Isothermal Continuous Stirred Tank Reactor Research Articles

On positive systems with multivariable positive control: Process stabilization

The process stabilization is tackled from the perspective of a family of positive systems with multivariable positive control. This framework is consistent because each of the n state variables and m control inputs in chemical processes is in R+. The aim is to provide multivariable positive feedback control schemes which take values in the hyperbox U∈R+m; while sufficient conditions are satisfied to ensure the stabilization process. Two approaches are discussed: the first is a sliding mode control based on hyperplanes intersecting a (n−m)-dimensional segment of the state space which comprises the equilibrium point; such scheme guarantees robust stability. The second is a continuous feedback stabilizer based on Lyapunov functions, which enhances convergence rate while stabilization is ensured. The process stabilization is shown with two examples: a heat exchanger and an isothermal continuous stirred tank reactor. Both examples illustrate the usefulness of the proposed framework of multivariable positive control.

Approximating Multiple Roots of Applied Mathematical Problems Using Iterative Techniques

In this study, we suggest a new iterative family of iterative methods for approximating the roots with multiplicity in nonlinear equations. We found a lack in the approximation of multiple roots in the case that the nonlinear operator be non-differentiable. So, we present, in this paper, iterative methods that do not use the derivative of the non-linear operator in their iterative expression. With our new iterative technique, we find better numerical results of Planck’s radiation, Van Der Waals, Beam designing, and Isothermal continuous stirred tank reactor problems. Divided difference and weight function approaches are adopted for the construction of our schemes. The convergence order is studied thoroughly in the Theorems 1 and 2, for the case when multiplicity p≥2. The obtained numerical results illustrate the preferable outcomes as compared to the existing ones in terms of absolute residual errors, number of iterations, computational order of convergence (COC), and absolute error difference between two consecutive iterations.

Minimum time controller in a class of chemical reactors based on Lagrangian approach

Abstract The main goal of this work is the construction of a class of controller, which employs directly a Lagrangian formulation to resolve the classical brachistochrone problem, this allows to obtain an optimal controller which reaches in a minimum time the stabilization of an isothermal continuous stirred tank reactor, whose chemical kinetic model is based on the power law. The proposed methodology is compared with an input/output linearizing which achieve asymptotic and exponential closed-loop convergence, sliding-mode controller with a finite time convergence and an exact gradient optimal control to compare the time convergence performance. Numerical experiments show the satisfactory performance of the proposed controller, despite sustained disturbances in the concentration input feed.

Computer-Aided Nonlinear Frequency Response Method for Investigating the Dynamics of Chemical Engineering Systems

The Nonlinear Frequency Response (NFR) method is a useful Process Systems Engineering tool for developing experimental techniques and periodic processes that exploit the system nonlinearity. The basic and most time-consuming step of the NFR method is the derivation of frequency response functions (FRFs). The computer-aided Nonlinear Frequency Response (cNFR) method, presented in this work, uses a software application for automatic derivation of the FRFs, thus making the NFR analysis much simpler, even for systems with complex dynamics. The cNFR application uses an Excel user-friendly interface for defining the model equations and variables, and MATLAB code which performs analytical derivations. As a result, the cNFR application generates MATLAB files containing the derived FRFs in a symbolic and algebraic vector form. In this paper, the software is explained in detail and illustrated through: (1) analysis of periodic operation of an isothermal continuous stirred-tank reactor with a simple reaction mechanism, and (2) experimental identification of electrochemical oxygen reduction reaction.

Design and analysis of an intensified column with side reactor configuration for ethylene glycol production from ethylene oxide

The side reactor concept (SRC), comprising a non-reactive column coupled with side reactors, is proposed for the synthesis of ethylene glycol (EG) by ethylene oxide (EO) hydration. Parallel isothermal continuous stirred tank reactors are employed and the combinations of the reactors and the column are cautiously examined. A preliminary analysis of the operating conditions for the reaction and separation and the detailed SRC configurations (such as the side stream rate, side stream location) are performed to generate the suggested values or ranges for their further optimization. The decisive variables are subsequently optimized by a sequential design procedure. The significant effects of the key decision variables on the trade-off between the reaction cost and the separation cost are well investigated. The reactor–separation unit, configured with a total liquid withdrawn and recycled back to the column stage below, demonstrates better system performance. Furthermore, the reaction at elevated temperature (150 °C) conductive for reaction and the separation occurring at the distillation temperature range of 106∼204 °C (124 kPa) gives out improved economic performance. Flexibility in the operating conditions in SRC makes it a competitive alternative to the RD process for EG production.

