Numerical Solution of the Two-dimensional Model

In this work, the commercial software package Comsol Multiphysics 3.2 (Corn-sol AB, Stockholm, Sweden) was used to solve the system of nonlinear partial differential equations. Apart from predefined built-in applications, this tool enables the user to define and solve PDE systems by finite-element discretization. Taking advantage of the axial symmetry of the problem, the model domain can be simplified to two dimensions. In this domain, a structured quadrilateral mesh 

In this section, steady-state two-dimensional velocity profiles in the PBMR will be analyzed, considering the relevance of boundary conditions (BC) and the effect of flow maldistribution. 

To demonstrate the above conclusion, selected results of a numerical study using the so-called one-domain approach to describe the momentum transfer at the fluid/porous interface in a membrane reactor will be presented. 

In the present study the membrane is treated as a homogeneous porous medium. Furthermore, for the sake of simplicity only single-gas permeation is considered. 

The extended Navier-Stokes equation is solved on the geometrical domain illustrated in Fig. 5.12. Since the packed bed, the membrane, and the shell side are included in the calculation, boundary conditions do not have to be formulated explicitly at the interface (Goyeau et al, 2003). To account for the friction forces the following coefficients were substituted in Eq. (5.6)  

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Differences in the Responses of the Different Types of Models. The basic differences that exist in the heat and mass balances for the different types of models determine deviations of the responses of types I and II with respect to type III. In a previous work (1) a method was developed to predict these deviations but for conditions of no increase in the radial mean temperature of the reactor (T0 >> Tw). In this work,the method is generalized for any values of T0 and Tw and for any kinetic equation. The proposed method allows the estimation of the error in the radial mean conversions of models I and II with respect to models III. Its validity is verified by comparing the predicted deviations with those calculated from the numerical solution of the two-dimensional models. A similar comparison could have been made with the numerical solution of the one-dimensional models. 

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