Math Concepts for Food Engineering Second Edition 55054_C000.indd 2 2/6/08 2:50:21 PM Math Concepts for Food Engineering Second Edition Richard W. Hyslop Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Group, an informa business 55054_C000.indd 3 2/6/08 2:50:22 PM CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2008 by Taylor and Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-13: 978-1-4200-5506-1 (Ebook-PDF) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use.
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For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.com and the CRC Press Web site at http://www.com Contents Chapter 1 Algebra.1 Variables and constants.2 Manipulation of equations.3 Rules of equations applied to engineering units.3 Linear and nonlinear equations.4 Multiple linear equations.1 Addition and subtraction. 17 Chapter 2 Interpolation of data in tables and charts.3 Interpolation in charts.
36 Chapter 3 Graphs and curve fitting.2 Polar coordinates and cylindrical coordinates.1 Semi-log graphs.2 Log-log graphs.indd 5 2/6/08 2:50:23 PM Chapter 4 Calculus.3 Rules of differentiation.2 Maxima and minima.3 Rules of integration.6 Areas by integration. 73 Chapter 5 Problem solving.77 Chapter 6 Gases and vapors.90 Chapter 7 Mass balances.1 Steady-state operation, no reaction.2 Steady-state operation, with chemical reaction.3 Unsteady-state operation, no reaction. 108 Chapter 8 Energy balances. 120 Chapter 9 Fluid mechanics.1 Modeling rheological behavior of fluid foods.2 Measurement of rheological properties.2 Determination of flow regime.3 Flow of a Newtonian fluid in a pipe.4 Effects of friction on fluid flow.5 Mechanical energy balance equation in fluid flow.3 Non-Newtonian fluid flow.
151 Chapter 10 Heat transfer.1 Conduction heat transfer.2 Convection heat transfer.3 Convective heat-transfer coefficients.2 Unsteady-state heat transfer.4 Radiation heat transfer. 182 Chapter 11 Mass transfer.1 Moist-air properties using thermodynamic principles and the gas laws.2 Moist-air properties on psychrometric chart.3 Using the psychrometric chart to find the state of air during processing.3 Convective mass transfer.4 Unsteady-state mass transfer. 202 Appendix 1: Common conversion factors for engineering units.203 Appendix 2: Answers to practice problems.209 Appendix 3b: Quiz with answers.indd 8 2/6/08 2:50:23 PM Preface to Second Edition For over 10 years now, we have been assessing the math skills of students entering the first food engineering course in our curriculum. There is a clear correlation between these skills and the student’s ability to do well in this course.
Students who enter with good math skills generally do well in engi- neering class. However, those students who need extra help with math can make up the gap through hard work and practice. This is the advantage that Math Concepts for Food Engineering brings to those students willing to work at improving their math skills. For this second edition, we have incorporated some simple food engineer- ing principles within the text.
Without going into the detail of a food engi- neering textbook, some of the more important technical principles have been included relative to the learning outcomes for our food engineering class. We feel that this will give our students a better perspective of the importance of the math skills, and help them better relate these simple problems with the principles they are learning in class. In this second edition, we have also made several other additions. First, we have incorporated various exercises throughout the text that use spread- sheets, a valuable tool for analyzing and manipulating data.
The use of spreadsheets to create mathematical tools of practical use for some applica- tions is developed in chapters 1 through 5, and these are used to help solve some of the examples in chapters 6 through 11, the second part of the text. The publisher will make the spreadsheet exercises seen in the book available on its Web site for those who purchase the book. We have also included a chapter on mass transfer, and added a simple units conversion page in the appendix. This offers a more complete reference for our students by provid- ing complete coverage of basic balance and transport principles used in food engineering.indd 9 2/6/08 2:50:24 PM Math Concepts for Food Engineering, second edition, is still intended as a supplemental reference to a standard textbook for a food engineering class.
Its purpose is to provide practice and experience in solving simple engineering problems so that students are better prepared to face the more rigorous problems presented in class.indd 10 2/6/08 2:50:24 PM The Authors Richard W. Hartel, PhD, currently a professor of food engineering, has been with the Department of Food Science at the University of Wisconsin–Madison since 1986. Hartel conducts a research program focused on crystallization and phase transitions in foods, including studies on ice crystallization, sugar crystallization and glass transition, and lipid-phase behavior and crystalli- zation. He teaches courses in food processing, phase transitions, and candy science.
