Packed Bed Reactor Models

We have chosen to concentrate on a specific system throughout the chapter, the methanation reaction system. Thus, although our development is intended to be generally applicable to packed bed reactor modeling, all numerical results will be obtained for the methanation system. As a result, some approximations that we will find to apply in the methanation system may not in other reaction systems, and, where possible, we will point this out. The methanation system was chosen in part due to its industrial importance, to the existence of multiple reactions, and to its high exothermicity. 

Mathematical models of packed bed reactors can be classified into two broad categories (1) one-phase, or pseudohomogeneous, models in which the reactor bed is approximated as a quasi-homogeneous medium and (2) two-phase, or heterogeneous, models in which the catalyst and fluid phases and the heat and mass transfer between phases are treated explicitly. Although the 

A significant step in the numerical solution of packed bed reactor models was taken with the introduction of the method of orthogonal collocation to this class of problems (Finlayson, 1971). Although Finlayson showed the method to be much faster and more accurate than that based on finite differences and to be easily applicable to two-dimensional models with both radial temperature and concentration gradients, the finite difference technique remained the generally accepted procedure for packed bed reactor model solution until about 1977, when the analysis by Jutan et al. (1977) of a complex butane hydrogenolysis reactor demonstrated the real potential of the collocation procedure. 

To retain consistency throughout this presentation, we will consider a general nonadiabatic, packed bed reactor, as shown in Fig. 1, with a central axial thermal well and countercurrent flow of cooling fluid in an exterior jacket.1 We focus on the methanation reaction since methanation is a reaction of industrial importance and since methanation exhibits many common difficulties such as high exothermicity and undesirable side reactions. 

1 Although this modeling description refers to a general, nonadiabatic packed bed reactor with an axial thermal well, the analysis easily extends to the consideration of adiabatic reactors and those without thermal wells. These are merely subsets of the more general case. 

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The application of CFD to packed bed reactor modeling has usually involved the replacement of the actual packing structure with an effective continuum (Kvamsdal et al., 1999 Pedernera et al., 2003). Transport processes are then represented by lumped parameters for dispersion and heat transfer (Jakobsen... 

This chapter presents an analysis of the development of dynamic models for packed bed reactors, with particular emphasis on models that can be used in control system design. Our method of attack will be first to formulate a comprehensive, relatively detailed packed bed reactor model next to consider the techniques available for numerical solution of the model then, utilizing... 

Numerical Solution Techniques for Partial Differential Equations Arising in Packed Bed Reactor Modeling... 

Although the finite difference technique is generally easily implemented and is quite robust, the procedure often becomes numerically prohibitive for packed bed reactor models since a large number of grid points may be required to accurately define the solution. Thus, since the early 1970s most packed bed studies have used one of the methods of weighted residuals rather than finite differences. 

These methods of weighted residuals are generally recommended for packed bed reactor modeling since solution computing time is usually low since the solution can usually be accurately defined with only a few grid points. 

Another potential solution technique appropriate for the packed bed reactor model is the method of characteristics. This procedure is suitable for hyperbolic partial differential equations of the form obtained from the energy balance for the gas and catalyst and the mass balances if axial dispersion is neglected and if the radial dimension is first discretized by a technique such as orthogonal collocation. The thermal well energy balance would still require a numerical technique that is not limited to hyperbolic systems since axial conduction in the well is expected to be significant. 

Table V shows some typical criteria used for evaluating the potential approximations for packed bed reactor modeling. Many other criteria are also used and some of these are referenced in this chapter.

De Wasch and Froment (1971) and Hoiberg et. al. (1971) published the first two-dimensional packed bed reactor models that distinguished between conditions in the fluid and on the solid. The basic emphasis of the work by De Wasch and Froment (1971) was the comparison of simple homogeneous and heterogeneous models and the relationships between lumped heat transfer parameters (wall heat transfer coefficient and thermal conductivity) and the effective parameters in the gas and solid phases. Hoiberg et al. (1971)... 

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 ... 

If the catalyst is dispersed throughout the pellet, then internal diffusion of the species within the pores of the pellet, along with simultaneous reaction(s) must be accounted for if the prevailing Thiele modulus > 1. This aspect gives rise to the effectiveness factor" problem, to which a significant amount of effort, summarized by Aris ( ), has been devoted in the literature. It is important to realize that if the catalyst pellet effectiveness factor is different from unity, then the packed-bed reactor model must be a heterogeneous model it cannot be a pseudohomogeneous model. 

With deactivation, the reactor model must immediately become a transient one, to account for change in catalyst activity with time. Among others, two successful instances of packed-bed reactor modeling, in the presence of catalyst deactivation and including comparisons with experiments, are found in the works of Weekman C37, 38) and Butt (39, 40). 

In the packed-bed reactor, the molar concentrations and temperature at the exterior of the catalyst particle are coupled to the respective fluid-phase concentrations and temperature through the interfacial fluxes given in Eqs. (3.3-1) and (3.3-2). Overall mass and heat transfer coefficients are often used to describe these interfacial fluxes, similar in structure to Eqs. (3.2-1) and (3.2-2). Complete solution of the packed-bed reactor model... 

In industrial reactors there are normally gradients in the species mass concentrations, temperature, pressure and velocity in all space directions. The fundamental microscopic equations give a detailed description of all the known mechanisms involved. In the chemical reactor engineering approach we desire to eliminate the mechanisms that is not essential for the reactor performance from the equations to reduce the computational demand. An appropriate engineering packed bed reactor model is thus tailored for its main purpose. It is as simple as possible, but still include a sufficient representation of essential mechanisms involved. 

Equations 7.S4-7.97 provide the full packed-bed reactor model given our assumptions. We next examine several packed-bed reactor problems that can be solved without solving this MI set of equations. Finally, we present Example 7.7, which requires numerical solution of the Ml set of equations. 

As an example, the mass and energy balances for a two-dimensional pseudo-homogeneous packed bed reactor model are represented by the following equations ... 

The most comprehensive packed bed reactor model is formulated accounting of ... 

Simultaneous solution of the so-called differential-algebraic equation (DAE) set requires coupling of the ODE and algebraic equation solvers, the latter which are not discussed here, but can be found in detail elsewhere [ 1 ]. Description of a DAE set and its solution in the context of a one-dimensional (ID) heterogeneous packed-bed reactor model for autothermal conversion of methane to hydrogen is available in the literature [7]. It is also worth noting that packages such as DASSL and DAEPACK are also available for the solution of coupled DAE sets. 

The same numerical methods as those used to solve the homogeneous reactor models (PFR, BR, and stirred tank reactor) as well as the heterogeneous catalytic packed bed reactor models are used for gas-Uquid reactor problems. For the solution of a countercurrent column reactor, an iterative procedure must be applied in case the initial value solvers are used (Adams-Moulton, BD, explicit, or semi-implicit Runge-Kutta). A better alternative is to solve the problem as a true boundary value problem and to take advantage of a suitable method such as orthogonal collocation. If it is impossible to obtain an analytical solution for the liquid film diffusion Equation 7.52, it can be solved numerically as a boundary value problem. This increases the numerical complexity considerably. For coupled reactions, it is known that no analytical solutions exist for Equation 7.52 and, therefore, the bulk-phase mass balances and Equation 7.52 must be solved numerically. 

In the present chapter, only the IMR shown in Fig. 12.1 is modelled. The SMR modelling is simply composed of the series of a packed bed reactor model (equations described in Section 12.2.2) followed by a membrane separation unit, where only the hydrogen stream permeated through the selective membrane (Equation [12.8]) has to be evaluated. 

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