Chapter 8 - Chemical systems with catalyst decay and regeneration

8 - Reactors With Catalyst Decay and Regeneration

This chapter provides an introduction to stochastic methods for modelling spatially homogeneous systems of chemical reactions. The Gillespie stochastic simulation algorithm and the chemical master equation are presented using simple examples of chemical systems. The chemical master equation is analysed for chemical systems containing zeroth-order, first-order and second-order chemical reactions. For zeroth-order and first-order chemical reactions, the average behaviour of the stochastic chemical system is described by the ordinary differential equations (ODEs) given by the standard deterministic model. However, when we consider higher-order chemical reactions, for which the deterministic description is nonlinear, the deterministic ODE model does not provide an exact description of the average behaviour of the stochastic system.

Most kinetic models are expressed in isothermal mathematics. This may lead unaware scientists either to the misconception that classical isothermal kinetic models cannot be used for any chemical process in an environment with a time-dependent temperature profile or, even worse, to a misuse of them. In reality, classical isothermal models can be employed to make kinetic predictions for reactions in environments with time-dependent temperature profiles, provided that there is a continuity/conservation in the reaction extent at every temperature-time step. In this article, fundamental analyses, illustrations, guiding tables, and examples are given to help the interested readers using either conventional isothermal reacted fraction curves or rate equations to make proper kinetic predictions for chemical reactions in environments with temperature profiles that vary, even arbitrarily, with time simply by the requirement of continuity/conservation of reaction extent whenever there is an external temperature change.

Sierra Pacific Power Company is building a 100 MW, IGCC power plant based on KRW fluid bed gasifier technology that utilizes transport reactors for hot gas desulfurization and sorbent regeneration. Use of a transport absorber avoids the need for pre-filtration of dust-laden gasifier effluent, while a transport regenerator allows for the use of 100% air without the need for heat exchange equipment. Selection of transport reactors for hot gas desulfurization using a proprietary sorbent, based on testing performed in a transport reactor test unit (TRTU) at the M. W. Kellogg Technology Development Center and in a fixed bed reactor at Morgantown Energy Technology Center (METC), is outlined. The results obtained in these two test facilities and reasons for selecting transport reactors for the IGCC power plant in preference to either fixed bed or fluidized bed reactors are discussed. This paper reviews the evolution of the hot gas desulfurization system designs and includes selected results on H{sub 2}S absorption and regeneration of sulfided sorbent over several absorption/regeneration cycles conducted in the TRTU and the METC fixed bed reactor. The original design for the Sierra Pacific Project was based on fixed bed reactors with zinc ferrite as the sorbent. Owing to the high steam requirements of this sorbent, zinc titanate was selected and tested in a fixed bed reactor and was found unacceptable due to loss of strength on cyclic absorption/regeneration operation. Another sorbent evaluated was Z-Sorb{reg_sign}, a proprietary sorbent developed by Phillips Petroleum Company, was found to have excellent sulfur capacity, structural strength and regenerability. Steam was found unsuitable as fixed bed regenerator diluent, this results in a requirement for a large amount of inert gas, whereas a transport regenerator requires no diluent. The final Sierra design features transport reactors for both desulfurization and regeneration steps using neat air. 3 refs., 3 figs., 2 tabs.

In order to decrease the fluid drag on an underwater robot manipulator, an optimal trajectory method based on the variational method is presented. By introducing the adjoint variables, which are Lagrange multipliers, we formulate a Lagrange function under certain constraints related to the target angle, target angular velocity, and dynamic equation of the robot manipulator. The state equation (the partial differentiation of the Lagrange function with respect to the state variables), adjoint equation (the partial differentiation of the Lagrange function with respect to the adjoint variables), and sensitivity equation (the partial differentiation of the Lagrange function with respect to torques) can be derived from the stationary conditions of the Lagrange function. Using the state equation, we can calculate the state variables (angles, angular velocities, and angular acceleration) at every time step in the forward time direction. These state variables are stored as data at every time step. Next, by using the adjoint equation, we can calculate the adjoint variables by using these state variables at every time step in the backward time direction. These adjoint variables are stored as data at every time step. Third, the sensitivity equation is calculated by using both the state variables and the adjoint variables. Finally, the optimal trajectory of the manipulator is obtained using the sensitivities. The proposed method is applied to the problem of two-link manipulators. It can obtain the optimal drag reduction trajectory of the manipulator under the constraints mentioned above.

