# Some Dynamic Features of Fluidised Bed Reactors for Partial Oxidation of o-Xylene.

код для вставкиСкачатьDev.Chem. Eng. Mineral Process., 11(3/4), pp. 349-361, 2003. Some Dynamic Features of Fluidised Bed Reactors for Partial Oxidation of o-Xylene M.E.E. Abashar Department of Chemical Engineering, King Saud University, P. 0. Box 800, Riyadh 11421, Saudi Arabia A dynamic analysis offuidised bed reactors for the partial oxidation of o-qlene to phthalic anhydride is presented. The theory of bfurcation, continuation and Poincare‘ techniques play key roles in the analysis. A new type of period-adding bifurcation of Poincare‘ return points is observed and is shown to generate a complex 4namic behaviour. The period adding is classified as a second kind of period adding according to the classrfication of Holden and Fan [15J It is found that relaration oscillations develop through distortion of limit cycles, which have fast and slow motion in the different regions of the phase space. Introduction Chemical reactors are known to exhibit fascinating phenomena in their dynamic behaviour, including steady-state multiplicity, periodic oscillations, quasi-periodicity, strange chaotic and non-chaotic behaviour [ 1-31. Such phenomena are caused by the inherent non-linearities of the reacting system, the most common of which are the Arrhenius-type dependence of the reaction rate on temperature and also the nonmonotonic kinetics. The nonlinear phenomena can be either very useful or harmful to chemical processes. For example chaos is known to develop through sequences of bifurcations. This nonlinear phenomenon can be utilized in combustion applications to enhance the mixing of air and fuel and thus leading to improved performance. Chaotic regions can be used to offer greater flexibility in the operation of chemical systems due to an infinite number of unstable attractors embedded in a chaotic attractor. These unstable attractors can be stabilized, e.g. by the OGY (Ott, Grebogi and York) method as required [4]. Unsteady-state operations of chemical reactors are also known to improve performance with regard to the conversion, stability and parameters sensitivity [ 5 ] . However, chaotic regions can be very harmful to the stability and control of chemical processes. For example the “butterfly effect”, which can be produced by small unintentional disturbances, can render our long-term predictions of the performance of chemical processes invalid. * Author for correspondence (tnuhnslinrir: Xsii. edii su). 349 M.E.E. Abashar Many industrial gas-solid reactions or solid-catalysed gas-phase reactions are carried out in fluidised bed reactors. The proper design of these reactors has always been hampered by our inability to correctly incorporate the highly complex kinetics and poorly understood hydrodynamics into mathematical models. However, recently many researchers have shown that the implementation of the modem theory of bifurcation and chaos gives a new tool for understanding, scaling, modelling, design and operation of fluidised bed reactors [6, 71.Abashar and co-workers analysed the phenomena of strange chaotic and non-chaotic behaviour of fluidised bed catalytic reactors [8,9]. The conventional approach of Van Heerden was used by Elshishini et al. [lo] to analyse the steady-state multiplicity in fluidised bed reactors for partial oxidation of o-xylene. The present investigation is an extension of the work by Elshishini et al. [101. The major difference between the present study and the earlier ones is that our analysis is based on the theory of bifurcation, continuation and Poincar6 techniques. The recent concepts and modes of thought have greatly increased our understanding of these types of complicated systems as shown in this preliminary investigation. Kinetics and Rate Expression For the partial oxidation reaction of o-xylene, the following simplified rdaction scheme given by Froment [I 11 is used: The corresponding rate expressions can be written in the form [ 101: kgmol kgcat s RA = C, aA exp(--) Y A x I x 3 9 XI R, = C , a, exp(--)Y w x 2 x t 9 x3 R, = C , a, exp(--)Yc x , x , 9 XI kgmol kgcat s kgmol kgcat s (3) where C, is the reference concentration (kgmoVm3); a, is the pre-exponential factor for reaction i (m3/kgcat.s); yi is the dimensionless activation energy for reaction i; x I and x 2 are the dimensionless concentration of o-xylene and phthalic anhydride respectively; and x is the dimensionlesstemperature. 