Finite difference formulation boundary conditions

Note that the boundary conditions have no effect on the finite difference formulation of interior nodes of the medium. This is not surprising since the control volume used in the development of the formulation does not involve any part of the boundary. You may recall that the boundary conditions had no effect on the differential equation of heat conduction in the medium either. 

The finite difference formulation is given above to demonstrate how difference equations ate obtained from differential equations. However, we use the energy balance approach in the following section.s to obtain the numerical formulation because it is more intuitive and can handle boundary conditions more easily. Besides, the energy balance approach does not require having the differential equation before the analysis. 

FIGURE 5-13 Finite difference formulation of specified temperature boundary conditions on both surfaces of a plane wall. 

Schematic for the finite difference formulation of the interface boundary condition for two mediums A and B that are in perfect thermal contact.

The development of finite difference formulation of boundary nodes in two- (or three-) dimensional problems is similar to the development in the one-dimensional case discussed earlier. Again, the region is partitioned between the nodes by forming volume elements around the nodes, and an energy balance is written for each boundary node. Various boundary conditions can be handled as discussed for a plane wall, except that the volume elements ill the two-dimensional case involve heat transfer in the y-direction as well as the x-direction. Insulated surfaces can still be viewed as mirrors, and the... 

Schematic for the explicit finite difference formulation of the convection condition at the left boundary of a plane wall.

The finite-difference formulation for inner nodes (2,3,4,5) is given by Eq. (4.7). A similar formulation for the boundary nodes needs to be developed. Here, the temperature of the base is specified and only the insulated tip (node 6) needs to be analyzed. Note that the finite-difference formulation of a boundary condition is related to a finite-difference volume. Accordingly, an application of the first four steps of our formulation procedure to the boundary (difference) system of length Ax/2 shown in Fig. 4.6 gives... 

As (hi + 02) - 0, Eq. (4.32) reduces to Eq. (4.24), as expected. Boundary conditions associated with Eq. (4.32) will not be elaborated here because of space considerations. Having studied the finite-difference formulation of steady, multidimensional problems, we illustrate now a numerical solution of this formulation in terms of an iteration method. 

These coupled second-order partial differential equations do not have a closed-form solution. Accordingly, the approximate numerical technique of finite differences is employed. First, however, the boundary conditions must be prescribed in order to complete the formulation of the problem. Symmetry of the laminate about several planes permits reduction of the region of consideration to a quarter of the laminate cross section in the y-z plane at any value of x as shown in Figure 4-52. There, along the stress-free upper surface. 

The exponential term which represents the effect of a point source is sometimes called the influence function or Green function of this diffusion problem. The method of sources and sinks easily produces solutions for an infinite medium or for systems of finite dimension when their boundary is kept at zero concentration. Different boundary conditions require a more elaborate formulation (Carslaw and Jaeger, 1959). 

Boundary conditions may be written in terms of a known inlet pressure p 0) = po a vanishing velocity at the closed end U(L) = 0. Using these boundary conditions, formulate a discrete form of the governing equations. Be careful with the sense of the finite differences, considering the order of each derivative, upwind differencing, and boundary-condition information. 

The analysis of polymer processing is reduced to the balance equations, mass or continuity, energy, momentum and species and to some constitutive equations such as viscosity models, thermal conductivity models, etc. Our main interest is to solve this coupled nonlinear system of equations as accurately as possible with the least amount of computational effort. In order to do this, we simplify the geometry, we apply boundary and initial conditions, we make some physical simplifications and finally we chose an appropriate constitutive equations for the problem. At the end, we will arrive at a mathematical formulation for the problem represented by a certain function, say / (x, T, p, u,...), valid for a domain V. Due to the fact that it is impossible to obtain an exact solution over the entire domain, we must introduce discretization, for example, a grid. The grid is just a domain partition, such as points for finite difference methods, or elements for finite elements. Independent of whether the domain is divided into elements or points, the solution of the problem is always reduced to a discreet solution of the problem variables at the points or nodal pointsinxxnodes. The choice of grid, i.e., type of element, number of points or nodes, directly affects the solution of the problem. 

Cook and Moore35 studied gas absorption theoretically using a finite-rate first-order chemical reaction with a large heat effect. They assumed linear boundary conditions (i.e., interfacial temperature was assumed to be a linear function of time and the interfacial concentration was assumed to be a linear function of interfacial temperature) and a linear relationship between the kinetic constant and the temperature. They formulated the differential difference equations and solved them successively. The calculations were used to analyze absorption of C02 in NaOH solutions. They concluded that, for some reaction conditions, compensating effects of temperature on rate constant and solubility would make the absorption rate independent of heat effects. 

Mathematical formulations of various boundary conditions were discussed in Section 2.3. These boundary conditions may be implemented numerically within the finite volume framework by expressing the flux at the boundary as a combination of interior values and boundary data. Usually, boundary conditions enter the discretized equations by suppression of the link to the boundary side and modification of the source terms. The appropriate coefficient of the discretized equation is set to zero and the boundary side flux (exact or approximated) is introduced through the linearized source terms, Sq and Sp. Since there are no nodes outside the solution domain, the approximations of boundary side flux are based on one-sided differences or extrapolations. Implementation of commonly encountered boundary conditions is discussed below. The technique of modifying the source terms of discretized equation can also be used to set the specific value of a variable at the given node. To set a value at... 

