Design of tubular reactors in recycle systems

The article presents an approach to design tubular reactors in recycle systems, based on non-linear analysis. A pseudo-homogeneous plug-flow reactor model is used. It is assumed that the separation unit delivers product and recycle streams with fixed composition. The stand-alone reactor has a unique stable steady state. The coupled reactor–separation–recycle system shows four types of conversion versus plant Damköhler number bifurcation diagrams. A feasible steady state exists only if the reactor volume exceeds a critical value. For isothermal reactor, the steady state is unique and stable. For non-isothermal reactor, one or two steady states are possible. In the second situation the low-conversion state is unstable. In some parameter regions, the unique state is unstable. The design should ensure state unicity and stability, which are favoured by large heat-transfer capacity, low coolant temperature and high reactor-inlet temperature. A case study demonstrates that these phenomena can be easily found in real plants.     

实验室绝热反应器的开发和设计

在绝热反应器中进行工程动力学的研究是一种有益的方法。本文介绍了实验室绝热反应器的设计方法及其结构。实验测定了反应器的绝热效果，并计算了在操作条件改变后，反应器的非稳态传热情况。计算结果表明，对本实验装置操作条件获得重新稳定所需时间为０．８４７ｈ。     

State multiplicity in PFR-separator-recycle polymerization systems

This article explores the non-linear behaviour of isothermal and non-isothermal plug-flow reactor (PFR)-separator-recycle systems, with reference to radical polymerization. The steady-state behaviour of six reaction systems of increasing complexity, from one-reactant first-order reaction to chain-growth polymerization, is investigated. In PFR-separator-recycle systems feasible steady states exist only if the reactor volume exceeds a critical value. For one-reaction systems, one stable steady state is born at a transcritical bifurcation. In case of consecutive-reaction systems, including polymerization, a fold bifurcation can lead to two feasible steady states. The transcritical bifurcation is destroyed when two reactants are involved. In addition, the thermal effects also introduce state multiplicity. When multiple steady states exist, the instability of the low-conversion branch sets a lower limit on the conversion achievable at a stable operating point. A low-density polyethylene process is presented as a real plant example.The results obtained in this study are similar to CSTR-separator-recycle systems. This suggests that the behaviour is dictated by the chemical reaction and flowsheet structure, rather than by the reactor type.     

THE DYNAMIC BEHAVIOR AND CHAOS FOR TWO PARALLEL REACTIONS IN A CONTINUOUS STIRRED TANK REACTOR

The dynamical behavior for two parallel reactions in a continuous stirred tank reactor (CSTR) is considered. The global maximum number of steady states is five. Stability conditions are obtained. When the steady state is unique and unstable, a circulating attractor is proven to exist. The orbit structure is numerically found to consist of periodic orbits and chaotic behavior.     

Fate of commensalistic cultures in identical coupled bioreactors

A comprehensive analysis of static and dynamic behavior of a mixed culture in two identical coupled bioreactors is presented considering anaerobic digestion involving acidogens (X) and methanogens (Y) as the example bioprocess. A single continuous culture may operate at up to seven steady states, including up to four coexistence steady states, with only one coexistence steady state being locally stable. The one-way interaction between X and Y allows for compartmentalization of the system for a stand-alone bioreactor and two coupled bioreactors into two subsystems, which facilitates the analysis of steady state types and stability characteristics of these and classification of dynamic behavior. The bioreactors in the two-reactor system are identical only in terms of feed composition and reactor space time. A two-reactor system may admit up to forty nine steady states, which are comprised of up to forty coexistence steady states, at least at very low interaction rate (R). The static and dynamic analysis of the two-reactor system is facilitated by appropriate grouping of large number of steady states arising for very low R into nine clusters. Numerical illustrations reveal the rich steady state structure of the bioprocess in coupled bioreactors. While a single bioreactor can operate at only one locally stable coexistence steady state, the coupled bioreactors can operate at up to five locally stable coexistence steady states over certain ranges of R. The two-reactor system is operationally more flexible and more robust vis-a-vis single reactor as concerns maintenance of mixed culture. Emergence of four additional steady state clusters and additional coexistence and partial washout steady states at intermediate R reveals that the coupled bioreactors are an example of a complex system.     

