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Unformatted text preview: First Edition – Volume 5 Formulas and Conversions Published by IDC Technologies, 982 Wellington Street WEST PERTH WA 6005 WESTERN AUSTRALIA Copyright 1994, 1995, 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004 IDC Technologies A.B.N. 003 263 189 ISBN 1 875955 09 7 US English. First Edition. All rights to this publication are reserved. No part of this publication may be copied, reproduced, transmitted or stored in any form or by any means (including electronic, mechanical, photocopying, recording or otherwise) without prior written permission from IDC Technologies Pty Ltd. Trademarks All terms noted in this publication that are believed to be registered trademarks or trademarks are listed below: • PC-DOS, IBM, IBM PC/XT, IBM PC/AT and IBM PS/2 are registered trademarks of International Business Machines Corporation. • Microsoft, MS-DOS and Windows are registered trademarks of Microsoft Corporation. • Intel is a registered trademark of the Intel Corporation Disclaimer Whilst all reasonable care has been taken to ensure that the description, opinions, listings and diagrams are accurate and workable, IDC Technologies does not accept any legal responsibility or liability to any person, organization or other entity for any direct loss, consequential loss or damage, however caused, that may be suffered as a result of the use of this publication. If you want further information or advice please contact our Engineering Division at tech@idc-online.com for further support. We would be delighted to assist you. A Message from IDC Technologies Technical Director, Steve Mackay Dear Colleague, Welcome to our latest engineering pocket guide focusing on engineering formulae and conversions. We have been providing practical training for over 12 years throughout the USA, Canada, United Kingdom, Ireland, Australia, Singapore, South Africa and New Zealand. Although we are one of the largest providers of this sort of training and have trained a remarkable 120,000 engineers and technicians in the past few years alone, we are not content with resting on our laurels and continue to achieve an amazing 99.8% satisfaction rating in which delegates indicated the course was "good", "very good" or "excellent". We want the course that you attend to be an outstanding, motivating experience where you walk away and say – "that was truly a great course with a brilliant instructor and we will derive enormous benefit from it". Our workshops are not academic but are rather designed to immediately provide you with the practical skills which will contribute to your productivity and your company's success. Our courses are vendor independent, free of bias and targeted solely at improving your productivity. We have a remarkable group of instructors whom we believe are among the best in the industry. Of greatest benefit is that they have real and relevant practical experience in both industry and training. Our policy is to continually re-examine and develop new training programs, update and improve them. Our aim is to anticipate the shifting and often complex technological changes facing everyone in engineering and business and to provide courses of the absolutely highest standards – helping you to improve your productivity. We put tremendous efforts into our documentation with award winning manuals which are well researched, practical and down to earth in support of the course; so much so that many delegates have remarked that the manual itself justifies the course fees. I would urge you to consider our courses and call us if you have any queries about them. We would be glad to explain in more detail what the courses entail and can even arrange for our instructors to give you a call to talk through the course contents with you and how it will benefit yourselves. Finally, thank you for being such tremendously supportive clients. We are blessed with having such brilliant people attending our courses who are enthusiastic about improving themselves and benefiting their companies with new insights and methods of improving their productivity. Your continual feedback is invaluable in making our courses even more appropriate for today's fast moving technology challenges. We want to be your career partner for life – to ensure that your work is both satisfying and productive and we will do whatever it takes to achieve this. Yours sincerely (C P Eng, BSEE, B.Sc(Hons), MBA) Technical Director P.S. Don't forget our no-risk guarantee on all our products – we give you a 100% guarantee of satisfaction or your money back. Other books in this series Volume 1 – INSTRUMENTATION Automation using PLCs, SCADA and Telemetry, Process Control and Data Acquisition Volume 2 – COMMUNICATIONS Data Communications, Industrial Networking, TCP/IP and Fiber Optics Volume 3 – ELECTRICAL Power Quality, Power Systems Protection and Substation Automation Volume 4 – ELECTRONICS Personal Computers, Digital Signal Processing and Analog/Digital Conversions 5.3.7 5.3.8 5.3.9 5.3.10 5.3.11 5.3.12 5.3.13 5.3.14 5.3.15 5.3.16 5.3.17 5.3.18 5.3.19 5.3.20 5.3.21 5.3.22 5.3.23 Table of Contents Chapter 1 Definition and Abbreviations for Physical Quantities ...........1 Chapter 2 Units of Physical Quantities .................................................3 Chapter 3 System of Units ..................................................................23 5.4 General Mathematical Formulae........................................27 4.1 4.2 4.3 4.4 4.5 4.6 Algebra................................................................................. 27 Geometry ............................................................................. 29 Trigonometry ........................................................................ 39 Logarithm ............................................................................. 40 Exponents ............................................................................ 42 Complex Numbers ............................................................... 42 Chapter 5 Engineering Concepts and Formulae ................................44 5.1 5.2 Electricity.............................................................................. 44 Applied Mechanics ............................................................... 57 5.2.1 5.2.2 5.2.3 5.2.4 5.2.5 5.3 Newton's laws of motion ..........................................................57 Linear Velocity And Acceleration .............................................60 Force........................................................................................61 Centripetal (Centrifugal) Force.................................................62 Stress, Strain And Modulus Of Elasticity..................................64 Thermodynamics.................................................................. 64 5.3.1 5.3.2 5.3.3 5.3.4 5.3.5 5.3.6 Laws of Thermodynamics ........................................................64 Momentum...............................................................................65 Impulse ....................................................................................65 Elastic and Inelastic collision ...................................................65 Center of Mass ........................................................................65 Angular Motion.........................................................................65 Fluid Mechanics ................................................................... 