Region Of Multiple Steady States Research Articles

Analysis of temperature patterns in shallow-bed autothermal catalytic reactors

Stationary transverse temperature patterns have been observed in packed-bed catalytic reactors used to carry out exothermic reactions in the autothermal mode of operation. In the literature, pattern formation was explained using physical property variations and different rates of heat and mass transport by utilizing continuum models but with less emphasis on the impact of thermokinetic multiplicity at the catalyst particle length scale. In this work, we show that continuum models with uniform averaged bed properties (e.g. effective radial conductivity) are unable to predict stable patterns whenever multiplicity exists at the cell or particle length scale. In such cases, we reason that cell models that incorporate and properly represent the appropriate physical phenomena can predict stable patterns that arise due to multiplicity at the particle or cell length scale. We analyze the bounds for stable pattern formation in shallow-bed reactors operated in the region of multiple steady-states and present a procedure for determining the bed conductivity needed for eliminating the patterned states. The general cell models analyzed incorporate physical property variations and approach the continuum limit for vanishingly small cell size. Our analysis shows that the inclusion of physical property variation compounds the impact of multiple mechanisms on pattern formation.

Multiple Steady and Oscillatory Solutions in a Collapsible Channel Flow

In this paper we study flow driven through a finite-length planar rigid channel by a fixed upstream flux, where a segment of one wall is replaced by a pre-stressed elastic beam subject to uniform external pressure. The steady and unsteady systems are solved using a finite element method. Previous studies have shown that the system can exhibit three steady states for some parameters (termed the upper, intermediate and lower steady branches, respectively). Of these, the upper and lower are stable to non-oscillatory perturbations, while the intermediate branch is always unstable. These upper and lower steady branches can (independently) become unstable to self-excited oscillations. We show that for some parameter combinations the system is unstable to both upper and lower branch oscillations simultaneously. However, we show that these two instabilities eventually merge together for large enough Reynolds numbers, exhibiting a nonlinear limit cycle which retains characteristics of both the upper and lower branches of oscillations. Furthermore, we show that increasing the beam pre-tension suppresses the region of multiple steady states but preserves the onset of oscillations. Conversely, increasing the beam thickness (a proxy for increasing bending stiffness) suppresses both multiple steady states and the onset of oscillations.

Study of reaction\u2013diffusion problem: modeling, exact analytical solution, and experimental verification

Nonlinear diffusion–reaction problem was investigated experimentally for the reference reaction (hydrogenation of propylene under isothermal conditions; a slab of catalyst pellet i.e., disks of large diameter/width ratio were applied). The diffusion–reaction model in the catalyst pellet with external mass-transfer resistances was solved analytically. Dependently on parameters values, two separate solutions were found: dead zone inside the pellet does not exist or it appears. In the first case, a common model is acceptable (regular model i.e., boundary value problem), in the second one, a model includes additional condition (dead zone model i.e., free boundary problem). Analysis of the solution presented indicated that either regular or “dead zone” model correctly describes the process for specific parameter values (with the only exception—multiple steady-state region—where the correct interpretation requires the combined application of the both). This result shows that the full description of the real process includes solutions of two different BVPs. Experimental research confirmed results anticipated by theory. It allowed to draw conclusions that go beyond this particular example i.e., the regular model, commonly applied in heterogeneous catalysis, does not adequately recognize dead zone problem. If “dead zone” appears, free boundary problem has to be consider, otherwise, process simulations will be incorrect. The conclusions drawn are valid also for biofilms.

Revealing regions of multiple steady states in heterogeneous catalytic chemical reaction networks using Gröbner basis

The Chemical Reaction Network Theory (CRNT) formalism classifies any CRN using a nonnegative index called deficiency. If the deficiency is equal to or greater than one, and the CRN fulfills some structural conditions, then it is possible to establish whether or not the set of Mass Action Kinetics Ordinary Differential Equations (MAK-ODEs) induced by the CRN can support Multiple Positive Steady States (MPSS) within each stoichiometric compatibility class. If MPSS exists the CRNT provides a set of kinetic constants for such steady states. Most of the heterogeneous catalytic CRNs are of deficiency one, thus the CRNT applies straightforwardly. Moreover, the balance of catalytic sites defines a stoichiometric compatibility class. In this contribution, CRNT is used along with Gröbner basis to derive analytical polynomial expressions depending on a measurable chemical species (e.g. the product of interest) and the catalytic site concentration. This methodology reveals the exact regions where MPSS are supported. Putative heterogeneous catalytic CRNs, such as Eley–Rideal and Langmuir–Hinshelwood, are analyzed using this methodology.

Mechanism for multiplicity of steady states with distinct cell concentration in continuous culture of mammalian cells.

