HOMOCLINIC ORBITS IN A POROUS PELLET

Transcription

1 HOMOCLINIC ORBITS IN A POROUS PELLET Andrzej Burghrdt, Mrek Berezowski Institute of Chemicl Engineering, Polish Acdemy of Sciences, Gliwice, Polnd E-mil: mrek.berezowski@polsl.pl Abstrct The nlysis performed s well s extensive numericl simultions hve reveled the possibility of the genertion of homoclinic orbits s result of homoclinic bifurction in porous pellet. A method hs been proposed for the development of specil type of digrms - the soclled bifurction digrms. These digrms comprise the locus of homoclinic orbits together with the lines of limit points bounding the region of multiple stedy sttes s well s the locus of the points of Hopf bifurction. Thus, they define set of prmeters for which homoclinic bifurction cn tke plce. They lso mke it possible to determine conditions under which homoclinic orbits re generted. Two kinds of homoclinic orbits hve been observed, nmely semistble nd unstble orbits. It is found tht the chrcter of the homoclinic orbit depends on the stbility fetures of the limit cycle which is linked with the sddle point. Very interesting dynmic phenomen re ssocited with the two kinds of homoclinic orbits; these phenomen hve been illustrted in the solution digrms nd phse digrms. Keywords: Dynmic simultion, Nonliner dynmics, Periodic solutions, Stbility. 1. INTRODUCTION The coupling of physicl nd chemicl processes such s diffusionl nd convective mss nd het trnsport nd chemicl rections my led, in given system, to number of interesting dynmic phenomen which cn be rther difficult to predict without thorough mthemticl nlysis. These phenomen include the multiplicity of stedy sttes nd the ccompnying hysteresis, s well s the oscilltory destbiliztion of the system, which cn lso led to the ppernce of hysteresis. All these phenomen re of primry importnce in the design, strt-up nd control of chemicl rectors. If these phenomen re insufficiently described either qulittively (no informtion bout the possible existence of multiple stedy sttes) or quntittively (unknown mplitude nd frequency of self-excited oscilltions of the system), the rector my not operte t its optimum or cn even be severely dmged (Vleeshouver et l., 199, Morud et l., 1998, Mncusi et l., ). In heterogeneous ctlytic rectors the chemicl trnsformtion of mtter tkes plce in ctlyst pellet. Consequently, ll the processes nd phenomen ssocited with this trnsformtion re generted within the pellet. The fluid flowing through the rector supplies rectnts for the rections occurring in the pellet nd tkes wy the products formed therein. Obviously, the fluid ffects the processes within the pellet, but only in n indirect wy, through the pproprite boundry conditions t the fluid-pellet interfce. Thus, the process occurring in heterogeneous ctlytic rector is determined by the phenomen within the ctlyst pellet. A precise quntittive description of these phenomen is therefore of utmost importnce. The problem of defining the exct criteri determining the conditions of the occurrence of stedy-stte multiplicity in porous ctlyst pellets hs received gret del of ttention nd my

