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Generation of chaotic oscillations in a system with flow reversal
Chemical Engineering Science 55 (2000) 339}343
Generation of chaotic oscillations in a system with #ow reversal Witold Z ukowski , Marek Berezowski * Cracow University of Technology, Institute of Inorganic Chemistry and Technology, 30-155 Cracow, ul. Warszawska 24, Poland Polish Academy of Sciences, Institute of Chemical Engineering, 44-100 Gliwice, ul. Baltycka 5, Poland Received 30 March 1999; accepted 1 April 1999
Abstract The paper is devoted to examining the e!ect of a reverse #ow on a cascade of two nonadiabatic continuously stirred tank reactors (CSTR) connected with stream #ow. A suitable numerical method was used to achieve data reduction, particularly in the region of transition between periodic oscillations and chaos. Computation results gave period doubling leading to chaos. Sensitivity to initial conditions has also been investigated. It has been shown that there is a dependence between oscillations of the cascade without reverse #ow and dynamic phenomena, which can occur with #ow reversal. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Reactor; Chaos; Reverse #ow
1. Introduction In recent years the reverse #ow process has been one of the commonly used systems in chemical reactor engineering. In particular, reverse #ow is often employed in heterogeneous catalytic reactors (Khinast & Luss, 1997; Khinast, Gurumoorthy & Luss, 1998; Rehacek, Kubicek & Marek, 1998). In these reactors a multiplicity of steady states is sometimes observed, some of which may be unstable, but at the same time favourable from the process point of view. Reverse #ow can be used for stabilising the operational parameters of a system (temperature and concentrations) around unstable steady state values. The paper presents a theoretical analysis of the dynamics of a simple model of a cascade made up of two CSTR with reverse #ow (Fig. 1). A substrate is continuously fed to the system, but inlets and outlets of the reactors are periodically exchanged. When the substrate is supplied to reactor A, the system product is obtained from reactor B and vice versa. Time intervals between #ow reversals are kept constant. Residence time of the reagents in the switching elements and lines connecting the reactors are neglected. A single chemical reaction APB of order n and heat exchange between reactors and the surroundings is as* Corresponding author. Fax: #48-32-231-03-18. E-mail addresses: [email protected] (W. Z ukowski), [email protected] (M. Berezowski)
sumed. This leads to the mathematical model presented below da #a !IO ) a " , dq
(1)
dH #H !IO ) H " #d (H !H ), & dq
(2)
da #a !(1!IO)a " , dq
(3)
dH #H !(1!IO)H " #d (H !H ), & dq
(4)
where a is the degree of substrate conversion, and H is the dimensionless temperature. Volumes of the reactors, the heat exchange coe$cients between the reactors and the surroundings are assumed equal for both reactors. The rate of reaction is given by the equation
bH G , where i"A, B.
"Da(1!a )L exp c (5) G G 1#bH G IO is the discontinuous rectangular wave-type function of unit amplitude. If q denotes the time interval between N #ow reversals then IO is de"ned by
q q !2 int , (6) q 2q N N where int(x) denotes the integral part of the variable x. IO"int
0009-2509/00/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 9 9 ) 0 0 3 2 9 - 2
340
W. ZQ ukowski, M. Berezowski / Chemical Engineering Science 55 (2000) 339}343
Fig. 1. Schematic representation of the cascade of CSTR reactors (simpli"ed). Fig. 2. Bifurcation diagram of the steady states (***) stable, (- - -) unstable.
The degree of conversion and the temperature of the product leaving system are described as follows: a "IOa #(1!IO)a , (7) H "IOH #(1!IO)H . (8) Fig. 2 shows the bifurcation diagram for the cascade of two CSTR without #ow reversal (IO"0, with q'0). The computations were made using the values of the parameters given in Section 3, below. The multiplicity of steady states, stable and unstable ones has a complex structure. This justi"es #ow reversal as a method of stabilisation.
