Numerical models, computer programs

There are special numerical analysis techniques for solving such differential equations. New issues related to the stabiUty and convergence of a set of differential equations must be addressed. The differential equation models of unsteady-state process dynamics and a number of computer programs model such unsteady-state operations. They are of paramount importance in the design and analysis of process control systems (see Process control). 

Additionally, solutions to problems are presented in the text and the accompanying CD contains computer programs (Microsoft Excel spreadsheet and software) for solving modeling problems using numerical methods. The CD also contains colored snapshots on computational fluid mixing in a reactor. Additionally, the CD contains the appendices and conversion table software. 

Particle trajectories can be calculated by utilizing the modern CFD (computational fluid dynamics) methods. In these calculations, the flow field is determined with numerical means, and particle motion is modeled by combining a deterministic component with a stochastic component caused by the air turbulence. This technique is probably an effective means for solving particle collection in complicated cleaning systems. Computers and computational techniques are being developed at a fast pace, and one can expect that practical computer programs for solving particle collection in electrostatic precipitators will become available in the future. 

Danek and his group have independently proposed a quite similar model, which they call the dissociation modeV - For this model Olteanu and Pavel have presented a versatile numerical method and its computing program. However, they calculated only the electrical conductivity or the molar conductivity of the mixtures, and the deviation of the internal mobilities of the constituting cations from the experimental data is consequently vague. 

The next two steps after the development of a mathematical process model and before its implementation to "real life" applications, are to handle the numerical solution of the model s ode s and to estimate some unknown parameters. The computer program which handles the numerical solution of the present model has been written in a very general way. After inputing concentrations, flowrate data and reaction operating conditions, the user has the options to select from a variety of different modes of reactor operation (batch, semi-batch, single continuous, continuous train, CSTR-tube) or reactor startup conditions (seeded, unseeded, full or half-full of water or emulsion recipe and empty). Then, IMSL subroutine DCEAR handles the numerical integration of the ode s. Parameter estimation of the only two unknown parameters e and Dw has been described and is further discussed in (32). 

For each system considered, an appropriate computer program was written for solving the equations involved in the modeling presented in Section 4.1. The first-order nonlinear differential equations were solved by numerical integration using the Runge-Kutta procedure [141]. 

The reaction set was numerically modeled using the computer program CHEMK (9) written by G. Z. Whitten and J. P. Meyer and modified by A. Baldwin of SRI to run on a MINC laboratory computer. CHEMK numerically Integrates a defined set of chemical rate equations to reproduce chemical concentration as a function of time. Equilibria can be modeled by Including forward and reverse reaction steps. Forward and reverse reaction rate... 

With these five equations (Eqs. 23-42 to 23-46), two of them partial differential equations, the limits of the analytical approach and the goals of this book are clearly exceeded. However, at this point we take the occasion to look at how such equations are solved numerically. User-friendly computer programs, such as MAS AS (Modeling of Anthropogenic Substances in Aquatic Systems, Ulrich et al., 1995) or AQUASIM (Reichert, 1994), or just a general mathematical tool like MATLAB and MATHE-MATICA, can be used to solve these equations for arbitrary constant or variable parameters and boundary conditions. 

At first glance the appearing equations seem to be very complex. But the numerical solution of the equations is a process which can be done with a computer program. The analytical model offers several advantages compared to simulations. Since such a theoretical ansatz needs only a small amount of computing time, more complex systems can be studied. Moreover our models are not restricted to small lattices which are inavoidably used in computer... 

The computer program for the material balance contains several parts. First, a description ofeach item of equipment in terms of the input and output flows and the stream conditions. Quite complicated mathematical models may be required in order to relate the input and output conditions (i.e. performance) of complex units. It is necessary to specify the order in which the equipment models will be solved, simple equipment such as mixers are dealt with initially. This is followed by the actual solution of the equations. The ordering may result in each equation having only one unknown and iteration becomes unnecessary. It may be necessary to solve sets of linear equations, or if the equations are non-linear a suitable algorithm applying some form of numerical iteration is required. 

Once a finite element formulation has been implemented in conjunction with a specific element type — either 1D, 2D or 3D — the task left is to numerically implement the technique and develop the computer program to solve for the unknown primary variables — in this case temperature. Equation (9.19) is a form that becomes very familiar to the person developing finite element models. In fact, for most problems that are governed by Poisson s equation, problems solving displacement fields in stress-strain problems and flow problems such as those encountered in polymer processing, the finite element equation system takes the form presented in eqn. (9.19). This equation is always re-written in the form... 

This statistical method is used for the estimation of parameters of -> electrode reaction by fitting a theoretical relationship to experimentally obtained data. This is achieved by minimizing the sum of squared deviations of the observed values for the dependent variable from those predicted by the model. Nonlinear least-square estimations can not be performed algebraically and numerical search procedures are used [i]. The Mar-quardt algorithm is commonly applied in the calculations performed by commercially available computer programs [ii]. 

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