2 now be regrded s solved (Hu, Blkotih nd Luss, 1985, 1985b, 1986, Burghrdt nd Berezowski, 1989, 199, 199b). Consequently, the question rises bout the stbility of the sttionry sttes thus determined, especilly in the multiplicity region. Anlysis of the stbility of stedy sttes of ctlyst pellet ws performed by mens of the vrious pproximtion methods such s the lineriztion technique (Hlvcek et l., 1969, 1969b) or the verging technique (Luss nd Lee, 1971, Lee et l., 197 nd Ry nd Hstings, 198). Burghrdt nd Berezowski (1991) nlysed the stbility of sttionry solutions of the system considered with respect to sttic bifurctions, i.e., the nlysis concerned the cse of smll Lewis numbers (Le ). As result of prtil integrtion nd further trnsformtion, the system of two prtil differentil equtions hs been reduced to one ordinry differentil eqution. The stbility nlysis of this eqution led to the chrcteristics of the sttic bifurction by mens of developing nlyticl equtions for the locl nd globl stbility. A number of uthors (Sony, Schmidt nd Aris, 1991, Vn den Bosch nd Pdmnbhn, 1974, Subrmnin nd Blkotih, 1996) hve plced emphsis on deriving method whereby the prmeter region could be determined in which the bifurction to periodic solutions is possible. The most importnt result of the studies by Burghrdt nd Berezowski (1993) is set of nlyticl reltions between the prmeters of the system which define the limits of oscilltory instbility (Hopf bifurction) nd sddle type instbility s well s pproximte formule to determine the criticl vlues of the Lewis number (i.e. the vlue bove which n oscilltory destbiliztion of the system becomes possible). The possibility of determining the limits of oscilltory instbility without resorting to tedious nd time-consuming computtions is especilly useful in modelling processes tht occur in heterogeneous chemicl rectors. Of specil interest re the conclusions of the study by Burghrdt nd Berezowski (1997), where extensive numericl results hve reveled three scenrios for the genertion of self-excited oscilltions of the temperture nd composition inside the pellet. Besides the well-known supercriticl nd subcriticl Hopf bifurctions to oscilltory solutions the uthors lso observed homoclinic bifurction in the region of multiple stedy sttes. To the best of our knowledge the possibility of the homoclinic bifurction occurring in the dynmic processes tking plce in porous ctlyst pellet hs not so fr been reported in the existing literture. The genertion of homoclinic orbit proceeds ccording to the following scenrio. Chnging the bifurction prmeter, prt of n existing limit cycle moves closer nd closer to sddle point. At the homoclinic bifurction the limit cycle touches the sddle point nd becomes homoclinic orbit. According to Strogtz (1994) nd Wiggins (1988) this phenomenon represents nother type of the infinite-period or globl bifurction. The key feture of this bifurction is the fct tht the stble nd unstble mnifolds of the sddle point join the existing limit cycle, nd thus the homoclinic orbit connects the sddle point to itself, developing limit cycle of n infinite period (Guckenheimer nd Holmes, 1983, Thompson nd Stewrt, 1993, Wiggins, 1998). The im of the nlysis crried out in the present study is to determine conditions under which homoclinic bifurction cn develop, using the model of the processes occurring in the ctlyst pellet. The next step is to investigte the dynmics of the formtion of the homoclinic orbit. It hs to be borne in mind tht these gols cnnot be chieved bsed solely on the nlysis of the dynmics of the genertion of the homoclinic orbit following the chnge in selected bifurction prmeter of the model. Rther, this nlysis should be crried out in reltion with other bifurction phenomen, such s bifurction to multiple stedy stte solutions nd Hopf bifurction.

3 3. MODEL OF THE PROCESS AND A METHOD FOR DEVELOPING BIFURCATION DIAGRAMS AND PHASE DIAGRAMS The bsic equtions describing the non-sttionry process of mss nd het trnsport with simultneous chemicl rection in porous ctlyst pellet re s follows: C A 1 C A = D e ρ + ν Ar( C A,T) (1) t ρ ρ ρ T 1 T ( ρ c) k = λe ρ + ( H) r( C A,T) () t ρ ρ ρ The trnsport cross the boundry is determined by the following reltionships between the fluxes: C A t ρ = ρ k ( CA C A ) = De (3) ρ α ( T T ) form: T = λ e (4) ρ If we restrict ourselves to symmetricl profiles, the remining boundry conditions tke the C t ρ = A T = nd = (5) ρ ρ The trnsient eqs. (1) nd () s well s boundry conditions (3)-(5) cn be expressed in the following dimensionless form: 1 = τ x y Aφ x x y + ν x R ( y,z) (6) 1 z 1 z = x βφr( y,z) + Le' τ x x x subject to boundry conditions: y z t x = = nd = x x (7) (8) t x = 1 y = 1 1 Bi M y x (9) 1 z nd z = 1 (1) Bi H x The dimensionless vribles together with the dimensionless numbers re defined in Nottion. In our previous nlyses concerning the stbility of stedy-stte solutions (Burghrdt nd Berezowski, 1993, 1997) we employed model of porous ctlyst pellet which tkes into ccount the concentrtion grdients inside the pellet nd ssumes tht the pellet temperture is uniform but different from tht of the surrounding fluid. Thus, the resistnce to het trnsfer is concentrted in boundry lyer round the pellet, while the resistnce to mss trnsfer occurs only inside the pellet.

