Optimization by distributed control of reactors with decaying catalyst Part II: First-order catalyst deactivation

The problem of optimally choosing the temperature T(z, t) as a function of time t and position z in a tubular fixed bed chemical reactor, so as to minimize the total yield of product over a fixed time period for a reaction-deactivation system with a slow decaying catalyst has been formulated for the case where the rate of catalyst decay is linearly dependent upon activity. Several characteristics of extremal control policies which supplement the theoretical characterization obtained for the case of general reaction-deactivation kinetics using Sirazetdinov and Degtyarev's maximum principle are indicated. Results are that the supplementary properties of the extremal controls for linear catalyst deactivation kinetics may be used to advantage to reduce the computational dimensionality in synthesizing the control policies. Numerical calculations are presented to illustrate this fact in the case of initial catalyst activity profiles which are not uniform nor continuous along the reactor bed.     

Optimization by lumped control of reactors with langmuir-hinshelwood catalyst deactivation

A method is proposed for solving the problem of temperature optimal control in tubular fixed-bed reactors with reaction systems described by Langmuir-Hinshelwood-Hougen-Watson kinetic equations. The optimization problem is formulated by N state differential equations corresponding to the N differential fixed-bed reactors in which the integral reactor is divided. It is solved using the control vector parameterization computational technique. The proposed method when applied to a simple reaction system reported previously in the literature gives analogous results, and thus validates the theory. This theory is applied to the dehydrogenation of benzyl alcohol to benzaldehyde. An analysis of optimality problem shows a strong influence of the temperature dependence of the ratio of reaction rate to deactivation reaction rate on the optimal policy.     

Optimal temperature policies by distributed control for reactors with lhhw catalyst deactivation

An approximate procedure for solving the problem of temperature optimal control distributed in both space and time coordinates is presented. It is applied to chemical reactors suffering from catalyst decay whose model is described by a complex second order partial differential equation deduced from LHHW kinetics. The proposed algorithm uses a finite-difference approximation method to solve the state equation, and the control vector parametrization technique to obtain the optimal control. Numerical examples are computed and the results obtained show that distributed control reaches significantly higher production than lumped control.     

The optimization of reactors with decaying catalyst: The effect of macromixing

The following general problem is considered: in a catalytic reactor with uniform temperature and a particular residence-time distribution (RTD), given the kinetics of the reaction and the kinetics of decay of the catalyst activity, how should the temperature be varied with time so that the total conversion over a fixed period of time is maximized? The solution to this problem is presented for reactors with various RTD's, and the effect of macromixing on the optimal policy is discussed. For reactors in segregated flow, with decay rate independent of conversion and a single irreversible reaction, the policy for any RTD is formed from three components: a stationary arc of constant conversion and two constrained arcs at the lower and upper temperature limits. In all but exceptional cases, the policy must end on the upper limit of temperature. By comparing the optimal average conversion in a single CSTR with that obtained by applying the optimal plug-flow policy to the CSTR one can find the maximum loss in conversion to be suffered by using the optimal plug-flow policy in a reactor of arbitrary, perhaps unknown, RTD. This loss was found in a numerical example to be negligible, which suggests that a reactor may be operated almost optimally without knowing its RTD.     

Optimization by catalyst packing in tubular reactors with catalyst decay

The optimal initial distribution of catalyst activity along the axis of a tubular fixed bed reactor is examined for a class of reaction-deactivation problems. A general set of simultaneous reactions is considered and a quasi-steady state approximation is used in describing the reaction kinetics. The rate of decay of the catalyst is expressed as a function of temperature, concentration or degree of conversion and catalyst activity. The influence of various initial catalyst activity distributions upon the reactor performance is studied theoretically for several types of decay rate expressions. Numerical results are presented for a single irreversible reaction with a conversion-dependent decay rate.     

Optimization by catalyst packing in tubular reactors with catalyst decay

The optimal initial distribution of catalyst activity along the axis of a tubular fixed bed reactor is examined for a class of reaction-deactivation problems. A general set of simultaneous reactions is considered and a quasi-steady state approximation is used in describing the reaction kinetics. The rate of decay of the catalyst is expressed as a function of temperature, concentration or degree of conversion and catalyst activity. The influence of various initial catalyst activity distributions upon the reactor performance is studied theoretically for several types of decay rate expressions. Numerical results are presented for a single irreversible reaction with a conversion-dependent decay rate.     

