Robust temperature control for batch chemical reactors

Since batch chemical reactors exhibit an integrating response, temperature control for these systems can be a real problem for conventional PID controllers. Tuning can be extremely difficult due to the reduced stability margins proved for this type of processes. In this work, a simple robust control strategy for temperature regulation in batch and semi-batch chemical reactors is proposed. The feedback controller is composed by an approximate I/O linearizing feedback equipped with a calorimetric balance estimator. Based on standard results from singular perturbations, it is proven that the proposed feedback controller (i) can track a bounded temperature trajectory as close as desired (i.e., practical stability) by adjusting a single estimation parameter, and (ii) after a short transient, the performance of the exact I/O linearizing feedback can be recovered as the calorimetric balance estimation rate is increased.     

Some aspects of perturbation solutions arising in non-isothermal tubular chemical flow reactors

A non-isothermal tubular chemical flow reactor with general reaction function has been considered in this paper. Conductive heat losses are taken into account in the system. The coupled conservation-equations are examined by the methods of perturbation theory for low and high Bodenstein numbers. Solutions for a very weak chemically reactive system in the case of high Bodenstein numbers are obtained by two methods of singular perturbation theory, and these results are compared to the results obtained by an exact calculation. This comparison shows excellent agreement.     

Some general considerations of reversible chemical reactions in batch and tubular reactors

The possibility of change in the algebraic sign of the net reaction rate of a single reversible reaction occurring in adiabatic or non-adibatic batch and tubular reactors is investigated. This change is shown not to occur for adiabatic reactors; it may depending on the operating conditions and the system parameters for non-adiabatic reactors. These and the related aspects of approach to system equilibrium are analyzed by an examination of the mathematical models describing these chemical reactors. Important results are summarized throughout the paper as Conclusions 1 - 10. The results have direct applications in reactor design and operation. From a practical point of view it means that yield may be reduced in some cases by allowing too long a residence time even for a single reaction. Numerical examples are given to illustrate the results.     

Studies in the control of tubular reactors—II Stabilization by modal control

The lumped parameter model that was developed in Part I is linearized to obtain the linear dynamical model of the system near an unstable steady state. When concentration and temperature measurements are possible along the reactor length and their number is the same as the number of collocation points, modal state-feedback controllers are designed to relocate the largest eigenvalues to negative values and thus locally stabilize an unstable steady state. Transient calculations of the non-linear system equations are preformed and the domain of attraction of the stabilized steady state is examined for different locations of the eigenvalues of the closed loop system. It is seen that for both problems I and II the domain of attraction becomes very large when the unstable and one stable eigenvalue are shifted near the third largest one. This is true in spite of the large differences in their dynamical characteristics.     

Modeling of chemical reactors—XXXI: The one-phase backflow cell model used for simulation of tubular adiabatic reactors

The backflow cell model is used to simulate steady state operation of a tubular adiabatic reactor. The model proposed embraces different mechanism of axial dispersion of heat and mass. It will be shown that the backflow cell model may be used for approximation of the dispersion model. While there are differences in qualitative behavior of simple cell and dispersion models, the backflow cell model gives results which are in agreement with the dispersion model. The model may be used for simulation of steady-state behavior of tubular homogeneous and heterogeneous reactors.     

Dispersion models of unsteady tubular reactors

Axial dispersion in time-variable laminar flow in a tubular reactor is analyzed using an exact procedure for the case of a homogenous first-order reaction. For the first time since the Taylor Dispersion model was originally introduced for the modeling of reactors, its validity is examined over a wide range of the reaction rate parameter by comparison against an exact analysis. It is shown that a constant coefficient dispersion model can be obtained from first principles for large values of time only for initial distribution problems; however, this simple approximate model also is reasonably good for describing concentration distributions for the present inlet distribution problem for slow reactions and for axial locations sufficiently far away from the inlet. For rapid reactions, while the dispersion model is inaccurate in describing axial concentration distributions, it is surprisingly good for predicting the reactor length required for complete conversion. In contrast to the conclusion of a recent article, it will be shown that the dispersion coefficient is independent of the reaction rate constant.     

