First-order Exothermic Reaction Research Articles

Nonstationary one-dimensional flows of a two-temperature dissociating gas: Group analysis and invariant solutions

Modern problems such as detonation engines and hypersonic reentry are reviving interest in the point explosion theory, which gives simple gasdynamics models of similar processes. This work is devoted to the study, using group analysis methods of mathematical properties of gasdynamics equations with thermochemical nonequilibrium, which is a typical problem in the theory of a point explosion. For this purpose, the system of equations under study was slightly corrected without distorting the physical meaning. This made it possible to obtain self-similar solutions. Solutions of two different models describing: a second-order endothermic dissociation reaction with vibrational exiting and a first-order exothermic reaction with spontaneous decay, are given in the paper. The problem of strong explosion of linear charge was considered. As in the classical case, the solution of the problem is reduced to integrating a system of ordinary differential equations in self-similar variables. The calculations show that the solutions to the modified system of equations is physically consistent and adequately describes the known effects of thermochemical nonequilibrium behind the wave front.

Adaptive Control for Narrow Bandwidth Input and Output Disturbance Rejection for a Non-Isothermal CSTR System

This paper presents an adaptive control scheme to face the challenge of rejecting input and output disturbances. The research is put on a layer of the design and start-up of chemical plants. The emphasis is on handling disturbances appearing in a narrow band of frequencies, which illustrates standard forms of disturbances in the alluded kind of systems. The controller is made up of a central RS structure that stabilizes the closed-loop plant. A second layer boosts the control law performance by adding the Youla–Kucera (YK) filter or Q parametrization and taking advantage of the internal model principle (IMP). This practice aids in modeling unknown disturbances with online control adjustment. We probe the resultant compensator for three non-isothermal continuous stirred tank reactors connected in series. The plant should conduct a first-order exothermic reaction consuming reactant A, while an isothermal operation stays and the outlet concentration is close to its nominal value. The primary concerns are open-loop instability and steady-state multiplicity in the plant’s first unit. The control objective is to reject input and output disturbances in a band of frequencies of 0.0002Hz to 0.007Hz, whether there are variants or not in time. We test the controller with input signals depicting both variations in the auxiliary services and abrupt changes. We then compare the executions of the resultant control law with a model-based predictive control (MPC). We find comparable responses to multiple disturbances. However, the adaptive control offers an effortless control input. We also conclude that the adaptive controller responds well to reference changes, while the MPC fails due to input constraints.

Investigation of the effect of DamKohler and Stanton non-dimensional numbers on the stability of continuous stirred tank reactors

The fully mixed continuous stirred tank reactor is an important type of industrial reactors mainly used to produce high volume products such as petrochemicals, detergents, sanitary products and products that are in demand in the market. Knowing the dynamic behavior of chemical reactors is of great importance in setting up, designing, controlling and stopping reactors. In this paper, the effect of non-dimensional numbers Damkohler and Stanton on the stability of a continuous stirred tank reactor in which a first-order exothermic reaction takes place is investigated. First, a mathematical model of the system's dynamic behavior was presented. Then, by simultaneous solving of the equations of mass and energy around the fixed point in MATLAB software, the effect of the mentioned numbers was investigated. The results show that the continuous stirred tank reactor shows different behaviors in different ranges of Damkohler and Stanton numbers. This reactor behaves unstable in small and large ranges of Damkohler and Stanton numbers due to the presence of mixed or positive and negative eigenvalues. The best range for Damkohler and Stanton numbers is close to 1, because in this range the reactor shows stable behavior due to having two negative eigenvalues. In this range, in addition to the stability, the conversion is also 100%. Finaly the ratio of Stanton to DamKohler was investigated as St / Da˃1 and St / Da = 1. If St / Da = 1, the system is in steady state, but in St / Da˃1, the system moves away from steady state.

Impact of delay on robust stable optimization of a CSTR with recycle stream

We demonstrate that the normal vector method for robust optimization of nonlinear systems with uncertain parameters can be extended to systems with delays. A first-order exothermic irreversible reaction carried out in a CSTR with recycle stream serves as example for the broad class of nonlinear delay differential equations (DDE) with uncertain parameters. The stability boundaries that must be taken into account in the robust optimization consist of Hopf bifurcation points in this case. We show that (i) an unstable steady state of operation results if stability boundaries are neglected and (ii) a conservative optimal steady state results if the delay is ignored.

