Steady-state multiplicity and bifurcation analysis via the newton polyhedron approach

A method based on the Newton polyhedron determines bifurcation of solution branches for simultaneous algebraic equations. This method has broad applicability in multiple steady-state analysis in chemical engineering. Reduction of the steady-state equations to a single nonlinear equation is not required. The method provides in nondegenerate cases the normal form of the algebraic equations—the leading nonlinear terms which alone determine steady-state multiplicity. Bifurcation conditions are developed for isothermal reaction between two adsorbed species in a catalytic CSTR and for two and three parallel reactions of arbitrary order in a nonisothermal CSTR.     

Steady-state multiplicity analysis of two-stage-riser catalytic pyrolysis processes

New two-stage-riser fluidized catalytic pyrolysis (TSRFCP) for maximizing propylene yield technology is considered as an efficient route to moderate the propylene demand/supply gap and to lower the propylene price. The possibility of existence of complex nonlinear behavior associated with the TSRFCP process puts limitations on the supervision of this system. Based on the developed and validated model for the TSRFCP process, this paper focuses on the elucidation of multiple steady states and relevant (in) stability characteristic over a wide range of operating condition. First, graphic analysis of heat generation/removal curves demonstrates that the TSRFCP process has at least one steady state and a maximum of three output steady states under the considered operating conditions and uncertainties such as cooling water flow rates and Conradson carbon residue. Then, operating maps revealing topologies between important input and output variables can disclose detailed nonlinear behavior (input/output multiplicity). Moreover, depending on the choice of the input variable and the relevant operating/design condition, input multiplicity may exist. In short, these results can guide the succeeding control structure selection for realistic TSRFCP processes.     

Steady-state multiplicity in a solid oxide fuel cell: Practical considerations

Steady-state multiplicity in a solid oxide fuel cell (SOFC) in three modes of operation, constant ohmic external load, potentiostatic and galvanostatic, is studied using a detailed first-principles lumped model. The SOFC model is derived by accounting for heat and mass transfer as well as electrochemical processes taking place inside the fuel cell. Conditions under which the fuel cell exhibits steady state multiplicity are determined. The effects of operating conditions such as convection heat transfer coefficient and inlet fuel and air temperatures and velocities on the steady state multiplicity regions are studied. Depending on the operating conditions, the cell exhibits one or three steady states. For example, it has three steady states: (a) at low external load resistance values in constant ohmic external load operation and (b) at low cell voltage in potentiostatic operation.     

Dynamic models of metabolism: Review of the cybernetic approach

The cybernetic approach to metabolic modeling tracing its progress from its early beginnings to its current state with regard to its relationship to other modeling approaches, applications to bioprocess modeling, metabolic engineering, and future prospects are described. The framework is shown to handle large metabolic networks in making dynamic predictions from limited data with looming prospects of extending to genome scale networks. © 2012 American Institute of Chemical Engineers AIChE J, 2012     

Steady-state multiplicity in the autohumidification polymer electrolyte membrane fuel cell

“Ignition/extinction” phenomena and steady-state multiplicity were discovered in an autohumidification polymer electrolyte membrane fuel cell. At steady state, the water produced by the fuel cell reaction is balanced by water removal by the flowing reactant gas streams. Ignition, corresponding to a high fuel cell current, arises from positive feedback between the water produced by the reaction and the transport of protons in the membrane. A critical level of membrane hydration is required for ignition; insufficient membrane hydration will extinguish the fuel cell current. This new autocatalytic mechanism has an interesting analogy to the autothermal reactor.     

Growth processes in a cascade of bioreactors: Mathematical models

Mathematical modeling of continuous multiple bioreactors is complicated by two factors. First, the chemical environments differ in different reactors, so biomass receives an environmental shock when it is transferred between reactors. Second, biomass occurs as discrete cells, which of course are and remain segregated from one another, so an element of biomass that enters a reactor does not mix with the biomass already present. This differs from the behavior of an element of liquid which enters a reactor in that such an element quickly mixes with the liquid already present. The biomass in a bioreactor receiving biomass from an external source is therefore heterogeneous with respect to the history of environmental conditions and composition. This article shows how to construct a mathematical model of multiple bioreactor apparatus that accounts for these complications. It also describes simpler models that do not account for both of them.     

