This page intentionally left blank

Relative Atomic Mass Values and Atomic Numbers

Atomic mass values apply to naturally-occurring isotopes based on the atomic mass of1 2C=12

Element Actinium Aluminum Americium Antimony Argon Arsenic Astatine Barium Beryllium Bismuth Boron Bromine Cadmium Calcium Carbon Cerium Cesium Chlorine Chromium Cobalt Copper Dysprosium Erbium Europium Fluorine Francium Gadolinium Gallium Germanium Gold Hafnium Helium Holmium Hydrogen Indium Iodine Iridium Iron Krypton Lanthanum

Symbol Ac Al

Am Sb Ar As At Ba Be Bi В Br Cd Ca С Ce Cs CI Cr Co Cu Dy Er Eu F Fr Gd Ga Ge Au Hf He Ho H In I Ir Fe Kr La

Atomic Number

89 13 95 51 18 33 85 56 4

83 5

35 48 20

6 58 55 17 24 27 29 66 68 63

9 87 64 31 32 79 72 2

67 1

49 53 77 26 36 57

Atomic Mass [227]

26.9815 [243]

121.7601 39.9481 74.9216 [210]

137.3277 9.0122

208.9804 10.812

79.9041 112.4118 40.0784 12.0108

140.1161 132.9055 35.4532 51.9962 58.9332 63.5463

162.5001 167.2593 151.9641 18.9984 [223]

157.253 69.7231 72.641

196.9666 178.492 4.0026

164.9303 1.00795

114.8183 126.9045 192.2173 55.8452 83.7982

138.9055

Element Lead Lithium Lutetium Platinum Plutonium Polonium Potassium Praseodymium Promethium Protactinium Radium Radon Rhenium Rhodium Rubidium Ruthenium Samarium Scandium Selenium Silicon Silver Sodium Strontium Sulfur Tantalum Technetium Tellurium Thallium Terbium Thorium Thulium Tin Titanium Tungsten Uranium Vanadium Xenon Ytterbium Yttrium Zinc Zirconium

Symbol РЬ Li Lu Pt Pu Po К Pr Pm Pa Ra Rn Re Rh Rb Ru Sm Sc Se Si Ag Na Sr S Та Тс Те TI Tb Th Tm Sn Ti w и V Xe Yb Y Zn Zr

Atomic Number

82 3

71 78 94 84 19 59 61 91 88 86 75 45 37 44 62 21 34 14 47 11 38 16 73 43 52 81 65 90 69 50 22 74 92 23 54 70 39 30 40

Atomic Mass 207.21 6.9412

174.9671 195.0782

[244] [209]

39.0983 140.9077

[145] 231.0359

[226] [222]

186.2071 102.9055 85.4678 101.072 150.363 44.9559 78.963 28.0855

107.8682 22.9898 87.621 32.0655

180.9479 [98]

127.603 204.3833 158.9253 232.0381 168.9342 118.7107 47.8671 183.841

238.0289 50.9415

131.2936 173.043 88.9059 65.4094 91.2242

Summary Descriptions of Nine Quantities that are Quotients of Amount of Substance, Volume, or Mass

Quantity in numerator

и о

fi ε о fi

fi fi

О

Amount of substance

Symbol: n

SI unit: mol

Volume

Symbol: V

SI unit: m

Mass

Symbol: m

SI unit: kg

Amount of substance

Symbol: n

SI unit: mol

Volume

Symbol: V

SI unit: m

Mass

Symbol: m

SI unit: kg

amount-of-substance fraction

xB = nB/n

SI unit: mol/mol = 1

amount-of-substance concentration

C B = H B / F

SI unit: mol/m

molality

bB = nB/mA

SI unit: mols/kg

molar volume

Vm = Win

SI unit: m /mol

volume fraction * х в Vm,B

Фв - * Σ *AVm,A..

SI unit: m /m = 1

specific volume

v = V/m

SI unit: m /kg

molar mass

M = т /л

SI unit: kg/mol

mass density

p = m/V

SI unit: kg/m

mass fraction

wB = mB/m

SI unit: kg/kg = 1

The volume fraction фв refers to substance В in a mixture of А, В, С, etc., and V* refers to the molar volume of the pure substance.

The concentration of components is based on mass or amount of substance per unit volume. The above diagram specifies these as mass density (kg/m3), and amount-of-substance concentration (moles per m3). The composition of the components is defined as a part-per-part dimension. These are specified as mass fraction (kilograms of component В divided by the total kilograms), amount-of-substance fraction (moles of component В divided by the total number of moles), molality (moles of solute В in solution divided by the mass of the solvent in kg), or volume fraction (m3 of component В divided by the total m3).

In SI, the amount-of-substance fraction term replaces the mole fraction term. Since mole fraction is still in common use, this Handbook will use both terms.

Conversion Factors and Units to SI*

To convert from

LENGTH mile foot inch angstrom (A)

VOLUME cubic yard barrel (petroleum, 42 U.S. gal.) m cubic foot gallon (U.S.) quart (U.S., liquid) fluid ounce (U.S.)

TIME year (tropical) year (365 days) day hour

MASS ton (long) ton (short) tonne pound (avoirdupois) ounce (troy) ounce (avoirdupois) grain

FORCE ton (short)-force kilogram-force pound-force poundal dyne

to

m m m m

m 3

m m3

m3

m3

m3

™ 3

m

s s s s

kg kg kg kg kg kg kg

N N N N N

Multiply by

1.609 344 3.048 2.54 1.0

7.645 549 1.589 873 2.831 685 3.785412 9.463 529 2.957 353

3.155 693 3.1536 8.64 3.6

1.016 047 9.071 847 1.0 4.535 924 3.110 348 2.834 952 6.479 891

8.896 443 9.806 65 4.448 222 1.382 550 1.0

E+03 E-01 E-02 E-10

E-01 E-01 E-02 E-03 E-04 E-05

E+07 E+07 E+04 E+03

E+03 E+02 E+03 E-01 E-02 E-02 E-05

E+03 E+00 E+00 E-01 E-05

* Exact conversion factors indicated in bold font.

To convert from to Multiply by

PRESSURE atmosphere (standard) bar psi inch of Hg (conventional) foot of water (conventional) inch of water (conventional) mm Hg (torr) (conventional) micron of Hg (conventional)

TEMPERATURE degree Celsius degree Fahrenheit degree Rankine

ENERGY AND WORK therm (U.S.) kilowatt hour kilocalorieth British thermal unitth horsepower hour (electric) liter atmosphere foot pound-force erg

POWER horsepower (electric) foot pound-force/sec kilocalorieth/h British thermal unitth/hr

1.013 25 1.0 6.894 757 3.386 389 2.989 067 2.490 889 1.333 224 1.333 224

T,K = t,

E+05 E+05 E+03 E+03 E+03 E+02 E+02 E-01

°C + 273.15 T, К = (t, °F + 459.67/1.8

T, K = = (T, R)/1.8

w w w w

1.054 804 3.6 4.184 1.054 350 2.6856 1.013 250 1.355 818 1.0

7.46 1.355 818 1.162 222 2.928 750

E+08 E+06 E+03 E+03 E+06 E+02 E+00 E-07

E+02 E+00 E+00 E-01

THE GAS CONSTANT R 8.314 47 J/(mol · K); m3 · Pa/(mol · K) 1.987 21 cal(mol · K); BTU/(lb-mol · °R) 0.730 241 ft3 · atm/(lb-mol · °R) 0.0820 575 liter · atm/(mol · K)

HANDBOOK ON MATERIAL AND ENERGY BALANCE CALCULATIONS IN MATERIALS PROCESSING

This page intentionally left blank

HANDBOOK ON MATERIAL AND ENERGY BALANCE CALCULATIONS IN MATERIALS PROCESSING

THIRD EDITION

Arthur E. Morris Gordon Geiger H. Alan Fine

TIMS )WILEY

A JOHN WILEY & SONS, INC., PUBLICATION

Copyright © 2011 by The Minerals, Metals, & Materials Society. All rights reserved

Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada.