77 5.4.1 5.4.2 5.4.3 Chapter 4 Conditions of Equilibrium .........................................................65 Gravity .....................................................................................66 Vibrations & Waves .................................................................66 Standing Waves.......................................................................66 Beats........................................................................................66 Temperature and Heat.............................................................67 Ideal Gases..............................................................................67 Elastic Deformation..................................................................68 Temperature Scales ................................................................68 Sensible Heat Equation ...........................................................68 Latent Heat ..............................................................................68 Gas Laws.................................................................................68 Specific Heats Of Gases..........................................................69 Efficiency of Heat Engines .......................................................70 Heat Transfer by Conduction ...................................................71 Thermal Expansion of Solids ...................................................72 Chemical Heating Value of a Fuel ...........................................72 Discharge from an Orifice ........................................................77 Bernoulli’s Theory ....................................................................78 Actual pipe dimensions ............................................................78 Chapter 6 References.........................................................................80 6.1 6.2 Periodic Table of Elements .................................................. 80 Resistor Color Coding .......................................................... 81 Formulas and Conversions Formulas and Conversions Symbol Definition and Abbreviations for Physical Quantities Prefix Factor by which unit is multiplied k Chapter 1 Kilo 103 h Hecto 102 da Deca 10 Quantity d Deci 10-1 meter Length c Centi 10-2 kilogram Mass m Milli 10-3 s second Time µ Micro 10-6 A ampere Electric current n Nano 10-9 K kelvin Thermodynamic temp p Pico 10-12 cd candela Luminous intensity Symbol Unit m kg Quantity Unit Symbol Equivalent Plane angle radian rad - Force newton N kg · m/s2 Work, energy heat joule J·N·m Power watt W J/s Frequency hertz Hz s-1 Viscosity: kinematic - m2/s 10 c St (Centistoke) Viscosity: Dynamic - Ns/m2 103 cP (Centipoise) Pressure - Pa or N/m2 pascal, Pa Symbol Prefix Factor by which unit is multiplied Quantity Electrical unit Symbol Derived unit Potential Volt V W/A Resistance Ohm Ώ V/A Charge Coulomb C A·s Capacitance Farad F A·s/V Electric field strength - V/m - Electric flux density - C/m2 - Quantity Magnetic unit Symbol Derived unit Magnetic flux Weber Wb V·s = N·m/A Inductance Henry H V·s/A = N·m/A2 T Tera 1012 A/m - Giga 109 Magnetic field strength - G M Mega 106 Magnetic flux density Tesla T Wb/m2 = (N)/(Am) -1- -2- Formulas and Conversions Formulas and Conversions Chapter 2 Multiply by Name lb·s2/in4 kg/m3 3 3 Divide by 1.069E+07 9.357E-08 Density 1.000E+07 1.602E-19 6.242E+18 Ft·lbf J 1.3557 0.7376 kiloton TNT J 4.187E+12 2.388E-13 KW·hr J 3.600E+06 2.778E-07 Megaton TNT J 4.187E+15 2.388E-16 Dyne N 1.000E-05 1.000E+05 Lbf N 4.4484 0.2248 Ozf N 0.2780 3.5968 BTU/lbm · °F J/kg·°C 4188 2.388E-04 Heat transfer coefficient Degrees Fahrenheit = (Degrees Celsius) (9/5) + 32 1.000E-07 J Heat capacity Degrees Celsius = (Degrees Fahrenheit - 32) (5/9) J eV Force 1 pound per square inch = 2.31 feet of water erg Force 1 pound = 0.454 kilograms 0.2389 Force 1 million gallons per day = 694 gallons per minute 4.1859 Energy 1 horsepower = 0.746 kilowatts J Energy 1 grain per gallon = 17.1 mg/L cal Energy 1 gallon = 8.34 pounds 9.478E-04 Energy 1 gallon = 3.79 liters 1.940E-03 1055 Energy 1 foot = 0.305 meters 515.40 J Energy 1 cubic foot = 7.5 gallons kg/m BTU Energy 1 acre = 43,560 square feet slug/ft Energy Conversion Factors (general): BTU/hr·ft2·°F W/m2·°C 5.6786 0.1761 Length To convert from 2 Multiply by To 2 Divide by AU m 1.496E+11 6.685E-12 Length 1% = 10,000 mg/L Name To Density Units of Physical Quantities To convert from ft m 0.3048 3.2810 Length in m 2.540E-02 39.3700 Length mile m 1609 6.214E-04 Nautical mile m 1853 5.397E-04 Acceleration ft/sec m/s 0.3048 3.2810 Length Area acre m2 4047 2.471E-04 Length parsec m 3.085E+16 3.241E-17 Area ft2 m2 9.294E-02 10.7600 Mass amu kg 1.661E-27 6.022E+26 Area hectare m2 1.000E+04 1.000E-04 Mass lbm kg 0.4535 2.2050 Area in2 m2 6.452E-04 1550 Mass lb·s2/in kg 1200.00 5.711E-03 Density g/cm3 kg/m3 1000 1.000E-03 Mass slug kg 14.59 6.853E-02 Density lbm/ft3 kg/m3 16.02 6.243E-02 Mass flow rate lbm/hr kg/s 1.260E-04 7937 Density lbm/in3 kg/m3 2.767E+04 3.614E-05 -3- -4- Formulas and Conversions Name To convert from lbm/sec Multiply by To Mass flow rate Formulas and Conversions kg/s Divide by Name To convert from To Multiply by Divide by sidereal year S 3.156E+07 3.169E-08 0.4535 2.2050 Time 2 2 Moment of inertia ft·lb·s kg·m 1.3557 0.7376 Torque ft·lbf N·m 1.3557 0.7376 Moment of inertia in·lb·s2 kg·m2 0.1130 8.8510 Torque in·lbf N·m 0.1130 8.8504 Moment of inertia oz·in·s2 kg·m2 7.062E-03 141.60 Torque In·ozf N·m 7.062E-03 141.61 Power BTU/hr W 0.2931 3.4120 Velocity ft/min m/s 5.079E-03 196.90 Power hp W 745.71 1.341E-03 Velocity ft/s m/s 0.3048 3.2810 Power tons of refrigeration W 3516 2.844E-04 Velocity Km/hr m/s 0.2778 3.6000 Pressure bar Pa 1.000E+05 1.000E-05 Velocity miles/hr m/s 0.4470 2.2370 2 2 Pressure dyne/cm Pa 0.1000 10.0000 Viscosity – absolute centipose N·s/m 1.000E-03 1000 Pressure in. mercury Pa 3377 2.961E-04 Viscosity – absolute g/cm·s N·s/m2 0.1000 10 2 47.87 2.089E-02 2 Pressure in. water Pa 2 248.82 4.019E-03 Viscosity – absolute 2 lbf/ft ·s N·s/m Pressure kgf/cm Pa 9.807E+04 1.020E-05 Viscosity – absolute lbm/ft·s N·s/m 1.4881 0.6720 Pressure lbf/ft2 Pa 47.89 2.088E-02 Viscosity – kinematic centistoke m2/s 1.000E-06 1.000E+06 Pressure 2 lbf/in Pa 6897 1.450E-04 Viscosity – kinematic Pressure mbar Pa 100.00 1.000E-02 Volume Pressure microns mercury Pa 0.1333 7.501 Pressure mm mercury Pa 133.3 Pressure std atm Pa Specific heat BTU/lbm·°F Specific heat cal/g·°C ft /sec 2 m /s 9.294E-02 10.7600 ft3 m3 2.831E-02 35.3200 Volume in3 m3 1.639E-05 6.102E+04 7.501E-03 Volume Liters m3 1.000E-03 1000 1.013E+05 9.869E-06 Volume U.S. gallons m3 3.785E-03 264.20 J/kg·°C 4186 2.389E-04 Volume flow rate ft3/min m3/s 4.719E-04 2119 J/kg·°C 4186 2.389E-04 Volume flow rate U.S. gallons/min m3/s 6.309E-05 1.585E+04 Temperature °F °C 0.5556 1.8000 Thermal conductivity BTU/hr·ft·°F W/m·°C 1.7307 0.5778 Thermal conductivity BTU·in/hr·ft2·°F W/m·°C 0.1442 6.9340 Thermal conductivity cal/cm·s·°C W/m·°C 418.60 2.389E-03 2 Thermal conductivity cal/ft·hr·°F W/m·°C 6.867E-03 day S 8.640E+04 1.157E-05 DISTANCE (Length) Conversions Multiply By To obtain LENGTH 145.62 Time A. 0.03280840 foot Centimeter -5- Centimeter 0.3937008 inch -6- Formulas and Conversions Multiply By To obtain Foot Foot Foot 304.8 Inch 0.0254* 2.54 25.4 Inch Kilometer * * Centimeters Yards 0.01093613 Centimeters Feet 0.0328084 Centimeters Inches 0.3937008 meter(m) * 0.01 millimeter(mm) 30.48* Multiply By Meters meter(m) 0.3048* To Centimeters centimeter(cm) 1.8288* To Convert meter(m) Chains, (Surveyor's) Rods 4 centimeter(cm) Fathom Inch Formulas and Conversions Chains, (Surveyor's) Meters 20.1168 millimeter(mm) 0.6213712 Meter 39.37008 Meter 0.54680066 Chains, (Surveyor's) Feet 66 mile(USstatute) Fathoms Meters 1.8288 Inch Fathoms Feet 6 Fathom Feet Statute Miles 0.00018939 Meter 3.280840 Foot Feet Kilometers 0.0003048 Meter 0.1988388 Rod Feet Meters 0.3048 Meter 1.093613 Yard Feet Yards 0.3333333 Meter 0.0006213712 Microinch micrometer(micron) mile(USstatute) Feet Inches 12 micrometer(micron)(µm) 0.0254* Feet Centimeters 30.48 39.37008 Microinch Furlongs Statute Miles 0.125 mile(USstatute) 1,609.344* meter(m) Furlongs Meters 201.168 mile(USstatute) * kilometer(km) Furlongs Yards 220 millimeter 0.003280840 1.609344 Foot Furlongs Feet 660 millimeter 0.0397008 Inch Furlongs Inches 7920 Rod 5.0292 * meter(m) Hands (Height Of Horse) Inches 4 Yard 0.9144* meter(m) Hands (Height Of Horse) Centimeters 10.16 Inches Meters 0.0254 Inches Yards 0.02777778 Inches Feet 0.08333333 Inches Centimeters 2.54 Inches Millimeters 25.4 To Convert To Multiply By Cables Fathoms 120 Cables Meters 219.456 Cables Yards 240 -7- -8- Formulas and Conversions Formulas and Conversions To Convert To Multiply By To Convert To Multiply By Kilometers Statute Miles 0.621371192 Miles, Statute Centimeters 160934.4 Kilometers Meters 1000 Millimeters Inches 0.039370079 Leagues, Nautical Nautical Miles 3 Mils Inches 0.001 Leagues, Nautical Kilometers 5.556 Mils Millimeters 0.0254 Leagues, Statute Statute Miles 3 Paces (US) Inches 30 Leagues, Statute Kilometers 4.828032 Paces (US) Centimeters 76.2 Links, (Surveyor's) Chains 0.01 Points (Typographical) Inches 0.013837 Links, (Surveyor's) Inches 7.92 Points (Typographical) Millimeters 