Mesh points

All stage-to-stage methods that work from both ends of the column toward the middle suffer from two other disadvantages. First, the top-down and the bottom-up calculations must me somewhere in the column. Usually the mesh is made at a feed stage, and if more than one feed stage exists, a choice of mesh point must be made for each component. When the components vary widely in volatility, the same mesh point cannot be used for all components if serious numerical difficulties are to be avoided. Second, arbitrary procedures must be set up to handle nondlstrihuted components. (A nondistributed component is one whose concentration in one of the end-product streams is smaller than the smallest number carried by the computer.) In the LM and TG equations, the concentrations for these components do not natur ly take on nonzero values at the proper point as the calculations proceed through the column. 

A staggered temporal mesh can also be constructed from the normal temporal mesh in a way similar to that described for the spatial temporal mesh, as shown in Fig. 9.7. The staggered temporal mesh points are at the midpoints of the mesh intervals. Some codes integrate the momentum balance equation, (9.3), on the staggered temporal mesh while the normal temporal mesh is used to integrate the other governing equations [18], [20], [21]. 

As shown by the flow chart in Fig. 49, at first a wave number plane is divided into a mesh with 128 x 128 rectangular mesh points. Then the amplitude of the concentration fluctuation at t = 0 corresponding to each... 

A typical computation such as the ones described here used about 100 adaptively placed mesh points and required about 5 minutes on a Cray 1-S. Of course, larger reaction mechanisms take more time. Also, simpler transport models can be used to reduce computation time. Since the solution methods are iterative, the computer time for a certain simulation can be reduced by starting it from the solution of a related problem. For example, it may be efficient to determine the solution to a problem with a susceptor temperature of 900 K starting from a converged solution for a reactor with a susceptor temperature of 1000 K. In fact, it is typical to compute families of solutions by this type of continuation procedure. 

Once a solution is obtained on an initial mesh, we adapt the grid in regions where the dependent solution components exhibit high spatial activity. We determine the mesh by subequidistributing the difference in the components of the discrete solution and its gradient between adjacent mesh points (10). Upon denoting the vector of N dependent solution components by U = [Ui, U2 we seek a mesh Af such that... 

This then provides a physical derivation of the finite-difference technique and shows how the solution to the differential equations can be propagated forward in time from a knowledge of the concentration profile at a series of mesh points. Algebraic derivations of the finite-difference equations can be found in most textbooks on numerical analysis. There are a variety of finite-difference approximations ranging from the fully explicit method (illustrated above) via Crank-Nicolson and other weighted implicit forward. schemes to the fully implicit backward method, which can be u.sed to solve the equations. The methods tend to increase in stability and accuracy in the order given. The difference scheme for the cylindrical geometry appropriate for a root is... 

A convenient and systematic way to represent fj (rtj) (r is the distance between particles i and j) as a linear function of unknowns is to employ cubic splines [48], as shown in Figure 8-3. The advantage of using cubic splines is that the function is continuous not only across the mesh points, but also in the first and second derivatives. This ensures a smooth curvature across the mesh points. The distance is divided into 1-dimensional mesh points, thus, fj rij) in the Mi mesh (r < rq < r +i) is described by Eqs. (8-4), (8-5) and (8-6) [48],... 

A multipass marching solution is used in COBRA IIIC (Rowe, 1973). The inlet flow division between subchannels is fixed as a boundary condition, and an iterated solution is obtained to satisfy the other boundary solution of zero pressure differential at the channel exit. The procedure is to guess a pattern of subchannel boundary pressure differentials for all mesh points simultaneously, and from this to compute, without further iteration, the corresponding pattern of crossflows using a marching technique up the channel. The pressure differentials are updated during each pass, and the overall channel iteration is completed when the fractional change in subchannel flows is less than a preset amount. 

It is important to place the first near-wall grid node far enough away from the wall at yP to be in the fully turbulent inner region, where the log law-of-the-wall is valid. This usually means that we need y > 30-60 for the wall-adjacent cells, for the use of wall functions to be valid. If the first mesh point is unavoidably located in the viscous sublayer, then one simple approach (Fluent, 2003) is to extend the log-law region down to y — 11.225 and to apply the laminar stress-strain relationship U — y for y < 11.225. Results from near-wall meshes that are very fine using wall functions are not reliable. 