No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permission.

Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages.

For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002.

Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic formats. For more information about Wiley products, visit our web site at www.wiley.com.

Library of Congress Cataloging-in-Publication Data:

Morris, Arthur E., 1935-Handbook on material and energy balance calculations in material processing / Arthur E. Morris, Gordon Geiger, H. Alan Fine. — 3rd ed.

p. cm. Rev. ed. of: Handbook on material and energy balance calculations in metallurgical processes. 1979.

Includes bibliographical references and index. ISBN 978-1-118-06565-5 (hardback) 1. Chemical processes—Mathematical models—Handbooks, manuals, etc. 2. Manufacturing processes—Mathematical models—Handbooks, manuals, etc.

3. Chemical processes—Mathematical models—Handbooks, manuals, etc. 4. Materials—Handbooks, manuals, etc. 5. Phase rule and equilibrium— Handbooks, manuals, etc. 6. Heat balance (Engineering)—Mathematics—Handbooks, manuals, etc. 7. Conservation laws (Physics)—Mathematics— Handbooks, manuals, etc. I. Fine, H. Alan. II. Geiger, Gordon Harold, 1937- III. Fine, H. Alan. Handbook on material and energy balance calculations in metallurgical processes. IV. Title.

TP155.7.M66 2011 660'.28—dc22 2011010947

Printed in Singapore.

10 9 8 7 6 5 4 3 2 1

Preface to the First Edition xiv

Preface to the Third Edition xvii

Acknowledgements xix

Chapter 1. Dimensions, Units, and Conversion Factors 1 1.1 The SI System of Units 1

1.1.1 Derived Units 2 1.1.2 Units Outside the SI 3 1.1.3 Comments on Some Quantities and Their Units 4

1.2 The American Engineering System (AES) of Units 4 1.3 Conversion of Units 6

1.3.1 Conversion Factor Tables 6 1.3.2 The Dimension Table 7 1.3.3 Conversion Equations — Temperature and Pressure 8

1.4 Unit Conversions Using the U-Converter Program 11 1.5 Amount of Substance — the Mole Unit 11 1.6 Density and Concentration 13

1.6.1 Density 13 1.6.2 Composition and Concentration 16 1.6.3 Composition of Gases 18

1.7 Electrical Units 20 1.8 Calculation Guidelines 21 1.9 Summary 22 References and Further Reading 23 Exercises 23

Chapter 2. Thermophysical and Related Properties of Materials 26 2.1 State of a System and Properties of a Substance 26 2.2 The Gibbs Phase Rule 27

2.2.1 Consequences of the Phase Rule for Non-Reactive and Reactive Systems 2.2.2 Application of the Phase Rule to One-Phase Non-Reactive Systems 28 2.2.3 Application of the Phase Rule to Multi-Phase Non-Reactive Systems 28 2.2.4 Application of the Phase Rule to Reactive Systems 29

2.3 The Gas Phase 30 2.3.1 The Ideal Gas Law 30 2.3.2 Non-Ideal Gas Behavior 32

2.4 Condensed Phases 34 2.5 Vapor-Liquid Equilibrium (VLE) 35

2.5.1 Mixtures of Condensable and Non-Condensable Gases 3 8 2.5.2 Software for Making Dew Point and Humidity Calculations 39

2.6 Effect of Pressure on Phase Transformation Temperatures 42 2.7 Steam and Air Property Calculators 44

v

Contents

vi Contents

2.8 Properties of Solutions 44 2.8.1 Ideal Solutions — Raoult's Law 44 2.8.2 Non-Ideal Solutions — Activity Coefficients 46 2.8.3 Solutions of Gases in Condensed Phases 47 2.8.4 The Solubility Limit 48 2.8.5 The Solubility of Ionic Species in Water; the Solubility Product 49

2.9 Summary 50 References and Further Reading 51 Exercises 52

Chapter 3. Statistical Concepts Applied to Measurement and Sampling 54 3.1 Basic Statistical Concepts and Descriptive Tools 55

3.1.1 Histograms and Frequency Distributions 56 3.1.2 Mean, Standard Deviation, and Variance 59 3.1.3 Median, Percentile and Quantile 60

3.2 Distributions of Random Variables 62 3.2.1 The Uniform Distribution 62 3.2.2 The Normal Distribution 70

3.3 Basic Applications of Inferential Statistics to Measurement 75 3.3.1 Sampling Distributions of the Mean and the Central Limit Theorem 78 3.3.2 Confidence Intervals 82 3.3.3 Treatment of Errors 86 3.3.4 Error Propagation 92

3.4 Curve Fitting 95 3.4.1 Simple Linear Regression and Excel's Trendline Tool 96 3.4.2 Using Solver to Develop Single-Variable Regression Models 101 3.4.3 Multiple Linear and Non-linear Regression 102 3.4.4 Using Solver and Excel's SSD Tool to Find Equation Coefficients 105 3.4.5 Choosing Among Models 106 3.4.6 Polynomial vs. Rational Function Models 116 3.4.7 Outliers 117 3.4.8 Warnings 119

3.5 Experimental Design 119 3.5.1 Factorial Design 120 3.5.2 Fractional Factorial Design 130

3.6 Summary 137 References and Further Reading 137 Exercises 138

Chapter 4. Fundamentals of Material Balances with Applications to Non-Reacting Systems 144 4.1 System Characteristics 144 4.2 Process Classifications 145 4.3 Flowsheets 146 4.4 The General Balance Equation 150 4.5 Material Balances on Simple Non-Reactive Systems 151 4.6 Strategy for Making Material Balance Calculations 154

4.6.1 Guidelines for Setting up a Materials Balance 155 4.6.2 Guidelines for Resolving a Set of Equations 156 4.6.3 Objectives of a Material Balance 157

Contents vii

4.7 Degree-of-Freedom Analysis 158 4.7.1 DOF Concepts 159 4.7.2 DOF Calculation Strategy for a Single Non-Reactive Device 160 4.7.3 A Washing Process Having Zero Degrees of Freedom 162 4.7.4 A Washing Process Having a DOF = +1 168 4.7.5 A Leaching Process Having a DOF = -1 171

4.8 Using Excel-based Calculational Tools to Solve Equations 174 4.8.1 Goal Seek and Solver as Calculational Aids 174 4.8.2 Software for Conversion of Stream Units: MMV-C 178

4.9 Balances on Systems with Multiple Devices 179 4.10 Extension of Excel's Calculational Tools for Repetitive Solving 194

