Cover Page



Half Title page

Title page

Copyright page



Part One: Introduction

Chapter 1: History of Chemical Engineering and Mass Transfer Operations


Chapter 2: Transport Phenomena vs Unit Operations Approach


Chapter 3: Basic Calculations


Units and Dimensions

Conversion of Units

The Gravitational Constant GC

Significant Figures and Scientific Notation(3)


Chapter 4: Process Variables




Moles and Molecular Weight

Mass, Volume, and Density


Reynolds Number


Vapor Pressure

Ideal Gas Law(7)


Chapter 5: Equilibrium vs Rate Considerations




Chemical Reactions


Chapter 6: Phase Equilibrium Principles


Gibb’s Phase Rule

Raoult’s Law

Henry’s Law

Raoult’s Law Vs Henry’s Law(7)

Vapor-Liquid Equilibrium in Nonideal Solutions(1)

Vapor-Solid Equilibrium

Liquid–Solid Equilibrium


Chapter 7: Rate Principles


The Operating Line

Fick’s Law

Mass Transfer Coefficients

Overall Mass Transfer Coefficient


Part Two: Applications: Component and Phase Separation Processes

Chapter 8: Introduction to Mass Transfer Operations


Classification of Mass Transfer Operations

Mass Transfer Equipment

Characteristics of Mass Transfer Operations


Chapter 9: Distillation


Flash Distillation

Batch Distillation

Continuous Distillation With Reflux


Chapter 9: Absorption and Stripping


Description of Equipment

Design and Performance Equations–Packed Columns

Design and Performance Equations–Plate Columns


Packed vs Plate Tower Comparison

Summary of Key Equations


Chapter 11: Adsorption(1,2)


Adsorption Classification

Adsorption Equilibria

Description of Equipment

Design and Performance Equations



Chapter 12: Liquid–Liquid and Solid–Liquid Extraction


Liquid–Liquid Extraction

Solid-Liquid Extraction (Leaching)(4)


Chapter 13: Humidification and Drying


Psychrometry and The Psychrometric Chart




Chapter 14: Crystallization


Phase Diagrams

The Crystallization Process

Crystal Physical Characteristics


Describing Equations

Design Considerations


Chapter 15: Membrane Separation Processes


Reverse Osmosis



Gas Permeation


Chapter 16: Phase Separation Equipment


Fluid–Particle Dynamics

Gas–Solid (G–S) Equipment

Gas–Liquid (G–L) Equipment

Liquid–Solid (L–S) Equipment

Liquid–Liquid (L–L) Equipment

Solid-Solid (S-S) Equipment


Part Three: Other Topics

Chapter 17: Other and Novel Separation Processes

Freeze Crystallization

Ion Exchange

Liquid Ion Exchange

Resin Adsorption


Foam Fractionation

Dissociation Extraction


Vibrating Screens


Chapter 18: Economics and Finance


The Need For Economic Analyses


Principles of Accounting(4)



Chapter 19: Numerical Methods




Chapter 20: Open-Ended Problems


Developing Students’ Power of Critical Thinking(4)



Inquiring Minds

Failure, Uncertainty, Success: Are They Related?

Angels On A Pin(11)



Chapter 21: Ethics


Teaching Ethics

Case Study Approach


Moral Issues(8)


Engineering and Environmental Ethics(11)

Future Trends(11)



Chapter 22: Environmental Management and Safety Issues


Environmental Issues of Concern(1)

Health Risk Assessment(2–4)

Hazard Risk Assessment(2–4, 8)




Appendix A: Units

A.1 The Metric System

A.2 The SI System

A.3 Seven Base Units

A.4 Two Supplementary Units

A.5 SI Multiples And Prefixes

A.6 Conversion Constants (SI)

A.7 Selected Common Abbreviations

Appendix B: Miscellaneous Tables

Appendix C: Steam Tables


Mass Transfer Operations for the Practicing Engineer

Title Page

To Ann Cadigan and Meg Norris:
for putting up with me (LT)


To my mother Laura, my father Joseph, and my brother Joseph Jr: for reasons which need not be spoken (FR)


Mass transfer is one of the basic tenets of chemical engineering, and contains many practical concepts that are utilized in countless industrial applications. Therefore, the authors considered writing a practical text. The text would hopefully serve as a training tool for those individuals in academia and industry involved with mass transfer operations. Although the literature is inundated with texts emphasizing theory and theoretical derivations, the goal of this text is to present the subject from a strictly pragmatic point-of-view.

