Introduction to Mathematical Methods for Environmental Engineers and Scientists

By Charles Prochaska and Louis Theodore

The material in this book attempts to address mathematical calculations common to both the environmental science and engineering professionals. The book provides the reader with nearly 100 solved illustrative examples. The interrelationship between both theory and applications is emphasized in nearly all of the 35 chapters. One key feature of this book is that the solutions to the problems are presented in a stand-alone manner. Throughout the book, the illustrative examples are laid out in such a way as to develop the reader’s technical understanding of the subject in question, with more difficult examples located at or near the end of each set.

In presenting the text material, the authors have stressed the pragmatic approach in the application of mathematical tools to assist the reader in grasping the role of mathematical skills in environmental problem-solving situations. The book is divided up into five (V) parts:

1. Introduction
2. Analytical Analysis
3. Numerical Analysis
4. Statistical Analysis

Optimization

• Language: English
• Publisher: Wiley
• Release date: May 31, 2018
• ISBN: 9781119364146

Book preview

Preface

It is no secret that in recent years the number of people entering the environmental field has increased at a near exponential rate. Some are beginning college students and others had earlier chosen a non-technical major/career path. A large number of these individuals are today seeking technical degrees in environmental engineering or in the environmental sciences. These prospective students will require an understanding and appreciation of the numerous mathematical methods that are routinely employed in practice. This technical steppingstone to a successful career is rarely provided at institutions that award technical degrees. This introductory text on mathematical methods attempts to supplement existing environmental curricula with a sorely needed tool to eliminate this void.

The question often arises as to the educational background required for meaningful analysis capabilities since technology has changed the emphasis that is placed on certain mathematical subjects. Before computer usage became popular, instruction in environmental analysis was (and still is in many places) restricted to simple systems and most of the effort was devoted to solving a few derived elementary equations. These cases were mostly of academic interest, and because of their simplicity, were of little practical value. To this end, a considerable amount of time is now required to acquire skills in mathematics, especially in numerical methods, statistics, and optimization. In fact, most environmental engineers and scientists are given courses in classical mathematics, but experience shows that very little of this knowledge is retained after graduation for the simple reason that these mathematical methods are not adequate for solving most systems of equations encountered in industry. In addition, advanced mathematical skills are either not provided in courses or are forgotten through sheer disuse.

As noted in the above paragraph, the material in this book was prepared primarily for beginning environmental engineering and science students and, to a lesser extent, for environmental professionals who wish to obtain a better understanding of the various mathematical methods that can be employed in solving technical problems. The content is such that it is suitable both for classroom use and for individual study. In presenting the text material, the authors have stressed the pragmatic approach in the application of mathematical tools to assist the reader in grasping the role of mathematical skills in environmental problem solving situations.

In effect, this book serves two purposes. It may be used as a textbook for beginning environmental students or as a reference book for practicing engineers, scientists, and technicians involved with the environment. The authors have assumed that the reader has already taken basic courses in physics and chemistry, and should have a minimum background in mathematics through elementary calculus. The authors’ aim is to offer the reader the fundamentals of numerous mathematical methods with accompanying practical environmental applications. The reader is encouraged through references to continue his or her own development beyond the scope of the presented material.

As is usually the case in preparing any text, the question of what to include and what to omit has been particularly difficult. The material in this book attempts to address mathematical calculations common to both the environmental engineering and science professionals. The book provides the reader with nearly 100 solved illustrative examples. The interrelationship between both theory and applications is emphasized in nearly all of the chapters. One key feature of this book is that the solutions to the problems are presented in a stand-alone manner. Throughout the book, the illustrative examples are laid out in such a way as to develop the reader’s technical understanding of the subject in question, with more difficult examples located at or near the end of each set.

