Complex nonlinear behaviour of a fixed bed reactor with reactant recycle

The fixed bed reactor with reactant recycle investigated in this paper can exhibit periodic solutions. These solutions bifurcate from the steady state in a Hopf bifurcation. The Hopf bifurcation encountered at the lowest value of the inlet concentration turns the steady state unstable and marks the emergence of a stable periodic solution. This periodic solution in turn undergoes a period doubling leaving it unstable and giving rise to a stable period 2 solution. It is know that if the system possesses one period doubling it often also has the possibility of posessing a chaotic attractor. It is shown, that the dynamic behaviour of a fixed bed reactor with reactant recycle is much more complex than previously reported.     

Some general observations on tubular reactor stability

A sufficient condition under which a tubular reactor with axial diffusion is globally stable is derived by use of the maximum principle for parabolic equations. It is shown that by use of some topological concepts the asymptotic stability of a steady state can be determined without the necessity of performing any computations. If a unique steady state exists it is asymptotically stable. If three steady states exist, the high and low temperature steady states are asymptotically stable, while the intermediate one is unstable.     

An Approach for Achieving Unstable Convergence for Non‐Isothermal CSTRs

The nonlinearity introduced by the temperature variation in the non‐isothermal continuous stirred tank reactor (CSTR) exhibits concentration multiplicity for certain parameter ranges. The dynamics of reactor temperature under these conditions would be very difficult to observe and maintain. In this work, a novel algorithm is proposed to stabilize the system by designing a cascade of CSTRs that are capable of achieving this inaccessible steady state. Optimization of reactors parameters is performed in an iterative manner to achieve this solution. For a first‐order reaction rate, this method was successful in achieving the inaccessible steady state temperature of 312.5 K using three CSTRs cascades. The suggested algorithm is presented both graphically as well as using computational optimization techniques. The transient simulation studies using the above three CSTRs showed that the unstable steady state is achieved. The newly designed cascade meets the design criteria and achieves the locally unstable steady state temperature to a high degree of accuracy.     

The effect of particle size distribution on the olefin polymer reactor dynamics

Mass and energy balances in a reactor have been derived to study the effect of particle size distribution (PSD) for each reaction mechanism on the reactor dynamics. It was observed that the PSD affects both bed height and particle volume. A feasible region for reactor operation has been calculated by using physical constraints. In a nonisothermal polymerization system, the reactor temperature does not change appreciably as catalyst injection rate increases. A unique steady state solution is found in a gas-phase continuous stirred-bed propylene polymerization reactor. The eigenvalues of the system of equations indicate that the steady state is unstable. A comparison with published data allows the observation that the actual reactor dynamics may be readily explained by using only the PSD derived from a simple reaction mechanism.     

Experimental study and simulation of the polymerization of methyl methacrylate at high temperature in a continuous reactor

The bulk polymerization of MMA at high temperature (120–180°C) in a continuous pilot‐plant reactor has been studied. The polymerization is initiated by diterbutyle peroxide and the chain transfer agent is 1‐butanethiol. A simulation program has been developed to predict the steady state behavior of the reactor. The particular features of the kinetic at above‐T_g temperature are included in the model, especially the thermal initiation of the reation and the attenuation of the autoacceleration effect. For the flow and mixing model, the actual vessel cannot be approximated to a single ideal reactor because of its design and of the moderate agitation imposed by the high viscosity of the reacting fluid. A tanks in series model with a recycle stream between tanks is proposed to evaluate the backmixing caused by the special design of the agitator. The parameters of the model are determined with the help of the experimental residence time distribution measured on the reactor. The data collected on the actual reactor, i.e., operation, conversion, molecular weight, temperature, are compared to the calculated one. The agreement is satisfactory but the tendencies are slightly underestimated. The program is a tool to evaluate the effect of modifications of the design of the reactor or changes on the operation parameters like input rate, temperature, and agitation on its behavior. © 2001 John Wiley & Sons, Inc. J Appl Polym Sci 79: 2038–2051, 2001     

