Mediating Operation of Catalytic CSTR : A Novel Possibility for

A novel approach for stabilizing intermediate steady state of CSTR is proposed by using a special type of periodic forced operation, socalled mediating operation. Mediating operation enables to produce new additional steady states for which one of new intermediate steady states is stable. Thus proposed approach employs a periodic forcing for stabilization of a selected steady state by control of a steady state multiplicity. Feed ow rate is considered as a manipulated variable. Changing of CSTR multiplicity is investigated for two{step control inputs. It is shown, analytically, that under two{step control CSTR can exhibit at most ve steady states having stable intermediate steady state. A constructive procedure is proposed for nding control parameters corresponded to maximal multiplicity, i.e. for stabilizing intermediate steady state by two{step control. INTRODUCTION A novel approach for achievement of stable intermediate steady state of a catalytic Continuous Stirred Tank Reactor (CSTR) is considered by using a special type of periodic forcing, so-called mediating operation. Mediating operation as a new periodic operation approach for catalytic CSTR with widely separated time scales has been proposed by Gol'dshtein et al. (1996). Mediating operation is introduced as a periodic intermediate mode operation with respect to di erent reactor time scales. The main goal of operating method is to control two system responses by manipulating a single control input. The study aims to demonstrate that mediating operation by feed ow rate inputs can produce additional steady states of CSTR for which one of new intermediate steady states is stable. Thus reactor behavior for intermediate temperature area can be stabilized. The proposed novel approach can be formulated as an application of mediating operation to control a steady state multiplicity for the achievement of a stable steady state in the selected area. The stabilization of chemical reactor operation can be viewed as a potential bene t for forced periodic operation approach. Unforced classical CSTR with single reaction may have single stable steady state as a minimum or three steady states as a maximum. For maximal three multiplicity the lower and upper steady states are stable and intermediate one is unstable (see, for example, Uppal et al., 1974; 1976). Traditionally, stabilizing reactor operation in intermediate domain is considered as moving stable steady state 1 (lower or upper or single) of periodically forced CSTR in an unstable operation domain of the unforced reactor. In this sense, vibrational control by feed ow rate manipulation can change multi{stability to a single stable steady state (Bellman et al., 1983) or can move an upper steady state in some intermediate operation domain (Cinar et al., 1987a,b). Mediating operation by feed ow rate manipulation can result also to a single stability or can move a stable lower steady state in the vicinity of unstable intermediate steady state of the unforced reactor (Gol'dshtein et al., 1996). A novel possibility for stabilizing intermediate steady state arises due to ability of mediating operation to control a steady state multiplicity. Gol'dshtein et al. (1990) have shown that two{step periodic forcing of intermediate mode may lead to ve steady states for which one of intermediate steady states is stable. Hence mediating operation enables to achieve a stable steady state for intermediate temperature area by creating additional multi{stability of CSTR. For this purpose two{step mediating operation is examined by using singular perturbation and averaging methods. Multiplicity features of averaged steady state system are studied by using catastrophe theory (see, for example, Br ocker and Lander, 1975; Poston and Stewart, 