Numerical solution procedure

The variational energy principles of classical elasticity theory are used in Section 3.3.2 to determine upper and lower bounds on lamina moduli. However, that approach generally leads to bounds that might not be sufficiently close for practical use. In Section 3.3.3, all the principles of elasticity theory are invoked to determine the lamina moduli. Because of the resulting complexity of the problem, many advanced analytical techniques and numerical solution procedures are necessary to obtain solutions. However, the assumptions made in such analyses regarding the interaction between the fibers and the matrix are not entirely realistic. An interesting approach to more realistic fiber-matrix interaction, the contiguity approach, is examined in Section 3.3.4. The widely used Halpin-Tsai equations are displayed and discussed in Section 3.3.5. 

Without a solution, formulated mathematical systems (models) are of little value. Four solution procedures are mainly followed the analytical, the numerical (e.g., finite different, finite element), the statistical, and the iterative. Numerical techniques have been standard practice in soil quality modeling. Analytical techniques are usually employed for simplified and idealized situations. Statistical techniques have academic respect, and iterative solutions are developed for specialized cases. Both the simulation and the analytic models can employ numerical solution procedures for their equations. Although the above terminology is not standard in the literature, it has been used here as a means of outlining some of the concepts of modeling. 

The most representative characteristics are given, since for example statistical formulations can be subject to statistical analytic or numerical solution procedures. 

In the gas/vapour phase the dimensionless distance tj ranges from 0 to 1, where tj — 1 corresponds to the position of the interface. In the liquid phase this parameter ranges from 0 to 1 for the mass transfer film and from 0 to Le for the heat transfer film. Hence, rj = 0 corresponds to the position of the interface and rj = I and t] = Le correspond, respectively, to the boundaries of the mass and heat transfer film. The mass and energy fluxes can now be calculated by solving the differential equations (4) and (8)-(12) subject to the boundary conditions (15). Due to the non-linearities a numerical solution procedure has been used which will be discussed subsequently. 

A numerical solution procedure is reasonably flexible in accommodating variations of problems. For example, the Graetz problem could be solved easily for velocity profiles other than the parabolic one. Also variable properties can be incorporated easily. Either of these alternatives could easily frustrate a purely analytical approach. The Graetz problem can also be worked for noncircular duct cross sections, as long as the velocity distribution can be determined as outlined in Section 4.4. 

In Section III, a number of nonconsistent IETs is first introduced together with their numerical solution procedures or, when available, with their analytical solutions. The accuracy of each envisaged theory is also shortly summarized. Then, the problem of the thermodynamic consistency preliminarily addressed in Section II is fully developped together with the SCIETs. Reference is made to very recent works. 

While the boundary layer equations that were derived in the preceding sections are much simpler than the general equations from which they were derived, they still form a complex set of simultaneous partial differential equations. Analytical solutions to this set of equations have been obtained in a few important cases. For the majority of flows, however, a numerical solution procedure must be adopted. Such solutions are readily obtained today using modest modem computing facilities. This was, however, not always so. For this reason, approximate solutions to the boundary layer equations have in the past been quite widely used. While such methods of solution are less important today, they are still used to some extent. One such approach will, therefore, be considered in the present text. 

Solutions to the boundary layer equations are, today, generally obtained numerically [6],[7],[8],[9],[10],[11],[12]. In order to illustrate how this can be done, a discussion of how the simple numerical solution procedure for solving laminar boundary layer problems that was outlined in Chapter 5 can be modified to apply to turbulent boundary layer flows. For turbulent boundary layer flows, the equations given earlier in the present chapter can, because the fluid properties are assumed constant, be written as ... 

A number of computer programs are discussed in this book. These are all based on relatively simple finite-difference procedures that are developed in the book. While the numerical methods used are relatively simple, it is believed that if the students gain a good understanding of these methods and are exposed to the power of even simple numerical solution procedures, they will have little difficulty in understanding and using more advanced numerical methods. Examples of the use of the computer programs are included in the text. 

Numerical method. The choice of the proper numerical solution procedure should be left to specialists. Even when a particular CFD package has been chosen, the user can usually choose different solution strategies for the linearized equations (point or line relaxation techniques versus whole field solution techniques) and associated values of the relaxation parameters. The proper choice of relaxation factors to obtain converged solutions (at all) within reasonable CPU constraints is a matter of experience where cooperation between the engineer and the specialist is required. This is especially true for new classes of fluid flow problems where previous experience is nonexistent. 