气相法聚乙烯工艺冷凝态操作模式的稳定性和动态行为

气相法聚乙烯工艺冷凝态操作模式由于显著提高了循环气移热能力和反应器时空产率,已成为流化床乙烯聚合工艺的主流操作模式。建立了气相法聚乙烯工艺冷凝态操作模式的数学模型,包括流化床反应器模型,多级换热器模型和反应温度、压力以及循环气组成的控制模型。基于此,采用流程模拟方法,计算了系统在反应器温度采用闭环控制时的稳态解;根据系统对小扰动的动态响应特点,定性判断了反应器温度采用开环控制和闭环控制时聚合反应系统的稳定性;考察了系统对1-己烯分压和催化剂进料速率的阶跃响应特性。结果表明,反应器温度采用闭环控制时,聚合反应系统在所考察操作条件下均是稳定的,而采用开环控制时,解曲线被分叉点分割为稳定区域和不稳定区域。反应器温度对1-己烯分压阶跃变化的动态响应表明聚合反应系统存在长、短周期两类振荡,表明冷凝态操作模式下乙烯聚合反应过程是一个多控制回路耦合的复杂过程。     

Bifurcation Analysis of the Fluidized Bed Polyethylene Reactor with Internal Cooler

A mathematical model has been developed to simulate a gas‐phase ethylene polymerization reactor with internal cooler. The model was analyzed to determine the effects of reactor operating conditions on dynamics and stability. The reactor model employed assumed that both the gas and polymer phase in the reactor are well mixed. Comparing the present model to one with external heat exchanger confirms that, in either form, gas‐phase polyethylene reactors are prone to show unstable steady states, limit cycles and excursions toward unacceptably high temperature steady states. It was also observed that, with internal cooler, minor design changes in the cooler area available for heat transfer and in the inlet temperature of the coolant have a significant effect on the low stable steady state range of catalyst feed rates. With internal cooler, the suitable operating range increased with the increase in the area available for heat transfer. This effect is insignificant in the case of a reactor with external heat exchanger. Manipulating the reactor coolant inlet temperature and/or gas velocity can increase the stability range in the reactor with internal cooler as against one with external heat exchanger.     

Bulk copolymerization of styrene-acrylic esters: Some analysis and design considerations

A common framework for the analysis of styrene-acrylic ester systems has been developed by analyzing the pertinent kinetic information. This is shown to lead to a well defined strategy for the design of copolymerization reactors especially in the industrially relevant high conversion region. The existence of stable steady states and its influence on the system parameters has been illustrated for the case of a continuous stirred tank reactor (CSTR). A novel strategy of a CSTR operated with a recycle is proposed. This is shown to lead to an operation in the unique steady state with the added advantage of a high conversion and uniform copolymer composition. This would seem to be the first such analysis in the high conversion region.     

The stability of fixed bed catalytic reactors

A proper control design for safe and economic operation of industrial catalytic reactors requires a detailed study of transient and steady state behaviour. The present paper discusses contributions in this area with an emphasis on the stability problem. In addition, an attempt has been made to relate the stability of a fixed bed catalytic reactor to that of individual catalyst particles. Sufficient conditions for local stability of a two-dimensional and a one dimensional model for the reactor are obtained using the extension of Lyapunov stability theory to distributed parameter systems. For the two-dimensional model, the sufficient conditions are in the form of inequalities which require steady-state information. For the one dimensional model, the reactor is locally stable if and only if the slope of the heat removal line is greater than the slope of the heat generation curve at the steady state for small perturbations in temperature and concentration.     

Studies in the control of tubular reactors—III stabilization by observer design

The stabilization of an unstable nonlinear distributed chemical reactor system is examined when concentration measurements are not possible. The linearized form of the finite dimensional approximate model developed in Parts I[1] and II[2] is used to show that the observability index is equal to two. Furthermore it is shown that the dynamical characteristics of the reactor are such that, by a proper design of the Luenberger observer, the concentration estimation at a given collocation point can be made independently of the estimation at other collocation points.For purposes of control a one-dimensional observer is designed to directly estimate the control variable. Simulations of the nonlinear model of the reactor show that the observer design is quite successful in the stabilization of an unstable steady state when only temperature measurements along the reactor are available.     

