Combined convection numerical solution

The overall clearance of hemofiltration is more difficult to calculate than the diffusive clearance or the HF clearance, as it combines diffusive and convective transfers. An approximate equation for this clearance, obtained from an exact numerical solution has been given by Jaffrin et al. [12] as... 

The above equations can be solved using numerical methods, i.e., using the same basic procedures as used with forced convection. There is, however, one major difference between the procedures used in forced convection and in mixed convection. In forced convection, the velocity field is independent of the temperature field because fluid properties are here being assumed constant. Thus, in forced convection it is possible to first solve for the momentum and continuity equations and then, once this solution is obtained, to solve for the temperature distribution in tike flow. However, in combined convection, because of the presence of the temperature-dependent buoyancy force term in the momentum equation, all of the equations must be solved simultaneously. Studies of flows for which the boundary layer equations are not applicable are described in [24] to [43]. 

All the electrode kinetic methodology described until now has assumed a steady state (or quasi-steady state in the case of the DME). Many techniques at stationary electrodes involve perturbation of the potential or current in combination with forced convection, this offers new possibilities in the evaluation of a wider range of kinetic parameters. Additionally, we have the possibility of modulating the material flux, the technique of hydrodynamic modulation which has been applied at rotating electrodes. Unfortunately, the mathematical solution of the convective-diffusion equation is considerably more complex and usually has to be performed numerically. 

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... 

Kietzmann et al. (1998) discussed discretization schemes for the solution of Eq. 8.25. The equation is a typical convection equation and an upwinding is required for numerical stability. However, the upwind discretization may result in numerical diffusion that blurs the flow front interface. A hybrid implicit scheme that combines upwind differencing and central differencing is therefore suggested. 

The surface temperature of the sheet during the time of heating will reach (0.7 = (T - To)/(T2 - To)) 824.5 °F, which may degrade the sheet surface. In practice we would use forced convection to cool the surface of the sheet. We could then use the combined heat transfer coefficient and resolve the problem. A more accurate solution to this problem can be obtained numerically, and this is done in Problem 5C.5. ... 

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