Tubular Chemical Reactor Research Articles

A Minimum Entropy Production Approach to Optimization of Tubular Chemical Reactors with Nature-Inspired Design

The problem of the shape optimization of tubular-type plug-flow chemical reactors equipped with a fluid flow-based cooling system is considered in this work. The hydraulic radius Rh(z) = 2A(z)/P(z) and an equivalent surface area-based radius Rs = P(z)/(2π) were computed from the cross-sectional area A(z) and perimeter P(z) measured along the nasal duct of Northern reindeer and used for shape optimization as nature-inspired design. The laminar flow in the cooling system was modeled using the Navier–Stokes equations for an incompressible liquid. In the central tube, a set of chemical reactions with temperature-dependent rates was considered. The temperature and flow velocity fields, pumping pressure, mass flow rate, and total heat flux Jth were obtained by numerical methods. Comparative analyses of the efficiency of different geometries were conducted on Pareto frontiers for hydraulic resistivity Zh, thermal resistivity Zth, thermal inlet length Lth, and entropy production Sirr as a sum of contributions from chemical reactions, thermal, and viscous dissipation. It was shown that the tube with Rs(z) as an interface between the reactor and cooler has the best Pareto efficiency using the (Zh,Zth,Lth) objective functions. Surprisingly, this design also exhibits the lowest Sirr and a more uniform distribution Sirr(z) (i.e., equipartition) among other designs. This geometry is suggested for densely packed tubular reactors.

Determination of steady states of tank and recycle tubular reactors using homotopy and parametric continuation methods

Abstract This work concerns the application of the homotopy method to solve the mathematical model of a non-adiabatic chemical continuous stirred tank reactor (CSTR) and tubular reactor with mass recycle (TRR) (Berezowski 2000. Spatio-temporal chaos in tubular chemical reactors with the recycle of mass, Chaos, Solitons & Fractals, vol. 11, no. 8, pp. 1197–1204). This method was associated with the parametric multivariable continuation algorithm. Thanks to this, this method can automatically find all the multiple steady states of the reactor model without the need to use any iteration. The parametric continuation method is used to determine a curve whose each point is a solution of the tested model. Therefore, the starting point must be very precisely designated so that it lies on this curve. Otherwise, the result is a graph that deviates from the correct graph. However, this condition is not required when the homotopy method is also introduced into the calculations. The starting point can then be a point with any coordinates. Different curves are also obtained, but the homotopy method ensures that each of them passes through the point where the parameter p = 1. The solution we are looking for in the model is just such a point. This is undoubtedly a big advantage resulting from the combination of both above-mentioned methods.

Finite-Time sliding mode control for singularly perturbed PDE systems

This article proposes a sliding mode control strategy for hyperbolic PDE systems under the requirement of finite-time boundedness. First, the singular perturbation theory is introduced to model multi-time scales phenomena, and a quantized measurement method is employed to save the communication resources in network. In addition, by considering the effect of the singular perturbation phenomenon in PDE systems, a sliding surface dependent on spatial position and singular perturbation parameter is constructed, then a sliding mode control law is developed to drive state trajectories to the designed sliding surface in finite time. Moreover, a partitioning strategy is introduced to ensure that the system is finite-time bounded in the reaching phase and the sliding motion phase, respectively. Finally, some sufficient conditions are given to ensure that the system is finite-time bounded in both reaching phase and sliding motion phase, and a simulation example of the chemical tubular reactor demonstrates the effectiveness of the proposed method.

An Effective Numerical Technique for Boundary Value Problems Arising from an Adiabatic Tubular Chemical Reactor Theory

Mathematical models for an adiabatic tubular chemical reactor which forms an irreversible exothermic reaction are investigated by an efficient numerical technique, Fibonacci Collocation method in this study. The reaction's steady-state temperature is calculated for several values of three parameters, namely, Peclet and Damkohler numbers and the dimensionless adiabatic temperature increment. When the generated outcomes are compared with the other numerical approaches, it has been sighted that the presented method produces reliable results for this type of problems.