Fischer–Tropsch Synthesis: Computational Sensitivity Modeling for Series of Cobalt Catalysts

Nearly a century ago, Fischer and Tropsch discovered a means of synthesizing organic compounds ranging from C1 to C70 by reacting carbon monoxide and hydrogen on a catalyst. Fischer–Tropsch synthesis (FTS) is now known as a pseudo-polymerization process taking a mixture of CO as H2 (also known as syngas) to produce a vast array of hydrocarbons, along with various small amounts of oxygenated materials. Despite the decades spent studying this process, it is still considered a black-box reaction with a mechanism that is still under debate. This investigation sought to improve our understanding by taking data from a series of experimental Fischer–Tropsch synthesis runs to build a computational model. The experimental runs were completed in an isothermal continuous stirred-tank reactor, allowing for comparison across a series of completed catalyst tests. Similar catalytic recipes were chosen so that conditional comparisons of pressure, temperature, SV, and CO/H2 could be made. Further, results from the output of the reactor that included the deviations in product selectivity, especially that of methane and CO2, were considered. Cobalt was chosen for these exams for its industrial relevance and respectfully clean process as it does not intrinsically undergo the water–gas shift (WGS). The primary focus of this manuscript was to compare runs using cobalt-based catalysts that varied in two oxide catalyst supports. The results were obtained by creating two differential equations, one for H2 and one for CO, in terms of products or groups of products. These were analyzed using sensitivity analysis (SA) to determine the products or groups that impact the model the most. The results revealed a significant difference in sensitivity between the two catalyst–support combinations. When the model equations for H2 and CO were split, the results indicated that the CO equation was significantly more sensitive to CO2 production than the H2 equation.

Design of robust <i>H</i><SUB align="right">∞ fuzzy output feedback controller for affine nonlinear systems: fuzzy Lyapunov function approach

In this paper, we propose a new systematic approach based on non-quadratic Lyapunov function and technique of introducing slack matrices, for a class of affine nonlinear systems with disturbance. To achieve the goal, first, the affine nonlinear system is represented via Takagi-Sugeno (T-S) fuzzy bilinear model. Subsequently, the robust H∞ controller is designed based on parallel distributed compensation (PDC) scheme. Then, the stability conditions are derived in terms of linear matrix inequalities (LMIs) by utilising Lyapunov function. Moreover, some slack matrices are proposed to reduce the conservativeness of the LMI stability conditions. Finally, for illustrating the merits and verifying the effectiveness of the proposed approach, the application of an isothermal continuous stirred tank reactor (CSTR) for Van de Vusse reactor is discussed in details.

Design of robust H_ fuzzy output feedback controller for affine nonlinear systems: Fuzzy Lyapunov function approach

In this paper, we propose a new systematic approach based on non-quadratic Lyapunov function and technique of introducing slack matrices, for a class of affine nonlinear systems with disturbance. To achieve the goal, first, the affine nonlinear system is represented via Takagi-Sugeno (T-S) fuzzy bilinear model. Subsequently, the robust H∞ controller is designed based on parallel distributed compensation (PDC) scheme. Then, the stability conditions are derived in terms of linear matrix inequalities (LMIs) by utilising Lyapunov function. Moreover, some slack matrices are proposed to reduce the conservativeness of the LMI stability conditions. Finally, for illustrating the merits and verifying the effectiveness of the proposed approach, the application of an isothermal continuous stirred tank reactor (CSTR) for Van de Vusse reactor is discussed in details.

Reaction flux versus reaction force: easy to stabilize?

Chemical reaction rate, also known as the reaction flux involved in chemical reactors, plays a central role as the source generating the abnormal dynamics characteristics. This paper proposes a structural approach for the stabilization of such systems through the control of the reaction flux by considering the Lyapunov stability theory within a standard thermodynamic framework. More precisely, the reaction flux is structurally considered as a nonlinear function of conjugated reaction force. The thermodynamic constraint of such a relationship is that the inherent non-negative definiteness property of the irreversible entropy production due to chemical reaction has to be fulfilled. Consequently, it allows to reexpress a large class of reaction rates described by the mass-action-law and more interestingly, the operation of the reaction system at a desired set-point consists in controlling the reaction force on the basis of an affinity-related storage function. Numerical simulations for a non isothermal continuous stirred tank reactor (CSTR) involving one reversible reaction operated with multiple steady states illustrate the application of the theoretical developments.

Fischer-Tropsch synthesis. Evaluation of an aluminum small channel reactor.