The book is divided into three parts: Introduction, Applications, and Other Topics. The first part provides a series of chapters concerned with principles that are required when solving most engineering problems, including those in mass transfer operations. The second part deals exclusively with specific mass transfer operations e.g., distillation, absorption and stripping, adsorption, and so on. The last part provides an overview of ABET (Accreditation Board for Engineering and Technology) related topics as they apply to mass transfer operations plus novel mass transfer processes. An Appendix is also included. An outline of the topics covered can be found in the Table of Contents.

The authors cannot claim sole authorship to all of the essay material and illustrative examples in this text. The present book has evolved from a host of sources, including: notes, homework problems and exam problems prepared by several faculty for a required one-semester, three-credit, “Principles III: Mass Transfer” undergraduate course offered at Manhattan College; L. Theodore and J. Barden, “Mass Transfer”, A Theodore Tutorial, East Williston, NY, 1994; J. Reynolds, J. Jeris, and L. Theodore, “Handbook of Chemical and Environmental Engineering Calculations,” John Wiley & Sons, Hoboken, NJ, 2004, and J. Santoleri, J. Reynolds, and L. Theodore, “Introduction to Hazardous Waste Management,” 2nd edition, John Wiley & Sons, Hoboken, NJ, 2000. Although the bulk of the problems are original and/or taken from sources that the authors have been directly involved with, every effort has been made to acknowledge material drawn from other sources.

It is hoped that we have placed in the hands of academic, industrial, and government personnel, a book that covers the principles and applications of mass transfer in a thorough and clear manner. Upon completion of the text, the reader should have acquired not only a working knowledge of the principles of mass transfer operations, but also experience in their application; and, the reader should find himself/herself approaching advanced texts, engineering literature and industrial applications (even unique ones) with more confidence. We strongly believe that, while understanding the basic concepts is of paramount importance, this knowledge may be rendered virtually useless to an engineer if he/she cannot apply these concepts to real-world situations. This is the essence of engineering.

Last, but not least, we believe that this modest work will help the majority of individuals working and/or studying in the field of engineering to obtain a more complete understanding of mass transfer operations. If you have come this far and read through most of the Preface, you have more than just a passing interest in this subject. We strongly suggest that you try this text; we think you will like it.

Our sincere thanks are extended to Dr. Paul Marnell at Manhattan College for his invaluable help in contributing to Chapter 9 on Distillation and Chapter 14 on Crystallization. Thanks are also due to Anne Mohan for her assistance in preparing the first draft of Chapter 13 (Humidification and Drying) and to Brian Bermingham and Min Feng Zheng for their assistance during the preparation of Chapter 12 (Liquid–Liquid and Solid–Liquid Extraction). Finally, Shannon O’Brien, Kathryn Scherpf and Kimberly Valentine did an exceptional job in reviewing the manuscript and page proofs.

April 2010


NOTE: An additional resource is available for this text. An accompanying website contains over 200 additional problems and 15 hours of exams; solutions for the problems and exams are available at for those who adopt the book for training and/or academic purposes.