Model predictive control of a second‐order hyperbolic transport‐reaction process

The model predictive controller (MPC) design is developed for a tubular chemical reactor, considering a second-order hyperbolic partial differential equation as the model of the transport-reaction process with boundary actuation. Without loss of generality, closed–closed boundary conditions and relaxed total flux are assumed. At the same time, the model is discretized in time by the Cayley–Tustin method, and, under the assumption that only the reactor's output is measurable, the observer design for the state reconstruction is addressed and integrated with the MPC design. The Luenberger observer gain is obtained by solving the operator Ricatti equation in the discrete-time setting, while the MPC accounts for constrained and optimal control. The simulations show that the output-based MPC design stabilizes the system under the input and output constraints satisfaction. In addition, to address the models' disparities, the results for both parabolic and hyperbolic equations are presented and discussed.

State Estimation for Coupled Reaction-Diffusion PDE Systems Using Modulating Functions.

Many systems with distributed dynamics are described by partial differential equations (PDEs). Coupled reaction-diffusion equations are a particular type of these systems. The measurement of the state over the entire spatial domain is usually required for their control. However, it is often impossible to obtain full state information with physical sensors only. For this problem, observers are developed to estimate the state based on boundary measurements. The method presented applies the so-called modulating function method, relying on an orthonormal function basis representation. Auxiliary systems are generated from the original system by applying modulating functions and formulating annihilation conditions. It is extended by a decoupling matrix step. The calculated kernels are utilized for modulating the input and output signals over a receding time window to obtain the coefficients for the basis expansion for the desired state estimation. The developed algorithm and its real-time functionality are verified via simulation of an example system related to the dynamics of chemical tubular reactors and compared to the conventional backstepping observer. The method achieves a successful state reconstruction of the system while mitigating white noise induced by the sensor. Ultimately, the modulating function approach represents a solution for the distributed state estimation problem without solving a PDE online.

Adaptive Boundary Observer Design for Coupled Parabolic PDEs With Different Diffusions and Parameter Uncertainty

This paper focuses on establishing an adaptive boundary observer for fully coupled reaction-diffusions partial differential equations (PDEs) containing unknown parameters. Firstly, the state error system is transformed into an intermediate system by finite dimensional backstepping-like transformation, and then we convert the intermediate system to a target system by infinite dimensional backstepping transformation. Secondly, a least-squares type parameter adaptive law is given. Thirdly, exploiting an ad hoc persistent excitation condition, the exponentially convergent of the observer will be established by applying Lyapunov functional. Finally, we use simulation analysis for Chemical Tubular Reactor model to demonstrate the validity of the results.

Improved event-triggered control for a chemical tubular reactor with singular perturbations

This study proposes an improved event-triggered controller based on spatiotemporally sampled data for the temperature and concentration profiles of a chemical tubular reactor subject to singular perturbations. The resulting system contains not only the spatial location characteristics but also multiple-time-scale features, which results in problems in the system design. To reduce the network communication burden, a spatiotemporal data approach is employed, based on which an improved spatiotemporal-event-triggered mechanism is proposed. Furthermore, a point-control strategy is developed to reduce the system-control costs for limited actuators. Finally, some conditions that guarantee stability of the chemical tubular reactor system are given, and the simulation results are presented to demonstrate the validity and applicability of the proposed design.

Spectral, Entropy and Bifurcation Analysis of the Dynamics of a Catalyst Chemical Reverse-Flow Tubular Reactor

Oscillations, including chaotic ones, can spontaneously appear in chemical reactors or lean premixed combustors. Such behavior of the system is undesirable and should be identified at the stage of its modeling. This article analyzes the behavior of reverse-flow tubular chemical reactor with longitudinal dispersion in terms of chaotic oscillations. The purpose of using reverse flow is to increase the conversion degree. For the analysis of the reactor, among others, spectral analysis, entropy, and bifurcation analysis were used. The obtained results show the chaotic behavior of the reactor in a wide range of changes in the parameter’s values.