4 4 The ssumption concerning the mss-trnsfer resistnce Bi M trnsforms boundry condition (9) to the following form: t x = 1 y 1 (11) Before introducing the second ssumption (concerning the het trnsfer), we verge Eq. (7) over the volume of the pellet: 1 1 z 1 1 z 1 dv x dv R( y,z)dv Le' V = V + βφ x x x V (1) τ V V V Introducing the following definitions: 1 R( y,z) dv = R( y,z) V (13) V nd 1 V V z dv = τ τ 1 V V z zdv = τ nd expressing the vlue of dv V for the three pellet shpes (slb, cylinder, sphere) s dv = ( + 1) x dx (15) V we cn rewrite Eq. (1) in the following form: 1 z z z = ( + 1) R( y,z) Le' x x 1 x + βφ (16) τ = x= Using boundry conditions (8) nd (1) we finlly obtin n eqution defining the men temperture of the ctlyst pellet: 1 z = ( + 1) Bi H ( 1 z) x= 1 + βφr( y,z) (17) Le' τ Introducing the second ssumption, ccording to which the intrprticle temperture grdient is negligible in comprison with the temperture difference in the fluid film surrounding the pellet, we cn write z = z = z( τ) (18) x = 1 Finlly, the system of equtions describing the process ccording to the model ssumed tkes the form: y 1 y = x AφR( y,z) + ν (19) τ x x x 1 z = ( + 1) Bi ( 1 z) H + βφ R( y,z) () Le' τ subject to boundry conditions: t x = 1 y = 1 (1) y t x = = () x nd initil conditions: = x = y x (3) τ ( ) ( ) y τ = z = z (4) Introducing, s previously, the modified Thiele modulus vlid for power lw kinetics (14)

5 5 1 φ n + 1 θ = (5) + 1 nd trnsforming the energy blnce eqution (), the following set of equtions cn be obtined: y 1 y = x q(,n) θr( y,z) + (6) τ x x x 1 dz = 1 z + β θ R( y,z) (7) Le dτ subject to boundry conditions (1) nd () nd initil conditions (3) nd (4). The modified prmeters Le nd β re defined s follows: αl ( ) ( + 1) Le = Le'Bi H + 1 = (8) ( ρc) k De nd β( + 1) ( H) C ADe ( + 1) β = = (9) Bi H ( n + 1) αlt n + 1 where ( + 1) q(,n) = ν A (3) n + 1 Assuming the following reltion describing the rection rte 1 n R( y,z) = exp γ 1 y z (31) we replce z with new stte vrible θ = θ 1 exp γ 1 (3) z which is more convenient in further nlysis. For the pproprite representtion of the formtion of homoclinic orbits nd the region of their existence it is of crucil importnce to illustrte, in single bifurction digrm (Kubiĉek nd Mrek, 1983), the coexistence of three bsic bifurction phenomen occurring in the ctlyst pellet. In the present study these coexisting phenomen re defined s bifurction to multiple stedy stte solutions for the pellet (LP), Hopf bifurction (HB) nd homoclinic bifurction (Hcl). Also, solution digrms (Kubiĉek nd Mrek, 1983) re derived which show the brnches of both stedy nd unstedy sttes of the pellet, the points of the Hopf bifurction nd the brnches of periodic solutions long with their complete mplitudes. These digrms illustrte, for strictly defined set of the model prmeters, the process of generting the homoclinic bifurction. The detiled loction of the homoclinic orbit reltive to the points determining stedy sttes of the pellet nd the possible limit cycles is shown in the phse digrm in the system of coordintes which define stte vribles of the system nlysed s functions of time. The pproch used to derive these three types of digrms is s follows. Using the method of orthogonl colloction the model equtions for the pellet re discretized, thus leding to n ordinry differentil equtions with temporl derivtives. This system is then nlysed numericlly. Bsed on the one-prmeter continution procedure solution digrms of the stedy sttes re derived long with the points of the Hopf bifurction (HB) situted on their brnches. These digrms form the bsis for the construction of bifurction digrms. The bifurction digrms illustrte the regions of multiple stedy sttes bounded by the lines of sttic bifurctions (LP), s

