Numerical techniques software

Chemometrics, in the most general sense, is the art of processing data with various numerical techniques in order to extract useful information. It has evolved rapidly over the past 10 years, largely driven by the widespread availability of powerful, inexpensive computers and an increasing selection of software available off-the-shelf, or from the manufacturers of analytical instruments. 

For these reasons, we include examples and problems that require numerical techniques for their solution together with suitable computer software (described below). 

Precisely how the AI techniques discussed above might improve the current, numerically-based software is still somewhat speculative. 

A large number of chemical/biological processes will be presented, modeled and efficient numerical techniques will be developed and programmed using MATLAB 2. This is a sophisticated numerical software package. MATLAB is powerful numerically through its built-in functions and it allows us to easily develop and evaluate complicated numerical codes that fulfill very specialized tasks. Our solution techniques will be developed and discussed from both the chemical/biological point of view and the numerical point of view. 

Warner [176] has given a comprehensive discussion of the principal approaches to the solution of stiff differential equations, including a hundred references among the most pertinent books, papers and application packages directed at simulating kinetic models. Emphasis has been put not only on numerical and software problems such as robustness, improving the linear equation solvers, using sparse matrix techniques, etc., but also on the availability of a chemical compiler, i.e. a powerful interface between kineticist and computer. 

FORTRAN language (FORTRAN IV, F66) could only be used on large mainframe systems that accepted programs in batch. Turnaround was slow interaction was nil. The availability of microcomputers (IBM personal computers or compatible) has made it easier for numerical techniques, simulation, and design problems to be developed. Computer software can now be used interactively on such systems. 

With these additional relationships, one observes that if the rate law is given and the concentrations can be expressed as a function of conversion, ilwn in fact tee have — as a ftmcrhn of X and this is all ihal is needed to ewiuaie ihe design ecjuatimis. One can u.se either the numerical techniques described in Chapter 2. or, as we shall see in Chapter 4, a table of integrals, and/or software programs (e.g.. Polymath). 

With todays computers and the state of the art regarding numerical techniques, it does not seem that the numerical solution of the model equations presents any serious problems. With the fast development of computer hardware and software, this problem will become almost trivial in the near future. 

One simplifying approximation can be made for all non-zero concentrations of NaOH, the pH should be basic and we can omit from the charge balance (19.21). The above 7 equations can then be reduced to 1 non-linear equation in 1 unknown, which can be solved by a numerical technique such as Newton-Raphson iteration. Suitable numerical equation solvers are now available as software for personal computers. The range of solutions to these equations for different NaOH concentrations and temperatures is illustrated in Figure 19.1. At very high NaOH concentrations we would also have to consider the doubly deprotonated species H2Si04. ... 

The present chapter provides an overview of several numerical techniques that can be used to solve model equations of ordinary and partial differential type, both of which are frequently encountered in multiphase catalytic reactor analysis and design. Brief theories of the ordinary differential equation solution methods are provided. The techniques and software involved in the numerical solution of partial differential equation sets, which allow accurate prediction of nonreactive and reactive transport phenomena in conventional and nonconventional geometries, are explained briefly. The chapter is concluded with two case studies that demonstrate the application of numerical solution techniques in modeling and simulation of hydrocar-bon-to-hydrogen conversions in catalytic packed-bed and heat-exchange integrated microchannel reactors. 

The pressure drop in a fluted mixing section can be calculated for a Newtonian fluid. The first theoretical analysis was performed by Tadmor and Klein [51]. Their final equation for the pressure drop contains five dimensionless numbers, which makes determination of the effect of certain design variables rather indirect. A non-iso-thermal and non-Newtonian analysis was performed by Lindt et al. [52]. This analysis requires numerical techniques to solve the equations. Therefore, this analysis can only be used if one develops the computer software to perform the calculations. A simpler analysis was made by the author [53], leading to closed form analytical solutions from which the effect of the most important design variables can be easily evaluated. 

The solution of the steady state problem described above was performed using the commercial software COMSOL Multiphysics v 3.5a. The numerical technique used by that software is the Finite Element Method (FEM). The shape functions, chosen for the simulation, are Lagrange quadratic shape functions. 

Voltage drop in an odd-shaped geometry can be calculated accurately only with numerical techniques. The easiest way is to use software tools (such as ANSYS Thermal Analysis System (TAS) Thermal Modeling Software) that are designed for this type of problem. 

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