Convective diffusion problem

Petera, J., Nassehi, V. and Pittman, J.F.T., 1989. Petrov-Galerkiii methods on isoparametric bilinear and biquadratic elements tested for a scalar convection-diffusion problem. Ini.. J. Numer. Meth. Heat Fluid Flow 3, 205-222,... 

Morton, K. W., 1996. Numerical Solution of Convection Diffusion Problems, Chapman Hall, London. 

Vabishchevich, P. (1994) Monotone difference schemes for convection-diffusion problems. Differential Equations, 30, 503-531 (in Russian). 

Two examples will now be given of solution of the convective diffusion problem, transport to a rotating disk as a stationary case and transport to a growing sphere as a transient case. Finally, an engineering approach will be mentioned in which the solution is expressed as a function of dimensionless quantities characterizing the properties of the system. 

The first step in solving convective diffusion problems is the derivation of the velocity profile. In this case, the flow arriving with velocity v is modified by... 

Figure 4. Schematic representation of the convective-diffusion problem for an active plane parallel to the direction of flow dealt with in Section 4.1. The liquid flow extends up to a — oo, where its free velocity is v in the direction of increasing y. The leading edge of the plane is the segment x — 0, y — 0, 0 < z

One dimensional convection-diffusion problem. One problem illustrating issues that arise with combinations of conduction and convection is the one-dimensional problem in Fig. 8.15. Here, we have a heat transfer convection-diffusion problem, where the conduction which results from the temperature gradient and the flow velocity are both in the rr-direction. 

Figure 8.16 Temperature profile for the ID convection-diffusion problem.

J. Li and C.S. Chen. Some observations on unsymmetric RBF collocation methods for convection-diffusion problems. Inter. Journal for Numerical Methods in Eng., 57 1085-1094, 2003. 

P 42] A further extension of the above-mentioned approach [P 41] for design optimization is described in [38], Here, the convection-diffusion problem was also reduced to a pure diffusion problem. Further details can be found in [38],... 

Solve a convection-diffusion problem in the fluid phase surrounding (granules) or penetrating (porous media) the object of interest. 

To obtain numerically the mass transfer coefficient, a porous medium is stochastically constructed in the form of a sphere pack. Specifically, the representation of the biphasic domains under consideration is achieved by the random deposition of spheres of radius Rina box of length L. The structure is digitized and the phase function (equal to zero for solid and unity for the pore space) is determined in order to obtain the porosity and to solve numerically the convection- diffusion problem. The next for this purpose is to obtain the detailed flow field in the porous domain through the solution of the Stokes equations ... 

Fig. 18. Convective diffusion problem geometry and boimdary conditions. (Figure and caption reprinted from Jordan and Tobias [57] by permission of the publisher. The Electrochmical Society, Inc.).

The present section reviews the concepts behind the Generalized Integral Transform Technique (GITT) [35-40] as an example of a hybrid method in convective heat transfer applications. The GITT adds to tiie available simulation tools, either as a companion in co-validation tasks, or as an alternative approach for analytically oriented users. We first illustrate the application of this method in the full transformation of a typical convection-diffusion problem, until an ordinary differential system is obtained for the transformed potentials. Then, the more recently introduced strategy of... 

As an illustration of the formal integral transform procedure, a transient convection-diffusion problem of n coupled potentials (velocity, temperature or concentration) is considered. These parameters are defined in tiie region V with boundary surface S and including non-linear effects in the convective and source terms as follows ... 

The LSMs are thus generally best suited solving convection dominated problems. However, in reactor engineering a few successful attempts have been made applying this method to solve convection-diffusion problems, by reformulating the second order derivatives into a set of two first order derivative equations. 

Step 2 For this problem, the flow and convection-diffusion problem must be solved together. Choose Solve/Solver manager. Choose the Solve For tab and select both the Navier-Stokes equation and the convective diffusion equation. Use the re-start button to solve. Do this and you will find that the flow field does not change very much, and neither does the concentration profile, as shown in Figure 11.9. While the concentration profile does not change much, now you have quantified the effect. 