4.10.1 SuperGS 194 4.10.2 SuperSolver 196

4.11 Special Multiple-Device Configurations I — Recycle and Bypass 197 4.12 Special Multiple-Device Configurations II — Counter-Current Flow 205 4.13 Using FlowBal for Material Balance Calculations 216

4.13.1 FlowBal Example #1: Mixer/Splitter 217 4.13.2 FlowBal Example #2: Evaporation/Condensation Process 219 4.13.3 FlowBal Example #3: Systems with Multi-Phase Streams 221

4.14 Continuous-Mixing Devices 223 4.14.1 Steady-State Processes 223 4.14.2 Unsteady-State Systems 226 4.14.3 Inert Gas Flushing 229

4.15 Graphical Representation of Material Balances 232 4.16 Measures of Performance 232 4.17 Controllers 234 4.18 Summary 239 References and Further Reading 240 Exercises 241

Chapter 5. Stoichiometry and the Chemical Equation 248 5.1 Atomic and Molecular Mass 248 5.2 Composition of Compounds and the Gravimetric Factor 249 5.3 Writing and Balancing Chemical Equations 251

5.3.1 Chemical Reaction Concepts 252 5.3.2 Writing and Balancing Chemical Reactions for Simple Processes 253

5.4 Calculations Involving Excess and Limiting Reactants 256 5.5 Progress of a Reaction 258

5.5.1 Extent of Species Reaction and Rate of Reaction Terminology 258 5.5.2 Chemical Reaction Kinetics 259 5.5.3 Reaction Progress and Кщ 263 5.5.4 Кщ Values from FREED 264 5.5.5 Guidelines for Using Кщ to Determine Maximum Reaction Extent 265 5.5.6 Application of Equilibrium Limitations for Gas-Condensed Phase Reactions 266 5.5.7 Application of Equilibrium Limitations to Gas-Phase Reactions 268

5.6 Practical Indicators of the Progress of Reactions and Processes 269 5.7 Parallel, Sequential and Mixed Reactions 273 5.8 Independence of Chemical Reactions 274

viii Contents

5.9 Practical Examples of Reaction Writing and Stoichiometry 274 5.9.1 Calculations in Gas-Condensed Phase Processes 275 5.9.2 Calculations in Gas-Phase Processes 277

5.10 Use of Chemical Reactions in FlowBal 279 5.10.1 FlowBal's Extent of Reaction Tool 280 5.10.2 FlowBal's Insert Equation Tool 282

5.11 Balancing Aqueous (Ionic) Reactions 283 5.12 Summary 286 References and Further Reading 288 Exercises 288

Chapter 6. Reactive Material Balances 294 6.1 The General Material Balance Procedure for a Reactive System 294

6.1.1 Independent Chemical Reactions, Independent Species, and Independent Elements 295 6.1.2 Molecular Species Material Balance Method 296 6.1.3 Atomic Species Method 300 6.1.4 Atomic and Molecular Species Balance Examples 302

6.2 The Use of Excel-based Computational Tools in Reactive System Balances 306 6.2.1 Application of SuperSolver 306 6.2.2 Reactive System Material Balances Using FlowBal 309

6.3 Combustion Material Balances 321 6.3.1 Material Balance for the Combustion of a Gaseous Fuel 322 6.3.2 Combustion of Liquid Fuels 327 6.3.3 Combustion of Solid Fuels 327 6.3.4 Use of Feed-Forward and Stack Gas Analysis for Combustion Control 329 6.3.5 Use of FlowBal for Combustion Calculations 333 6.3.6 Trace Combustion Products 335

6.4 The Production of a Reducing Gas 340 6.5 Gas-Solid Oxidation-Reduction Processes 345

6.5.1 Oxidation-Reduction During Calcination 345 6.5.2 The Reduction of Iron Ore Concentrate 348 6.5.3 The Chemistry of Fluidized Bed Reduction of Iron Ore by Hydrogen 349 6.5.4 Excel Simulation of the Fluidized Bed Reduction of Hematite 351 6.5.5 FlowBal Simulation of the Fluidized Bed Reduction of Hematite 355 6.5.6 Shaft Furnace Reduction of Iron Ore Concentrate 359 6.5.7 The Roasting of a Sulfide Concentrate 366

6.6 The Production of Gases with Controlled Oxygen and Carbon Potential 370 6.7 Processes Controlled by Chemical Reaction Kinetics 371 6.8 The Reconciliation of an Existing Materials Balance 372 6.9 The Use of Distribution Coefficients in Material Balance Calculations 377

6.9.1 Use of Tabulated Distribution Coefficients 377 6.9.2 Thermodynamic Databases as a Source of Distribution Coefficient Data 381

6.10 Time-Varying Processes 384 6.11 Systems Containing Aqueous Electrolytes 389

6.11.1 The Stability of Ions 390 6.11.2 Aqueous Processes 392 6.11.3 The Solubility of Ionizable Gases in Water 398

6.12 Summary 402 References and Further Reading 404 Exercises 405

Contents

Chapter 7. Energy and the First Law of Thermodynamics 410 7.1 Principles and Definitions 410 7.2 General Statement of the First Law of Thermodynamics 413 7.3 First Law for an Open System 415 7.4 Enthalpy, Heat Capacity, and Heat Content 416 7.5 Enthalpy Change of Phase Transformations 418 7.6 Enthalpy Change of Chemical Reactions 420 7.7 Thermodynamic Databases for Pure Substances 421 7.8 Effect of Temperature on Heat of Reaction 426

7.8.1 Application of Kirchhoff s Equation to Chemical Reactions 426 7.8.2 Heat of Transformation for Non-Standard and Non-Physical States 428

7.9 The Properties of Steam and Compressed Air 431 7.9.1 Properties of Steam 431 7.9.2 Properties of Compressed Air 432 7.9.3 Temperature Change for Free Expansion of a Gas 433 7.9.4 Cooling by Steam Venting 435 7.9.5 Enthalpy of Psychrometry 43 6

7.10 The Use of FREED in Making Heat Balances 437 7.11 Heat of Solution 441

7.11.1 Formation of Non-ideal Metallic Solutions 441 7.11.2 Polymeric Solutions 442 7.11.3 Aqueous Solutions 444

7.12 Summary 445 References and Further Reading 446 Exercises 447

Chapter 8. Enthalpy Balances in Non-Reactive Systems 450 8.1 Combined Material and Heat (System) Balances 450 8.2 Heat Balances for Adiabatic Processes 458 8.3 Psychrometric Calculations 462 8.4 Energy Efficiency 468 8.5 Recovery and Recycling of Heat 469

8.5.1 Heat Exchange Between Fluids 469 8.5.2 Heat Exchange between Solids and Fluids 474 8.5.3 Application of Heat Recovery Techniques to Aluminum Melting 474 8.5.4 Heat Exchange Accompanied by Material Transfer 478

8.6 Multiple-Device System Balances 483 8.7 Use ofFlowBal for System Balances 488 8.8 Heat Balances Involving Solution Phases 494 8.9 Enthalpy Change During Dissolution of an Electrolyte 496 8.10 Graphical Representation of a Heat Balance 499 8.11 Summary 500 References and Further Reading 501 Exercises 502

X Contents

Chapter 9. System Balances on Reactive Processes 505 9.1 Thermal Constraints on a Material Balance 505

