A. Numerical Techniques

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A. Numerical Techniques

Lake Michigan—unsalted and shark-free.

A.1 Useful Integrals in Chemical Reactor Design

$\begin{matrix} \int_{0}^{χ} \frac{d x}{1 - x} = \ln \frac{1}{1 - x} & (A-1) \end{matrix}$

$\begin{matrix} \int_{x_{1}}^{x_{2}} \frac{d x}{(1 - x)^{2}} = \frac{1}{1 - x_{2}} - \frac{1}{1 - x_{1}} & (A-2) \end{matrix}$

$\begin{matrix} \int_{0}^{x} \frac{d x}{(1 - x)^{2}} = \frac{χ}{1 - x} & (A-3) \end{matrix}$

$\begin{matrix} \int_{0}^{x} \frac{d x}{1 + ɛ x} = \frac{1}{ɛ} \ln (1 + ɛ x) & (A-4) \end{matrix}$

$\begin{matrix} \int_{0}^{x} \frac{(1 + ɛ x) d x}{1 - x} = (1 + ɛ) \ln \frac{1}{1 - x} - ɛ x & (A-5) \end{matrix}$

$\begin{matrix} \int_{0}^{x} \frac{(1 + ɛ x) d x}{(1 - x)^{2}} = \frac{(1 + ɛ) x}{1 - x} - ɛ \ln \frac{1}{1 - x} & (A-6) \end{matrix}$

$\begin{matrix} \int_{0}^{x} \frac{(1 + ɛ x)^{2} d x}{(1 - x)^{2}} = 2 ɛ (1 + ɛ) \ln (1 - x) + ɛ^{2} x + \frac{(1 + ɛ)^{2} x}{1 - x} & (A-7) \end{matrix}$

$\begin{matrix} \begin{matrix} \int_{0}^{x} \frac{d x}{(1 - x) (Θ_{B} - x)} = \frac{1}{Θ_{B} - 1} In \frac{Θ_{B} - x}{Θ_{B} (1 - x)} & Θ_{B} \neq 1 \end{matrix} & (A-8) \end{matrix}$

$\begin{matrix} \int_{0}^{W} (1 - α W)^{1 / 2} d W = \frac{2}{3 α} [1 - (1 - α W)^{3 / 2}] & (A-9) \end{matrix}$

$\begin{matrix} \int_{0}^{x} \frac{d x}{a x^{2} + b x + c} = \frac{- 2}{2 a x + b} + \frac{2}{b} & for b^{2} = 4 a c (A-10) \end{matrix}$

$\begin{matrix} \int_{0}^{x} \frac{d x}{a x^{2} + b x + c} = \frac{1}{a (p - q)} \ln (\frac{q}{p} \cdot \frac{x - p}{x - q}) & for b^{2} > 4 a c (A-11) \end{matrix}$

where p and q are the roots of the equation.

$\begin{matrix} \begin{matrix} \begin{matrix} {a x}^{2} + b x + c = 0, & that is, p, q \end{matrix} = \frac{- b \bar{+} \sqrt{b^{2} - 4 a c}}{2 a} \\ \int_{0}^{x} \frac{a + b x}{c + g x} d x = \frac{b x}{g} + \frac{a g - b c}{g^{2}} In \frac{c + g x}{c} \end{matrix} & (A-12) \end{matrix}$

A.2 Equal-Area Graphical Differentiation

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

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

TABLE A-1 FINDING DIFFERENTIALS $(\frac{d y}{d x})_{}$ FROM DISCRETE DATA

x_i	y_i	Δx	Δy	$\frac{Δ y}{Δ x}$	$\frac{d x}{d y}$
x₁	y₁				$(\frac{d y}{d x})_{1}$
		x₂ – x₁	y₂ – y₁	$(\frac{Δ y}{Δ x})_{2}$
x₂	y₂				$(\frac{d y}{d x})_{2}$
		x₃ – x₂	y₃ – y₂	$(\frac{Δ y}{Δ x})_{3}$
x₃	y₃				$(\frac{d y}{d x})_{3}$
x₄	y₄		and so on.