Part One

The purpose of this Part can be found in its title. The book itself offers the reader the fundamentals of mass transfer operations with appropriate practical applications, and serves as an introduction to the specialized and more sophisticated texts in this area. The reader should realize that the contents are geared towards practitioners in this field, as well as students of science and engineering, not chemical engineers per se. Simply put, topics of interest to all practicing engineers have been included. Finally, it should also be noted that the microscopic approach of mass transfer operations is not treated in any required undergraduate Manhattan College offering. The Manhattan approach is to place more emphasis on real-world and design applications. However, microscopic approach material is available in the literature, as noted in the ensuing chapters. The decision on whether to include the material presented ultimately depends on the reader and/or the approach and mentality of both the instructor and the institution.

A general discussion of the philosophy and the contents of this introductory section follows.

Since the chapters in this Part provide an introduction and overview of mass transfer operations, there is some duplication due to the nature of the overlapping nature of overview/introductory material, particularly those dealing with principles. Part One chapter contents include:

1 History of Chemical Engineering and Mass Transfer Operations

2 Transport Phenomena vs Unit Operations Approach

3 Basic Calculations

4 Process Variables

5 Equilibrium vs Rate Considerations

6 Phase Equilibrium Principles

7 Rate Principles

Topics covered in the first two introductory chapters include a history of chemical engineering and mass transfer operations, and a discussion of transport phenomena vs unit operations. The remaining chapters are concerned with introductory engineering principles. The next Part is concerned with describing and designing the various mass transfer unit operations and equipment.

Chapter 1

History of Chemical Engineering and Mass Transfer Operations

A discussion on the field of chemical engineering is warranted before proceeding to some specific details regarding mass transfer operations (MTO) and the contents of this first chapter. A reasonable question to ask is: What is Chemical Engineering? An outdated, but once official definition provided by the American Institute of Chemical Engineers is:

Chemical Engineering is that branch of engineering concerned with the development and application of manufacturing processes in which chemical or certain physical changes are involved. These processes may usually be resolved into a coordinated series of unit physical “operations” (hence part of the name of the chapter and book) and chemical processes. The work of the chemical engineer is concerned primarily with the design, construction, and operation of equipment and plants in which these unit operations and processes are applied. Chemistry, physics, and mathematics are the underlying sciences of chemical engineering, and economics is its guide in practice.

The above definition was appropriate up until a few decades ago when the profession branched out from the chemical industry. Today, that definition has changed. Although it is still based on chemical fundamentals and physical principles, these principles have been de-emphasized in order to allow for the expansion of the profession to other areas (biotechnology, semiconductors, fuel cells, environment, etc.). These areas include environmental management, health and safety, computer applications, and economics and finance. This has led to many new definitions of chemical engineering, several of which are either too specific or too vague. A definition proposed here is simply that “Chemical Engineers solve problems”. Mass transfer is the one subject area that somewhat uniquely falls in the domain of the chemical engineer. It is often presented after fluid flow(1) and heat transfer,(2) since fluids are involved as well as heat transfer and heat effects can become important in any of the mass transfer unit operations.

A classical approach to chemical engineering education, which is still used today, has been to develop problem solving skills through the study of several topics. One of the topics that has withstood the test of time is mass transfer operations; the area that this book is concerned with. In many mass transfer operations, one component of a fluid phase is transferred to another phase because the component is more soluble in the latter phase. The resulting distribution of components between phases depends upon the equilibrium of the system. Mass transfer operations may also be used to separate products (and reactants) and may be used to remove byproducts or impurities to obtain highly pure products. Finally, it can be used to purify raw materials.

Although the chemical engineering profession is usually thought to have originated shortly before 1900, many of the processes associated with this discipline were developed in antiquity. For example, filtration operations were carried out 5000 years ago by the Egyptians. MTOs such as crystallization, precipitation, and distillation soon followed. During this period, other MTOs evolved from a mixture of craft, mysticism, incorrect theories, and empirical guesses.

