Numerical solution method

In this chapter several numerical methods frequently employed in reactor engineering are introduced. To simulate the important phenomena determining single- and multiphase reactive flows, mathematical equations with different characteristics have to be solved. The relevant equations considered are the governing equations of single phase fluid mechanics, the multi-fluid model equations for multiphase flows, and the population balance equation. 

Other methods may be more appropriate for equations with particular mathematical characteristics or when more accurate, robust, stable and efficient solutions are required. The alternative spectral methods can be classified as sub-groups of the general approximation technique for solving differential equations named the method of weighted residuals (MWR) [51]. The relevant spectral methods are called the collocation Galerkin, Tan- and Least squares methods. These methods can also be applied to subdomains. The subdomain 

Jakobsen, Chemical Reactor Modeling, doi 10.1007/978-3-540-68622-4 12, Springer-Verlag Berlin Heidelberg 2008 

The CFD market is currently dominated by four codes CFX, FLUENT, PHOEN-ICS and STAR-CD, that are all based on the finite volume method [248]. A few commercial CFD codes based on the finite element method have entered the market 

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The relativistic Schrodinger equation is very difficult to solve because it requires that electrons be described by four component vectors, called spinnors. When this equation is used, numerical solution methods must be chosen. 

Broadly speaking, this model seeks to predict temperature and species concentrations, in both the gas and solid phases, as a function of time and axial position along the monolith length. The numerical solution method employed involves a uniform-mesh spatial discretization and subsequent time-integration for the PDE using a standard, robust software (such as LSODI found in ODEPACK), and x-integration by LSODl for the DAE system [6]. 

Eq. (122) represents a set of algebraic constraints for the vector of species concentrations expressing the fact that the fast reactions are in equilibrium. The introduction of constraints reduces the number of degrees of freedom of the problem, which now exclusively lie in the subspace of slow reactions. In such a way the fast degrees of freedom have been eliminated, and the problem is now much better suited for numerical solution methods. It has been shown that, depending on the specific problem to be solved, the use of simplified kinetic models allows one to reduce the computational time by two to three orders of magnitude [161],... 

Though this new algorithm still requires some time step refinement for computations with highly inelastic particles, it turns out that most computations can be carried out with acceptable time steps of 10 5 s or larger. An alternative numerical method that is also based on the compressibility of the dispersed particulate phase is presented by Laux (1998). In this so-called compressible disperse-phase method the shear stresses in the momentum equations are implicitly taken into account, which further enhances the stability of the code in the quasi-static state near minimum fluidization, especially when frictional shear is taken into account. In theory, the stability of the numerical solution method can be further enhanced by fully implicit discretization and simultaneous solution of all governing equations. This latter is however not expected to result in faster solution of the TFM equations since the numerical efforts per time step increase. 

When simple graphical, closed form or empirical solution methods arc not appropriate or do not provide sufficient information, the numerical time integration method can be used. This method is also known as the time history method. Most texts on structural dynamics (Biggs 1964, Clough 1993, Paz 1991) provide extensive coverage on numerical solution methods for nonlinear, SDOF systems. 

Equation (18) is a partial differential equation which, for given initial and boundary conditions, can be solved for contaminant concentration as a function of space and time. Both analytical and numerical solution methods will be discussed below. 

The little book, Differential Equations in Applied Chemistry (Fig. 6), has been unsung for years, though all through an era when few engineers truly understood calculus it showed off powerful mathematical tools, among them (in the 1936 second edition) numerical solution methods. It connected with chemist J. W. Mellor s 1902 Higher Mathematics for Students of Chemistry... 

Computational fluid dynamics involves the analysis of fluid flow and related phenomena such as heat and/or mass transfer, mixing, and chemical reaction using numerical solution methods. Usually the domain of interest is divided into a large number of control volumes (or computational cells or elements) which have a relatively small size in comparison with the macroscopic volume of the domain of interest. For each control volume a discrete representation of the relevant conservation equations is made after which an iterative solution procedure is invoked to obtain the solution of the nonlinear equations. Due to the advent of high-speed digital computers and the availability of powerful numerical algorithms the CFD approach has become feasible. CFD can be seen as a hybrid branch of mechanics and mathematics. CFD is based on the conservation laws for mass, momentum, and (thermal) energy, which can be expressed as follows ... 

