Computations and MATLAB

The history of human mathematical computations goes back for several millenia. The need for numerical computations has increased since the age of enlightenment and the industrial revolution three centuries ago. For the last 50 years, the human race has become more and more dependent on numerical computations and digital computers. Computational techniques have developed from early hand computations, through table look up, mechanical adding and multiplying devices, the slide rule etc, to programmable electronic computers, mainframes, PCs, laptops, and notebooks. 

One ideally suited software for engineering and numerical computations is MATL AET-7 1. This acronym stands for Matrix Laboratory . Rs operating units and principle are vectors and matrices. By their very nature, matrices express linear maps. And in all modern and practical numerical computations, the methods and algorithms generally rely on some form of linear approximation for nonlinear problems, equations, and phenomena. Nowadays all numerical computations are therefore carried out in linear, or in matrix and vector form. Thus MATLAB fits our task perfectly in the modern sense. 

1 MATLAB is a registered trade mark of The MathWorks, Inc., 3 Apple Hill Drive, Natick, MA 01760 http //www.mathworks.com 

MATLAB comes as one main body of built-in functions and codes, and there are many additional specialized MATLAB toolboxes for various applications. As this book is primarily directed towards undergraduate and beginning graduate students, we have restricted ourselves deliberately to using the main body of MATLAB only in our codes and none of its many toolboxes. 

---

The holistic thermodynamic approach based on material (charge, concentration and electron) balances is a firm and valuable tool for a choice of the best a priori conditions of chemical analyses performed in electrolytic systems. Such an approach has been already presented in a series of papers issued in recent years, see [1-4] and references cited therein. In this communication, the approach will be exemplified with electrolytic systems, with special emphasis put on the complex systems where all particular types (acid-base, redox, complexation and precipitation) of chemical equilibria occur in parallel and/or sequentially. All attainable physicochemical knowledge can be involved in calculations and none simplifying assumptions are needed. All analytical prescriptions can be followed. The approach enables all possible (from thermodynamic viewpoint) reactions to be included and all effects resulting from activation barrier(s) and incomplete set of equilibrium data presumed can be tested. The problems involved are presented on some examples of analytical systems considered lately, concerning potentiometric titrations in complex titrand + titrant systems. All calculations were done with use of iterative computer programs MATLAB and DELPHI. 

We have given up the pretense that we can cover controller design and still have time to do all the plots manually. We rely on MATLAB to construct the plots. For example, we take a unique approach to root locus plots. We do not ignore it like some texts do, but we also do not go into the hand sketching details. The same can be said with frequency response analysis. On the whole, we use root locus and Bode plots as computational and pedagogical tools in ways that can help to understand the choice of different controller designs. Exercises that may help such thinking are in the MATLAB tutorials and homework problems. 

Equation (27) expresses an error in the dynamic matrix element Lij obtained from full matrix analysis if the error in peak volumes is Aa [50]. It also assumes that volume errors are equal for all peaks and are uncorrelated Aa is volume error normalized to the volume of a single spin at Tm = 0. Modem computer programs (Matlab, Mathematica, Mapple) can calculate the dynamic matrix from eq. (11) directly. 

In order to solve chemical/biological problems of differing levels we rely throughout on well tested numerical procedures for which we include MATLAB codes and test files. Moreover, a large part of this book is dedicated to explain the workings of our algorithms on an intuitive level and thereby we give a valuable introduction to the world of scientific computation and numerical analysis. 

The MATLAB function multiconversionDa.m is designed to compute and display... 

Choose a reaction from the literature that is similar to the reaction of this reactor, such as catalytic hydrogenation, and perform the above computations using MATLAB to obtain the concentrations, temperatures, and effectiveness factor profiles along the length of the reactor. 

The furnace equation (7.136) that describes the heat transfer in the side fired furnace is a single transcendental equation used to compute and iterate Tt o and Qr. It can be solved using fzero of MATLAB. 

Matlab computer codes for the helix-coil model are given below. First, we introduce a function that computes and returns the vector [1, v, v, u2] M, given the inputs v, w, and k ... 

W. Gander and J. Hrebfcek, Solving Problems in Scientific Computing Using Maple and MATLAB. Springer, New York, 2004. 

MATLAB High-Performance Numeric Computation and Visualization Software, User s Guide, The MathWorks, Natick, MA, 1992. 

---