The overall objective of this program is to develop a preliminary assessment of the potential of an advanced gasification process based upon char, steam, lime and metal salt additives and catalyzed carbon, oxygen, and steam reactions. Work was initiated on restoring the 2.5-cm-diam, pressure fluid bed reactor system. Some modifications to the system were made in order to make operation of the system more flexible and efficient, and to solve particulate carry-over problems. The char screening study effort was planned. A new fixed bed reactor was designed, constructed and preliminarly checked out. A preliminary study of the effects which major process parameters have on hydrogen product costs was undertaken for the conceptual, catalytic hydrogen process design developed under the previous contract. The eight parameters selected for study were: reaction extent in the hydrogen production reactor, shift converter catalyst selection and operation temperature, catalyst recyclability performance, steam utilization rate in the hydrogen production reactor, gasifier and acceptor regenerator residence times, gasifier and acceptor regenerator operational temperature, acceptor activity, and energy recovery from the spent solids bleed stream. (LK)

The objective of the paper is to develop a boundary control method for the Black–Scholes PDE which describes option dynamics. It is shown that the procedure for numerical solution of Black–Scholes PDE results into a set of nonlinear ordinary differential equations (ODEs) and an associated state equations model. For the local subsystems, into which a Black–Scholes PDE is decomposed, it becomes possible to apply boundary-based feedback control. The controller design proceeds by showing that the state-space model of the Black–Scholes PDE stands for a differentially flat system. Next, for each subsystem which is related to a nonlinear ODE, a virtual control input is computed, that can invert the subsystem’s dynamics and can eliminate the subsystem’s tracking error. From the last row of the state-space description, the control input (boundary condition) that is actually applied to the Black–Scholes PDE system is found. This control input contains recursively all virtual control inputs which were computed for the individual ODE subsystems associated with the previous rows of the state-space equation. Thus, by tracing the rows of the state-space model backwards, at each iteration of the control algorithm, one can finally obtain the control input that should be applied to the Black–Scholes PDE system so as to assure that all its state variables will converge to the desirable setpoints.

We prove the maximum principle for the problem outlined in the title with the aid (or more precisely in the class) of so-called sliding variations. INTRODUCTION The by-now classical maximum principle for the simplest nonlinear optimal control problem was formulated by Pontryagin and proved by Boltyanskii over 30 years ago (see [1, 2, 3] and also [8]). Since then, the maximum principle has been derived for various optimal control problems in an enormous number of studies. This is well-known and we do not give a full list of references here; we only mention [4-7, 9-24, 27-30]. In our view, the deepest and most thorough studies are those of Milyutin and Dubovitskii [9-14]. These authors advance very far in their development of the maximum . principle for problems with ordinary differential equations - first to problems with phase constraints [9] and then to the general problem with phase and arbitrary (in general, nonregular) mixed constraints [10-14]. These authors completely resolve the issue of deriving the necessary first-order conditions for these problems. Despite these achievements, we are still witnessing the publication of studies that derive the maximum principle for various problems which are particular cases of the general Dubovitskii-Milyutin problem. The authors of some of these studies possibly do not realize that they are rederiving well-established results.** This is partly due to the fact that, on the one hand, optimal control results are primarily interesting for applied engineers, while the references [9-14] are far from being accessible to every engineer, because they use a fairly complex mathematical apparatus which is not familiar to engineers; on the other hand, optimal control theory as such so far has failed to attract a sufficiently wide followign of mathematicians. "Optimizers" still feel that if a problem originates in engineering practice, then its solution (and preferably the process of its analysis) should remain in the domain of concepts which are familiar in the engineering environment (compare this situation with the totally different state of things in the theory of partial differential equations or in probability theory). This, in our view, accounts for repeated attempts to simplify the proof of the maximum principle. However, even if some studies achieve a certain simplification of the proof, this is the result of extreme simplification of the problem, and not an achievement of the particular method of analysis. The positive impact of these attempts is that they highlight the complexity of the problem. They have shown that the problem cannot be solved by elementary techniques, without invoking new methods (specifically, methods of functional analysis). We are thus forced to admit that the complexity of the work of Dubovitskii-Milyutin is not contrived: it is inherent to the complexity of the underlying problem. Yet there is a fairly wide class of problems for which the maximum principle can be derived relatively simply. These are the problems with pure phase constraints and so-called "regular" mixed constraints. They cover most of the problems considered in the literature. The maximum principle for these problems was derived by Milyutin and Dubovitskii back in the 1960s, immediately after their derivation of the maximum principle for problems with phase constraints [9]. However, it has never been published because, in the authors' opinion, the addition of regular mixed constraints to phase constraints did not constitute a significant advance.