0.3514598 Links, (Surveyor's) Centimeters 20.1168 Rods Meters 5.0292 Meters Statute Miles 0.000621371 Rods Yards 5.5 Meters Kilometers 0.001 Rods Feet 16.5 Meters Yards 1.093613298 Spans Inches 9 Meters Feet 3.280839895 Spans Centimeters 22.86 Meters Inches 39.370079 Yards Miles 0.00056818 Meters Centimeters 100 Yards Meters 0.9144 Meters Millimeters 1000 Yards Feet 3 Microns Meters 0.000001 Yards Inches 36 Microns Inches 0.0000394 Yards Centimeters 91.44 Miles, Nautical Statute Miles 1.1507794 Miles, Nautical Kilometers 1.852 Miles, Statute Kilometers 1.609344 Miles, Statute Furlongs 8 Miles, Statute Rods 320 Miles, Statute Meters 1609.344 Miles, Statute Yards 1760 Miles, Statute Feet 5280 Miles, Statute Inches 63360 -9- Conversion Length 1 ft = 12 in 1 yd = 3 ft 1 cm = 0.3937 in 1 in = 2.5400 cm 1 m = 3.281 ft 1 ft = 0.3048 m 1 m = 1.0936 yd 1 yd = 0.9144 m 1 km = 0.6214 mile 1 mile = 1.6093 km 1 furlong = 40 rods 1 fathom = 6 ft - 10 - Formulas and Conversions Formulas and Conversions Conversion Conversion 1 statute mile = 8 furlongs 1 rod = 5.5 yd Dry Volume 1 statute mile = 5280 ft 1 in = 100 mils 1 quart = 2 pints 1 quart = 67.2 in3 1 nautical mile = 6076 ft 1 light year = 9.461 x 1015 m 1 peck = 8 quarts 1 peck = 537.6 in3 1 bushel = 4 pecks 1 bushel = 2150.5 in3 -5 1 league = 3 miles 1 mil = 2.540 x 10 m Area 1 ft2 = 144 in2 1 acre = 160 rod2 1 yd2 = 9 ft2 Area Conversions 1 acre = 43,560 ft2 2 1 rod = 30.25 yd 2 Multiply 1 mile2 = 640 acres 1 cm2 = 0.1550 in2 2 B. 1 in2 = 6.4516 cm2 2 1 m = 10.764 ft 1 ft2 = 0.0929 m2 1 km2 = 0.3861 mile2 1 mile2 = 2.590 km2 1 cm3 = 0.06102 in3 1 in3 = 16.387 cm3 1 m3 = 35.31 ft3 1 ft3 = 0.02832 m3 1 Litre = 61.024 in3 1 in3 = 0.0164 litre 1 Litre = 0.0353 ft3 1 ft3 = 28.32 litres 1 Litre = 0.2642 gal. (U.S.) 1 yd3 = 0.7646 m3 1 Litre = 0.0284 bu (U.S.) 1 gallon (US) = 3.785 litres 1 Litre = 1000.000 cm3 1 gallon (US) = 3.785 x 10-3 m3 1 Litre = 1.0567 qt. (liquid) or 0.9081 qt. (dry) 1 bushel (US) = 35.24 litres 1 oz (US fluid) = 2.957 x 10-5 m3 1 stere = 1 m3 2 centimeter2 2 hectare 0.1550003 inch2 0.001076391 * foot2 meter2 (m2) foot 0.09290304 foot2 929.03042 centimeter2 (cm2) 2 foot 92,903.04 millimeter2 (mm2) hectare 2.471054 acre inch2 645.16* millimeter2 (mm2) inch2 6.4516 centimeter2 (cm2) inch2 0.00064516 meter2 (m2) 2 meter 1,550.003 inch2 meter2 10.763910 foot2 1.195990 yard2 meter 1 gill = 4 fluid ounces 1 barrel = 31.5 gallons meter2 1 pint = 4 gills 1 hogshead = 2 bbl (63 gal) millimeter2 1 quart = 2 pints 1 tun = 252 gallons millimeter 1 gallon = 4 quarts 1 barrel (petrolum) = 42 gallons yard2 - 11 - meter2 (m2) 0.4046856 acre 2 Liquid Volume To obtain 4,046.856 acre centimeter Volume By AREA 0.0002471054 2 acre 0.00001076391 foot2 0.001550003 inch2 0.8361274 - 12 - meter2 (m2) Formulas and Conversions C. Formulas and Conversions Volume Conversions Metric Conversion Factors: Volume (including Capacity) To Multiply By Carat Multiply To Convert Milligrams 200 Drams, Avoirdupois Avoirdupois Ounces 0.06255 Drams, Avoirdupois Grams 1.7718452 inch3 Drams, Avoirdupois Grains 27.344 meter3 (m3) Drams, Troy Troy Ounces 0.125 liter Drams, Troy Scruples 3 meter3 (m3) Drams, Troy Grams 3.8879346 litre Drams, Troy Grains 60 meter3 (m3) By To obtain VOLUME (including CAPACITY) centimeter3 0.06102376 foot3 0.028311685 foot3 28.31685 gallon (UK liquid) 0.004546092 gallon (UK liquid) 4.546092 gallon (US liquid) 0.003785412 Grains Kilograms 6.47989E-05 gallon (US liquid) 3.785412 liter Grains Avoirdupois Pounds 0.00014286 inch3 16,387.06 millimeter3 (mm3) Grains Troy Pounds 0.00017361 inch3 16.38706 centimeter3 (cm3) Grains Troy Ounces 0.00208333 inch3 0.00001638706 meter3 (m3) Grains Avoirdupois Ounces 0.00228571 Liter 0.001* meter3 (m3) Grains Troy Drams 0.0166 Liter 0.2199692 gallon (UK liquid) Grains Avoirdupois Drams 0.03657143 Liter 0.2641720 gallon (US liquid) Grains Pennyweights 0.042 Liter 0.03531466 foot3 Grains Scruples 0.05 meter3 219.9692 gallon (UK liquid) Grains Grams 0.06479891 3 meter 264.1720 gallon (US liquid) Grains Milligrams 64.79891 meter3 35.31466 foot3 Grams Kilograms 0.001 meter3 1.307951 yard3 Grams Avoirdupois Pounds 0.002204623 meter3 1000.* liter Grams Troy Pounds 0.00267923 meter3 61,023.76 inch3 Grams Troy Ounces 0.032150747 0.00006102376 inch3 Grams Avoirdupois Ounces 0.035273961 meter3 (m3) Grams Avoirdupois Drams 0.56438339 Grams Grains 15.432361 millimeter3 Yard3 D. 0.7645549 Mass and Weight Conversions - 13 - - 14 - Formulas and Conversions Formulas and Conversions To Convert To Multiply By To Convert To Multiply By Grams Milligrams 1000 Ounces, Avoirdupois Avoirdupois Drams 16 Hundredweights, Long Long Tons 0.05 Ounces, Avoirdupois Grams 28.34952313 Hundredweights, Long Metric Tons 0.050802345 Ounces, Avoirdupois Grains 437.5 Hundredweights, Long Short Tons 0.056 Ounces, Troy Avoirdupois Pounds 0.06857143 Hundredweights, Long Kilograms 50.802345 Ounces, Troy Troy Pounds 0.0833333 Hundredweights, Long Avoirdupois Pounds 112 Ounces, Troy Avoirdupois Ounces 1.097143 Hundredweights, Short Long Tons 0.04464286 Ounces, Troy Troy Drams 8 Hundredweights, Short Metric Tons 0.045359237 Ounces, Troy Avoirdupois Drams 17.55429 Hundredweights, Short Short Tons 0.05 Ounces, Troy Pennyweights 20 Hundredweights, Short Kilograms 45.359237 Ounces, Troy Grams 31.1034768 Hundredweights, Short Avoirdupois Pounds 100 Ounces, Troy Grains 480 Kilograms Long Tons 0.0009842 Pennyweights Troy Ounces 0.05 Kilograms Metric Tons 0.001 Pennyweights Grams 1.55517384 Kilograms Short Tons 0.00110231 Pennyweights Grains 24 Kilograms Short Hundredweights 0.02204623 Pounds, Avoirdupois Long Tons 0.000446429 Kilograms Avoirdupois Pounds 2.204622622 Pounds, Avoirdupois Metric Tons 0.000453592 Kilograms Troy Pounds 2.679229 Pounds, Avoirdupois Short Tons 0.0005 Kilograms Troy Ounces 32.15075 Pounds, Avoirdupois Quintals 0.00453592 Kilograms Avoirdupois Ounces 35.273962 Pounds, Avoirdupois Kilograms 0.45359237 Kilograms Avoirdupois Drams 564.3834 Pounds, Avoirdupois Troy Pounds 1.215278 Kilograms Grams 1000 Pounds, Avoirdupois Troy Ounces 14.58333 Kilograms Grains 15432.36 Pounds, Avoirdupois Avoirdupois Ounces 16 Milligrams Grains 0.015432358 Pounds, Avoirdupois Avoirdupois Drams 256 Ounces, Avoirdupois Kilograms 0.028349523 Pounds, Avoirdupois Grams 453.59237 Ounces, Avoirdupois Avoirdupois Pounds 0.0625 Pounds, Avoirdupois Grains 7000 Ounces, Avoirdupois Troy Pounds 0.07595486 Pounds, Troy Kilograms 0.373241722 Ounces, Avoirdupois Troy Ounces 0.9114583 Pounds, Troy Avoirdupois Pounds 0.8228571 - 15 - - 16 - Formulas and Conversions Formulas and Conversions To Convert To Multiply By To Convert To Multiply By Pounds, Troy Troy Ounces 12 Tons, Short Long Tons 0.8928571 Pounds, Troy Avoirdupois Ounces 13.16571 Tons, Short Metric Tons 0.90718474 Pounds, Troy Avoirdupois Drams 210.6514 Tons, Short Long Hundredweights 17.85714 Pounds, Troy Pennyweights 240 Tons, Short Short Hundredweights 20 Pounds, Troy Grams 373.2417216 Tons, Short Kilograms 907.18474 Pounds, Troy Grains 5760 Tons, Short Avoirdupois Pounds 2000 Quintals Metric Tons 0.1 Quintals Kilograms 100 Quintals Avoirdupois Pounds 220.46226 Scruples Troy Drams 0.333 Scruples Grams 1.2959782 Scruples Grains 20 Tons, Long (Deadweight) Metric Tons 1.016046909 Tons, Long (Deadweight) Short Tons 1.12 Tons, Long (Deadweight) Long Hundredweights 20 Tons, Long (Deadweight) Short Hundredweights 22.4 Tons, Long (Deadweight) Kilograms 1016.04691 Tons, Long (Deadweight) Avoirdupois Pounds 2240 Tons, Long (Deadweight) Avoirdupois Ounces 35840 Tons, Metric Long Tons 0.9842065 Tons, Metric Short Tons 1.1023113 Tons, Metric Quintals 10 Tons, Metric Long Hundredweights 19.68413072 Tons, Metric Short Hundredweights 22.04623 Tons, Metric Kilograms 1000 Tons, Metric Avoirdupois Pounds 2204.623 Tons, Metric Troy Ounces 32150.75 - 17 - E. Density Conversions To Convert To Multiply By Grains/imp. Gallon Parts/million 14.286 Grains/US gallon Parts/million 17.118 Grains/US gallon Pounds/million gal 142.86 Grams/cu. Cm Pounds/mil-foot 3.405E-07 Grams/cu. Cm Pounds/cu. in 0.03613 Grams/cu. Cm Pounds/cu. ft 62.43 Grams/liter Pounds/cu. ft 0.062427 Grams/liter Pounds/1000 gal 8.345 Grams/liter Grains/gal 58.417 Grams/liter Parts/million 1000 Kilograms/cu meter Pounds/mil-foot 3.405E-10 Kilograms/cu meter Pounds/cu in 0.00003613 Kilograms/cu meter Grams/cu cm 0.001 Kilograms/cu meter Pound/cu ft 0.06243 