Nonlinear analysis of stationary patterns in convection-reaction-diffusion systems

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<ul><li><p>PHYSICAL REVIEW E MARCH 2000VOLUME 61, NUMBER 3Nonlinear analysis of stationary patterns in convection-reaction-diffusion systems</p><p>Olga A. Nekhamkina,1 Alexander A. Nepomnyashchy2,3, Boris Y. Rubinstein1,3, and Moshe Sheintuch1,31Department of Chemical Engineering, TechnionIIT, Technion City, Haifa 32 000, Israel</p><p>2Department of Mathematics, TechnionIIT, Technion City, Haifa 32 000, Isreal3Minerva Center for Research in Nonlinear Phenomena, TechnionIIT, Technion City, Haifa 32 000, Israel</p><p>~Received 16 June 1999; revised manuscript received 11 November 1999!</p><p>Stationary spatially inhomogeneous patterns which appear due to the interaction of reaction and convectionin a packed-bed cross-flow reactor are observed and analyzed. A linear stability analysis was performed for thecase of unbounded system, and an analytical expression for the amplification threshold was determined. Abovethis threshold stationary patterns could be sustained in bounded systems. A weakly nonlinear analysis is usedin order to derive the governing amplitude equation. The results of linear and nonlinear analyses were verifiedby direct numerical simulations.</p><p>PACS number~s!: 82.40.Ck, 47.54.1ra</p><p>ma</p><p>onw</p><p>rnndnds</p><p>nti</p><p>tua</p><p>fingwadu</p><p>meif-talo</p><p>antaetembb</p><p>-</p><p>apso</p><p>a</p><p>ctsin</p><p>xo-ted</p><p>ingcentlyt</p><p>by</p><p>r-ly</p><p>s</p><p>alitysed</p><p>ionsrns</p><p>hedcingo-pat-ero</p><p>oftsgauoesho-butcon-</p><p> INTRODUCTION</p><p>In this work we describe the emergence of stationary pterns due to interaction of convection, conduction~or diffu-sion!, and reaction in a realistic model of a catalytic mebrane reactor. Aside from describing the phenomena weto derive a general complex Ginzburg-Landau~CGL!-likemodel. While such models that account for both diffusiand convection have been studied by several groups,highlight the importance of boundary conditions for patteselection, and point to the problem of translating real bouary conditions to those of the CGL model. It was demostrated in several studies that pattern selection in a bounregion is significantly affected by the boundary conditionDeissler@1# was probably the first to point out the importarole of boundary conditions, but he was interested mainlythe effect of boundaries as a source through which disbances are injected into the flow and propagate downstreIn subsequent works@24# the effect of various types oboundary conditions including reflective and absorbboundaries was investigated using the CGL equation. Itestablished that close to threshold the dynamics of a bounsystem is reasonably well described by the amplitude eqtion derived for the infinite system@5#.</p><p>Spatiotemporal patterns in reaction-diffusion systehave been studied extensively, and typically emerge duthe interaction of a short-range activator with a highly dfusing inhibitor. Oscillations in high- and low-pressure calytic systems are accounted for by a fast activator and a sbut localized~nondiffusing! inhibitor. Typically, the fast setin catalytic reactors accounts for reactant concentrationthe catalyst temperature, while the slow inhibitor is the calytic activity, as described above. The identity and the kinics of the latter are still debated. Patterns in such systmay emerge due to long-range interaction imposed by glocontrol or by gas-phase mixing and were recently studiedour group as well as other groups~see the review by Sheintuch and Schvartsman@6#!. When the activity is fixed thesystem decays into a steady state due to the large heat city of the catalyst. The model considered here assumefixed activity, and patterns emerge due to the interactionconvection and reaction.</p><p>Commercial catalytic reactors are typically organizedPRE 611063-651X/2000/61~3!/2436~9!/$15.00t-</p><p>-im</p><p>e</p><p>--ed.</p><p>nr-m.