For most numerically solved models, a control-volume approach is used. This approach is based on dividing the modeling domain into a mesh. Between mesh points, there are finite elements or boxes. Using Taylor series expansions, the governing equations are cast in finite-difference form. Next, the equations for the two half-boxes on either side of a mesh point are set equal to each other hence, mass is rigorously conserved. This approach requires that all vectors be defined at half-mesh points, all scalars at full-mesh points, and all reaction rates at quarter-mesh points. The exact details of the numerical methods can be found elsewhere (for example, see ref 273) and are not the purview of this review article. The above approach is essentially the same as that used in CFD packages (e.g.. Fluent) or discussed in Appendix C of ref 139 and is related to other numerical methods applied to fuel-cell modeling. ... 

The various layers of the fuel-cell sandwich described above are linked to each other through boundary conditions, which apply at the mesh point between two regions. There are two main types of boundary conditions, those that are internal and those that are external. The internal boundary conditions occur between layers inside the modeling domain, and the external ones are the conditions at the boundary of the entire modeling domain. 

For the purposes of presentation, the velocity held is usually presented at regular intervals. This new scheme is very efficient and incorporates a vector validation procedure, making it independent of operator intervention. The time it takes to compute a vector field depends on the computer hardware and it ranges from 350 mesh points per second on a PC 150 MHz Pentium to 1400 mesh points per second on a 200-megahertz dual Pro. 

The spatial domain is divided into discrete volumes defined by a mesh. The values of the independent variable, f are given at the mesh points, or nodes, by fj. The value of the dependent variable w in the volume surrounding the node is presumed to be represented by the value at the node, wj. The volume surrounding each node extends midway to the neighboring node that is, the radial extent of the volume extends from fj-1/2 to 77+1/2, where fj+i/2 = fj + /7+1). 

Following the very brief introduction to the method of lines and differential-algebraic equations, we return to solving the boundary-layer problem for nonreacting flow in a channel (Section 7.4). From the DAE-form discretization illustrated in Fig. 7.4, there are several important things to note. The residual vector F is structured as a two-dimensional matrix (e.g., Fuj represents the residual of the momentum equation at mesh point j). This organizational structure helps with the eventual software implementation. In the Fuj residual note that there are two timelike derivatives, u and p (the prime indicates the timelike z derivative). As anticipated from the earlier discussion, all the boundary conditions are handled as constraints and one is implicit. That is, the Fpj residual does not involve p itself. 

The pressure p(z) is a function of z alone. Thus it could be carried as a single scalar dependent variable, rather than defined as a variable at each mesh point. However, analogous to the reasoning used in Section 16.6.2 for one-dimensional flames, carrying the extra variables has the important benefit of maintaining a banded Jacobian structure in the differential-equation solution. 

Discretize the system on a uniform mesh of J =21 points. Write the discrete system in the residual form at the boundaries and for the interior mesh points. 

The convective terms carry information in the direction of the flow, and by their hyperbolic-like character, they cannot know about information ahead of the flow. In general, one needs to check the direction of the velocity to determine the sense of the convective differencing. In the premixed flames here, however, the velocity is always positive (i.e., flowing away from the burner) and never changes direction. Therefore the difference formula uses mesh points j and j — 1. 

The thermal conductivity is evaluated using the average dependent variables between mesh points. That is,... 

The mass flux is a constant, which would not seem to require any differencing at all. However, there are computational benefits to defining a mass flux at every mesh point, then demanding that they are all the same, namely... 

A difference form of each steady-state governing equation is written in residual form at each mesh point. For example, take the species equation at mesh point j,... 

The burner-face temperature is an element in the dependent-variable vector and determined through the Newton iteration just as is the temperature at any other mesh point. Even though the implicit imposition of boundary conditions has relatively little benefit for the simple example just shown, it has great benefit in more complex boundary conditions that are frequently needed in chemically reacting flow problems. For example, as will be discussed later, surface chemistry can result in boundary conditions that are far too difficult to impose explicitly. 

The residual vector F has a component for every equation at every mesh point, and there is a dependent variable for every residual equation at every mesh point. The dependent variable and the residual vectors are arranged similarly as... 

The Jacobian of the system is a square matrix, but importantly, because the residuals at any mesh point depend only on variables at the next-nearest-neighbor mesh point, the Jacobian is banded in a block-tridiagonal form. Figure 16.10 illustrates the structure of the Jacobian in the form used by the linear-equation solution at a step of the Newton iteration,... 

Fig. 16.10 Illustration of the block-tridiagonal structure of the Jacobian matrix. The structure on the right would result if the mass flux were not defined as a variable at each mesh point.

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