350 Fluidised Bed Reactorsfor Partial Oxidation of o-Xylene Model Development A fluidised bed reactor with a simple linear-proportional feedback control system is considered in this investigation. The mathematical model is developed based on the two-phase bubbling bed reactor model. The following simplifying assumptions are used in the derivation of the conservation equations of the model [lo]: 1. The emulsion phase is assumed to be perfectly mixed. 2. The gas in the bubble phase is assumed as plug flow. 3. The bubble size is taken as uniform throughout the reactor. 4. The bubble phase is solids free. 5. The reactor is adiabatic. 6. Due to the low concentration of o-xylene in the feed ( a1%) and the large excess of oxygen in the air, the partial pressure of oxygen is considered to be constant. The unsteady-state mass balance equations for the emulsion phase are thus given by: For o-xylene: For phthalic anhydride: The unsteady-state energy balance for the emulsion is given by: (6) F, 'T, +K(T, - T ) (7) 351 M.E.E. Abmhar The bubble-phase mass and energy balance equations are at pseudo steady-state because of their negligible mass and heat capacities and are given by: - Equations (8) (10) can be solved analytically and the resulting solutions used to evaluate analytically the integrals in Equations (4) - (6). Assuming (K,,), = (K,,)2 = H, and casting the equations in dimensionless forms gives: dx I d.t= G ( X , - X l ) - L dB dt O-=G((8 (1 1) -- @ + A i +acPcexp -k X , ( 6 ) b e , + K ( e , -e) , whereX and X are the dimensionless emulsion-phase concentrations of o-xylene and phthalic anhydride respectively, and 6 is the dimensionless ernulsion-phase temperature. The data used for the present system is given in Table 1. 352 Fluidised Bed Reactors for Partial Oxidation of o-Xylene Table 1. Data usedfor thejluidised-bed catalytic reactor [ I 01. Clf 2.031 I X lo4 kgmol/m3 aE CZf 0.00 kgmovm3 a, D 0.618 m YA H 3 m YE Tf 600 K u, 0.175 m/S - M A umf 0.079 m/S m B Yc 23.484 0.488294 aA I 11.832 X lo5 m3/kgcat.s I 8, I 0.929 Results and Discussion Figure 1(a) shows the dimensionless concentration of phthalic anhydride (X2) versus the dimensionless emulsion-phase temperature (0) for the uncontrolled adiabatic case (K=O). It is clear that the maximum concentration of phthalic anhydride corresponds to a middle unstable static steady state (saddle-type) as shown in Figure l(b). The feed and reactor temperatures at this optimal condition are 8 , =0.032806 (-253.47"C) and 8 = 0.929 (284.25"C) respectively. It is obvious that this feed temperature is of no practical use. A practical feed temperature of about 20°C (8 = 0.488294) which corresponds to 35.7% of the phthalic anhydride optimal value was selected for this study to investigate some dynamical features of this important industrial system. In order to operate the reactor at this unstable steady-state, a simple linear proportional feedback control system is used. The autonomous model of the fluidised bed reactor (Equations 1 1 - 13) includes a large number of parameters (G,K,XIF,X~F,a, ,aE PA PE PC 3 Y A Y E ~ Y ce1 8, , A ,0 ). It is a difficult task to investigate the entire static and dynamic behaviour of this system over seventeen dimensional parameter space. Instead the dynamic behaviour is investigated by considering two parameters of the control system (K, 0, ) with all other parameters kept constant. The richness and complexity of the behaviour associated with the one parameter investigation reported in this study strongly justifies this severe reduction in the parameter space dimensions. Figure 2(a) shows a bifbrcation diagram