The book is divided up into five (V) parts (see also the Table of Contents):

I.     Introduction

II.    Analytical Analysis

III.  Numerical Analysis

IV.  Statistical Analysis

V.    Optimization

Most chapters contain a short introduction to the mathematical method in question, which is followed by developmental material, which in turn, is followed by one or more illustrative examples. Thus, this book offers material not only to individuals with limited technical background but also to those with extensive environmental industrial experience. As noted above, this book may be used as a text in either a general introductory environmental engineering/ science course and (perhaps) as a training tool in industry for challenged environmental professionals.

Hopefully, the text is simple, clear, to the point, and imparts a basic understanding of the theory and application of many of the mathematical methods employed in environmental practice. It should also assist the reader in helping master the difficult task of explaining what was once a very complicated subject matter in a way that is easily understood. The authors feel that this delineates this text from the numerous others in this field.

It should also be noted that the authors have long advocated that basic science courses – particularly those concerned with mathematics – should be taught to engineers and applied scientists by an engineer or applied scientist. Also, the books adopted for use in these courses should be written by an engineer or an applied scientist. For example, a mathematician will lecture on differentiation – say dx/dy – not realizing that in a real-world application involving an estuary y could refer to concentration while x could refer to time. The reader of this book will not encounter this problem.

The reader should also note that parts of the material in the book were drawn from one of the author’s notes of yesteryear. In a few instances, the original source was not available for referencing purposes. Any oversight will be corrected in a later printing/edition.

The authors wish to express appreciation to those who have contributed suggestions for material covered in this book. Their comments have been very helpful in the selection and presentation of the subject matter. Special appreciation is extended to Megan Menzel for her technical contributions and review, Dan McCloskey for preparing some of the first draft material in Parts II and III, and Christopher Testa for his contributions to Chapters 13 and 14. Thanks are also due to Rita D’Aquino, Mary K. Theodore, and Ronnie Zaglin.

Finally, the authors are especially interested in learning the opinions of those who read this book concerning its utility and serviceability in meeting the needs for which it was written. Corrections, improvements and suggestions will be considered for inclusion in later editions.

Chuck Prochaska

Lou Theodore

April 2018

Part I

INTRODUCTORY PRINCIPLES

Webster defines introduction as … the preliminary section of a book, usually explaining or defining the subject matter … And indeed, that is exactly what this Part I of the book is all about. The chapters contain material that one might view as a pre-requisite for the specific mathematical methods that are addressed in Parts II–V.

There are seven chapters in Part I. The chapter numbers and accompanying titles are listed below.

Chapter 1: Fundamentals and Principles of Numbers

Chapter 2: Series Analysis

Chapter 3: Graphical Analysis

Chapter 4: Flow Diagrams

Chapter 5: Dimensional Analysis

Chapter 6: Economics

Chapter 7: Problem Solving

Chapter 1

Fundamentals and Principles of Numbers

The natural numbers, or so-called counting numbers, are the positive integers: 1, 2, 3, … and the negative integers: –1, –2, –3, … The following applies to real numbers:

(1.1)

(1.2)

(1.3)

(1.4)

(1.5)

(1.6)

(1.7)

(1.8)

(1.9)

(1.10)

(1.11)

(1.12)

(1.13)

(1.14)

(1.15)

(1.16)

(1.17)

(1.18)

(1.19)

Based on the above, one may write

Given any quadratic equation of the general form

(1.20)

a number of methods of solution are possible depending on the specific nature of the equation in question. If the equation can be factored, then the solution is straightforward. For instance, consider

(1.21)

Put into the standard form,

(1.22)

this equation can be factored as follows:

(1.23)

This condition can be met, however, only when the individual factors are zero, i.e., when x = 5 and x = –2. That these are indeed the solutions to the equation may be verified by substitution.

If, upon inspection, no obvious means of factoring an equation can be found, an alternative approach may exist. For example, in the equation

(1.24)

the expression

(1.25)

could be factored as a perfect square if it were

(1.26)

which equals

(1.27)

This can easily be achieved by adding 9 to the left side of the equation. The same amount must then, of course, be added to the right side as well, resulting in:

(1.28)

so that,

(1.29)

This can be reduced to

(1.30)

or

(1.31)

and

(1.32)

Since Graphic above has two solutions, i.e., +4 and –4, the first equation leads to the solution x = 0.5 while the second equation leads to the solution x = –7/2, or x = –3.5.

