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.     

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.     

The bifurcation behavior of tubular reactors

Methods for studying the bifurcation behavior of tubular reactors have been developed. This involves the application of static and Hopf bifurcation theory for PDE's and the very precise determination of steady state profiles. Practical computational methods for carrying out this analysis are discussed in some detail. For the special case of a first order, irreversible reaction in a tubular reactor with axial dispersion, the bifurcation behavior is classified and summarized in parameter space plots. In particular the influence of the Lewis and Peclet numbers is investigated. It is shown that oscillations due to interaction of dispersion and reaction effects should not exist in fixed bed reactors and moreover, should only occur in very short “empty” tubular reactors. The parameter study not only brings together previously published examples of multiple and periodic solutions but also reveals a hitherto undiscovered wealth of bifurcation structures. Sixteen of these structures, which come about by combinations of as many as four bifurcations to multiple steady states and four bifurcations to periodic solutions, are illustrated with numerical examples. Although the analysis is based on the pseudohomogeneous axial dispersion model, it can readily be applied to other reaction diffusion equations such as the general two phase models for fixed bed reactors.     

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.     

Dynamics of a cascade of two continuous stirred tank polymerization reactors with a binary initiator mixture

The dynamic behavior of two continuous stirred tank reactors in series has been investigated for free radical solution polymerization of styrene with a binary mixture of two initiators having different thermal decomposition activities. For a wide range of initiator feed composition, both reactors exhibit quite complex nonlinear steady state and transient behavior. When the reactor residence time is used as a bifurcation parameter, the second reactor can have up to five steady states. For certain range of reactor operating conditions, bifurcations to various types of periodic solutions have been observed, such as Hopf bifurcation, isolas, period doubling, period-doubling cascade, and homoclinics. The effects of other reactor variables, such as total initiator concentration, coolant temperature, and reactor volume ratio on the reactor dynamics, are illustrated to show the complex dynamic behavior of the two-reactor system catalyzed by a mixture of t-butyl perbenzoate and benzoyl peroxide.     

Numerical computation of periodic solution branches and oscillatory dynamics of the stirred tank reactor with A → B → C reactions

We use a continuation technique for branches of periodic solutions to investigate the oscillatory behavior of a continuously stirred tank reactor with consecutive A → B → C reactions. This continuation technique allows the computation of entire periodic solution branches, including those with limit points and asymptotically unstable solutions. Our computations reveal dynamic phenomena not seen in previous studies of this reactor. The results include response diagrams exhibiting stable and unstable periodic branches that contain multiple limit points. The presence of these points indicates that the reactor may jump from a steady state to a periodic orbit or from one orbit to another. The computations also illustrate interactions of multiple steady state limit points, Hopf bifurcations and infinite periodic bifurcations.     

Development of a bifurcation analysis approach based on gPROMS platform☆

A bifurcation analysis approach is developed based on the process simulator gPROMS platform, which can automatically trace a solution path, detect and pass the bifurcation points and check the stability of solutions. The arclength continuation algorithm is incorporated as a process entity in gPROMS to overcome the limit of turning points and get multiple solutions with respect to a user-defined parameter. The bifurcation points are detected through a bifurcation test function τ which is written in C ++ routine as a foreign object connected with gPROMS through Foreign Process Interface. The stability analysis is realized by evaluating eigenvalues of the Jacobian matrix of each steady state solution. Two reference cases of an adiabatic CSTR and a homogenous azeotropic distillation from literature are studied, which successfully validate the reliability of the proposed approach. Besides the multiple steady states and Hopf bifurcation points, a more complex homoclinic bifurcation behavior is found for the distillation case compared to literature.     

Two‐Parameter Continuation and Bifurcation Strategies for Oscillatory Behavior Elimination from a Zymomonas mobilis Fermentation System

Two‐parameter continuation and bifurcation analysis strategies were applied to deal with the oscillatory phenomena of a Zymomonas mobilis ethanol fermentation system. A structured verified non‐linear mathematical model considering the physiological limitations of microorganisms for a single continuous fermenter for ethanol production using Z. mobilis was built to identify the Hopf bifurcation (HB) points, which indicate the oscillatory behavior, using the inlet substrate concentration and the dilution rate as bifurcation parameters. The path of the HB points can be determined with different controlling operating parameters. It was found that with the addition of a small amount of cells or ethanol to the feed stream or by increasing the dilution rate, the oscillations could be eliminated and steady‐state behavior was attained. Using a two‐parameter continuation strategy, the Z. mobilis fermentation system could operate at steady state without oscillatory behavior.     