2. The computational method When the aim is to de"ne the attractors associated with the systems dynamics, direct, numerical solution of the system equations can be used. Neglecting data for the initial period, a phase portrait can then be constructed. Stable sink, node, limit cycle and other forms of the attractors of system dynamics can be seen in the phase portrait, but parametric dependency analysis is di$cult. A large body of data must be collected, particularly near the transition from simple limit cycle to the chaotic dynamics. In the transition region small parameter steps must be used to obtain a satisfactory picture of the evolution of the system dynamics. It is therefore an advantage to use a method, which can reduce the number of dynamic data. In the present case, #ow reversal has been used with discrete sampling at regular time intervals q . N The system variables can thus obtain only at switch-over times. By using a discrete method we can avoid problems connected with arbitrarily assumed discretization when the numerical method of solving ODEs is used, since
unsuitable calculation steps can lead to errors in the image of the dynamics. Let q denote the time between #ow reversals and N x"[a , H , a , H ] the vector of state variables. The discrete dynamics equations can then be written as x "F(x , IO ), IO "1!IO L> L L L> L with (x , IO) "(c, 0) as the initial condition. (9) The function F results from the solution of equation set (1}4), with initial condition (x , IO ) in [0, q ] time L L N interval. The discrete sequence of x can converge to L a steady solution, give the limit cycle solution or be aperiodic. If the sets of points representing solutions are plotted on a plain, de"ned by the parameter axis and the state variable axis, then the Feigenbaum bifurcation diagram results (Fig. 3).
3. Results It should be noted that the model indicates that there is a 7-dimensional parametric plain, is de"ned by the axes of h , q , Da, n, d, b, c. The calculations were carried out & N for given parameters: h "0, Da"0.04776, n"1.5, & d"3, b"2.7, c"15. The dimensionless time interval between #ow reversals, denoted as q , was used as the N control parameter. It was found that changes in each parameter values can lead to the evolution in system dynamics from stationary states through the periodic solutions to chaos. To observe general dependence between the system dynamics and the values of the reverse#ow intervals the Feigenbaum diagram was investigated in the q range from 0 to 40 (Fig. 3). Solutions of N the Monodromy matrix given by mixed, analytical and
W. ZQ ukowski, M. Berezowski / Chemical Engineering Science 55 (2000) 339}343
Fig. 3. Feigenbaum diagram.
numerical method con"rmed the existence of the bifurcation, which can be seen in Fig. 3. The Monodromy matrix is given by the di!erential equation (Seydel, 1994) M "AM, M(0)"I
(10)
A is the Jacobian matrix of set (1)}(4), which is updated at every solution step of Eq. (10). If Eq. (10) is integrated in the range 0)q)2nq , then the eigenvalues of the matrix N M(2nq ) determine the nth multiple limit cycle, with basic N period of 2q . These solutions are also the eigenvalues of N Eq. (9). Fig. 3 shows chaos alternating with regions of periodicity, but with some regularity. There is a limit cycle, as a state of the system dynamics, for a small q (4.9, then N there is chaos in the range (4.9}8.4) with narrow windows of periodicity, in which period doubling cascades can be seen. The "rst chaos range di!ers from all the others. It has been veri"ed, with an accuracy of the numerical method used to solve the model, that the forms of the consecutive windows of chaos (accept "rst one) is identical. In these windows, patterns of strips of cascades of period doubling and alternating strips of chaos are obviously identical. It is necessary only to shift them on the axis of time between #ow reversals to achieve complete overlapping. The reproducibility of chaos windows is associated with two facts:
oscillation of the cascade of the two CSTR generates periodicity. This occurs in the time of #ow reversals, and can be seen in Feigenbaum diagram (Fig. 3). This phenomenon is characteristic for systems with external cyclic forcing. If oscillations for the system without reverse #ow had more complicated form, then chaos and periodic windows in the Feigenbaum diagram would also have more complicated patterns. Chaotic oscillations for cascade without #ow reversals would lead to chaotic pattern in the Feigenbaum diagram. Such relationships are not analyzed in detail and the problem remains open. Fig. 4 shows an enlarged portion of the Feigenbaum diagram. Period doubling and intermittency manifesting itself as a cloud of points lying in the lower part of the "gure should be noted. One of the methods of investigating the system dynamics is testing values of state variables at the instant of #ow reversals. In this way a PoincareH plot is constructed. Fig. 5 shows such a plot for q "16.8, the blurred cloud N of points refers to chaotic behaviour of the system. It is interesting to examine an enlargement of a part of the PoincareH plot (see Fig. 6). Layers can be seen, which attract points in the PoincareH diagram cross-section, through which the trajectory passes. It looks as if there was some stretching of the layers, with simultaneous folding, one layer upon another that could be described as a pastry e!ect. The dynamics of the system analysed can change over long periods of time. Figs. 5 and 6 show results obtained at simulation time 3;10 and with stabilization period equal to 4;10 (the results obtained for the stabilization time is discarded). But for stabilization time of 10 a different form of the PoincareH cross-section appears (Fig. 7). This shows that stabilization time, from the practical point of view, can be very long. Di!erent results obtained and shown in Figs. 5 and 7 mean that the dynamics of the
1. without #ow reversal the system has stable oscillations, 2. the time of stabilization of these oscillations is shorter than the interval between #ow reversal q . N Thus the interval between the windows of chaotic behaviour is equal to the period of the oscillations of the system without reverse-#ow. We can say, that stable
341
Fig. 4. Enlargement of the Feigenbaum diagram (see Fig. 3).
342
W. ZQ ukowski, M. Berezowski / Chemical Engineering Science 55 (2000) 339}343
Fig. 5. PoincareH cross-section for #ow reversal q "16.8; stabilization N time q "4;10. Q
system depends on the initial condition of the process, although in these cases the behaviour of the system is chaotic. In Fig. 8, the degree of conversion a as the time sequence, at moments of #ow reversal is presented. The sensitivity of the system to initial conditions are evident. This is characteristic of chaos. Fig. 9 shows two di!erent regions (limited by the parameters: Da and b) in which generation of periodic oscillations occurs in the system without reverse #ow. The bordering lines are the sets of points of Hopf bifurcation. With "nite time of #ow reversal, chaotic dynamics could appear in these regions. This type of behaviour could also emerge for parameter values outside the regions shown in Fig. 9. During our work, we found that the existence of the stable focuses is an essential condition for the generation of chaos in the system with #ow reversal. Fig. 9a represents the HB points for the values of parameters used in this work. When the value of c is increased to 40 and d is decreased to 1.5, the region limited by HB points shifts in the direction of signi"cantly lower values of the b parameter, which means the existence of chaos for low values of b. This is shown in Fig. 9b.
4. Conclusions
Fig. 6. A fragment of the PoincareH cross-section, from Fig. 5.
Reverse #ow leads to important complication of the dynamics of the cascade of two CSTRs. Parametric analysis of the system with #ow reversals shows recurrent regions of dynamic behaviour in which steady state evolves into chaos (Fig. 3). The course of the evolution depends on the complexity of the dynamics of the system without reverse #ow. The most complicated dynamics obtained is chaotic. From the practical point of view, this is undesirable but it
Fig. 7. PoincareH cross-section for #ow reversal q "16.8; stabilization N time q "4;10. Q
Fig. 8. Sampled time dependence of the degree of conversion. Sensitivity to initial conditions; q "16.8. N
W. ZQ ukowski, M. Berezowski / Chemical Engineering Science 55 (2000) 339}343
k k O n R r t ¹ < 0
343
Arrhenius constant (" k exp(!E/R¹)), heat exchange coe$cient, kJ/(msK) order of reaction gas constant, kJ/(kmol K) rate of reaction, ("kCL kmol/(ms) time, s temperature, K volume of the reactor, m
Greek letters a b d c H q q N q Q h ,r! &
dimensionless degree of conversion, ((C !C)/C ) dimensionless number related to adiabatic increase of temperature, ("(!*H)C /¹ oC ) N dimensionless heat exchange coe$cient, ("A K /oC Fb) O O N dimensionless number related to activation energy, ("E/(R¹ )) dimensionless temperature, ("(¹!¹ )/b¹ ) dimensionless time, ("F/< t) 0 dimensionless time between #ow reversals dimensionless stabilization time parameters
Subscripts Fig. 9. Regions of periodic oscillations in the model without #ow reversal: (a) c"15, d"3;(b) c "40, d "1.5.