9.1.1 Uncoupled System Balances 506 9.1.2 Strategy for Coupled System Balance 508

9.2 Combustion of Fuels 511 9.2.1 Heat of Combustion Calculations 512 9.2.2 Use of Wobbe Index for Combustion Burner Control 514 9.2.3 Combustion of Fuels of Uncertain Molecular Composition 516

9.3 Adiabatic Processes 517 9.3.1 System Balances Involving Combustion Reactions 518 9.3.2 ART for Condensed-Phase Reaction Processes 531

9.4 System Balances Using FlowBal 533 9.5 Quality of Heat and Thermal Efficiency 542 9.6 System Balances with Heat Exchangers 548 9.7 Aqueous Processes 565 9.8 Electrolytic Processes 571

9.8.1 Energy Requirement for Electrorefming 571 9.8.2 Energy Requirement for Electrowinning 572

9.9 Summary 573 References and Further Reading 574 Exercises 574

Chapter 10. Case Studies 577 10.1 Material Balance for an H-Iron Reduction Process with Gas Tempering and Recycle 577 10.2 Mass and Heat Balance Simulation for the Use of DRI in EAF Steelmaking 581 10.3 Natural Gas Combustion Control and the Wobbe Index 588

10.3.1 The Stoichiometry of NG Combustion with Excess Air 588 10.3.2 The Wobbe Index as a Natural Gas Combustion Control Parameter 591

10.4 Reduction of Hematite to Magnetite 592 10.4.1 Preliminary Calculations — Single Reactor 593 10.4.2 Simulation of Hematite Reduction by a Multi-Stage Process 595

10.5 Conversion of Quartz to Cristobalite in a Fluidized Bed 598 10.5.1 Process Characteristics 598 10.5.2 Device Sizing and Heat Loss Calculation 598 10.5.3 Material and Heat Balance Calculations 599

Exercise 600

Appendix. Computational Tools for Making Material and Heat Balance Calculations 601 A.l U-Converter 601 A.2 Thermophysical Properties of Steam and Air 602 A.3 Stream Units Conversion Calculator (MMV-C) 602 A.4 Extension of Excel Tools for Repeat Calculation 603 A.5 Thermodynamic Database Programs 604 A.6 Flowsheet Simulation and System Balancing 604

General References 605

Index 606

Contents

List of Examples

Chapter 1. Dimensions, Units, and Conversion Factors 1.1 Mass and Weight of Aluminum 5 1.2 Kinetic Energy 6 1.3 Energy of Lifting 6 1.4 Units of Energy 7 1.5 Dimensions for Flowrate 8 1.6 Conversion of Temperature 9 1.7 Conversion Formula 9 1.8 Conversion of Pressure - 1 10 1.9 Conversion of Pressure - II 10 1.10 Pressure in a Liquid 11 1.11 The SI and AES Mole 13 1.12 The Density of a Slurry 14 1.13 Bulk Density of a Solid-I 16 1.14 Bulk Density of a Solid - II 17 1.15 Concentration Conversion 17 1.16 Composition of a Gas on a Wet and Dry Basis 19 1.17 Electrical Flow in a Wire 20 1.18 Electrical Energy for Metal Deposition 20 1.19 Number of Significant Figures 22

Chapter 2. Thermophysical and Related Properties of Material 2.1 Removal of Air by a Vacuum Pump 31 2.2 Gas Volume and Flowrate 31 2.3 Compressibility of Steam 32 2.4 Thermal Expansion of Titanium 34 2.5 Evaporation of Water in a Closed Vessel 36 2.6 Humidity and Dew Point 38 2.7 Moisture Content of Clay Dryer Streams 40 2.8 Effect of Pressure on the Freezing Point of Water. 42 2.9 Effect of Pressure on the Vapor Pressure of Water 43 2.10 Vapor and Liquid Phase Composition for the Cu - Ni System 45 2.11 Evaporation from Liquid Cd - Mg Alloys at 700 °C 47 2.12 Volumetric Solubility of C02 in Water 48 2.13 The Solubility of CaF2 in Water 49

Chapter 3. Statistical Concepts Applied to Measurement and Sampling 3.1 A Histogram of Ceramic Strength Measurements 57 3.2 Percentiles of the %Cu Data using Excel 61 3.3 Uniformity of Vermiculite Particles 66 3.4 Evaluation of the Normal Distribution for Ceramic Strength Data 74 3.5 Finding a 90% Confidence Interval 84 3.6 Relationship Between Sample Size and Interval Width Using the Si2ON2 Example

xii Contents

3.7 Heat Capacity Systematic Error 88 3.8 Ore Assay 89 3.9 Improving Measurement Precision 91 3.10 The Professor Tries Again 92 3.11 Linear Random Error Propagation 93 3.12 Multiplicative Random Error Propagation 94 3.13 Other Random Error Propagation 94 3.14 The Difference Between Propagation of Random and Systematic Errors 95 3.15 Modeling the Heat Capacity of TiOx 99 3.16 Non-linear Models for the CpTiOx Data 100 3.17 An Asymptotic Model for the Heat Capacity of TiOx 102 3.18 Using the Regression Tool to Find Non-Linear Models 104 3.19 Removing the Thickness Variable from the Galvanized Corrosion Example 107 3.20 Hypothesis Testing for the Galvanized Steel Model with Three Independent Variables 108 3.21 Selecting a Model for the Hydrogen Reduction of NiO 114 3.22 Calculating the Pressure-Catalyst Interaction 124 3.23 Effect of a Fractional Factorial Design: How Much Information is Lost? 135

Chapter 4. Fundamentals of Material Balances with Applications to Non-Reacting Systems 4.1 Distillation ofaCd-Zn Alloy 151 4.2 Charge Calculation for Feed to a Brass Melting Furnace 153 4.3 Vacuum De-Zincing of Lead 153 4.4 Leaching of Salt Cake from Aluminum Recycling 177 4.5 Refining Crude Boric Acid by a Two-Stage Aqueous Process 181 4.6 Recovery of KMn04 by Evaporation 184 4.7 Removing Dust and S02 from a Roaster Gas 187 4.8 Absorption of HC1 197 4.9 Preparation of a Pigment Precursor 202 4.10 Removal of CuS04 from a Pollution Control Residue 208 4.11 Catalyst Reactivation 225 4.12 Dissolution of ZnCl2 228 4.13 Removal of Hydrogen from Steel 230 4.14 Vacuum Refining of a Cd-Zn Alloy 230 4.15 Control Strategy for Upgrading Spent Reducing Gas 235

Chapter 5. Stoichiometry and the Chemical Equation 5.1 Use of the Gravimetric Factor for Silicon 250 5.2 Mineralogical Constituents of a Concentrate 250 5.3 Reduction ofWustite by CO 253 5.4 Production of Molybdenum Carbide 255 5.5 Production of Titanium by the Kroll Process 257 5.6 The Reaction Between Oxygen and Carbon 272 5.7 Reduction of Molybdenum Oxide with Hydrogen 275 5.8 Carbothermic Reduction of Zinc Oxide 276 5.9 Steam Reforming of Methane 278