Milligrams/liter Parts/million 1 Pounds/cu ft Pounds/mil-foot 5.456E-09 Pounds/cu ft Pounds/cu in 0.0005787 - 18 - Formulas and Conversions Formulas and Conversions To Convert To Multiply By Pounds/cu ft Grams/cu cm 0.01602 Pounds/cu ft Kgs/cu meter 16.02 Pounds/cu in Pounds/mil-foot 0.000009425 Pounds/cu in Gms/cu cm 27.68 Pounds/cu in Pounds/cu ft 1728 Pounds/cu in Kgs/cu meter 27680 Substance Relative Density Sand (dry) 1.42 Carbon (graphite) 2.3 Silicon 2.6 Carbon (charcoal) 1.8 Relative Density (Specific Gravity) Of Various Substances Substance Relative Density 10.57 Chromium F. Silver 6.5 Slate 2.1-2.8 Clay 1.9 Sodium 0.97 1.36-1.4 7.87 Water (fresh) 1.00 Coal Mica 2.9 Steel (mild) Water (sea average) 1.03 Cobalt 8.6 Nickel 8.6 Sulphur 2.07 Aluminum 2.56 Copper 8.77 Oil (linseed) 0.94 Tin 7.3 Antimony 6.70 Cork 0.24 Oil (olive) 0.92 Tungsten 19.1 Bismuth 9.80 Glass (crown) 2.5 Oil (petroleum) 0.76-0.86 Wood (ash) 0.75 Brass 8.40 Glass (flint) 3.5 Oil (turpentine) 0.87 Wood (beech) 0.7-0.8 Brick 2.1 Gold 19.3 Paraffin 0.86 Wood (ebony) 1.1-1.2 Calcium 1.58 Iron (cast) 7.21 Platinum 21.5 Wood (elm) 0.66 3.4 Iron (wrought) 7.78 Carbon (diamond) - 19 - - 20 - Formulas and Conversions Substance Relative Density Wood (lignum-vitae) 1.3 Formulas and Conversions Name Lower Case Upper Case Eta η Η Lead 11.4 Theta θ Θ Magnesium 1.74 Iota ι Ι Manganese 8.0 Kappa κ Κ Mercury 13.6 Lambda λ Λ Lead 11.4 Mu µ Μ Magnesium 1.74 Nu ν Ν Manganese 8.0 Xi ξ Ξ Omicron ο Ο Pi π Π Wood (oak) 0.7-1.0 Wood (pine) 0.56 Wood (teak) 0.8 Rho ρ Ρ Zinc 7.0 Sigma σ and ς Σ Wood (oak) 0.7-1.0 Tau τ Τ Wood (pine) 0.56 Upsilon υ Υ Wood (teak) 0.8 Phi φ Φ Zinc 7.0 Chi χ Χ Mercury 13.6 Psi ψ Ψ Omega ω Ω G. Greek Alphabet Name Lower Case Upper Case Alpha α Α Beta β Β Gamma γ Γ Delta δ ∆ Epsilon ε Ε Zeta ζ Ζ - 21 - - 22 - Formulas and Conversions Formulas and Conversions Chapter 3 Multiply by Into Milli Into Centi Into Deci Into MGL* Into Deca Into Hecto Into Kilo System of Units To convert Hecto 105 104 103 102 101 1 10-1 To convert Deca 104 103 102 101 1 10-1 10-2 To convert MGL* 103 102 101 1 10-1 10-2 10-3 To convert Deci 102 101 1 10-1 10-2 10-3 10-4 To convert Centi 101 1 10-1 10-2 10-3 10-4 10-5 To convert Milli 1 10-1 10-2 10-3 10-4 10-5 10-6 The two most commonly used systems of units are as follows: • SI • Imperial SI: The International System of Units (abbreviated "SI") is a scientific method of expressing the magnitudes of physical quantities. This system was formerly called the meter-kilogramsecond (MKS) system. Imperial: A unit of measure for capacity officially adopted in the British Imperial System; British units are both dry and wet Metric System Exponent value Numerical equivalent Representation Example Tera 1012 1000000000000 T Thz (Tera hertz) MGL = meter, gram, liter Giga 109 1000000000 G Ghz (Giga hertz) Example: Mega 106 1000000 M Mhz (Mega hertz) Unit quantity 1 1 Micro 10-6 0.001 µ µF (Micro farads) Nano 10-9 0.000001 n nF (Nano farads) p pF (Pico farads) Pico -12 10 hz (hertz) F (Farads) To convert Kilogram Into Milligram → (1 Kilo X 106 ) Milligrams Physical constants Conversion Chart Symbolic Representation Numerical Equivalent Avogadro's number N 6.023 x 1026 /(kg mol) Bohr magneton B 9.27 x 10-24 Am 252 Boltzmann's constant k 1.380 x 10-23 J/k Stefan-Boltzmann constant 0.000000000001 Name d 5.67 x 10-8 W/(m2K4) Multiply by Into Milli Into Centi Into Deci Into MGL* Into Deca Into Hecto Into Kilo Characteristic impedance of free space Zo (µo/Eo)1/2=120ΠΩ To convert Kilo 106 105 104 103 102 101 1 Electron volt eV 1.602 x 10-19 J Electron charge e 1.602 x 10-19 C - 23 - - 24 - Formulas and Conversions Formulas and Conversions Name Symbolic Representation Numerical Equivalent Name Symbolic Representation Numerical Equivalent Electronic rest mass me 9.109 x 10-31 kg Acceleration due to gravity on Earth g 9.80 m s-2 Electronic charge to mass ratio e/me 1.759 x 1011 C/kg Acceleration due to gravity on the Moon gM 1.62 m s-2 Faraday constant F 9.65 x 107 C/(kg mol) Radius of the Earth RE 6.37 x 106 m Permeability of free space µ0 4Π x 10-7 H/m Mass of the Earth ME 5.98 x 1024 kg Permittivity of free space Eo 8.85 x 10-12 F/m Radius of the Sun RS 6.96 x 108 m Planck's constant h 6.626 x 10-34 J s Mass of the Sun MS 1.99 x 1030 kg Radius of the Moon RM 1.74 x 106 m Proton mass mp 1.672 x 10-27 kg Mass of the Moon MM 7.35 x 1022 kg Proton to electron mass ratio mp/me 1835.6 Earth-Moon distance - 3.84 x 108 m Standard gravitational acceleration g 9.80665 m/s2, 9.80665 N/kg Earth-Sun distance - 1.50 x 1011 m Speed of light in air c 3.00 x 108 m s-1 Universal constant of gravitation G 6.67 x 10-11 N m2/kg2 Electron charge e -1.60 x 10-19 C Universal gas constant Ro 8.314 kJ/(kg mol K) Mass of electron me 9.11 x 10-31 kg 2.9979 x 10 m/s Planck's constant h 6.63 x 10-34 J s C 5/9(0F - 32) Universal gravitational constant G 6.67 x 10-11 N m2 kg-2 K 5/9(0F + 459.67), 5/90R, 0C + 273.15 Electron volt 1 eV 1.60 x 10-19 J Mass of proton mp 1.67 x 10-27 kg Acceleration due to gravity on Earth g 9.80 m s-2 Acceleration due to gravity on the Moon gM 1.62 m s-2 Ton 1 ton 1.00 x 103 kg Velocity of light in vacuum Temperature Temperature C 0 8 Speed of light in air c 3.00 x 108 m s-1 Electron charge e -1.60 x 10-19 C Mass of electron me 9.11 x 10-31 kg Planck's constant h 6.63 x 10-34 J s Universal gravitational constant G 6.67 x 10-11 N m2 kg-2 Electron volt 1 eV 1.60 x 10-19 J Mass of proton mp 1.67 x 10-27 kg - 25 - - 26 - Formulas and Conversions Formulas and Conversions Identity a+0 = 0+a = a Inverse a + (-a) = 0, a(1/a) = 1 Cancellation If a+x=a+y, then x=y Zero-factor a0 = 0a = 0 Negation -(-a) = a, (-a)b= a(-b) = -(ab), (-a)(-b) = ab Chapter 4 General Mathematical Formulae 4.1 Algebra A. Expansion Formulae Square of summation • (x + y) 2 = x2 + 2xy + y2 Square of difference Algebraic Combinations • (x – y) 2 = x2 – 2xy + y2 Factors with a common denominator can be expanded: a+b a b =+ c cc Difference of squares • x2 – y2 = (x + y) (x – y) Cube of summation • (x + y) 3 = x3 + 3x2y + 3xy2 + y3 Fractions can be added by finding a common denominator: a b ad + bc += cd cd Summation of two cubes • x3 + y3 = (x + y) (x2 - xy + y2) Products of fractions can be carried out directly: a b ab ×= c d cd Cube of difference • (x – y) 3 = x3 – 3x2y + 3xy2 – y3 Difference of two cubes • x3 – y3 = (x – y) (x2 + xy + y2) Quotients of fractions can be evaluated by inverting and multiplying: a b = a × d = ad c b c bc d B. Quadratic Equation • If ax2 + bx + c = 0, Then x = −b ± b 2 − 4ac 2a Radical Combinations The basic algebraic properties of real numbers a, b and c are: a + b = b + a, ab = ba Associative a = a1/ n n a = b a + b and ab are real numbers Commutative ab = n a n b Description Closure n n Property (a+b) + c = a + (b+c), (ab)c = a(bc) (a+b)c = ac+bc b am = a n nm - 27 - a n m n Distributive n a = mn a - 28 - 2 (L + B) Circumference / Perimeter s1 + s2 + s3 where s1, s2, s3 are the 3 sides of the triangle Rectangle Item Triangle s1 + s2 + s3 4s Square Right triangle Circumference / Perimeter Item 4.2 Geometry 2 1 × B× H 2 1 × B× H 2 Area - 29 - NA NA Surface Area - 30 - NA NA Surface Area NA NA Volume NA NA Volume Formulas and Conversions (Length)(Breadth) = L·B s Area Formulas and Conversions Figure Figure Circle C = 2πr C = πd where Ө and Φ are the 2 base angles Circumference / Perimeter Item Trapezoid 3s where s is the length of each side s1 + s2 + s3 Circumference / Perimeter Equilateral triangle Generic triangle Item A = πr2 ⎛a +b⎞ A=⎜ ⎟h ⎝2⎠ Area 1 bh 2 a+b+c 2 A= s= where - 31 - NA NA Surface Area - 32 - NA NA Surface Area NA NA Volume NA NA Volume Formulas and Conversions s ( s − a)( s − b)( s − c) Area Formulas and Conversions Figure Figure Circumference / Perimeter Sum of all sides 6s Trapezoid Hexagon where D and d are the two axis Item Ellipse 2r + (arc length) Circle Sector (1/4)·D·d·∏ Circumference / Perimeter Item 4 π Dd 2 θ °r 2 1 (b1 + b2 )h 2 A = 2.6s2 Where s is the length of 1 side A= Area - 33 - NA NA Surface Area - 34 - NA NA Surface Area NA NA Volume NA NA Volume Formulas and Conversions D is the larger radius and d is the smaller radius A= A= A= arc × r 2 θ° A= × πr 2 360 Area Formulas and Conversions Figure Figure NA Area NA Circumference / Perimeter NA NA Cube Item Rectangular solid Right cylinder NA NA NA Volume 2 - 35 - 6s s3 NA Surface Area - 36 - S = 2πrh + 2πr2 2 l h + 2wh + 2 