Tuning of PID Controllers for Continuous Stirred Tank Reactors

Tuning of proportional integral and derivative (PID) controllers for stable SOPTDZ (second order plus time delay with a zero) systems is proposed. The PID controller was designed based on internal model control and stability analysis principles. The proposed controllers were applied to the various stable transfer function models of jacketed CSTR, isothermal CSTR carrying out Van de Vusse reaction and a bioreactor. Simulation results on various transfer function models and on non-linear model equations of jacketed CSTR carrying first-order exothermic reaction, isothermal constant volume CSTR carrying out Van de Vusse reaction and a bioreactor are given to show the effectiveness of the proposed PID controllers. The performance under model uncertainty is also studied considering perturbation in one parameter at a time. The performance of the proposed controllers is compared with the controller designed by direct synthesis method.

Self-oscillations in a flow ideal mixing reactor in the region of multiplicity of stationary equilibrium states: a consecutive reaction

A model of a flow ideal mixing reactor for a consecutive two-stage exothermic first-order reaction was considered. The presence of a region of multiplicity of stationary states was established. This multiplicity region can contain regions of three and five stationary states; the region of five states always appears inside the region of three stationary states. Changes in the type of stationary state stability in each of these regions are analyzed. Numerical calculations of phase trajectories in the regions of stationary state stability and instability were performed. A mechanism of the creation of a stable limiting cycle was suggested for the case of three stationary states.

Are 3-D models necessary to simulate packed bed reactors? analysis and 3-D simulations of adiabatic and cooled reactors

Simulations and analysis of transversal patterns in a homogeneous three-dimensional (3-D) model of adiabatic or cooled packed bed reactors (PBRs) catalyzing a first-order exothermic reaction were presented. In the adiabatic case the simulation verify previous criteria, claiming the emergence of such patterns when (ΔTad/ΔTm)/(PeC/PeT) surpasses a critical value larger than unity, where ΔTad and ΔTm are adiabatic and maximal temperature rise, respectively. The reactor radius required for such patterns should be larger than a bifurcation value, calculated here from the linear analysis. With increasing radius new patterned branches, corresponding to eigenfunction of the problem emerge, whereas other branches become unstable. The maximal temperature of the 3-D simulations may exceed the 1-D prediction, which may affect design procedures. Cooled reactor may exhibit patterns, usually axisymmetric ones that can be characterized by two anomalies: the peak temperature may exceed the corresponding value of an adiabatic reactor and may increase with wall heat-transfer coefficient, and the peak temperature in a sufficiently wide reactor need not lie at the center but rather on a ring away from it. In conclusions, we argue that transversal patterns are highly unlikely to emerge in practical adiabatic PBRs with a single exothermic reaction, as in practice PeC/PeT > 1. That eliminates patterns in stationary and downstream-moving fronts, whereas patterns may emerge in upstream-moving fronts, as shown here. This conclusion may not hold for microkinetic models, for which stationary modes may be established over a domain of parameters. This suggests that a 1-D model may be sufficient to analyze a single reaction in an adiabatic reactor and a 2-D axisymmetric model is sufficient for a cooled reactor. The predictions of a 2-D cylindrical thin reactor with those of a 3-D reactor were compared, to show many similarities but some notable differences. © 2012 American Institute of Chemical Engineers AIChE J, 2012

The method of phase trajectories in nonstationary thermal explosion theory

It is shown that consideration of the heating rate-temperature dependence, an analysis of self-heating for first-order exothermic reactions, and strict inclusion of kinetic deceleration puts forward fresh ideas about the diversity of reaction regimes on the Todes criterion-Semenov criterion parameter plane. From this point of view, classic Semenov theory is a particular case of the projection of two-dimensional regions onto the direction of Semenov criterion changes. An analysis of the mutual arrangement of reaction region boundaries with different kinetics and the shape of the representation point movement trajectory when the initial temperature of the system changed allowed us to determine the thermal explosion degeneration region, transition region, and regions of possible explosive reaction modes.

Dynamics of a Reactive Thin Film

Consider the dynamics of a thin film flowing down an inclined plane under the action of gravity and in the presence of a first-order exothermic chemical reaction. The heat released by the reaction induces a thermocapillary Marangoni instability on the film surface while the film evolution affects the reaction by influencing heat/mass transport through convection. The main parameter characterizing the reaction-diffusion process is the Damköhler number. We investigate the complete range of Damköhler numbers. We analyze the steady state, its linear stability and nonlinear regime. In the latter case, long-wave models are compared with integral-boundary-layer ones and bifurcation diagrams for permanent solitary wave solutions of the different models are constructed. Time-dependent computations with the integral-boundary-layer models show that the system approaches a train of coherent structures that resemble the solitary pulses obtained in the bifurcation diagrams.

