Nonlinear Reaction Mechanisms Research Articles

Modelling of the thyroid hormone synthesis as a part of nonlinear reaction mechanism with feedback

The synthesis of thyroid hormones in the hypothalamic-pituitary-thyroid (HPT) axis was studied. For this purpose, a reaction model for HPT axis with stoichiometric relations between the main reaction species was postulated. Using the law of mass action, this model has been transformed into a set of nonlinear ordinary differential equations. This new model has been examined by stoichiometric network analysis (SNA) with the aim to see if it possesses the ability to reproduce oscillatory ultradian dynamics founded on the internal feedback mechanism. In particular, a feedback regulation of TSH production based on the interplay between TRH, TSH, somatostatin and thyroid hormones was proposed. Besides, the ten times larger amount of produced T4 with respect to T3 in the thyroid gland was successfully simulated. The properties of SNA in combination with experimental results, were used to determine the unknown parameters (19 rate constants of particular reaction steps) necessary for numerical investigations. The steady-state concentrations of 15 reactive species were tuned to be consistent with the experimental data. The predictive potential of the proposed model was illustrated on numerical simulations of somatostatin influence on TSH dynamics investigated experimentally by Weeke et al. in 1975. In addition, all programs for SNA analysis were adapted for this kind of a large model. The procedure of calculating rate constants from steady-state reaction rates and very limited available experimental data was developed. For this purpose, a unique numerical method was developed to fine-tune model parameters while preserving the fixed rate ratios and using the magnitude of the experimentally known oscillation period as the only target value. The postulated model was numerically validated by perturbation simulations with somatostatin infusion and the results were compared with experiments available in literature. Finally, as far as we know, this reaction model with 15 variables is the most dimensional one that have been analysed mathematically to obtain instability region and oscillatory dynamic states. Among the existing models of thyroid homeostasis this theory represents a new class that may improve our understanding of basic physiological processes and helps to develop new therapeutic approaches. Additionally, it may pave the way to improved diagnostic methods for pituitary and thyroid disorders.

SUPG-YZβ computation of chemically reactive convection-dominated nonlinear models

In this computational study, we deal with the stabilized finite element solutions of convection-dominated models having nonlinear reaction mechanisms. The existence of advection terms in these models causes the numerical solutions obtained by standard discretization methods to exhibit nonphysical oscillations. The Galerkin finite element method is stabilized using the Streamline-Upwind/Petrov–Galerkin formulation to avoid such spurious oscillations. The stabilized formulation is also complemented with the YZβ shock-capturing technique to resolve steep gradients and discontinuities accurately. The nonlinear equation systems arising from the spatial discretizations are solved with the Newton–Raphson method supplemented with the ILU-preconditioned GMRES search technique. The proposed methods are tested on a comprehensive set of nonlinear reactive models. Numerical experiments show that the presented methods and techniques eliminate spurious oscillations significantly and resolve strong gradients accurately. All computations are performed in the FEniCS environment.

TAP analysis of single and double peak responses during CO oxidation over Pt

Critical phenomena in heterogeneous catalysis have attracted a considerable interest throughout history due to their intriguing dynamical complexity. The coordinated interplay of various phenomena, such as surface interactions, nonlinear reaction mechanisms and the transport of energy and mass, determine the likelihood of dynamic patterns in certain catalytic systems. In this work, pulse experiments performed with a Temporary Analysis of Products (TAP) reactor showcase the complex dynamic behavior of CO oxidation over a Pt catalyst and exhibit its dependence on the pulsed CO/O2 ratio and the local bed temperature. Pulse responses exhibit a dynamic transition between two critical values of the reaction rate for feed mixtures richer in O2 due to the controlled interactions of the adsorbed species on the surface of the catalyst.