1978; Golubitsky and Schae er, 1985). In general chemically reacting systems may exhibit complicated steady state multiplicity. Steady state multiplicity and possible types of bifurcation diagrams for CSTR as a multi{reaction system have been researched in detail 2 by Balakotaiah and Luss (1982, 1983, 1984, 1988). Dynamic features of systems with widely separated time scales have been studied by Sheintuch and Luss (1987) for limiting case. Multiplicity problem of CSTR-s coupled in series has been considered by Dangenlmayr and Stewart (1985), Retzlo et al. (1992). This communication addresses a new multiplicityproblem of CSTR caused by periodic manipulation of external input as a feed ow rate. It is shown, analytically, that for two{step control multi{parameter CSTR problem is described by standard butter y singularity. As a result CSTR can exhibit at most ve steady states having stable intermediate steady state. The systematic numerical procedure is presented for nding amplitude parameters of two{step control which provide ve steady states and enable stabilizing intermediate steady state by such a way. In particular, shown, that only stroboscopical control can lead to desired stabilization. It is shown also that correlation between a coolant temperature and a feed ow temperature is essential for realization of maximal multiplicity. 1 REACTOR MODEL AND MEDIATING OPERATION Chemical reactor is considered as a two{phase well{stirred tank reactor with a gas mixture reacting on the surface of a solid catalyst. A thermal equilibrium is assumed between di erent phases in the reactor at all times. For a single rst{order reaction the heat and mass balance equations of the 3 gas{solid CSTR have the form (V cp +mcs)dT dt0 = cpF (T Tf) hA(T Tc) +V ( H)Ck0 exp( E RT ) V dC dt0 = F (C Cf) V Ck0 exp( E RT ) (1) For reducing to dimensionless form it would be desirable that F as a control variable should be a linear parameter of a dimensionless system as it was in the initial system (1). So following to Gol'dshtein et al. (1996) the system (1) has the following dimensionless form: d dt = u ( c) +B(1 )k( ) d dt = u + (1 )k( ) (2) The ratio of a gas heat capacity to the total reactor heat capacity, is considered as a small parameter, because the solid density is 103 times greater than the gas density. Thus system (2) is a singularly perturbed system with a small parameter 1. Further following asymptotic analysis scheme (see, for example, Gol'dshtein and Sobolev 1988,1992) zero{approximation, = 0, of system (2) is considered, d dt = u ( c) +B(1 )k( ) 0 = u + (1 )k( ) =) = k( ) k( )+u (3) Gol'dshtein et al. (1996) have shown that heat processes are approximately 1= times slower than concentration processes in gas{solid CSTR, conc: temp:. Mediating operation is introduced as an intermediate mode 4 operation with respect to slow and fast reactor processes, conc: temp:. Taking into account the considered type of operation, traditional averaging method (Bogoliubov and Mitropolsky, 1961) can be used with period of averaging de ned by control variable period, . Note that temperature as a slow variable can be regarded as a constant during the interval of averaging. The averaged equations of (3) become d dt = û ( c) +B(1 ̂)k( ) ̂= 1 R 0 k( ) k( )+u dt (4) Steady states of the averaged system (4) are de ned by the relation d dt = 0 which leads 0 = û ( c) +B(1 ̂)k( ) ̂= 1 R 0 k( ) k( )+u dt (5) 2 MAXIMAL MULTIPLICITY OF TWO{STEP MEDIATING OPERATION Introducing two{step periodic input of intermediate mode can lead to essential change of CSTR dynamic behavior. In Gol'dshtein et al. (1990) was demonstrated existence of ve steady states for two{step mediating operation. In this part of paper we aim to show analytically that multiplicity features of system (5) are described by so-called butter y singularity (see, for