The high Schmidt numbers (i.e., the ratio of kinematic viscosity v to the relevant diffusion coefficient D, that arise in electrochemical systems) suggest that many convection-diffusion problems may be characterized by large Peclet numbers,... 

For a convection-diffusion problem, where the electrical potential is not relevant to the prediction of current distribution, a semi-infinite domain poses no conceptual problems however, the treatment of an electrical potential in a two-dimensional, semi-infinite domain is problematic. When comparing simulation with experiment, the potential drop between the outer edge of the computational domain and the actual position of the reference or counter-electrode must be estimated. 

In general in complex media, the convection and diffusion of a solute is a difficult problem to analyze at the microscale and the moment method enables the analysis to be carried out at the macroscale leading to the replacement of the convective-diffusion problem by an effective global velocity and an effective dispersion tensor as in Eq. (4.6.30). Brenner s procedure analyzes the time evolution of the spatial moments (cf. Eq. 4.6.36) of the conditional probability density that a Brownian particle is located at a given position at a specific time knowing the position from which it was initially released into the fluid. 

The ADI method was reported by Peace-man and Rachford [24] in 1955 and introduced into electrochemical applications by Heinze [25-27]. A fully implicit procedure is used to solve multidimensional diffusion or convective-diffusion problems. 

The ADI method has also been apphed to convective-diffusion problems to assess the influence of convective transport on the voltammetric response of microdisc electrodes [38] in channel cells. The convective component may be routinely introduced to Eq. (28) in an analogous manner to that noted for the EFD Eq. (18). The three-dimensional transport problem was addressed in an analogous manner to the two-dimensional problem by splitting each full time increment into three time-steps of one third, to allow the x, y, and z directions to be solved implicitly. 

The fully implicit BI method was used by Anderson [45, 46] to simulate the steady state current response to flow rate within a channel flow cell. Subsequently, this was implemented by Fisher and Compton [47] to evaluate the time-dependent convective-diffusion problem... 

This approach permits the efficient iterative solution of the matrix equations using a standard NAG routine (NAG FORTRAN Library (D03EBF)). The approach has been comparedto the ADI and HS methods with the authors concluding that the SIP provides a highly efficient competitor to these strategies in both diffusion and convective-diffusion problems [75]. 

In electrochemistry, a wide variety of complex electrode geometries and flow patterns can be used, and few are amenable to quantitative treatment. Hence the normal approach to the convective diffusion problem is to treat the cell as a uniform or averaged entity, and to seek expressions in terms of space averaged quantities which permit some insight into mass transport conditions in the cell. 

The convective-diffusion problem for the RRDE may be solved analytically for the case where the species formed at the disc is completely stable. As a result, the collection efficiency can also be calculated for specified disc and ring dimensions the analytical solution N = /(r, T2, is, however, a complex equation, and the interested reader is referred to the book by Albery Hitchman (see further reading). There is very good agreement between calculated and experimentally determined collection efficiencies, as can be seen from the examples shown in Table 4.1 (also taken from the book by Albery Hitchman). It should be emphasised that N is not a function of rotation rate since increasing cj will increase the rate of transport to the disc as well as across the gap to the ring. 

This chapter discusses finite-difference techniques for the solution of partial differential equations. Techniques are presented for pure convection problems, pure diffusion or dispersion problems, and mixed convection-diffusion problems. Each case is illustrated with common physical examples. Special techniques are introduced for one- and two-dimensional flow through porous media. The method of weighted residuals is also introduced with special emphasis given to orthogonal collocation. 

Experience has shown that low-order weighted residual approximations work well when the solution varies smoothly over the entire domain of the trial function. When there are rapid changes over a varying portion of the domain, then the method requires high-order approximations. In that case, the finite-difference approach is more effective. Convective-diffusion problems are particularly difficult for the w eighted residual method, whereas pure conduction problems can be treated very efficiently by this approach. 

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