Hydrodynamic methods convective-diffusion equation, solution

Chapter 8 ignored axial diffusion, and this approach would predict reactor performance like a PFR so that conversions would be generally better than in a laminar flow reactor without diffusion. However, in microscale devices, axial diffusion becomes important and must be retained in the convective diffusions equations. The method of lines ceases to be a good solution technique, and the method of false transients is preferred. Application of the false-transient technique to PDFs, both convective diffusion equations and hydrodynamic equations, is an important topic of this chapter. 

Many of the mixing simulations described in the previous section deal with the modeling of mass transfer between miscible fluids [33, 70-77]. These are the simulations which require a solution of the convection-difliision equation for the concentration fields. For the most part, the transport of a dilute species with a typical diSusion coeflEcient 10 m s between two miscible fluids with equal physical properties is simulated. It has already been mentioned that due to the discretization of the convection-diffusion equation and the typically small diffusion coefficients for liquids, these simulations are prone to numerical diffiision, which may result in an over-prediction of mass transfer efficiency. Using a lattice Boltzmann method, however, Sullivan et al. [77] successfully simulated not only the diffusion of a passive tracer but also that of an active tracer, whereby two miscible fluids of different viscosities are mixed. In particular, they used a coupled hydrodynamic/mass transfer model, which enabled the effects of the tracer concentration on the local viscosity to be taken into account. 

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