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     

SUDDEN DESTABILIZATION OF CONTROLLED CHEMICAL PROCESSES

Input multiplicity occurs when more than one set of inputs can produce the same set of outputs. Input multiple steady states are divided into compatible steady states having process gains of similar sign, and opposed steady stales with process gains of opposite sign. For a system controlled with reset action, only the compatible steady states satisfy the necessary condition for stability. Any disturbance which drives the controlled system from the designed steady state to a less stable or unstable compatible steady state can cause sudden destabilization of the process. Several examples are given of the possible types of behavior resulting from this phenomenon.

Multiple steady states also occur for systems with proportional controllers. For single-input-single-output systems with continuous process characteristics, whether or not reset action is used, two steady states positioned next to each other cannot both be stable under closed-loop control. However, under proportional control, opposed steady states for which 1 + K_cK_p is positive can be stable.     

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.     

Undamped oscillations in bacterial glycolysis models

The exotic dynamical behaviors exhibited in chemical reaction systems, such as multiple steady states, undamped oscillations, chaos, and so on, often result from unstable steady states. A bacterial glycolysis model is studied, which involves the generation of adenosine triphosphate (ATP) in a flow system and consists of eight species and ten reactions. A minimum subnetwork of the bacterial glycolysis model is determined to exhibit an unstable steady state with a positive real eigenvalue, which gives rise to undamped oscillations for a small perturbation. A set of rate constants and the corresponding unstable steady state are computed by using a positive real eigenvalue condition. The phenomena of oscillations and bifurcation are discussed. These results are extended to the bacterial glycolysis model and several parent networks.     

Multiple steady states in continuous distillation and realization of a selected steady state

A general strategy for a column startup is suggested which enables one to attain all the feasible steady states. By numerical calculations, the whole space of the starting concentrations was partitioned into regions corresponding to each of the steady states.     

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.     

高效再生催化裂化装置多稳态分析：反应温度开/闭环控制条件对热反馈机制的影响

针对催化裂化反应-再生系统在提升管反应温度开环和闭环控制条件下的输出与输入多稳态问题,分析了烧焦罐式高效再生催化裂化反应-再生系统在两种条件下随着CO助燃剂添加量变化时的多稳态分布。在反应温度开环条件下,因再生温度与反应温度的耦合程度较低,使系统移热曲线呈单调递增,导致了系统出现3个稳态操作点。在反应温度闭环控制条件下,提升管反应器和再生器间热反馈机制发生改变,由于再生剂循环量可以作为额外的自由度对再生温度和反应温度之差进行补偿,再生器和提升管反应器的耦合程度增强,使得系统只会在助燃剂添加量极低时才会出现多个稳态点,而在基准操作条件下只有一个稳态点,规避了系统在提升管反应温度开环时的多个稳态点的问题。     

Stability and dynamics of isothermal CMSMPR crystallizers

Detailed analysis of stability and bifurcation properties and of the dynamic behaviour of isothermal continuous mixed suspension, mixed product removal (CMSMPR) crystallizers is presented using the moment equation model. It is shown that isothermal CMSMPR crystallizers may exhibit not only limit cycle oscillations but, in the case of magma-dependent nucleation, also saddle point instabilities. Applying the Mikhailov stability criterion, two equivalent sets of necessary and sufficient conditions are derived for checking the stability of steady states.Stability maps are presented in the planes of different pairs of system parameters. It is shown that saddle node and boundary bifurcation, as well as bifurcation at infinity may arise in the case of magma-dependent secondary nucleation. Oscillations of the population density and distribution functions are also presented and analysed, approximating the size distribution by gamma distribution. In dynamic states, the crystal size distribution may differ significantly from the exponential one, while the mean values of oscillations of supersaturation are lower than the corresponding steady state values. A causal loop diagram between the variables reveals that instabilities are created by the interactions of the autoinhibition generated negative feedback and a positive feedback between the four leading moments of crystal size.     

Determination of multiple steady states in oxidation of monophenols by tyrosinase with enzymatic-enzymatic-chemical model

A family of an enzymatically catalyzed reaction network is studied, which involves the oxidation of monophenols by tyrosinase with enzymatic-enzymatic-chemical model in an isothermal continuous flow stirred tank reactor (CFSTR). This system consists of 11 coupled non-linear equations and is determined to have the capacity to exhibit computational multiple steady states. A set of rate constants and two corresponding steady states are computed. The phenomena of bistability, hysteresis and bifurcation are discussed. Moreover, the capacity of steady state multiplicity is extended to its family of reaction networks.     

