Example MATLAB Programs

Polymath and MATLAB program examples can be loaded from the DVD-ROM (see ihe Introduction). 

The MATLAB program Example ].m finds the friction factor from the Colebrook equation by three different methods of root finding. The program first asks the user to input the values for and e/D. It then asks for the method of solution of the Colebrook equation, name of the m-file that contains the Colebrook equation, and the initial value(s) to start the method. The program then calculates the fiiction factor by the selected method and continues asking for the method of solution until the user enters 0. 

The example below shows a short Matlab program that fits the function y =tan(x) with a polynomial of degree 3 defined by 4 linear parameters, i.e. the elements of a. 

We subtract the mean spectrum from each measured spectrum yp and as a result, the origin of the system of axes is moved into the mean. In the above example, it is into the plane of all spectral vectors. This is called meancentring. Mean-centring is numerically superior to subtraction of one particular spectrum, e.g. the first one. The Matlab program, Main MeanCenter, m, performs mean-centring on the titration data and displays the resulting curve in such a way that we see the zero us,3-component, i.e. the fact that the origin (+) lies in the (us ,i,us >2)-plane. 

This example gives a good overview of the kind of problems that chemical/biological engineers encounter daily with numerical computations. Besides, our numerical codes will introduce the reader to more advanced aspects of MATLAB programming and plotting. 

Several important types of reactions are considered in the following sections. The equations describing each of these systems are developed. The steady-state design of CSTRs with these reactions are discussed, using Matlab programs for hypothetical chemical examples and the commercial software Aspen Plus for a real chemical example. 

The function in Example 4.4 can be used to autoscale a data matrix. The function determines the size of the argument, its mean vector, and its standard deviation vector. On the last fine, a MATLAB programming trick is used to extend the mean vector and standard deviation vector into matrices having the same number of rows as the original argument prior to subtraction and division. The expression ones < r, i) creates an r x 1 column vector of ones. When used as an index in the statement mn (ones(r,1), ), it instructs MATLAB to replicate the mean vector r times to give a matrix having the dimensions r x c. 

You can try this above analysis yourself using the simulated data shown in Figure 4.16 and Figure 4.17 and the MATLAB program in Example 4.8. The contaminated spectra are stored in the variable called au in the data file called residvar.mat . The training spectra are saved in the variable called a. 

Table E4-8.1 shows the POUYMATH input used to solve the above equations. The MATLAB program is given as a living example problem on the CD-ROM.

The parameters cmt also be easily varied in the example problems by loading the POLYMATH or MATLAB programs ftom the CD onto a computer to explore and answer What if,.. questions. 

Program examples POLYMATH, MatLab can be loaded from the CD-ROM (see tbe Introduction)... 

We see that for Fq =70°F the reactor has dropped below the extinetion temperature and can no longer operate at the upper steady state. In Problem P9-16, we will see it is not always necessary for the temperature to drop below the extinetion temperature in order to fall to the lower steady state. The equations deseribing the dynamic drop from the upper steady state to the lower steady state are identieal to those given in Example 9-4 only the initial conditions and entering temperature are different. Consequently, the same POLYMATH and MATLAB programs ean be used with these modifications. (See CD-ROM)... 

These matrix operations are easily programmed with mathematical software packages. In MATLAB, for example, only two statements, M = A A and [V,S] = eig(M) , are required... 

Program examples Polymath. MATLAB can be loaded Trom the CD-ROM (see the Introduction I. 

The commercial software MATLAB makes it easy to generate root locus plots. The Control Toolbox contains programs that aid in this analysis. We illustrate in the following example the use of some simple MATLAB programs to generate root locus 1 plots. Similar programs will be used in Chapter 11 to compute frequency response results. 

The MATLAB Control Toolbox makes it quite easy to generate frequency response plots for both openloop and closedloop systems. Table 11.3 gives a MATLAB program that generates Bode plots and Nichols plots. The process example is the three-heated-tank system with the openloop transfer function... 

Table 12.1 gives a MATLAB program that generates a W plot for the Wood and Berry column. After the four transfer functions are formed for the process and the two transfer functions are formed for the controllers, they are evaluated at each frequency using the polyval command. The identity matrix is formed by using the eye siie(g)) command. Then the W function is calculated at each frequency using the wnyquist nw)-— I +det eye(size(g))+g gc) command. This calculation is a good example of how easy it is to handle complex matrix calculations in MATLAB. 

The equations to calculate the capital cost of all the equipment and the energy cost of the energy are given in the Matlab program shown in Table 4.2. The numerical example is for the 32-stage column studied in Chapter 3. 

An example of a MATLAB program [33] that was used to solve the problem described above is given in Appendix B. 

Additional changes have been made in Chapter 3 and Chapter 9. In particular. Chapter 3 has been expanded to include systems of first-order differential equations. Chapter 9 has been expanded to include the numerical method of lines. As usual, an example using the numerical method of lines is provided which extends into Appendix B, where an actual MATLAB program is given. 

Table 14.1 MATLAB Program to Calculate and Plot the Frequency Response of a Complex Process in Example 14.3...

In Section 20.6.1, MPC was applied to the Wood-Berry distillation column model. A MATLAB program for this example and constrained MPC is shown in Table E20.9. The design parameters have the base case values (Case B... 

The MATLAB program Examplel 4.m is written to solve the particular problem of this example. This program simply takes the required input data from the user and calls Newton.m to solve the set of equadons. The program allows the user to repeat the calculation and tiy new initial values and relaxation factor without changing the problem parameters. 

To solve the problem posed in this example, three more MATLAB programs are written. The main program, named Example4 i.m, does the necessary calculations and plots the results. The function Ex4 l profile.m represents the concentration profile of this problem [Eqs. (2) and (3)J. The independent variable of this function z and other variables necessary to evaluate the function are entered as parameters. The function Ex4 l j hi.m is the nonlinear function from which the value of (p is calculated [Eq. (4)]. 

In the technical literature, some tutorial papers are published, for example, Wang (2001) and Smith et al. (2003). Kumar (2006) has given a tutorial coimected with a MATLAB program. [An Excel worksheet you find in the folder Elastic-mechanical, Eluid replacement on the website (see also Section 11.2) http //booksite.elsevier.eom/9780081004043/.]... 

Files with Excel/VBA or MATLAB programs for all of the examples in the book, any concluding comments for those programs, and solutions to all of the end-of-chapter exercises are available on a DVD from the publisher with a qualifying course adoption. Additionally, solutions to all of the end-of-chapter exercises are provided. When possible, concluding comments are included along with the programs. 

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. 

Excel does not provide functions for the factor analysis of matrices. Further, Excel does not support iterative processes. Consequently, there are no Excel examples in Chapter 5, Model-Free Analyses. There are vast numbers of free add-ins available on the internet, e.g. for the Singular Value Decomposition. Alternatively, it is possible to write Visual Basic programs for the task and link them to Excel. We strongly believe that such algorithms are much better written in Matlab and decided not to include such options in our Excel collection. 

Many of the fields in the structure s now contain two entries. These are arranged as cell arrays e.g. the field s. Y contains the arrays s. Y 1 and s.Y 2, the field s. t contains the two vectors s.t l and s. t 2, etc. Naturally, more than two data sets can be arranged in this way. A new field, s. nm, contains the number of measurements nm (i.e. data sets). Recall that Matlab requires curly brackets when referring to elements of a cell array. This natural expansion of the structure requires veiy few changes in the other programs. As an example, the central fitting function nglm3. m is not affected at all. 

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