A. Numerical Techniques

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where p and q are the roots of the equation.

This method finds use in Chapter 5.

1. Tabulate the (y_i, x_i) observations as shown in Table A-1.

2. For each interval, calculate Δx_n = x_n – x_n–1 and Δy_n = y_n – y_n–1.

TABLE A-1

3. Calculate Δy_n/Δx_n as an estimate of the average slope in an interval x_n–1 to x_n.

4. Plot these values as a histogram versus x_i. The value between x₂ and x₃, for example, is (y₃–y₂)/(x₃ – x₂). Refer to Figure A-1.

Figure A-1 Equal-area differentiation.

5. Next, draw in the smooth curve that best approximates the area under the histogram. That is, attempt in each interval to balance areas such as those labeled A and B, but when this approximation is not possible, balance out over several intervals (as for the areas labeled C and D). From our definitions of Δx and Δy, we know that

The equal-area method attempts to estimate dy/dx so that

that is, so that the area under Δy/Δx is the same as that under dy/dx, everywhere possible.

6. Read estimates of dy/dx from this curve at the data points x₁, x₂, ... and complete the table.

An example illustrating the technique is given on the CRE Web site, Appendix A.

Differentiation is, at best, less accurate than integration. This method also clearly indicates bad data and allows for compensation of such data. Differentiation is only valid, however, when the data are presumed to differentiate smoothly, as in rate-data analysis and the interpretation of transient diffusion data.

Using integrating factor , the solution is

Example A–1 Integrating Factor for Series Reactions

are given in Web Appendix A.3 on the CRE Web site.

can be found in such texts as Applied Differential Equations by M. R. Spiegel (Upper Saddle River, NJ: Prentice Hall, 1958, Chapter 4; a great book even though it’s old) or in Differential Equations by F. Ayres (New York: Schaum Outline Series, McGraw-Hill, 1952). Solutions of this type are required in Chapter 15. One method of solution is to determine the characteristic roots of

which are

The solution to the differential equation is

where A₁ and B₁ are arbitrary constants of integration. It can be verified that Equation (A-18) can be arranged in the form

Equation (A-19) is the more useful form of the solution when it comes to evaluating the constants A and B because sinh(0) = 0 and cosh(0) = 1.0. Also see Appendix A.3.B.

1. Trapezoidal rule (two-point) (Figure A-2). This method is one of the simplest and most approximate, as it uses the integrand evaluated at the limits of integration to evaluate the integral

when h = X₁ – X₀.

2. Simpson’s one-third rule (three-point) (Figure A-3). A more accurate evaluation of the integral can be found with the application of Simpson’s rule:

where

Methods to solve in Chapters 2, 4, 12, and in Chapter 17

Figure A-2 Trapezoidal rule illustration.

Figure A-3 Simpson’s three-point rule illustration.

3. Simpson’s three-eighths rule (four-point) (Figure A-4). An improved version of Simpson’s one-third rule can be made by applying Simpson’s three-eighths rule:

where

Figure A-4 Simpson’s four-point rule illustration.

4. Five-point quadrature formula.

5. For N + 1 points, where (N/3) is an integer

6. For N + 1 points, where N is even

These formulas are useful in illustrating how the reaction engineering integrals and coupled ODEs (ordinary differential equation(s)) can be solved and also when there is an ODE solver power failure or some other malfunction.

For the ordinary differential equation solver (ODE solver), contact:

Polymath Software
P.O. Box 523
Willimantic, CT 06226-0523
Web site: www.polymathsoftware.com/fogler

MATLAB
The Math Works, Inc.
20 North Main Street, Suite 250
Sherborn, MA 01770

Aspen Technology, Inc.
10 Canal Park
Cambridge, MA 02141-2201
Email: [email protected]
Web site: www.aspentech.com

COMSOL, Inc.
1 New England Executive Park
Burlington, MA 01803
Tel: +1-781-273-3322
Fax: +1-781-273-6603
Email: [email protected]
Web site: www.comsol.com