Prediction of 3D Transversal Patterns in Packed-Bed Reactors Using a Reduced 2D Model: Oscillatory Kinetics

We have recently showed the formation of transversal patterns in a 3D cylindrical reactor in which an exothermic first-order reaction of Arrhenius kinetics occurs with variable catalytic activity. Under these oscillatory kinetics, the system exhibits a planar front (1D) solution, with the front position oscillating in the axial direction, while in the 3D case, three types of transversal patterns can emerge: rotating fronts, oscillating fronts with superimposed transversal (nonrotating) oscillations, and mixed rotating−oscillating fronts. In the present study, we analyze the possible reduction of the 3D model to a 2D cylindrical shell model to predict patterns. We map bifurcation diagrams showing domains of different modes using the reactor radius (R) as a bifurcation parameter and show that the front divergence and the domains of the kn-mode pattern in the 3D model [corresponding to the transversal eigenfunction Jk(μknr) exp(ikθ), in which Jk is the Bessel function of the first kind] can be predicted by those of the one wave in the 2D model using the linear transformation R3D = μknR2D.

Steady-state behaviour of PFR-separation-recycle systems with simultaneous exothermic and endothermic, first-order reactions

A systematic investigation of plug-flow reactor (PFR)-separation-recycle systems where first-order exothermic and endothermic reactions are simultaneously performed is presented. The nonlinear behaviour is analyzed for two flowsheet alternatives and four plantwide control structures. It is shown that the system can exhibit a complex nonlinear behaviour. For the parameter values used, regions of unfeasibility, two or three multiple steady states and branches of isolated solutions were found. The undesired nonlinear phenomena can be avoided by fixing the reactor-inlet flow rates of each reactant or, when this is impossible due to the flowsheet structure, by providing sufficient cooling capacity.

On the dynamical instability of packed-bed reactors in the presence of catalyst deactivation

Due to the thermal instability of the packed-bed reactor running an exothermic reaction, unsteady-state operation (for example a fluctuating inflow temperature) can result in a variety of thermal responses. These include the amplification of input temperature perturbations and high-temperature pre-extinction waves. Catalyst deactivation adds further dynamical features to these scenarios. We explore them numerically, using a first-order exothermic reaction and a pseudo-homogeneous (single phase) model of the PBR together with a first-order deactivation model of the catalyst. At low deactivation rate, moving hot spots are found, as well as a non-uniform activity profile of the catalyst. At high deactivation rate, however, high-temperature waves (so-called pre-extinction waves) are followed by the complete extinction of the reactor. The amplification of input temperature perturbations is generally enhanced by the presence of catalyst deactivation. Finally, a power-law model is derived numerically that predicts the resonance frequency for amplification as a function of operating parameters.

Modeling of the dynamics of a well-stirred reactor with parallel exothermic reactions

The dynamics of a well-stirred reactor in which two parallel exothermic first-order reactions occur was modeled. Phase portraits of the system were plotted, the possibility of realization of oscillatory regimes was revealed, and the stability of the steady states involved was examined. Two types of limit cycle extinction, smooth and sharp, were demonstrated to exist.

Stabilization of a CSTR in an Oscillatory State by Varying the Thermal Characteristics of the Reactor Vessel

One of the most thoroughly studied systems in chemical engineering is the classic problem of a first-order exothermic reaction in a diabatic CSTR. In this paper, an extension of the classic problem, in which the reactor vessel temperature varies independently of its contents, is considered. A bifurcation analysis showed that a CSTR in an oscillatory state may be stabilized by the use of a reactor with the appropriate combination of thermal characteristics. Stabilization of an oscillatory state was demonstrated for a variety of scenarios that had been previously shown by Uppal, Ray and Poore to be representative cases of phase diagrams that exhibit limit-cycle behavior.

EFFECT OF REACTOR HEAT CAPACITY ON THE STABILITY AND START-UP TIME OF A DIABATIC CONTROLLED-CYCLED STIRRED TANK REACTOR

ABSTRACT A simple model was written to simulate a first-order exothermic reaction taking place in a diabatic controlled-cycled stirred tank reactor (CCTR) in which the reactor jacket heat capacity is not infinitely larger than the reacting fluid heat capacity. The simulations have shown tpthat including the reactor heat capacity in the model tends to stabilize oscillatory states to a unique steady state. A further result is that a smaller reactor heat capacity results in a shorter transition time to the final steady state. The study shows the possibility that reactor stability and short start-up times may be achieved by simple changes in the reactor design rather than through complex control systems.