can be used to isolate (using frequency "ltering) a speci"c orbit, which would give practical advantages. In chaotic behaviour provides an in"nite number of orbits. This type of approach has already been considered in the literature (Chien-Chong Chen, 1996).
Notation A O c N C Da E F *H IO
heat exchange area, m heat capacity, kJ/(kg K) concentration, kmol/m DamkoK hler number VR r /(F C ) activation energy, kJ/kmol volumetric #ow rate, m/s heat of reaction, kJ/kmol switching function
A, B H o out
refers to reactor A, B refers to heat exchanger feed refers to the outlet from the system with reverse #ow
References Chien-Chong Chen (1996). Stabilized chaotic dynamics of coupled nonisothermal CSTRs. Chemical Engineering Science, 51(23), 5159}5169. Khinast, J., & Luss, D. (1997). Mapping regions with di!erent bifurcation diagrams of reverse-#ow reactor. A.I.Ch.E. Journal, 43(8), 2034}2047. Khinast, J., Gurumoorthy, A., & Luss, D. (1998). Complex dynamic features of a cooled reverse-#ow reactor. A.I.Ch.E. Journal, 44(5), 1128}1140. Rehacek, J., Kubicek, M., & Marek, M. (1998). Periodic, quasiperiodic and chaotic spatiotemporal patterns in a tubular catalytic reactor with periodic #ow reversal. Computer Chemical Engineering, 22(1-2), 283}297. Seydel, R. (1994). Practical bifurcation and stability analysis. From equilibrium to chaos. New York: Springer.
Generation of chaotic oscillations in a system with #ow reversal Witold Z ukowski , Marek Berezowski * Cracow University of Technology, Institute of Inorganic Chemistry and Technology, 30-155 Cracow, ul. Warszawska 24, Poland Polish Academy of Sciences, Institute of Chemical Engineering, 44-100 Gliwice, ul. Baltycka 5, Poland Received 30 March 1999; accepted 1 April 1999
Abstract The paper is devoted to examining the e!ect of a reverse #ow on a cascade of two nonadiabatic continuously stirred tank reactors (CSTR) connected with stream #ow. A suitable numerical method was used to achieve data reduction, particularly in the region of transition between periodic oscillations and chaos. Computation results gave period doubling leading to chaos. Sensitivity to initial conditions has also been investigated. It has been shown that there is a dependence between oscillations of the cascade without reverse #ow and dynamic phenomena, which can occur with #ow reversal. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Reactor; Chaos; Reverse #ow
1. Introduction In recent years the reverse #ow process has been one of the commonly used systems in chemical reactor engineering. In particular, reverse #ow is often employed in heterogeneous catalytic reactors (Khinast & Luss, 1997; Khinast, Gurumoorthy & Luss, 1998; Rehacek, Kubicek & Marek, 1998). In these reactors a multiplicity of steady states is sometimes observed, some of which may be unstable, but at the same time favourable from the process point of view. Reverse #ow can be used for stabilising the operational parameters of a system (temperature and concentrations) around unstable steady state values. The paper presents a theoretical analysis of the dynamics of a simple model of a cascade made up of two CSTR with reverse #ow (Fig. 1). A substrate is continuously fed to the system, but inlets and outlets of the reactors are periodically exchanged. When the substrate is supplied to reactor A, the system product is obtained from reactor B and vice versa. Time intervals between #ow reversals are kept constant. Residence time of the reagents in the switching elements and lines connecting the reactors are neglected. A single chemical reaction APB of order n and heat exchange between reactors and the surroundings is as* Corresponding author. Fax: #48-32-231-03-18. E-mail addresses: [email protected] (W. Z ukowski), [email protected] (M. Berezowski)
sumed. This leads to the mathematical model presented below da #a !IO ) a " , dq
(1)
dH #H !IO ) H " #d (H !H ), & dq
(2)
da #a !(1!IO)a " , dq
(3)
dH #H !(1!IO)H " #d (H !H ), & dq
(4)
where a is the degree of substrate conversion, and H is the dimensionless temperature. Volumes of the reactors, the heat exchange coe$cients between the reactors and the surroundings are assumed equal for both reactors. The rate of reaction is given by the equation
bH G , where i"A, B.