Analysis of the lumped and distributed optimal temperature trajectories for packed bed reactors with concentration dependent catalyst deactivation

Algorithms published previously by the authors for solving the problem of spatially uniform and non-uniform temperature optimal control in tubular fixed bed reactors have been applied to three systems with series catalyst deactivation and different ratios between the activation energies of the main and deactivation reactions. The optimal temperature and conversion trajectories as well as the maximum production obtained have been compared at the same final duration of operation. Systems with slow deactivation obtain maximum production with spatially uniform control, whilst for systems with rapid deactivation the temperature trajectory distributed both in space and time improves the production obtained with lumped control.     

Optimization of reactors with catalyst decay I — single tubular reactor with uniform temperature

The optimal choice of temperature with time is sought so as to maximize the total amount of reaction in a fixed time in a tubular reactor with uniform temperature and decaying catalyst. The single reaction is assumed to be irreversible with a rate expressible as a product of separate functions of temperature, activity and conversion. The rate of decay of activity is also a product of separate functions of temperature and activity but independent of conversion. To each total reaction time, there is a unique optimal policy. The policies are derived for various cases, distinguished by the ratio of activation energies for reaction and decay. Any optimal policy must end on the upper temperature constraint except for certain special cases. The stationary sub-policy is shown to be one of constant conversion when the inlet conversion is constant. The stationary sub-policy for a variable inlet conversion is also derived. Numerical calculations are presented to illustrate the optimal policies and a comparison of the present results is made with those of previous workers.     

Optimal inlet temperature control of reactors with catalyst decay

The constant conversion policy which has been shown to be optimal in certain classes of optimal control problems for reactors with decaying catalyst is examined here for the inlet temperature control in a tubular reactor where the catalyst decay is a function of the composition. A single irreversible reaction is considered where the rate expression is a product of separate functions of temperature, concentration or conversion and catalyst activity. The catalyst decay rate expression is also a product of separate functions of the same variables. A proof of the constant exit conversion property is given for problems where the decay rate is of first-order with respect to the catalyst activity.     

Optimal inlet temperature control of reactors with catalyst decay

The constant conversion policy which has been shown to be optimal in certain classes of optimal control problems for reactors with decaying catalyst is examined here for the inlet temperature control in a tubular reactor where the catalyst decay is a function of the composition. A single irreversible reaction is considered where the rate expression is a product of separate functions of temperature, concentration or conversion and catalyst activity. The catalyst decay rate expression is also a product of separate functions of the same variables. A proof of the constant exit conversion property is given for problems where the decay rate is of first-order with respect to the catalyst activity.     

A learning fedforward control strategy for the concentration control of recycle reactors. Part I: Set-point control

Unknown reaction rates make concentration control in laboratory recycle reactors a difficult task, particularly when gas chromatographs or other analytical equipment with long analysis times from part of the control system. For such time-delay systems, a model-based feedforward control technique is developed which uses some kind of learning linear mapping to provide estimates of the reaction rates. Regarding the number of time steps necessary to satisfy a new set point, in a variety of simulation studies, this technique has proven superior to similar but simpler control strategies.     

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.     

Optimization of reactors with catalyst decay and the constant conversion policy

For continuous stirred-tank and plug-flow catalytic reactors in which the catalyst activity decreases with time, Szepe [1] showed that the optimal temperature-time policy can lead to a policy of constant exit conversion under specific conditions. These conditions included a single irreversible reaction with a separable rate equation and a rate of decay of catalyst activity also separable and independent of composition.It is the purpose of this paper to extend the results of Szepe and to establish: (a) A necessary and sufficient condition for a constant-conversion optimal policy in a continuous stirred-tank reactor, and (b) A proof of the constant-conversion policy for a plug-flow reactor with distributed control of temperature for a general irreversible reaction with separable kinetics.In both cases, the rate of decay is assumed to depend on composition.     

