Grid-Peclet number

Numeric dispersion can be eliminated largely by a high-resolution discretisation. The Grid-Peclet number helps for the definition of the cell size. Pinder and Gray (1977) recommend the Pe to be < 2. The high resolution discretisation, however, leads to extremely long computing times. Additionally the stability of the numeric finite-differences method is influenced by the discretisation of time. The Courant number (Eq. 104) is a criterion, so that the transport of a particle is calculated within at least one time interval per cell. 

Numerical dispersion or oscillation effects can occur as accidental source of error when using finite differences and finite element methods while modeling mass transport. Utilizing the criteria of numerical stability (Grid-Peclet number or Courant number) or the random walk procedure, these errors can be either reduced or even eliminated. 

The upwind scheme described here is first-order accurate in space while the central difference scheme is second-order accurate. Hence a central-difference scheme is preferred whenever possible. Since it is the grid Peclet number that decides the behavior of the numerical schemes, it is, in principle, possible to refine the grids until the grid Peclet is smaller than 2. This strategy, however, is often limited by the required computing time. With sufficiently fine meshes, the two schemes should give essen-... 

Assuming K is directly proportional to Pn and equating Eq. [4] and [5] results in 0 = K/v. For this special case, the 0 parameter is commonly referred to as the dispersivity. In numerical transport modeling, the grid size is used as 0 in order to define a grid Peclet number, which must be kept small to minimize numerical dispersion. 

The coupled set of nonlinear differential equations (equations 1 and 4) are solved by the alternating direction implicit (ADI) method f9-10) on an evenly spaced grid. The advective transport of a solute species was solved using the Lax-Wendroff two-step method (10). To ensure that numerical dispersion is avoided, a grid spacing was chosen such that the grid Peclet number (defined by < 2 fll). The computational expense involved in using a... 

It can be seen that the central differencing scheme discussed above is conservative. The coefficients of CDS satisfy the Scarborough criterion. However, for uniform grid (Ae = 0.5), when the Peclet number is higher than 2, the coefficients oe will become negative [Fg > This violates the boundedness requirements and may... 

Figure 3.4. Contours of log(req), which is equal to log(Xeq), for a range of Damkohler and Peclet numbers. The dashed line separates a region where req and Xeq depend only on Da (reaction dominated) from a region where they depend only on Pe (transport or advection dominated). The shaded areas are discussed in the text. The arrows point to the region where the modeling grid size can be less than 100 meters.

This means that as Pe increases, the mesh size must decrease. Since the mesh size decreases, it takes more elements or grid points to solve the problem, and the problem may become too big. One way to avoid this is to introduce some numerical diffusion, which essentially lowers the Peclet number. If this extra diffusion is introduced in the flow direction only, the solution may still be acceptable. Various techniques include upstream weighting (finite difference [10]) and Petrov-Galerkin (finite element [11]). Basically, if a numerical solution shows imphysical oscillations, either the mesh must be refined, or some extra diffusion must be added. Since it is the relative convection and diffusion that matter, the Peclet number should always be calculated even if the problem is solved in dimensional units. The value of Pe will alert the chemist, chemical engineer, or bioengineer whether this difficulty would arise or not. Typically, is an average velocity, x is a diameter or height, and the exact choice must be identified for each case. 

Value may be comparable with the hydrodynamic dispersion D. and even greater than it. When D., modeling results to a greater extent reflect the influence of numerical and not hydrodynamic dispersion. If in equation (4.4) is used Peclet number (Pe) of the discretization grid equal to... 

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