6 6 well s the regions of oscilltions bounded by the lines of the Hopf bifurction (HB). The two sets of bounding lines re obtined employing the two-prmeter continution procedure. The clcultions were performed using the Auto 97 pckge (Doedel et l., 1997). In order to obtin full picture of the brnches of periodic solutions (i.e. mximums nd minimums of the mplitudes) necessry in further nlysis, the pckge hd to be modified (originlly, it generted only mximums of these solutions). The most complex tsk ws to determine numericlly the lines of homoclinic bifurctions (Hcl) in the bifurction digrm. Therefore, the following procedure ws dopted. The solution digrm ws determined in which the brnch of periodic solutions ws linked with the sddle point on n unstble brnch of stedy sttes thus forming the homoclinic orbit. Prmeters of single point of the homoclinic bifurction were thus obtined. Then, strting t this point of the homoclinic bifurction the line of the homoclinic bifurction ws determined (Hcl) in the bifurction digrm by using the two-prmeter continution procedure. Of gret help were suggestions mde by one of the uthors of Auto 97, Prof. Eusebius Doedel. In order to develop the homoclinic orbit in the phse digrm it ws necessry to nlyse the full dynmics of its genertion by mens of numericl integrtion of the system of nonliner equtions (6) nd (7) subject to boundry conditions (1) nd () nd different initil conditions (3) nd (4). The integrtion led to the temperture trnsients for the ctlyst pellet nd to the trnsient concentrtion profiles inside the pellet. The clssifiction of the dynmic behviour of given system my be best done using two-dimensionl phse digrms. Therefore the concentrtion profile hs been replced with the sptil integrl verge of this profile: 1 ( τ) = y( x, τ) x ( 1)dx η (33) + If, then, the vribles η i θ re used in the phse digrm, it is possible to crry out rigorous nlysis of the dynmic behviour of the system nd to determine the homoclinic orbit which connects the sddle point to itself, s well s the loction of other sttionry points nd possible limit cycles. The exceptionlly strong sensitivity of the loction of the homoclinic orbit in the phse digrm to the initil conditions used in numericl simultions led to serious problems. 3. RESULTS AND DISCUSSION The bifurction digrms discussed erlier were derived using two model prmeters s coordintes, nmely the prmeter γ nd the Thiele modulus θ, t constnt vlues of the prmeters β nd Le. In the solution digrms the dimensionless pellet temperture θ is selected s stte vrible, while the Thiele modulus θ is bifurction prmeter. The phse digrms, which illustrte the dynmic behviour of the process nlysed, re drwn in the system of coordintes η, θ representing stte vribles of the system. The nlysis ws performed for two different vlues of the Lewis number: Le = 1 nd Le = 5, t constnt vlue of β =. 5. In Tble 1 the vlues of the Lewis numbers re listed for the vrious operting conditions typicl of ctul processes in fixed-bed rectors. This tble fully confirms the prcticl significnce of the choice of the two Lewis numbers employed in our nlysis.