Surface Area V = πr2h l ×w ×h Volume Formulas and Conversions A = 4.83 s2 Where s is the length of 1 side 8s Octagon Area Circumference / Perimeter Item Formulas and Conversions Figure Figure NA Area NA Circumference / Perimeter NA NA Pyramid Item Rectangular prism Cone NA NA NA NA Sphere Area Circumference / Perimeter Item perpendicular height 1 base area· 3 - 38 - pi·r(r+sh) 2lh+2lw+2wh Surface Area 43 πr 3 Volume 12 πr h 3 V = lwh Volume Formulas and Conversions - 37 - ½.perimeter· slant height + B S = 4πr2 Surface Area Formulas and Conversions Figure Figure Formulas and Conversions Formulas and Conversions Tangent, Secant and Co-Secant 4.3 Trigonometry A. Pythagoras' Law sin θ cosθ 1 secθ = cosθ 1 cscθ = sin θ tan θ = c2 = a2 + b2 B. Basic Ratios • Sin θ = a/c • Cos θ = b/c • Tan θ = a/b • Cosec θ = c/a • Sec θ = c/b • Cot θ = b/a c a θ b C. Trigonometric Function Values Euler’s Representation e jθ = cos(θ ) + j sin(θ ) Degrees versus Radians • A circle in degree contains 360 degrees • A circle in radians contains 2π radians e− jθ = cos(θ ) − j sin(θ ) e jnθ = cos(nθ ) + j sin(nθ ) hypotenuse θ cosθ = e jθ + e − jθ 2 sin θ = opposite e jθ − e − jθ 2j adjacent 4.4 Logarithm Sine, Cosine and Tangent sin θ = opposite hypotenus cosθ = adjacent hypotenus Sine, Cosine and the Pythagorean Triangle [sin θ ] + [cosθ ] 2 2 = sin 2 θ + cos 2 θ = 1 tan θ = opposite adjacent Definition The logarithm of a number to a particular base is the power (or index) to which that base must be raised to obtain the number. The number 8 written in index form as 8 = 23 The equation can be rewritten in logarithm form as log 2 8 = 3 Logarithm laws The logarithm laws are obtained from the index laws and are: • loga x + loga y = loga xy - 39 - - 40 - Formulas and Conversions • loga x – loga y = loga (x/y) Formulas and Conversions 4.5 Exponents Summary of the Laws of Exponents • loga xy = y loga x Let c, d, r, and s be any real numbers. • loga (1/x) = -loga x c r ⋅ c s = c r+s cr = c r−s , c ≠ 0 cs • loga a = 1 • a (log a x ) cr ⎛c⎞ ⎜ ⎟ = r , d ≠0 d⎠ d ⎝ ( c r ) s = c r ⋅s • loga 1 = 0 (c ⋅ d ) r = c r ⋅ d r c −r = r =x Note: It is not possible to have the logarithm of a negative number. All logarithms must have the same base. Euler Relationship The trigonometric functions are related to a complex exponential by the Euler relationship: e jx = cos x + j sin x − jx e = cos x − j sin x From these relationships the trig functions can be expressed in terms of the complex exponential: Basic Combinations Since the raising of a number n to a power p may be defined as multiplying n times itself p times, it follows that n p1 + p 2 = n p1 n p 2 The rule for raising a power to a power can also be deduced (na)b = nab (ab)n = anbn am/an = am-n e jx + e − jx cos x = 2 e jx − e − jx sin x = 2 where a not equal to zero 4.6 Complex Numbers A complex number is a number with a real and an imaginary part, usually expressed in Cartesian form Hyperbolic Functions The hyperbolic functions can be defined in terms of exponentials. Hyperbolic sine = sinh x = 1 cr e x − e− x 2 Hyperbolic cosine = cosh x = e x + e− x 2 Hyperbolic tangent = tanh x = sinh x e x − e − x = cosh x e x + e x a + jb where j = √-1 and j · j = -1 Complex numbers can also be expressed in polar form Aejθ where A = √a2 +b2 and θ = tan-1 (b/a) The polar form can also be expressed in terms of trigonometric functions using the Euler relationship ejθ = cos θ + j sin θ Euler Relationship The trigonometric functions are related to a complex exponential by the Euler relationship ejx = cos x + j sin x - 41 - - 42 - Formulas and Conversions Formulas and Conversions Chapter 5 e-jθ = cos x - j sin x Engineering Concepts and Formulae From these relationships the trigonometric functions can be expressed in terms of the complex exponential: 5.1 Electricity e jx + e − jx cos x = 2 e jx − e − jx sin x = 2 Ohm's Law I= This relationship is useful for expressing complex numbers in polar form, as well as many other applications. Polar Form, Complex Numbers The standard form of a complex number is a + jb where j = √-1 But this can be shown to be equivalent to the form Aejθ where A = √a2 +b2 and θ = tan-1 (b/a) which is called the polar form of a complex number. The equivalence can be shown by using the Euler relationship for complex exponentials. Ae jθ b⎤ b⎤ ⎡ ⎡ = a + b (cos ⎢ tan −1 ⎥ + j sin ⎢ tan −1 ⎥ ) a⎦ a⎦ ⎣ 2 2 Ae jθ = a 2 + b 2 ( a a2 + b2 +j b a2 + b2 ) = a + jb V R Or V = IR Where I = current (amperes) E = electromotive force (volts) R = resistance (ohms) Temperature correction Rt = Ro (1 + αt) Where Ro = resistance at 0ºC (.) Rt = resistance at tºC (.) α = temperature coefficient which has an average value for copper of 0.004 28 (Ω/Ω ºC) R2 = R1 (1 + αt2 ) (1 + αt1 ) Where R1 = resistance at t1 R2 = resistance at t2 Values of alpha Copper 0.00428 Platinum 0.00358 Nickel 0.00672 Tungsten - 43 - Ω/Ω ºC 0.00450 - 44 - Formulas and Conversions Aluminum Current, I = Formulas and Conversions 0.0040 Where EG = generated e.m.f. EB = generated back e.m.f. Ia = armature current Ra = armature resistance nqvtA = nqvA t Alternating Current Conductor Resistivity R= ρL a Where ρ = specific resistance (or resistivity) (ohm meters, Ωm) L = length (meters) a = area of cross-section (square meters) Quantity Equation Resistance R of a uniform conductor L R=ρ A Resistors in series, Rs Rs = R1 + R2 + R3 Resistors in parallel, R p 1 1 1 1 = + + R p R1 R2 R3 Power dissipated in resistor: Potential drop across R P = VI = I 2 R = Slip of Induction Motor [(Slip speed of the field – Speed of the rotor) / Speed of the Field] × 100 V2 R V=IR Dynamo Formulae 2ϕNpZ Average e.m.f. generated in each conductor = 60c Where Z = total number of armature conductors c = number of parallel paths through winding between positive and negative brushes Where c = 2 (wave winding), c = 2p (lap winding) Φ = useful flux per pole (webers), entering or leaving the armature p = number of pairs of poles N = speed (revolutions per minute) Generator Terminal volts = EG – IaRa Motor Terminal volts = EB + IaRa - 45 - RMS value of sine curve = 0.707 of maximum value Mean Value of Sine wave = 0.637 of maximum value Form factor = RMS value / Mean Value = 1.11 pN cycles per second Frequency of Alternator = 60 Where p is number of pairs of poles N is the rotational speed in r/min Inductors and Inductive Reactance Physical Quantity Inductors and Inductance Equation VL = L di dt Inductors in Series: LT = L1 + L2 + L3 + . . . . Inductor in Parallel: 1 1 1 1 = + + + ..... L T L1 L 2 L 3 Current build up (switch initially closed after having been opened) At v L ( t) = E e t τ − t v R ( t) = E(1 - e τ ) i(t) = E R L τ= R Current decay (switch moved to a new position) - (1 − e - - t τ ) t i(t) = I o e τ ′ vR(t) = R i(t) vL(t) = − RT i(t) - 46 - Formulas and Conversions Physical Quantity Formulas and Conversions Equation Quantity L τ' = RT Current Divider Rule Equation Alternating Current f = 1/T ϖ=2πf Two impedance values in parallel Complex Numbers: C=a+jb C = M cos θ + j M sin θ I x = IT ZT = Capacitance M = a 2 + b2 ⎛b⎞ θ = tan -1 ⎜ ⎟ ⎝a⎠ Capacitors Polar form: C=M∠θ Capacitor in Series Inductive Reactance |XL| = ω L 1 1 1 1 = + + + ..... C T C1 C 2 C 3 Capacitive Reactance |XC| = 1 / (ω C) Capacitors in Parallel C T = C1 + C 2 + C 3 + ..... Resistance R Charging a Capacitor Impedance Resistance: ZR = R ∠0° Inductance: ZL = XL ∠90° = ω L ∠90° Capacitance: ZC = XC ∠-90° = 1 / (ωC) ∠-90° C= Q V [F] (Farads) t i(t) = E - RC e R v R ( t) = E e - t RC v C ( t) = E(1 - e - t RC ) τ = RC Quantity Equation Ohm’s Law for AC Discharging a Capacitor V=IZ v(t) = Vm sin (ω t ± φ) i(t) = Im sin (ω t ± φ) Phasor Notation V = Vrms ∠ φ V = Vm ∠ φ Components in Series ZT = Z1 + Z2 + Z3 + . . Vx = VT Components in Parallel Vo - τ ′ e R v R ( t) = − Vo e Time Domain Voltage Divider Rule t i(t) = − Zx ZT - - t τ′ t v C ( t) = Vo e τ ′ τ' = RTC Quantity Capacitance Equation C= Q V 1 1 1 1 = + + + ... Z T Z1 Z 2 Z 3 - 47 - - 48 - ZT Zx Z1 Z 2 Z1 + Z 2 Formulas and Conversions Quantity Equation Capacitance of a Parallel-plate Capacitor C= Isolated Sphere Current in AC Circuit RMS Current In Cartesian form εA d E= Formulas and Conversions V d C = 4πεr I= V 1 ⎞⎤ ⎡ ⎛ ⋅ ⎢ R − j ⎜ ωL − ⎟ 2 ωC ⎠ ⎥ ⎡2 ⎛ ⎝ ⎦ 1⎞⎤ ⎣ ⎟⎥ ⎢ R + ⎜ ωL − ωC ⎠ ⎥ ⎝ ⎢ ⎣ ⎦ Amperes In polar form V I= 2 1⎞ ⎛ [ R + ⎜ ωL − ⎟] ωC ⎠ ⎝ ∠ − φ s Amperes 2 Capacitors in parallel Capacitors in series