for the controlled fluidised bed reactor. The bifurcation diagram is obtained by using the software package AUTO86 of Doedel and Kerntvez [12]. The bifurcation diagram is characterized by a region of multiplicity of steady-state and three Hopf bifurcation points (HBI at K=6.070374, HB2 at K=6.622175, and HB3at Kg.541273). The Hopf bifurcation point connects a steady-state branch with a periodic branch (limit cycle); exchange of stability occurs 9 9 9 9 9 353 M.E.E. Abashar I.so I.20 0.80 0.40 2.00 0 - 1.oo (b) 0.50 0.00 -1.50 - ---- Stable Unstable /'Sl I I I I I I t I - \ 0.40 0.80 1.20 I.60 2.00 0 Figure 1. Uncontrolled adiabatic case (K=O): (a) dimensionless concentration of phthalic anhydride (Xd vs the dimensionless emulsion-phase temperature (0); (b) one-parameter bifurcation diagram of q v s 0for K=O. 354 Fluidised Bed Reactorsfor Partial Oxidation of o-Xylene 0.60 0.40 x" 0.20 0.00 160.0 120.0 -(ID U .-o 80.0 tl a 40.0 0.0 2.0 4.0 6.0 8.0 10.0 12.0 K Figure 2. Controlledfluidised bed reactor: (a) one-parameter bifurcation diagram of X , versus K for Of= 0.488294; (b) period of oscillation versus K. 355 M.E.E.Abashar as shown in Figure 2(a). At the Hopf bifurcation point a complex conjugate pair of eigenvalues crosses the imaginary axis transversally. The periodic branches originating from these Hopf bifurcation points terminate homoclinically, i.e. forming homoclinic orbits with infinite periods as shown in Figure 2(b). This global phenomenon requires a saddle type of unstable steady-state. The eigenvalues for the unstable saddle steady-state are pure and real of opposite sign (-,+,-).The negative values correspond to two stable manifolds that enter the saddle point and are called the separatrices; the positive eigenvalue corresponds to the unstable manifold. When the stable and unstable manifolds of the same saddle coincide (connecting the saddle to irself), we have the homoclinic orbit (saddle loop) that marks the end of the periodic branch. The homoclinic orbits strongly influence the nature of other trajectories passing near them. The region between HB2 and HB3 forms a region between two homoclinic orbits and contains three unstable attractors of steady states as shown in Figure 2(a). Even though such global homoclinic bihcations involve the saddle-type steady state, they cannot be detected through the local information (e.g. linearization) around this steady state. Due to this difficulty, and the lack of the starting point for the computation, it is not possible to use AUTO86 for this region. Therefore, we developed a Poincard bifurcation diagram to study this complex region. Figure 3 shows a one parameter Poincard bifurcation diagram for the region between the two-homoclinic orbits. This technique is based on the reduction of the dimension of the problem fiom three to two [8]. Instead of following the whole trajectory, discrete points are taken when the trajectory crosses a certain hyperplane of two dimensions transversally and in the same direction. These discrete points are called return points. The plotting of one of the co-ordinate of the return points (in this case X,) versus one of the free parameters (in this case K) is known as the Poincard bikcation diagram as shown in Figure 3. Figure 3 shows clearly that with increasing K the system bifurcates through period adding mechanism. A periodic branch with period 3 (P3) originates fiom the homoclinic orbit and goes through a sequence of period adding. The periodic sequence of periods are: P3,P4, P5, P6, P7etc, until accumulation point with very high periodicity after which the system bifurcates in a reverse way through period decreasing e.g. ...P16,PIS, Pi4,..., P4 up to the end of this region. The stability of the periodic solutions is determined by the eigenvalues of certain monodromy matrix called the characteristic or Floquet multipliers. The multiplier with largest absolute value is called the principle Floquet multiplier (PFM). When the PFM crosses a unit circle the periodic orbit loses stability and a dynamic