One-dimensional Pseudohomogeneous Model

A ID model can be formulated to describe both the FBR and the PBMR using the following simpHfying assumptions  

Based on these assumptions, the mass-balance equation for species i in the tube side can be written as  

The component balance equation is expressed with mass fractions of the individual species, cOt and the density of the reaction mixture, Pg. In Eq. (5.1) Ft is the total volumetric flow rate, = zjL the dimensionless reactor length. The first term on the right-hand side corresponds to sources by chemical reactions the second term quantifies the local mass fluxes, jj( ), entering through the membrane. In the FBR model the flux term is omitted. The catalyst density is denoted by p t and e is the averaged porosity of the catalyst bed. Vr is the reactor volume. Mi the molecular mass of species i, the surface area of the membrane wall. The reaction rate rj represents the jth rate of the overall Nr reactions taking place. 

To solve the mass-balance equation, the feed composition and the flow need to be defined at the reactor entrance as the initial condition of the system of ordinary differential equations given with Eq. (5.1)  

In packed-bed reactors typically temperature profiles develop that influence the reactor performance considerably. Axial temperature profiles can be calculated using the following simplified ID energy balance  

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The one-dimensional pseudohomogeneous model is the most used model to describe PB MRs, especially for laboratory-scale applications. In its simplest form, namely the plug flow steady state model, the model describes only axial profiles of radially averaged temperatures and concentrations. 

Design of a Eixed Bed Reactor According to the One-Dimensional Pseudohomogeneous Model... 

De facto Equation A9.4 is the one-dimensional, pseudohomogeneous model for fixed beds presented in Chapter 5. 

A one-dimensional pseudohomogeneous plug flow reactor model assuming isothermality was used to simulate experimental results. The continuity and kinetic expressions used were as follows ... 

The process is described by an one-dimensional, pseudohomogeneous, non-steady state dispersion model for an adiabatic fixed bed reactor. The kinetics are modelled by a re-versibll reaction system where each reaction step follows a power law with a reaction order of one in the gas and in the solid component. The temperature dependency of the reaction rate constant follows the Arrhenius law. The equilibrium constant is set to be independent of temperature. 

A broad classification of the various quasi-continuum models is presented in Table 12.1. The simplest is clearly the one-dimensional pseudohomogeneous plug-flow model (Al-a) in which the radial gradients of heat and mass within a tube are neglected. Then complicating factors can be added, one at a time, to allow for increasing reality,... 

One-dimensional pseudohomogeneous nonisothermal nonadiabatic plug flow (model A1-a)... 

Reactor models accounting for radial porosity profile were compared with models using the averaged bed-porosity value. Isothermal conditions were applied in order to rule out thermal effects on the concentration profiles. To check the need for two-dimensional models the results were compared with that obtained by using the pseudohomogeneous one-dimensional reactor model (Eqs. (5.1)-(5.4)). 

The objective of the present study is to show that the agreement between one and two dimensional pseudohomogeneous models is satisfactory in the PO regime, and that this agreement is still valid when the intrapartiele diffusional effects are considered. 

In this chapter, we first cite examples of catalyzed two-phase reactions. We then consider types of reactors from the point of view of modes of operation and general design considerations. Following introduction of general aspects of reactor models, we focus on the simplest of these for pseudohomogeneous and heterogeneous reactor models, and conclude with a brief discussion of one-dimensional and two-dimensional models. 

Assume steady-state, adiabatic operation, and use the pseudohomogeneous, one-dimensional plug-flow model. 

Wei [107] in 1982 was the first to come up with a continuous pseudohomogeneous model which allowed to simulate shape-selective effects observed during the alkylation of toluene using methanol to yield xylene isomers on a HZSM-5 catalyst. He treated diffusion and reaction of the xylene isomers inside the pores in a one-dimensional model. The isomer concentration at the pore mouth was set to zero, as a boundary condition. This allowed the model equations to be solved analytically, but it also limited the application of the results to small conversions. 

The development of mathematical models for the simulation of non-adiabatic fixed-bed catalytic reactors has received considerable attention. In previous work, we have analyzed the two-dimensional and one-dimensional versions of the models (1, 2) which, in turn, were classified as (I) pseudohomogeneous, (II) heterogeneous, but conceptually wrong, and (III) heterogeneous, written in the correct way (Table I). Model equations are in the Appendix. 

For the coefficient of the one-dimensional models 1-0 and II-O we have used the well known expression in terms of the two-dimensional pseudohomogeneous parameters 9) ... 

In the following section, a two-dimensional model will be described that is used for the computation of temperature and concentration profiles inside a PB MR for hydrogen production. For simplicity, only a pseudohomogeneous model will be described. The extension of the heterogeneous model is analogous to the one-dimensional model. 

Figure 3.8 Cross-sectional average concentration (a) and temperature (b) profiles according to a pseudohomogeneous model (full lines two-dimensional dashed lines one-dimensional). After Ref [133] with tj = 160 C and Tw = 100 C.

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