Combined convection series solution

Before turning to a discussion of other methods of solving the laminar boundary layer equations for combined convection, a series-type solution aimed at determining the effects of small forced velocities on a free convective flow will be considered. In the analysis given above to determine the effect of weak buoyancy forces on a forced flow, the similarity variables for forced convection were applied to the equations for combined convection. Here, the similarity variables that were previously used in obtaining a solution for free convection will be applied to these equations for combined convection. Therefore, the following similarity variable is introduced ... 

In cases where hydrodynamic dispersion and the corresponding broadening of residence-time distributions deteriorate the performance of a process, the question arises as to which channel design minimizes dispersion. Already from the analysis of Taylor and Aris it becomes clear that an enhanced mass transfer perpendicular to the main flow direction reduces the broadening of concentration tracers. Such a mass-transfer enhancement can be achieved by the secondary fiow occurring in a curved channel. This aspect was investigated by Daskopoulos and Lenhoff [78] for ducts of circular cross section. They assumed the diameter of the duct to be small compared to the radius of curvature and solved the convection-diffusion equation for the concentration field numerically. More specifically, a two-dimensional problem defined on the cross-sectional plane of the duct was solved based on a combination of a Fourier series expansion and an expansion in Chebyshev polynomials. The solution is of the general form... 

Then, d cjdx = 0 if the diffusion in the x direction is negligible compared to that by convection. Combining Eqs. (7.3-14) and (7.3-15), the final solution (SI) is a complex series similar to the Graetz solution for heat transfer and a parabolic velodty profile. 

Typically, an electrochemical simulation models the transport of the reactant from the bulk to the electrode surface, the transfer of electrons at the electrode/solution interface and the transport of the product away from the electrode. Depending on the complexity of the electrochemical process, the simulation may account for the rate of electron transfer kinetics, the possibility and rate of preceding or following chemical reactions, the possibility and rate of adsorption processes and even combinations of different forms of mass transport (planar or spherical diffnsion, convection and migration). In other words, the simulation performs a series of actions which mimic the sequence of events thought to occur in the electrode reaction. 

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