If the methods of factoring or completing the square are not possible, any quadratic equation can always be solved by the quadratic formula. This provides a method for determining the solution of the equation if it is in the form

(1.33)

In all cases, the two solutions of x are given by the formula

(1.34)

For example, to find the roots of

(1.35)

the equation is first put into the standard form of Equation (1.33)

(1.36)

As a result, a = 1, b = –4, and c = 3. These terms are then substituted into the quadratic formula presented in Equation (1.34).

(1.37)

(1.38)

The practicing environmental engineer and scientist occasionally has to solve not just a single equation but several at the same time. The problem is to find the set of all solutions that satisfies both equations. These are called simultaneous equations, and specific algebraic techniques may be used to solve them. For example, a simple solution exists given two linear equations and two unknowns:

(1.39)

(1.40)

The variable y in Equation (1.40) is isolated (y = 5 – 2x), and then this value of y is substituted into Equation (1.39).

(1.41)

This reduces the problem to one involving the single unknown x and it follows that

or

(1.42)

so that

(1.43)

When this value is substituted into either equation above, it follows that

(1.44)

A faster method of solving simultaneous equations, however, is obtained by observing that if both sides of Equation (1.40) are multiplied by 4, then

(1.45)

If Equation (1.39) is subtracted from Equation (1.45), then 5x = 10, or x = 2. This procedure leads to another development in mathematics, i.e., matrices, which can help to produce solutions for any set of linear equations with a corresponding number of unknowns (refer also to Chapter 13).

Four sections compliment the presentation of this chapter. Section numbers and subject titles follow:

1.1: Interpolation and Extrapolation

1.2: Significant Figures and Approximate Numbers

1.3: Errors

1.4: Propagation of Errors

1.1 Interpolation and Extrapolation

Experimental data (and data in general) in environmental engineering and science may be presented using a table, a graph, or an equation. Tabular presentation permits retention of all significant figures of the original numerical data. Therefore, it is the most numerically accurate way of reporting data. However, it is often difficult to interpolate between data points within tables.

Tabular or graphical presentation of data is usually used if no theoretical or empirical equations can be developed to fit the data. This type of presentation of data is one method of reporting experimental results. For example, heat capacities of benzene might be tabulated at various temperatures. This data may also be presented graphically. One should note that graphs are inherently less accurate than numerical tabulations. However, they are useful for visualizing variations in data and for interpolation and extrapolation.

Interpolation is of practical importance to the environmentalist because of the occasional necessity of referring to sources of information expressed in the form of a table. Logarithms, trigonometric functions, water properties of steam, liquid water and ice vapor pressures, and other physical and chemical data are commonly given in the form of tables in the standard reference works. Although these tables are sometimes given in sufficient detail so that interpolation may not be necessary, it is important to be able to interpolate properly when the need arises.

Assume that a series of values of the dependent variable y are provided for corresponding tabulated values of the independent variable x. The goal of interpolation is to obtain the correct value of y at any value of x. (Extrapolation refers to a value of x lying outside the range of tabulated values of x.) Clearly, interpolation or extrapolation may be accomplished by using data for x and y to develop a linear relationship between the two variables. The general method would be to fit two points (y1, x1), and (y2, x2) by means of

(1.46)

and then employ this equation to calculate y for some value of x lying between x1 and x2. Most practitioners do this mentally when reading values from a table, e.g., steam tables. If a number of points are used, a polynomial of a correspondingly higher degree may be employed. Thus, interpolation may be viewed as the process of finding the value of a function at some arbitrary point when the function is not known but is represented over a given range as a table of discrete points. (See also Table 1.1 where y represents a reservoir’s height as a function of time in days during a rainy season.) Interpolation is thus necessary to find y when x is some value not given in the table. For instance, one may be interested in finding y when x = 11. (The process of finding x when y is known is referred to as inverse interpolation). Given a table such as Table 1.1, one can draw a picture and write the equation of the straight line through the points (x1, y1) and (x2, y2) for y.