Fischer-Tropsch synthesis was conducted in a small channel compact heat exchange reactor that was constructed of aluminum. While limited to lower temperature-pressure regions of the Fischer-Tropsch synthesis, the reactor could be operated in an isothermal mode with nearly a constant temperature along the length of the channel. The results obtained with the compact heat exchange reactor were similar to those obtained in the isothermal continuous stirred tank reactor, with respect to both activity and selectivity. Following a planned or unplanned shutdown, the reactor could be restarted to produce essentially the same catalytic activity and selectivity as before the shutdown.

Tuning of PID Controllers for Unstable Systems Using Direct Synthesis Method

Tuning of proportional integral derivative (PID) controllers for unstable first-order plus time delay systems and unstable second-order plus time delay systems is proposed using the direct synthesis (DS) method. In this method, the mode of PID controller is selected and the characteristic equation of the unstable process with the PID controller is compared with the desired characteristic equation to derive PID controller parameters. The performance of the proposed controller is compared with the recently reported DS method, internal model control method and setpoint overshoot method (SOM) to demonstrate the efficiency of the controller. The performance comparison is made in terms of integral absolute error (IAE). The smooth functioning of controller is determined in terms of total variation (TV) (input usage). The controllers designed by the present method are applied to many transfer function models and the nonlinear model of isothermal continuous stirred tank reactor (CSTR) carrying out an enzymatic reaction with nonlinear kinetics and jacketed CSTR carrying out a first-order reaction. The robustness of the controller is given in terms of Ms (maximum magnitude of the sensitivity function), phase margin and gain margin.

Robust feedback linearization of an isothermal continuous stirred tank reactor: H∞ mixed-sensitivity synthesis and DK-iteration approaches

This paper studies the problem of designing a robust controller for the nonlinear multi-input multi-output continuous stirred tank reactor (CSTR) in the presence of uncertainties in parameters of the process model. For this purpose, by first using the feedback linearization method, the equivalent linearized model of the CSTR is obtained. In the second step, to cope with uncertainties, two robust controllers are designed; one using H∞ mixed-sensitivity and the other using DK-iteration method. In this step, the required performance and uncertainties are expressed in terms of the suitable weight functions. Finally, the performance of the resulting feedback system is verified through numerical simulations.

Control of Continuous Stirred Tank Reactor using Neural Networks

Background/Objectives: This paper presents the design of neuro controller NARMA-L2 for composition control in an isothermal Continuous Stirred Tank Reactor (CSTR) by manipulating the input feed composition. Methods/Statistical Analysis: The NARMA-L2 controller design is implemented in two stages in which the first stage is system identification to model the process and the second stage is designing the process controller. For controlling the product composition in the CSTR, the neuro controller NARMA-L2 is implemented in MATLAB Simulink environment. Findings: The simulation results show the superiority of the NARMA-L2 in accurately tracking the composition set-point changes in the CSTR and control the system better as compared to that of the conventional PID. The neuro controller NARMA-L2 can handle non-linear aspects of the CSTR by transforming its non-linear dynamic into an implicit algebraic model which can control the trajectory of the CSTR efficiently. Application/Improvements: The advantage of using the neuro controller NARMA-L2 is that it requires the minimal online computation compared to other neural network architecture for control such as model reference control and model predictive control.

On the equivalence of storage functions in controlled thermodynamic systems

Motivated by thermodynamic concepts strongly related to the second law of thermodynamics, this paper deals with the mathematical foundations of thermodynamics to reconstruct some storage functions usable both for the stability analysis and control of homogeneous chemical processes through the case study of non isothermal continuous stirred tank reactors (CSTRs). Besides, the equivalence of the storage functions in the thermodynamics framework is also shown.

On the relaxing dissipation of dissipative pseudo Hamiltonian models

This paper further explores the link between irreversible thermodynamics and system theory, and its use for port-based modeling for reaction systems. More specifically we show here that a pseudo Hamiltonian representation with R(x) > 0 can be obtained by considering the Brayton-Moser formulation via a unified potential function that verifies a thermodynamic evolution criterion. As a consequence, it gives additional degrees of freedom (i.e. to construct alternate pseudo Hamiltonian models with new passive outputs) usable for further studies on the control design. A representative example of irreversible processes via the non isothermal continuous stirred tank reactor model is used to illustrate the theoretical developments.

Optimal Choice of Weighting Factors in Adaptive Linear Quadratic Control

Most of the processes in the industry have nonlinear behavior. Control of such processes with conventional control methods could lead to unstable, suboptimal, etc., results. On the other hand, the adaptive control is a technique widely used for controlling of nonlinear systems. The approach here is based on the recursive identification of the external linear model as a linear representation of the originally nonlinear system. The controller then reacts to the change of the working point or disturbances which could occur by the change of the parameters, structure, etc. The polynomial synthesis together with the linear quadratic (LQ) approach is employed here for the controller synthesis. These techniques satisfy basic control requirements such as the stability, the reference signal tracking and the disturbance attenuation. Resulted controller could be tuned with the choice of weighting factors in LQ approach. This work investigates the effect of these factors on control results. Proposed methods are tested on the mathematical model of the isothermal continuous stirred-tank reactor and simulated results are also verified on the real model of the continuous stirred tank reactor.