example, Poston and Stewart, 1978). Hence for two{step mediating operation CSTR may exhibit ve steady states as a maximum. Let us underline that 5 for maximal ( ve) multiplicity of butter y singularity at least one of three intermediate solutions is a stable one. Thus stabilization of an intermediate steady state is achieved by realization of maximal multiplicity. Bifurcation set of butter y singularity is a four{dimension space described by four bifurcation parameters. So, we also determine here four critical (bifurcation) parameters to which the reactor steady state system (5) characterized by a large number of parameters may be reduced. For investigating an in uence of two{step mediating operation on dynamic behavior of CSTR it would be desirable to separate two{step control parameters from other bifurcation parameters. In particular the separation enables to nd out two{step control values for which ve steady states with stable intermediate one can be achieved (see next part). Let us consider zero{average oscillations of feed ow rate, û= u0 in form of two{step periodic input (see Fig.1) ~ u(t) = ( u1 for n t (n+ ) u2 for (n+ ) < t (n+ 1) (6) where is duty fraction and n = 0, 1, 2, ... . Note that the selected forcing function gives the same average output as widely used nonsymmetric rectangular waveform (see, for example, Meerkov, 1980). Equations describing steady states of averaged system (5) under two{step periodic control are 0 = u0 ( c) +B(1 ̂)k( ) ̂= k( )(k( ) + u2 + u1 u0)=[(k( ) + u1)(k( ) + u2)] (7) 6 Equations of system (7) can be combined to give a single equation (u0 + ) c = Bk( ) k( )u0 + k( )u1u2 (k( ) + u1)(k( ) + u2) (8) For reducing (8) de ne new variable z, new reactor parameters z = 1+ c u0+ + c r = c u0+ + c q = B u0 u0+ + c s = 1 c u0+ + c = 1 (1 r) (9) and new forcing parameters n1 = u1=u0 n2 = u2=u0 (10) Denominator of equation (8) is positive, so the equation can be reduced [(1 + q)z sq]e2(z+r)+ u0[(n1 + n2 + n1n2q)z n1n2sq]ez+r + u20n1n2z = 0 (11) Note that feasible region of described above parameters can be determined as follows. Reactor parameters q, s, u0 must be positive, parameter r can be negative or positive but close to zero. Control parameters satisfy, for example, the condition, 0 n1 1 n2, because relative amplitudes of control pulses are symmetric variables for averaged reactor system (7). Presence of non{zero mediating operation means that in (11), u20n1n2 = u1u2 6= 0. Under the condition equation (11) can be rewritten F (z;A0; B0; A1; B1) def = A1(z A0)e2z + (B1z B0)ez + z = 0 (12) 7 where A0 = A0(s; q) = sq=(1 + q) A1 = A1(n1; n2; u0; q; r) = 1 n1n2 (1 + q) e2r u20 B0 = B0(u0; s; q; r) = sq er u0 B1 = B1(n1; n2; u0; q; r) = [( 1 n1 + 1 n2 ) + q] er u0 (13) The set of equations F = dF dz = d2F dz2 = d3F dz3 = d4F dz4 = 0 has a unique solution (for details see Appendix A) z = 3 A0 = 6 A1 = 1=e6 B0 = 12=e3 B1 = 4=e3 (14) Note that d5F dz5 6= 0. So following Poston and Stewart (1978) the point (14) is a single butter y point of butter y singularity. By Thom theorem (see, for example, Poston and Stewart, 1978) steady state equation of CSTR (12) de nes locally butter y singularity. Hence maximal multiplicity of CSTR under two{step mediating operation is ve (see, also, Appendix B). Furthermore multi{dimension space of CSTR parameters can be reduced to four{dimension space, (A0; B0; A1; B1) where bifurcation behavior is described by classical bifurcation diagram of butter y singularity (see, for example, Brocker and Lander, 1975). Below a situation of maximal ( ve) multiplicity having stable intermediate steady state is investigated, numerically, in detail. 