Higher Order Implicit Scheme for Nonlinear Time-Dependent Convection-Diffusion- Reaction Equation

A mathematical model comprising of nonlinear reaction, diffusion, and convection mechanisms seen in natural and anthropogenic processes is numerically investigated here. It is proposed that a higher order numerical scheme of finite difference method be used in conjunction with an iterative approach in order to solve the nonlinear one dimensional convection-diffusion-reaction equation. To account for the wide variety of physical characteristics and boundary conditions, an iterative approach is presented that yields a reliable and precise solution every time. We examined the accuracy and operational efficiency of two distinct finite difference approaches. The efficiency of the system is determined by comparing the estimated results to the appropriate analytical solution by adhering to established norms. Coherence and convergence were analyzed for each approach. The simulation results demonstrate the efficacy and accuracy of these methods in solving nonlinear convection- diffusion-reaction equations. Convection-diffusion-reaction equation modeling is critical for employing the offered results in heat and mass transport processes.

Space-time adaptive ADER discontinuous Galerkin schemes for nonlinear hyperelasticity with material failure

We are concerned with the numerical solution of a unified first order hyperbolic formulation of continuum mechanics that goes back to the work of Godunov, Peshkov and Romenski [65,68,97] (GPR model) and which is an extension of nonlinear hyperelasticity that is able to describe simultaneously nonlinear elasto-plastic solids at large strain, as well as viscous and ideal fluids. The proposed governing PDE system also contains the effect of heat conduction and can be shown to be symmetric and thermodynamically compatible, as it obeys the first and second law of thermodynamics. In this paper we extend the GPR model to the simulation of nonlinear dynamic rupture processes, which can be achieved by adding an additional scalar to the governing PDE system. This extra parameter describes the material damage and is governed by an advection-reaction equation, where the stiff and highly nonlinear reaction mechanisms depend on the ratio of the local equivalent stress to the yield stress of the material. The stiff reaction mechanisms are integrated in time via an efficient exponential time integrator. Due to the multiple spatial and temporal scales involved in the problem of crack generation and propagation, the model is solved on space–time adaptive Cartesian meshes using high order accurate discontinuous Galerkin finite element schemes endowed with an a posteriori subcell finite volume limiter. A key feature of our new model is the use of a twofold diffuse interface approach that allows the cracks to form anywhere and at any time, independently of the chosen computational grid, which is simply adaptive Cartesian (AMR). This is substantially different from many fracture modeling approaches that need to resolve discontinuities explicitly, such as for example dynamic shear rupture models used in computational seismology, where the geometry of the rupture fault needs to prescribed a priori. We furthermore make use of a scalar volume fraction function α that indicates whether a given spatial point is inside the solid (α=1) or outside (α=0), thus allowing the description of solids of arbitrarily complex shape. We show extensive numerical comparisons with experimental results for stress-strain diagrams of different real materials and for the generation and propagation of fracture in rocks and pyrex glass at low and high velocities. Overall, a very good agreement between numerical simulations and experiments is obtained. The proposed model is also naturally able to describe material fatigue.

Identification of Hammerstein-Weiner models for nonlinear MPC from infrequent measurements in batch processes

Here we introduce a new approach for the identification of Hammerstein-Wiener (H–W) models from infrequent measurements in batch processes. Because concentration measurements during each batch run are very infrequent, such models cannot be effectively estimated directly from the infrequent measurements. This difficulty is bypassed by first developing a dynamic response surface methodology (DRSM) model. One can also calculate the optimal trajectories using this DRSM model and identify the local H-W model around such optimal or any other operational trajectory for control purposes. This is obtained by frequently “sampling” the DRSM model around these trajectories. We demonstrate the efficacy of the proposed approach using the simulation of an isothermal semi-batch reactor with nonlinear reaction mechanism. The MPC controller using the nonlinear H-W model is able to outperform the same type of controller using a linear recursive model identified directly from the original infrequent measurements.