In a very real sense, the chemical industry dates back to prehistoric times when people first attempted to control and modify their environment. The chemical industry developed as did any other trade or craft. With little knowledge of chemical science and no means of chemical analysis, the earliest chemical “engineers” had to rely on previous art and superstition. As one would imagine, progress was slow. This changed with time. The chemical industry in the world today is a sprawling complex of raw-material sources, manufacturing plants, and distribution facilities which supply society with thousands of chemical products, most of which were unknown over a century ago. In the latter half of the nineteenth century, an increased demand arose for engineers trained in the fundamentals of chemical processes. This demand was ultimately met by chemical engineers.

The first attempt to organize the principles of chemical processing and to clarify the professional area of chemical engineering was made in England by George E. Davis. In 1880, he organized a Society of Chemical Engineers and gave a series of lectures in 1887 which were later expanded and published in 1901 as A Handbook of Chemical Engineering. In 1888, the first course in chemical engineering in the United States was organized at the Massachusetts Institute of Technology by Lewis M. Norton, a professor of industrial chemistry. The course applied aspects of chemistry and mechanical engineering to chemical processes.(3)

Chemical engineering began to gain professional acceptance in the early years of the twentieth century. The American Chemical Society had been founded in 1876 and, in 1908, it organized a Division of Industrial Chemists and Chemical Engineers while authorizing the publication of the Journal of Industrial and Engineering Chemis ry. Also in 1908, a group of prominent chemical engineers met in Philadelphia and founded the American Institute of Chemical Engineers.(3)

The mold for what is now called chemical engineering was fashioned at the 1922 meeting of the American Institute of Chemical Engineers when A. D. Little’s committee presented its report on chemical engineering education. The 1922 meeting marked the official endorsement of the unit operations concept and saw the approval of a “declaration of independence” for the profession.(3) A key component of this report included the following:

Any chemical process, on whatever scale conducted, may be resolved into a coordinated series of what may be termed “unit operations,” as pulverizing, mixing, heating, roasting, absorbing, precipitation, crystallizing, filtering, dissolving, and so on. The number of these basic unit operations is not very large and relatively few of them are involved in any particular process… An ability to cope broadly and adequately with the demands of this (the chemical engineer’s) profession can be attained only through the analysis of processes into the unit actions as they are carried out on the commercial scale under the conditions imposed by practice.

It also went on to state that:

Chemical Engineering, as distinguished from the aggregate number of subjects comprised in courses of that name, is not a composite of chemistry and mechanical and civil engineering, but is itself a branch of engineering…

A time line diagram of the history of chemical engineering between the profession’s founding to the present day is shown in Figure 1.1.(3) As can be seen from the time line, the profession has reached a crossroads regarding the future education/curriculum for chemical engineers. This is highlighted by the differences of Transport Phenomena and Unit Operations, a topic that is treated in the next chapter.

Figure 1.1 Chemical engineering time line.(3)


1. P. ABULENCIA and L. THEODORE, “Fluid Flow for the Practicing Engineer,” John Wiley & Sons, Hoboken, NJ, 2009.

2. L. THEODORE, “Heat Transfer for the Practicing Engineer” John Wiley & Sons, Hoboken, NJ, 2011 (in preparation).

3. N. SERINO, “2005 Chemical Engineering 125th Year Anniversary Calendar,” term project, submitted to L. Theodore, 2004.

4. R. BIRD, W. STEWART, and E. LIGHTFOOT, “Transport Phenomena,” 2nd edition, John Wiley & Sons, Hoboken, NJ, 2002.

NOTE: Additional problems are available for all readers at Follow links for this title. These problems may be used for additional review, homework, and/or exam purposes.

Chapter 2

Transport Phenomena vs Unit Operations Approach

The history of Unit Operations is interesting. As indicated in the previous chapter, chemical engineering courses were originally based on the study of unit processes and/or industrial technologies. However, it soon became apparent that the changes produced in equipment from different industries were similar in nature, i.e., there was a commonality in the mass transfer operations in the petroleum industry as with the utility industry. These similar operations became known as Unit Operations. This approach to chemical engineering was promulgated in the Little report discussed earlier, and has, with varying degrees and emphasis, dominated the profession to this day.