Contents

5.10 Controlled Oxidation of Pyrite 284 5.11 Dissolution of Gold in Cyanide Solution 285

Chapter 6. Reactive Material Balances 6.1 Production of Sulfur by Reduction of Sulfur Dioxide 302 6.2 Chlorination of Silicon 304 6.3 Application of FlowBal to Stack Gas Desulfurization 314 6.4 Combustion of Natural Gas with XSA 325 6.5 Effect of Oxygen Enrichment on the Oxidant Required for Complete Combustion 326 6.6 Stack Gas Composition and Dew Point for Coal Combustion with Dry Air 328 6.7 Calculation of % Excess Air from Stack Gas Analysis 332 6.8 Calculation of CO, H2 and NO content in Hot Stack Gas 338 6.9 Calculation of Reformer Gas Composition 342 6.10 Calculation of dpt from Gas Analysis 344 6.11 Calcination of Wet Pickling Cake 347 6.12 Simulation of a Pre-Reduction Fluidized Bed Process 357 6.13 Material Balance on Shaft Furnace Reduction of Hematite 363 6.14 Roasting a Zinc Sulfide Concentrate 369 6.15 Material Balance for BOF Steelmaking 378 6.16 Leaching of Scrubber Dust 395 6.17 The Optimum Precipitation of CaC03 by C02 398

Chapter 7. Energy and the First Law of Thermodynamics 7.1 Work and Heat During the Compression of an Ideal Gas 415 7.2 Heat Capacity and Enthalpy for a Flux 417 7.3 Heat of Fusion of Lead 419 7.4 Standard Heat of the Water-Gas Shift Reaction from 800 to 1500 К 427 7.5 Supercooling Liquid Tin 430 7.6 Combustion of CO with Preheated Air 437 7.7 Adiabatic Compression of Steam 438 7.8 Enthalpy Change During Reduction of NiO with С 439 7.9 Temperature Change of an Adiabatic Reaction 440

Chapter 8. Enthalpy Balances in Non-Reactive Systems 8.1 Heat Balance for Melting Aluminum 454 8.2 Heat Balance for Spray Cooling of Hot Air 455 8.3 Fog Cooling of Ceramic Parts 457 8.4 Atomization of a Molten Metal 459 8.5 Dehumidifying Spent Gas from an Iron Ore Reducing Furnace 463 8.6 Using Stack Gas to Dry Cadmium Powder 466 8.7 Heat Exchange in a Waste Heat Boiler 471 8.8 Preheating HC1 in a Pebble-Bed Vertical Shaft Heat Exchanger 474 8.9 Lowering the Water Temperature from a Crystallizer 481 8.10 Condensation of Zinc Vapor from a Gas 486 8.11 Production of Distilled Water 492

Xl l l

xiv Contents

Chapter 9. System Balances on Reactive Processes 9.1 Heat of Combustion of a Spent Gas from a Reduction Process 513 9.2 Effect of Preheating Combustion Air on AFT 519 9.3 Production of a Reducing Gas 521 9.4 Adiabatic Reforming of Fuel Oil 523 9.5 Oxidation of S02 to S03 for Sulfuric Acid Production 525 9.6 Metallothermic Reduction of Uranium Tetrafluoride 531 9.7 Lime-Assisted Reduction of Magnetite 539 9.8 Calcination of Magnesium Carbonate 552 9.9 Formation of Nickel Ferrite by Spray Roasting 561

CD Contents

1 CD Content Descriptions 2 Air 3 Atmospheres 4 Charts 5 Combustion Documents 6 Copper Smelting 7 FlowBal and MMV-C 8 Material and Heat Balance Notes 9 NG Combust & Wobbe Index 10 Statistics 11 Steel 12 SuperGoalSeek 13 SuperSolver 14 Thermodynamic Database 15 Unit Conversions

Preface to the First Edition

We live in a day and age when realization of the "limits to growth" and the finite extent of all of our natural resources have finally hit home. Yet our economy and our livelihoods depend on successful operation of industries that require and consume raw materials and energy. This success depends, in turn, on efficient use of the available resources, which not only allows industry to conserve materials and energy, but also allows it to compete successfully in the world markets that exist today.

The duties of the metallurgical engineer include, among many other things, development of information concerning the efficiency of metallurgical processes, either through calculation from first principles, or by experimentation. The theory of the construction of material and energy balances, from which such knowledge is derived, is not particularly complicated or difficult, but the practice, particularly in pyrometallurgical operations, can be extremely difficult and expensive.

In this Handbook, we have tried to review the basic principles of physical chemistry, linear algebra, and statistics, which are required to enable the practicing engineer to determine material and energy balances. We have also tried to include enough worked examples and suggestions for additional reading that a novice to this field will be able to obtain the necessary skills for making material and energy balances. Some of the mathematical techniques, which can be used when a digital computer is available, are also presented. The user is cautioned, however, that the old computing adage "garbage in, garbage out" is particularly true in this business, and that great attention must still be paid to setting up the proper equations and obtaining accurate data. Nevertheless, the computer is a powerful ally and gives the engineer the tool to achieve more accurate solutions than was possible just twenty-five years ago.

It is hoped that readers, particularly those who are out of practice at these kinds of calculations will ultimately be able to perform energy balances in processes for which they are responsible, and as a result be able to improve process efficiencies. A bibliography of past work on this subject is presented in an appendix to provide reference material against which results of studies can be checked. Hopefully, results reported in the future will reflect increases in efficiency.

H. Alan Fine University of Kentucky Lexington, Kentucky

Gordon H. Geiger University of Arizona Tucson, Arizona

December, 1979

XV

This page intentionally left blank

Preface to the Third Edition

Because the fundamental bases on which the laws of conservation of mass and energy depend remain the same, users of this edition will find an essential similarity between this edition and the slightly revised (second) edition of 1993. Two noteworthy changes in the professional engineer's practice have occurred since 1993, however. First, in the last 25 years, a dramatic shift has occurred away from metallurgical engineering and the extractive industry towards materials engineering. A large and growing number of recent graduates are employed in such fields as semiconductor processing, environmental engineering and the production and processing of advanced and exotic materials for aerospace, electronic and structural applications. Second, in the same time frame the advance in computing power and software for the desktop computer has significantly changed the way engineers make computations.

This edition of the text reflects these changes. The text now includes examples that involve environmental aspects, processing and refining of semiconductor materials, and energy-saving techniques for the extraction of metals from low-grade ores. However, the biggest change comes from the computational approach to problems. The spreadsheet program Excel is used extensively throughout the text as the main computational "engine" for solving material and energy balance equations, and for statistical analysis of data. A large thermodynamic database (FREED) replaces the thermodynamic tables in the back of the previous Handbook. A number of specialized add-in Excel programs were developed specifically to enhance Excel's problem-solving capability. Finally, on-line versions of two commercial programs for steam table and psychrometric calculations were identified and incorporated in the text examples. These programs simplify the rather difficult calculations commonly required in making material and heat balances. The use of Excel and the introduction of the add-in programs have made it possible to study the effect of a range of variables on critical process parameters. More emphasis is now placed on multi-device flowsheets with recycle, bypass and purge streams whose material and heat balance equations were previously too complicated to solve by the normally-used hand calculator. The Appendix has a brief description of these programs.