Entropy-based stabilizing feedback law under input constraints of a CSTR

The paper deals with nonlinear control under input constraints of a non isothermal Continuous Stirred Tank Reactor (CSTR) using thermodynamic concepts. More precisely, the paper presents an extension of the previous work (Hoang et al. (2012)) where the jacket temperature is used as the only control input. Constrained input control strategy is based on the observation that the chemical reactor operates at an unique stationary temperature when the lower or upper bound of the input variable is imposed. This control design results in a global asymptotic stabilizing feedback law that provides a locally exponentially stable behaviour of the overall system.

Evaluation of periodic processes with two modulated inputs based on nonlinear frequency response analysis. Case study: CSTR with modulation of the inlet concentration and flow-rate

In our previous work, a new, fast and easy, nonlinear frequency response method for analysing potential improvements of reactor performance by forced periodic operation was presented. This method, which is based on Volterra series, generalized Fourier transform and the concept of higher-order frequency response functions (FRFs), gives an approximate value of the average process performance directly, without numerical simulation of the complete process. It was shown that the asymmetrical second order frequency response function, (G2(ω,−ω)) corresponds to the dominant term of the non-periodic (DC) component of the periodic steady-state response and determines the average performance of the periodic process. Thus, in order to evaluate the potential of a periodic reactor operation, it is enough to derive and analyse this function.In this work this method is extended to evaluating periodic operations with forced oscillations of two modulated inputs. In this case the nonlinear system has to be defined by three sets of frequency response functions, two of them correlating the output to each of the inputs and one set of cross-FRFs. The general methodology for this case is developed. It is further used to analyse the time-average performance of an isothermal continuous stirred tank reactor (CSTR) with forced periodic modulation of the inlet concentration and flow-rate, for a simple nth order homogeneous reaction. The analysis is performed for cases when the inlet concentration and flow-rate are modulated simultaneously. The optimal choice of the phase shift between the two inputs is discussed in detail.

The reduction of NO by CO under oxygen-rich conditions in a fixed-bed catalytic reactor: A mathematical model that can explain the peculiar behavior

A kinetic model of the NO+CO+O2 catalytic reaction in the isothermal continuous stirred tank reactor (CSTR) has been introduced previously in [A.G. Makeev, N.V. Peskov, H. Yanagihara, Appl. Catal. B: Environ. 119–120 (2012) 273]. The present study implements the same kinetic mechanism but now it is combined with a more realistic model of the catalytic flow reactor – the axially dispersed plug flow reactor model. The resulting new model is compared with the CSTR-based model. The calculations demonstrate that back-mixing in the isothermal reactor can lead to the following effect. At sufficiently high temperatures, when the conversion of CO is close to 100%, an increase in NO-to-N2 conversion can be induced by: (1) the decrease of the active catalyst surface area and (2) the increase of the space velocity of the feed gas. These items just represent the peculiar behavior, which seems to contradict with the physical intuition. In fact, such results have been observed experimentally, but probably they have not been properly explained. We demonstrate that these peculiar effects can be caused by the kinetics of competing reactions in a catalytic flow reactor. The vital step in the reaction mechanism appears to be the reversible dissociation of the adsorbed NO species. In particular, the calculations show that the overall reactor efficiency can decrease significantly when the catalyst surface area is increasing. This effect disappears if the ideal plug flow model (i.e., without dispersion/diffusion) is applied.

Robust Controller Design using Quantitative Feedback Theory (QFT)

of the practical systems are characterized by high uncertainty which makes it difficult to maintain good stability margins and performance properties for closed loop system. In case of conventional control, if plant parameter changes we cannot assure about the system performance hence it is necessary to design robust control for uncertain plant. Among the various strategies proposed to tackle this problem, Quantitative Feedback Theory (QFT) has proved its superiority especially in the face of significant parametric uncertainty. The feature of QFT is that it can take care of large parametric uncertainty along with phase information. For the purpose of QFT, the feedback system is normally described by the two-degrees-of freedom structure. A chemical system common to many chemical processing plants, known as a continuous stirred tank reactor (CSTR).For an isothermal CSTR the product concentration can be controlled by manipulating the feed flow rate,which changes the residence time. Keywords—