A Mathematical Software

Software Packages Relevant to Mathematical Calculations in Chemical Engineering 

CFD Solvers CFX and FLUENT FLOW-3D PHOENIX SPIKE, TINA, FINS 

Continuation and bifurcation problems Computational fluid dynamics 

Finite element analysis, free equation-based modeling, general periodic boundary conditions, evaluation of material, energy balances 

Nonlinear equations, discrete algebraic equations, continuation and stability analysis, parameter analysis, and parameter estimation 

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Subscript 1 indicates simple problems that require application of equations provided in the text. Subscript 2 indicates problems whose solutions require some more in-depth analysis and modifications of given equations. Subscript 3 indicates problems whose solutions require more comprehensive analysis and involve application of several concepts. Subscript 4 indicates problems that require the use of a mathematical software or the writing of a computer code to obtain numerical solutions. 

We solve (f) numerically for different values of t using a mathematical software. Figure E8.6.1 shows the reaction curve for various values of 2Co/ i- Note that the curve for 2Co/ i = 0 represents the solution of the irreversible reaction. Once we have the reaction operating curve, we use Eq. 2.7.8 to obtain the species curves... 

This matrix equation can be easily solved using a mathematical software package such as Matlab or Maple. 

A force field does not consist only of a mathematical eiqjression that describes the energy of a molecule with respect to the atomic coordinates. The second integral part is the parameter set itself. Two different force fields may share the same functional form, but use a completely different parameterization. On the other hand, different functional forms may lead to almost the same results, depending on the parameters. This comparison shows that force fields are empirical there is no "correct form. Because some functional forms give better results than others, most of the implementations within the various available software packages (academic and commercial) are very similar. 

Once the objective and the constraints have been set, a mathematical model of the process can be subjected to a search strategy to find the optimum. Simple calculus is adequate for some problems, or Lagrange multipliers can be used for constrained extrema. When a Rill plant simulation can be made, various alternatives can be put through the computer. Such an operation is called jlowsheeting. A chapter is devoted to this topic by Edgar and Himmelblau Optimization of Chemical Processes, McGraw-HiU, 1988) where they list a number of commercially available software packages for this purpose, one of the first of which was Flowtran. 

For some reactions, the equation for x in terms of K ma> be a higher-order polynomial. If an approximation is not valid, one approach to solving the equation is to use a graphing calculator or mathematical software to find the roots of the equation. 

When rearranged, this equation is a cubic equation (an equation in x3), which can be solved with a graphing calculator or mathematical software. However, because K is very small, we suppose that x will turn out to be so small that we can use the approximation procedure ... 

This is a cubic equation in [H3Q+]. To solve this equation, it is best to use a graphing calculator or mathematical software like that on the Web site for this text. 

Cubic equations are often very tedious to solve exactly, and so it is better to use mathematical software, a graphing calculator, or a plotter, such as the one on the Web site for this book, and to identify the locations where the graph of y(x) against x passes through y = 0 (Fig. 2). 

Figure 8 depicts our view of an ideal structure for an applications program. The boxes with the heavy borders represent those functions that are problem specific, while the light-border boxes represent those functions that can be relegated to problem-independent software. This structure is well-suited to problems that are mathematically either systems of nonlinear algebraic equations, ordinary differential equation initial or boundary value problems, or parabolic partial differential equations. In these cases the problem-independent mathematical software is usually written in the form of a subroutine that in turn calls a user-supplied subroutine to define the system of equations. Of course, the user must write the subroutine that defines his particular system of equations. However, that subroutine should be able to make calls to problem-independent software to return many of the components that are needed to assemble the governing equations. Specifically, such software could be called to return in-... 

Consider Equations (6-10) that represent the CVD reactor problem. This is a boundary value problem in which the dependent variables are velocities (u,V,W), temperature T, and mass fractions Y. The mathematical software is a stand-alone boundary value solver whose first application was to compute the structure of premixed flames.Subsequently, we have applied it to the simulation of well stirred reactors,and now chemical vapor deposition reactors. The user interface to the mathematical software requires that, given an estimate of the dependent variable vector, the user can return the residuals of the governing equations. That is, for arbitrary values of velocity, temperature, and mass fraction, by how much do the left hand sides of Equations (6-10) differ from zero ... 

Optimisation may be used, for example, to minimise the cost of reactor operation or to maximise conversion. Having set up a mathematical model of a reactor system, it is only necessary to define a cost or profit functionOptimisation and then to minimise or maximise this by variation of the operational parameters, such as temperature, feed flow rate or coolant flow rate. The extremum can then be found either manually by trial and error or by the use of a numerical optimisation algorithms. The first method is easily applied with ISIM, or with any other simulation software, if only one operational parameter is allowed to vary at any one time. If two or more parameters are to be optimised this method however becomes extremely cumbersome. 

Non-linear parameter estimation is far from a trivial task, even though it is greatly simplified by the availability of user-friendly program packages such as a) SIMUSOLV (Steiner et al., 1986), b) ESL, c) a set of BASIC programs (supplied with the book of Nash and Walker-Smith, 1987) or d) by mathematical software (MATLAB). ISIM itself does not supply these advanced features, but ISIM programs can easily be translated into other more powerful languages. 

Leis, J.R., and Kramer, M.A., "The Simultaneous Solution and Sensitivity Analysis of Systems Described by Ordinary Differential Equations", ACM transactions on Mathematical Software, 14,45-60 (1988). 

A widely used mathematical tool for PCA is SVD which is a standard method implemented in many mathematical software packages (see also Appendix A.2.7). According to SVD, any matrix X (size nxm) can be decomposed into a product of three matrices (Figure 3.11). 

Modern mathematical software, such as Mathematica, allows us to compute symbolically the mean square deviation of this approximation from the exact acceleration, integrated over the feasible region, differentiate the resulting expression symbolically with respect to the parameters a and b, set the results to zero and solve the equations symbolically, and simplify the whole lot to find the following remarkably simple expressions... 

A mathematical analysis of equilibrium behavior in a polydisperse system leads to the conclusion that from the slope at any point on curve C, we can obtain a weight average molecular weight at that local concentration of solute i. Several software programs are available for carrying out the necessary calculations [5]. 

A fire model is a physical or mathematical representation of burning or other processes associated with fires. Mathematical models range from relatively simple formula that can be solved analytically to extensive hybrid sets of differential and algebraic equations that must be solved numerically on a computer. Software to accomplish this is referred to as a computer fire model. 

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