"Da(1!a )L exp c (5) G G 1#bH G IO is the discontinuous rectangular wave-type function of unit amplitude. If q denotes the time interval between N #ow reversals then IO is de"ned by
q q !2 int , (6) q 2q N N where int(x) denotes the integral part of the variable x. IO"int
0009-2509/00/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 9 9 ) 0 0 3 2 9 - 2
340
W. ZQ ukowski, M. Berezowski / Chemical Engineering Science 55 (2000) 339}343
Fig. 1. Schematic representation of the cascade of CSTR reactors (simpli"ed). Fig. 2. Bifurcation diagram of the steady states (***) stable, (- - -) unstable.
The degree of conversion and the temperature of the product leaving system are described as follows: a "IOa #(1!IO)a , (7) H "IOH #(1!IO)H . (8) Fig. 2 shows the bifurcation diagram for the cascade of two CSTR without #ow reversal (IO"0, with q'0). The computations were made using the values of the parameters given in Section 3, below. The multiplicity of steady states, stable and unstable ones has a complex structure. This justi"es #ow reversal as a method of stabilisation.
2. The computational method When the aim is to de"ne the attractors associated with the systems dynamics, direct, numerical solution of the system equations can be used. Neglecting data for the initial period, a phase portrait can then be constructed. Stable sink, node, limit cycle and other forms of the attractors of system dynamics can be seen in the phase portrait, but parametric dependency analysis is di$cult. A large body of data must be collected, particularly near the transition from simple limit cycle to the chaotic dynamics. In the transition region small parameter steps must be used to obtain a satisfactory picture of the evolution of the system dynamics. It is therefore an advantage to use a method, which can reduce the number of dynamic data. In the present case, #ow reversal has been used with discrete sampling at regular time intervals q . N The system variables can thus obtain only at switch-over times. By using a discrete method we can avoid problems connected with arbitrarily assumed discretization when the numerical method of solving ODEs is used, since
unsuitable calculation steps can lead to errors in the image of the dynamics. Let q denote the time between #ow reversals and N x"[a , H , a , H ] the vector of state variables. The discrete dynamics equations can then be written as x "F(x , IO ), IO "1!IO L> L L L> L with (x , IO) "(c, 0) as the initial condition. (9) The function F results from the solution of equation set (1}4), with initial condition (x , IO ) in [0, q ] time L L N interval. The discrete sequence of x can converge to L a steady solution, give the limit cycle solution or be aperiodic. If the sets of points representing solutions are plotted on a plain, de"ned by the parameter axis and the state variable axis, then the Feigenbaum bifurcation diagram results (Fig. 3).
3. Results It should be noted that the model indicates that there is a 7-dimensional parametric plain, is de"ned by the axes of h , q , Da, n, d, b, c. The calculations were carried out & N for given parameters: h "0, Da"0.04776, n"1.5, & d"3, b"2.7, c"15. The dimensionless time interval between #ow reversals, denoted as q , was used as the N control parameter. It was found that changes in each parameter values can lead to the evolution in system dynamics from stationary states through the periodic solutions to chaos. To observe general dependence between the system dynamics and the values of the reverse#ow intervals the Feigenbaum diagram was investigated in the q range from 0 to 40 (Fig. 3). Solutions of N the Monodromy matrix given by mixed, analytical and
W. ZQ ukowski, M. Berezowski / Chemical Engineering Science 55 (2000) 339}343
Fig. 3. Feigenbaum diagram.