An analysis of chemical reactor stability and control—XVI: The stochastic stirred pot

A continuous stirred tank reactor subject to random fluctuations in flow rate and inlet temperature was simulated on a hybrid computer. The stochastic response of the reactor about unique stable deterministic steady states was studied as a function of the damping coefficient and response time of the deterministic system and the power spectrum of the stochastic input. It was found that the stochastic response could be classified into categories similar to those used for forced periodic systems according to the relationship between the deterministic system response time and the 90% cut-off frequency of the stachastic input. The nature of the stochastic response is predictable for relatively low frequency inputs but unexpected results may occur at intermediate frequencies. The magnitude of reactor state fluctuations was seen to be dependent on the deterministic damping coefficient. The distribution of reactor states was studied as a function of input process variance and it was found that the distribution can become bimodal even when the associated deterministic steady state is unique.The concept of stochastic stability is discussed and several practical stochastic stability definitions are proposed. The stochastic stability of the random systems was seen to be well described by the stochastic regions of operation predicted by the input process power spectrum and the deterministic system response time. The input variance levels necessary to produce stochastic instability can be estimated in the Quasi Steady region of operation. It was found that exposure of an autonomous limit cycle about a unique unstable deterministic steady state to high frequency random inputs may lead to effective stabilization.     

Mathematical modelling of fluidized bed reactors—III: Axial dispersion model

An axial dispersion model has been developed for a continuous fluidized bed catalytic reactor with a cocurrent flow of the emulsion phase gas and the catalyst particles. The influence of some parameters on multiplicity of steady states has been reported. Several examples illustrating the transient behavior of the system are presented. In cases where three steady states are possible it appears that the intermediate steady state is unstable, while the lower and the upper steady states are locally stable. It was noted that the initial temperature of the emulsion phase is a predominant factor in determining which steady state will be approached.     

Absorption with exothermic reaction in a falling film column

A gaseous reactant is absorbed in a liquid with a positive heat of absorption. It reacts with the liquid by an exothermic, irreversible first order reaction. Analysis of a CSTR for this process has shown that up to 5 steady state solutions may exist even though the system appears to be quite simple with only one reaction. The existence of an extraordinary large number of steady state solutions is explained by the coupling between absorption and reaction: A low temperature gives no reaction, a high temperature prevents absorption of the gas and hence leads to mass transfer control of the reaction, while an intermediate temperature may permit operation at a third (stable) steady state. In between the stable steady states one finds (as us two unstable steady states.We extend the analysis of the CSTR to the distributed system of a falling film reactor, and we treat both the transient situation in the entrance partAn eigenanalysis of the steady states shows that temperature rise across the gas film may lead to 3 stable steady states even if an isothermal model is     

Design and Control of an Integrated 1,4‐Butanediol Dehydrogenation and Furfural Hydrogenation Plant

The design and plantwide control of an integrated plant, where the endothermic dehydrogenation of 1,4‐butanediol leading to γ‐butyrolactone and the exothermic hydrogenation of furfural leading to furfural alcohol and 2‐methyl furan are simultaneously performed in a single reactor, is presented. Analysis of the reactor‐separation‐recycle system shows that the reactions can be carried out using small hydrogen excess, in an adiabatic reactor, with reduced parameter sensitivity. The plant is flexible, allowing different production rates and a wide range of ratios between the furfural alcohol and 2‐methyl furan products. The additional separation of γ‐butyrolactone from the reaction mixture is easy. The conclusions are supported by rigorous steady‐state and dynamic simulations performed in AspenPlus and AspenDynamics.     

Liquefaction of coal by SRC-II process: Part V: Dynamic thermal behavior of the reactor

Criteria for identification of the unstable or stable nature of the steady state based on the reactor temperature response to step changes in feed temperature, and a dynamic thermal simulation of SRC-II reactors are developed. These are used in the analysis of a dynamic SRC-II reactor experiment to confirm its unstable operation under normal process conditions. The simulation is used further to study the sensitivity of reactor temperature to changes in feed temperature and to variations in the total heat capacity of the reactor vessel and insulation materials. It is shown that, under normal SRC-II process conditions, it would take about an hour for the reactor temperature to change by ± 10°C, if no controls were used. With quench gas manipulation, the reactor temperature could be maintained within ± 1°C even for a sustained small perturbation in the process conditions. Therefore, in spite of its unstable character, the SRC-II reactors can be readily operated with quench gas manipulation without incurring any major temperature control problems.     