Energy stored in a charged capacitor C = C1 + C2 + C3 Modulus Q 2 1 1 W= = CV 2 = QV 2C 2 2 W= Q If the capacitor is connected to a battery W= 1 CV 2 2 Charging a capacitor Discharging a capacitor R ⎢ ⎣ 1 1 1 1 = + + C C1 C 2 C 3 If the capacitor is isolated For R C circuits ⎡ ⎢ ωL − where φ s = tan −1 ⎢ I= V 1⎞ ⎛ R + ⎜ ωL − ⎟ ωC ⎠ ⎝ 2 1⎤ ωC ⎥ ⎥ ⎥ ⎦ Amperes 2 2 2C Q = Qo (1 - e-t/RC); V = Vo (1 - e-t/RC) Complex Impedance In Cartesian form In polar form Q = Qo e- t/RC V = Vo e-t/RC • If the capacitor is isolated, the presence of the dielectric decreases the potential difference between the plates • If the capacitor is connected to a battery, the presence of the dielectric increases the charge stored in the capacitor. • The introduction of the dielectric increases the capacitance of the capacitor - 49 - Modulus 1⎞ ⎛ Z = R + j ⎜ ωL − ⎟ Ohms ωC ⎠ ⎝ 2 1⎞ ⎛ Z = R 2 + ⎜ ωL − ⎟ ∠φ s Ohms ωC ⎠ ⎝ 1⎤ ⎡ ωL − −1 ⎢ ωC ⎥ Where φ s = tan ⎢ ⎥ R ⎥ ⎢ ⎦ ⎣ 2 1⎞ ⎛ Z = [ R 2 + ⎜ ωL − ⎟ ] Ohms ωC ⎠ ⎝ - 50 - Formulas and Conversions Formulas and Conversions Power dissipation Three Phase Alternators Average power, P = VI cos φ Watts Power dissipation in a resistor P = I R Watts 2 Rectification Controlled half wave rectifier Average DC voltage = Volts Controlled full wave rectifier Average DC voltage = Volts Vm (1 + cos α ) 2π Vm π (1 + cos α ) Star connected Line voltage = 3 · phase voltage Line current = phase current Delta connected Line voltage = phase voltage Line current = 3 · phase current Three phase power P = 3 EL IL cos Φ EL = line voltage IL = line current cos Φ = power factor Electrostatics Quantity Power Factor Instantaneous current, DC Power AC Power Pdc = VI = I 2 R = Equation I= 2 V R Permittivity of free space Pac = Re(V .I ) = VI cos φ Quantity The mean power = P = Irms Vrms = Irms2 R Inductance The instantaneous power = (Io sin wt) (Vo sin (wt + π) The mean power P =0 Capacitance P =0 Formula for a.c. power The mean power = P = Irms Vrms cos φ = 1 CV 2 Joules 2 The instantaneous power = (Io sin (wt + π/2)) (Vo sin wt ) The mean power 10 −9 = 8.85 × 10 −12 Farads 36π Equation Resistance ε0 = (meters)-1 Energy stored in a capacitor Power in ac circuits dv dq Amperes =C dt dt Quantity Equation Coulomb’s law F =k Electric fields E= Due to a conducting sphere carrying charge Q Inside the sphere - 52 - F q E= Due to a point charge - 51 - Q1Q2 r2 Q 4πε o r 2 E=0 Formulas and Conversions Quantity Formulas and Conversions Equation Outside the sphere Quantity E= Just outside a uniformly charged conducting sphere or plate Q 4πε o r 2 Relation between E and V E= σ εo Equation For uniform electric field • An electric field E is a vector • The electric field strength is directly proportional to the number of electric field lines per unit cross-sectional area, • The electric field at the surface of a conductor is perpendicular to the surface. • The electric field is zero inside a conductor. Quantity Equation Suppose a point charge Q is at A. The work done in bringing a charge q from infinity to some point a distance r from A is Electric potential Qq 4πε o r W= V= Due to a point charge W q V= Due to a conducting sphere, of radius a, carrying charge Q: Inside the sphere Outside the sphere Q 4πε o r V= Q 4πε o a V= E=− E= Physical Quantity Equation Magnetic flux density (also called the Bfield) is defined as the force acting per unit current length. B= Force on a current-carrying conductor in a magnetic field Force on a moving charged particle in a magnetic field Circulating Charges Work done in bringing charge q from A of potential VA to point B of potential VB W = q (VB – VA) F = I l BF = I l · B And Magnitude of F = F = I l B sin θ F=q v · B mv 2 r Calculation of magnetic flux density Physical Quantity Equation Magnetic fields around a long straight wire carrying current I B= µo I 2πa where a = perp. distance from a very long straight wire. Magnetic fields inside a long solenoid, carrying current I: B = µo n I, where n = number of turns per unit length. Hall effect At equilibrium Q The current in a material is given by - 53 - F Il qvB = 4πε o r U = qV V d Magnetostatics Q If the potential at a point is V, then the potential energy of a charge q at that point is dV dx VH = QvB and d I = nQAv - 54 - VH = B v d Formulas and Conversions Formulas and Conversions Physical Quantity Equation Quantity The forces between two current-carrying conductors µIIl F21 = o 1 2 2πa Energy stored in an inductor: Equation Transformers: Physical Quantity Equation The torque on a rectangular coil in a magnetic field T = F b sin θ = N I l B b sinθ = N I A B sinθ If the coil is in a radial field and the plane of the coil is always parallel to the field, then T = N I A B sin θ = N I A B sin 90o =NIAB Magnetic flux φ φ = B A cos θ and Flux-linkage = Current Sensitivity SI = θ I = Lenz's law The direction of the induced e.m.f. is such that it tends to oppose the flux-change causing it, and does oppose it if induced current flows. I= When a great load (or smaller resistance) is connected to the secondary coil, the flux in the core decreases. The e.m.f., εp, in the primary coil falls. Vp -εp = I R; I = d φ dt Equation Power E.m.f. induced in a straight conductor E.m.f. induced between the center and the rim of a spinning disc ε = B πr2f E.m.f. induced in a rotating coil Ε = N A B w sin wt P= W = VI t I= q t ε =BLv ε dI / dt Nφ =LI - 55 - Electric current Work W = qV Ohm’s Law V = IR Resistances in Series Equation L=− R Kirchoff's second law (Loop Theorem) The net e.m.f. round a circuit is equal to the sum of the p.d.s round the loop. Physical Quantity Self-induction VP − ε p Kirchoff's first law (Junction Theorem) At a junction, the total current entering the junction is equal to the total current leaving the junction. EMF Equations Quantity E (1 − e − Rt / L ) R Kirchoff’s laws NAB c ε = −N VS N S = VP N P The L R (d.c.) circuit: Nφ 12 LI 2 U= R T = R1 + R 2 K Resistances in Parallel 1 1 1 = + K R T R1 R 2 Magnetic flux Φ = BA - 56 - Formulas and Conversions Formulas and Conversions Impulse = force · time = change of momentum Ft=mv–mu (Φ 2 − Φ 1 ) t emf = l v B Electromagnetic induction Emf = − N Magnetic force F=I l B Transformer turns ratio Vs = Vp Newton's third law of motion When two objects interact, they exert equal and opposite forces on one another. "Third-law pair" of forces act on two different bodies. Universal Law F = Gmsmp/d2 Ns Np ms is the mass of the sun. mp is the mass of the planet. The Universal law and the second law must be consistent Electromagnetic spectrum Newton’s Laws of Motion and Their Applications Wavelength 102 λ (m) 10 10-1 10-2 10-3 10-4 10-5 10-6 10-7 10-8 10-9 10-10 10-11 1 radio frequencies Physical Quantity v av = Average velocity X-rays Area of Spectrum Equations Acceleration visible microwaves ultraviolet radiation s v+u = t 2 a= v-u t Momentum Force f(Hz) 6 10 7 10 10 8 9 10 10 10 10 11 12 13 10 10 10 Frequency 14 15 10 16 10 10 17 18 10 19 10 10 20 Note: 1. Shaded areas represent regions of overlap. 2. Gamma rays and X-rays occupy a common region. 5.2 Applied Mechanics F = ma Weight gamma rays infrared radiation p = mv weight = mg Work done W = Fs Kinetic energy 1 E k = 2 mv 2 Gravitational potential energy E p = mgh Equations of motion a= v−u ; t 1 s = ut + 2 at 2 ; 5.2.1 Newton's laws of motion Centripetal acceleration a= Newton' first law of motion The inertia of a body is the reluctance of the body to change its state of rest or motion. Mass is a measure of inertia. v2 r F = ma = Newton’s Law of Universal Gravitation Newton's second law of motion mv-mu ; F= ∆t Centripetal force F=G F=ma - 57 - - 58 - mv 2 r m1m2 r2 v 2 = u 2 + 2as Formulas and Conversions Physical Quantity Formulas and Conversions Equations Gravitational field strength g=G Physical Quantity Moment of a force M r2 Equations Stress 1 ft m = 3.28 2 s2 s Acceleration due to gravity, g is 9.81 m/s2 5.2.2 Linear Velocity and Acceleration Quantity Equations If u initial velocity and v final velocity, then displacement s, M = rF ∑M = 0 Principle of moments Conversion: ⎛v+u⎞ s=⎜ ⎟ ⎝2⎠ Stress = Young’s Modulus If t is the elapsed time Strain = Strain F A ∆l l If a is the acceleration Y= F/A ∆ l/ l s = ut + 12 at 2 v 2 = u 2 + 2as Angular Velocity and Acceleration Quantity Vector: a property described by a magnitude and a direction θ= • ω angular velocity (radians/s); ω1 = initial, ω2 = final Velocity: vector property