The Unit Operations approach was adopted by the profession soon after its inception. During the 130 years (since 1880) that the profession has been in existence as a branch of engineering, society’s needs have changed tremendously and so has chemical engineering.

The teaching of Unit Operations at the undergraduate level has remained relatively unchanged since the publication of several early- to mid-1900 texts. However, by the middle of the 20th century, there was a slow movement from the unit operation concept to a more theoretical treatment called transport phenomena or, more simply, engineering science. The focal point of this science is the rigorous mathematical description of all physical rate processes in terms of mass, heat, or momentum crossing phase boundaries. This approach took hold of the education/curriculum of the profession with the publication of the first edition of the Bird et al. book.(l) Some, including both authors of this text, feel that this concept set the profession back several decades since graduating chemical engineers, in terms of training, were more applied physicists than traditional chemical engineers. There has fortunately been a return to the traditional approach to chemical engineering, primarily as a result of the efforts of ABET (Accreditation Board for Engineering and Technology). Detractors to this pragmatic approach argue that this type of theoretical education experience provides answers to what and how, but not necessarily why, i.e., it provides a greater understanding of both fundamental physical and chemical processes. However, in terms of reality, nearly all chemical engineers are now presently involved with the why questions. Therefore, material normally covered here has been replaced, in part, with a new emphasis on solving design and open-ended problems; this approach is emphasized in this text.

The following paragraphs attempt to qualitatively describe the differences between the above two approaches. Both deal with the transfer of certain quantities (momentum, energy, and mass) from one point in a system to another. There are three basic transport mechanisms which can potentially be involved in a process. They are:

1 Radiation

2 Convection

3 Molecular Diffusion

The first mechanism, radiative transfer, arises as a result of wave motion and is not considered, since it may be justifiably neglected in most engineering applications. The second mechanism, convective transfer, occurs simply because of bulk motion. The final mechanism, molecular diffusion, can be defined as the transport mechanism arising as a result of gradients. For example, momentum is transferred in the presence of a velocity gradient; energy in the form of heat is transferred because of a temperature gradient; and, mass is transferred in the presence of a concentration gradient. These molecular diffusion effects are described by phenomenological laws.(1)

Momentum, energy, and mass are all conserved. As such, each quantity obeys the conservation law within a system:


This equation may also be written on a time rate basis:


The conservation law may be applied at the macroscopic, microscopic, or molecular level.

One can best illustrate the differences in these methods with an example. Consider a system in which a fluid is flowing through a cylindrical tube (see Fig. 2.1) and define the system as the fluid contained within the tube between points 1 and 2 at any time. If one is interested in determining changes occurring at the inlet and outlet of a system, the conservation law is applied on a “macroscopic” level to the entire system. The resultant equation (usually algebraic) describes the overall changes occurring to the system (or equipment). This approach is usually applied in the Unit Operation (or its equivalent) courses, an approach which is highlighted in this text and its two companion texts.(2,3)

Figure 2.1 Flow system.

In the microscopic/transport phenomena approach, detailed information concerning the behavior within a system is required; this is occasionally requested of and by the engineer. The conservation law is then applied to a differential element within the system that is large compared to an individual molecule, but small compared to the entire system. The resulting differential equation is then expanded via an integration in order to describe the behavior of the entire system.

The molecular approach involves the application of the conservation laws to individual molecules. This leads to a study of statistical and quantum mechanics—both of which are beyond the scope of this text. In any case, the description at the molecular level is of little value to the practicing engineer. However, the statistical averaging of molecular quantities in either a differential or finite element within a system can lead to a more meaningful description of the behavior of a system.

Both the microscopic and molecular approaches shed light on the physical reasons for the observed macroscopic phenomena. Ultimately, however, for the practicing engineer, these approaches may be valid but are akin to attempting to kill a fly with a machine gun. Developing and solving these equations (in spite of the advent of computer software packages) is typically not worth the trouble.

