Linear reactor models

This paper extends previous studies on the control of a polystyrene reactor by including (1) a dynamic lag on the manipulated flow rate to improve dynamic decoupling, and (2) pole placement via state variable feedback to improve overall response time. Included from the previous work are optimal allocation of resources and steady state decoupling. Simulations on the non-linear reactor model show that response times can be reduced by a factor of 6 and that for step changes in desired values the dynamic decoupling is very satisfactory. 

A GENERAL THEOREM FOR SIMPLE, LINEAR REACTOR MODELS... 

APPLICATION OF THE GAMMA DISTRIBUTION, 107 A GENERAL THEOREM FOR SIMPLE LINEAR REACTOR MODELS, 108 APPLICATION TO A MODEL OF THE BUBBLING FLUIDIZED BED, 109 THE DAMKOHLER NUMBER, 111... 

The initial value of the reactor temperature is 413 K. The non-linear and linear reactor model are shown in Fig. 8.10. 

The linear reactor model may require some additional explanation. It is chosen from Simulink Continuous Transfer Fen. The block is edited by double clicking it and entering the data as shown in Fig. 8.11. 

The Control and Estimation Tools Manager menu should look like Fig. 9.8. As ean be seen, the input point is the feed flow and the output point the concentration from the mass balance. On this display, now select Linearize Model , this will produce the Bode diagram for the linearized reactor model as shown in Fig. 9.9. 

Fig. 9.9. Bode diagram for the linearized reactor model from feed to outlet concentration.

The PBL reactor considered in the present study is a typical batch process and the open-loop test is inadequate to identify the process. We employed a closed-loop subspace identification method. This method identifies the linear state-space model using high order ARX model. To apply the linear system identification method to the PBL reactor, we first divide a single batch into several sections according to the injection time of initiators, changes of the reactant temperature and changes of the setpoint profile, etc. Each section is assumed to be linear. The initial state values for each section should be computed in advance. The linear state models obtained for each section were evaluated through numerical simulations. 

The methods concerned with differential equation parameter estimation are, of course, the ones of most concern in this book. Generally reactor models are non-linear in their parameters and therefore we are concerned mostly with nonlinear systems. 

A control algorithm has been derived that has improved the dynamic decoupling of the two outputs MW and S while maintaining a minimum "cost of operation" at the steady state. This algorithm combines precompensation on the flow rate to the reactor with state variable feedback to improve the overall speed of response. Although based on the linearized model, the algorithm has been demonstrated to work well for the nonlinear reactor model. 

This chapter is organized in the following way. First, the general model of the CSTR process, based on first principles, is derived. A linearized approximate model of the reactor around the equilibrium points is then obtained. The analysis of this model will provide some hints about the appropriate control structures. Decentralized control as well as multivariable (MIMO) control systems can be designed according to the requirements. 

The equations describing the concentration and temperature within the catalyst particles and the reactor are usually non-linear coupled ordinary differential equations and have to be solved numerically. However, it is unusual for experimental data to be of sufficient precision and extent to justify the application of such sophisticated reactor models. Uncertainties in the knowledge of effective thermal conductivities and heat transfer between gas and solid make the calculation of temperature distribution in the catalyst bed susceptible to inaccuracies, particularly in view of the pronounced effect of temperature on reaction rate. A useful approach to the preliminary design of a non-isothermal fixed bed catalytic reactor is to assume that all the resistance to heat transfer is in a thin layer of gas near the tube wall. This is a fair approximation because radial temperature profiles in packed beds are parabolic with most of the resistance to heat transfer near the tube wall. With this assumption, a one-dimensional model, which becomes quite accurate for small diameter tubes, is satisfactory for the preliminary design of reactors. Provided the ratio of the catlayst particle radius to tube length is small, dispersion of mass in the longitudinal direction may also be neglected. Finally, if heat transfer between solid cmd gas phases is accounted for implicitly by the catalyst effectiveness factor, the mass and heat conservation equations for the reactor reduce to [eqn. (62)]... 