This perspective illustrates the electromagnetic induction heating technology for a rational heat control in catalytic heterogeneous processes. It mainly focuses on the remarkable advantages of this approach in terms of process intensification, energy efficiency, reactor setup simplification, and safety issues coming from the use of radio frequency heated susceptors/catalysts in fixed-bed reactors under flow operational conditions. It is a real enabling technology that allows a catalytic process to go beyond reactor bounds, reducing inefficient energy transfer issues and heat dissipation phenomena while improving reactor hydrodynamics. Hence, it allows pushing catalytic processes to the limits of their kinetics. Undoubtedly, inductive heating represents a twist in performing catalysis. Indeed, it offers unique solutions to overcome heat transfer limitations (i.e. slow heating/cooling rates, nonuniform heating environments, low energy efficiency) to those endo- and exothermic catalytic transformations that make use of conventional heating methodologies.

The problem of applying the various computational methods of mathematical programming in the design of an optimal control system is discussed. A general case of non-linear, non-autonomous, state equations, subject to inequality constraints on both state and control variables, is considered. Both continuous and discrete time systems are investigated. In case of discrete time systems, the sampling intervals are assumed generally unequal and aperiodic, with inequality constraints imposed upon them. Systems like these impose considerable computational difficulties when treated by the maximum principle or dynamie programming. Using mathematical programming, one may simplify a wide class of those computational problems. Several examples of applying mathematical programming to particular control problems are presented.

PANSYM: a symbolic equation generator for mathematical modelling, analysis and control of metabolic and pharmacokinetic systems

A Regional Model for the Distribution of Industrial Activities

One of the major obstacles against succesful chemotherapy of cancer is the emergence of resistance of cancer cells to cytotoxic agents. Applying optimal control theory to mathematical models of cell cycle dynamics can be a very efficient method to understand and, eventually, overcome this problem. Results that have been hitherto obtained have already helped to explain some observed phenomena, concerning dynamical properties of cancer populations. Because of recent progress in understanding the way in which chemotherapy affects cancer cells, new insights and more precise mathematical formulation of control problem, in the meaning of finding optimal chemotherapy, became possible. This, together with a progress in mathematical tools, has renewed hopes for improving chemotherapy protocols. In this paper we consider a population of neoplastic cells stratified into subpopulations of cells of different types. Due to the mutational event a sensitive cell can acquire a copy of the gene that makes it resistant to the agent. Likewise, the division of resistant cells can result in the change of the number of gene copies. We convert the model in the form of an infinite dimensional system of ordinary differential state equations discussed in our previous publications (see e.g. Swierniak etal., 1996b; Polariski etal., 1997; Swierniak etaL, 1998c), into the integro‐differential form. It enables application of the necessary conditions of optimality given by the appropriate version of Pontryagin′s maximum principle, e.g. (Gabasov and Kirilowa, 1971). The performance index which should be minimized combines the negative cumulated cytotoxic effect of the drug and the terminal population of both sensitive and resistant neoplastic cells. The linear form of the cost function and the bilinear form of the state equation result in a bang‐bang optimal control law. To find the switching times we propose to use a special gradient algorithm developed similarly to the one applied in our previous papers to finite dimensional problems (Duda 1994; 1997).

Phenomena treated by non-equilibrium thermodynamics can be very effectively described by thermodynamic variational principles. The remarkable advantage of such an approach consists in possibility to account for an arbitrary number of constraints among state or kinetic variables stemming, e.g., from conservation laws or balance equations. As shown in the current paper, the variational principles can provide original evolution equations for the state variables implicitly respecting the constraints. Moreover, the variational approach allows formulating the problem directly in discrete state variables and deriving their evolution equations without the necessity to solve partial differential equations. The variational approach makes it also possible to use different kinetic variables in formulation of dissipation and dissipation function.

Models of complex systems often consist of state variables with structurally similar dynamics that differ in the specific values of some parameters. Examples are multi-class epidemiological models, chemical reaction networks describing multiple components (e.g., binding sites) with equivalent functional behavior, and models of electric circuits with replicated designs. In these cases, the analysis may be expensive due to the model size. Here we consider models defined as systems of polynomial ordinary differential equations (ODEs) with positive solutions. We present an algorithm to reduce the computational cost by transforming the original ODE model into one for which we can compute an appropriate over-approximation on a smaller set of state variables. The algorithm is based on the theory of differential inequalities and consists of two steps. The first step computes a differential hull, which is an ODE system providing lower and upper bounds for each state variable. The hull is constructed such that variables with structurally similar dynamics but originally different parameters may now be represented by the same lower and upper bounds. Based on this, the second step exploits already developed notions of exact model reduction for ODEs to lump such variables. The algorithm is showcased on several case studies and its results are favourably compared against CORA, a well-known tool for reachability analysis of dynamical systems.KeywordsOrdinary differential equationsModel reductionReachability analysis