Occurrence Of Multiple Steady States Research Articles

Study of Static and Dynamic Behavior of a Membrane Reactor for Hydrogen Production

The paper investigates the stability and bifurcation phenomena that can occur in membrane reactors for the production of hydrogen by ammonia decomposition. A simplified mixed model of the membrane reactor is studied and two expressions of hydrogen permeation are investigated. The effect of the model design and operating parameters on the existence of steady state multiplicity is discussed. In this regard, it is shown that the adsorption-inhibition effect caused by the competitive adsorption of ammonia can lead to the occurrence of multiple steady states in the model. The steady state multiplicity exists for a wide range of feed ammonia concentration and reactor residence time. The effect of the adsorption constant, the membrane surface area and its permeability on the steady state multiplicity is delineated. The analysis also shows that no Hopf bifurcation can occur in the studied model.

Critical role of Prandtl number on multiple steady states during natural convection in square enclosures: Analysis of heat transfer rates, flow and thermal maps

The effect of Prandtl number (Pr) on the multiplicity of the steady states (flow and temperature maps) is investigated for the natural convection of Newtonian fluids within a square enclosure involving hot bottom wall, cold side walls and adiabatic top wall. The analysis is based on the steady state governing equations, which are solved by penalty Galerkin finite element method and Newton–Raphson solution scheme. A novel perturbation technique has been developed, which is applied in conjunction with an in-house continuation scheme for the initiation of various symmetric as well as asymmetric steady states. The current computation strategy involving perturbation and continuation schemes leads to a rich bifurcation diagram in the parameter space of Pr (0.01⩽Pr⩽10) for Ra (Rayleigh number) =106. It is found that the occurrence of multiple steady states depends on Pr. Only one symmetric solution is obtained in the high–Pr regime (Pr>0.42), while a highly populated bifurcation diagram consisting of 6 symmetric branches, 18 asymmetric branches and 4 isolas is obtained in the low–Pr regime (especially for Pr≲0.05). The solution branches correspond to a wide spectrum of flow structures involving single cell, multiple cells, reverse-cells, corner-cells, horizontally stacked cells, vertically stacked cells etc., which are reported for the first time. These flow structures result in a wide range of Nusselt numbers (average heat transfer rates) at a given Pr, especially for the low–Pr fluids (such as liquid metals). The flow structures are critically analyzed in terms of the location of the flow intensification and its impact on the overall heat transfer rates. It has been shown that the overall heat transfer rates for the low–Pr fluids can be significantly enhanced (80 –100% in few cases) based on corner-cells or single cell driven flow.

Effect of diffusion limitation and substrate inhibition on steady states of a biofilm reactor treating a single pollutant

ABSTRACTThe occurrence of multiple steady states in a toluene biodegrading, diffusion-limited biofilm under aerobic conditions was investigated by computer models: one steady-state, and one nonsteady-state. Two stable and one unstable intermediate steady-state were identified in a narrow set of combinations of parameters values. The nonsteady-state model predicts conditions that evolve to a steady state that is within 0.02–1% of the solution of the steady-state model, depending on the number of grid points used, confirming the algorithms are valid. Multiple steady states occur if, (1) a biofilm is exposed to a constant gas-phase pollution concentration, which exceeds or undershoots a certain threshold, (2) in a narrow range of parameter values and (3) provided that the pollutant degradation follows Haldane kinetics. Such a biofilm displays half-saturation (i.e., Michaelis-Menten)-like apparent (“falsified”) kinetics from a concentration range starting at zero up to the occurrence of a second steady state. Multiple steady states and falsified kinetics can negatively affect a biofilter and the experimental determination of kinetic parameters, respectively.Implications: The occurrence of multiple steady states in a VOC treating biofilm, shows the significant impact of degradation kinetics and diffusion limitation on the biofilm behavior. Moreover, the implied possible sudden drop of removal efficiency of a biofilter, based on the occurrence of multiple steady states lead to possible bottle-necks in biofilter application and operation.

Identifying the parametric occurrence of multiple steady states for some biological networks

We consider a problem from biological network analysis of determining regions in a parameter space over which there are multiple steady states for positive real values of variables and parameters. We describe multiple approaches to address the problem using tools from Symbolic Computation. We describe how progress was made to achieve semi-algebraic descriptions of the multistationarity regions of parameter space, and compare symbolic and numerical methods.The biological networks studied are models of the mitogen-activated protein kinases (MAPK) network which has already consumed considerable effort using special insights into its structure of corresponding models. Our main example is a model with 11 equations in 11 variables and 19 parameters, 3 of which are of interest for symbolic treatment. The model also imposes positivity conditions on all variables and parameters.We apply combinations of symbolic computation methods designed for mixed equality / inequality systems, specifically virtual substitution, lazy real triangularization and cylindrical algebraic decomposition, as well as a simplification technique adapted from Gaussian elimination and graph theory. We are able to determine semi-algebraic conditions for multistationarity of our main example over a 2-dimensional parameter space. We also study a second MAPK model and a symbolic grid sampling technique which can locate such regions in 3-dimensional parameter space.