In simplifying the packed bed reactor model, it is advantageous for control system design if the equations can be reduced to lit into the framework of modern multivariable control theory, which usually requires a model expressed as a set of linear first-order ordinary differential equations in the so-called state-space form ... 

In a linear well-mixed reactor model the flushing rate is kv = 0.5 h-1, the total reaction rate constant of a specific chemical, ot =1.5 h-1. What is the retention factor of the reactor for the considered chemical, that is, what percentage of the chemical is reacting in the reactor How does this percentage change when the input of the chemical is doubled ... 

A tubular reactor model that may apply to viscous fluids such as polymers has a radial distribution of linear velocities represented by... 

By a reactor model, we mean a system of equations (algebraic, ordinary, or partial differential, functional or integral) which purports to represent a chemical reactor in whole or in part. (The adequacy of such a representation is not at issue here.) It will be called linear if all its equations are linear and simple if its input and output can be characterized by single, concentration-like variables, Uo and u. The relation of input and output will also depend on a set of parameters (such as Damkohler number. Thiele modulus, etc.) which may be denoted by p. Let A(p) be the value of u when w0 = 1. Then, if the input is a continuous mixture with distribution g(x) over an index variable x on which some or all of the parameters may depend, the output is distributed as y(x) = g(x)A(p(jc)) and the lumped output is... 

A non-linear mathematical model, which is a set of ordinary differential equations, for the process in the SPBER was developed.19 The model accounts for the heterogeneous electrochemical reaction and homogeneous reaction in the bulk solution. The lateral distributions of potential, current density and concentration in the reactor were studied. The potential distribution in the lateral dimension, x, of the packed bed was described by a one dimensional Poisson equation as ... 

Model. A difference equation for the material balance was obtained from a discrete reactor model which was devised by dividing the annulus into a two dimensional array of cells, each taken to be a well stirred batch reactor. The model supposes that axial motion of the mobile phase and bed rotation occur by instantaneous discontinuous jumps, between cells. Reaction occurs only on the solid surface, and for the reaction type A B + C used in this work, -dn /dt = K n - n n. Linear isotherms, n = BiC, were used, and while dispersion was not explicitly included, it could be simulated by adjusting the number of cells. The balance is given by Eq. 2, where subscript n is the cell index in the axial direction, and subscript m is the index in the circumferential direction. 

On the basis of the reactor model (Eq. 1 or 2), the estimator is designed by a geometric non-linear approach [5]. The approach follows a detectability property evaluation of the reactor motion to underlie the construction, tuning and convergence conditions of the estimator. 

Not all of the balance equations are independent of one another, thus the set of equation used to solve particular problems is not solely a matter of convenience. In chemical reactor modeling it is important to recall that all chemical species mass balance equations or all chemical element conservation equations are not independent of the total mass conservation equation. In a similar manner, the angular momentum and linear momentum constraints are not independent for flow of a simple fluid . 

Due to the strong coupling and the non-linearity of the transport equations determining a reactor model, the usefulness of the numerical methods are conditional on being able to solve the set of PDE s accurately. This is difficult for most flows of engineering interest. 

A numerical solution method is said to be stable if the method does not magnify the errors that appear during the numerical solution process. This property is relevant as a consistent discretization scheme provides no guarantee that the solution of the discretized equation system will become an accurate solution of the differential equation in the limit of small step size. The stability of low order numerical schemes applied to idealized problems can be analyzed by the von Neumann s method. However, when solving relevant, non-linear and coupled reactor model equations with complex boundary... 

In most investigations concerning the reactor modelling, simple pseudohomogeneous (t = 1) reactor models were used. The effect of external and internal mass and heat transfer resistances on the effectiveness factors using realistic complex reaction network has not been widely investigated. The simple linear kinetics proposed by... 

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