Including real fuel chemistry in LES of turbulent spray combustion

Large Eddy Simulation (LES) is progressively becoming a crucial design tool for the next generation of aeronautical combustion chambers. However, further improvements of the predictive capability of LES is required especially for predictions of pollutant formation. In general, the exact description of real fuel combustion requires to take into account thousands of unique chemical species involved in complex and highly non-linear chemical reaction mechanisms, and the direct integration of such chemistry in LES is not a viable path because of excessive computational demands and numerical stiffness. Modeling of real aeronautical transportation fuel is further complicated by the fact that kerosenes are complex blends of a large number of hydrocarbon compounds and their exact composition is very difficult to determine. In this work, we propose a new framework relying upon the Hybrid Chemistry (HyChem) approach and Analytically Reduced Chemistry (ARC) to allow a direct integration of real fuel chemistry in the compressible LES solver AVBP. The HyChem-ARC model is coupled with the Dynamically Thickened Flame LES model (DTFLES) and a Lagrangian description of the spray to investigate the turbulent two-phase flow flame in a lean direct injection combustor, fueled with Jet-A. The LES results are compared to experimental data in terms of gas velocity, temperature and species (CO2, H2O, CO, NO) mass fractions. It is found that the proposed methodology leads to very satisfying predictions of both the flow dynamics and the NOx levels. Additionally, the refined level of chemistry description enables to gain valuable insights into flame/spray interactions as well as on the NOx formation mechanism in such complex flame configurations. To improve further the results, a more detailed experimental characterization of the liquid fuel injection should be provided.

Time-space adaptive discontinuous Galerkin method for advection-diffusion equations with non-linear reaction mechanism

In this work, we apply a time-space adaptive discontinuous Galerkin method using the elliptic reconstruction technique with a robust (in P\'eclet number) elliptic error estimator in space, for the convection dominated parabolic problems with non-linear reaction mechanisms. We derive a posteriori error estimators in the $L^{\infty}(L^2)+L^2(H^1)$-type norm using backward Euler in time and discontinuous Galerkin (symmetric interior penalty Galerkin (SIPG)) in space. Numerical results for advection dominated reactive transport problems in homogeneous and heterogeneous media demonstrate the performance of the time-space adaptive algorithm.

Discontinuous Galerkin finite element methods with shock-capturing for nonlinear convection dominated models

In this paper, convection-diffusion-reaction models with nonlinear reaction mechanisms, which are typical problems of chemical systems, are studied by using the upwind symmetric interior penalty Galerkin (SIPG) method. The local spurious oscillations are minimized by adding an artificial viscosity diffusion term to the original equations. A discontinuity sensor is used to detect the layers where unphysical oscillations occur. Finally, the proposed method is tested on various single- and multi-component problems.

Automatic network coupling analysis for dynamical systems based on detailed kinetic models

We introduce a numerical complexity reduction method for the automatic identification and analysis of dynamic network decompositions in (bio)chemical kinetics based on error-controlled computation of a minimal model dimension represented by the number of (locally) active dynamical modes. Our algorithm exploits a generalized sensitivity analysis along state trajectories and subsequent singular value decomposition of sensitivity matrices for the identification of these dominant dynamical modes. It allows for a dynamic coupling analysis of (bio)chemical species in kinetic models that can be exploited for the piecewise computation of a minimal model on small time intervals and offers valuable functional insight into highly nonlinear reaction mechanisms and network dynamics. We present results for the identification of network decompositions in a simple oscillatory chemical reaction, time scale separation based model reduction in a Michaelis-Menten enzyme system and network decomposition of a detailed model for the oscillatory peroxidase-oxidase enzyme system.