7 7 Tble 1. Lewis numbers for the vrious operting conditions P(MP) Re α ,576 11,453 [ ( m K) ] L = 1 (cm) Re α Le = αl ρc m W ( ) e D k D ,88 56,6 [ ( K) ] L =.5 (cm) Le = αl ρc W ( ) e e ( m s) D e ( m s) k D It is seen from Fig. 1, which is the bifurction digrm for Le = 1, tht below γ = only single stble stedy sttes exist for ny vlue of θ. Over the rnge γ = the loci of the Hopf bifurction determine such intervls of the Thiele modulus over which periodic solutions exist. These loci form the limits of the brnches of periodic solutions in the solution digrms, i.e. the points of the Hopf bifurction (HB). Fig.1. Bifurction digrm: =, n = 1, β =,5, Le = 1. LP locus of limit points, HB locus of Hopf bifurction points, Hcl locus of homoclinic bifurction points. An interesting feture from the stndpoint of the dynmics of the system studied is the moment t which the locus of the Hopf bifurctions begins to touch the lines limiting the region of multiple solutions (the so-clled limit points), nd then enters this region. It cn be seen from Fig. (which is n enlrgement of prt of Fig. 1) tht the line of the points of the Hopf bifurctions, once inside the region of multiple solutions, becomes utomticlly the locus of the homoclinic

8 8 bifurction. This is direct proof tht for the vlues of γ nd θ which lie on this line the limit cycle hs indeed joined the sddle point s result of the so-clled infinite-period bifurction nd the formtion of homoclinic orbit. Fig.. Bifurction digrm: enlrgement of prt of Fig. 1. For the vlues of γ = the brnch of periodic solutions extends from the lower line of the points of the Hopf bifurctions up to the line of homoclinic bifurctions inside the region of multiple solutions. To crry out detiled nlysis of the genertion of homoclinic bifurction nd the formtion of homoclinic orbit, solution digrms were prepred (Figs 3 nd 4). In Fig. 3 ( γ = 7.939) two points of the Hopf bifurction still exist which constitute the boundry of the brnches of periodic solutions; the region of multiple solutions is only being formed, s demonstrted by two brely visible limit points (LP). However, lredy for γ = the limit cycle clerly meets the sddle point of the unstble brnch of stedy sttes nd the homoclinic orbit is formed.

9 9 Fig.3. Solution digrm: =, n = 1, β =,5, γ = 7,939, Le = 1. stble stedy sttes, unstble stedy sttes, stble limit cycles, unstble limit cycles. Consequently, over the rnge γ = solution digrms exist of the form presented in Fig. 4.

10 1 Fig.4. Solution digrm: =, n = 1, β =,5, γ = 7,95, Le = 1. stble stedy sttes, unstble stedy sttes, stble limit cycles, unstble limit cycles. Prticulr chrcteristics of the homoclinic orbit hve now to be discussed. This orbit is formed s result of the stble limit cycle joining both stble nd unstble mnifolds of the sddle, nd surrounds n unstble sttionry point. Therefore, the homoclinic orbit hs to be internlly stble; otherwise, the re inside the orbit would not possess ny ttrctor, which is impossible on both mthemticl nd physicl grounds. The nture of the externl side of the homoclinic orbit ws nlysed using the phse digrm (Fig. 5).

11 11 Fig.5. Phse digrm: =, n = 1, β =,5, γ = 7,95, Le = 1, θ =, stble stedy stte, unstble stedy stte, trjectories. It is obvious from this digrm tht this orbit is externlly unstble. Consequently, the homoclinic orbit formed is semistble, i.e., it is stble on the inside nd unstble on the outside (Fig. 5). It cn be seen from the phse digrm tht the trjectories originting inside the homoclinic orbit, when moving closer to the orbit finlly meet the orbit nd thus tend towrds the sddle point (obviously, for n infinitely long period of time). This is quite surprising conclusion, s region hs thus been creted with specil property: trjectories originting from this region cn rech n unstble sddle point (or, strictly speking, cn pproch this point t n infinitely short distnce). This phenomenon would be bsolutely impossible without the existence of the bove mentioned homoclinic orbit, s otherwise the reching of the sddle would be fesible only long the stble mnifolds (which form the lines). The dynmic system formed hs therefore two ttrctors, one ttrctor proper (i.e. the stble sttionry point on the upper brnch of the solution digrm) nd the other ttrctor which is reched indirectly vi the homoclinic orbit (i.e. the sddle point). The ltter ttrctor cn be regrded s such only for trjectories originting from the inside of the region bounded by the homoclinic orbit. For the Thiele moduli θ = (Fig. 4) region exists chrcterized by two unstble stedy sttes ( sddle nd the lower stedy stte) nd one stble stedy stte (the upper stedy stte). This is n interesting exmple of the existence of region of multiple stedy sttes with only one stble stedy stte. It follows from the bifurction digrm (Fig. ) tht with further increse in γ bove γ = the loci of the Hopf bifurction enter the region of multiple solutions, crossing the line of the limit points. Strting from the point t which the line of the Hopf bifurctions crosses tht of the