3 MAPPING OF PARAMETER REGION PROVIDED FIVE STEADY STATES WITH STABLE INTERMEDIATE ONE 8 This part aims to illustrate a successive procedure for nding control parameters, u1, u2 which lead to an appearance of the stable intermediate steady state. For the purpose parameter area provided maximal multiplicity of ve steady states is searched in (A0; B0; A1; B1) space. A systematic procedure for dividing parameter space into regions with di erent number of solutions for standard polynomial singularities is described, for example, in Brocker and Lander (1975), Poston and Stewart (1978). We follow approach which has been developed by Balakotaiah and Luss (1984, 1988) for global parameter space of nonlinear CSTR problem. Firstly, in parametric space (A0, B0) we trace the locus of the swallowtail points de ned by F = dF dz = d2F dz2 = d3F dz3 = 0 (15) By elimination A1, B1 from (15) the next equations (parameterized by z) can be obtained A0 = 2(z2 6)=(2z 5) B0 = 4(3 3z + z2)=ez (16) Calculation shows that positiveness of denominator, 2z 5 > 0 is required for tracing feasible A0, B0 values. Figure 2 shows the graph of equations (16) which divides the (A0, B0) plane into two regions. For all A0, B0 values inside the cusp equation (12) has a maximum of ve solutions for some A1, B1. For all A0, B0 values outside the cusp equation (12) has at most three solutions for some A1, B1. 9 It should be underlined that only parameters, A1 and B1 depend on control parameters u1 and u2 (see (13)). So if reactor dimensionless parameters u0, , B, , c provide a location of A0, B0 values inside the cusp in Fig.2, then control parameters, u1 and u2 can be chosen by determining (A1, B1) set which leads to ve multiplicity having stable intermediate steady state. So secondly, for A0, B0 xed inside the cusp in Figure 2, A1, B1 values are searched for which equation (12) has ve solutions. For this purpose the locus of the fold points can be constructed in parametric space (A1(A0=B0)2, B1(A0=B0)). For any A0, B0 inside the cusp in Figure 2 this locus separates the (A1(A0=B0)2, B1(A0=B0)) plane into three regions having either 1, 3, 5 solutions. The locus of the fold points is de ned by F = dF dz = 0 (17) A solution of the system (17) gives the next equations parameterized by z (for xed A0 and B0 values) A1(A0=B0)2 = 1 n1n2 =(1 + q) = (A0=B0)2(z2 B0ez)=[e2z(A0 A0z + z2)] B1(A0=B0) = [( 1 n1 + 1 n2 ) + q]=(1 + q) = (A0=B0)[B0(1 A0 + z)ez A0 + 2A0z 2z2]=[ez(A0 A0z + z2)] (18) The calculation shows that feasible region of A1, B1 corresponds to z values belonging to an interval between roots of denominator of (18). Figures 3 show the locus of the fold points for various values of A0, B0 parameters. In Fig.3 10 a locality of the ve multiplicity region can be detached and corresponding control parameters, u1 and u2 can be chosen inside the region. Thus, if reactor dimensionless parameters u0, , B, , c de ne a point, (A0, B0) located inside cusp in Fig.2 then control parameters (10), n1 and n2 provided ve steady states can be determined by plotting the cusp of the fold points in (A1(A0=B0)2, B1(A0=B0)) plane (for example, Fig.3). So stabilization of intermediate steady state can be achieved by described above strategy. Figure 4 shows average steady state characteristics obtained by applying described above procedure. Positive slope of each curve corresponds to stable steady state, so curves 2, 3 demonstrate domain of stabilized operation in intermediate zone. The e ect is more pronounced for curve 2. It should be underlined that Fig.4 indicates higher reactor productivity that can be expected for intermediate steady state. Bene cial ability of mediating operation for increasing average conversion at a given temperature has been researched earlier by Gol'dshtein et al. (1996) for lower and upper steady states. So Fig.4 illustrates that this desirable e ect can be achieved, also, for a stable intermediate steady state. 