Dynamic process behavior and model validation of reactive dividing wall columns

Reactive dividing wall columns (RDWC) are a highly integrated column type. The coupling of a reactive distillation and a dividing wall column leads to complex interactions between vapor-liquid mass transfer, chemical kinetics and component split around the dividing wall. In simulation studies such interactions have been shown to lead to complex dynamics and the occurrence of multiple steady states. However, experimental investigations are still missing to verify the theoretical findings. In this work the first comprehensive experimental study of the dynamic behavior of RDWCs is presented. The reference system of enzymatic catalyzed butyl acetate transesterification with hexanol has been employed to investigate the start-up and open loop behavior under different operating conditions. Two different start-up strategies are tested and compared. The experiments demonstrate the reliable and secure start-up and stable operation of a RDWC. Additionally, a developed rigorous dynamic RDWC model is presented that considers the dynamic influence on the vapor distribution in the dividing wall section. A detailed model validation is carried out using the obtained experimental data. The comparison of simulation results and experimental values show good agreement over a wide range of operating conditions.

Continuation of probability density functions using a generalized Lyapunov approach

Techniques from numerical bifurcation theory are very useful to study transitions between steady fluid flow patterns and the instabilities involved. Here, we provide computational methodology to use parameter continuation in determining probability density functions of systems of stochastic partial differential equations near fixed points, under a small noise approximation. Key innovation is the efficient solution of a generalized Lyapunov equation using an iterative method involving low-rank approximations. We apply and illustrate the capabilities of the method using a problem in physical oceanography, i.e. the occurrence of multiple steady states of the Atlantic Ocean circulation.

Stability and bifurcation in plant–pathogens interactions

We consider a plant–pathogen interaction model and perform a bifurcation analysis at the threshold where the pathogen-free equilibrium loses its hyperbolicity. We show that a stimulatory–inhibitory host response to infection load may be responsible for the occurrence of multiple steady states via backward bifurcations. We also find sufficient conditions for the global stability of the pathogen-present equilibrium in case of null or linear inhibitory host response. The results are discussed in the framework of the recent literature on the subject.

Multiscale Modeling of Biopolymer Production in Multicellular Systems

AbstractIn this contribution we present a multiscale modeling approach to systematically explore the heterogeneity of biopolymer production in multicellular systems. The first step is a dynamic single cell model which is based on the hybrid cybernetic modeling approach to include cell internal regulation. The single cell model is used for nonlinear analysis and the occurrence of multiple steady states is discussed. The single cell model is then reduced by using the lumped hybrid cybernetic approach and by approximation of enzyme levels. The reduced single cell model is used to develop a population balance model for predicting heterogeneity in multicellular production processes. Thus, we have combined the cybernetic modeling approach with population balance modeling, which consequently includes cell internal regulation. The multiscale modeling approach is first developed for the microorganism Ralstonia eutropha, which serves as a model and benchmark organisms. Good agreement between model predictions and experimental data is shown. An extension to Rhodospirillum rubrum is discussed, which shows interesting transient bimodal dynamics.

Experimental and theoretical analysis of poly(β-hydroxybutyrate) formation and consumption in Ralstonia eutropha

Abstract In this paper a mathematical model is presented to describe poly(β-hydroxybutyrate) (PHB) formation and consumption in Ralstonia eutropha. The model is based on the hybrid cybernetic modeling approach, which was introduced by Kim et al. [1] and which allows a systematic derivation of the model equations from elementary mode analysis. An extension of this approach is presented to allow for non quasi-stationary metabolites, i.e. PHB. The model is shown to be in good agreement with experimental data for PHB formation and consumption. The model is used afterwards to discuss the occurrence of multiple steady states in a continuous bio reactor. It is shown that the multiplicity region predicted by the model is rather small and it is argued that multiple steady states are therefore unlikely to occur in practice for this specific system. Due to various desirable features such as accounting for cellular regulation at network level and dynamics of intracellular metabolites with a moderate complexity, it is believed that the constructed model is most suitable for control, optimization and monitoring of industrial PHB production processes.