12 1 homoclinic bifurctions inside the region of multiple solutions, bsic chnge in the shpe of the solution digrm cn be seen (Fig. 6). Fig.6. Solution digrm: =, n = 1, β =,5, γ = 8,, Le = 1. stble stedy sttes, unstble stedy sttes, unstble limit cycle. Due to the smll distnce between the point of the Hopf bifurction nd the line of sddle points, the brnch of periodic solutions is unble to develop fully nd is lcking prt of the brnch with stble limit cycles. Therefore, upon meeting the sddle point it forms n unstble homoclinic orbit illustrted in the phse digrm (Fig. 7).

13 13 Fig.7. Phse digrm: =, n = 1, β =,5, γ = 8,, Le = 1, θ =, stble stedy stte, unstble stedy stte, trjectories. Consequently, the dynmic system shown in this digrm hs two stble stedy sttes (on both the upper nd the lower brnch of the stedy sttes), long with sddle point. The unstble homoclinic orbit plys here the role of closed seprtrix which divides the whole region into two subregions with different ttrctors (inside nd outside the homoclinic orbit). A further increse in the Thiele modulus leds to the brek-up of the homoclinic orbit nd the formtion of n unstble limit cycle surrounding stble stedy stte (Fig. 6). This cycle grdully diminishes nd finlly meets the Hopf bifurction point ( θ =.5437 ), thus destbilizing this stedy stte. For θ > two unstble stedy sttes exist in the region of multiple solutions ( sddle nd the lower stedy stte) long with single stble stte (the upper stedy stte), similrly s in Fig. 4. With n increse in γ the Hopf bifurction point, together with the homoclinic bifurction point, moves towrds the limit point (Fig. 1). When the meeting with the limit point (LP) occurs (this tkes plce t γ = 8. 55), the Hopf bifurction point disppers due to the so-clled double-zero singulrity. For Le = 5 the system (whose dynmics is illustrted in Figs 8-15) becomes by fr more complex thn tht nlysed previously, nd revels some dditionl interesting bifurction phenomen. It follows from the nlysis of the bifurction digrms (Figs 8 nd 9) tht for γ less thn 7.95 region of single stedy sttes exists in which two lines of the Hopf bifurction points occur. These lines define the rnges of the Thiele moduli ssocited with the fesible periodic solutions, similrly s in the previous cse.

14 14 Fig.8. Bifurction digrm: =, n = 1, β =,5, Le = 5. LP locus of limit points, HB locus of Hopf bifurction points, Hcl locus of homoclinic bifurction points. Fig.9. Bifurction digrm: enlrgement of prt of Fig. 8.

15 15 For γ equl to bout 7.94 region of multiple stedy sttes is formed entirely surrounded by stble limit cycle. Such system is illustrted in the solution digrm (Fig. 1) vlid for γ = Fig.1. Solution digrm: =, n = 1, β =,5, γ = 7,95, Le = 5. stble stedy sttes, unstble stedy sttes, stble limit cycles, unstble limit cycles. With n increse in γ the Hopf bifurction point, locted on the upper brnch of stedy sttes, moves towrds the upper limit point, while the Hopf bifurction point locted on the lower brnch of stedy sttes moves towrds the lower limit point. An importnt moment is tht when the limit cycle meets the lower limit point (due to the bove-mentioned shift of the upper point of the Hopf bifurction), nd then when the Hopf bifurction point enters the region of multiple solutions (which occurs t γ 7. 99). At this point, s cn be seen from the bifurction digrm (Fig. 9), homoclinic bifurction line is initited which, with incresing γ, is contined within the region of multiple stedy sttes. The solution digrm typicl of the rnge of γ = is shown in Fig. 11.