Cubic autocatalysis with Michaelis-Menten kinetics: semi-analytical solutions for the reaction-diffusion cell

Cubic-autocatalysis with Michaelis-Menten decay is considered in a one-dimensional reaction-diffusion cell. The boundaries of the reactor allow diffusion into the cell from external reservoirs, which contain fixed concentrations of the reactant and catalyst. The Galerkin method is used to obtain a semi-analytical model consisting of ordinary differential equations. This involves using trial functions to approximate the spatial structure of the reactant and autocatalyst concentrations in the reactor. The semi-analytical model is then obtained from the governing partial differential equations by averaging. The semi-analytical model allows steady-state concentration profiles and bifurcation diagrams to be obtained as the solution to sets of transcendental equations. Singularity theory is then used to determine the regions of parameter space in which the four main types of bifurcation diagram occur. The region of parameter space, in which Hopf bifurcations can occur, is found by a local stability analysis of the semi-analytical model. An example of a stable limit-cycle is also considered. Comparison with numerical solutions of the governing partial differential equations shows that the semi-analytical solutions are very accurate.     

The bifurcation behavior of continuous free-radical solution loop polymerization reactors

The bifurcation behavior of continuous free-radical solution loop polymerization reactors is analyzed in this work. A mathematical model is developed in order to describe the impact of the recycling pump and other external reactor parts upon the process dynamics and stability. Stability analysis is performed using bifurcation theory and continuation methods. It is shown that under certain operational conditions as many as seven steady states are predicted for the loop polymerization reactor. Oscillatory behavior is observed for a wide range of process parameters and onset of oscillations is observed during the transition from operation without material recycling to operation with partial recirculation of the polymer solution. Besides, at certain constrained range of operation conditions, complex dynamics can be observed, including the onset of chaotic behavior. It is also shown that the thermal parameters of the reactor and recycling pump exert a profound effect upon the process stability. For this reason it is shown that oscillatory behavior is very unlikely to occur in actual industrial reactors.     

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.     

THE EFFECT OF MACROMIXING ON OSCILLATORY BEHAVIOR IN RECOMBINANT FERMENTATION IN A CONTINUOUS STIRRED TANK REACTOR

A general model is presented for recombinant fermentation in a chemostat in which micromixing may be imperfect. Using the production of tryptophan synthetase as an example and the dilution rate, D, as the bifurcation parameter, conditions are first derived for Hopf bifurcation in a perfectly mixed chemostat. The bifurcation surfaces in three-dimensional space are such that oscillatory solutions exist above the surfaces and washout conditions below. A similar analysis is done for imperfect mixing. The bifurcation region is a concave hyperboloid in the [D,f, n]-space, where f and n ; are model parameters. It is argued that poor macromixing reduces the range of D over which bifurcation to periodic solutions is possible.     

Semi-analytical solutions for a Gray-Scott reaction-diffusion cell with an applied electric field

An ionic version of the Gray-Scott chemical reaction scheme is considered in a reaction-diffusion cell, with an applied electric field, which causes migration of the reactant and autocatalyst in a preferred direction. The Galerkin method is used to reduce the governing partial differential equations to an approximate model consisting of ordinary differential equations. This is accomplished by approximating the spatial structure of the reactant and autocatalyst concentrations. Bifurcation analysis of the semi-analytical model is performed by using singularity theory to analyse the static multiplicity and a stability analysis to determine the dynamic multiplicity. The application of the electric field causes variation in the parameter regions, in which multiple steady-state and oscillatory solutions occur. Moreover, as the reactor is not symmetric, reversal of the direction of the electric field can cause bifurcation in the reactor between high and low conversion states. Comparisons with numerical solutions of governing partial differential equations confirms the accuracy and usefulness of the semi-analytical model.     

The application of dynamic modeling for thermal risks analysis of the acid‐catalyzed hydrolysis of glycidol

The aim of this work was to determine the limits of safe operation of continuous flow stirred‐tank reactor (CSTR) for the acid‐catalyzed hydrolysis of glycidol. The stability analysis was performed by dynamic modelling. The obtained results were compared with the experimental data reported in the open literature. For this purpose, dimensionless variables and parameters were introduced and unstable material and energy balances were defined. The system equations were solved using Matcont (Matlab^® software). Thus, bifurcation diagrams (in one and two dimensions) were mapped. All different dynamic states were identified and studied (thermal stability and instability, with unique and multiple solutions; Hopf bifurcations; turning points and envelope of periodic solutions). Finally, the intrinsic thermal unstable and cycle behavior of the acid‐catalyzed hydration of glycidol to produce glycerol was identified. The appropriate conditions to guarantee safe operation of CSTR were found. © 2016 American Institute of Chemical Engineers AIChE J, 62: 4418–4426, 2016     