equal to displacement / time The magnitude of velocity may be referred to as speed In SI the basic unit is m/s, in Imperial ft/s Other common units are km/h, mi/h Conversions: 1m/s = 3.28 ft/s 1km/h = 0.621 mi/h ω1 + ω 2 2 ×t 1 2 θ = ω 1t + αt 2 α angular acceleration (radians/s2) ω 2 2 = ω 1 2 + 2αθ Linear displacement s=rθ Linear velocity v=rω Linear, or tangential acceleration Speed of sound in dry air is 331 m/s at 0°C and increases by about 0.61 m/s for each °C rise. Speed of light in vaccum equals 3 x 108m/s Equations θ angular displacement (radians) Scalar: a property described by a magnitude only aT = r α Tangential, Centripetal and Total Acceleration 2 In SI the basic unit is m/s In Imperial ft/s2 - 59 - Quantity Equations Tangential acceleration aT is due to angular acceleration α Acceleration: vector property equal to change in velocity time. aT = rα - 60 - Formulas and Conversions Formulas and Conversions Centripetal (Centrifugal) acceleration ac is due to change in direction only ac = v2/r = r ω2 Total acceleration, a, of a rotating point experiencing angular acceleration is the vector sum of aT and ac a = aT + ac Kinetic Energy 1 mk 2ω 2 2 Where k is radius of gyration, ω is angular velocity in rad/s ER = Kinetic Energy of Rotation Er = 5.2.3 Force Vector quantity, a push or pull which changes the shape and/or motion of an object In SI the unit of force is the newton, N, defined as a kg m In Imperial the unit of force is the pound lb Conversion: 9.81 N = 2.2 lb Weight The gravitational force of attraction between a mass, m, and the mass of the Earth In SI weight can be calculated from Weight = F = mg, where g = 9.81 m/s2 In Imperial, the mass of an object (rarely used), in slugs, can be calculated from the known weight in pounds weight m= g ft g = 32.2 2 s Torque Equation T = I α where T is the acceleration torque in Nm, I is the moment of inertia in kg m2 and α is the angular acceleration in radians/s2 Momentum Vector quantity, symbol p, p = mv [Imperial p = (w/g)v, where w is weight] in SI unit is kgm / s Work Scalar quantity, equal to the (vector) product of a force and the displacement of an object. In simple systems, where W is work, F force and s distance W=Fs In SI the unit of work is the joule, J, or kilojoule, kJ 1 J = 1 Nm In Imperial the unit of work is the ft-lb Energy Energy is the ability to do work, the units are the same as for work; J, kJ, and ft-lb 1 Iω 2 2 Where I = mk2 is the moment of inertia 5.2.4 Centripetal (Centrifugal) Force mv 2 r Where r is the radius Where ω is angular velocity in rad/s Fc = Potential Energy Quantity Equation Energy due to position in a force field, such as gravity Ep = m g h In Imperial this is usually expressed Ep = w h Where w is weight, and h is height above some specified datum Thermal Energy In SI the common units of thermal energy are J, and kJ, (and kJ/kg for specific quantities) In Imperial, the units of thermal energy are British Thermal Units (Btu) Conversions 1 Btu = 1055 J 1 Btu = 778 ft-lb Electrical Energy In SI the units of electrical energy are J, kJ and kilowatt hours kWh. In Imperial, the unit of electrical energy is the kWh Conversions 1 kWh = 3600 kJ 1 kWh = 3412 Btu = 2.66 x 106 ft-lb Power - 61 - - 62 - Formulas and Conversions A scalar quantity, equal to the rate of doing work In SI the unit is the Watt W (or kW) J 1W = 1 s In Imperial, the units are: Mechanical Power – (ft – lb) / s, horsepower h.p. Thermal Power – Btu / s Electrical Power - W, kW, or h.p. Formulas and Conversions • 1 atmosphere (atm) = 101.3 kPa = 14.7 psi Simple Harmonic Motion Velocity of P = ω R 2 − x 2 m s 5.2.5 Stress, Strain And Modulus Of Elasticity Young’s modulus and the breaking stress for selected materials Conversions 746W = 1h. p. 1h. p. = 550 Material ft − lb s Breaking stress x 108 Pa Aluminium 0.70 2.4 Copper 1.16 4.9 Brass Btu 1kW = 0.948 s 0.90 4.7 Iron (wrought) In Imperial, the basic unit is the pound per square inch, psi Atmospheric Pressure At sea level atmospheric pressure equals 101.3 kPa or 14.7 psi Pressure Conversions 1 psi = 6.895 kPa Pressure may be expressed in standard units, or in units of static fluid head, in both SI and Imperial systems Common equivalencies are: • 1 kPa = 0.294 in. mercury = 7.5 mm mercury • 1 kPa = 4.02 in. water = 102 mm water • 1 psi = 2.03 in. mercury = 51.7 mm mercury • 1 psi = 27.7 in. water = 703 mm water • 1 m H2O = 9.81 kPa Other pressure unit conversions: • 1 bar = 14.5 psi = 100 kPa • 1 kg/cm2 = 98.1 kPa = 14.2 psi = 0.981 bar - 63 - 11.0 Glass 0.55 10 4.10 20 Bone N m2 3.0 2.10 Tungsten A vector quantity, force per unit area In SI the basic units of pressure are pascals Pa and kPa 1.93 Mild steel Pressure 1Pa = 1 Young modulus x 1011 Pa 0.17 1.8 5.3 Thermodynamics 5.3.1 Laws of Thermodynamics • W = P∆V • ∆U = Q – W • W= nRT lnVf/Vi • Q = Cn∆T • Cv= 3/2R • Cp= 5/2R • Cp/Cv = γ= 5/3 • e = 1 – Qc/Qh = W/Qh • ec = 1 – Tc/Th • COP = Qc/W (refrigerators) • COP = Qh /W (heat pumps) • Wmax= (1-Tc/Th)Qh • ∆S = Q/T - 64 - Formulas and Conversions Formulas and Conversions • ∑ Fy = 0 • ∑τ = 0 5.3.2 Momentum • p = mv • ∑F = ∆p/∆t (any axis) 5.3.8 Gravity 5.3.3 Impulse I = Fav∆ t = mvf – mvi 5.3.4 Elastic and Inelastic collision • miv1i + m2v2i = m1v1f + m2v2f • (½) miv1i2 + (½) m2v2i2 = ½ m1v1f2 + ½ m2v2f2 • miv1i + m2v2i = (m1 + m2)vf 5.3.5 Center of Mass • xcm = ∑mx/M • Vcm = ∑mv/M • Acm = ∑ma/M • MAcm = Fnet • F = Gm1m2/r2 • T = 2π / √r3 /GMs • G = 6.67 x 10-11N-m2/kg2 • g = GME / R2E • PE = - Gm1m2 / r • ve = √2GME / RE • vs = √GME / r • ME = 5.97 x 1024 kg • RE = 6.37 x 106 m 5.3.9 Vibrations & Waves • F = -kx • PEs = ½kx2 • x = Acosθ = Acos(ωt) • v = -Aωsin(ωt) • a = -Aω2cos(ωt) • ω = √k / m •f = 1 / T • T = 2π√m / k • E = ½kA2 • T = 2π√L / g • vmax = Aω • amax = Aω2 •v = λ f v = √FT/µ • µ = m/L • I = P/A • β = 10log(I/Io) • Io = 1 x 10-12 W/m2 • f’ = f[(1 ± v0/v)/(1 m vs/v)] • Surface area of the sphere = 4πr2 • Speed of sound waves = 343 m/s 5.3.6 Angular Motion • s = rθ • vt = rω • at = rα • ac = vt2/r = rω2 • ω = 2π/T • 1 rev = 2π rad = 360o For constant α • ω = ω o + αt • ω2 = ωo2 +2αθ • θ = ω o t + ½α t 2 • θ = (ωo + ω)·t/2 • I = ∑mr2 • KER = ½Iω2 • τ = rF • ∑τ = Iα • WR = τθ • L = Iω • ∑τ = Iα • WR = τθ • L = Iω • Li = Lf 5.3.10 Standing Waves • fn = nf1 • fn = nv/2L (air column, string fixed both ends) n = 1,2,3,4……. • fn = nv/4L (open at one end) n = 1,3,5,7……… 5.3.11 Beats 5.3.7 Conditions of Equilibrium • fbeats = | f1 – f2 | • ∑ Fx = 0 • Fluids - 65 - - 66 - Formulas and Conversions • ρ = m/V • P = F/A • P2 = P1 + ρgh • Patm = 1.01 x 105Pa = 14.7 lb/in2 • FB = ρfVg = Wf (weight of the displaced fluid) • ρo/ρf = Vf /Vo (floating object) 3 • ρwater = 1000 kg/m • Wa=W-FB Equation of Continuity: Av = constant Bernoulli’s equation: P + ½ ρv2 + ρgy = 0 5.3.12 Temperature and Heat • TF= 9/5TC+32 • TC= 5/9(TF-32) • ∆TF = 9/5∆TC • T= TC+273.15 • ρ= m/v • ∆L = αLo∆T • ∆A = γAo∆T • ∆V = βVo∆T β=3α • Q = mc∆T • Q = mL • 1 kcal = 4186 J • Heat Loss = Heat Gain • Q = (kA∆T)t/L, • H = Q/t =(kA∆T)/L • Q = eσT4At • P = Q/t • P = σAeT4 • P net= σAe(T4-TS4) • σ = 5.67 × 10-8 W/m 2K4 Formulas and Conversions 5.3.14 Elastic Deformation • P = F/A • Y = FLo/A∆L • S = Fh/A∆x • B = –Vo∆F / A∆V • Volume of the sphere = 4πr3/3 • 1 atm = 1.01 × 105 Pa 5.3.15 Temperature Scales • °C = 5/9 (°F – 32) • °F = 5/9 (°C + 32) • °R = °F + 460 (R Rankine) • K = °C + 273 (K Kelvin) 5.3.16 Sensible Heat Equation • Q=mc∆T • M=mass • C=specific heat • ∆T=temperature chance 5.3.17 Latent Heat • Latent heat of fusion of ice = 335 kJ/kg • Latent heat of steam from and at 100°C = 2257 kJ/kg • 1 tonne of refrigeration = 335 000 kJ/day = 233 kJ/min 5.3.18 Gas Laws Boyle’s Law When gas temperature is constant PV = constant or P1V1 = P2V2 Where P is absolute pressure and V is volume Charles’ Law 5.3.13 Ideal Gases When gas pressure is constant, V = const. T or • PV = nRT • R = 8.31 J/mol K • PV = NkT • NA = 6.02 × 1023 molecules/mol • k = 1.38 × 10-23 J/K • M=NAm • (KE)av=(1/2mv2 )av= 3/2kT • U= 3/2NkT = 3/2nRT V1 V2 = T1 T2 where V is volume and T is absolute temperature - 67 - - 68 - Formulas and Conversions Formulas and Conversions Gay-Lussac's Law When gas volume is constant, P = const. T GAS Specific Heat at Constant Volume kJ/kgK or kJ/kg oC Ratio of Specific γ= cp / cv Helium 1.105 0.85 1.30 2.177 1.675 1.30 1.043 0.745 1.40 Oxygen 0.913 0.652 1.40 Sulphur Dioxide 0.632 0.451 1.40 5.3.20 Efficiency