Separation of Organic Chlorine Compounds through Batch Distillation Process

In the present work simulation studies were carried out focusing on the separation of a mixture of chlorinated hydrocarbons of an industrial process stream by batch distillation process. The effect of different process variables on the performance of the batch separation process was evaluated. Based on the simulation results, it was observed that the column pressure has a significant effect on the condenser operation, indicating that operating pressure equal to 3 bar is more favorable to the process. It was shown that high values of the reboiler heat duty lead to a decrease in the operation time necessary to attain the steady state regime. Regarding the influence of the column reflux, it was also observed that reflux values exceeding 1.0 reduce the separation process performance, leading to a poor recovery of the components in the product tank. It was illustrated the occurrence of multiple steady states as a consequence of nonlinear response of the process to the reboiler holdup, characterized by complex nonlinear phenomena.

Theoretical investigation of steady state multiplicities in solid oxide fuel cells*

The nonlinear steady state behaviour of solid oxide fuel cells (SOFCs) is investigated. It is found that the temperature dependence of the electrolyte conductivity has a very strong influence on the occurrence of multiple steady states, instabilities and the formation of hot spots. Two correlations from the literature for the electrolyte conductivity are studied in a lumped model and in a 1D spatially distributed model of a SOFC. The cases of galvanostatic operation, potentiostatic operation, and operation under a constant ohmic load are considered. The lumped model possesses a unique steady state under galvanostatic operation and up to three steady states under potentiostatic operation or under constant load. In the distributed model, three steady states may coexist under galvanostatic operation and up to five under potentiostatic operation.

Steady State Analysis of Reactive Distillation Using Homotopy Continuation

Homotopy continuation is applied to the industrially important MTBE and methyl acetate reactive distillation (RD) columns to analyse for steady state multiplicities. Continuation is applied in two distinct ways, with respect to tray catalyst mass and with respect to column inputs. Catalyst mass continuation provides a tool for convergence to a steady state solution of the highly non-linear governing equations. Also, the solution diagram reveals the occurrence of multiple steady states. For the MTBE column, up to five steady states in the kinetically controlled regime are detected. Of these, only three remain in the limit of reaction equilibrium on the trays. For the methyl acetate column, three steady states in both the kinetically controlled and reaction equilibrium regimes are obtained at fixed reflux rate and reboiler duty. If the column specification is changed to fixed reflux ratio and reboiler duty, only a single steady state solution is observed. The column operating policy thus substantially impacts the existence of multiple steady states. Column input continuation is used to obtain the complex relationships between input variables (reboiler duty, reflux rate, and so on) and output variables (tray temperatures, conversion, and so on). Steady state multiplicities are seen in both the MTBE and methyl acetate columns. A thorough understanding of these multiplicities is essential for devising robust control strategies.

Investigations of Multiple Steady-States in Continuous Cultivation of Saccharomyces cerevisiae

During aerobic continuous cultivation experiments with Saccharomyces cerevisiae (CEN.PK 113-7D), the occurrence of multiple steady-states were demonstrated using closed loop control of the dilution rate. By specifying the ethanol concentration as a setpoint for the control loop, steady-states were achieved for various ethanol concentrations. The tuning of the dilution rate controller was basedon a structured model of S. cerevisiae. Experimental results showed that when the ethanol setpoint was increased, the corresponding dilution rate, at steady-state, decreased. Thus, it was shown that multiple steady-states do exist for a region of dilution rates just below the critical dilution rate.

Occurrence of Multiple Steady States in the Biphasic Alkaline Hydrolysis of C-4 to C-8 Ethyl Alkanoates

The autocatalytic biphasic alkaline hydrolysis of C-4 to C-8 ethyl alkanoates was studied in a CSTR. In our experimental conditions, the two shorter esters (C-4 and C-5) showed monostable behavior ...

A DYNAMICAL MODEL OF B–T CELL REGULATION

We present a minimal regulatory model for the dynamics of the humoral immune response of two lymphocytes populations (B and T helper) interacting with a specific antigen pool (bacterium). Stability analysis reveals that the system accounts for the occurrence of multiple steady states in the absence as well as in the presence of the antigen population. The model exhibits (i) a state of immune memory, (ii) one state with high antigen and low helper concentration (disease), and (iii) one state with low antigen and high helper concentration (tolerance). The latter state allows oscillatory behavior. Injection of high antigen doses as well as minimal changes of structural parameters provoke the system to jump from the state of disease to the state of tolerance. This is reminiscent of therapies where the patient is treated with allergen, immuno-suppressants or drugs.