4 INFLUENCE OF SOME PARAMETERS ON FEASIBILITY OF A FIVE MULTIPLICITY To determine global requirements to control parameters, n1, n2 for which ve solutions are possible the locus of the swallowtail points can be traced 11 in (n1, n2) plane. A simultaneous solution of equations (15) to n1n2, n1+n2 variables gives n1n2 = 1=[A1(A0=B0)2(1 + q)] = 4 1+q (2z 5)(z2 3z+3)2 (z2 6)2 n1 + n2 = [B1(A0=B0) q=(1 + q)]=[A1(A0=B0)2] = 4(z2 3 z+3) z2 6 [2(z 2) q 1+q (2z 5)(z2 3z+3) z2 6 ] (19) By xing q the locus of swallowtail points de ned by (19) can be plotted in (n1, n2) plane (see Fig.5). For all n1, n2 inside the cusp (of xed q) equation (12) may have ve solutions for some parameters r, s, u0 (not necessarily feasible). Figure 5 illustrates that non{zero control, solely, can lead to ve multiplicity, i.e. used above condition, u20n1n2 = u1u2 6= 0 is a necessary (see, also, Appendix B). Figure 5 indicates that ve multiplicity can be attained, practically, by control with relative amplitude of one pulse greater than ve, n1 or n2 > 5. The type of control was designated as stroboscopical control in Gol'dshtein et al. (1996). Thus for achievement of stable intermediate steady state by two{step input stroboscopical control can be viewed as a necessary control strategy. Let us estimate now in uence of reactor cooling (as internal parameter of CSTR system) on feasibility of maximal multiplicity. In Fig.6 the locuses of the swallowtail points de ned by (16) are shown in (A0, B0=er) plane for various xed values r. Five steady states are possible for a (A0, B0=er) area located inside each cusp in Fig.6. For xed r the area range depends on other (internal) reactor parameters, u0, q, s, essentially, on u0 (see, (13), 12 (9)). So Figure 6 indicates that intense cooling with negative r ( c < 0, i.e. Tc < Tf) enables a ve multiplicity for a more wide range of average feed ow rate values, u0 = 1=Da (i.e. for a more Damkohler number range too). The opposite relation, Tc > Tf decreases feasibility of ve multiplicity. CONCLUSION In this communication a novel approach for stabilizing CSTR behavior in intermediate temperature area has been proposed. The approach is based on introduction of mediating operation for creating additional steady states in intermediate temperature area. CSTR is considered as a simple model of gas{solid catalytic reactor with widely separated slow temperature processes and fast concentration ones. Mediating operation has been introduced as a periodic operation in intermediate mode between fast and slow reactor processes. We analyzed here the ability of mediating operation to control steady state multiplicity for achievement of an additional stable intermediate steady state. It was shown, analytically, that for two{step mediating operation CSTR multiplicity is described by a classical butter y singularity. Therefore CSTR multiplicity can exhibit ve steady states as a maximum for which at least one intermediate steady state is stable. A systematic numerical procedure for nding two control amplitude parameters corresponded to maximal multiplicity has been presented in the third part of the paper. The results of the fourth part have demonstrated that only stroboscopical control of two{step 13 input can lead to desired appearance of a stable intermediate steady state. Concerning internal reactor parameters it has been shown that correlation between a coolant temperature, Tc and a feed temperature, Tf is critical for realization of maximal multiplicity. For intense cooling, if Tc < Tf , ve multiplicity can be attained for a more wide range of average feed ow rate values, u0 (i.e. Damkohler numbers). The opposite relation, Tc > Tf leads to an opposite result which allows to avoid an unexpected appearance of additional steady states (if this situation is undesirable). Control of steady state multiplicity by periodic forced (external) inputs can be considered as a promising novel application for stabilizing desired steady states. Our study aims to demonstrate possibilities of a new mediating operation approach for the purpose. The ability to change in a wide range dynamic behavior of system with widely separated time scales using a single control variable can be viewed as a major advantage of mediating operation. ACKNOWLEDGMENT We are grateful to Prof. V. Balakotaiah for helpful comments. 