</p><p>seda-</p><p>sto</p><p>-w</p><p>d--s</p><p>aly</p><p>ac-af</p><p>s</p><p>packed beds with the flow of the reactants and produthrough the bed of catalytic pellets. Pattern formationconvection-conduction-reaction systems with a simple ethermic and activated catalytic reaction has recently attracconsiderable attention. Convection will affect the emergpatterns in oscillatory systems, as demonstrated in restudies @7#. A reactor with flow reversal was extensivestudied and installed commercially@8#. Patterns due to fronfeedback in a loop reactor were recently demonstratedGilles and co-workers in Refs.@9,10#.</p><p>We consider a membrane~or cross-flow! reactor with re-alistic parameters@high heat capacity and small axial dispesion ~diffusion!#: physically it requires a continuous suppof reactants along the bed, so that a homogeneous~space-independent! solution may be attained. A similar model waused in recent works by Yakhnin and co-workers@1113#devoted to differential flow induced chemical instability inone-dimensional tubular cross-flow reactor. Linear stabianalysis around a spatially homogeneous solution was uto obtain the dispersion relation, and some useful estimatof the possible wave numbers were obtained. The patteexcited above the convective instability threshold are wasout in bounded system, so the authors used temporal forat the reactor inlet in order to excite nonvanishing inhomgeneous structures. It should be underlined that theseterns are periodic in time, and characterized by a nonzamplitude growth rate.</p><p>In the model discussed below we consider two typesboundary conditionsperiodic, and realistic Danckwertype. For the former type we found well known travellinwave solutions described by a complex Ginzburg-Landequation. For the latter the homogeneous state typically dnot satisfy the boundary conditions, and the emerging inmogeneous structure is not the result of the bifurcationcan be viewed as the system response to the boundaryditions.</p><p>The structure of this work is as follows. The reactmodel in both dimensional and dimensionless forms is intduced in Sec. II. In Sec. III a neutral curve for an unboundsystem with a minimum corresponding to an oscillato~Hopf! bifurcation is obtained by means of linear analysThis minimum represents the convective instability threold. Periodic boundary conditions may lead to a shift of t2436 2000 The American Physical Society</p></li><li><p>e-a</p><p>manoiseecais</p><p>hedar</p><p>ninn</p><p>se</p><p>ethicfhafrais</p><p>e</p><p>mn</p><p>te</p><p>ed</p><p>tono</p><p>fa</p><p>rxcon</p><p>s</p><p>on</p><p>ti-i-ess</p><p>l-it</p><p>ater</p><p>dardam-ns-</p><p>sed</p><p>PRE 61 2437NONLINEAR ANALYSIS OF STATIONARY PATTERNS . . .critical wave number, but the type of bifurcation is prserved. It is shown that the phase velocity is not constalong the neutral curve; moreover, it may change its signsome point on the curve. This point corresponds to the aplification threshold, so that it appears to be very importfor further analysis of the system. In a semi-infinite regiona finite region, traveling waves decay if the instabilityconvective. Danckwerts boundary conditions produce a ctain stationary regime instead of traveling waves. Its propties are essentially different below and above the amplifition threshold. Only after the amplification thresholdpassed does one obtain new patterns.</p><p>The nonlinear analysis is performed in Sec. IV and tgoverning equation for a perturbation amplitude is derivFor periodic boundary conditions the result is quite stand@it leads to the complex Ginzburg-Landau equation~CGLE!#,and it is not presented. In the vicinity of the amplificatiothreshold, in contrast to the standard CGLE, the governamplitude equation lacks the diffusional term, while the covective term is present, and the latter have a complex~veloc-ity! coefficient. The cases where this equation is well poare found and discussed.</p><p>The stability analysis results are checked against dirnumerical simulations presented in Sec. V. They confirmexcitation of traveling waves in a system with periodboundary conditions, and the decay of these solutionsrealistic boundary conditions below the amplification thresold. Numerics also show that above this threshold stationstructures emerge, these patterns remain stable even farthreshold, while the profile changes significantly. Note ththe developed theory is valid if the critical wavelengthsignificantly smaller than the size of the system.