Computing CHEMKIN Sensitivities Using Complex Variables

This paper discusses an accurate numerical approach based on complex variables for the computation of the Jacobian matrix of complex chemical reaction mechanisms. The Jacobian matrix is required in the calculation of low dimensional manifolds during kinetic chemical mechanism reduction. The approach is suitable for numerical computations of large-scale problems and is more accurate than the finite difference approach of computing Jacobians. The method is demonstrated via a nonlinear reaction mechanism for the synthesis of Bromide acid and a H2/Air mechanism using a modified CHEMKIN package. The Bromide mechanism consisted of five species participating in six elementary chemical reactions and the H2/Air mechanism consisted of 11 species and 23 reactions. In both cases it is shown that the method is superior to the finite difference approach of computing derivatives with an arbitrary computational step size h.

Solvable model of kinetic control

A two-step nonlinear sequential reaction mechanism is studied as a dimensionless model of kinetic control. In this mechanism the unimolecular step is: A→B1, and the bimolecular step is: A+B→Cε, with normalized initial conditions. For ε≪1, little C is formed, whereas for ε≫1, the concentration of B approaches a steady state and C is the main product. Because of the coupling between the two reaction steps the steady-state approximation (SSA) is both a nullcline and a solution of the ordinary differential equations (ODEs) for the model. All concentrations can be expressed in terms of a timelike, real phase variable φ, whose evolution is described by a first-order, nonlinear ODE. Algorithms for the systematic perturbation solution of this ODE for large and small ε are given.

Finite difference solutions of reaction diffusion equations with continuous time delays

This paper is an extension of the monotone iterative methods for finite difference equations with discrete time delays to a class of nonlinear finite difference system with continuous time delays. The system under consideration is a finite difference approximation of a class of reaction diffusion equations with continuous time delays in the nonlinear reaction under either Dirichlet or Neumann-Robin boundary conditions. Various monotone iterative schemes, which depend on the property of the nonlinear reaction mechanism, are developed for the computation of numerical solutions. It is shown by the method of upper and lower solutions that the two sequences obtained from each iterative scheme converge monotonically from above and below, respectively, to a unique solution of the finite difference system. Applications are given to two model problems known as the diffusive logistic equation and the Fisher's diffusion equation in population genetics.

Recent advance in catalysis for solving energy and environmental problems

Recent advances in catalysis for solving the energy and environmental problems are summarized. For these purposes, rapid conversion and selective reaction even under conditions deviating extremely from reaction stoichiometry must be indispensable requisites. In order to realize these requisites, changes in the state of catalyst surface during the reaction were studied, and the catalyst structures on which the optimum reaction performance occurs were determined. An ultra-rapid reforming of methane to syngas with a space–time yield (STY) of 25 000 mol/l h was achieved by using a Rh-modified Ni–Ce 2O 3–Pt catalyst in which the Rh played the role of portholes for hydrogen spillover and prevents coke deposition on the catalyst surface. As a result, a stable state of the catalyst and the high reaction rate were exhibited. A new catalyst composed of Cu–Zn–Cr–Al–Ga oxides modified with supported Pd exerted a high activity with a high STY of methanol, 6700 g/l h. The catalyst components, Pd and Ga, controlled the reduction state of the catalyst surface by their role on normal and inverse spillover of hydrogen, respectively. The methanol thus produced was then totally converted selectively on a metallosilicate catalyst containing Ga or Fe into an aromatics-lean gasoline using an STY of 1860 g/l h. Finally, non-linear reaction mechanism is used to explain the elimination of NO on metallosilicate catalysts under O 2-excess conditions.

Control of spatiotemporal chaos: A study with an autocatalytic reaction-diffusion system

The characterization of chaotic spatiotemporal dynamics has been studied for a representative nonlinear autocatalytic reaction mechanism coupled with diffusion. This has been carried out by an analysis of the Lyapunov spectrum in spatiallylocalised regions. The linear scaling relationships observed in the invariant measures as a function of thesub-system size have been utilized to assess the controllability, stability and synchronization properties of the chaotic dynamics. The dynamical synchronization properties of this high-dimensional system has been analyzed using suitable Lyapunov functionals. The possibility of controlling spatiotemporal chaos for relevant objectives using available noisy scalar time-series data with simultaneous self-adaptation of the control parameter(s) has also been discussed.