14 NOTATION Parameters and variables A heat transfer area A0; A1 bifurcation parameters de ned by (13) B = ( H)Cf= cpTf (RTf=E) B0; B1 bifurcation parameters de ned by (13) cp; cs heat capacity of reacting mixture and solid phase, respectively C;Cf concentration, feed concentration Da Damkohler number = k0 exp( 1= )V=F0 = 1=u0 E activation energy F;F0 feed ow rate and its average value h heat transfer coe cient H heat of reaction k0 reaction rate constant k( ) = exp[ =(1 + )] m mass of solid catalyst n integer number n1 = u1=u0 de ned by (10) n2 = u2=u0 de ned by (10) q de ned by (9) r de ned by (9) R universal gas constant s de ned by (9) t dimensionless time = t0 k0 exp( 1= ) = t0 DaF=V t0 time T; Tf ; Tc reactor temperature, feed temperature, coolant temperature u dimensionless feed ow rate = F=V k0 exp( 1= ) u0 average value of dimensionless feed ow rate = F0=V k0 exp( 1= ) = 1=Da 15 u1; u2 amplitude parameters of two{step function,de ned under (6)V volume of gas reaction mixturez de ned by (9)Greek letters = hA=V cpk0 exp( 1= )= RTf=Esmall parameter= V cp=(V cp +mcs)dimensionless concentration or conversion= (Cf C)=Cfdimensionless reactor temperature and its average value= (T Tf)=Tf (RTf=E)c dimensionless coolant temperature= (Tc Tf)=Tf (RTf=E)duty fraction parameter of two{step function,de ned by (6), (0 < 1)density of the reacting mixturedimensionless period of control variable isequal to period of averagingconc:; temp: dimensionless characteristic response times of subsystemsfor concentration and heat processes, respectivelySuperscripts ̂ averaged value~u(t) two{step periodic function, de ned under (6)16 REFERENCESBalakotaiah, V. and D. 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Mathemat-ical Problems of Chemical Kinetics. Nauka, Novosibirsk. (In Russian).(1990).18 Gol'dshtein, V., V. Pan lov and I. Shreiber, Mediating operation of hetero-geneous CSTR. AIChE J., in press, (1996).Golubitsky, M. and D. G. Schae er, Singularities and Groups in BifurcationTheory, Volume 1. Springer-Verlag, New York, (1985).Meerkov, S. M., Principle of vibrational control: theory and applications.IEEE Trans. Automat. Control, AC{25, 755{762, (1980).Poston, T. and I. Stewart, Catastrophe Theory and its Applications. Pit-man, London, (1978).Retzlo , D. G., P. C.-H. Chan, C. Chicone and I. Parick, Maximal multi-plicity for sequential bifurcations of a rst-order reaction occurring inContinuous Stirred Tank Reactors coupled in series. SIAM J. Appl.Math., 52, 1136{1147, (1992).Sheintuch, M. and D. Luss, Dynamic features of two ordinary di erentialequations with widely separated time scales. Chem. Eng. Sci., 40,1653{1664, (1985).Uppal, A., W. H. Ray and A. B. Poore, On the dynamic behavior of Con-tinuous Stirred{Tank Reactors. Chem. Eng. Sci., 29, 967{985, (1974).Uppal, A., W. H. Ray and A. B. Poore, The classi cation of the dynamicbehavior of Continuous Stirred{Tank Reactors | in uence of reactorresidence time. Chem. Eng. Sci., 31, 205{214, (1976).19 Appendix ADetermination of the singular point de ned by equation (12)Equation (12) isF (z;A0; B0; A1; B1) def= A1(z A0)e2z + (B1z B0)ez + z = 0 (A1)The locus of the butter y points is de ned by set of equationsF = dFdz = d2Fdz2 = d3Fdz3 = d4Fdz4 = 0(A2)Solution of system (A2) has been obtained by Gol'dshtein et al. (1990). Forcompleteness we reproduce here solution calculation following procedure byBalakotaiah and Luss (1982).Let us de ne new