bifurcation occurs. In fact, the appearance of P3 represents a type of “bench-mark” in this process because it is the first member of Sharkovsky’s T set, so that all types of periodic solutions are possible [ 131. The behaviour of the system in this region is similar to behaviour of the autonomous three-dimensional Rose-Hindmarsh model for neuronal activity which exhibits a new type of period adding bifurcation [14]. Holden and Fan in a series of papers [ 15- 181, investigated the complex dynamic behaviour of the Rose-Hindmarsh model of action potential. This model was proposed as a mathematical representation of the bursting behaviour of neurons and is expected to simulate the repetitive, patterned and irregular activity in molluscan neurons. Using this model, Holden and Fan discovered a new period-adding type of bifurcation on the Poincard bifurcation diagram. They classified the bifbrcation behaviour into three 356 Fluidised Bed Reactorsfor Partial Oxidation of o-Xylene kinds of period adding. It is clear that the period-adding bifimation presented in Figure 3 belongs to the second kind where the lines of periodicity on the Poincard bifurcation diagram are discontinuous at the period-adding point. It is interesting to observe that the sizes of the periodic regions (windows) are different and there is no clear relationship between their sizes. 0.36 0.34 0.32 0.30 Qf 0.28 6.0 = 0.488294 I I I 7.0 8.0 9.0 10.0 K Figure 3. Poincare' bifurcation diagram of X , versus K. Figures 4(a - d) show the time traces and phase planes at different values of K. It is clear that as K increases, the limit cycle is distorted (see Figures 4b - 4 4 and becomes different from the circular form (Figure 4a). The distorted limit cycle exhibits regions of fast and slow motion in the phase space and the initial smoothness diminishes gradually and gives large and sharp oscillations of a saw-tooth shape. This distortion of the limit cycle produces large amplitude oscillations called relaxation oscillations [ 131. Figure 5 shows various bifbrcation diagrams obtained by changing the feed temperature (0, ). The nature of these diagrams depends upon the value of 0, and whether it is smaller or larger than 0, =0.488294. It is clear that there exist complex regions of unstable multiple steady states. The problem in these regions becomes too stiff for the software AUTO86 to complete the periodic branches, as shown in Figures 5(b) and 5(c). Further investigation for these regions needs to be conducted. 357 M.E.E. Abashar " . ( I42 !' ' ' . ,' ' . ;- I 1147L1 l161lJ I -1.1 "'1"' I-8.18 1.01, 1 1.11 - 4 1.11 lMm.0 lOI10.0 1 0 ~ ~ 0 llEla.0 1124a.P 0.20 0.40 a.81 XZ I(4 1 Figure 4. Time traces andphase planes at different values of K (6.65, 6.9, 8.00, 9.5). 358 Fluidised Bed Reactorsfor Partial Oxidation of o-Xylene 0.4 s 0.2 0.0 2.0 4.0 8.0 8.0 10.0 120 1.0 K 0.40 i ef 0.6 0.00 0.0 4.0 8.0 K 120 8.0 Figure 5. Bijiurcation diagrams at different values of 0, (0.46, 0.SO, 0.64). Conclusions We have shown that for a specific parameter space the dynamic behaviour of the fluidised bed reactor for partial oxidation of o-xylene is quite complex. The periodadding phenomenon is observed. It may be too early to comment on the practical implications of this type of strange phenomenon on the design and control of such reactors. The results presented in this investigation suggest that such strange dynamic behaviour is to be avoided (or suppressed) when it is harmful to the system, and exploited to the maximum when it is beneficial. Further study is needed to characterize the period-adding bihcation &om a mathematical standpoint. 