Table 1.1 Reservoir height vs. time in days.

(1.47)

Equation (1.47) can be solved for y in terms of x

(1.48)

or

(1.49)

Illustrative Example 1.1

Refer to Table 1.1. Find y at x = 11.

Solution

Proceed as follows. Set up calculations as shown in Table 1.2

Table 1.2 Information for Illustrative Example 1.1.

Apply Equation (1.48). Therefore,

As noted above, inverse interpolation involves estimating x which corresponds to a given value of y and extrapolation involves estimating values of y outside the interval in which the data x0, …, xn fall. It is generally unwise to extrapolate any empirical relation significantly beyond the first and last data points. If, however, a certain form of equation is predicted by theory and substantiated by (other) available data, reasonable extrapolation is ordinarily justified.

1.2 Significant Figures and Approximate Numbers [1]

Significant figures provide an indication of the precision with which a quantity is measured or known. The last digit represents in a qualitative sense, some degree of doubt. For example, a measurement of 8.32 nm (nanometers) implies that the actual quantity is somewhere between 8.315 and 8.325 nm. This applies to calculated and measured quantities; quantities that are known exactly (e.g., pure integers) have an infinite number of significant figures. Note, however, that there is an upper limit to the accuracy with which physical measurements can be made.

The method for counting the significant digits of a number follows one of two rules depending on whether there is or is not a decimal point present. The significant digits of a number always start from the first nonzero digit on the left to either:

the last digit (whether it is nonzero or zero) on the right if there is a decimal point present, or

the last nonzero digit on the right of the number if there is no decimal point present.

For example:

Whenever quantities are combined by multiplication and/or division, the number of significant figures in the result should equal the lowest number of significant figures of any of the quantities. In long calculations, the final result should be rounded off to the correct number of significant figures. When quantities are combined by addition and/or subtraction, the final result cannot be more precise than any of the quantities added or subtracted. Therefore, the position (relative to the decimal point) of the last significant digit in the number that has the lowest degree of precision is the position of the last permissible significant digit in the result. For example, the sum of 3702, 370, 0.037, 4, and 37 should be reported as 4110 (without a decimal). The least precise of the five number is 370, which has its last significant digit in the tens position. Therefore, the answer should also have its last significant digit in its tens position.

Unfortunately, environmental engineers and scientists rarely concern themselves with significant figures in their calculations. However, it is recommended that the reader attempt to follow the calculational procedure set forth in this section.

In the process of preforming engineering/scientific calculations, very large and very small number are often encountered. A convenient way to represent these numbers is to use scientific notation. Generally, a number represented in scientific notation is the product of a number and 10 raised to an integer power. For example,

A positive feature of using scientific notation is that only the significant figures need appear in the number.

Thus, when approximate numbers are added, or subtracted, the results are presented in terms of the least precise number. Since this is a relatively simple rule to master, note that the answer in Equation (1.50) follows the aforementioned rule of precision.

(1.50)

(The result is 4.667L.) The expressions in Equation (1.50) have two, one, and three decimal places respectively. The least precise number (least decimal places) in the problem is 2.8, a value carried only to the tenths position. Therefore, the answer must be calculated to the tenths position only. Thus, the correct answer is 4.7L. (The last 6 and the 7 are dropped from the 4.667L, and the first 6 is rounded up to provide 4.7L.)

In multiplication and division of approximate numbers, finding the number of significant digits is used to determine how many digits to keep (i.e., where to truncate). One must first understand significant digits in order to determine the correct number of digits to keep or remove in multiplication and division problems. As noted earlier in this section, the digits 1 through 9 are considered to be significant. Thus, the numbers 123, 53, 7492, and 5 contain three, two, four and one significant digits respectively. The digit zero must be considered separately.