Traditionally, the applied mathematician has developed differential equations describing the detailed behavior of systems by applying the appropriate conservation law to a differential element or shell within the system. Equations were derived with each new application. The engineer later removed the need for these tedious and error-prone derivations by developing a general set of equations that could be used to describe systems. These have come to be referred to by many as the transport equations. In recent years, the trend toward expressing these equations in vector form has gained momentum (no pun intended). However, the shell-balance approach has been retained in most texts where the equations are presented in componential form, i.e., in three particular coordinate systems—rectangular, cylindrical, and spherical. The componential terms can be “lumped” together to produce a more concise equation in vector form. The vector equation can be, in turn, re-expanded into other coordinate systems. This information is available in the literature.(1,4)


Explain why the practicing engineer/scientist invariably employs the macroscopic approach in the solution of real world problems.

SOLUTION: The macroscopic approach involves examining the relationship between changes occurring at the inlet and the outlet of a system. This approach attempts to identify and solve problems found in the real world, and is more straightforward than and preferable to the more involved microscopic approach. The microscopic approach, which requires an understanding of all internal variations taking place within the system that can lead up to an overall system result, simply may not be necessary.


1. R. BIRD, W. STEWART, and E. LIGHTFOOT, “Transport Phenomena,” John Wiley & Sons, Hoboken, NJ, 1960.

2. L. THEODORE, “Heat Transfer for the Practicing Engineer” John Wiley & Sons, Hoboken, NJ, 2011 (in preparation).

3. P. ABULENCIA and L. THEODORE, “Fluid Flow for the Practicing Engineer” John Wiley & Sons, Hoboken, NJ, 2009.

4. L. THEODORE, “Introduction to Transport Phenomena” International Textbook Co., Scranton, PA, 1970.

NOTE: Additional problems are available for all readers at Follow links for this title. These problems may be used for additional review, homework, and/or exam purposes.

Chapter 3

Basic Calculations


This chapter provides a review of basic calculations and the fundamentals of measurement. Four topics receive treatment:

1 Units and Dimensions

2 Conversion of Units

3 The Gravitational Constant, gc

4 Significant Figures and Scientific Notation

The reader is directed to the literature in the Reference section of this chapter(1–3) for additional information on these four topics.


The units used in this text are consistent with those adopted by the engineering profession in the United States. For engineering work, SI (Système International) and English units are most often employed. In the United States, the English engineering units are generally used, although efforts are still underway to obtain universal adoption of SI units for all engineering and science applications. The SI units have the advantage of being based on the decimal system, which allows for more convenient conversion of units within the system. There are other systems of units; some of the more common of these are shown in Table 3.1. Although English engineering units will primarily be used, Tables 3.2 and 3.3 present units for both the English and SI systems, respectively. Some of the more common prefixes for SI units are given in Table 3.4 (see also Appendix A.5) and the decimal equivalents are provided in Table 3.5. Conversion factors between SI and English units and additional details on the SI system are provided in Appendices A and B.

Table 3.1 Common Systems of Units

Table 3.2 English Engineering Units

Physical quantity Name of unit Symbol for unit
Length foot ft
Time second, minute, hour s, min, h
Mass pound (mass) lb
Temperature degree Rankine °R
Temperature (alternative) degree Fahrenheit °F
Moles pound mole lbmol
Energy British thermal unit Btu
Energy (alternative) horsepower ˙ hour hp ˙ h
Force pound (force) lbf
Acceleration foot per second square ft/s2
Velocity foot per second ft/s
Volume cubic foot ft3
Area square foot ft2
Frequency cycles per second, Hertz cycles/s, Hz
Power horsepower, Btu per second hp, Btu/s
Heat capacity British thermal unit per (pound mass ˙ degree Rankine) Btu/lb ˙ °R
Density pound (mass) per cubic foot lb/ft3
Pressure pound (force) per square inch psi
pound (force) per square foot psf
atmospheres atm
bar bar

