MATLAB program

The accuracy of the integration was evaluated by comparing the values computed from the MATLAB program to the tables available ([14], p. 3) at several selected values of X (where X represents the number of standard deviations at which the truncated SD was evaluated from) as a function of the integration interval. The results are presented in Table 53-5. 

FIGURE 58-10 This image shows the volume of interest (VOI) drawn on MRI (left) matches the VOI on the PET scan (right) with C-rclopride using MatLab program. 

Mathematical software, for process control, 20 694-695. See also Software MATLAB program, 20 695 MatML extensible mark-up language, 75 759... 

In the Matlab programs, every single absolute difference between known and computed total component concentrations defines the termination criterion. In Excel, the Solver termination criterion is the sum of all the absolute... 

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. 

The Matlab program Main NG2. m has implemented the additions for a termination criterion and numerical derivatives. Refer to the Matlab Help Desk for information on the while end loop and also the break command. 

Depending on the change of the sum of squares the Marquardt parameter is adjusted the parameter is reduced upon convergence and increased otherwise. There are no general rules on how exactly this should be done in detail it depends on the specific case. Usually convergence occurs with no Marquardt parameter at all in the Matlab program Ma in Chrom. m, it is thus initialised as zero. 

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. 

Here is the appropriate Matlab program Ma in FSW EFA. m. The data, generated by Data eqAH4a. m, are mimicking a spectrophotometric titration of a tetra-protic acid AH4. with log(K) values of 8, 7, 6 and 2. 

The calculation process has been implemented by using a Matlab program throughout the following steps. 

Dut04b] M. Dutour, TorusDraw, a Matlab program for toroidal maps, www.liga.ens. fr/ dutour/TorusDraw, 2004. 

Our MATLAB programs generally carry many comments after a % sign so that the reader can better understand what we do when and why. This will help our users to adapt our printed programs for other purposes. 

The last part of this section lists a few standard MATLAB operations, functions, and commands, collected into groups, together with short descriptions. This may help our readers to more easily find and use built-in MATLAB functions in their own MATLAB program codes. Please note that our MATLAB function descriptions below are very few and very short by necessity. The user should use the help. . . command to find the full length MATLAB reference guide entry for each MATLAB function when the need arises. This will help our readers use the full power and functionality of MATLAB commands and will enable them to browse for and find related built-in MATLAB functions. 

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. 

Choose a set of physically feasible parameters for the system and write a MATLAB program to find the steady state. 

When the equilibrium constants for the reactions (A) and (B) are expressed in terms of the partial pressure of the various species (in atm), the equilibrium constants for these reactions have the values KpA = 0.046 and KpB = 0.034. Determine the number of independent reactions, and then determine the equilibrium composition of the mixture, making use of a simple MATLAB program that you develop for this purpose. 

The reaction is exothermic. For (3 = 1.2 the exothermicity factor, and 7 the dimensionless activities factor, find the range of a, the dimensionless preexponential factor, that gives multiplicity of steady states using a suitable MATLAB program with Kc = 0, the dimensionless heat transfer coefficient of the cooling jacket. Choose a value of a in the multiplicity region and obtain the multiple steady-state dimensionless temperatures and concentrations. 

Develop a MATLAB program that gives you the values of a and Kc that result in multiple steady states. 

If it is unstable, develop and use a MATLAB program for a nonadiabatic CSTR and find the cooling jacket parameters Kc and yc that will stabilize the unstable steady state. 

Here is our MATLAB program, specialized for acetone cracking, but easily adaptable for other chemico-physical processes. 

Here is the MATLAB program neurocycle. m, which upon the user s specification and using MATLAB s (un)commenting feature, which places the symbol % at the start of comment lines, either plots all eight profiles, or one profile and one phase plot, or only one phase plot for relatively high values of the time parameter t, when the system has reached its periodic limit cycle. 

Next we display 13 single, double, or multiple plots drawn by our MATLAB program neurocycle.m of (a) the acetylcholine concentration profile in compartment (II) above the phase plot of the acetylcholine concentration in compartment (I) versus that in compartment (II), or (b) the limit cycle plot, or (c) the plot of all 8 profiles. We include interpretative comments on the solution s behavior in each case. 

Let us investigate how MATLAB handles boundary value problems such as (5.24) with the boundary conditions (5.25) and (5.26) for first- and second-order reactions. The MATLAB program linquadbvp.m has been designed for this purpose. 

In this section we have presented and solved the BVPs associated with the diffusion and reaction that take place in the pores of a porous catalyst pellet. The results were expressed graphically in terms of the effectiveness factor rj versus the Thiele modulus d> for two cases One with negligible external mass and heat transfer resistances, i.e., when Sh and Nu —> oo, and another with finite Sh and Nu values. This problem is very important in the design of fixed-bed catalytic reactors. The sample results presented here have shown that for exothermal reactions multiple steady states may occur over a range of Thiele moduli d>. Efficient numerical techniques have been presented as MATLAB programs that solve singular two-point boundary value problems. 

This model leads to a set of nonlinear equations that can be solved numerically by using the multi-dimensional Newton method of Section 1.2. It is this more realistic and more accurate nonlinear case which we concentrate on in this book. We now present an efficient MATLAB program to solve this nonlinear and more general case. 

The MATLAB program threeCSTR.m contains a few lines of numerical calculations at its top that generate the numerical solution for the IVP in equation (6.144), followed by more than two pages of plotting and graph annotation code, and it finishes with the right-hand side DE equation evaluator for the nine DEs collected in the equations (6.135) to (6.143). 

In this chapter we have presented multistage systems with special emphasis on absorption processes. We have studied multitray countercurrent absorption towers with equilibrium trays for both cases when the equilibrium relation is linear and when it is nonlinear. This study was accompanied by MATLAB codes that can solve either of the cases numerically. We have also introduced cases where the trays are not efficient enough to be treated as equilibrium stages. Using the rate of mass transfer RMT in this case, we have shown how the equilibrium case is the limit of the nonequilibrium cases when the rate of mass transfer becomes high. Both the linear and the nonlinear equilibrium relation were used to investigate the nonequi-librium case. We have developed MATLAB programs for the nonequilibrium cases as well. 

Additionally, countercurrent packed bed absorption column has been studied and a MATLAB program involving IVPs for this problem has been developed and explained. 

Specifically, verify that the data used for Figure 4.55 on p. 238 leads to chaotic behavior for the neurocycle enzyme system. Do so by changing the MATLAB integrator ode.. . and the tolerances used by them (as explained on p. 462) in the MATLAB program neurocycle.m of Chapter 4. 

We have developed a detailed two-phase model for the UNIPOtr process. This model was used to investigate the steady and dynamic characteristics of this important industrial process that creates polymers directly from gaseous components. The reader should develop MATLAB programs to solve for the steady state and the unsteady-state equations of this model. This will enable him or her to investigate... 

Create a MATLAB program to simulate steam reformers and methanators of... 

As usual in this industrial problems chapter, we do not include actual MATLAB programs for this model. The readers should, however, be well prepared to create such programs and try to verify our included graphical results. 

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. 

Figure 2.10 Matlab program for CSTR with jacket.

---