Optimization of reactors with catalyst decay iii-tubular reactor with several beds of uniform temperature

The temperature of a series of beds are chosen with time so as to maximize the overall production of an irreversible reaction, over a fixed total time. Each bed has uniform temperature and activity at any instant and the catalyst activity decays at a rate which is independent of conversion. It is shown that the temperature of each bed should finally be at the upper limit. Further, if no bed temperature is constrained, for constant inlet composition the conversion out of every bed should be held constant. Under certain conditions, it is best to shut some beds down temporarily, to save their activities for later. Numerical examples are given.     

Pore network model of deactivation of immobilized glucose isomerase in packed-bed reactors. Part III: Multiscale modelling

A widely used method for converting glucose to fructose is by enzymatic isomerization. This process, which uses immobilized glucose isomerase, takes place in a packed-bed reactor that consists of microporous particles with a range of pore sizes, characterized by a pore size distribution. The micropores are also interconnected, giving rise to a three-dimensional (3D) network of pores with distributed sizes and connectivities. The particles themselves generate a 3D pore network at the reactor level with distributed pore sizes, but with a fixed connectivity. In this paper, Part III of a series, we develop a multiscale modelling approach to this problem, beginning with the relevant phenomena at the scale of the micropores, and integrating them into the particle and reactor length scales. As the efficiency of the process is significantly affected by deactivation of the microporous particles, we take this phenomenon into account at all the relevant length scales. We use a real random packing of particles, originally constructed by Finney (Proc. R. Soc. Lond. A, 319 (1970) 479), and map its pore space onto an equivalent 3D Voronoi network in which the pores are represented by the edges of the Voronoi polyhedra. The flow field in the Voronoi network is determined, and the convection-diffusion-reaction equation is then solved in the Voronoi network, taking into account the gradual deactivation of the microporous particles. Several plausible mechanisms of deactivation of the microporous particles are considered, and their effect on the performance of the reactor is investigated. Good agreement is found between the results of the computer simulations and the relevant experimental data.     

Analysis of deactivation disguised kinetics in transport reactors by periodic operation

Deactivation disguised kinetic schemes in an isothermal transport reactor are differentiated by means of periodic operation of the reactor. A periodic rectangular pulse is assumed for the inlet concentration. Three numerical examples are given to demonstrate that the deactivation disguised kinetic models give different conversions under periodic operation.     

Low-density polyethylene vessel reactors: Part I: Steady state and dynamic modelling

By the use of a perfectly mixed model and an imperfectly mixed one for lowdensity polyethylene vessel reactors, we show that increases in the initiator consumption with polymerization temperature are due to mixing limitations at the initiator feed. With all its parameters independently estimated, the imperfectly mixed model provides an excellent agreement with experimental data for several initiators, feed flow rates and polymerization pressures. In the temperature region of industrial interest for each type of initiator, the open-loop reactor dynamics drastically change from open-loop unstable, at low temperatures, to open-loop stable at high polymerization temperatures.     

Fixed bed reactors with catalyst poisoning: First order kinetics

The long time performance of an isothermal fixed bed reactor undergoing catalyst poisoning is theoretically analyzed using the dispersion model. First order reaction with dth order deactivation is assumed and the model equations are solved by matched asymptotic expansions for large Peclet number. Simple closed-form solutions, uniformly valid in time, are obtained.     

A learning feedforward control strategy for the concentration control of recycle reactors. Part II: Disturbance control

A disturbance control scheme for the concentration control of differential recycle reactors is developed which is based on the learning feedforward control strategy. Steady-state decoupling eliminates interactions between the closed loops of this non-linear, multivariable control system, thereby significantly improving the response to disturbances. Since the process model is iteratively updated, the control strategy can be regarded as a model identification adaptive control. Simulation experiments show the performance of the overall control concept.     

An integrated dynamic model for reaction kinetics and catalyst deactivation in fixed bed reactors: skeletal isomerization of 1-pentene over ferrierite

Catalyst deactivation in fixed beds is a critical issue in many industrial processes. A general dynamic model for fixed bed reactors involving catalyst deactivation was presented and simplified. The simplifications were based on the pseudo-steady state hypothesis for the fluid phase. Catalyst deactivation was treated by a linear model with respect to surface intermediates. A semi-analytical solution was obtained for the surface intermediates. The approach was applied to skeletal isomerization of 1-pentene over ferrierite. The comparison of experimental results and model predictions revealed that the model is able to describe the essential features of catalyst deactivation in skeletal isomerization and it can be useful for process scale-up.