Math Concepts for Food Engineering

234 Pages·2008·5.463 MB·\234

by  Richard W. Hartel (Author), D.B. Hyslop (Author), D.B. Hyslop (Author), T.A. Howell Jr. (Author)

Math Concepts
for
Food Engineering
Second Edition
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Math Concepts
for
Food Engineering
Second Edition
Richard W. Hartel
Robin K. Connelly
Terry A. Howell, Jr.
Douglas B. Hyslop
Boca Raton London New York
CRC Press is an imprint of the
Taylor & Francis Group, an informa business
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CRC Press
Taylor & Francis Group
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Boca Raton, FL 33487-2742
© 2008 by Taylor and Francis Group, LLC
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Contents
Chapter 1 Algebra ...............................................................................................1
1.1 Variables and constants ......................................................................1
1.1.1 Variables .....................................................................................1
1.1.2 Constants ....................................................................................2
1.2 Equations ...............................................................................................2
1.2.1 Functions ....................................................................................2
1.2.2 Manipulation of equations ......................................................3
1.2.3 Rules of equations applied to engineering units ..................6
1.3 Linear and nonlinear equations ........................................................8
1.4 Multiple linear equations ..................................................................10
1.4.1 Algebraic manipulation .........................................................10
1.4.2 Substitution ..............................................................................13
1.4.3 Matrices (Kramer’s rule).........................................................15
1.4.3.1 Addition and subtraction ........................................16
1.4.3.2 Multiplication ...........................................................17
Chapter 2 Interpolation of data in tables and charts ................................27
2.1 Single interpolation ...........................................................................27
2.2 Double interpolation .........................................................................31
2.3 Interpolation in charts .......................................................................36
Chapter 3 Graphs and curve fitting ..............................................................39
3.1 Coordinate systems ...........................................................................39
3.1.1 Cartesian coordinates .............................................................39
3.1.2 Polar coordinates and cylindrical coordinates ...................41
3.2 Linear graphs......................................................................................44
3.3 Logarithmic transformations ...........................................................49
3.3.1 Semi-log graphs .......................................................................52
3.3.2 Log-log graphs .........................................................................54
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Chapter 4 Calculus ...........................................................................................57
4.1 Differentiation ....................................................................................57
4.1.1 Derivatives ...............................................................................57
4.1.2 Formal differentiation ............................................................61
4.1.3 Rules of differentiation ...........................................................62
4.1.4 Second derivatives ..................................................................64
4.1.5 Partial derivatives ...................................................................64
4.1.6 Applications .............................................................................65
4.1.6.1 Time derivatives .......................................................65
4.1.6.2 Maxima and minima ...............................................65
4.2 Integration ...........................................................................................67
4.2.1 The antiderivative ...................................................................67
4.2.2 Formal integration ..................................................................68
4.2.3 Rules of integration .................................................................69
4.2.4 Closed integrals .......................................................................70
4.2.5 Trapezoidal rule ......................................................................71
4.2.6 Areas by integration ...............................................................72
4.3 Differential equations .......................................................................73
Chapter 5 Problem solving .............................................................................77
Chapter 6 Gases and vapors ...........................................................................83
6.1 Pressure ...............................................................................................83
6.2 Gas laws...............................................................................................86
6.3 Gas mixtures.......................................................................................89
6.3.1 Partial pressure .......................................................................89
6.3.2 Partial volume .........................................................................90
Chapter 7 Mass balances .................................................................................93
7.1 Steady-state operation, no reaction .................................................95
7.2 Steady-state operation, with chemical reaction ...........................105
7.3 Unsteady-state operation, no reaction ..........................................108
Chapter 8 Energy balances ...........................................................................111
8.1 Steam tables ......................................................................................113
8.1.1 Saturated steam .....................................................................113
8.1.2 Superheated steam ................................................................117
8.2 Enthalpy balances ............................................................................120
Chapter 9 Fluid mechanics ...........................................................................127
9.1 Rheology ...........................................................................................127
9.1.1 Modeling rheological behavior of fluid foods ..................128
9.1.2 Measurement of rheological properties .............................131
9.2 Fluid flow ..........................................................................................135
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9.2.1 Continuity equation ..............................................................135
9.2.2 Determination of flow regime .............................................137
9.2.3 Flow of a Newtonian fluid in a pipe ..................................138
9.2.4 Effects of friction on fluid flow ...........................................143
9.2.5 Mechanical energy balance equation in fluid flow ..........146
9.3 Non-Newtonian fluid flow .............................................................149
References ...................................................................................................151
Chapter 10 Heat transfer ................................................................................153
10.1 Steady state .......................................................................................154
10.1.1 Conduction heat transfer ...................................................154
10.1.2 Convection heat transfer ....................................................160
10.1.3 Convective heat-transfer coefficients ...............................166
10.1.4 Heat exchangers ..................................................................170
10.2 Unsteady-state heat transfer ...........................................................174
10.3 Freezing .............................................................................................179
10.4 Radiation heat transfer ....................................................................182
Chapter 11 Mass transfer ...............................................................................185
11.1 Psychrometrics .................................................................................185
11.1.1 Moist-air properties using thermodynamic
principles and the gas laws................................................186
11.1.2 Moist-air properties on psychrometric chart ..................187
11.1.3 Using the psychrometric chart to find
the state of air during processing .....................................189
11.2 Molecular diffusion .........................................................................191
11.3 Convective mass transfer ................................................................196
11.4 Unsteady-state mass transfer .........................................................199
References ....................................................................................................202
Appendix 1: Common conversion factors for engineering units ..........203
Appendix 2: Answers to practice problems ...............................................205
Appendix 3a: Quiz ..........................................................................................209
Appendix 3b: Quiz with answers ................................................................215
Index .....................................................................................................................219
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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.
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