effects of perturbation period on the catalytic parametersFig. 3 shtions gives aThis gives uthe perturbtime characmize the cavery large cthe processtry.4.2. EffectsparametersFigs. 4 anin the casean effect onmodulationbut the selebation is inFig. 3. EffeFig. 4. Effects of the concentration modulation on the catalytic parameters.5. Effects of the temperature modulation on the catalytic parameters.fects of the Lewis number on the catalytic parametersresults represented in Fig. 6 enabled us to determine thel values for the Lewis number (relationship between massows that the increase in the period of the perturba-n signicant improvement of the catalytic parameters.s an idea on the order of magnitude of the period ofations which must be very important compared withteristic of the system R2/De,1. Thus if one wants to opti-talytic output it is necessary to work with periods ofyclic perturbations compared to time characteristic ofbeing held in a catalytic pellet of unspecied geome-of the amplitudes of the perturbations on the catalyticd 5 show that only the modulation of the temperatureof a consecutive-parallel chemical reaction scheme hasthe catalytic performances. In the case of temperature, a good enhancement is obtained for the effectivenessctivity decreases slightly if the amplitude of the pertur-creased.Fig.4.3. EfTheoptimacts of the period of the perturbations on the catalytic parameters. Fig. 6. Effects of the Lewis number on the catalytic parameters.1094 Y. Benguerba, B. Djellouli / Chemical Engineering Journal 166 (2011) 10901094diffusivity and thermal diffusivity). These values are given in theinterval Le [1, 10]. This can be explained by the fact that in thisinterval the heat transfer and the mass diffusion rather favoredwith rather ideal proportions and are balanced what created thebest conditions for an optimal catalytic yield. For too large Lewisnumber, there is a very great mass diffusion with a weak thermaldiffusion processwhat created an imbalance between themass andthe quantity of energy available for the conversion of this mass.For weak Lewis number there is too much energy which diffusescompared to the mass available for conversion.5. ConclusionThis second part of our study concerning the case where thetime of the disturbances (period) is of the same order of magnitudeas 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 of Dirac. The creation of the nonstationary conditions must be accompa-nied by a control of the catalytic activity prole. The location ofthe optimal distribution of the active sites, given by the function of Dirac, for a given moment, is variable with time. 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[1] Y. Benguerba, B. Djellouli, L. Chibane, L. Bencheikh, Numerical analysis of theoptimal catalyst distribution in createdunsteady state conditions, J. Iran. Chem.Res. 2 (2009) 121131.[2] Y. Benguerba, B. Djellouli, Enhancement of the Catalytic Performances in theCase of a Consecutive-Parallel Reaction Scheme, vol. 8, A. A64, 2010, pp. 120.[3] P.L. Silveston, R.R. Hudgins, A. Renken, Periodic operation of catalyticreactorsintroduction and overview, Catal. Today 25 (1995) 91112.[4] V.V. Andreev, Conditions required to maximize the productivity of porous cat-alyst granules with a controlled activity prole, Mendeleev Commun. 2 (1998)4382.[5] V.V. Andreev, A mathematical treatment of the use of ultrasound in homoge-neous and heterogeneous catalysis, Ultrason. Sonochem. 6 (1999) 2124.[6] R. Aris, TheMathematical Theory of Diffusion and Reaction in Permeable Catal-ysis, vol. 1, Clarendon Press, Oxford, 1975.[7] J.B. Wang, A. Varma, On shape normalization for nonuniformly active catalystpellets, Chem. Eng. Sci. 35 (1980) 613617.[8] H. Wu, A. Brunovska, M. Morbidelli, A. Varma, Optimal catalyst activity inpellets. VIII. General nonisothermal reacting systems with arbitrary kinetics,Chem. Eng. Sci. 45 (1990) 18551862.[9] J. Villadsen, M.L. Michelsen, Solution of Differential Equation Models by Poly-nomial Approximation, Prentice-Hall, Englewood Cliffs, NJ, 1978.[10] B.A. Finlayson, Nonlinear Analysis in Chemical Engineering, McGraw-Hill, NewYork, 1980.Enhancement of the catalytic performances under non-steady state conditionsIntroductionMathematical formulation of the problemNumerical procedureResults and discussionEffects of perturbation period on the catalytic parametersEffects of the amplitudes of the perturbations on the catalytic parametersEffects of the Lewis number on the catalytic parametersConclusionReferences