The extension of this approach to higher Reynolds numbers and more complex geometries is very important and deserves further attention in the future but depends critically on the future advances in computer hardware and numerical solution methods. As far as LES or partially resolved simulations are concerned, similar limitations exist although these type of simulations can be carried to (much) higher Reynolds numbers. DNS can also be used to obtain insight in turbulence producing mechanisms as shown in the study of Lyons et al. (1989). 

Since the equations are nonlinear, a numerical solution method is required. Weisz and Hicks calculated the effectiveness factor for a first-order reaction in a spherical catalyst pellet as a function of the Thiele modulus for various values of the Prater number [P. B. Weisz and J. S. Hicks, Chem. Eng. Sci., 17 (1962) 265]. Figure 6.3.12 summarizes the results for an Arrhenius number equal to 30. Since the Arrhenius number is directly proportional to the activation energy, a higher value of y corresponds to a greater sensitivity to temperature. The most important conclusion to draw from Figure 6.3.12 is that effectiveness factors for exothermic reactions (positive values of j8) can exceed unity, depending on the characteristics of the pellet and the reaction. In the narrow range of the Thiele modulus between about 0.1 and 1, three different values of the effectiveness factor can be found (but only two represent stable steady states). The ultimate reaction rate that is achieved in the pellet... 

A recent contribution to mass-balance modeling of weathering (Bowser and Jones, 2002) utilizes a spreadsheet graphical method to interpret mass balance in watershed systems in place of the strictly numerical solution methods. Key to the approach is to solve the mass-balance equation for a 10 X 10 matrix as a function of mineral composition (specifically in terms of plagioclase feldspar and smectite compositions). Exploration by means of models that cover mineral compositional space limited the range of possible compositions and restricted the range of possible mass-balance solutions. Figure 1 (after Bowser and Jones, 2002) is an example of the spreadsheet approach applied to the Sierra Nevada ephemeral... 

C How do numerical solution methods differ from analytical ones What ace the advantages and disadvantages of numerical and analytical melhods ... 

Before discussing the finite volume method, it is worthwhile to examine the desired properties of the numerical solution method, which are summarized below ... 

Boundedness Numerical solution methods must respect the physically consis-... 

If A and eg change with temperature, cf. section 2.1.4, a closed solution to the heat conduction equation cannot generally be found, which only leaves the possibility of using a numerical solution method. We will show how temperature dependent properties are accounted for by using the example of the plate, m = 0 in (2.274). The transfer of the solution to a cylinder or sphere (m = 1 or 2 respectively) is... 

This chapter starts with an introduction to modeling of chromatographic separation processes, including discussion of different models for the column and plant peripherals. After a short explanation of numerical solution methods, the next main part is devoted to the consistent determination of the parameters for a suitable model, especially those for the isotherms. These are key issues towards achieving accurate simulation results. Methods of different complexity and experimental effort are presented that allow a variation of the desired accuracy on the one hand and the time needed on the other hand. Appropriate models are shown to simulate experimental data within the accuracy of measurement, which permits its use for further process design (Chapter 7). Finally, it is shown how this approach can be used to successfully simulate even complex chromatographic operation modes. 

Detailed discussion of discretization techniques and numerical solution methods is beyond the scope of this book and, therefore, only some general procedures are presented. Numerical methods are discussed in detail by, for example, Finlayson (1980), Davis (1984) and Du Chateau and Zachmann (1989). Summaries of different discretization methods applied in the simulation of chromatography are given... 

Significant improvements of the numerical methods have been obtained during the last decade, but the present algorithms are still far from being sufficiently robust and efficient. Further work on the numerical solution methods in the framework of FVMs should proceed along the paths sketched in the sequel. 

The important ingredients of a numerical solution method are outlined in this section [141, 49, 201]. 

A numerical solution method is said to be stable if the method does not magnify the errors that appear during the numerical solution process. This property is relevant as a consistent discretization scheme provides no guarantee that the solution of the discretized equation system will become an accurate solution of the differential equation in the limit of small step size. The stability of low order numerical schemes applied to idealized problems can be analyzed by the von Neumann s method. However, when solving relevant, non-linear and coupled reactor model equations with complex boundary... 

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