16 16 Fig.11. Solution digrm:, n = 1, β =,5, γ = 8,5, Le = 5 unstble stedy sttes, stble limit cycles, unstble limit cycles. =. stble stedy sttes, A more detiled illustrtion of the phenomenon of homoclinic bifurction over the bove rnge of the prmeter γ is given in the phse digrm for θ = (Fig. 1). Fig.1. Phse digrm: =, n = 1, β =,5, γ = 8,5, Le = 5, θ =, stble stedy stte, unstble stedy stte, trjectories.

17 17 The nlysis of this digrm revels the existence of lrge stble limit cycle, inside which n unstble homoclinic orbit is locted tht surrounds stble sttionry point. Outside the homoclinic orbit (obviously, over the region of the lrge stble limit cycle) n unstble sttionry point is situted. The system discussed hs therefore two ttrctors: stble limit cycle nd stble sttionry point inside the homoclinic orbit. The homoclinic orbit thus divides the region inside the stble limit cycle into two subregions. Over certin distnce the stble limit cycle nd the unstble homoclinic orbit run so close to one nother tht, bsed on the phse digrm (Fig. 1), one might hve the impression tht they hve merged ltogether which, obviously, does not tke plce. Similrly, the visible trjectory clerly tends towrds the stble sttionry point; this is fully corroborted by numericl clcultions. Around the vlue of γ (cf. Fig. 9) the Hopf bifurction point moving long the lower brnch of stedy sttes enters the region of multiple solutions, thus leding to the formtion of second line of the points of homoclinic bifurction inside this region. Consequently, in the solution digrm (Fig. 13) two homoclinic orbits form for the given vlues of the Thiele modulus. Fig.13. Solution digrm: =, n = 1, β =,5, γ = 8,175, Le = 5. stble stedy sttes, unstble stedy sttes, unstble limit cycles. A further increse in γ (Fig. 9) leds to the merger of the Hopf bifurction point locted on the upper brnch of stedy sttes with the upper limit point ( γ 8.37), to the disppernce of this point of the Hopf bifurction nd, finlly, to the disppernce of periodic solutions. On the other hnd, the Hopf bifurction point moving long the lower brnch of stedy sttes towrds the lower limit point genertes brnches of periodic solutions of the form illustrted in Fig. 14.

18 18 Fig.14. Solution digrm: =, n = 1, β =,5, γ = 8,3, Le = 5. stble stedy sttes, unstble stedy sttes, unstble limit cycles. The phse digrm (Fig. 15) corresponding to this solution digrm shows the system dynmics for θ =.597, i.e., for such vlue of the Thiele modulus for which homoclinic bifurction tkes plce.

19 19 Fig.15. Phse digrm: =, n = 1, β =,5, γ = 8,3, Le = 5, θ =, 597. stble stedy stte, unstble stedy stte, trjectories. In this digrm, prt from two stble stedy sttes n unstble homoclinic orbit cn be seen. The trjectory shown in this digrm psses outside the unstble homoclinic orbit nd pproches this orbit so closely tht it seems to glide long the orbit; then, fter covering certin distnce, it leves the orbit nd tends towrds the stble sttionry point. It follows from the nlysis of the solution digrms (Figs 11, 13 nd 14) nd the corresponding phse digrms (Figs 1 nd 15) tht the homoclinic orbits for the system studied re unstble. This is due to the fct tht the brnches of periodic solutions, which correspond to the subcriticl Hopf bifurction include only prt of the brnch contining unstble limit cycles s, before the prt of the brnch with the stble limit cycles becomes fully developed, it meets the sddle point to form homoclinic orbit. For γ 8 the Hopf bifurction point locted on the lower brnch of stedy sttes joins the limit point nd vnishes ltogether. 4. CONCLUSIONS The nlysis performed nd the simultions crried out in the present study revel the possibility of the occurrence of the homoclinic bifurction which forms homoclinic orbit during the phenomen tking plce in ctlyst pellet. A method is developed which mkes it possible to determine, in the bifurction digrm, the loci of the homoclinic bifurction (Hcl), together with the lines of the limit points (LP) nd the loci of the Hopf bifurction. It is clerly seen tht the formtion of the homoclinic bifurction depends upon the coexistence of periodic solutions nd the region of multiple stedy sttes nd, in prticulr,