Steady-state multiplicity of a packed-bed reactor operating under a fixed pressure drop

The change in the physical properties of the fluid in a packed-bed reactor operating under a fixed pressure drop may lead to steady-state multiplicity, with each state corresponding to a different inlet velocity. Using a pseudo-homogeneous plug flow model with no axial dispersion we determine the conditions under which multiplicity occurs for a zeroth-order reaction.     

Use of bifurcation analysis for development of nonlinear models for control applications

Nonlinear system identification poses challenging questions because a closed general theory is not available for this field. Particularly, nonlinear models based on neural networks (NN) may present incompatible general dynamic process behavior, leading to improper closed-loop responses, even when they allow for satisfactory one step ahead prediction of process dynamics, as required by traditional validation methods. It is shown here that performing detailed bifurcation and stability analysis may be very helpful for the adequate development and implementation of nonlinear models and model based controllers. The study of many parameters that are defined a priori during the training of the NN shows that the spurious dynamic behavior is related mostly to the use of incomplete data sets during the learning process. This is an indication that, for each kind of process, the number, range and distribution of the data points in the operation region of interest are of paramount importance for proper training of the nonlinear model. Strategies to improve the quality of the training procedure are provided and analyzed both theoretically and experimentally, using the solution polymerization of styrene in a tubular reactor as a case study.     

Hopf bifurcation and solution multiplicity in a model for destabilized Bridgman crystal growth

Flow instabilities are analyzed within a destabilized vertical Bridgman crystal growth system, first studied experimentally by Kim et al. (J. Electrochem. Soc. 119(1972) 1218), using a distributed-parameter model consisting of balance equations for energy and momentum transport. Numerical solution of the governing equations via a Galerkin finite element method reveals multiple operating states and dynamic phenomena. Bifurcation analysis shows that the onset of time-periodic flows occurs in the model system via a supercritical Hopf bifurcation, consistent with prior experimental observations on the dynamics of flow in similar systems.     

Steady-state multiplicity of lumped-parameter systems in which two consecutive or two parallel irreversible first-order reactions occur

This work examines and classifies the steady-state multiplicity features of lumped-parameter systems in which either two consecutive or two parallel, irreversible, first-order reactions are carried out. Several new and surprising features were discovered such as the occurrence of steady-state multiplicity for all Damköhler numbers in certain cases. The multiplicity may occur only if at least one of the reactions is exothermic. Simple criteria are presented for a priori prediction of the conditions which guarantee steady-state uniqueness or multiplicity.     

Parameter estimation in some diffusion and reaction models: An application of bifurcation theory

We consider a class of nonlinear diffusion and reaction models involving an unknown parameter, e.g. activation energy in the case of classical thermal combustion phenomena[1], or the Thiele modulus in the case of enzyme kinetics[2]. We investigate how the knowledge of a critical (bifurcation) value leads to an estimate of the parameter. In practical situations, the critical value usually corresponds to a jump in the solution diagram when hysteresis is observed, and can thus be easily determined. We give an algorithm to estimate the parameter and test it on two model problems.     

Steady-state analysis of high temperature fuel cells

A mathematical model is presented to describe the steady state behavior of high temperature solid electrolyte fuel cells. The resulting equations are solved for the case of anodic oxidation of hydrogen and carbon monoxide. Similar to chemical reactors, fuel cells are found to exhibit steady-state multiplicity over a wide range of parameters. The paper discusses the relative importance of the pertinent design and operating parameters in order to maintain ignited steady states corresponding to high current densities and nearly complete fuel conversion.     