numerical method con"rmed the existence of the bifurcation, which can be seen in Fig. 3. The Monodromy matrix is given by the di!erential equation (Seydel, 1994) M "AM, M(0)"I
(10)
A is the Jacobian matrix of set (1)}(4), which is updated at every solution step of Eq. (10). If Eq. (10) is integrated in the range 0)q)2nq , then the eigenvalues of the matrix N M(2nq ) determine the nth multiple limit cycle, with basic N period of 2q . These solutions are also the eigenvalues of N Eq. (9). Fig. 3 shows chaos alternating with regions of periodicity, but with some regularity. There is a limit cycle, as a state of the system dynamics, for a small q (4.9, then N there is chaos in the range (4.9}8.4) with narrow windows of periodicity, in which period doubling cascades can be seen. The "rst chaos range di!ers from all the others. It has been veri"ed, with an accuracy of the numerical method used to solve the model, that the forms of the consecutive windows of chaos (accept "rst one) is identical. In these windows, patterns of strips of cascades of period doubling and alternating strips of chaos are obviously identical. It is necessary only to shift them on the axis of time between #ow reversals to achieve complete overlapping. The reproducibility of chaos windows is associated with two facts:
oscillation of the cascade of the two CSTR generates periodicity. This occurs in the time of #ow reversals, and can be seen in Feigenbaum diagram (Fig. 3). This phenomenon is characteristic for systems with external cyclic forcing. If oscillations for the system without reverse #ow had more complicated form, then chaos and periodic windows in the Feigenbaum diagram would also have more complicated patterns. Chaotic oscillations for cascade without #ow reversals would lead to chaotic pattern in the Feigenbaum diagram. Such relationships are not analyzed in detail and the problem remains open. Fig. 4 shows an enlarged portion of the Feigenbaum diagram. Period doubling and intermittency manifesting itself as a cloud of points lying in the lower part of the "gure should be noted. One of the methods of investigating the system dynamics is testing values of state variables at the instant of #ow reversals. In this way a PoincareH plot is constructed. Fig. 5 shows such a plot for q "16.8, the blurred cloud N of points refers to chaotic behaviour of the system. It is interesting to examine an enlargement of a part of the PoincareH plot (see Fig. 6). Layers can be seen, which attract points in the PoincareH diagram cross-section, through which the trajectory passes. It looks as if there was some stretching of the layers, with simultaneous folding, one layer upon another that could be described as a pastry e!ect. The dynamics of the system analysed can change over long periods of time. Figs. 5 and 6 show results obtained at simulation time 3;10 and with stabilization period equal to 4;10 (the results obtained for the stabilization time is discarded). But for stabilization time of 10 a different form of the PoincareH cross-section appears (Fig. 7). This shows that stabilization time, from the practical point of view, can be very long. Di!erent results obtained and shown in Figs. 5 and 7 mean that the dynamics of the
1. without #ow reversal the system has stable oscillations, 2. the time of stabilization of these oscillations is shorter than the interval between #ow reversal q . N Thus the interval between the windows of chaotic behaviour is equal to the period of the oscillations of the system without reverse-#ow. We can say, that stable
341
Fig. 4. Enlargement of the Feigenbaum diagram (see Fig. 3).