20 upon the existence of the unstble brnch of stedy sttes with ll the chrcteristics of sddle points in the neighbourhood of the brnch of periodic solutions. It is only under these conditions tht the limit cycle cn join the sddle point to form the homoclinic orbit. Furthermore, the bifurction digrms define such set of prmeters for which homoclinic bifurctions cn pper. It is found tht two types of homoclinic orbits cn form, nmely, n unstble orbit nd semistble orbit. The unstble homoclinic orbit ppers s result of the merger between n unstble limit cycle nd sddle point nd surrounds the stble sttionry point. This orbit plys therefore the role of closed seprtrix, dividing the region of stte vribles in the phse digrm into two subregions with two ttrctors. Two stble sttionry points or stble sttionry point inside the orbit nd stble limit cycle outside the homoclinic orbit cn be these ttrctors. On the other hnd, the merger of stble limit cycle nd sddle point leds to the formtion of semistble homoclinic orbit which is stble on the inside nd unstble on the outside. The internl stbility of the orbit is due to the fct tht the orbit surrounds n unstble sttionry point. The bsence of stbility of the internl side of the homoclinic orbit would men tht the region inside the orbit does not hve ny ttrctor, which is impossible form the stndpoint of the principles of dynmics. A further highly interesting dynmic phenomenon is ssocited with the existence of semistble homoclinic orbit. The internl stbility of this orbit leds to the fct tht the trjectories originting from the region surrounded by the semistble homoclinic orbit tend to this orbit nd, consequently, to the unstble sddle point (obviously, fter n infinitely long time). A region of stte vribles is thus creted with specil property: the trjectories issuing from this region cn rech the unstble sttionry point phenomenon bsolutely impossible without the existence of the homoclinic orbit. It hs lso to be stressed tht ll the points of the Hopf bifurction determined in the present study re the points of subcriticl bifurction. This leds to the possibility of the occurrence, under strictly defined conditions, of unstble homoclinic orbits. NOTATION geometric fctor defining the shpe of the pellet ( =, slb; = 1, cylinder; =, sphere) αl Bi H = λe Biot number for het trnsfer kl Bi M = D Biot number for mss trnsport A e C concentrtion of species A, D e effective diffusivity, m s 1 kmol m E 1 ctivtion energy, kj kmol HB Hopf bifurction point Hcl homoclinic bifurction point H 1 het of rection, kj kmol k 1 mss trnsfer coefficient, m s L chrcteristic dimension of the pellet, m αl( + 1) Le = ( ρ c ) modified Lewis number k e λ e Le' = ( ρc) k De Lewis number LP limit point 3

21 1 n rection order 3 1 r( CA,T) rection rte, kmol m s ra ( ) ( CA,T) R y,z = dimensionless rte of rection ra ( CA,T ) T temperture, K t time, s V 3 volume of the pellet, m ρ x = ρ dimensionless coordinte in the pellet C A y = C A dimensionless concentrtion T z = dimensionless temperture T GREEK LETTERS α het trnsfer coefficient, W m ( H) D C Tλ e β( + 1) β = dimensionless prmeter Bi H ( n + 1) E γ = dimensionless ctivtion energy R g T e A β = dimensionless Prter number 1 φ n + 1 θ = modified Thiele modulus θ λ e + 1 = θ 1 exp γ 1 z K 1 trnsformed dimensionless temperture effective conductivity, W m 1 K ν stoichiometric coefficient ( ν = ) ( ρ c) k het cpcity of ctlyst, Det τ = dimensionless time L L r( CA,T ) φ = Thiele modulus D C e A SUBSCRIPTS 1 A 1 3 kj m K refers to conditions in bulk phse z refers to surfce conditions 1

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