Non-Newtonian flows Reasonable More complex materials and more complex geometries -t- improved numerical solution procedures + extension to multiphase flow... 

We have remarked earlier that the treatment given above is based on an assumption for the case of that is, they are in an effective parallel combination. This is not strictly correct for a number of conditions, so the logarithmic potential-decay slopes in relation to Tafel slopes must be worked out from the full kinetic equations of Harrington and Conway (104) referred to earlier, based on the relevant mechanism of the electrode reaction. Numerical solution procedures, using computer simulation calculations, are then usually necessary for comparison with observed experimental behavior. 

Once the possibility of non-physical values of volume fraction is eliminated, solving phase continuity equations does not exhibit any other peculiarities, and the methods discussed in the previous chapter can be applied. One more point that must be mentioned while discussing the solution of phase continuity equations is of numerical or false diffusion. Numerical diffusion or false diffusion is not specific to multiphase flows and is related to any fixed-grid numerical solution procedure. However, it becomes very important in simulating multiphase flows. For example, suppose that in a field of uniform velocity, a front exists across which phase volume fraction exhibits a discontinuity. In the absence of diffusion, such a front will move within the... 

The generalized numerical solution procedure involves the following steps ... 

The linear approach described here is expandable to multienzyme electrodes as well as multilayer electrodes. At least for the stationary case, multilayer models of bienzyme electrodes may be easily treated, too. The whole system is readily adaptable to potentiometric electrodes (Carr and Bowers, 1980). It must be noted, however, that the superiority over purely numerical solution procedures decreases with increasing number of enzyme species and in the multilayer model. The advantage in calculation speed using the sum formulas described (e.g., in Section 2.5.2) amounts to about two orders of magnitude. With multilayer electrodes and formulas containing double and triple sums it is reduced to one order of magnitude. 

In complex and realistic situations, the material balance for the batch reactor must be solved numerically. However, if the reactor is isothermal, and the rate laws are assumed to be quite simple, then analytical solutions of the material balance are possible. Analytical solutions are valuable for at least two reasons. First, due to the closed form of the solution, analytical solutions provide insight that is difficult to achieve with numerical solutions. The effect of parameter values on the solution is usually more transparent, and the careful study of analytical solutions can often provide insight that is hard to extract from numerical computations. Secondly, even if one must compute a numerical solution for a problem of interest, the solution procedure should be checked for errors by comparison to known solutions. Comparing a numerical solution procedure to an analytical solution for a simplified problem provides some assurance that the numerical procedure has becm constructed correctly. Then the yerified numerical procedure can... 

Figure 4.2 Scheme of classification and numerical solution procedure of hydrogeochemical models. 

Introduction of the dimensionless coordinate y instead of r results in that the external area R numerical solution procedure. 

The finite element method, an approximate numerical solution procedure, was... 

The solution of this coupled system of molar mass balances (Equation 5.29) and the energy balance (Equation 5.147) always needs to be conducted numerically analytical solutions cannot be applied, since the energy and mass balances are coupled through concentrations in the reaction rate expressions and through the exponential temperature dependencies of the rate constants. The numerical solution procedure is discussed in Ref. [10]. 

Only for the case of model <24> with constant gas phase mole fraction (pressure can be variable), an easy to handle analytical solution is available which can successfully be used for evaluation of experimental profiles, for instance, in oxygen mass transfer measurements [25, 32]. Usually, the assumption of constant gas concentration is not at all permitted, and if gas flow variations are also considered, the balance equations are nonlinear even for first order rate processes. Therefore, only numerical solution procedures are appropriate to solve the resulting nonlinear boundary value... 

In the numerical analysis the radial and axial derivatives of the parabolic equations are replaced by the central difference approximations and the backward difference approximations, respectively. Thus N-1 sets of finite parabolic difference equations are obtained, N being the number of radial steps. The number of equidistant grid points in radial direction amounted to 40, while for the finite difference increment of ( 2z/Re) a value of 1.25 10 was used. Details of the numerical solution procedure are given elswhe-re (J 2). Figure I is a graphical representation of the development of the axial velocity obtained by the numerical solution procedure. This result agrees quite well with that obtained by Vrentas et.al. 

Details of the numerical solution procedure are given elswhere (12)... 

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