</p><p>II. REACTOR MODEL</p><p>Consider the main physical phenomena that take placa one-dimensional membrane~or cross-flow! reactor inwhich a first-order exothermic reaction occurs. By mebrane reactors we imply a fixed bed, in which the reactaflow and react, while a small stream of highly concentrareactants is supplied continuously~by diffusion or flow!through the reactor wall. Dispersion of a reactant alongpacked-bed reactor, rather than supplying it with the femay be advantageous in several classes of reactions~see thediscussion in Ref.@14#!. A continuous supply of a reactanmay also be effectively achieved in a sequence of two csecutive reactions, when the first one is reacts at an almconstant rate. The mathematical model accounts for theand localized concentrationC and the slow and conductingtemperatureT, which can also be viewed as the activatoThe mass and energy balances are conventional ones efor the mass supply term; they account for accumulaticonvection, axial dispersion, chemical reactionr (C,T), heatloss due to cooling@ST5aT(T2Tw)#, and mass supplythrough a membrane wall@SC5aC(C2Cw)#. For the one-dimensional case the appropriate system of equations hafollowing forms:</p><p>]C</p><p>]t1u</p><p>]C</p><p>]z2eD f</p><p>]2C</p><p>]z252~12e!r ~C,T!1SC ,</p><p>~1!ntat-tr</p><p>r-r--</p><p>e.d</p><p>g-</p><p>d</p><p>cte</p><p>or-ryomt</p><p>in</p><p>-tsd</p><p>a,</p><p>-stst</p><p>.ept,</p><p>the</p><p>~rcp!e]T</p><p>]t1~rcp! fu</p><p>]T</p><p>]z2ke</p><p>]2T</p><p>]z2</p><p>5~2DH !~12e!r ~C,T!1ST .</p><p>Danckwerts boundary conditions are typically imposedthe model:</p><p>z50, eD f]C</p><p>]z5u~C2Cin!, ke</p><p>]T</p><p>]z5~rcp! fu~T2Tin!;</p><p>~2!</p><p>z5L,]T</p><p>]z5</p><p>]C</p><p>]z50.</p><p>Mass dispersion is typically negligible. A first-order acvated kineticsr 5Aexp(2E/RT)C is assumed. The approprate mathematical model may be written in the dimensionlform of Eqs.~1! and ~2! ~see Ref.@7# for a derivation!:</p><p>]x</p><p>]t1</p><p>]x</p><p>]j5~12x!G~y!2~x2xw!5 f ~x,y!,</p><p>Le]y</p><p>]t1</p><p>]y</p><p>]j2</p><p>1</p><p>Pe</p><p>]2y</p><p>]j25B~12x!G~y!2a~y2yw!5g~x,y!,</p><p>~3!</p><p>G~y!5Da expS gyg1yD .First Eqs.~3! will be treated in an unbounded region. Athough this solution can hardly be realized, technicallyprovides some insight into the bounded system case. Lwe supply it with Danckwerts boundary conditions</p><p>j50, x5xin ,]y</p><p>]j5Pe~y2yin!, j5jex ,</p><p>]y</p><p>]j50.</p><p>~4!</p><p>Here the dependent variables are normalized in a stanway, but independent variables and nondimensional pareters~Da, Pe! are normalized with respect to the mass trafer coefficientac :</p><p>x5Cin2C</p><p>Cin, y5g</p><p>T2TinTin</p><p>, j5aCz</p><p>L, t5</p><p>aCtu</p><p>L,</p><p>g5E</p><p>RTin, B5g</p><p>~2DH !Cin~rcp! fTin</p><p>, Le5~12e!~rcp!s</p><p>~rcp! f,</p><p>~5!</p><p>Pe5~rcp! fLu</p><p>keaC, Da5</p><p>AL</p><p>uaCe2g,</p><p>aC5kgPL</p><p>u, a5</p><p>hTPL</p><p>~rcp! fuaC.</p><p>In order to reduce the number of free parameters we uTw5Tin , i.e., yin5yw50; note thatxw,0 corresponds tothe caseCw.Cin . The length of the reactorjex was chosento resolve the structure of emerging patterns.</p></li><li><p>le-</p><p>d</p><p>-r</p><p>tonndei</p><p>t t</p><p>th-</p><p>aro-th</p><p>omlr-</p><p>a</p><p>or-</p><p>is</p><p>g toonc--</p><p>Foriti-nd,ve</p><p>inityan</p><p>tens</p><p>on-in-pli-</p><p>tiontion</p><p>2438 PRE 61OLGA A. NEKHAMKINA et al.III. LINEAR ANALYSIS</p><p>System~3! in an unbounded region may admit multiphomogeneous solutionsxs andys of the corresponding algebraic systemf (x,y)5g(x,y)50. The rootsxs and ys arelinearly dependent, and theys values may be determinefrom the equation</p><p>H~ys!G~ys!5a~ys2yw!, ~6!</p><p>where</p><p>H~ys!5B~12xw!2a~ys2yw!.</p><p>The left hand side of Eq.~6! is a sigmoidal curve, and upto three intersections with the straight line@the right handside of Eq.~6!# may exist. The multiplicity of the homogeneous solutions as well as the dynamic behavior of thelated system</p><p>]x</p><p>]t5 f ~x,y!,</p><p>]y</p><p>]t5g~x,y! ~7!