The system of linear equations originating from the difference equation (2.308) has to be supplemented by the difference equations for the points around the boundaries where the decisive boundary conditions are taken into account. As a simplification we will assume that the boundaries run parallel to the x- and y-directions. Curved boundaries can be replaced by a series of straight lines parallel to the x- and y-axes. However a sufficient degree of accuracy can only be reached in this case by having a very small mesh size Ax. If the boundaries are coordinate lines of a polar coordinate system (r, differential equation and its boundary conditions are formulated in polar coordinates and then the corresponding finite difference equations are derived. 

It is obvious that we obtain a stability condition that is not much different from the stability condition of the initial value equation. If At is larger than 2 y/MfK, (the cosine is smaller than —1), the solution grows exponentially and is numerically unstable. Hence, in the straightforward boundary value formulation of classical mechanics, we gain very little in terms of stability and step size compared to the solution of the initial value differential equation. The difficulty is not in the philosophical view (global or local) but in the estimate of the time derivative, which is approximated by a local finite difference expression. 

The Galerkin weighted residual method is employed to formulate the finite element discretisation. An implicit mid-interval backward difference algorithm is implemented to achieve temporal discretisation. With appropriate initial and boundary conditions the set of non-linear coupled governing differential equations can be solved. 

Numerically, it is now a common practice to calculate within the dielectric continuum formulation but employing cavities of realistic molecular shape determined by the van der Waals surface of the solute. The method is based upon finite-difference solution of the Poisson-Boltzmann equation for the electrostatic potential with the appropriate boundary conditions [214, 238, 239]. An important outcome of such studies is that even in complex systems there exists a strong linear correlation between the calculated outer-sphere reorganization energy and the inverse donor-acceptor distance, as anticipated by the Marcus formulation (see Fig. 9.6). More... 

A surface heat transfer coefficient h can be defined as the quantity of heat flowing per unit time normal to the surface across unit area of the interface with unit temperature difference across the interface. When there is no resistance to heat flow across the interface, h is infinite. The heat transfer coefficient can be compared with the conductivity the conductivity relates the heat flux to the temperature gradient the surface heat transfer coefficient relates the heat flux to a temperature difference across an unknowm distance. Some theoretical work has been done on this subject [8], but since it is rarely possible to achieve in practice the boundary conditions assumed in the mathematical formulation, it is better to regard it as an empirical factor to be determined experimentally. Some typical values are given in Table 2. Cuthbert [9] has suggested that values greater than about 6000 W/m K can be regarded as infinite. The spread of values in the Table is caused by mold pressure and by different fluid velocities. Heat loss by natural convection also depends on whether the sample is vertical or horizontal. Hall et al. [10] have discussed the effect of a finite heat transfer coefficient on thermal conductivity measurement. 

Difference quotients are then used to approximate the derivatives in Equation 23.25. We note that if linear basis functions are utilized in Reference 23.25, one obtains a formulation which corresponds exactly with the standard finite difference operator. Regardless of the difference scheme or order of basis function, the approximation results in a linear system of equations of the form A = b, subject to the appropriate boundary conditions. 

The assumption about the upstream conditions is crucial. If we assume instead that the fluid enters with parallel streamlines and that the axial temperature derivative at X = 0 is zero, we obtain the temperature map shown in Figure 8.14, which is very different qualitatively and quantitatively from the one in Figure 8.11. These boundary conditions are equivalent to assuming that the temperature profile is fully developed at the entrance (i.e., that the upstream channel is infinitely long), with the subsequent distortion in the final section caused by the recirculating flow at the exit. The computed temperature at the bottom plate at x = 0 is 196 °C, which is essentially the value obtained from Equation 3.34a with the parameters used here and in Chapter 3 if the heat transfer coefficient U is set to zero. (The limit corresponds to the case Bi oo and 11 0, with the product TlBi remaining finite.) This computation emphasizes the importance of the proper problem formulation the output from a computer simulation is only as meaningful as the input. 

Some, but not all, of the above considerations were incorporated within finite difference schemes that included a range of boundary and interfacial conditions intended to cover the range of circumstances prevailing during absorption, desorption, and resorption. That scheme contained a number of adjustable parameters that enabled to match weight gain data. The major drawback of that approach is that in the presence of capillaries, a detailed solution requires the implementation of at least a two-dimensional finite difference scheme, while the present formulation is essentially one dimensional. It is worth mentioning that the above formulation required the development of several novel finite difference schemes for hitherto unencountered diffusion boundary value problems. 

A simple scheme. Now suppose that a numerical solution for the pressure field is available, for example, the finite difference solutions presented later in Chapter 7. The solution, for instance, may contain the effects of arbitrary aquifier and solid wall no-flow boundary conditions we also suppose that this pressure solution contains the effects of multiple production and injection wells. How do we pose the streamline tracing problem using T without dealing with multivalued functions The solution is obvious subtract out multivalued effects and treat the remaining single-valued formulation using standard methods. Let us assume that there exist N wells located at the coordinates (Xn,Yn), having... 

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