Diffusion numerical scheme

Ambrosini, W., Ferreri, J.C., 2003. Prediction of stabiUty of one-dimensional natural circulation with a low diffusion numerical scheme. Annals of Nuclear Energy 30, 1505—1537. 

Long, P. E., and Pepper, D. W., A comparison of six numerical schemes for calculating the advection of atmospheric pollution, in "Proceedings of the Third Symposium on Atmospheric Turbulence, Diffusion and Air Quality." American Meteorological Societv, Boston, 1976, pp. 181-186. 

In problems of heat convection, the most complex equations to solve are the fluid flow equations. Often times, the governing equations for the fluid flow are the Navier-Stokes equations. It is useful, therefore, to study a model equation that has similar characteristics to the Navier-Stokes equations. This model equation has to be time-dependent and include both convection and diffusion terms. The viscous Burgers equation is an appropriate model equation. In the first few sections of this chapter, several important numerical schemes for the Burgers equation will be discussed. A simple physical heat convection problem is solved as a demonstration. 

Kinetic ripening Adatom emission, diffusion, capture diffusion control numerical scheme. CSD,dasf(t) computation not elaborate large local loadings possible good for redispersion. Results depend on emission/ci rture model need initial conditions. 

For a poiydisperse aerosol, the number of particles deposited up to any point in the system can be calculated from the theory for monodisperse aerosols and then integrating over the initial. size distribution, which is the quantity sought- The experimental measure ments made with the condensation nuclei counter gives the number concentration of the poiydisperse aerosol as a function of the distance from the inlet to the diffusion battery. The recovery of the size distribution function from the measured decay In particle concentration can be accomplished in an approximate way. Various numerical schemes based on plausible approximations have been developed to accomplish the inversion (Cheng, 1993). The lower detection limit for the diffusion battery is 2 to 5 nm. Systems are not difficult to build for specific applications or can be purchased commercially. 

The first class is a generalization of the advection-diffusion problem discussed in Section 8.3, and much of the material developed there can be reformulated to develop a realizable scheme for the GPBE. The second class is a generalization of the kinetic equation considered in Section 8.4, and has been referred to in the literature as the semi-kinetic model (Laurent et al, 2004 Laurent Massot, 2001). In the following, we will treat each class separately, although the reader will undoubtedly note many similarities between the numerical schemes. A third class of GPBE, lying between the two listed above, can also be identified wherein the scalar-dependent convection velocity has a parametric form such as... 

A flat plate with thickness l = 10 cm and diffusivity or = 1 x 10"4 m2/s is initially at temperature of 0 °C. One surface of the plate is kept at a temperature of 0 °C while the other is suddenly raised to 100 °C. We wish to determine the transient temperature distribution within the plate by using an explicit numerical scheme. 

In general, boundary conditions are difficult to specify and oftentimes difficult to incorporate into the numerical scheme. Typical boundary conditions used are given in [92, 97, 129, 130, 136]. Boundary conditions for the mass continuity Eq. (22) specify a zero electron density at the wall, or an electron flux equal to the local thermal flux multiplied by an electron reflection coefficient. The ion diffusion flux is set to... 

Once again, remember that these mass balance equations are nonlinear (because of the nonlinear adsorption expression) hence, they can not be solved by direct analytical means. Generally, numerical methods must be used to solve this problem. However, when the adsorption rate is much faster than the diffusion rate (i.e., ix < 1), conventional numerical methods will run into some difficulty owing to instability. This arises because of the steepness of the profiles generated. To achieve convergence with conventional numerical schemes, a large number of discretization points must be used for the spatial domain. The fine mesh is necessary to observe such a very sharp change in the profile. 

In mathematical terms, the adsorption being diffusion-limited means that the variation of the free energy with respect to 0o can be taken to zero at all times whereas the variation with respect to (j> x> 0) cannot. This has two consequences. The first is that the relation between 0o and (j)i is given at all times by the equilibrium adsorption isotherm [(3) in our model]. The solution of the adsorption problem in the non-ionic, diffusion-limited case amounts, therefore, to the simultaneous solution of the Ward-Tordai equation (8) and the adsorption isotherm. Exact analytical solution exists only for the simplest, linear isotherm, °c 0i [19]. For more realistic isotherms such as (3), one has to resort to numerical techniques (useful numerical schemes can be found in refs. [2, 8]). The second consequence of the vanishing of 5Ay/5 o is that the dynamic surface tension, Ay t), approximately obeys the equilibrium equation of state (4). These two consequences show that the validity of the schemes employed by previous theories is essentially restricted to diffusion-limited cases. 

A subtle problem with serious consequences, pointed out by Hanafey et al (1978), arises from the way we normally tackle the numerical solution of an equation such as Eq. 7.8. It has the continuous-time variable which, during the time interval 6T, varies both due to diffusion and the chemical reaction these two effects are coupled. In the simplest numerical schemes, we pretend that they are separate, compute their separate contributions during 6T and add them together. A crude scheme... 

Numerical results from the LOW Tables show how the saturation potential depends on Ka. Many analytical approximations have been developed for the spherical diffuse layer, but the advent of fast, efficient numerical schemes (HNC, hypemet-ted chain equation), as well as that of the mean spherical approximation (MSA) has diminished their utility (see further parts of this monograph). Moreover MSA and HNC approximations can be applied to the evaluation of most practical transport coefficients of electrolytes in a large variety of experimental situations, as it will be seen in other parts of this book. 

For such cases, in which separated phases have to be considered, special numerical tools were developed to keep the interface sharp, like interface capturing, interface tracking, or some approximate numerical methods like surface sharpening. In case of dispersed flows special highly accurate numerical schemes would be required to avoid or at least minimize numerical diffusion, but most codes use standard lower order schemes thus one still has to live with more or less strong numerical diffusion. 

Fig. 7. Comparison of various transport schemes for advecting a cone-shaped puff in a rotating windfield after one complete rotation (a), the exact solution (b), obtained by an accurate numerical technique (c), the effect of numerical diffusion where the peak height of the cone has been severely tmncated and (d), where the predicted concentration field is very bumpy, showing the effects of artificial dispersion. In the case of (d), spurious waves are...

FIG. 16-9 General scheme of adsorbent particles in a packed bed showing the locations of mass transfer and dispersive mechanisms. Numerals correspond to mimhered paragraphs in the text 1, pore diffusion 2, solid diffusion 3, reaction kinetics at phase boundary 4, external mass transfer 5, fluid mixing. 

Errors in advection may completely overshadow diffusion. The amplification of random errors with each succeeding step causes numerical instability (or distortion). Higher-order differencing techniques are used to avoid this instability, but they may result in sharp gradients, which may cause negative concentrations to appear in the computations. Many of the numerical instability (distortion) problems can be overcome with a second-moment scheme (9) which advects the moments of the distributions instead of the pollutants alone. Six numerical techniques were investigated (10), including the second-moment scheme three were found that limited numerical distortion the second-moment, the cubic spline, and the chapeau function. 

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