Continuous culture for the production of biopharmaceutical proteins offers the possibility of steady state operations and thus more consistent product quality and increased productivity. Under some conditions, multiplicity of steady states has been observed in continuous cultures of mammalian cells, wherein with the same dilution rate and feed nutrient composition, steady states with very different cell and product concentrations may be reached. At those different steady states, cells may exhibit a high glycolysis flux with high lactate production and low cell concentration, or a low glycolysis flux with low lactate and high cell concentration. These different steady states, with different cell concentration, also have different productivity. Developing a mechanistic understanding of the occurrence of steady state multiplicity and devising a strategy to steer the culture toward the desired steady state is critical. We establish a multi-scale kinetic model that integrates a mechanistic intracellular metabolic model and cell growth model in a continuous bioreactor. We show that steady state multiplicity exists in a range of dilution rate in continuous culture as a result of the bistable behavior in glycolysis. The insights from the model were used to devise strategies to guide the culture to the desired steady state in the multiple steady state region. The model provides a guideline principle in the design of continuous culture processes of mammalian cells.

Multiple steady states and transitional regimes in a cylindrical fixed-bed catalytic reactor

Heterogeneous catalytic reactions in a cylindrical axial-flow fixed-bed reactor have been investigated using a two-phase mathematical model. Regions of multiple steady states have been mapped for basic process parameters, namely, the flow velocity and coolant temperature. Transitions from one steady state to another and possible pathways of the establishment of various steady states have been examined.

Operating strategies for Fischer-Tropsch reactors: A model-directed study

A comprehensive parametric study for a Fischer-Tropsch (FT) synthesis process has been conducted to investigate the relation between process parameters and reactor characteristics such as conversion, selectivity, multiplicity, and stability. A flexible model was employed for this purpose, featuring the dependence of Anderson-Shultz-Flory (ASF) factor on composition and temperature. All variable process parameters in industrial FT reactors were subject to variation, including reaction temperature, reactor pressure, feed ratio, inlet mass flux, feed temperature, heat transfer coefficient, catalyst concentration, catalyst activity, etc. While typical trade-off was encountered in most cases, i.e., the change of a parameter in one direction enhances one aspect but deteriorating another, the change of feed conditions gave some promising results. It has been found that decreasing the feed rate (or increasing the residence time) and/or lowering the feed concentration can successfully enhance the conversion up to more than 90% for our specific case, without hurting the product selectivity as well as effectively condense the region of multiple steady states. The benefits and limitations accompanied with the variation of the parameters were discussed in detail and a rational start-up strategy was proposed based on the preceding results. It is shown that the decrease of inlet mass flux (say, 85% decrease of the feed rate or 60% decrease of the feed concentration from the nominal condition chosen here) or the decrease of H2/CO ratio (specifically, below about 0.25), or their combination can eliminate multiple steady states. The resulting unique relation between temperature and manipulated variable (i.e., coolant flow rate) appears to assure a safe arrival at the target condition at the start-up stage.

Approximate model for diffusion and reaction in a porous pellet and an effectiveness factor

In the present work the approximate model for the single reaction with a non-linear kinetic equation is utilized for the calculation of an effectiveness factor value. The values obtained were compared with those obtained by numerical solution of the exact model and by using approximate methods reported in the literature. The accuracy of calculations of effectiveness factor values using the approximate model is good and the method handles the multiple steady-state region.

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.

Parametric Analysis of the Simplest Model of the Theory of Thermal Explosion—the Zel'dovich—Semenov Model

Parametric analysis is performed for the Zel'dovich–Semenov model, a basic model of the theory of thermal explosion that describes the dynamics of an exothermic reaction with arbitrary kinetics in a continuous stirred tank reactor. Particular attention is given to the cases of first‐ and nth‐order reactions, oxidation reaction, and reaction with arbitrary kinetics. Parametric dependences of steady states on dimensionless parameters, curves of multiplicity and neutrality of steady states, and parametric and phase portraits of the system are constructed. Regions of multiple steady states and self‐oscillations and a region of technological safety are distinguished.

Parametric Analysis of the Models of a Stirred Tank Reactor and a Tube Reactor

A parametric analysis is performed for a basic model of the combustion theory with dimensional parameters. The model describes the dynamics of a first‐order exothermic reaction in a well‐stirred tank reactor. Parametric dependences of steady states on dimensional parameters, curves of multiplicity and neutrality of steady states, and parametric and phase portraits of the system are constructed. Regions of multiple steady states and self‐excited oscillations and a region of technological safety are distinguished. Combustion regimes in a continuous stirred tank reactor are compared to those in a tube reactor.

Generation of oscillatory solutions in a porous catalyst pellet

A stability analysis concerning bifurcation to periodic solutions of steady-state solutions has been performed for a model which describes transport phenomena and chemical reaction in a porous catalyst pellet. Extensive numerical simulations have revealed the existence of three scenarios for the generation of self-excited oscillations of the mixture temperature and composition inside the pellet. These are the supercritical and subcritical bifurcations to oscillatory solutions, which occur in the region of single steady-states, and the homoclinic bifurcation in the region of multiple steady-states.