The bifurcation behavior of tubular reactors

Methods for studying the bifurcation behavior of tubular reactors have been developed. This involves the application of static and Hopf bifurcation theory for PDE's and the very precise determination of steady state profiles. Practical computational methods for carrying out this analysis are discussed in some detail. For the special case of a first order, irreversible reaction in a tubular reactor with axial dispersion, the bifurcation behavior is classified and summarized in parameter space plots. In particular the influence of the Lewis and Peclet numbers is investigated. It is shown that oscillations due to interaction of dispersion and reaction effects should not exist in fixed bed reactors and moreover, should only occur in very short “empty” tubular reactors. The parameter study not only brings together previously published examples of multiple and periodic solutions but also reveals a hitherto undiscovered wealth of bifurcation structures. Sixteen of these structures, which come about by combinations of as many as four bifurcations to multiple steady states and four bifurcations to periodic solutions, are illustrated with numerical examples. Although the analysis is based on the pseudohomogeneous axial dispersion model, it can readily be applied to other reaction diffusion equations such as the general two phase models for fixed bed reactors.     

Studies in the control of tubular reactors—III stabilization by observer design

The stabilization of an unstable nonlinear distributed chemical reactor system is examined when concentration measurements are not possible. The linearized form of the finite dimensional approximate model developed in Parts I[1] and II[2] is used to show that the observability index is equal to two. Furthermore it is shown that the dynamical characteristics of the reactor are such that, by a proper design of the Luenberger observer, the concentration estimation at a given collocation point can be made independently of the estimation at other collocation points.For purposes of control a one-dimensional observer is designed to directly estimate the control variable. Simulations of the nonlinear model of the reactor show that the observer design is quite successful in the stabilization of an unstable steady state when only temperature measurements along the reactor are available.     

Parametric sensitivity in tubular polymerization reactors

A recent technique for studying the parametric sensitivity of chemical reactors is applied to tubular chain homopolymerization reactors. The sensitivities of the temperature maxima with respect to various parameters of the model are computed. Using conditions typically encountered for high-pressure polyethylene systems, it is found that the temperature sensitivities with respect to all the nine parameters have their maxima at approximately the same value of the feed initiator concentration, thus leading to a generalized sensitivity-based constraint for design. It is also found that, under usual conditions of operation, no significant design constraints on the feed temperature are indicated. Detailed sensitivity plots are presented, which could be used to obtain “safe” operating conditions. The effects of changing the most important parameters, the dimensionless heat of reaction and the dimensionless activation energy (ϵ), on the sensitivity envelope are also investigated. Our studies reveal that better estimates of ϵ than are available presently are required. Sensitivities of the number-average chain length maxima with respect to the same nine parameters are also computed. Under conditions where the steady-state hypothesis applies, estimates of these sensitivities can be obtained analytically. However, for the usual values of the parameters, and close to “sensitive” values of the feed initiator concentration, this hypothesis does not apply, and the chain length sensitivities need to be obtained numerically. In the absence of the gel effect, chain length sensitivities do not usually provide design constraints because of the very low monomer conversions encountered.     

Radial transport in tubular polymerization reactors

The importance of radial convective flows in tubular polymerization reactors is examined by analysing a model problem which contains many of the features of an actual tubular reactor. The transport equations for the model problem are simplified using lubrication theory, and a convergent finite-difference method is formulated to obtain solutions to the simplified equation set. The computed solutions show that significant errors can be introduced in the velocity and concentration fields if the radial convective term in the species continuity equation for the monomer is neglected when the ratio of polymer to monomer viscosities is sufficiently large. It is also shown that a fine finite-difference mesh is needed to obtain accurate solutions for high viscosity ratios.     

Robust discrete control of nonlinear processes: Application to chemical reactors

Trajectory tracking or rejecting persistent disturbances with digital controllers in nonlinear processes is a class of problems where classical control methods breakdown since it is very difficult to describe the dynamic behavior over the entire trajectory. In this paper, a model-based robust control scheme is proposed as a potential solution approach for these systems. The proposed control algorithm is a robust error feedback controller that allows us to track predetermined operation profiles while attenuating the disturbances and maintaining the stability conditions of the nonlinear processes. Various numerical simulation examples demonstrate the effectiveness of this robust scheme. Two examples deal with effective trajectory tracking in chemical reactors over a wide range of operating conditions. The third example analyses the attenuation of periodic load in a biological reactor. All examples illustrate the ability of the robust control scheme to provide good control in the face of parameter uncertainties and load disturbances.     