Optimization of the active component distribution through the catalyst bed for the case of adiabatic reactor

This article is the next one in the previous series of publications concerning the optimal active component distribution. The aim of the present work is to extend the optimization problem taking into account the energy balance. The variational problem on searching for the optimal active component distribution profile along the catalyst bed was formulated for the case of an adiabatic reactor at an arbitrary form of reaction rate expression. The Euler differential equation was derived and the existence conditions for the solution of variational problem were obtained. A numerical algorithm was suggested and the optimal profiles of the active component distribution were calculated for the first-order reaction. Under isothermal conditions with linear dependence of the reaction rate on reactant concentration at a constant mass transfer coefficient, the uniform distribution is optimal. As opposed to the case of an isothermal reactor with a first-order catalytic reaction, in an adiabatic reactor considerable economy in the active component loading might be achieved due to optimization of its distribution. It was shown that the optimal active component distribution profiles were axially decreasing for the first-order exothermic reactions and increasing for endothermic reactions. The optimal active component distribution profile was calculated for the case of methane combustion on catalytic monoliths.

Bifurcation Analysis of Catalytic Monoliths with Nonuniform Catalyst Loading

A one-dimensional two-phase model with position-dependent heat- and mass-transfer coefficients and a nonuniform catalyst distribution along the channel is used to present a complete steady-state bifurcation analysis of a catalytic monolith for the case of an exothermic first-order reaction when the feed temperature is taken as the bifurcation variable. It is found that, for typical operating conditions, washcoat diffusion and the species Lewis number have the largest influence on the light-off boundary (in the sense that the metal loading needed to obtain an ignition point in the bifurcation diagram of exit cup-mixing conversion versus inlet fluid temperature varies by a factor of 100 or more as these parameters are varied), followed by solid conduction (factor of 50), nonuniform catalyst distribution (factor of 10), and channel geometry (factor of 5). A nonuniform catalyst distribution with three zones of catalyst loading (with high activity in the middle zone) appears to be the best configuration considering important factors such as the exit fluid conversion, inlet fluid temperature required for ignition, fouling, and transient time required for the monolith to attain the mass-transfer-controlled regime.

An LMI Approach to H.INF. PI Controller Design.

A Linear Matrix Inequality (LMI) approach for designing the H∞ Proportional-Integral (PI) controller for a nonlinear dynamic system is studied. A local linear model containing time-varying norm-bounded uncertain parameters is identified at first. The robust PI control design problem based on the norm-bounded uncertain linear model is then transformed into a standard H∞ control problem, where the latter is further formulated as LMIs. By adopting the LMI expression, a symmetric positive definite matrix P with guaranteed overall system’s quadratic stability can be easily determined. The matrix P can finally be used to infer the robust PI controller parameters. One chemical process, a non-isothermal continuous-stirred-tank reactor (CSTR) with a first-order exothermic reaction, is illustrated to demonstrate the effectiveness of the proposed LMI-based H∞ PI controller design method for nonlinear dynamic processes.

Spatiotemporal patterns in models of cross-flow reactors

We review the behavior of stationary and moving spatially periodic patterns in a simple cross-flow fixed-bed reactor with a first-order exothermic reaction subject to the Danckwert’s boundary conditions and realistically high Le and Pe. Spatiotemporal patterns emerge due to the interaction of concentration and temperature balances, much like dynamic patterns in a CSTR. Moving waves emerge in an unbounded system, but they transform into stationary spatially inhomogeneous patterns in a bounded system above certain Pe threshold. The critical parameters of this threshold are derived analytically. A weakly nonlinear analysis is used in order to derive the governing amplitude equation. The spatial behavior in the bounded system with Pe →∞ is analogous to the temporal behavior of the simple thermokinetic CSTR problem and the behavior of the distributed system is classified according to that of the lumped one. Both regular kinetics and oscillatory one (with reversible catalytic activity) are considered. Suggestions for experimental realization of these phenomena are discussed.

Nonlinear behavior of an ideal reactor separator network with mass recycle

In this work the nonlinear behavior of a coupled reactor–separator network is analyzed using singularity and bifurcation theory. The reactor is modeled as a CSTR, which sustains an exothermic first-order reaction. The separator is modeled as a flash process. The effluent from the reactor is fed to the separator. The reactant-rich stream (assumed to be the bottoms) from the separator is recycled back to the reactor. Focus is on pure mass recycle, as the two units are decoupled energetically via heat exchangers. Two different modes of operation are considered. These are compared with each other and with the well-known stand alone reactor with a first-order exothermic reaction. For the first mode of operation, the feed rate is fixed and for the second the recycle rate is fixed. In practice, these different modes of operation can be achieved by a suitable control strategy. It is shown that the behavior crucially depends on the mode of operation. Fixing the feed rate can lead to severe operational problems including monotonic and oscillatory unstable steady states over a wide range of operating conditions. Further, parameter regions are identified, where no steady state exists at all. On the other hand when the recycle rate is fixed the coupled system admits at least one stable steady-state solution for a fixed set of operating conditions.