化工过程稳态检验的区间拓展法

介绍了组合统计检验法的基本思想,提出了区间拓展法用于实际化工过程稳态检验的新策略。实例考核表明,新方法显著地降低了第一类错误的概率,可有效地用于实际化工过程的稳态检验。     

Bifurcation analysis of two continuous bioreactors operated in series with recycle

In this paper, bifurcation analysis has been carried out for two continuous bioreactors operated in series with recycle from the second reactor. The existence of multiplicity of steady states is analyzed by considering Contois growth kinetics in the process model. It was observed that there exist two possible steady states of which one is trivial (wash out condition). Stability analysis is carried out to determine the stability of these steady states and it was observed that both these steady states are unstable in nature. Bifurcation analysis has been carried out for substrate and biomass concentration with dilution rate as the bifurcation parameter. Effect of recycle ratio, substrate separation factor and biomass separation factor is studied and analyzed. It was observed that Hopf bifurcation occurs at a dilution rate of 1.0208 with purely imaginary Eigen values which showed that sustained oscillatory behavior exists in the substrate concentration of the second reactor. The significance of different bifurcation points and the operating conditions by considering biomass and substrate concentrations in each reactor is studied and it was observed that the bioreactors need to be operated at intermediate dilution rates to obtain improved conversion and yield.     

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.     

KINETICS OF SUCROSE INVERSION BY INVERTASE-MULTIPLE STEADY STATES IN A CSTR

The kinetics of a substrate-inhibited enzymatic reaction-sucrose inversion by invertase was studied in this work. For this system, the existence of two stable steady states in a CSTR was verified experimentally under the condition predicted by previous theoretical analysis. Mathematical relations were derived to enable one to use multiple steady states information For parameter estimation. The values obtained by this method were compared with that obtained by other methods and discussed.     

KINETICS OF SUCROSE INVERSION BY INVERTASE-MULTIPLE STEADY STATES IN A CSTR

The kinetics of a substrate-inhibited enzymatic reaction-sucrose inversion by invertase was studied in this work. For this system, the existence of two stable steady states in a CSTR was verified experimentally under the condition predicted by previous theoretical analysis. Mathematical relations were derived to enable one to use multiple steady states information For parameter estimation. The values obtained by this method were compared with that obtained by other methods and discussed.     

State multiplicity in CSTR-separator-recycle polymerisation systems

This article continues earlier work (Comput. Chem. Eng. 24 (2000) 209) concerning the design and control of isothermal reactor-separator-recycle systems. The multiplicity behaviour of six reaction systems of increasing complexity, from one-reactant, first-order reaction to chain-growth polymerisation, is investigated. Below a critical value of the plant Damkohler number, Da<Da^cr, the only steady state involves infinite flow rates. Feasible steady states become possible if the critical value is exceeded, Da>Da^cr. For one-reaction systems, one stable steady state is born at a transcritical bifurcation. For consecutive-reaction systems, including polymerisation, a fold bifurcation can lead to two feasible steady states. Moreover, the transcritical bifurcation is destroyed when two reactants are involved. If the gel-effect is included, a maximum of four steady states are possible. When multiple steady states exist, the achievable conversion is constrained by the instability of the low-conversion branch. This has practical importance for polymerisation systems when the radicals’ quasi-steady state assumption is not valid or the gel effect is significant.     

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.     

STABILITY OF THE CSTR BY CORRELATION OF LIAPUNOV FUNCTIONS

A general Liapunov function for a homogeneous CSTR is formally developed from available Liapunov functions which guarantee the global asymptotic stability of steady states associated with the separate mechanisms (flow mixing and chemical reaction) which compete in a CSTR. The Liapunov functions for the asymptotic systems (mixing tank and batch reactor) are combined via a one parameter correlation form to provide a one parameter family of Liapunov functions for the general CSTR. The single parameter allows for convenient optimization of regions of asymptotic stability surrounding CSTR steady states.     

Spatially patterned states in systems of interacting catalyst particles

Considering an assemblage of identical catalyst particles exposed to uniform surroundings, we show in this theoretical study that interparticle interactions give rise to spatially patterned steady states under some conditions. In such patterned states, the rate of reaction on one particle differs from the rate on all or some of the neighboring particles. Results, which are based on simple geometric configurations and on the assumption that a single nonisothermal reaction occurs, show that spatially patterned states of two types exist. The first occurs only if a single isolated particle has multiple steady states. The second type, not inferable from the behavior of a single isolated particle, is attributable to the instability of a spatially uniform state caused by particle-particle interactions. This type may occur in cases where the steady state of an isolated particle is unique and globally stable. Though the study is not extended here to complex assemblages, the results suggest that these phenomena may occur in fixed bed reactors and lead to insidious nonuniformities in reactor temperature and catalyst activity.     

Steady States of the Reactor–Distillation Column System for an A + B ⇄ C Reaction

The steady states of the reactor-distillation column recycle system are analyzed for the reaction A + B C at various feed compositions. The analysis is performed for an infinite column operated at total reflux. If the lightest component, A, is in excess, the complete conversion of the medium-boiling component, B, is impossible. If reactant B is in excess, there are three steady states, of which two are characterized by complete conversion of reactant A. In the case of a recycle system with a finite column, there is no continuum of steady states.