Positive feedback circuits and memory

The concept of regulatory feedback circuit refers to oriented cyclic interactions between elements of a system. There are two classes of circuits, positive and negative, whose properties are in striking contrast. Positive circuits are a prerequisite for the occurrence of multiple steady states (multistationarity), and hence, they are involved in all processes showing hysteresis or memory. Endogenous or exogenous perturbations can lead the system to exhibit or to evoke one particular stable regime. The role of positive circuits in cell differentiation and in immunology is well documented. Negative circuits are involved in homeostatic regulation, with or without oscillations. The aim of this paper is to show: a) that positive circuits account for many features of memory stricto sensu (i.e., neural memory and mnesic evocation) as well as largo sensu (e.g. differentiation or immunological memory); and b) that simple combinations of positive and negative circuits provide powerful regulatory modules, which can also be associated in batteries. These entities have vast dynamical possibilities in the field of neurobiology, as well as in the fields of differentiation and immunology. Here we consider a universal minimal regulatory module, for which we suggest to adopt the term ‘logical regulon’, which can be considered as an atom of Jacob's integron. It comprises a positive and a negative circuit in its interaction matrix, and we recall the main results related to the simultaneous presence of these circuits. Finally, we give three applications of this type of interaction matrix. The first two deal with the coexistence of multiple stable steady states and periodicity in differentiation and in an immunological system showing hysteretic properties. The third deals with the dual problems of synchronization and desynchronization of a neural model for hippocampus memory evocation processes.

Steady‐State Multiplicity Phenomena in the Deposition of Silicon Carbide

An experimental investigation of the occurrence of multiple steady states in the chemical vapor deposition of silicon carbide from methyltrichlorosilane, hydrogen, and hydrogen chloride mixtures is presented in this study. Experiments have been carried out over broad temperature, pressure, and composition ranges in a hot‐wall chemical vapor deposition reactor coupled with an electronic microbalance. It has been found that in the absence of hydrogen chloride in the feed, multiple steady states appear at temperatures around 850°C, and that the addition of HCl moves the multiplicity region toward higher temperatures. Multiple steady states have been observed over the whole temperature (800–1400°C) and pressure (20–760 Torr) region covered in our experiments, with the difference in the deposition rate between the upper and lower stable branches exceeding in some cases two orders of magnitude. Since the key parameter involved in the appearance of multiple steady states is the concentration of hydrogen chloride, which depends on the residence time of the reacting mixture in the reactor, the temperature profile, and the reactor geometry, the poor agreement among deposition rate data reported in the literature by different groups may be partly due to the multiple steady‐states phenomenon. The appearance of multiplicity is explained qualitatively in terms of a simplified model of SiC deposition.

Uniqueness and multiplicity of steady states in monocyclic enzyme cascades: a graph-theoretic analysis

Monocyclic enzyme cascades are important regulators of biochemical reactions in living cells. The reaction network may have either one steady state or many, depending on its structure. The occurrence of multiple steady states has important biological implications. A simple graph-theoretic method has been applied to five reaction mechanisms—which together cover many common monocyclic cascades—to determine which mechanisms generate just one steady state and which ones allow more than one state. It is shown that an unstable steady state in a multiplicity region can be usefully exploited and that in some cases transitions may occur between uniqueness and multiplicity regions. The possible effects of such transitions are discussed.

Flow Field and Non-Isothermal Effects on Diffusion, Convection, and Reaction in Permeable Catalysts

We analyze non-isothermal effects on the efficiency of large-pore catalyst particles, where intraparticle convection may be important. The particle permeability is taken into account by studying the flow field within the particle simultaneously with the mass and the energy transport processes. Two particle geometries, slab and sphere, are considered. Non-isothermal effects are observed on the occurrence of multiple steady states due to reaction ignition. It is found that, when the flow field equations are taken into account, the minimum values of the efficiency and the minimum temperatures reached inside the particle are lower than the ones estimated considering just the mass and energy transport equations

ON THE DYNAMICS OF THE COMBUSTION OF SINGLE CHAR PARTICLES

Diffusion and reaction models of varying complexity are used to study the stability of the combustion of single char particles. In contrast to past studies which have considered only the reaction of carbon with oxygen, both the heterogeneous reactions of carbon with oxygen and carbon dioxide and the homogeneous oxidation of carbon monoxide are taken into account in the formulation of the mathematical models. Emphasis is placed on the investigation of the feasibility of occurrence of oscillatory combustion. Our results show that high rates of the C-O2 reaction and high concentration of O2 in the ambient favor the occurrence of multiple steady states and oscillatory solutions, but the appearance of oscillatory instability is suppressed by the homogeneous reaction in the gas phase and the presence of CO and CO2 in the ambient. The parametric investigation of the problem reveals, in agreement with the results of past studies, that the heat capacity of the porous solid, the Lewis number, and the thickness of t...