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

A graph shows an equal-area differentiation. X-values are marked on the horizontal axis and derivative of y with respect to x values are marked on the vertical axis. The graph resembles a right-skewed histogram in an interval of x1 to x2, x2 to x3, x3 to x4, x4 to x5, x5 to x6, and so on. A negative sloping curve passes over the bars. For an interval of x 2 minus x 3, a slope is calculated using delta y over delta x the whole cubed. Points A and B are marked on the curve.
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

$\begin{matrix} y_{n} - y_{1^{=}} Σ_{i = 2}^{n} \frac{Δ y}{Δ x_{i}} Δ x_{i} & (A-13) \end{matrix}$

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

$\begin{matrix} y_{n} - y_{1} = \int_{x_{1}}^{x_{n}} \frac{d y}{d x} d x & (A-14) \end{matrix}$

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.

A.3 Solutions to Differential Equations

A.3.A First-Order Ordinary Differential Equations

See the CRE Web site, Appendix A.3.

$\begin{matrix} \frac{d y}{d t} + f (t) y = g (t) & (A-15) \end{matrix}$

Using integrating factor = exp $(\int f dt)$ , the solution is

$\begin{matrix} y = e^{- \int f dt} \int g (t) e^{\int f d t} d t + K_{1} e^{- \int f d t} & (A-16) \end{matrix}$

Example A–1 Integrating Factor for Series Reactions

$\frac{d y}{d x} + k_{2} y = k_{1} e^{- k_{1} t}$

Comparing the earlier equation with Equation (A-15) we note

f(t) = k₂

and the integrating factor $= \exp \int k_{2} d t = e^{k_{2} t}$

Multiplying both sides by the integrating factor

$\frac{d (y e^{k_{2^{t}}})}{d t} = e^{k_{2} t} k_{1} e^{- k_{1} t} = k_{1} e^{(k_{2} - k_{1}) t}$

Integrating

$\begin{matrix} \begin{matrix} \begin{matrix} e^{k_{2^{t}}} y = k_{1} e^{(k_{2} - k_{1}) t} d t = \frac{k_{1}}{k_{2} - k_{1}} e^{(k_{2} - k_{1}) t} + K_{1} \\ y = \frac{k_{1}}{k_{2} - k_{1}} e^{- k_{1} t} + K_{1} e^{- k_{2} t} \end{matrix} \\ t = 0 y = 0 \end{matrix} \\ y = \frac{k_{1}}{k_{2} - k_{1}} [e^{- k_{1} t} - e^{- k_{2} t}] \end{matrix}$

A.3.B Coupled Differential Equations

Techniques to solve coupled first-order linear ODEs such as

$\begin{matrix} \frac{d x}{d t} = a x + b y \\ \frac{d y}{d t} = c x + d y \end{matrix}$

are given in Web Appendix A.3 on the CRE Web site (http://www.umich.edu/~elements/6e/appendix/AppA.3_Web.pdf).

A.3.C Second-Order Ordinary Differential Equations

Methods of solving differential equations of the type

$\begin{matrix} \frac{d^{2} y}{d^{2} x} - β y = 0 & (A-17) \end{matrix}$

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

$(\frac{d^{2}}{d x^{2}} - β) y = (m^{2} - β) y$

which are

$m = \pm \sqrt{β}$

The solution to the differential equation is

$\begin{matrix} y = A_{1} e^{- \sqrt{β x}} + B_{1} e^{+ \sqrt{β x}} & (A-18) \end{matrix}$

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

$\begin{matrix} y = A \sinh \sqrt{β x} + B \cosh \sqrt{β x} & (A-19) \end{matrix}$

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.

A.4 Numerical Evaluation of Integrals

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

$\begin{matrix} \int_{X_{0}}^{X_{1}} f (X) d X = \frac{h}{2} [f (X_{0}) + f (X_{1})] & (A-20) \end{matrix}$

when h = X₁ = X₀.

Figure A-2 Trapezoidal rule illustration.

The graph illustrates the Trapezoidal rule (two-point). X values are marked on the horizontal axis and f(X) values are marked on the vertical axis. The curve starts from (X 0, f of X subscript 0), increases up to (X1, f of X 1) within a distance of 'h' from the vertical axis. The maximum point on the curve is connected using a horizontal and vertical dashed line at (X1, f of X 1),
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:

