Stability of the steady states and transient behavior for a non-isothermal bimolecular langmuir-hinshelwood reaction

The steady state stability and transient behavior of a catalytic reaction following the bimolecular Langmuir-Hinshelwood expression, phenomenologically adequate for carbon monoxide and hydrocarbon oxidation on noble metal catalysts, is describe. The corresponding steady state problem was discussed earlier. Some unique features of the isothermal and non-isothemal transient behavior with uniform initial conditions are identified.     

The non-adiabatic tubular reactor: Stability considerations

This paper is a continuation of four previous ones in this journal ^{(1, 2, 3, 4)} and is involved with the transient and stability behavior of a non-adiabatic unpacked tubular reactor. The steady states found earlier are tested for stability both by computing transient profiles and by computing the eigenvalues of the appropriate linearized problem. It is found in all cases that the two computations predict the proper stability behavior. It is shown that in some cases all steady states for a given set of parameters may be unstable and that a unique steady state may give rise to an oscillatory system.     

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.     

The dynamic behavior of continuous polymerization reactors—I: Isothermal solution polymerization in a CSTR

A mathematical model having no adjustable parameters is developed and compared with both dynamic and steady-states” in the neighborhood of unstable middle steady states is thought to be due to a combination of methacrylate with ethyl acetate solvent, in a CSTR, was shown to have ignition and hysteresis behavior characteristic of multiple steady states for sufficiently low solvent concentrations. The observation of “pseudosteady-states” in the neighborhood of unstable middle steady states is thought to be due to a combination of kinetic and hydrodynamic mechanisms.     

Bifurcation and its implications for a novel autothermal circulating fluidized bed membrane reformer for the efficient pure hydrogen production

The present investigation is a prelude to the experimental exploration of Static Bifurcation Behavior (SBB) in a novel Autothermal Circulating Fluidized Bed Membrane Reformer (ACFBMR) for pure hydrogen production by steam reforming of heavy hydrocarbons. Important impacts of a wide range of design and operating parameters on the reformer performance are explored with two reformer configurations. One is with the catalyst regeneration before the gas-solid separation and another is with the catalyst regeneration after the gas-solid separation. For both configurations there are three steady states (multiplicity of the steady states, static bifurcation behavior). The system behavior in the bifurcation region is quite complex and defies the simple logic of non-autothermal processes. For the first configuration, on the branch of upper temperature steady state the carbon formation and deposition on the nickel catalyst is highest, while the net hydrogen yield is highest on the branch of lower temperature steady state. For the second configuration, the conversion of heptane is always 100%. In the multiplicity region, the order of net hydrogen yield from high to low is the middle, upper and lower temperature steady states, while the order of reformer exit carbon flow rate from high to low is the lower, middle and upper temperature steady states. Efficient production of pure hydrogen for fuel cells requires fundamental and practical understanding of their bifurcation behaviors.     

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.     

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.     

Transient analysis of exothermic reactions within catalyst pellets. Effect of initial conditions

The transient behavior of exothermic, irreversible, first-order catalytic reaction inside a spherical, porous pellet is analyzed using mixed type boundary conditions to represent external resistances. Calculations are performed for Lewis numbers, ?C_pD_e/?λ_e, equal to unity and a pellet initially at uniform temperature and devoid of reactant, which is considered realistic. Phenomena occurring during the evolution of transients in the region of steady external transport limitation are described.In ranges where more than one steady state exists, the effect of the initial conditions on the attainment of the different steady states is analyzed. The results include information about transient behavior when the initial condition is at the boundary between stable domains of attraction. For certain ranges of parameters it is shown that the initial conditions studied may fall outside of the domain of attraction of one of the stable steady states, demonstrating that information about domains of attraction as well as steady states should be taken into account when designing catalytic systems.     

Dynamic Behavior of Industrial Gas Phase Fluidized Bed Polyethylene Reactors under PI Control

A mathematical model describing the UNIPOL process for the production of polyethylene in the gas phase using a Ziegler‐Natta catalyst in a bubbling fluidized bed is used to analyze the major processes determining the behavior and performance of these industrially important units. The investigation shows that both static bifurcation (multiplicity of the steady states) as well as dynamic bifurcation (stable/unstable periodic attractors) behavior cover wide regions of the design and operating parameter domain. A conventional proportional‐integral (PI) control policy is suggested to stabilize the behavior of the system. The control philosophy covers both aspects of stabilizing unstable steady states as well as compensating for external disturbances. It is shown that for some controller configurations and set points the controlled process can go through a period doubling sequence leading to chaotic strange attractors. The industrial implications of the phenomena discovered for both the open loop (uncontrolled) as well the closed‐loop (controlled) systems are analyzed.     

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.     

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.     

等温CMSMPR结晶器定态及瞬态特性分析(Ⅱ)分岔行为及瞬态特性

应用非线性方程组的延拓解法 ,对同时发生初级成核和二次成核的等温连续混合悬浮混合排料(CMSMPR)结晶器的定态方程组进行了求解 ,考察了结晶过程的定态在不同参数区域之间的分岔行为 ,确定了各参数区域内结晶过程的定态数目 ,并应用Routh -Hurwitz准则对各定态的稳定性进行了分析 .采用四阶Runge -Kutta法求解CMSMPR结晶器的动态方程组 ,分析了在多定态区域以及持续振荡区域中结晶过程的瞬态特性     

Dispersion and Phase Separation of Water‐Oil‐Amphiphile Systems in Stirred Tanks

Drop size distributions and phase separation behavior of water‐oil‐nonionic amphiphile systems are investigated using an in situ endoscope measurement technique and an external camera in stirred tanks in batch mode. The fitting procedure and the simulation results of a phase separation model are analyzed under the condition that either the swarm sedimentation speed or the mean drop size during sedimentation is known. The steady‐state drop size distributions are self‐similar over the whole range of process parameters, but not in the decaying turbulence field after agitation stop. The coalescence rate in the first seconds after agitation stop clearly affects the separation behavior, so that a prediction of the separation time based on the initial conditions in steady state is not trivial.     