A study on the states of the non isothermal tubular wall reactor

The thermal behaviour of a cooled tube wall reactor is analyzed.The main goal of this work is the characterization of the steady state. Three different steady state solutions, in which the control through the reactor differs from each other, are presented. The possibility of simultaneous existence of these states is analyzed as a function of the parameters of the model.For the zeroth order reaction here treated, it is possible to get isothermal operating conditions. Minimum thermal levels at the wall reactor for a given conversion are discussed and they result closely related to the isothermal conditions.An analytical solution is developed for the so called “highest steady state” (HISS). This expression makes possible to predict the “quenching” conditions of the reactor.     

Dynamics of a CSTR for styrene polymerization initiated by a binary initiator system

The steady state and dynamic behavior of a continuous stirred tank reactor has been analyzed for free radical solution polymerization of styrene initiated by a mixture of two initiators having different thermal stabilities. From the steady state analysis of the reactor model with a mean residence time as a bifurcation parameter, four unique regions of steady state solutions are identified in an operating parameter space for a given initiator feed composition. A variety of complex bifurcation behavior such as multiple steady states, Hopf bifurcation and limit cycles have been observed and their stability characteristics have been analyzed. The effects of feed initiator composition and the concentration of the initiator in the feed stream on the reactor dynamics are also presented.     

ON THE CONCEPT OF THE STEADY STATE IN CHEMICAL REACTOR ANALYSIS

The inlet and outlet streams of a chemical reactor may reach a steady state before the internal distributions of concentration, etc. have come to the steady state. Such a steady state, in which the reactor is viewed only as a system with certain inputs and outputs, may be called the black-box steady state (BBSS). The existence of a BBSS does not always imply that of a true SS, and the BBSS may be attained in a finite time even though the SS is only attained asymptotically. Indeed under some circumstances the BBSS can be achieved immediately upon start-up. Provided that the internal transients are not catastrophic, the practical, or operator's, concept of a steady state is really that of a BBSS.     

Multiplicity of steady states in fluidized bed reactors—V: The effect of catalyst decay

A simplified two-phase model is used to investigate the behaviour of non-isothermal fluidized bed reactors experiencing catalyst decay. The investigation shows that for highly exothermic reaction it is almost always desirable to operate the reactor at the middle unstable steady state, since it gives higher accumulative yield than both the high and the low temperature steady states. A simple feedback control scheme with time varying set point is suggested to stabilize the middle steady state. The dynamic behaviour and stability of the system is investigated for the open-loop reactor (uncontrolled) and the closed loop reactor (controlled).     

EFFECT OF REACTOR HEAT CAPACITY ON THE STABILITY AND START-UP TIME OF A DIABATIC CONTROLLED-CYCLED STIRRED TANK REACTOR

A simple model was written to simulate a first-order exothermic reaction taking place in a diabatic controlled-cycled stirred tank reactor (CCTR) in which the reactor jacket heat capacity is not infinitely larger than the reacting fluid heat capacity. The simulations have shown tpthat including the reactor heat capacity in the model tends to stabilize oscillatory states to a unique steady state. A further result is that a smaller reactor heat capacity results in a shorter transition time to the final steady state. The study shows the possibility that reactor stability and short start-up times may be achieved by simple changes in the reactor design rather than through complex control systems.     

Simultaneous process and control system design for grade transition in styrene polymerization

We present a comprehensive approach to the simultaneous design and control of a continuous stirred tank reactor (CSTR) for styrene solution polymerization that must be able to produce different polymer grades. The resulting tool allows simultaneous selection of the polymerization equipment, the multivariable feedforward-feedback controller's structure and tuning parameters, the steady states and the transition paths between them. For this purpose a multiobjective optimization is implemented to minimize the annualized reactor cost, the operating costs, the production of off-specification polymer and the transition time between steady states. Trade-offs between the sometimes conflicting objectives are dealt with by the optimization. Unlike many previous grade transition studies, steady states are not known a priori. The only known parameters are the target molecular weights to be produced at each steady state. We have analyzed three different scenarios, and propose practical criteria for selecting the most reasonable optimum when the solution is not unique.     

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.     

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.     

