Approaches to Multi-scale Averaging or Dimension Reduction

Different Approaches to Multi-scale Averaging or Dimension Reduction 

As stated in the introduction, dimension reduction of the governing partial differential equations describing reactors is necessary for the purpose of design, control, and optimization of chemical processes, and is typically achieved by three different approaches, as illustrated in Fig. 5. 

The first approach is the discretization of the convection and the diffusion operators of the PDEs, which gives rise to a large (or very large) system of effective low-dimensional models. The order of these low-dimensional models depend on the minimum mesh size (or discretization interval) required to avoid spurious solutions. For example, the minimum number of mesh points (Nxyz) necessary to perform a direct numerical simulation (DNS) of convective-diffusion equation for non-reacting turbulent flow is given by (Baldyga and Bourne, 1999) 

Bottom-up Approach Averaging Approach Top-down Approach 

Da is the Damkohler number, Pe the Peclet number based on the length of the reactor and is given by 

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