The Excel-based program FlowBal is the most important addition to this Edition. FlowBal helps the user set up material and heat balance equations for processes with multiple streams and units. FlowBal uses the thermodynamic database program FREED for molecular mass and enthalpy data. FlowBal's purpose is to introduce the increasingly important subject of flowsheet simulation. FlowBal and all other software and supplementary reference material is on a CD included with the Handbook. A text file on the CD describes its contents (CD Content Descriptions.doc).

Many changes have been made throughout the text. There are now ten chapters instead of six, which reflects a desire to organize the material in non-reactive vs. reactive material and energy balance sections. The concept of degree-of-freedom analysis has been introduced to provide a basis for analyzing the adequacy of information presented in a flowsheet. The concepts of extent-of-reaction and the equilibrium constant are presented as ways to designate how far a given chemical reaction will (or can) proceed. The introduction of the equilibrium constant requires the Handbook user to have completed a course in chemistry that covers the main principles of thermochemistry, or at least to have available a chemistry textbook typical of those used in the first year of a materials engineering program.

Chapter 3 has been completely revised to emphasize the statistical analysis of experimental data, while de-emphasizing the descriptive material on chemical analysis and techniques for sampling process streams. A final chapter has been added on case studies, showing the application of computational techniques and software to more complex processes.

This edition frequently uses web citations and Wikipedia as references and suggestions for further reading. Wikipedia has well-written articles on many Handbook topics, and more are being

xvii

xviii Prefaces

added. Wikipedia is a work in progress, so readers are encouraged to search it for additional information even if a Wikipedia reference is not listed in the text. You are also encouraged to improve any of its articles that are in your area of expertise. The web pages cited in the General References section and as Chapter references may disappear or change after publication, and other sites may appear.

A number of useful and interesting public domain articles were found during revision of the Handbook. These articles have been collected to the Handbook CD in various folders. Many of these articles give background information on processes that were used as Handbook examples. Some of the documents contain articles on processes described in the FlowBal User's Guide.

Finally, a web page has been created where changes and additions to the Handbook are posted. The web page contains updates to the Handbook software, error corrections, references to new software, and links to other sites having useful information on material/energy balances and process simulation. We encourage Handbook users to alert the authors to useful information and to submit material for posting on the page.

http ://thermart.net/

Arthur E. Morris Thermart Software San Diego, California

Gordon H. Geiger University of Arizona Tucson, Arizona

December, 2010

Acknowledgements

This Handbook was prepared under Subcontract 00014529, with Bechtel BWXT Idaho, LLC. We are grateful for the assistance of Simon Friedrich of OIT-DOE in obtaining the contract. Professor Geiger organized this effort with DOE, and made many valuable suggestions during the preparation of the manuscript. In particular, he prepared an outline for a greatly revised chapter on statistics, and contributed advice on the treatment of psychrometry and controllers.

Several people made substantial contributions to the Handbook. First, Chapter 3 is largely the work of two graduate students from Texas A and M University, Mr. Blair Sterba-Boatwright and Mr. Peng-lin Huang. Both made their contributions while they were candidates for a PhD. in statistics, and made what was a very rough draft into a polished product. Second, Mr. Knut Lindqvist wrote the code for Super Goal Seek and Super Solver, and Robert Baron wrote U-Converter. These three programs are on the Handbook CD.

Dr. Semih Perdahcioglu was a prime contributor to the Handbook by his development of FlowBal and MMV-C. FlowBal, in particular, became a mainstay of the computational tools used throughout the Handbook. As a result of his dedicated work, both students and process engineers now have a flowsheet simulation tool capable of dealing with complex multi-device flowsheets. Semih also revised FREED to include a reaction tool. He is now a post-doc research assistant at University of Twente, Netherlands.

We are obligated to three faculty members for miscellaneous advice with various topics. Professor Eric Grimsey (WASM Kalgoorlie) was an inspiration from the beginning, and provided valuable suggestions, encouragement, and help throughout, especially with the application of the degree-of freedom concept. Professor Grimsey also contributed a set of his course notes ("Basic Material and Heat Balances for Steady State Flowsheets") for inclusion on the Handbook CD. These notes provide a shorter (and somewhat different) approach to the construction of system balances, and are recommended without reservation as an adjunct to the Handbook approach.

Professor David Robertson (Missouri University of Science and Technology) cleared up a number of points regarding the material on continuously mixed and unsteady-state processes, and pointed out a number of text errors in Chapter 3. Some of his graduate course examples were adapted for use in FlowBal, and he provided suggestions for FlowBal changes to help the user. Professor Mark Schlesinger (Missouri University of Science and Technology) permitted use of several of his examples and exercises.

Finally, no text is generated in isolation. Items from the General References Section (page 605, just before the Index) provided background information and material data that was used for working out examples and exercises. This Section also cites texts that influenced the structure of this edition of the Handbook. Some of the problem-solving strategies of those texts were modified to fit the more computationally intensive approach adopted in this text.

One of us (AEM) also wishes to express his appreciation for the steadfast support of his wife Helen throughout the revision project.

xix

About the Authors

Arthur E. Morris joined the University of Missouri - Rolla (now the Missouri University of Science and Technol-ogy) in 1965 after receiving his PhD from the Pennsylvania State University in 1965. During his tenure on the fac-ulty of the Department of Metallurgical Engineering, he taught courses in extractive metallurgy, thermodynamics, and process simulation, and also carried out research that resulted in theses for several MS and PhD students. Dr. Morris was a consultant to several industrial corporations in their research laboratories, and was asked by the U.S. Bureau of Mines to organize a new research group at UMR called the Center for Pyrometallurgy. He was a Princi-pal Investigator at the Center until his retirement in 1996. While at UMR, Professor Morris published nearly 70 pa-pers on various aspects of extractive metallurgy and materials processing, and conducted short courses and sym-posia on the applications of computer modeling to metallurgical processes. He presently develops educational software and prepares CDs for materials-related textbooks.

Gordon H. Geiger earned his Bachelor of Engineering degree in Metallurgy at Yale University and his M.S. and Ph.D. degrees in Metallurgy and Materials Science at Northwestern University. He worked in the research depart-ments of Allis-Chalmers Mfg. Co. and Jones and Laughlin Steel Company before teaching process metallurgy at the University of Wisconsin, the University of Illinois at Chicago, and the University of Arizona. In addition to his teaching career, Dr. Geiger worked in industry as a technical officer for a major international bank, a multi-plant steel company and founded a new steel company. He is now retired and lives in Arizona, where he consults and where as Academic Director, he assisted the University of Arizona in establishing an Engineering Management de-gree program.

H.Alan Fine graduated with a PhD degree in Metallurgy from the Massachusetts Institute of Technology in 1974. He then joined the faculty of the University of Arizona's Metallurgical Engineering Department as an Assistant Pro-fessor. Dr. Fine remained on the faculty until 1981, when he joined the University of Kentucky as Associate Pro-fessor in the Department of Metallurgical Engineering and Materials Science. During his time at Kentucky, he also worked with the Environmental Protection Agency. Dr. Fine co-authored the first two editions of this handbook with Dr. Geiger, and he is now retired and living with his family in Florida.

