Solution Using MATLAB

To solve for the equilibrium in Eq. (4.14) using MATLAB, you create an m-file that will calculate 

Step 1 Construct an m-file that evaluates the function, given x. The name is equil eq.m and it is listed below. For testing puiposes, you leave off the of the commands that involve computation. Save this function in your desired workspace. (Note that the values for the in parameters have been set arbitrarily for the test.) 

This m-file acts like any function Give it an x and it comes back with the value of y. 

Step 2 To test this function, you evaluate it for a specific value of x. Issue the following 

This is the correct answer as determined by manual calculation. 

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A more recent book that stresses numerical solutions using Matlab is... 

It is clear that (11.6), (11.7), (11.10), and (11.13) form a set of state space equations in first-order form written in the Cauchy form. These set of equations derived by a conventional method such as applying Newton s equations can be solved also using conventional solutions using MATLAB and its tools tailored to first-order differential equations. These equations can also be arranged in matrix form. That is presented next as we compare the two methods. 

Plot the unit step response using just the first and second order Pade approximation in Eqs. (3.30) and (3-31). Try also the step response of a first order function with dead time as in Example 3.2. Note that while the approximation to the exponential function itself is not that good, the approximation to the entire transfer function is not as bad, as long as td x. How do you plot the exact solution in MATLAB ... 

Note 2 The iterative solution in solving the ultimate frequency is tricky. The equation has poor numerical properties—arising from the fact that tan9 "jumps" from infinity at 9 = (ir/2) to negative infinity at 9 = (ir/2)+. To better see why, use MATLAB to make a plot of the function (LHS of the equation) with 9 < co < 1. With MATLAB, we can solve the equation with the f zero () function. Create an M-file named f. m, and enter these two statements in it ... 

Solution. The coefficients in the two models were fitted using MATLAB, yielding the following results ... 

Solution. Using the Optimization Toolbox from MATLAB, the BFGS method requires 20 iterations before the search is terminated, as shown below. 

A large number of chemical/biological processes will be presented, modeled and efficient numerical techniques will be developed and programmed using MATLAB 2. This is a sophisticated numerical software package. MATLAB is powerful numerically through its built-in functions and it allows us to easily develop and evaluate complicated numerical codes that fulfill very specialized tasks. Our solution techniques will be developed and discussed from both the chemical/biological point of view and the numerical point of view. 

Then uses MATLAB s fzero to find the solution xsol. 

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. 

The computer program for solving the nonlinear programming problem (9) was developed using Matlab, applying the simplex method for searching the optimal solution. 

The simultaneous numerical solution of the ODE system was performed using Matlab (The MathWorks). The kinetic and model parameters are summarized in Table 1,... 

FEMLAB is treated in detail in Chapters 9-11 and Appendix D, but it can also be used to solve reactor problems. The advantage of FEMLAB is that you program with a GUI, so computer errors are less likely. It is still necessary to check your work, though. While the applications in this chapter are all one-dimensional (to compare with MATLAB solutions), it is easy to solve two-dimensional problems, as described in more detail in later chapters. We show here how to solve the same three problems already solved using MATLAB the simple exponential, Eq. (8.16) the isothermal flow reactor, Eqs. (8.21)—(8.22) and the nonisothermal reactor, Eqs. (8.24)-(8.26). 

Complications arose when the moles or total concentration changed, or when mass transfer effects were important, and you learned how to incorporate those complications. Finally, CSTRs were studied using both MATLAB and Excel in cases where the solution is steady, where multiple steady solutions exist, and you learned to use MATLAB when the solution is time-dependent and the problem is stiff, leading to limit cycles. 

Starting with a 10 cm X 10 cm sheet of paper, what is the largest volume you can create by cutting out cm X x cm from each corner of the sheet and then foldii up the sides See Figure 15.11. Use MATLAB to obtain the solution. 

In the previous sections, we discussed how to use MATLAB to solve eng eering problems with numerical values. In this section, we briefly explain the symbolic capabilities of MATLAB. In symbolic mathematics, as the name implies, the problem and the solution are presented using symbols, such as x instead of numerical values. We will demonstrate MATLAB s symbolic capabilities using Examples 15.7 and 15.8. 

In this section, we will show how you can use MATLAB s symbolic solvers to obtain solutions to a set of linear equations. 

Manually program the solution of the equations such as using MATLAB, ACSL, EASY5, SYSQUAKE or create your own program. 

For postprocessing the model solution in MATLAB, postinterp can be used to deliver COMSOL variables back to the MATLAB workspace where they can be plotted and manipulated using typical MATLAB functions. The postinterp function has the following syntax [matlabvarl,matlabvar2,...] = postinterp (fern, comsolvarlcomsolvar2grid, options), where com-solvar is the variable name to be used by COMSOL, and matlabvar is the variable name as defined in MATLAB. 

Figure 2.21 MATLAB M-file tangent. m for Solutions Using...

Figure 7.2 Plot of y vs. t as a solution for dy/dt =y using MATLAB ode45 solver with At = 0.0125.

Find the anal54ical solution using either the technique(s) for solving second-order, nonhomogeneous ODE, or MATLAB built-in dsolve. Calculate PRE at f = 1. 

Compare the result at versus that of the analytical solution that can be obtained by either solving a first-order ODE, or using MATLAB built-in dsolve. What is the steady-state value of Q (i.e., c ) ... 

Let us use MATLAB to solve such a system and see how a slight change in the coefficient matrix will result in a drastic change in the solution. 

Solve the aforementioned equations using MATLAB. Use Gaussian elimination, the inverse of the matrix of coefficients (inv(A) b), and the backslash operation (A b) to show that such a system does not have a solution. 

Let us use MATLAB to solve this ODE. Figure 10.6 shows the MATLAB code (ST ODE.m) for symbolically (analytically) and numerically solving the ODE FEq. 00.20 1 and Fig. 10.7 shows the output plot of r vs. f for both analytical and numerical solutions. 

Eq. riO.531 can be anal5 ically solved for given boundary and initial conditions using the method of separation of variables. Obviously, T =f(x,t) = T(x, t). The open-form (i.e., power series) solution will be shown here for the sake of enriching chemical engineering students math-based skills. The faculty or student may jump, however, to Eg. 00.701 and go from there for the sake of using MATLAB as a tool for expressing such an open-form solution. 

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