</p><p>was extensively investigated in Refs.@15,16#, as it describesthe temporal dynamics of a continuously stirred tank reac~CSTR!. This problem can exhibit a plethora of phase plaregimes, including a simple limit cycle, a pair of stable aunstable limit cycles around a stable state, the coexistenca limit cycle with a low-conversion stable state, and a limcycle surrounding three unstable states. We expect thaspatiotemporal behavior of the distributed system~3! is re-lated to the corresponding CSTR problem. Obviously, inlimit of large Peclet numbers Pe`, the steady state solutions of Eq. ~3!, if they exist, are fully equivalent to thetemporal solution of Eq.~7! ~with j instead oft). However,a comprehensive investigation of the behavior of system~3!is beyond the limits of this study. We shall consider a pticular case when system~3! admits three homogeneous slutions; the results presented below are obtained only forupper one. Other situations are considered elsewhere@17#.</p><p>The stability of the solution (xs ,ys) can be determined bymeans of a linear stability analysis. Denoting deviations frthe basic solution as (x1 ,y1), and linearizing the originaproblem, we arrive at the following system for the distubances:</p><p>]x1]t</p><p>52]x1]j</p><p>211G~ys!x11G8~ys!H~ys!</p><p>By1 ,</p><p>~8!]y1]t</p><p>52Le21]y1]j</p><p>11</p><p>Le Pe</p><p>]2y1</p><p>]j2</p><p>2B</p><p>LeG~ys!x11G8~ys!</p><p>H~ys!</p><p>Ley12</p><p>a</p><p>Ley1 .</p><p>Assuming disturbances to be harmonic in the space vable j, i.e., (x1 ,y1);e</p><p>ikj1st, where k is the disturbancewave number ands is its growth rate, and using it in Eq.~9!,we arrive at the dispersion relationD(s,k)50. This hasform of a quadratic equation withcomplexcoefficients,</p><p>s21~Ar1 iAi !s1~Nr1 iNi !50, ~9!e-</p><p>re</p><p>ofthe</p><p>e</p><p>-</p><p>e</p><p>ri-</p><p>where the coefficients are determined by the following fmulas:</p><p>Ar511G~ys!1k21a Pe</p><p>Le Pe2G8~ys!</p><p>H~ys!</p><p>Le, Ai5k</p><p>11Le</p><p>Le,</p><p>Nr52G8~ys!H~ys!</p><p>Le1k2</p><p>11G~ys!2Pe</p><p>Le Pe1</p><p>a11G~ys!</p><p>Le,</p><p>~10!</p><p>Ni52kG8~ys!H~ys!</p><p>Le1k</p><p>11a1G~ys!</p><p>Le1</p><p>k3</p><p>Le Pe.</p><p>It can be easily shown that the bifurcation condition Res50 is satisfied if</p><p>Ar2Nr1ArAiNi5Ni</p><p>2 , ~11!</p><p>while the frequency of the emerging oscillatory patterngiven by</p><p>vc5Im s52Ni /Ar . ~12!</p><p>The latter should be calculated at the point correspondinthe minimum of the neutral curve determined by relati~11!. This curve can be found numerically, and typically aquires the form shown in Fig. 1~the chosen bifurcation parameter is Pe!.</p><p>The minimum of the curve (kc ,Pec) corresponds, in anunbounded system, to the excitation of moving waves.bounded system with periodic boundary conditions the crcal wave number value may be shifted. On the other hathis minimum coincides with the threshold of the convectiinstability; hence any spatial nonuniformities are advectedthe direction determined by the sign of the phase veloc(dv/dk)c , and in the finite region these disturbances csurvive only for periodic boundary conditions. In a finiregion with realistic boundary conditions these perturbatiowould inevitably decay.</p><p>Because our main goal is to study the influence of statiary nonhomogeneous boundary conditions, we will alsovestigate the transition between nontransparency and am</p><p>FIG. 1. The neutral curve determined by relation~11! calculatedwith B530, g520, a520, xw5211, yw50, and Le5333; thevalue of Da corresponds to the upper branch of the basic soluys517.54. The dot on the curve corresponds to the amplificathreshold.</p></li><li><p>thh</p><p>her</p><p>n-</p><p>aeth</p><p>tu</p><p>are</p><p>o</p><p>ntcan</p><p>rrthhiqu</p><p>ra</p><p>of</p><p>t of</p><p>rothing</p><p>ue,hund</p><p>us</p><p>art</p><p>nrte-</p><p>. Itva-</p><p>PRE 61 2439NONLINEAR ANALYSIS OF STATIONARY PATTERNS . . .fication for stationary boundary disturbances@18#. We haveshown that this transition is connected with a point onneutral curve corresponding to the zero wave velocity. Tlocation of this point is deter...</p></li></ul>