Zeroes are significant when they occur between significant digits. In the following example, all zeroes are significant: 10001, 402, 1.1001, 500.09 with five, three, five, and four significant figures, respectively. Zeroes are not significant when they are used as place holders. When used as a place holder, a zero simply identifies where a decimal is located. For example, each of the following numbers has only one significant digit: 1000, 500, 60, 0.09, 0.0002. In the numbers 1200, 540, and 0.0032 there are two significant digits, and the zeroes are not significant. When zeroes follow a decimal and are preceded by a significant digit, the zeroes are significant. In the following examples, all zeroes are significant: 1.00, 15.0, 4.100, 1.90, 10.002, 10.0400. For 10.002, the zeroes are significant because they fall between two significant digits. For 10.0400, the first two zeroes are significant because they fall between two significant digits; the last two zeroes are significant because they follow a decimal and are preceded by a significant digit. As noted above, when approximate numbers are multiplied or divided, the result is expressed as a number having the same number of significant digits as the number in the problem with the least number of significant digits.

When truncating (removing final, unwanted digits), rounding is normally applied to the last digit to be kept. Thus, if the value of the first digit to be discarded is less than 5, one should retain the last retained digit with no change. If the value of the first digit to be discarded is 5 or greater, one should increase the last kept digit’s value by one. Assume, for example, only the first two decimal places are to be kept for 25.0847 (the 4 and 7 are to be dropped). The number is then 25.08. Since the first digit to be discarded (4) is less than 5, i.e., the 8 is not rounded up. If only the first two decimal places are to be kept for 25.0867 (the 6 and the 7 are to be dropped), it should be rounded to 25.09. Since the first digit to be discarded (6) is 5 or more, the 8 is rounded up to 9.

When adding or subtracting approximate numbers, a rule based upon precision determines how many digits are kept. In general, precision relates to the decimal significance of a number. When a measurement is given as 1.005 cm, one can say that the number is precise to the thousandth of a centimeter. If the decimal is removed (1005 cm), the number is precise to thousands of centimeters.

In some water pollution studies, a measurement in gallons or liters may be required. Although a gallon or liter may represent an exact quantity, the measuring instruments that are used are only capable of producing approximations. Using a standard graduated flask in liters as an example, can one determine whether there is exactly one liter? Not likely. In fact, one would be pressed to verify that there was a liter to within ±1/10 of a liter. Therefore, depending upon the instruments used, the precision of a given measurement may vary.

If a measurement is given as 16.0L, the zero after the decimal indicates that the measurement is precise to within 1/10L i.e., 0.1L. A given measurement of 16.00L, indicates precision to the 1/100L. As noted, the digits following the decimal indicate how precise the measurement is. Thus, precision is used to determine where to truncate when approximate numbers are added or subtracted.

1.3 Errors

This is the first of two sections devoted to errors. This section introduces the various classes of errors while the next section demonstrates the propagation of some of these errors. As one might suppose, numerous books have been written on the general subject of errors. Different definitions for errors appear in the literature but what follows is the authors’ attempt to clarify the problem [1].

Any discussion of errors would be incomplete without providing a clear and concise definition of two terms: the aforementioned precision and accuracy. The term precision is used to describe a state or system or measurement for which the word precise implies little to no variation; some refer to this as reliability. Alternatively, accuracy is used to describe something free from the matter of errors. The accuracy of a value, which may be represented in either absolute or relative terms, is the degree of agreement between the measured value and the true value.

All measurements and calculations are subject to two broad classes of errors: determinate and indeterminate. The error is known as a determinate error if an error’s magnitude and sign are discovered and accounted for in the form of a correction. All errors that either cannot be or are not properly allowed for in magnitude and sign are known as indeterminate errors.