Comparison of deterministic and stochastic kinetics for nonlinear systems

The deterministic kinetics of chemical reactions is compared with a stochastic description for the cubic Schlögl model with a single stable steady state, which has a nonlinear reaction mechanism. We solve numerically the birth-death master equation for this system for various numbers of particles (N=20–160). For small systems with tens of particles the deviation of the first moment of the stochastic distribution from the deterministic temporal variation of concentration can be substantial in the initial relaxation towards a stationary state. The relaxation of the master equation is faster than that of the deterministic equation. The maximum deviation in trajectories decreases as the parameters in the kinetic model are altered towards a linear mechanism. The maximum deviation differs from N1/2 as N decreases, but approaches N1/2 as N increases. Deviations from deterministic temporal evolution due to fluctuations depend on the extent of nonlinearity of the reaction. The variance of a stationary distribution of the master equation is shown to be significantly larger than the average for a nonlinear system.

Cooperative instability phenomena in arrays of catalytic sites

Cooperative instability phenomena are shown to arise in arrays of localized catalytic sites immersed in a bulk system. Nonlinear reaction mechanisms may occur on the catalytic sites; the reactions in the bulk are stable. From the partial differential reaction–diffusion equations for the total system we derive ordinary integral equations and obtain from these, by linearization, stability conditions as a function of catalytic site density. The theory is applied to one-dimensional lattices of catalytic sites with several model reaction mechanisms. For a product activated enzyme mechanism occurring on each catalytic site we show that there exist cooperative effects among sites due to the intersite coupling via the bulk reactions and diffusion. For given constraints, the number of stationary states available (one or three) depends on the density of catalytic sites, such that as the density increases, first three, then one, then three stationary states exist. For the Prigogine–Lefever mechanism critical concentrations necessary to produce chemical oscillations are shown to depend on the catalytic site density; the functional dependence has a maximum due to cooperative interactions among the sites. We study next a linear array of two types of alternating catalytic sites on which occur reactions of mutual activation of two species (a generalization of a model of Shymko and Glass). Multiple stationary states are found, and again the number of such states available depends on the site density.

Photoreduction of hydrogen peroxide by hydrogen

The photoreduction of hydrogen peroxide in water under hydrogen of up to 100 atm pressure has been investigated. The reaction involves a chain mechanism with the quantum efficiency for the disappearance of hydrogen peroxide being strongly dependent upon the concentration of hydrogen peroxide. A maximum quantum efficiency of about 35 mol/einstein occurs at a hydrogen peroxide concentration of about 5 x 10/sup -4/ M. At higher hydrogen peroxide concentrations the reaction (HO. + H/sub 2/O/sub 2/ ..-->.. H/sub 2/O + HO/sub 2/.), which leads to chain termination by subsequent HO/sub 2/. reactions, is competitive with the chain-propagating reaction (HO. + H/sub 2/ ..-->.. H/sub 2/O + H.) so that the quantum efficiency decreases with increasing (H/sub 2/O/sub 2/). At lower hydrogen peroxide concentrations the chain-propagating step (H. + H/sub 2/O/sub 2/ ..-->.. H/sub 2/O + HO.) cannot compete with chain termination by (H. + O/sub 2/ ..-->.. HO/sub 2/.) so the yield again decreases. Scavenging of hydrogen atoms at lower hydrogen peroxide concentrations is not well understood. The participation of HO/sub 2/. radical is inferred from the observation of a strong pH effect which shows an inflection point at a pH near the pK/sub a/ of HO/sub 2/. (4.88). Amore » complex mechanism for the reaction is proposed and analyzed by numerical integration of the resulting rate expressions. This part of the work is an example of the usefulness of recently developed numerical integration techniques in the analysis of complex nonlinear reaction mechanisms.« less