Homotopic Parametric Continuation Method for Determining Stationary States of Chemical Reactors with Dispersion

Physical processes occurring in devices with distributed variables and a turbulent tide with a dispersion of mass and heat are often modeled using systems of nonlinear equations. Solving such a system is sometimes impossible in an analytical manner. The iterative methods, such as Newton’s method, are not always sufficiently effective in such cases. In this article, a combination of the homotopy method and the parametric continuation method was proposed to solve the system of nonlinear differential equations. These methods are symmetrical, i.e., the calculations can be made by increasing or decreasing the value of the parameters. Thanks to this approach, the determination of all roots of the system does not require any iterative method. Moreover, when the solutions of the system are close to each other, the proposed method easily determines all of them. As an example of the method use a mathematical model of a non-adiabatic catalytic pseudohomogeneous tubular chemical reactor with longitudinal dispersion was chosen.

On using variable molecular masses in multicomponent lattice Boltzmann simulations

The option of varying the molecular mass in multicomponent lattice Boltzmann simulations is being explored. First, results are presented for droplet formation at an aperture in a second immiscible liquid medium in which the difference in density between the two media is achieved by introducing asymmetry in the EOS, via adding particularly intra-component interaction forces in a pseudo-potential LB model. The second application for models with variable molecular masses is a single-phase heterogeneous laminar-flow tubular chemical reactor, where the molecular masses of reactants and products differ. In this application, tuning the molecular mass requires modification of the standard equilibrium distribution function as well as the use of an extended velocity set, in our case D2Q13. The method is validated against analytical solutions for canonical 1-D diffusion-reaction cases. In both the droplet formation study and the chemical reactors, the results of the exploratory 2-D simulations look qualitatively correct.

Adaptive λ-tracking controller for an exothermic chemical plug flow tubular reactor

ABSTRACT This paper deals with the tracking of a prespecified profile temperature for exothermic chemical tubular reactor whose dynamics is described by a set of nonlinear partial differential equations where the state variables are the reactor temperature and the reactant concentration. The coolant temperature, the inlet temperature and the inlet concentration are considered as control actions. For practical reasons, it is preferable to consider a non distributed control law to achieve the control objective. In contrast to our previous work that considers fully distributed control actions where the three control inputs are assumed to be distributed along the reactor, here the last two control inputs are applied at the reactor inlet and only the coolant temperature is distributed along the reactor. We show that the temperature of the reactor tends asymptotically to a ball of arbitrary prescribed radius , centred at the given temperature profile.

Application of Lyapunov's Methods for Analyzing the Stability of a Tubular Chemical Reactor with Recycle

AbstractThe first and second Lyapunov methods were adapted to analyze the stability of a tubular chemical reactor with recycle. The criterion of local stability in the form of inequalities was derived using Lyapunov's first method. It is an explicit algebraic form, despite the fact that the mathematical model of the reactor is a partial differential problem with a boundary condition. As part of the second Lyapunov method, a mathematical criterion for non‐local stability was derived in the form of an increase of Lyapunov's function. Both criteria are presented on graphs.

Determination of catastrophic sets of a tubular chemical reactor by two-parameter continuation method

AbstractThe work relates to development and presentation a two-parameter continuation method for determining catastrophic sets of stationary states of a tubular chemical reactor with mass recycle. The catastrophic set is a set of extreme points occurring in the bifurcation diagrams of the reactor. There are many large IT systems that use the parametric continuation method. The most popular is AUTO’97. However, its use is sometimes not convenient. The method developed in this work allows to eliminate the necessity to use huge IT systems from the calculations. Unlike these systems, it can be inserted into the program as a short subroutine. In addition, this method eliminates time-consuming iterations from the calculations.

Chaos Predictability in a Chemical Reactor

The dynamics of the tubular chemical reactor with mass recycle was examined. In such a system, temperature and concentrations may oscillate chaotically. This means that state variable values are then unpredictable. In this paper, it has been shown that despite the chaos, the behavior of such a reactor can be predictable. It has been shown that this phenomenon can occur in two cases. The first case concerns intermittent chaos. It has been shown that intermittent outbursts can occur at regular intervals. The second case concerns transient chaos, i.e. a situation when chaos occurs only for a certain period of time, e.g. only during start-up. This phenomenon makes it impossible to predict what will occur in the reactor in the nearest time, but, makes it possible to precisely determine the values of the variables even in the distant future. Both of these phenomena were tested by numerical simulation of the mathematical model of the reactor.

