Optimization by distributed control of reactors with decaying catalyst Part I: Non-linear catalyst deactivation

The quasi-steady state optimization of a single tubular fixed bed reactor with a slowly decaying catalyst is considered. The optimal choice of temperature T(z, t) distributed in both the space of the reactor and in chronological time is sought so as to maximize the total amount of reaction in a fixed given period of time. A single irreversible reaction is considered with a rate expressible as a product of separate functions of temperature, activity and conversion. The rate of catalyst decay is also a product of separate functions of temperature and activity but independent of conversion. Upper and lower bounds are placed on the permitted temperature. Theoretical characterization of the optimal policy is obtained using Sirazetdinov and Degtyarev's maximum principle derived for first-order partial differential equations and the influence of the ratio of reaction activation energy to catalyst deactivation energy on the derived optimal policy is indicated. Numerical calculations are presented to illustrate the optimal policies.     

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.     

The optimization of axially dispersed packed bed reactors experiencing catalyst decay

Techniques for the optimization of axially dispersed packed bed reactors having catalyst decay are developed. A weak maximum principle is presented along with an efficient computational algorithm for synthesizing the optimal policies. Singular perturbation methods are used to solve the very stiff state and adjoint equations resulting at high Peclet numbers. Numerical examples are worked to illustrate the salient features of the optimization technique as well as to give insight into the nature of the optimal policies. A certain amount of axial dispersion is found to actually improve the performance of the reactors in some cases due to the “washing out” of the hot spots which cause accelerated catalyst deactivation.     

Multiplicity of steady states in fluidized bed reactors—V: The effect of catalyst decay

A simplified two-phase model is used to investigate the behaviour of non-isothermal fluidized bed reactors experiencing catalyst decay. The investigation shows that for highly exothermic reaction it is almost always desirable to operate the reactor at the middle unstable steady state, since it gives higher accumulative yield than both the high and the low temperature steady states. A simple feedback control scheme with time varying set point is suggested to stabilize the middle steady state. The dynamic behaviour and stability of the system is investigated for the open-loop reactor (uncontrolled) and the closed loop reactor (controlled).     

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.     

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.     

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.     

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.     

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.     

Deactivation of cracking catalyst in short contact time reactors: Alternative models

Several mathematical models have been used to describe the deactivation of cracking catalysts by coke. For the case of gas oil cracking under short contact times (less than 20 seconds) it was found, using data from two different experimental reactor units, that an exponential decay function or a power law function could equally represent the data. Both functions are forms of the general hyperbolic decay expression; however, the power law assumes the unrealistic limits of infinite catalyst activity at zero time-on-stream and requires two parameters to describe deactivation. This work shows that the simple first order decay function is an effective equation to be used in describing catalyst activity decay for short reaction times.     

Multiobjective dynamic optimization of semibatch copolymerization reactors

Polymerization reactor optimal design and control problems are inevitably multiobjective in nature. Copolymerization problems encompass an even larger set of objective functions than homopolymerization. This work describes in some detail a set of typical components of the vector of objective functionals for copolymerization reactor problems. A subset of these is selected for detailed examination employing multiobjective function optimization techniques. Noninferior, or Pareto, sets of optimal solutions have been determined for the dual objectives of narrowing both copolymer composition and molecular weight distributions in the styrene-acrylonitrile system. Noninferior solutions occur when the two objectives are in conflict and no objective can be improved without sacrifice of the other. We show for several sets of control variables that, in the S-AN system, one should seldom, if ever, control with respect to one objective functional exclusively. Compromise policies can be found which are close to utopian for both objectives. A better quantitative understanding of the trade-offs between objectives in copolymerization engineering has been achieved.     

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.     

Temperature excursions in packed bed reactors with an axial variation of catalyst activity

Research appears to be growing on a type of structured reactor in which catalyst activity varies or in which different catalysts are arranged in the reactant flow direction. These reactors offer improved selectivity for some classes of complex reactions under non-isothermal conditions or when composition modulation is employed. Our examination of the rather extreme case of alternating layers of inert and active catalyst indicate that this reactor structure accentuates wrong-way temperature excursions after a step-change in temperature and amplifies periodic input temperature disturbances. Experiments used a near adiabatic 2.5 cm diameter reactor packed with 3 mm particles of 0.2 wt.% Pt/Al₂O₃. Inert layers were just the 3 mm alumina particles. Step and triangular wave inputs of constant amplitude were used. Temperature response in the bed was measured by an axial array of computer-monitored thermocouples. Measurements were compared to those made on a homogeneous mixture of catalyst and support under identical input conditions. Simulation studies show that accentuation of the temperature excursions depends on layer thickness. Even first-order reaction kinetics show accentuated temperature excursions when layered beds are used.     

Modelling of fluidized bed reactors—II: Uniform catalyst temperature and concentration

A model based on the simple two phase theory of fluidization including the catalyst particles as a third phase has been developed for a nonisothermal fluidized bed catalytic reactor with continuous circulation of catalyst particles. The dilute phase is assumed to be in plug flow, the emulsion phase gas is considered to be perfectly mixed and the particles are assumed to be perfectly mixed and uniform.Exact criteria for uniqueness and multiplicity of the steady state solutions are presented and some conclusions derived therefrom. Several examples illustrating the influence of some parameters on the steady state multiplicity are reported. The steady states are analyzed for local asymptatic stability using Liapunov's direct method, but the sufficient conditions for stability are found to be rather conservative.Numerical examples illustrating the transient behavior of the system are presented, and it has been found that the initial temperature of the catalyst particles is a predominant factor in determining which steady state will be approached.     

Similar effect of carbon and silica catalyst support on the hydrogenation reaction rate in organic slurry reactors

This paper investigates the influence of the catalyst support type on mass transport and reaction rate for the case of hydrogenation of α-methylstyrene to cumene in a gas inducing stirred slurry reactor and in a slurry bubble column. The reaction is carried out in the presence of 3% Pd/carbon and 3% Pd/silica catalyst particles. The lyophobicity of the two catalyst supports in the cumene slurry is found to be similar. The overall rate of the hydrogenation reaction is described by the classical transport and reaction resistances-in-series model. The rate of gas-to-liquid mass transfer is somewhat larger during reaction than without reaction. This enhanced mass transfer points to particle-to-bubble adhesion as a result of the relative affinity of both catalyst supports to the gas phase. The observed reaction enhancements are similar for both Pd/carbon and Pd/silica catalyst/cumene slurries.