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∫x0dxax2+bx+c=-22ax+b+2bfor b2=4ac (A-10)
∫x0dxax2+bx+c=1a(p-q)ln(qp⋅x−px−q)for b2>4ac (A-11)
where p and q are the roots of the equation.
ax2+bx+c=0,that is, p,q=−b−+√b2−4ac2a∫x0a+bxc+gxdx=bxg+ag−bcg2 Inc+gxc(A-12)
There are many ways of differentiating numerical and graphical data (cf. Chapter 7). We shall confine our discussions to the technique of equal-area differentiation. In the procedure delineated here, we want to find the derivative of y with respect to x.
This method finds use in Chapter 5.
Tabulate the (yi, xi) observations as shown in Table A-1.
For each interval, calculate Δxn = xn – xn-1 and Δyn – yn – yn-1.
TABLE A-1 FINDING DIFFERENTIALS (dydx) FROM DISCRETE DATA
x2 – x1
y2 – y1
x3 – x2
y3 – y2
and so on.
Calculate Δyn/Δxn as an estimate of the average slope in an interval xn-1 to xn.
Plot these values as a histogram versus xi. The value between x2 and x3, for example, is (y3 – y2)/(x3 – x2). Refer to Figure A-1.
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.
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.
See the CRE Web site, Appendix A.3.
Using integrating factor = exp (∫fdt), the solution is
Comparing the earlier equation with Equation (A-15) we note
f(t) = k2
and the integrating factor =exp∫k2dt=ek2t
Multiplying both sides by the integrating factor
Techniques to solve coupled first-order linear ODEs such as
are given in Web Appendix A.3 on the CRE Web site (http://www.umich.edu/~elements/6e/appendix/AppA.3_Web.pdf).
Methods of solving differential equations of the type
can be found in such texts as Applied Differential Equations by M. R. Spiegel (Upper Saddle River, NJ: Pearson, 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
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
y=A sinh √βx+B cosh√βx(A-19)
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.
In this section, we discuss techniques for numerically evaluating integrals for solving first-order differential equations.
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.
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:
Methods to solve
in Chapter 17
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:
Five-point quadrature formula.
For N = 1 points, where (N/3) is an integer
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.
Instructions on how to use Polymath, MATLAB, Python, Wolfram, COMSOL, and Aspen can be found on the CRE Web site.
For the ordinary differential equation solver (ODE solver), contact:
P.O. Box 523
Willimantic, CT 06226-0523
100 Trade Center Drive
Champaign, IL 61820-7237
206 East 9th Street
Austin, TX 78701
The Math Works, Inc.
20 North Main Street, Suite 250
Sherborn, MA 01770
For ordinary and partial differentiating equation solvers, contact:
1 New England Executive Park
Burlington, MA 01803