Numerical Schemes and Algorithms

Significant improvements of the numerical methods have been obtained during the last decade, but the present algorithms are still far from being sufficiently robust and efficient. Further work on the numerical solution methods in the framework of FVMs should proceed along the paths sketched in the sequel. 

Several schemes and algorithms for solving the fluid dynamic part of the model have been published. This work has been concentrated on several items. Most important, one avoids using the very diffusive first order upwind schemes discretizing the convective terms in the multi-fluid transport equations. Instead higher order schemes that are more accurate have been implemented into the codes [62, 139, 140, 65, 105, 66[. The numerical truncation errors induced by the discretization scheme employed for the convective terms may severely alter the numerical solution and this can destroy the physics reflected 

The capabilities of alternative methods like FEMs (e.g., [19]) and fully spectral methods (e.g., [32]) are still rarely explored. 

The accuracy of the experimental data used for model validation was also questioned. 

In summary, one of the weakest links in modeling reactive systems operated in bubble columns is the fluid dynamic part considering multi-phase turbulence modeling, interfacial closures, and especially the impact and descriptions of bubble size and shape distributions. For reactive systems the estimates of the contact areas and thus the interfacial mass transfer rates are likely to contain large uncertainties. 

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