Study of Unsteady‐State Operation of Methanation by Modeling and Simulation

Methanation of CO under unsteady‐state operation conditions was studied systematically based on a simplified mathematical model for an integral reactor using steady‐state kinetics available in the literature. The inlet composition of CO and H₂ was changed stepwise and the step response of the system was monitored in order to study the dynamic behavior of the reactor. Furthermore, periodic changes were applied with different cycling times. It was observed that the time average reaction rate could not be improved by cycling the feed composition. Moreover, the reactor appears to be self‐stabilizing, since the amplitude at the outlet is reduced, leading to a steady state for high cycling frequencies. The results allow conclusions on principles to design a methanation reactor for unsteady‐state operation. However, it also becomes obvious that unsteady‐state kinetics is mandatory in order to describe the experimental results obtained under dynamic conditions.     

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。     

Branching by copolymerization of monovinyl and divinyl monomers in continuous-flow stirred reactors

The effect on molecular weight distribution of copolymerization with divinyl monomers, which leads to branch points in an otherwise linear polymer, is calculated for a continuous-flow stirred reactor. It is found that steady state operation of the reactor becomes impossible at a degree of branching equal to only half that which causes gellation in a batch reactor. In steady state operation the ratio of weight to number average degree of polymerization cannot exceed 7, this ratio being 2 in the absence of branching. The classical result that the fraction of mers coupled at the gel point is 1/(D?P_w)_o, where (D?P_w)_o is the weight-average degree of polymerization without branching, does not apply in a CFSR at any steady state conversion. Such deviations appear in batch reactions only when gelation occurs at high conversions.     

Dynamics of a continuous stirred tank reactor for styrene polymerization initiated by a binary initiator mixture. II: Effect of viscosity dependent heat transfer coefficient

The dynamic behavior of the solution polymerization of styrene in a continuous stirred tank reactor is analyzed with a mixture of tert-butyl perbenzoate and benzoyl peroxide as an initiator system. In the modeling of the reactor, a viscosity dependent reactor wall heat transfer coefficient is used to account for the changing heat transfer efficiency as monomer conversion and polymer molecular weight increase. The steady state and bifurcation behaviors have been investigated with the reactor residence time, initiator feed composition, initiator concentration, feed solvent volume fraction, and coolant temperature as bifurcation parameters. Unlike the reactors with constant heat transfer coefficient, the present system exhibits relatively simple steady state and dynamic bifurcation behaviors. Oscillatory behavior is observed only when the solvent volume fraction in the feed exceeds 0.2. The dynamic simulation of the reactor also indicates that a feedback temperature controller may fail to maintain the reactor temperature when the heat transfer coefficient changes as a result of process disturbances.     

Synthesis of model-based controllers for an autothermal reactor

Model-based controllers for a bench scale autothermal tubular packed-bed reactor have been formulated using the Internal Model Control (IMC) approach. The Structural Dominance Analysis technique has been used in developing the reduced-order models. Controller performance at robust and sensitive steady states have been assessed through simulations and experiments. Both PI and model-based controllers can regulate reactor operation at robust steady states, but only third order IMC controllers are able to regulate reactor operation at the sensitive steady state.     

A stochastic treatment of unimolecular reactions in an unsteady state continuous flow system

A stochastic approach, namely a continuous time Markov chain (Markov process), is employed to analyze and model, in a unified fashion, both the kinetics of unimolecular reactions and mixing accompanied by flow in a continuous flow reactor under an unsteady state operation; the results reduce to those under the corresponding steady state operation in the limit as t→∞. This approach can be applied to both the time homogeneous and heterogeneous processes. The transitional distributions of the number of molecules of each type inside the reactor as well as at the exit are formulated. The treatment leads to the general expressions for the transient internal and exit age distributions of molecules in the flow reactor. The life time distribution of molecules under steady state is also derived. The statistical basis of the residence time distribution theory for flow reactors is clarified. The approach is illustrated with two examples.     

Free‐radical solution polymerization of styrene in a tubular reactor—effects of recycling

The styrene free‐radical solution polymerization reaction in a tubular loop reactor is studied here both experimentally and through simulation. An attempt is made to compare the performances of tubular loop reactors when the recycle ratio is varied, based on steady‐state and dynamic responses and on the quality of the polymer produced at different conditions. It is shown here that steady‐state responses of loop reactors and traditional tubular reactors are very similar as far as the quality of the polymer obtained is concerned. Therefore, the recycle ratio cannot be used as a fundamental operation parameter for grade transitions at plant site. However, it is also shown that the recycling of polymer material is very important to accelerate the attainment of the final steady‐state in tubular reactor configurations, because recirculation of material homogenizes the distorted radial profiles of the axial flow velocities.