Design of tubular reactors in recycle systems

The article presents an approach to design tubular reactors in recycle systems, based on non-linear analysis. A pseudo-homogeneous plug-flow reactor model is used. It is assumed that the separation unit delivers product and recycle streams with fixed composition. The stand-alone reactor has a unique stable steady state. The coupled reactor–separation–recycle system shows four types of conversion versus plant Damköhler number bifurcation diagrams. A feasible steady state exists only if the reactor volume exceeds a critical value. For isothermal reactor, the steady state is unique and stable. For non-isothermal reactor, one or two steady states are possible. In the second situation the low-conversion state is unstable. In some parameter regions, the unique state is unstable. The design should ensure state unicity and stability, which are favoured by large heat-transfer capacity, low coolant temperature and high reactor-inlet temperature. A case study demonstrates that these phenomena can be easily found in real plants.     

Optimal location of measurements in tubular reactors

The optimal location of temperature measurements along the length of a non-adiabatic tubular reactor in which a first-order exothermic reaction is occu     

Scaleup factors of tubular emulsion polymerization reactors

This paper examines the important dimensionless numbers that control emulsion polymerization in a tubular reactor. It was found that the activation energy of polymerization was of major importance, while the role of monomer diffusion was not very significant. By selecting certain combinations of the dimensionless numbers, changes occurring during scaleup from a small tubular diameter to a larger diameter can be approximated.     

Control of chemical reactors in the subspace of reaction and control variants

Control of stirred tank reactors is studied. Possibilities of utilizing variant and invariant properties of a general reactor model for designing control systems are investigated. The studied optimal feedback controllers are based on linearized reactor models and are computed from the reaction and control variant part of the system only. One advantage of this approach, compared to state feedback, is that the dimension of the reaction and control variant space usually is smaller than the dimension of the state space of the reactor.Another advantage is that the reaction and control variants only have to be measured or estimated. This advantage is, however, not revealed by a straightforward implementation since the reaction and control variants are combinations of all states. In the paper it is shown how the control law can be modified so that just as many states as there are reaction and control variants have to be fed back, thus reducing the number of sensors needed.The methods discussed in the paper are illustrated through simulation of linear-quadratic control of a nonlinear reactor with heat exchange.     

Parametric sensitivity in co-currently cooled tubular reactors

An intrinsic criterion was derived for parametric sensitivity and runaway in co-currently cooled tubular reactors. It is essentially based on the approach of Van Welsenaere and Froment (1970) developed for the constant wall temperature case and which makes use of fundamental properties of the trajectory in the temperature vs. partial pressure plane. The criterion limits the maximum allowable temperature on an objective basis and permits the calculation of the corresponding critical inlet values for the operating variables, like partial pressure of the reactant, temperature of the reactor fluid, temperature of the cooling medium, coolant flow rate, reactor throughput ... . In addition to the rigorous treatment, a simple extrapolation procedure is presented, which proves to be very useful and accurate, provided the conditions are not too severe.     

Partial state reference model adaptive control of batch and fed-batch chemical reactors

This paper presents a study where an adaptive partial state reference model (PSRM) controller is designed and experimentally implemented for temperature control in batch and fed-batch chemical reactors. The PSRM control design is carried out using a linear quadratic control within the delta operator formulation. The control allows to specify the tracking and regulation dynamics independently, while the delta formulation is motivated by the convergence of its performances to their continuous counterpart as well as its intrinsic numerical robustness.     

Energy shaping plus damping injection control for a class of chemical reactors

Traditionally, stabilization of chemical reacting systems has been achieved with linear P or PD compensation schemes. Practical and numerical results have showed that classical linear compensation can yield acceptable performance. On the other hand, recent years have witnessed the emergence of systematic feedback control strategies based on energy and port-interconnected systems. These approaches exploit the physical structure of the chemical reactor to construct compensation schemes with physical appealing. The aim of this work is to show that traditional PD compensation for CSTRs can be interpreted in terms of mechanical system analogies. In the line of energy shaping plus damping injection for robotic systems, it is shown that proportional feedback is a type of potential energy shaping to accommodate a unique equilibrium point. On the other hand, derivative control acts as a damping injector for the energy balance within the chemical reactor. The stability proof uses a novel approach to convert the temperature dynamics into a second-order systems where the mechanical analogies become more evident. In this way, the stability analysis can be performed with singular perturbation methods with a Lyapunov function for the energy balance derived from a “potential plus kinetics” energy construction.