Performance of tower loop bioreactors—I: Model calculations for nonlimited growth

The behavior of tower loop reactors is described by a recycle-dispersion model assuming a nonlimited growth rate and negligible residence times of the cell suspension in the loop. For nonsterile feed the parameter ranges of possible steady states are evaluated. In general, recirculation leads to increased biomass concentration. This also causes that unlimited growth may occur even for high Bo numbers. If the feed is sterile, only wash-out and nonlimited growth are found as possible stable steady states.     

Modelling thixotropic behavior of fresh cement pastes from MRI measurements

From MRI measurements we show that in a flowing cement paste thixotropic effects dominate over short time scales while irreversible effects become significant over larger timescales. The steady and transient flows exhibit a yielding behavior which differs from usual yield stress model: the transition from the solid to the liquid regime is abrupt. We propose a simple thixotropic model based on these observations. The validation is done on the steady and transient state on local experimental tests. We build the “local” rheogram which is representative of the intrinsic rheological properties. Comparisons with “apparent” rheograms demonstrate that it is possible to use correction techniques from the literature to have access to the real behavior law of the material from standard measurements but that, in the case of the material studied in this paper, this would nevertheless lead to an underestimation of the yield stress.     

An analysis of the steady state behavior of a mixer-settler extraction system

The steady state behavior of a multi-component mixer-settler extraction system has been analyzed by adopting a mixing cell model with the individual Murphree stage efficiency parameters. Empirical equations for the chemical equilibrium of HNO₃-UO₂ (NO₃)2 -TBP system have been derived and incorporated into the model equations. Nonlinear Component Block Successive Relaxation (NCBSR) method executed through an Approximate Newton Raphson routine has been proposed for the solution of nonlinear steady state system equations and compared with the other methods. The model and the derived equilibrium relations were found to be successfully employed in depicting the steady state behavior and the proposed NCBSR method was proven for its effectiveness in comparsion with the other methods.     

Steady-state multiplicity in bioreactors: bifurcation analysis of cybernetic models

Biological systems have an additional level of complexity compared to other chemical systems because of the effects of metabolic regulation, a defining feature of biosystems. Metabolic regulation in the form of control of enzyme synthesis and activity leads to non-linear behavior in bioreactors. Mathematical models that take into account these control mechanisms can be very successful in capturing the peculiarities of bioreactors such as multiple steady states and periodic phenomena. Cybernetic models model the expression and activation of enzymes by the use of cybernetic control variables and have been used to explain multiplicities in hybridoma reactors. In particular Namjoshi et al. (Biotechnol. BioEng. (2002), accepted) have been able to predict the transition from batch and fed batch to continuous culture in hybridoma experiments (Biotechnol. BioEng. 67(1) (2000) 25). The resulting multiple steady states vary widely in cell mass and waste metabolites. The model captures this multiplicity and its bifurcation analysis has revealed additional steady-state branches, three unstable and one stable. The stability of the additional steady state (steady state 4) was confirmed by dynamic simulations using fed-batch strategy prior to initiation of continuous operation. Steady state 4, in view of its close proximity to steady state 3, appears to have little practical significance. The likelihood of additional steady states that may be significantly different can, however not be ignored. Thus, it seems possible to envisage other states of metabolic activity, displaying alternative flux distributions that could lead to steady states notably different from those already determined. Calculation of the most singular point of the system herein is rendered difficult by both its size and possession of non-differentiable variables. The bifurcation analysis reveals the steady-state behavior under a range of operating conditions and can be used to plan optimum bioreactor operation.     

Quasi-stable temperature of the steady state of microwave heated hematite

Microwave heated materials often reach a quasi-stable temperature resulting in thermal runaway. To control steady state in microwave processing, it is important to predict the quasi-stable temperature of the steady state. We demonstrated that the microwave heating behavior of hematite varies significantly with its initial temperature. In microwave heating, hematite samples could not be heated from room temperature, whereas hematite samples preheated to 410 °C or higher was heated to a temperature of 1020 °C. The microwave heating behavior can be accurately predicted by considering the steady-state energy balance.     

Multiplicity and stability of premixed laminar flames: an application of bifurcation theory

Numerical bifurcation techniques are used to describe multiple steady states for a premixed, laminar flame stabilized on a flat flame burner. The flame is assumed to be adiabatic, and the kinetic mechanism is approximated by a single reaction. The numerical methods make it possible to determine all steady states and eliminate computational difficulties near singular points. The possibility of defining solutions near singularities is particularly important in flame modeling for it is near such points that ignition and burn-out may occur. Three steady states are identified: a stable upper state corresponding to a flame burning at or near the adiabatic flame temperature, a lower solution representing an extinguished stable state, and an unstable intermediate state. Sensitivity of the solutions to changes in kinetic parameters is enhanced near burn-out. It is expected that the ability to predict flame behavior near such singular points will be particularly useful in the determination of flame kinetics.