Table 3.3 SI Units

Physical unit Name of unit Symbol for unit
Length meter m
Mass kilogram, gram kg, g
Time second s
Temperature Kelvin K
Temperature (alternative) degree Celsius °C
Moles gram mole gmol
Energy Joule J, kg ˙ m2/s2
Force Newton N, kg ˙ m/s2, J/m
Acceleration meters per second squared m/s
Pressure Pascal, Newton per square meter Pa, N/m2
Pressure (alternative) bar bar
Velocity meters per second m/s
Volume cubic meters, liters m3, L
Area square meters m2
Frequency Hertz Hz, cycles/s
Power Watt W, kg ˙ m2 ˙ s3, J/s
Heat capacity Joule per kilogram ˙ Kelvin J/kg ˙ K
Density kilogram per cubic meter kg/m3
Angular velocity radians per second rad/s

Table 3.4 Prefixes for SI Units

Multiplication factors Prefix Symbol
1,000,000,000,000,000,000 = 1018 exa E
1,000,000,000,000,000 = 1015 peta P
1,000,000,000,000 = 1012 tera T
1,000,000,000 = 109 giga G
1,000,000 = 106 mega M
1,000 = 103 kilo k
100 = 102 hecto h
10 = 101 deka da
0.1 = 10−1 deci d
0.01 = 10−2 centi c
0.001 = 10−3 milli m
0.000 001 = 10−6 micro μ
0.000 000 001 = 10−9 nano n
0.000 000 000 001 = 10−12 pico p
0.000 000 000 000 001 = 10−15 femto f
0.000 000 000 000 000 001 = 10−18 atto a

Table 3.5 Decimal Equivalents

Inch in fractions Decimal equivalent Millimeter equivalent
A. 4ths and 8ths
1/8 0.125 3.175
1/4 0.250 6.350
3/8 0.375 9.525
1/2 0.500 12.700
5/8 0.625 15.875
3/4 0.750 19.050
7/8 0.875 22.225
B. 16ths
1/16 0.0625 1.588
3/16 0.1875 4.763
5/16 0.3125 7.938
7/16 0.4375 11.113
9/16 0.5625 14.288
11/16 0.6875 17.463
13/16 0.8125 20.638
15/16 0.9375 23.813
C. 32nds
1/32 0.03125 0.794
3/32 0.09375 2.381
5/32 0.15625 3.969
7/32 0.21875 5.556
9/32 0.28125 7.144
11/32 0.34375 8.731
13/32 0.40625 10.319
15/32 0.46875 11.906
17/32 0.53125 13.494
19/32 0.59375 15.081
21/32 0.65625 16.669
23/32 0.71875 18.256
25/32 0.78125 19.844
27/32 0.84375 21.431
29/32 0.90625 23.019
31/32 0.96875 24.606

Two units that appear in dated literature are the poundal and slug. By definition, one poundal force will give a 1 pound mass an acceleration of 1 ft/s2. Alternatively, 1 slug is defined as the mass that will accelerate 1 ft/s2 when acted upon by a 1 pound force; thus, a slug is equal to 32.2 pounds mass.


Converting a measurement from one unit to another can conveniently be accomplished by using unit conversion factors; these factors are obtained from a simple equation that relates the two units numerically. For example, from

(3.1) equation

the following conversion factor can be obtained:

(3.2) equation

Since this factor is equal to unity, multiplying some quantity (e.g., 18 ft) by this factor cannot alter its value. Hence

(3.3) equation

Note that in Equation (3.3), the old units of feet on the left-hand side cancel out leaving only the desired units of inches.

Physical equations must be dimensionally consistent. For the equality to hold, each additive term in the equation must have the same dimensions. This condition can be and should be checked when solving engineering problems. Throughout the text, great care is exercised in maintaining the dimensional formulas of all terms and the dimensional homogeneity of each equation. Equations will generally be developed in terms of specific units rather than general dimensions, e.g., feet, rather than length. This approach should help the reader to more easily attach physical significance to the equations presented in these chapters.