A particularly important class of indeterminate errors is that of accidental errors. To illustrate the nature of these, consider the very simple and direct measurement of temperature. Suppose that several independent readings are made and that temperatures are read to 0.1 °F. When the results of the different readings are compared, it may be found that even though they have been performed very carefully, they may differ from each other by several tenths of a degree. Experience has shown that such deviations are inevitable in all measurements and that these result from small unavoidable errors of observation due to the sensitivity of measuring instruments and the keenness of the sense of perception. Such errors are due to the combined effect of a large number of undetermined causes and they can be defined as accidental errors.

Regarding the words precision and accuracy, it is also important to note that a result may be extremely precise and at the same time inaccurate. For instance, the temperature readings just mentioned might all agree within 1 °F. From this it would not be permissible to conclude that the temperature is accurate to 1 °F until it can be definitively shown that the combined effects of uncorrected constant errors and known errors are negligible compared with 1 °F. It is quite conceivable that the calibration of the thermometer might be grossly incorrect. Errors such as these are almost always present and can never be detected individually. Such errors can be detected only by obtaining the readings with several different thermometers and, if possible, several independent methods and observers.

It should also be understood at the onset that most numerical calculations are by their very nature inexact. The errors are primarily due to one of three sources: inaccuracies in the original data, lack of precision in carrying out calculations, or inaccuracies introduced by approximate or incorrect methods of solution. Of particular significance are the aforementioned errors due to round-off and the inability to carry more than a certain number of significant figures. The errors associated with the method of solution are usually the area of greatest concern [1]. These usually arise as a result of approximations and assumptions made in the development of an equation used to calculate a desired result and should not be neglected in any error analysis.

Finally, many list the following three errors associated with a computer (calculator).

Truncation error. With the truncation of a series after only a few terms, one is committing a generally known error. This error is not machine-caused but is due to the method.

Round-off error. The result of using a finite number of digits to represent a number. In reality, numbers have an infinite number of digits extending past the decimal point. For example, the integer 1 is really 1.000 … 0 and π is 3.14159 … but numbers are rounded to allow for calculation and representation. This rounding is a form of error known as round-off error.

Propagation or inherited error. This is caused by sequential calculations that include points previously calculated by the computer which already are erroneous owing to the two errors above. Since the result is already off the solution curve, one cannot expect any new values computed to be on the correct solution curve. Adding the round-off errors and truncation errors into the calculation causes further errors to propagate, adding more error at each step.

1.4 Propagation of Errors

When a desired quantity W is related to several directly measured independent quantities W1, W2, W3, …, Wn by the equation

(1.51)

W becomes an indirectly measured dependent quantity. In general, the true value of W cannot be known because the true values of W1, W2, W3, …, W are unknown, but the most probable value of W may be calculated by inserting the most probable values of W1, W2, W3, …, Wn, into Equation (1.51). The errors in the directly measured quantities of Wi will result in an error in the calculated quantity W, the value of which is important to ascertain. If the original measurements are available, a method referred to as the propagation-of-error could be employed to estimate in the resultant error. The general propagation-of-error for a function W = f(x1, x2) is described by

(1.52)

where s = error in the function W

s = x1 error in variable x1

s = x2 error in variable x2

Thus, if a linear function W = ax1 + bx2 is involved, it is found by direct application of Equation (1.52) that

(1.53)

where

(1.54)

(1.55)

For the case where W = x1x2 application of Equation (1.52) can be shown to give

(1.56)

Thus, the square of the fractional error is equal to the sum of the squares of fractional errors of the independent variables. Alternatively, taking the logarithm of a product c = xy reduces it to the form of Equation (1.57):

(1.57)

As would be expected, the greatest variation in the approach of different investigators lies in the details of how they propose to obtain values of the error measurement. Basically, these may be obtained by comparing either a series of pairs of estimates and true values from similar previous readings (i.e., looking at the record) or obtaining a number of independent estimates of the particular value needed.

Illustrative Example 1.2 [2]

Table 1.3 gives basic data on investment and production costs for a unit for coking a heavy crude to