Behaviour of an elementary oxidation reaction in a semi-batch reactor

The dynamics of a non-isothermal bi-molecular gas-phase reaction in a semi-batch reactor is investigated. It is assumed that one of the reactants flows into a reactor containing the second. A reduced model is obtained by making a ‘pool-chemical’ approximation on the concentration of the reactant initially in the reactor. The region in parameter space in which oscillations are observable in the full transient model is estimated by determining the Hopf bifurcation locus in the reduced system. The contribution of the current work is its comparative study of the behaviour of the full system to that of the pool-chemical model. Although the reaction scheme is symmetric with respect to the reactants the regions of oscillatory behaviour are not identical because the reactants have different heat capacities.     

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。     

Experimental and modelling investigations of steady-state and dynamic characteristics of ethylene hydrogenation on Pt/Al2O3

A robust strategy based on fluid-phase measurements is described for the testing and development of kinetic models for heterogeneous catalytic reactions. Steady-state, step-response, feedback-induced Hopf bifurcation and forced concentration cycling experiments were applied to ethylene hydrogenation over 0.05% Pt/Al₂O₃ in a CSTR at 80°C. Two versions of a reaction mechanism that differ in the order for hydrogen adsorption describe the experimental steady-state data. However, only one of these models adequately describes the system responses to step changes in the feed composition. Step-response experiments were used to identify a time scale of 5000 s which is associated with chemisorbed hydrogen. Conversely, feedback-induced Hopf bifurcation data indicate this time scale to be of the order of 1 s in magnitude. In the overall strategy of dynamic modelling, the two techniques are complementary since each inherently focuses on an opposite region in the spectrum of time scales for the reactor system. A detailed method for analysing feedback-induced Hopf bifurcation with a time delay is presented. The dynamic model based upon steady-state, step-response and bifurcation data was found to be inadequate for describing the results from cycled-feedstream experiments. Cycling the feedstream composition resulted in an improvement of the time-average reaction rate for the ethylene hydrogenation reaction compared to steady-state reactor operation. Rate improvement was observed for the entire range of ethylene feed concentrations that were tested, including conditions which give rise to a maximum in the steady-state reaction rate surface.     

Analysis of steady state of free radical solution polymerization in a continuous stirred tank reactor

The steady state structure of non-isothermal free radical solution polymerization in a continuous stirred tank reactor is analyzed. With mean residence time as a bifurcation parameter, five regions of steady states are identified. The effect of reactor operating conditions on the structure and the stability of steady state is discussed.     

The mixing effect on the free radical MMA solution polymerization

Mathematical models of reactors for the polymerization of methylmethacrylate (MMA) have been developed and analyzed to elucidate reactor dynamics and to determine conditions for improved operation. The effects of mixing and heat transfer in an MMA polymerization reactor system have been explored by the development of an imperfect mixing model. To model imperfect mixing in polymerization, a reactor configuration using two tanks in parallel was used. Bifurcation diagrams developed using numerical analysis of the model have been drawn with two variable parameters, an exchange ratio, σ, and a volume ratio, κ. We use feed and coolant temperatures as bifurcation parameters. If variable parameters are small, the lower solution branch of the steady state solutions is quite different from that of a simple model that assumes perfect macro-mixing as bifurcation parameters change. If σ increases (κ=0.1, σ=1.0), the shape of a steady state solution curve differs significantly from that of a simple model as the feed temperature decreases.     

Periodic solutions in a porous catalyst pellet—homoclinic orbits

The analysis performed as well as extensive numerical simulations have revealed the possibility of the generation of homoclinic orbits as a result of homoclinic bifurcation in the model which describes transport phenomena and chemical reaction in a porous catalyst pellet. A method has been proposed for the development of a special type of diagrams—the so-called bifurcation diagrams. These diagrams comprise the locus of homoclinic orbits together with the lines of limit points bounding the region of multiple steady states as well as the locus of the points of Hopf bifurcation. Thus, they define a set of parameters for which homoclinic bifurcation can take place. They also make it possible to determine conditions under which homoclinic orbits are generated.Two kinds of homoclinic orbits have been observed, namely semistable and unstable orbits. It is found that the character of the homoclinic orbit depends on the stability features of the limit cycle which is linked with the saddle point.Very interesting dynamic phenomena are associated with the two kinds of homoclinic orbits; these phenomena have been illustrated in the solution diagrams and phase diagrams.