CHAPTER 1

Dimensions, Units, and Conversion Factors

Most science and engineering calculations are performed using quantities whose magnitudes are expressed in terms of standard units of measure or dimensions. A dimension is a property that can be measured, such as length, time, mass or temperature, or obtained by manipulating other dimensions, such as length/time (velocity), length3 (volume), or electric current/area (current density). Dimensions are specified by giving the value relative to some arbitrary standard called a unit. Therefore, the complete specification of a dimension must consist of a number and a unit. Convention, custom, or law can specify which units are used, such that the volume of a substance may be expressed in cubic feet, liters, or gallons.

There are two common systems of units used in engineering calculations. One is the American engineering system (AES) based on the foot (ft) for length, the pound-mass (lbm) for mass, degrees Fahrenheit (°F) for temperature, and the second (s) for time. The two main drawbacks of this system are the occurrence of conversion factors which are not multiples of 10, and the unit of force, which will be discussed later. The other is Le System Internationale d'Unites or SI for short, which has gained widespread acceptance for all scientific and much engineering work. In 1991, the US Department of Commerce promulgated regulations for the required use of the SI system for all Federal agencies. Despite the nearly worldwide acceptance of the SI system, the AES system is still in use in many U.S. industries, and the last vestiges of its use may take decades to obliterate.

This text emphasizes the use of SI units with some exceptions. The calorie and atmosphere are used when dealing with thermodynamic data based on these units. Some non-SI units will be used in selected cases. Converting between units is made easier with a units conversion program (U-Converter, on the Handbook CD). Some of the Chapter examples require thermophysical data, which can be obtained from one of the General References (page 605).

1.1 The SI System of Units In 1960, the General Conference on Weights and Measures (CGPM, Conference General des

Poids et Mesures) established conventions to be used for a set of basic and derived units. The National Institute of Standards and Technology (NIST) is the Federal agency assigned responsibility for publishing guides for SI use. Revisions were made since the first guide was issued in 1960, culminating in the publication of three important NIST documents (Butcher 2006; Taylor 2008; Thompson 2008). These documents are described on the NIST web site. Another useful document is available from the U.S. Metric Association (Antoine 2001).

There are three classes of SI units:

— base units — derived units — supplementary units

which together form what is called "the coherent system of SI units". Table 1.1 gives the seven base quantities on which the SI is founded, and the names and symbols of their respective units, called "SI base units". One of the SI base units — the candela for luminous intensity — is not used in this Handbook.

2 Chapter 1 Dimensions, Units, and Conversion Factors

The quantity of a substance can be expressed in two ways: its mass or its amount. The mass unit is the kilogram (abbreviation kg). The amount unit is the mole (abbreviation mol), which specifies the amount of substance in a given mass. When defining quantity in terms of moles, the mole unit is defined as that amount of a substance containing as many elementary particles as there are atoms in 0.012 kg of the nuclide 12C. This number has been found to equal about 6.022 1367 x 1023, which is Avogadro 's number (NA). Section 1.4 discusses the mole unit in more detail. The distinction between the mass and mole units and conversions between them is an important part of this Handbook, and will be covered in detail in later sections of this Chapter.

Table 1.1 SI base units.

Base quantity length mass time electric current thermodynamic temperature amount of substance luminous intensity

Name meter

kilogram second ampere kelvin mole

candela

Symbol m kg s A К

mol cd

Prefixes are used in SI to form decimal multiples and submultiples of SI units. The most common of these prefixes and their abbreviations are giga (G) for 109, mega (M) for 106, kilo (k) for 103, cent (c) for 10~2, milli (m) for 10-3, micro (μ) for 10-6, and nano (n) for 10~9. There are also some less-common prefixes.

1.1.1 Derived Units

There are two classes of derived units. First, those obtained by mathematical operations of multiplication or division. For example, velocity as meter per second (m/s), and current density as ampere per square meter (A/m2). Second are similarly derived units with special names, such as force (newton, or N, as m · kg · s~2) and energy (joule, or J, as m2 · kg · s"2 or as N · m). NIST SP 811 (Thompson 2008) gives a complete list of derived units. Table 1.2 and Table 1.3 list the derived units used in the Handbook.

Another group of derived units are those expressed with special names. For example, the molar entropy or molar heat capacity is better expressed as joules per mol kelvin, J/(mol · K) rather than m2 · kg · s~2 · K"1 · mol-1. Table 1.4 lists some common examples of this type of unit. These special names exist for convenience, and such derived units can be expressed in different ways. Preference is given to customary use.

Table 1.2 Examples of SI derived units expressed in terms of SI base units.

Derived quantity area volume speed, velocity acceleration mass density (density) specific volume current density amount-of-substance concentration (concentration)

SI derived unit Name

square meter cubic meter meter per second meter per second squared kilogram per cubic meter cubic meter per kilogram amperes per square meter mole per cubic meter

Symbol m2

m3

m/s m/s2

kg/m3

m3/kg A/m2

mol/m3

Chapter 1 Dimensions, Units, and Conversion Factors 3

Table 1.3 Examples of SI derived units with special names and symbols.

SI derived unit

Derived quantity

frequency force pressure, stress energy, work, heat power quantity of electricity electric potential, emf electric resistance Celsius temperature

Special name

hertz newton pascal joule watt coulomb volt ohm degree Celsius

Special symbol

Hz N Pa J

W С V Ω °C

Expression in terms of other SI units

N/m2

N - m J/s

W/A V/A

Expression in terms of SI base unit s-1

m · kg · s~2

rrf ' · kg · s~2

m2 · kg · s~2

m2 · kg · s~3

s-A m3 · kg · s~3 · A-1

m2 · kg · s"3 · A"2

К

Table 1.4 Additional SI units without special names, expressed in terms of named units.

Derived quantity

heat flux density heat capacity, entropy specific heat capacity, entropy molar entropy, heat capacity thermal conductivity

Name

watt per square meter joule per kelvin joule per kilogram kelvin joule per mole kelvin watt per meter kelvin

SI derived unit Symbol

W/m2

J/K J/(kg · K) J/(mol · K) W/(m · K)

Expression in terms of SI base units

kg-s-3

m2 · kg · s-2 · K"1

m2 · s"2 · K"1

m2-kg-s-2-K-1-mor11

m · kg · s"3 · K"1

SI allows the expression of units as fractions, or expressed as negative exponentials. Thus it's equally appropriate to express heat flux density as W/m2 or W · m"2.

1.1.2 Units Outside the SI

There are three categories of units outside the SI:

- those units that are accepted for use with the SI;

- those units that are temporarily accepted for use with the SI;

- those units that are not accepted for use with the SI, and are to be avoided.

Accepted units include minute, hour, or day (min, h, d) instead of second; liter (symbol L) instead of m3; and metric ton (t) instead of 103 kg or 1 Mg. (The metric ton is called the tonne in many countries). The composition term % is acceptable in place of 0.01. For example, it is preferable to state, "the mass fraction of В is 0.02", or "wB = 0.02", but acceptable to state, "the mass fraction of В is 2 %", or "wB = 2 %". The temporarily accepted units include the pressure unit bar (bar), which is equivalent to 105 Pa. Some thermodynamic tables list the standard pressure as 1 bar. The standard atmosphere is approximately 1.013 bar.

