MATLAB curve

The parameters in the adsorption isotherms were estimated from the experimental equilibrium data using MATLAB Curve Fitting Toolbox. The comparison of experimental and estimated data by Langmuir, Freundlich, Redlich-Peterson and combined Langmuir-Freundlich models for the investigated systems are presented in Figures 1 to 3 for six investigated systems. 

NOTE We will show how to proceed with curve-fitting process showing both the old and new look of the MATLAB Curve Fitting Toolbox. In general. Fig. 5.2q will be reserved for the old look and Fig. 5.2b for the new look whenever there is a difference worth mentioning. 

Figure 5.15(a) The main window for the MATLAB Curve Fitting Toolbox (old look). 

Second, define the dependent (y) and independent (x) variables. Click the Data button (shown as a framed button in Fig. 5.15). Figure 5.16 shows the data window that defines thex andy variables. With the new look of MATLAB Curve Fitting Toolbox, you define the dependent variable (y), the independent variable (x), the model main category, and the method t e from the drop-down lists (shown as framed boxes in Fig. 5.15bL... 

Figure 5.17 The MATLAB Curve Fitting Toolbox is ready for the next step, i.e., the fitting step, after properly defining the x and y variables.

Figure 5.25(a) The old look of the MATLAB Curve Fitting Toolbox, where it is ready to start the process of curve fitting or nonlinear regression. 

Figure 5.25(b) The new look of the MATLAB Curve Fitting Toolbox. After successfully entering the syntax error-free model (i.e., y = f(x)), MATLAB will either start the process of curve fitting or wait for the user s command if the Auto fit option is de-selected. 

Based on the aforementioned results, the model with the estimated parameters is definitely a misfit, as also shown in Fig. 5.26. The same terrible situation will occur with the new look MATLAB Curve Fitting Toolbox (see Fig. 5.25bl. 

Figure 5.5Qq shows the 95% confidence interval which brackets (or sandwiches) the curve. From a statistics point of view, the 95% confidence interval means that out of 100 samples being measured fory, at the given x 95 of them will have a value ofy that lies within the range ofy, (i.e., between lower f(x,) and upper / (x,)) at the given x,. For the new look of the MATLAB Curve Fitting Toolbox (see Fig. 5.25bi. click on the Tools menu, followed by the Prediction Bounds submenu, and pick up, for example, a 95% confidence interval so that the 95% confidence envelope will sandwich the curve, as shown in Fig. 5.30b. 

Finally, from the File menu in the main window of the Curve Fitting Toolbox, you may choose Generate Code from the drop-down list to create the M-file, which when executed will create a plot similar to the plot in the main Curve Fitting Toolbox, using the data that you provide as input. You can use this function with the same data you used with the MATLAB Curve Fitting Toolbox or different data sets. You may want to edit the function to customize the code. 

Figure S.34 shows the default main window of the MATLAB Surface Fitting Toolbox, which is similar to that of Fig. 5.25b (a new look for the MATLAB Curve Fitting Toolbox).

It was indicated that the original method can be extended on systems where two or three analytes can be determined from a single titration curve. The shifts DpH affected by j-th PT addition should be sufficiently high it depends on pH value, a kind and concentration of the buffer chosen and its properties. The criterion of choice of the related conditions of analysis has been proposed. A computer program (written in MATLAB and DELPHI languages), that enables the pH-static titration to be done automatically, has also been prepared. 

How did we generate the solid curve We computed the result for the first order function and then shifted the curve down three time units (td = 3). The MATLAB statements are ... 

Even with MATLAB, we should still know the expected shape of the curves and its telltale features. This understanding is crucial in developing our problem solving skills. Thus doing a few simple hand constructions is very instructive. When we sketch the Bode plot, we must identify the comer (break) frequencies, slopes of the magnitude asymptotes and the contributions of phase lags at small and large frequencies. We ll pick up the details in the examples. 

With MATLAB, what you find is that the actual curves are very smooth it is quite different from hand sketching. Nevertheless, understanding the asymptotic features is important to help us check if the results are correct. This is particularly easy (and important) with the phase lag curve. 

