Dispersion reactor models homogeneous

The conventional two-dimensional pseudo-homogeneous reactor model consists of the continuity equation (11.1) and the simplified momentum equation (11.3) defined in connection with the pseudo-homogeneous dispersion model. The species mass and temperature equations are extended to 2D by adding postulated diffusion terms in the radial space dimension [3]. 

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). 

The units of rv are moles converted/(volume-time), and rv is identical with the rates employed in homogeneous reactor design. Consequently, the design equations developed earlier for homogeneous reactors can be employed in these terms to obtain estimates of fixed bed reactor performance. Two-dimensional, pseudo homogeneous models can also be developed to allow for radial dispersion of mass and energy. 

Equations 12.7.28 and 12.7.29 provide a two-dimensional pseudo homogeneous model of a fixed bed reactor. The one-dimensional model is obtained by omitting the radial dispersion terms in the mass balance equation and replacing the radial heat transfer term by one that accounts for thermal losses through the tube wall. Thus the material balance becomes... 

Equations 12.7.48 and 12.7.39 provide the simplest one-dimensional mathematical model of tubular fixed bed reactor behavior. They neglect longitudinal dispersion of both matter and energy and, in essence, are completely equivalent to the plug flow model for homogeneous reactors that was examined in some detail in Chapters 8 to 10. Various simplifications in these equations will occur for different constraints on the energy transfer to or from the reactor. Normally, equations 12.7.48 and 12.7.39... 

The physical situation in a fluidized bed reactor is obviously too complicated to be modeled by an ideal plug flow reactor or an ideal stirred tank reactor although, under certain conditions, either of these ideal models may provide a fair representation of the behavior of a fluidized bed reactor. In other cases, the behavior of the system can be characterized as plug flow modified by longitudinal dispersion, and the unidimensional pseudo homogeneous model (Section 12.7.2.1) can be employed to describe the fluidized bed reactor. As an alternative, a cascade of CSTR s (Section 11.1.3.2) may be used to model the fluidized bed reactor. Unfortunately, none of these models provides an adequate representation of reaction behavior in fluidized beds, particularly when there is appreciable bubble formation within the bed. This situation arises mainly because a knowledge of the residence time distribution of the gas in the bed is insuf-... 

We will consider a dispersed plug-flow reactor in which a homogeneous irreversible first order reaction takes place, the rate equation being 2ft = k, C. The reaction is assumed to be confined to the reaction vessel itself, i.e. it does not occur in the feed and outlet pipes. The temperature, pressure and density of the reaction mixture will be considered uniform throughout. We will also assume that the flow is steady and that sufficient time has elapsed for conditions in the reactor to have reached a steady state. This means that in the general equation for the dispersed plug-flow model (equation 2.13) there is no change in concentration with time i.e. dC/dt = 0. The equation then becomes an ordinary rather than a partial differential equation and, for a reaction of the first order ... 

Fixed-bed reactors may exhibit axial dispersion. If axial dispersion is important for reactor simulation, analysis, or design, a variant of the one-dimensional homogeneous model that contains an axial dispersion term may be used. Approximate criteria to determine if mass and heat axial dispersion have to be considered are available (see, e.g., Froment and Bischoff, Chemical Reactor Analysis and Design, Wiley, 1990). 

In this form, the two-mode model is identical to the classical steady-state two-phase model of a tubular catalytic reactor with negligible axial dispersion. There is also a striking structural similarity between the two-mode models for homogeneous reactions and two-phase models for catalytic reactions in the practical limit of Per 1. This could be seen more clearly when Eqs. (137) and (138) are rewritten as... 

In this model, the process of fuel pyrolysis is assumed to be achieved instantaneously once the fuel is introduced into the bed and then dispersed throughout the reactor due to the fluidization effect. The subdivision of the reactor into many compartments was adopted (apart from other reasons) to allow this homogeneous distribution of the fuel from the bottom to the top of the gasifier to be taken into account. Pyrolysis yields are assumed from [5]. 

A number of fine reviews have appeared recently which address in part the problems mentioned and the models employed. Rietema (R12) discusses segregation in liquid-liquid dispersion and its effect on chemical reactions. Resnick and Gal-Or (RIO) considered mass transfer and reactions in gas-liquid dispersions. Shah t al. (S16) reviewed droplet mixing phenomena as they applied to growth processes in two liquid-phase fermentations. Patterson (P5) presents a review of simulating turbulent field mixers and reactors in which homogeneous reactions are occurring. In Sections VI, D-F the use of these models to predict conversion and selectivity for reactions which occur in dispersions is discussed. 

To simulate the effects of reaction kinetics, mass transfer, and flow pattern on homogeneously catalyzed gas-liquid reactions, a bubble column model is described [29, 30], Numerical solutions for the description of mass transfer accompanied by single or parallel reversible chemical reactions are known [31]. Engineering aspects of dispersion, mass transfer, and chemical reaction in multiphase contactors [32], and detailed analyses of the reaction kinetics of some new homogeneously catalyzed reactions have been recently presented, for instance, for polybutadiene functionalization by hydroformylation in the liquid phase [33], car-bonylation of 1,4-butanediol diacetate [34] and hydrogenation of cw-1,4-polybutadiene and acrylonitrile-butadiene copolymers, respectively [10], which can be used to develop design equations for different reactors. 

Several models have been suggested to simulate the behavior inside a reactor [53, 71, 72]. Accordingly, homogeneous flow models, which are the subject of this chapter, may be classified into (1) velocity profile model, for a reactor whose velocity profile is rather simple and describable by some mathematical expression, (2) dispersion model, which draws analogy between mixing and diffusion processes, and (3) compartmental model, which consists of a series of perfectly-mixed reactors, plug-flow reactors, dead water elements as well as recycle streams, by pass and cross flow etc., in order to describe a non-ideal flow reactor. 

Vanden Bussche and Proment [13] simulated an adiabatic Bench Scale Reactor using a pseudo-homogeneous one-dimensional model. Using a similar pseudo-homogeneous axial dispersion model Jakobsen et al [5] obtained axial concentration and temperature profiles that were hardly distinguishable from the pseudo-homogeneous one-dimensional model results of Vanden Bussche and Proment [13]. 

The use of a full 2D pseudo-homogeneous axi-S3Tnmetric model has the advantage, compared to the conventional 2D pseudo-homogeneous dispersion model, that it enables an evaluation of the influence of a non-uniform void distribution in the reactor. 

---