Robin Kay Connelly, PhD, is currently an assistant professor of food engi- neering with dual appointments with the Departments of Food Science and Biological Systems Engineering at the University of Wisconsin–Madison since 2003. Connelly conducts research in the areas of computational fluid dynamics, simulation of food processing operations, and food rheology, with an emphasis on mixing applications. She teaches the introductory food engineering principles course for food scientists and biological concepts for engineers., received his BS and MS from Texas A&M University in agricultural engineering, with a PhD from the University of Wiscon- sin–Madison in food engineering. He served on the University of Arkansas Food Science faculty for three years, and now works for McKee Foods Corp.
(Collegedale, TN) as manager of research and new technology. Hyslop received his PhD in food chemistry in 1978 from the University of Wisconsin–Madison with an emphasis on physical surface and colloid chemistry. Since joining the staff at Madison in 1980, he has taught numerous food science courses, including food analysis, food chemistry, food physical chemistry, food colloid chemistry, food processing, and food engineering.indd 12 2/6/08 2:50:24 PM Introduction Mathematical reasoning is an important skill for a food scientist, but it is a skill that is not shared by all students at the same level. Although all food science students are required to have a comprehensive course in calculus (both integral and differential), often, for one reason or another, mathemati- cal reasoning skills have been lost or forgotten by the time a student reaches the food engineering course.
In the spirit of the Institute of Food Technologists education standards, it makes sense to assess a student’s mathematical reasoning skills prior to starting the food engineering course. The intention of this book is to help with that assessment and then to provide assistance for those students who need to brush up on their mathematical skills. The book is organized into sections that present different materials needed for developing mathematical reasoning skills. The first section (chapters 1–4) covers important mathematical skills needed by students in a food engineer- ing class.
These principles are primarily a review of previous math classes (algebra, calculus, etc. This preliminary section is followed by an important chapter on problem solving (chapter 5). The remaining chapters (6–11) cover food engineering topics likely to be found in a food engineering course for food scientists. In the back of the book is a short quiz, the screening test (appendix 3a), which can be used to assess mathematical reasoning skills prior to the start of a food engineering course.
Students are asked to read through the intro- ductory math sections to refresh their memory on the important concepts prior to taking the screening test. For those students who score low on the screening test or simply want to improve their quantitative skills, the prob- lems provided in Math Concepts for Food Engineering are intended to build mathematical confidence, as well as to bridge the simplest math concepts and the more complex engineering principles. As the semester unfolds, students should progress through chapters 6–11 as each topic is covered in class. Students should study each worked problem to make sure they under- stand the mathematical (and engineering) principles being demonstrated, and then work independently on the accompanying practice problems.
To help students apply the principles in Math Concepts for Food Engineering to 55054_C000.indd 13 2/6/08 2:50:24 PM food engineering class, this second edition has included a brief coverage of engineering principles important for the sample problems as a supplement to the main Food Engineering textbook. The following approach is recommended for working through this book: • Review chapters 1–4 to refresh your knowledge of some important math skills. • Take the screening test at the back of the book in appendix 3a. • Grade yourself with the answer key in appendix 3b (after you have completed the entire exam).
• Assess which mathematical skill areas you struggled with. • Review appropriate materials in chapters 1–4 based on your skills assessment. • Carefully read through the problem-solving approach in chapter 5. Consider your own approach to mathematical problem solving to see where you might find ways to improve.
For example, did you gener- ally follow the steps outlined in chapter 5 when you took the screen- ing test? If not, what might you do differently? • Work your way through chapters 6 to 11 as each section is covered in a food engineering class. We recommend that you work through the appropriate chapter during the first day or two as each topic is covered in class. In that way, you will be well prepared to solve the more complex problems required in class. If you get stuck on a prac- tice problem, be sure to meet with your instructor for assistance.
Although the answers for these practice problems are given in the back of the book, seeking help from your instructor can help you see where you are getting stuck. In our experience, those students who score low on the screening test but work hard at mastering the mathematical principles covered in Math Con- cepts for Food Engineering go on to do well in food engineering classes. As with most things, the likelihood of success is enhanced when the student is willing to put lots of hard work into learning the material. We hope that Math Concepts for Food Engineering provides a resource for you to improve your mathematical reasoning skills and, thereby, to attain greater success in learning food engineering principles.