To help understand MATLAB results, a sketch of the low and high frequency asymptotes is provided in Fig. E8.9. A key step is to identify the comer frequencies. In this case, the comer frequency of the first order lead is at 1/5 or 0.2 rad/s, while the two first order lag terms have their comer frequencies at 1/10, and 1/2 rad/s. The final curve is a superimposition of the contributions from each term in the overall transfer function. 

We ll see two curves. By default, MATLAB maps and plots also the image of the negative Im-axis. That can make the plot too busy and confusing, at least for a beginner. So we ll stay away from the default in the following exercises. 

We do not design our own algorithm here but use the fin Insearch. m function supplied by Matlab. It is based on the original Nelder, Mead simplex algorithm. As an example, we re-analyse our exponential decay data Data Decay. m (see p. 106], this time fitting both parameters, the rate constant and the amplitude. Compare the results with those from the linearisation of the exponential curve, followed by a linear least-squares fit, as performed in Linearisation of Non-Linear Problems, (p.127). 

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. 

Analysis of variance (ANOVA) analyses were performed using the general statistical package StatView 5.01 (SAS Institute, Cary, NC, USA). The ANOVAs were calculated as repeated-measures ANOVAs with wells as within factor for phase 1 and with plates as within factor for subsequent phases. Specialized statistics, such as comparison of fits of different calibration curves, were calculated in MATLAB 5.1 (MathWorks, Natick, MA, USA) using custom routines. 

Next we draw a grayscale contour map of this surface, as well as the level curve F(a,y) = 0 drawn in black using the MATLAB function adiabNisocolorcontour. m. On a computer screen the same coloring scheme as in Figure 3.6 is used. 

Our CD also contains the MATLAB function m file runNadiabNisokccurve. m. A call of runNadiabNisokccurve (180000,1,15,1,100,. 001, .7,1.2), for example, plots only the bifurcation curve with respect to Kc in Figure 3.16, i.e., it repeats the bottom plot in Figure 3.14. 

The lines that follow the initial % MATLAB comment lines in fixedbedreact.m set up default values for the seven optional parameters. Then we prepare for the MATLAB IVP solver ode.. . that solves our problem by using the function dydt to evaluate the right-hand side of our IVP (4.22). Having solved (4.22) we plot two curves of the solution to the two joint DEs. 

Next we study the effect of varying the heat transfer area A" by setting = 300, A2 = 240, As = 160, and A4 = 80 in the MATLAB call of AdprunC [300,240, 160,80], . 008). Again the curve for the first-mentioned value of A is denoted by A", and so forth. Note that the solutions are increasing more slowly as A decreases, denoting a slower reaction for smaller areas A of heat transfer. 

The respective MATLAB codes for multiple A, Tj, and Tf value curves are all quite similar to the one given in qfrun.m above. Therefore we do not print these out here, but rather refer the user to the accompanying CD. 

Figure 4.29 was obtained via the MATLAB code hetcontbifmultiK.m which we have derived from hetcontbif range. m. It features an auxiliary plot of the exponential curve to find the proper axes limits for the plot first. Note also the elaborate sequence of legend commands that we use. 

This code is typically called by a MATLAB command such as f luidbed(4000,. 4,. 9,1,1, 45/99,10 s8,10 sll,.4,.6,18,27,1,0,10), which uses a large time limit of 4000 dimensionless time units so that the individual solution curves of the IVPs have time to run to the global steady state at (0.17619, 0.72225, 0.86446). We do this with the original parameter data of p. 183, a controller gain of K = 10, and several different initial values. Since the value of K is greater than 4, the middle steady state with maximal x i> yield is both unique and (statically) stable. 

Create a MATLAB m file that draws the zero-contour curves F(y, K) = 0 of Figures 4.26 or 4.30 directly. 

Here is an auxiliary program for this task. It computes the solution curve y w) from an initial guess of the solution to the BVP and evaluates q using MATLAB s definite integral evaluator quad once the BVP has been solved successfully. 

These indicate the limit of our successful numerical BVP integrations near the bifurcation points. In between the x and o marks on the middle branch, the curve is drawn using interpolation of our successful BVP solutions data, while in between two adjacent x or two adjacent o marks, the curve is drawn by extrapolating nearby computed function data. This is done automatically by MATLAB s plot commands. 

The following calculations and curves drawing can be accomplished by mathematical software such as Mathcad2000 or Matlab. 

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