Dynamic Tubular Reactor Model

Mathematical models of tubular chemical reactor behaviour can be used to predict the dynamic variations in concentration, temperature and flow rate at various locations within the reactor. A complete tubular reactor model would however be extremely complex, involving variations in both radial and axial... 

The coupling of the component and energy balance equations in the modelling of non-isothermal tubular reactors can often lead to numerical difficulties, especially in solutions of steady-state behaviour. In these cases, a dynamic digital simulation approach can often be advantageous as a method of determining the steady-state variations in concentration and temperature, with respect to reactor length. The full form of the dynamic model equations are used in this approach, and these are solved up to the final steady-state condition, at which condition... 

Figure 4.11. Finite-differencing for a dynamic tubular reactor model.

In this chapter the simulation examples are described. As seen from the Table of Contents, the examples are organised according to twelve application areas Batch Reactors, Continuous Tank Reactors, Tubular Reactors, Semi-Continuous Reactors, Mixing Models, Tank Flow Examples, Process Control, Mass Transfer Processes, Distillation Processes, Heat Transfer, and Dynamic Numerical Examples. There are aspects of some examples which relate them to more than one application area, which is usually apparent from the titles of the examples. Within each section, the examples are listed in order of their degree of difficulty. 

This example models the dynamic behaviour of an non-ideal isothermal tubular reactor in order to predict the variation of concentration, with respect to both axial distance along the reactor and flow time. Non-ideal flow in the reactor is represented by the axial dispersion flow model. The analysis is based on a simple, isothermal first-order reaction. 

Chemical Kinetics, Tank and Tubular Reactor Fundamentals, Residence Time Distributions, Multiphase Reaction Systems, Basic Reactor Types, Batch Reactor Dynamics, Semi-batch Reactors, Control and Stability of Nonisotheimal Reactors. Complex Reactions with Feeding Strategies, Liquid Phase Tubular Reactors, Gas Phase Tubular Reactors, Axial Dispersion, Unsteady State Tubular Reactor Models... 

PACKED TUBULAR REACTOR MODELING AND CATALYST DESIGN USING COMPUTATIONAL FLUID DYNAMICS... 

Instead of the partial differential equation model presented above, the model is developed here in dynamic difference equation form, which is suitable for solution by dynamic simulation packages, such as MADONNA. Analogous to the previous development for tubular reactors and extraction columns, the development of the dynamic dispersion model starts by considering an element of tube... 

Use a dynamic model for a tank reactor to verify the values calculated by this program. Do the same for a steady state and dynamic tubular reactor model. 

Anthony G. Dixon, Michiel Nijemeisland, and E. Hugh Stitt, Packed Tubular Reactor Modeling and Catalyst Design Using Computational Fluid Dynamics... 

Bird, R. B., Stewart, W. E., and Lightfoot, E. N., Transport Phenomena. Wiley, New York, 1960. Bonvin, D., Dynamic Modeling and Control Structures for a Tubular Autothermal Reactor at an Unstable State. Ph.D. thesis, University of California, Santa Barbara (1980). 

Dixon, A. G., Nijemeisland, M. and Hugh Stitt, E., Packed tubular reactor modeling and catalyst design using computational fluid dynamics, in "Advances in Chemical Engineering", Vol. 31, pp. 307-389, Elsevier, Amsterdam (2006). 

Effect Of Number Of Lumps If the number of plotting points in Aspen Plus is set at 10 (the default), the resulting exit temperature from the reactors under steady-state conditions in Aspen Dynamics is 578 K. Remember that it should be 583 K from the rigorous integration of the ordinary differential equations describing the steady-state tubular reactor that are used in Aspen Plus. Changing the number of points to 20 produces an exit temperature of 580 K. Changing the number of points to 50 produces an exit temperature of 582 K, which is very close to the correct value. Therefore a 50-lump model should be used. 

The reaction considered is the gas-phase, irreversible, exothermic reaction A + B — C occurring in a packed tubular reactor. The reactor and the heat exchanger are both distributed systems, which are rigorously modeled by partial differential equations. Lumped-model approximations are used in this study, which capture the important dynamics with a minimum of programming complexity. There are no sharp temperature or composition gradients in the reactor because of the low per-pass conversion and high recycle flowrate. 

A simple model of lumped kinetics for supercritical water oxidation included in the partial differential equations for temperature and organic concentrations allows to qualitatively simulate the dynamic process behavior in a tubular reactor. Process parameters can be estimated from measured operational data. By using an integrated environment for data acquisition, simulation and parameter estimation it seems possible to perform an online update of the process parameters needed for prediction of process behavior. 