359 M.E.E. Abashar Nomenclature Cross-section areas of the bed occupied by bubble and dense phases respectively (m2) Concentrations of component i in bubble and emulsion phases respectively (kmovm3) Reference concentration (km0Vm3) Specific heat of gas and catalyst respectively (k.J/kg K) Bed diameter (m) Reciprocal of the modified residence time of the bed (s-’) Vertical coordinate measured from distributor plate (m) Bed height (in) Heat exchange coefficient between bubble and emulsion phases (s-’) Heat of reaction i (kJkm01) Dimensionless controller gain Mass exchange coefficient of component i (s-I) Bubble and emulsion phase flow rates respectively (m3/s) Rate of reaction i (km0Vkgcat.s) Time (s) Bubble and emulsion phase temperatures respectively (K) Desired temperature (set point) (K) Superficial and minimum fluidisation velocities respectively ( d s ) Dimensionless concentration of o-xylene and phthalic anhydride respectively Dimensionless temperature. Dimensionless emulsion phase concentrations of o-xylene and phthalic anhydride respectively Greek symbols Pre-exponential factor for reaction i (m3/kgcats) Dimensionless overall exothermicity factor for reaction i Dimensionless activation energy for reaction i Void fiaction in the bed at minimum fluidising conditions Dimensionless feed and emulsion phase temperatures respectively Set point for the controller Dimensionless parameter (= (1 -Ern/ ) p, / E,, ) Gas and solid densities respectively (kg/m3) Dimensionlesseffective mass capacity of the bed Fluidised Bed Reactorsfor Partial Oxidation of o-Xylene References I. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. Abashar, M.E. and Judd, M.R. 1998. Synchronizationof Chaotic Nonlinear Oscillators: Study of Two Coupled CSTRs. Chem. Eng. Sci., 53(21), 3741-3750. Ark, R. and Varma, A. 1980. The Mathematical Understanding of Chemical Engineering Systems, in Selected Papers of N.R. Amundson. Pergamon Press, Oxford. Planeaux, J.B. and Jensen, K.F. 1986. Bitkcation Phenomena in CSTR Dynamic: A System with Extraneous Thermal Capacitance. Chem. Eng. Sci., 41(6), 1497-1523. On,E., Grebogi, C. and York, J.A., 1990, Controlling Chaos. Phys. Rev. Letters, 64, 1196-1199. Bailey, J.E. 1973. Periodic Operation of Chemical Reactors. A review. Chem. Eng. Commun., I, I I I124. Daw, C.S. and Hallow, J.S. 1992. Modelling Deterministic Chaos in Gas Fluidized Beds. AIChE Symp. Series, 289 (88), 61-69. van den Bleek, C.M. and Schouten, J.C. 1993. Deterministic Chaos: A New Tool in Fluidized Bed Design and Operation. Chem. Eng. J., 53,75-87. Elnashaie, S.S.E.H., Abashar, M.E. and Teymour, F. 1995. Chaotic Behaviour of Fluidised Bed Catalytic Reactors with Consecutive Exothermic Chemical Reactions. Chem. Eng. Sci., 50(1), 49-67. Abashar, M.E., Elnashaie, S.S.E.H. and Hughes, R. 1997. Homoclinicity in the Dynamic Behaviour of Forced Fluidised Bed Catalytic Reactors. Chaos, Solitons and Fractals, 8(10), 1655-1684. Elshishini, S.S., Elnashaie, S.S.Elnashaie and EL-Rfaie, M.A. 1987. Multiplicity of the Steady State in Fluidised Bed Reactors-VIII. Partial Oxidation of 0-Xylene. Comput. Chem. Eng., I1(2), 95-100. Rase, H.F. 1977. Chemical Reactor Design for Process Plants, Vol. 2. pp.123-126. John Wiley & Sons, Inc., USA. Doedel, E. and Kemtvez, J.P. 1986. AUTO: Software for Continuation and Bifurcation Problems in Ordinary Differential Equations. California Institute of Technology, Pasedena, California, USA. Jackson, E.A. 1989. Perspective of Nonlinear Dynamics, Cambridge University Press, UK. Hindmarsh, J.L. and Rose, R.M. 1982. A Model of Neuronal Bursting Using Three Coupled First Order Differential Equations. Proc. Royal Society London, B221,87-102. Holden, A.V. and Fan, Y. 1992. From Simple to Simple bursting Oscillatory Behaviour Via Chaos in The Rose-Hindmarsh Model for Neuronal Activity. Chaos, Solitons and Fractals, 2(3), 221-236. Holden, A.V. and Fan, Y. 1992. From Simple to Complex Oscillatory Behaviour Via Intermittent Chaos in The Rose-Hindmarsh Model for Neuronal Activity. Chaos, Solitons and Fractals, 2(4), 349369. Holden, A.V. and Fan, Y. 1992. Crisis-Induced Chaos in The Rose-Hindmarsh Model for Neuronal Activity. Chaos, Solitons and Fractals, 2(6), 583-595. Fan, Y. and Holden, A.V. 1993. Bifurcation, Bursting, Chaos and Crises in The Rose-Hindmarsh Model for Neuronal Activity. Chaos, Solitons and Fractals, 4(4), 439-449. Received: 12 November 2001; Accepted afier revision: 1 April 2002. 361

1/--страниц