New wavelet method for solving boundary value problems arising from an adiabatic tubular chemical reactor theory

This paper displays an efficient numerical technique of realizing mathematical models for an adiabatic tubular chemical reactor which forms an irreversible exothermic chemical reaction. At a steady-state solution for an adiabatic rounded reactor, the model can be diminished to a conventional nonlinear differential equation which converts into a system of the nonlinear equation that can proceed numerically utilizing Newton’s iterative method. An operational matrix of coordination is derived and is utilized to decrease the model for an adiabatic tubular chemical reactor to an arrangement of algebraic equations. Simple execution, basic activities, and precise arrangements are the fundamental highlights of the proposed wavelet technique. The numerical solutions attained by the present technique have been contrasted and compared with other techniques.

Comparison of the method of variation of parameters to semi-analytical methods for solving nonlinear boundary value problems in engineering

AbstractSolutions to nonlinear boundary value problems modelling physical phenomena in engineering applications have traditionally been approximated using numerical methods. More recently, several semi-analytical methods have been developed and used extensively in diverse engineering applications. This work compares the method of variation of parameters to semi-analytical methods for solving nonlinear boundary value problems arising in engineering. The accuracy and efficiency of the method of variation of parameters are compared to those of two widely used semi-analytical methods, the Adomian decomposition method and the differential transformation method, for three practical engineering boundary value problems: (1) the deflection of a cantilevered beam with a concentrated load, (2) an adiabatic tubular chemical reactor which processes an irreversible exothermal reaction, and (3) the electrohydrodynamic flow of a fluid in an ion drag configuration in a circular cylinder conduit. The accuracy and convergence of each method is investigated using the error remainder function. The method of variation of parameters is significantly more efficient than both the semi-analytical methods and traditional numerical methods while maintaining comparable accuracy. Unlike the semi-analytical methods, the efficiency of variation of parameters is independent of the nonlinearity. Variation of parameters is shown to be an attractive alternative to semi-analytical methods and traditional numerical methods for solving boundary value problems encountered in engineering applications in which solution efficiency is important.

Approximate analytic solutions for a chemical tubular reactor model using the Adomian decomposition method

The purpose of this paper is to apply the Adomian decomposition method to acquire the approximate analytical solutions for a chemical tubular reactor model composed of a system of strongly nonlinear second-order partial differential equations with standard boundary conditions. By properly assigning the dimensionless parameters in the model, the objective description of the solutions of temperature and process component concentration distributions can be obtained.

B-spline Wavelet Method for Solving Fredholm Hammerstein Integral Equation Arising from Chemical Reactor Theory

AbstractMathematical model for an adiabatic tubular chemical reactor which processes an irreversible exothermic chemical reaction has been considered. For steady state solution for an adiabatic tubular chemical reactor, the model can be reduced to ordinary differential equation with a parameter in the boundary conditions. Again the ordinary differential equation has been converted into a Hammerstein integral equation which can be solved numerically. B-spline wavelet method has been developed to approximate the solution of Hammerstein integral equation. This method reduces the integral equation to a system of algebraic equations. The numerical results obtained by the present method have been compared with the available results.

Semi-analytical solutions for tubular chemical reactors

The one-dimensional tubular reactor model with advection and possibly axial diffusion is the classical model of distributed chemical reaction systems. This system is described by partial differential equations that depend on the time t and the spatial coordinate z. In this article, semi-analytical solutions to these partial differential equations are developed regardless of the complexity of their initial and boundary conditions and reaction kinetics. These semi-analytical solutions can be used to analyze the effect on the concentrations at the current coordinates z and t of (i) the initial and boundary conditions, and (ii) the reactions that took place at an earlier time. A case study illustrates the application of these results to tubular reactors for the two cases, without and with diffusion.