A dynamically distributed reactor model for identifying the flow fields in industrial loop propylene polymerization reactors

The use of distributed parameter model is becoming a common approach for simulating liquid–solid flow in loop polymerization reactors. However, there are still several issues with it. One of them is the absence of modeling of distributed pressure, as no thermodynamic state‐equation is incorporated into the model. In this work, inner pressure of the reactor was associated with temperature using a thermodynamic state‐equation for high‐pressure liquid. The thermodynamic state‐equation was solved together with a dynamically distributed reactor model based on the mass, energy, and momentum conservation as well as polymerization kinetics to predict the dynamic trajectories of component concentration, temperature, pressure, and bulk mass velocity in the reactor. Industrial steady‐state data were used for model validation. The application of the model was demonstrated by simulating the effect of recycle ratio on the above distributed reactor parameters. © 2012 Wiley Periodicals, Inc. J. Appl. Polym. Sci., 2013     

Autothermal reactors with internal heat exchang—I Numerical and experimental studies in the multiple steady-state region

Detailed numerical and experimental studies in the multiple steady-state region of an autothermal reactor with internal countercurrent heat exchange were conducted. The mildly exothermic water-gas shift reaction was used in the studies. In this distributed-parameter system multiple steady states were found experimentally. Although significant radial and longitudinal temperature gradients exist, one-dimensional (radially-lumped) simulations predict mean behavior quite well. For two-dimensional simulations, Legendre polynomial trial functions give faster convergence than Jacobi polynomials when using orthogonal collocation methods. Inflections in the radial temperature gradients were measured near the hot spot although not predicted by the simulations. Significant temperature jumps at the inner wall upstream of the hot spot were also measured. Production of hydrogen could be conducted at surprisingly mild conditions if the reactor could be controlled in a stable mode essentially identical to the middle unstable steady state.     

Studies in the control of tubular reactors—II Stabilization by modal control

The lumped parameter model that was developed in Part I is linearized to obtain the linear dynamical model of the system near an unstable steady state. When concentration and temperature measurements are possible along the reactor length and their number is the same as the number of collocation points, modal state-feedback controllers are designed to relocate the largest eigenvalues to negative values and thus locally stabilize an unstable steady state. Transient calculations of the non-linear system equations are preformed and the domain of attraction of the stabilized steady state is examined for different locations of the eigenvalues of the closed loop system. It is seen that for both problems I and II the domain of attraction becomes very large when the unstable and one stable eigenvalue are shifted near the third largest one. This is true in spite of the large differences in their dynamical characteristics.     

Backstepping control of chemical tubular reactors

In this paper, a globally stabilizing boundary feedback control law for an arbitrarily fine discretization of a nonlinear PDE model of a chemical tubular reactor is presented. A model that assumes no radial velocity and concentration gradients in the reactor, the temperature gradient described by use of a proper value of the effective radial conductivity, a homogeneous reaction, the properties of the reaction mixture characterized by average values, the mechanism of axial mixing described by a single parameter model, and the kinetics of the first order is considered. Depending on the values of the nondimensional Peclet numbers, Damköhler number, the dimensionless adiabatic temperature rise, and the dimensionless activation energy, the coupled PDE equations for the temperature and concentration can have multiple equilibria that can be either stable or unstable. The objective is to stabilize an unstable steady state of the system using boundary control of temperature and concentration on the inlet side of the reactor. We discretize the original nonlinear PDE model in space using finite difference approximation and get a high order system of coupled nonlinear ODEs. Then, using backstepping design for parabolic PDEs we transform the original coupled system into two uncoupled target systems that are asymptotically stable in l²-norm with appropriate homogeneous boundary conditions. In the real system, the designed control laws would be implemented through small variations of the prescribed inlet temperature and prescribed inlet concentration. The control design is accompanied by a simulation study that shows the feedback control law designed with sensing only on a very coarse grid (using just a few measurements of the temperature and concentration fields) can successfully stabilize the actual system for a variety of different simulation settings (on a fine grid).     

Multiplicity of the steady state in fluidized bed reactors—VIII. Partial oxidation of o-xylene

A model is derived for the case of a fluidized bed reactor in which partial oxidation of o-xylene occurs. The use of fluidized beds instead of fixed bed reactors allows for higher feed concentrations and lower feed temperatures. However, multiple steady states arise and it is shown that the maximum yield is obtained when the reactor is operated at the unstable middle steady state. However, for maximum productivity, the reactor must be operated at a lower temperature than that corresponding to maximum yield. This is due to the pathological dependence of the middle steady state on feed temperature and concentration.