Periodic behaviour of a class of unstructured kinetic models for continuous bioreactors

The periodic behaviour of a large dass of unstructured kinetic models for continuous bioreactors is analyzed using elementary concepts of singularity theory and continuation techniques. The class consists of models for which the utilization rate of the limiting substrate is linearly related to the rates of cell growth and product formation. The model kinetics are allowed, on the other hand, to depend on substrate, biomass and product. The stability analysis allows the derivation of general analytical conditions for the occurrence of periodic behaviour in these models. It is shown that for a number of important cases, the occurrence of oscillatory behaviour is conditioned mainly by the kinetics of product formation. The singularity theory also allows the construction of a useful picture in the multidimensional parameter space delineating the different behaviour these models can predict including bistability and stable oscillatory behaviour.     

Comparison of two IT DIR-SOFC models: Impact of variable thermodynamic, physical, and flow properties. Steady-state and dynamic analysis

This paper compares two dynamic, one-dimensional models of a planar anode-supported intermediate temperature (IT) direct internal reforming (DIR) solid oxide fuel cell (SOFC): one where the flow properties (pressure, gas stream densities, heat capacities, thermal conductivities, and viscosity) and gas velocities are taken as constant throughout the system, based on inlet conditions, and one where this assumption is removed to focus on the effect of considering the variation of local flow properties on the prediction of the fuel cell performance. The refined model consists of mass, energy, and momentum balances, and of an electrochemical model that relates the fuel and air gas compositions and temperatures to voltage, current density, and other relevant fuel cell variables. Simulations for steady-state and dynamic conditions have been carried out and the results obtained from the two models compared. For a co-flow SOFC operating on a 10% pre-reformed methane fuel mixture, with 75% fuel utilisation, inlet fuel and air temperatures of 1023 K, average current density of , and an air ratio of 8.5, the results show that, although the error incurred in the prediction of the flow properties in the first model is significant, there is good agreement between both models in terms of the overall cell performance: the maximum difference in the local temperature values is about 7 K and the cell efficiency differs by less than 1%. However, the discrepancies between the two models increase, especially in the fuel channel, when higher current density values are assigned to the cell.     

Dynamic optimization of bioreactors using probabilistic tendency models and Bayesian active learning

Due to the complexity of metabolic regulation, first-principles models of bioreactor dynamics typically have built-in errors (structural and parametric uncertainty) which give rise to the need for obtaining relevant data through experimental design in modeling for optimization. A run-to-run optimization strategy which integrates imperfect models with Bayesian active learning is proposed. Parameter distributions in a probabilistic model of bioreactor performance are re-estimated using data from experiments designed for maximizing information and performance. The proposed Bayesian decision-theoretic approach resorts to probabilistic tendency models that explicitly characterize their levels of confidence. Bootstrapping of parameter distributions is used to represent parametric uncertainty as histograms. The Bajpai & Reuss bioreactor model for penicillin production validated with industrial data is used as a representative case study. Run-to-run convergence to an improved policy is fast despite significant modeling errors as long as data are used to revise iteratively posterior distributions of the most influencing model parameters.     

Coupling of biokinetic and population balance models to account for biological heterogeneity in bioreactors

The development of a population balance model accounting for cell adaptation to a fluctuating environment is focussed in this article. In a bioreactor, the substrate concentration field and the bioreaction kinetics are strongly coupled. The latter are determined by the intensity and magnitude of concentration fluctuations encountered along the cell trajectory. Modeling these interactions between hydrodynamics and biology in heterogeneous bioreactors is a major challenge. This model is based on a previous work regarding the dynamic response of bioreactors. It is shown that a simplified population balance equation considering only growth and adaptation is sufficient to reproduce the population growth rate dynamics in batch and continuous cultures. Finally, a validation of the model implementation in a computational fluid dynamics software is proposed. The model developed in the homogeneous case now allows the numerical scale‐up of a bioreactor, for it connects the population state to the concentration changes experienced. © 2012 American Institute of Chemical Engineers AIChE J, 59: 369–379, 2013     

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.