Theoretical and experimental results on deactivation have been summarized in two reviews by Butt (.1,2). Previous work of particular interest to the present study has been done by Blaum (3) who used a one-dimensional two-phase model to explore the dynamic behavior of a deactivating catalyst bed. Butt and cowotkers (b,5,6)have performed deactivation studies in a short tubular reactor for benzene hydrogenation for both adiabatic and nonadiabatic arrangements. They experimentally observed both the standing (6) and travelling (4) deactivation wave. Hlavacek... 

Jensen and Ray (50) have recently tabulated some 25 experimental studies which have demonstrated steady state multiplicity and instabilities in fixed-bed reactors many of these (cf., 29, 51, 52) have noted the importance of using a heterogeneous model in matching experimental results with theoretical predictions. Using a pseudohomogeneous model, Jensen and Ray (50) also present a detailed classification of steady state and dynamic behavior (including bifurcation to periodic solutions) that is possible in tubular reactors. 

For ease of fabrication and modular construction, tubular reactors are widely used in continuous processes in the chemical processing industry. Therefore, shell-and-tube membrane reactors will be adopted as the basic model geometry in this chapter. In real production situations, however, more complex geometries and flow configurations are encountered which may require three-dimensional numerical simulation of the complicated physicochemical hydrodynamics. With the advent of more powerful computers and more efficient computational fluid dynamics (CFD) codes, the solution to these complicated problems starts to become feasible. This is particularly true in view of the ongoing intensified interest in parallel computing as applied to CFD. 

Although it does not now require the use of dynamic programming, it will be convenient to notice the especially simple case of reactors of equal (or prescribed) size. The importance of this special case lies in the fact that it is sometimes necessary to use identical reactors, and also in the value of such a sequence as a model of the tubular reactor with diffusion. 

The influence of activity changes on the dynamic behavior of nonisothermal pseudohomogeneoiis CSTR and axial dispersion tubular reactor (ADTR) with first order catalytic reaction and reversible deactivation due to adsorption and desorption of a poison or inert compound is considered. The mathematical models of these systems are described by systems of differential equations with a small time parameter. Thereforej the singular perturbation methods is used to study several features of their behavior. Its limitations are discussed and other, more general methods are developed. 

Mathematical models of tubular chemical reactor behaviour can be used to predict the dynamic variations in concentration, temperature and flow rate at various locations within the reactor. A complete tubular reactor model would however be extremely complex, involving variations in both radial and axial positions, as well as perhaps spatial variations within individual catalyst pellets. Models of such complexity are beyond the scope of this text, and variations only with respect to both time and axial position are treated here. Allowance for axial dispersion is however included, owing to its very large influence on reactor performance, and the fact that the modelling procedure using digital simulation is relatively straightforward. 

Instead of the partial differential equation model presented above, the model is developed here in dynamic difference equation form, which is suitable for solution by dynamic simulation packages, such as Madonna. Analogous to the previous development for tubular reactors and extraction columns, the development of the dynamic dispersion model starts by considering an element of tube length AZ, with a cross-sectional area of Ac, a superficial flow velocity of V and an axial dispersion coefficient, or diffusivity D. Convective and diffusive flows of component A enter and leave the liquid phase volume of any element, n, as indicated in Fig. 4.24 below. Here j represents the diffusive flux, L the liquid flow rate and and Cla the concentration of any species A in both the solid and liquid phases, respectively. 

Under steady-state conditions in a plug-flow tubular reactor, the onedimensional mass transfer equation for reactant A can be integrated rather easily to predict reactor performance. Equation (22-1) was derived for a control volume that is differentially thick in all coordinate directions. Consequently, mass transfer rate processes due to convection and diffusion occur, at most, in three coordinate directions and the mass balance is described by a partial differential equation. Current research in computational fluid dynamics applied to fixed-bed reactors seeks a better understanding of the flow phenomena by modeling the catalytic pellets where they are, instead of averaging or homogenizing... 

For a practically useful tubular reactor model, the reacting (polymerizing) mixture is considered homogeneous and only axial dispersion is considered. At each specific position along the tube, perfect radial mixing and a uniform velocity profile are assumed. The tube, therefore, can be modeled as a one-dimensional tubular reactor. Also, instantaneous fluid dynamics are assumed because of the incompressibility of the liquid mixture (hence the calculation of the velocity profile is simplified). 

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