Consider now the example of calculating the perimeter, P, of a rectangle with length, L, and height, H. Mathematically, this may be expressed as P = 2L + 2H. This is about as simple a mathematical equation that one can find. However, it only applies when P, L, and H are expressed in the same units.

Terms in equations must be consistent from a “magnitude” viewpoint.(3) Differential terms cannot be equated with finite or integral terms. Care should also be exercised in solving differential equations. In order to solve differential equations to obtain a description of the pressure, temperature, composition, etc., of a system, it is necessary to specify boundary and/or initial conditions (B a/o IC) for the system. This information arises from a description of the problem or the physical situation. The number of boundary conditions (BC) that must be specified is the sum of the highest order derivative for each independent differential equation. A value of the solution on the boundary of the system is one type of boundary condition. The number of initial conditions (IC) that must be specified is the highest order time derivative appearing in the differential equation. The value for the solution at time equal to zero constitutes an initial condition. For example, the equation

(3.4) equation

requires 2 BCs (in terms of the position variable z). The equation

(3.5) equation

requires 1 IC. And finally, the equation


requires 1 IC and 2 BCs (in terms of the position variable y).


Convert units of acceleration in cm/s2 to miles/yr2.

SOLUTION: The procedure outlined on the previous page is applied to the units of cm/s2.

Thus, 1.0 cm/s2 is equal to 6.18 × 109 miles/yr2.


The momentum of a system is defined as the product of the mass and velocity of the system:

(3.7) equation

A commonly employed set of units for momentum are therefore lb ˙ ft/s. The units of the time rate of change of momentum (hereafter referred to as rate of momentum) are simply the units of momentum divided by time, i.e.,

(3.8) equation

The above units can be converted to units of pound force (lbf) if multiplied by an appropriate constant. As noted earlier, a conversion constant is a term that is used to obtain units in a more convenient form; all conversion constants have magnitude and units in the term, but can also be shown to be equal to 1.0 (unity) with no units (i.e., dimensionless).

A defining equation is

(3.9) equation

If this equation is divided by lbf, one obtains

(3.10) equation

This serves to define the conversion constant gc. If the rate of momentum is divided by gc as 32.2 lb ˙ ft/lbf ˙ s2—this operation being equivalent to dividing by 1.0—the following units result:


One can conclude from the above dimensional analysis that a force is equivalent to a rate of momentum.


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 inches implies that the actual quantity is somewhere between 8.315 and 8.325 inches. This applies to calculated and measured quantities; quantities that are known exactly (e.g., pure integers) have an infinite number of significant figures.

The significant digits of a number are the digits from the first nonzero digit on the left to either (a) the last digit (whether it is nonzero or zero) on the right if there is a decimal point, or (b) the last nonzero digit of the number if there is no decimal point. For example:

370 has 2 significant figures
370. has 3 significant figures
370.0 has 4 significant figures
28,070 has 4 significant figures
0.037 has 2 significant figures
0.0370 has 3 significant figures
0.02807 has 4 significant figures

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 numbers is 370, which has its last significant digit in the tens position. The answer should also have its last significant digit in the tens position.

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

In the process of performing engineering calculations, very large and very small numbers 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 (< 10 but > or = 1) 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.


1. D. GREEN and R. PERRY (eds), “Perry’s Chemical Engineers’ Handbook,” 8th edition. McGraw-Hill, New York City, NY, 2008.

2. J. REYNOLDS, J. JERIS, and L. THEODORE, “Handbook of Chemical and Environmental Engineering Calculations,” John Wiley & Sons, Hoboken, NJ, 2004.

3. J. SANTOLERI, J. REYNOLDS, and L. THEODORE, “Introduction to Hazardous Waste Incineration,” 2nd edition, John Wiley & Sons, Hoboken, NJ, 2000.

NOTE: Additional problems are available for all readers at Follow links for this title. These problems may be used for additional review, homework, and/or exam purposes.