Unacceptable units of course include those of the AES, such as ft and lb. Other "metric" unacceptable units are the dyne and erg (left over from the CGS system), the torr and atmosphere (atm) as units of pressure, the kilogram-force (kgf) as a unit of force, and the calorie (cai, in various dimensions) as a unit of energy. Similarly, composition terms such as ppm or ppb (parts per million, parts per billion) are unacceptable unless required by law. As mentioned earlier, some thermodynamic data may be available only in units of cai and atm, in which case, they will be used in this Handbook with no further explanation.

4 Chapter 1 Dimensions, Units, and Conversion Factors

1.1.3 Comments on Some Quantities and Their Units

Temperature. The quantity Celsius temperature (symbol /) is used in addition to the thermodynamic temperature expressed in the unit kelvin. It is defined by the equation:

T-To [1.1]

where T0 = 273.15 К by definition, and is exactly 0.01 К below the triple point of water. The degree unit Celsius is equal in magnitude to the degree unit kelvin. An interval or difference of °C can be expressed as well in the unit kelvin. Note that the centigrade temperature scale is obsolete; the degree centigrade is almost, but not quite, equal to the degree Celsius. This Handbook uses an upper-case T to denote temperature in unit kelvin (К), and a lower-case t to denote temperature in degree Celsius. Furthermore, to avoid confusion, we will spell out the word tonne instead of using t to designate the metric ton.

Weight. In science and technology, we define the weight of a body as the force that gives the body acceleration equal to the local acceleration of free fall. We define the acceleration of gravity as exactly 9.806 65 m/s2 at the standard location of sea level and 45° north latitude. Thus, the SI unit of the quantity weight is the newton (N). In commercial and everyday use, weight is usually used as a synonym for mass. Thus, the SI unit used in this way is the kilogram (kg). In order to avoid confusion, the term weight should be avoided, and mass used instead.

Amount of Substance, Concentration, Molality, etc. Concentration and fractional amount terms require special attention for SI usage (Section 1.6, Thompson 2008). In particular, the terms "molarity" and "normal" are not acceptable for SI, and "amount-of-substance fraction of B" is preferred to "mole fraction of B". Section 1.6.2 discusses this issue in more detail.

Printing and Using Symbols and Numbers. This Handbook follows NIST recommendations (Thompson 2008) in naming and formatting the various symbols used. Digits for numbers should be separated in groups of three. For example, the conversion factor for cubic feet (ft3) to cubic meters (m3) is 2.831 685 x 10~2. However, the conversion factor from kilo-calorieth to joule is 4184, which is an exact factor. Try to avoid using a number like 2400 because it is unclear if the last two zeros are significant figures. Instead, use 2.4 x 103 or 2.400 x 103 as appropriate to indicate the number of significant figures. The exception to this policy occurs in the use of Excel, which does not accept spaces between digits.

1.2 The American Engineering System (AES) of Units As much as we would like to see units such as horsepower, Btu, and °F disappear, they have

not. The AES is still used industrially, and even some documents from Federal agencies intended for use by industry. The policy of this Handbook is to illustrate AES use in selected cases, point out some of the difficulties in its use, and illustrate the use of conversion factors from the AES to SI. A brief list of the more common conversion factors is given on the inside cover, and a more extensive list is in NIST SP811 (Thompson 2008). The U-Converter program has the most comprehensive list.

A notable difference in SI and the AES system is the derived unit of force. In SI, the derived force unit is the newton (N), based on the natural force unit of kg · m/s2. In the AES, a choice can be made to select an arbitrary unit of force or an arbitrary unit of mass. Newton's law automatically fixes the other unit:

Force = mass · acceleration [1.2]

If the pound is chosen as the mass unit (lbm), it may be expressed in terms of the kilogram; the lbm has 0.4536 times the mass of a kg. Then the fundamental derived unit of force is that which produces an acceleration in 1 lbm of 1 ft/s2. This unit is the poundal, with dimensions of ft/s2.

If the pound is chosen as the fundamental unit of force, the lbf is the unit of force that will give a lbm an acceleration of 32.174 ft/s2. It is also the force of gravity between the lbm and earth

Chapter 1 Dimensions, Units, and Conversion Factors 5

existing at sea level and 45° latitude (the standard location). When the lbf is selected as the unit of force, the derived unit of mass is that mass which will be accelerated at the rate of 1 ft/s2 when acted on by 1 lbf. This derived unit of mass is known as the slug, with a mass of 14.5939 kg, and units of lbf · sec2/ft. A lbf gives a lbm an acceleration of 32.174 ft/s2.

Unfortunately, engineers have selected the lbf as the unit of force, and the lbm as the unit of mass. When these are substituted in Equation [1.2], the resulting equation is neither algebraically or dimensionally correct. To avoid this incongruity, Equation [1.2] must be rewritten as:

Force = mass · acceleration/gc [1.3]

where gc is a constant equal to 32.174 lbm · ft/(lbf · sec2), and is independent of location.

The weight of a body is the force of gravity existing between the body and earth, and since weight is a force, we express it in terms of the lbf when the AES is used. Fortunately, the variations in weight produced by latitude or elevation are small, rarely exceeding 0.25%. Thus, Equation [1.3] is rewritten as:

F=W-a/g [1.4]

where F = force acting on a body in any direction in lbf; W is the weight of the body in lbf; g = acceleration of gravity at the location, in ft/s2; a = acceleration of the body in the direction of the force, in ft/s2.

Note that the weight of a body at standard location, expressed in lbf, is numerically equal to the mass of the object, expressed in lbm. Thus at that location, a kilogram of water weighs 2.2046 lb (i.e., lbf), and the mass of water is 2.2046 lb (i.e., lbm). This numerical equality is a close approximation at other locations.

EXAMPLE 1.1 — Mass and Weight of Aluminum.

Calculate the mass of a block of aluminum with a volume of 0.1500 m3 (5.297 ft3) at the standard location. Calculate the gravitational force acting on the block, and the stress in a 0.500 cm (0.1969 in) diameter wire suspending the block. Make the calculations in SI and AES units.

Data. The mass density of aluminum is 2702 kg/m3 (168.7 lbm/ft3). Stress has the units of force per unit area (or pressure), expressed as Pa or lbf/in2. Assume the mass of the wire and air buoyancy can be neglected.

Solution. The SI mass of the block is (2702 kg/m3)(0.15 m3) - 405.3 kg. The SI weight is: W— m · g

Weight (SI) - (405.3 kg)(9.8066 m/s2) - 3975 kg · m/s2 = 3975 newton.

The mass of the body in AES units is (5.297 ft3)(168.7 lbm/ft3) = 893.6 lbm. According to Equation [1.4], the numerical value of the weight in lbf = the weight in lbm when the acceleration of gravity equals gc so:

Weight (AES) - (5.297 ft3)(168.7 lbm/ft3) = 893.6 lbm - 893.6 lbf.

Stress has units of force/area. The SI area of the wire is π(0.25/100)2 = 1.964 x IO-5 m2. The AES area of the wire is π(0.09845)2 = 0.03043 in2. The stress σ is calculated as:

Q = FIA

σ (SI) = 3975 N/1.964 x 10-5 m2 = 2.024 x 108 Pa.

σ (AES) = 893.6 lbf/0.03043 in2 = 2.937 x 104 lbf/in2.

Assignment. Calculate the force (in SI and AES) that would accelerate the aluminum block to a speed of 10 ft/s if applied over a time of 5 s.