variablesx0 = A1(z A0)e2z x1 =A1e2zy0 = (B1z B0)ez y1 =B1ez(A3)Using new variables (A3) system (A2) can be written as a linear system ofequationsx0 + y0 + z = 02x0 + x1 + y0 + y1 = 14x0 + 4x1 + y0 + 2y1 = 08x0 + 12x1 + y0 + 3y1 = 016x0 + 32x1 + y0 + 4y1 = 0(A4)Solution of system (A4) isz = 3x0 = 3x1 = 1y0 = 0y1 = 4(A5)Substituting (A5) in (A3) we can get (14).20 Appendix BMaximal multiplicity of equation (11)Equation (12) has been obtained from equation (11) under conditionu20n1n2 =u1u2 6= 0, it has been shown that equation (12) has unique ve multiple solu-tion (see (14)). We consider here equation (11) in a more common form forthe purpose to prove that condition, u1u2 6= 0 is a necessary for achievementof maximal ( ve) number of solutions. Existence of ve multiple solution forequation (11) has been proved, earlier, by Gol'dshtein et al. (1990).Let us consider equation (11) in a common form (see, also Gol'dshtein etal., 1990)G(z; a0; b0; c0; a1; b1; c1) def= (a1z a0)e2z + (b1z b0)ez + c1z c0 = 0 (B1)where ai, bi, ci (i = 0; 1) are parameters.If a1 6= 0 and c1 6= 0 than equation (B1) can have ve solutions as amaximum.Proof. Let us introduce new variable, y = ez > 0. Equation (B1) can bewrittenG(y; a0; b0; c0; a1; b1; c1) = (a1 ln ya0)y2 + (b1 ln y b0)y + c1 ln y c0 = 0(B2)and d3Gdy3 = 2a1 1y b1 1y2 + c1 1y3 = 1y3 (2a1y2 b1y + c1).If a1 6= 0 and c1 6= 0 equation d3Gdy3 = 0 has two solutions (not necessarilyfeasible), hence G = 0 has at most ve solutions in the global parameterspace. 221 In equation (11) a1 = (1 + q)e2r > 0 for all feasible values of parameterq > 0. Hence condition c1 =u20n1n2 = u1u2 6= 0 is a necessary for equation(11) to have maximal ( ve) multiplicity.Thus used above condition, u1u2 6= 0 meaning presence of non{zero con-trol, is a critical condition.22 List of captions to Figures.Fig.1. Two{step periodic forcing function.Fig.2. Locus of the swallowtail points in (A0, B0) plane.Fig.3. Locus of the fold points in(A1(A0=B0)2, B1(A0=B0)) plane forA0, B0 located inside cusp in Figure 2.Fig.4. Steady state characteristics vs. Damkohler number.Fig.5. In uence of control parameters on a feasibility of the ve mul-tiplicity.Fig.6. In uence of reactor cooling on a feasibility of the ve multi-plicity. ν τ τDIMENSIONLESS TIMEu1u0u2 DIMENSIONLESSFEEDFLOWRATE t~uFigure 1:Dashed line corresponds average value of feed ow rate, u0. 610141800.20.40.6 Figure 2:For A0, B0 inside cusp 5 steady states are possible, for A0, B0 outside cuspthe equation (12) may have no more than 3 solutions. 0.060.10.141.92.12.3 A = 8.00B = 0.550*1.92.12.3 A = 7.50B = 0.550*1.92.12.3 A = 7.00B = 0.550* Figure 3:The locus divides(A1(A0=B0)2, B1(A0=B0)) plane into regions having 1, 3,5 steady states. By is marked a region of ve steady states. Inside theregion control parameters, n1 and n2 can be chosen because for expressions,A1(A0=B0)2 = 1=n1n2(1 + q), B1(A0=B0) = [(1=n1 + 1=n2) + q]=(1 + q),parameter q is de ned by xing A0, B0 values. 0.02 0.04 0.06 0.08 0.1DAMKOHLER NUMBER, Da = 1/u000.20.40.60.81CONVERSION12 3024681012 DIMENSIONLESSTEMPERATURE123 Figure 4:Reactor parameters are: = 2, B = 12, = 0:04, c = 0. (1) = unforcedreactor. Control parameters provided stable intermediate steady state, n1 =u1=u0 and n2 = u2=u0 are: (2) = u1=u0 = 0:42, u2=u0 = 15:6; (3) = u1=u0 =0:385, u2=u0 = 12:8. 0.20.40.60.810510152025 123Figure 5:Loci of the swallowtail points in (n1, n2) plane for various reactor parameterq. For n2, n1 inside each cusp 5 steady states are possible. For each locusparameter q is : (1) = q = 0:3; (2) = q = 1; (3) = q = 2. 6 8 10 12 14 16 18 2000.20.40.60.8112 3Figure 6:Loci of the swallowtail points in (A0, B0=er) plane for various reactor pa-rameter r. Inside each cusp 5 steady states are possible. For each locusparameter r is : (1) = r = 0, (Tc = Tf); (2) = r = 0:5, (Tc < Tf); (3) =r = 0:5, (Tc > Tf).