Microscopic simulation of chemical bistability in homogeneous systems

Microscopic simulation is used to clarify the status of stochastic theories of homogeneous chemical systems operating in the multiple steady state region. The results demonstrate the failure of the Langevin approach, but show excellent agreement with the master equation formulation.

Multiplicity pattern in the nonisothermal nonadiabatic packed bed chemical reactor

The using of the continuation methods for studying the static bifurcation behaviour of the two-dimensional one phase model of the packed tubular reactors is discussed. Practical computational aspects for carrying out this analysis are presented in some detail. The mathematical model is discretized using a compact fourth order finite difference scheme. Three continuation algorithms are tested. As linear solver in the continuation algorithm, the GMRES method is used. The analysis of the region of multiple steady states is made for the nonisothermal-nonadiabatic regime. The key parameter is the Damkohler number. Some results concerning the influence of the dimensionless adiabatic temperature rise and that of the Biot number are presented. Two values of the axial mass Peclet number greater than 100 are considered.

Multiple steady states in binary distillation—Theoretical and experimental results

Steady-state multiplicity and instability in distillation is studied theoretically and experimentally for columns with volumetric reflux. Using nonlinear wave propagation theory for distillation processes, we derive simple expressions for the prediction of multiplicity regions in parameter space with paper and pencil. This method also gives further insight into qualitative column dynamics in the region of multiple steady states. The results are illustrated for ideal binary mixtures and further conclusions are drawn for nonideal binary mixtures and ideal or nonideal multicomponent mixtures. With regard to experimental work, a rigorous, numerical bifurcation analysis of more detailed models is presented for the separation of methanol-1-propanol. The interaction between the boundary set of feasible operation and the bifurcation set is shown to be of great importance for the understanding of multiplicity and stability features for this type of column configuration. Finally, these multiple steady states are confirmed experimentally.

Effect of biofilm thickness distribution on substrate-inhibited kinetics

The effect of biofilm thickness distribution (BTD) on substrate-inhibited kinetics in anaerobic filters was evaluated. The modeling results show that the kinetic behavior of normal BTD is different from that of uniform biofilm thickness when the substrate utilization rate of some biofilms is limited by diffusion or reaction. However, when all biofilms are either diffusion-limited or reaction-limited, kinetic behaviors of normal BTD and uniform biofilm thickness are identical. In addition, the BTD can reduce the apparent maximum substrate utilization rate and narrow the multiple steady-state region of substrate-inhibited kinetics. In this study, phenol was used as an inhibitory substrate. A general agreement was found between the numerical simulation results using the normal BTD and experimental results for anaerobic filters. On the contrary, numerical simulation using uniform biofilm thickness failed to predict the outcome of the experiments. This indicates that an anaerobic filter is composed of a biofilm with varying thickness and thus, the BTD of a filter is of great importance, especially for treating wastewater containing inhibitory substrates.

A generalized analytical method for determining multiplicity features in chemical reactors with recycle

An analytical method is developed and illustrated for determining the regions of multiple steady states of a tubular adiabatic reactor with the recycle of the product. The procedure is general (it allows for a single reaction described by any kinetic expression) and does not require an iterative matching of the boundary conditions resulting from the recycle. The method enables the calculation of the so-called hysteresis varieties and the catastrophic sets determining the occurence of multiple steady states.

The effect of hydrogen ion on the steady-state multiplicity of substrate-inhibited enzymatic reactions. II. Transient behavior.

In this paper we concentrate our attention on the stability and transient behavior of the isothermal system (CSTR) with a substrate-inhibited enzyme reaction producing hydrogen ions. Our investigation covers the region of multiple steady states uncovered previously (1) (ordinary hysteresis and isola). We investigate the local stability characteristics of the different steady states, the effect of the initial condition on the transient behavior and the response of the system to feed disturbances of various magnitudes and durations.

Isothermal diffusion—reaction in a slab catalyst with bimolecular Langmuir-Hinshelwood kinetics: Connections with negative first-order kinetics

The problem of isothermal diffusion-reaction in a slab catalyst with bimolecular Langmuir—Hinshelwood kinetics is studied. Analytic solutions for the concentration profile and the catalytic effectiveness factor are reported for the full range of Thiele modulus values, in the limit of large adsorption inhibition constant. In this limit, connections with negative first-order kinetics are shown, and rather good explicit estimates of the region of multiple steady states are provided. Comparisons with numerical calculations are also made.

Explicit multiplicity criteria for first-order catalytic reactions with external transport limitations

Explicit approximate bounds on the region of multiple steady states for a non-isothermal first-order reaction ocurring in infinite slab or spherically Two different approximations are used. In the case of the slab, both approximations provide rigorous lower bounds on the values of Thiele modulus withi