342
W. ZQ ukowski, M. Berezowski / Chemical Engineering Science 55 (2000) 339}343
Fig. 5. PoincareH cross-section for #ow reversal q "16.8; stabilization N time q "4;10. Q
system depends on the initial condition of the process, although in these cases the behaviour of the system is chaotic. In Fig. 8, the degree of conversion a as the time sequence, at moments of #ow reversal is presented. The sensitivity of the system to initial conditions are evident. This is characteristic of chaos. Fig. 9 shows two di!erent regions (limited by the parameters: Da and b) in which generation of periodic oscillations occurs in the system without reverse #ow. The bordering lines are the sets of points of Hopf bifurcation. With "nite time of #ow reversal, chaotic dynamics could appear in these regions. This type of behaviour could also emerge for parameter values outside the regions shown in Fig. 9. During our work, we found that the existence of the stable focuses is an essential condition for the generation of chaos in the system with #ow reversal. Fig. 9a represents the HB points for the values of parameters used in this work. When the value of c is increased to 40 and d is decreased to 1.5, the region limited by HB points shifts in the direction of signi"cantly lower values of the b parameter, which means the existence of chaos for low values of b. This is shown in Fig. 9b.
4. Conclusions
Fig. 6. A fragment of the PoincareH cross-section, from Fig. 5.
Reverse #ow leads to important complication of the dynamics of the cascade of two CSTRs. Parametric analysis of the system with #ow reversals shows recurrent regions of dynamic behaviour in which steady state evolves into chaos (Fig. 3). The course of the evolution depends on the complexity of the dynamics of the system without reverse #ow. The most complicated dynamics obtained is chaotic. From the practical point of view, this is undesirable but it
Fig. 7. PoincareH cross-section for #ow reversal q "16.8; stabilization N time q "4;10. Q
Fig. 8. Sampled time dependence of the degree of conversion. Sensitivity to initial conditions; q "16.8. N
W. ZQ ukowski, M. Berezowski / Chemical Engineering Science 55 (2000) 339}343
k k O n R r t ¹ < 0
343
Arrhenius constant (" k exp(!E/R¹)), heat exchange coe$cient, kJ/(msK) order of reaction gas constant, kJ/(kmol K) rate of reaction, ("kCL kmol/(ms) time, s temperature, K volume of the reactor, m
Greek letters a b d c H q q N q Q h ,r! &
dimensionless degree of conversion, ((C !C)/C ) dimensionless number related to adiabatic increase of temperature, ("(!*H)C /¹ oC ) N dimensionless heat exchange coe$cient, ("A K /oC Fb) O O N dimensionless number related to activation energy, ("E/(R¹ )) dimensionless temperature, ("(¹!¹ )/b¹ ) dimensionless time, ("F/< t) 0 dimensionless time between #ow reversals dimensionless stabilization time parameters
Subscripts Fig. 9. Regions of periodic oscillations in the model without #ow reversal: (a) c"15, d"3;(b) c "40, d "1.5.
can be used to isolate (using frequency "ltering) a speci"c orbit, which would give practical advantages. In chaotic behaviour provides an in"nite number of orbits. This type of approach has already been considered in the literature (Chien-Chong Chen, 1996).
Notation A O c N C Da E F *H IO
heat exchange area, m heat capacity, kJ/(kg K) concentration, kmol/m DamkoK hler number VR r /(F C ) activation energy, kJ/kmol volumetric #ow rate, m/s heat of reaction, kJ/kmol switching function
A, B H o out
refers to reactor A, B refers to heat exchanger feed refers to the outlet from the system with reverse #ow
References Chien-Chong Chen (1996). Stabilized chaotic dynamics of coupled nonisothermal CSTRs. Chemical Engineering Science, 51(23), 5159}5169. Khinast, J., & Luss, D. (1997). Mapping regions with di!erent bifurcation diagrams of reverse-#ow reactor. A.I.Ch.E. Journal, 43(8), 2034}2047. Khinast, J., Gurumoorthy, A., & Luss, D. (1998). Complex dynamic features of a cooled reverse-#ow reactor. A.I.Ch.E. Journal, 44(5), 1128}1140. Rehacek, J., Kubicek, M., & Marek, M. (1998). Periodic, quasiperiodic and chaotic spatiotemporal patterns in a tubular catalytic reactor with periodic #ow reversal. Computer Chemical Engineering, 22(1-2), 283}297. Seydel, R. (1994). Practical bifurcation and stability analysis. From equilibrium to chaos. New York: Springer.