$\begin{matrix} \int_{X_{0}}^{X_{2}} f (X) d X = \frac{h}{3} [f (X_{0}) + 4 f (X_{1}) + f (X_{2})] & (A-21) \end{matrix}$

where

$h = \frac{X_{2} - X_{0}}{2} X_{1} = X_{0} + h$

Methods to solve

$\int_{0}^{X} \frac{F_{A 0}}{- r_{A}} d X$

in Chapters 2, 4, 5, 12, and $\int_{0}^{\infty} X (t) E (t) d t$

in Chapter 17

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

The graph illustrates Simpson's three-point rule. X values are marked on the horizontal axis and f(X) values are marked on the vertical axis. The curve that starts from the vertical y-axis rises with the increasing values of X. Two values are marked on the curve using the vertical and horizontal dashed lines from (X 1, f (X 1)) and (X2, f of X 2). The distance between the two vertical dashed lines is h.
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:

$\begin{matrix} \int_{X_{0}}^{X_{3}} f (X) d X = \frac{3}{8} h [f (X_{0}) + 3 f (X_{1}) + 3 f (X_{2}) + f (X_{3})] & (A-22) \end{matrix}$

where

$h = \frac{X_{3} - X_{0}}{3} X_{1} = X_{0} + h X_{2} = X_{0} + 2 h$

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

The graph illustrates Simpson's four-point rule. X values are marked on the horizontal axis and f(X) values are marked on the vertical axis. The curve that starts from the y-axis rises with the increasing values of X. Three values are marked on the curve using the vertical dashed lines from X1, X2, and X3. The distance between the vertical dashed lines is h. Each point on the curve has a left arrow indicated on it.
Five-point quadrature formula.

$\begin{matrix} \int_{X_{0}}^{X_{4}} f (X) d X = \frac{h}{3} (f_{0} + 4 f_{1} + 2 f_{2} + 4 f_{3} + f_{4}) & (A-23) \end{matrix}$

where

$h = \frac{X_{4} - X_{0}}{4}$
For N = 1 points, where (N/3) is an integer

$\begin{matrix} \begin{matrix} \int_{X_{0}}^{X_{N}} f (X) d X = \frac{3}{8} h [f_{0} + 3 f_{1} + 3 f_{2} + 2 f_{3} \\ {\begin{matrix} + 3 f \end{matrix}}_{4} + 3 f_{5} + 2 f_{6} + ... + 3 f_{N - 1} + f_{N}] \end{matrix} & (A-24) \end{matrix}$

where $h = \frac{X_{N} - X_{0}}{N}$
For N = 1 points, where N is even

$\begin{matrix} \int_{X_{0}}^{X_{N}} f (X) d X = \frac{h}{3} (f_{0} + 4 f_{1} + 2 f_{2} + 4 f_{3} + 2 f_{4} + \dots + 4 f_{N - 1} + f_{N}) & (A-25) \end{matrix}$

where $h = \frac{X_{N} - X_{0}}{N}$

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.

A.5 Semi-Log Graphs

Review how to take slopes on semi-log graphs on the Web. Also see https://bolide.cs.uoguelph.ca/tutorials/GLP. Also see Web Appendix A.5, Using Semi-Log Plots for Data Analysis.

A.6 Software Packages

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:

Polymath Software

P.O. Box 523

Willimantic, CT 06226-0523

https://www.polymath-software.com/fogler

Wolfram

100 Trade Center Drive

Champaign, IL 61820-7237

Tel: +1-217-398-0700
+1-800-WOLFRAM (965-3726)

Fax: +1-217-398-0747

www.wolfram.com

Python Software

206 East 9th Street

Floor 18

Austin, TX 78701

Tel: +1-512-222-5440

E-mail: [email protected]

www.python.org

MATLAB

The Math Works, Inc.

20 North Main Street, Suite 250

Sherborn, MA 01770

For ordinary and partial differentiating equation solvers, contact:

COMSOL, Inc.

1 New England Executive Park

Burlington, MA 01803

Tel: +1-781-273-3322

Fax: +1-781-273-6603

E-mail: [email protected]

www.comsol.com

For simulators, contact:

Aspen Technology, Inc.

10 Canal Park

Cambridge, MA 02141-2201

Email: [email protected]

www.aspentech.com

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Table of Contents for A. Numerical Techniques

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A. Numerical Techniques

A.1 Useful Integrals in Chemical Reactor Design

A.2 Equal-Area Graphical Differentiation

A.3 Solutions to Differential Equations

A.3.A First-Order Ordinary Differential Equations

A.3.B Coupled Differential Equations

A.3.C Second-Order Ordinary Differential Equations

A.4 Numerical Evaluation of Integrals

A.5 Semi-Log Graphs

A.6 Software Packages

Table of Contents for
A. Numerical Techniques