of Heat Engines Carnot Cycle T1 − T2 T1 where T1 and T2 are absolute temperatures of heat source and sink Also PV = nRoT where P = absolute pressure (kPa) V = volume (m3) T = absolute temperature K N = the number of kmoles of gas Ro = the universal gas constant 8.314 kJ/kmol/K η= Air Standard Efficiencies 5.3.19 Specific Heats Of Gases Specific Heat at Constant Pressure kJ/kgK or kJ/kg oC Specific Heat at Constant Volume kJ/kgK or kJ/kg oC Ratio of Specific γ= cp / cv Air 1.005 0.718 1.40 Ammonia 2.060 1.561 1.32 Carbon Dioxide 0.825 0.630 1.31 Carbon Monoxide 1.051 0.751 1.40 - 69 - 1.41 Nitrogen P1V1 P2V 2 = = const. T1 T2 P V = m R T where P = absolute pressure (kPa) V = volume (m3) T = absolute temp (K) m = mass (kg) R = characteristic constant (kJ/kgK) 1.66 10.096 Methane General Gas Law 3.153 14.235 Hydrogen Sulphide where P is absolute pressure and T is absolute temperature 5.234 Hydrogen or P1 P2 = T1 T2 GAS Specific Heat at Constant Pressure kJ/kgK or kJ/kg oC Spark Ignition Gas and Oil Engines (Constant Volume Cycle) 1 η = 1 − (γ −1) rv rv= compression ratio γ = specific heat (constant pressure) / Specific heat (constant volume) Diesel Cycle η =1− Rγ − 1) γ −1 rv γ ( R − 1) Where r = ratio of compression R = ratio of cut-off volume to clearance volume High Speed Diesel (Dual-Combustion) Cycle η =1 kβ γ − 1 rv γ −1 [(k − 1) + γk ( β − 1)] - 70 - Formulas and Conversions Where rv= cylinder volume / clearance volume k = absolute pressure at the end of constant V heating (combustion) / absolute pressure at the beginning of constant V combustion β = volume at the end of constant P heating (combustion) / clearance volume Gas Turbines (Constant Pressure or Brayton Cycle) 1 η =1− r ⎛ γ −1 ⎞ ⎜ ⎜γ⎟ ⎟ ⎠ p⎝ where rp = pressure ratio = compressor discharge pressure / compressor intake pressure 5.3.21 Heat Transfer by Conduction Material Coefficient of Thermal Conductivity W/m °C Formulas and Conversions 5.3.22 Thermal Expansion of Solids Increase in length = L α (T2 – T1) Where L = original length α = coefficient of linear expansion (T2 – T1) = rise in temperature Increase in volume = V β (T2 – T1) Where V = original volume β = coefficient of volumetric expansion (T2 – T1) = rise in temperature Coefficient of volumetric expansion = Coefficient of linear expansion × 3 β = 3α 5.3.23 Chemical Heating Value of a Fuel Chemical Heating Value MJ per kg of fuel = 33.7C + 144( H 2 − C is the mass of carbon per kg of fuel H2 is the mass of hydrogen per kg of fuel O2 is the mass of oxygen per kg of fuel S is the mass of sulphur per kg of fuel Air 0.025 Brass 104 Concrete 0.85 Cork 0.043 Glass 1.0 Iron, cast 70 Steel 60 Wallboard, paper 0.076 Aluminum 206 Brick 0.6 Copper 380 Boiler Efficiency Felt 0.038 m s (h1 − h2 ) mf × (calorificvalue) Theoretical Air Required to Burn Fuel ⎡8 ⎤ 100 Air (kg per kg of fuel) = ⎢ C + 8( H 2 − O2 ) + S ⎥ ⎣3 ⎦ 23 Air Supplied from Analysis of Flue Gases Air in kg per kg of fuel = N2 ×C 33(CO2 + CO) Boiler Formulae m s (h1 − h2 ) 2257 kj / kg (h1 − h2 ) Factor of evaporation = 2257 kj / kg Equivalent evaporation = Glass, fibre 0.04 Plastic, cellular 0.04 Where Wood 0.15 ms = mass flow rate of steam h1 = enthalpy of steam produced in boiler h2 = enthalpy of feedwater to boiler mf = mass flow rate of fuel - 71 - - 72 - O2 ) + 9.3S 8 0 1 Constant pressure P=Pressure Isothermal T=Constant Polytropic PVn = Constant γ -- T1 P =1 T2 P2 T-P P ⎡V2 ⎤ 1 =⎢ ⎥ P2 ⎣ V1 ⎦ n T1 ⎡ P ⎤ = ⎢ 1⎥ T2 ⎣ P2 ⎦ T1 ⎡ P ⎤ = ⎢ 1⎥ T2 ⎣ P2 ⎦ P V2 1 = P2 V1 P ⎡V2 ⎤ 1 =⎢ ⎥ P2 ⎣ V1 ⎦ -- -- -- P-V n −l n γ −l γ -- T1 V1 = T2 V2 -- T-V P(V2-V1) 0 Work done - 73 - mc n (T2 − T1 ) mR (T1 − T2 ) n −1 mc v (T1 − T2 ) ⎛P⎞ mRT log e ⎜ 1 ⎟ ⎜P ⎟ ⎝ 2⎠ 0 ⎛P⎞ mRT log e ⎜ 1 ⎟ ⎜P ⎟ ⎝ 2⎠ mc p (T2 − T1 ) mc v (T2 − T1 ) Heat added - 74 - ⎛γ − n⎞ cm = Specific heat for polytropic process = cv ⎜ ⎟kJ / kgK ⎝ 1− n ⎠ H = Enthalpy, kJ γ = Isentropic Exponent, cp/cv n = polytropic exponent P = Pressure, kPa R = Gas content, kJ/kgK S = Entropy, kJ/K T = Absolute Temperature, K = 273+˚C U = Internal Energy, kJ V = Volume, m3 m = Mass of gas, kg Formulas and Conversions *Can be used for reversible adiabatic processes cv = Specific heat at constant volume, kJ/kgK cp = Specific heat at constant pressure, kJ/kgK n −1 γ −1 T1 ⎡V2 ⎤ =⎢ ⎥ T2 ⎣ V1 ⎦ T1 ⎡V2 ⎤ =⎢ ⎥ T2 ⎣ V1 ⎦ P-V-T Relationships Thermodynamic Equations for perfect gases n γ ∞ Constant Volume V=Constant Isentropic S=Constant Value of n Name of process Formulas and Conversions mc v (T2 − T1 ) mc v (T2 − T1 ) 0 mc v (T2 − T1 ) mc v (T2 − T1 ) Change in Internal Energy mc p (T2 − T1 ) mc p (T2 − T1 ) 0 mc p (T2 − T1 ) mc p (T2 − T1 ) Change in Enthalpy ⎞ ⎟ ⎟ ⎠ ⎛T mc n log e ⎜ 2 ⎜T ⎝1 ⎛T mc n log e ⎜ 2 ⎜T ⎝1 0 ⎞ ⎟ ⎟ ⎠ ⎛P⎞ mR log e ⎜ 1 ⎟ ⎜P ⎟ ⎝ 2⎠ ⎞ ⎟ ⎟ ⎠ ⎛T mc v log e ⎜ 2 ⎜T ⎝1 Change in Entropy 0.909 0.209 0.125 0.383 0.795 0.402 Aluminum Antimony Bismuth Brass Carbon Cobalt 0.130 Glass Gold 12.0 29.0 0.465 Iron (cast) Iron (wrought) 0.389 Zinc 1.800 4.183 1.633 Olive oil Water 0.139 Mercury Turpentine 3.643 Carbon Dioxide 2.135 1.138 Benzine 2.093 0.473 Ammonia Gasoline 2.470 Alcohal Petroleum Specific Heat (at 20 o C ) KJ/kgK or kJ/kg o C Liquid Specific Heat and Volume Expansion for Liquids - 76 - 16.5 26.7 12.0 8.6 3.7 9.4 12.0 1.80 1.82 12.4 11.0 Coefficient of Volume Expansion (Multiply by 10-4) Formulas and Conversions - 75 - 0.230 Tin 19.5 0.235 0.494 Silver 0.741 Silicon Steel (mild) 7.8 0.134 Platinum 13.0 0.131 0.452 Lead Nickel 10.4 0.544 Ice (between -20 C & 0 C ) 50.4 14.2 9.0 16.5 12.3 7.9 18.4 12.4 17.5 23.8 Coefficient of Linear Expansion between 0 o C and 100 o C (multiply by 10-6) 2.135 o 0.388 0.896 Copper o Mean Specific Heat between 0 o C and 100 o C kJ/kgK or kJ/kg o C Specific Heat and Linear Expansion of Solids Formulas and Conversions Formulas and Conversions Formulas and Conversions 5.4 Fluid Mechanics 5.4.1 Discharge from an Orifice Let A = cross-sectional area of the orifice = π 4 And Ac = cross-sectional area of the jet at the vena conrtacta Then Ac = CcA π 4 Where B = breadth (m) H = head (m above sill) Triangular Right Angled Notch: Q = 2.635 H5/2 Where H = head (m above sill) d2 5.4.2 Bernoulli’s Theory dc 2 Or C c = Ac ⎛ d c ⎞ =⎜ ⎟ A ⎝d⎠ H =h+ 2 P v2 + w 2g H = total head (meters) w = force of gravity on 1 m3 of fluid (N) h = height above datum level (meters) v = velocity of water (meters per second) P = pressure (N/m2 or Pa) Loss of Head in Pipes Due to Friction L v2 Loss of head in meters = f d 2g L = length in meters v = velocity of flow in meters per second d = diameter in meters f = constant value of 0.01 in large pipes to 0.02 in small pipes Where Cc is the coefficient of contraction 5.4.3 Actual pipe dimensions Nominal pipe size (in) 10.3 6.8 1.73 3.660 × 10-5 13.7 9.2 2.24 6717 × 10-5 17.1 12.5 2.31 1.236 × 10-4 1/2 - 77 - Flow area (m2) 3/8 • Rectangular notch: Q = 0.62 (B · H) 2/3 √2gh Wall thickness (mm) 1/4 • Or Q = C c AC v 2 gh • Typically, values for Cd vary between 0.6 and 0.65 • Circular orifice: Q = 0.62 A √2gh 3 2 • Where Q = flow (m /s) A = area (m ) h = head (m) Inside diameter (mm) 1/8 At the vena contracta, the volumetric flow rate Q of the fluid is given by • Q = area of the jet at the vena contracta · actual velocity = AcV Outside diameter (mm) 21.3 15.8 2.77 1.960 × 10-4 3/4 26.7 20.9 2.87 3.437 × 10-4 1 33.4 26.6 3.38 5.574 × 10-4 1¼ 42.2 35.1 3.56 9.653 × 10-4 1½ 48.3 40.9 3.68 1.314 ×10-3 2 60.3 52.5 3.91 2.168 × 10-3 - 78 - Formulas and Conversions Formulas and Conversions Nominal pipe size (in) Outside diameter (mm) Inside diameter (mm) Wall thickness (mm) Flow area (m2) 2½ 73.0 62.7 5.16 3.090 × 10-3 3 88.9 77.9 5.49 4.768 × 10-3 3½ 101.6 90.1 5.74 6.381 × 10-3 4 114.3 102.3 6.02 8.213 × 10-3 5 141.3 128.2 6.55 1.291 × 10-2 6 168.3 154.1 7.11 1.864 × 10-2 8 219.1 202.7 8.18 3.226 × 10-2 10 273.1 254.5 9.27 5.090 × 10-2 12 323.9 303.2 10.31 7.219 × 10-2 14 355.6 333.4 11.10 8.729 × 10-2 16 406.4 381.0 12.70 0.1140 18 457.2 428.7 14.27 0.1443 20 508.0 477.9 15.06 0.1794 24 609.6 574.7 17.45 0.2594 Chapter 6 References 6.1 Periodic Table of Elements A 1 1 H 1.00 8 8A 18 2A 2 3A 13 4 3 Li Be 6.94 9.01 1 2 11 12 Na Mg 22.9 24.3 9 1 4A 14 5A 15 6A 16 7A 17 2 He 4.00 3 5 6 7 8 9 10 B C N O F Ne 10.8 12.0 14.0 16.0 19.0 20.1 1 1 1 0 0 8 3B 3 4B 4 5B 5 6B 6 7B 7 8B 8 8B 9 8B 10 1B 11 2B 12 13 14 15 16 17 18 Al Si P S Cl Ar 26.9 28.0 30.9 32.0 35.4 39.9 8 9 7 7 5 5 19 31 32 33 34 35 36 20 21 22 23 24 25 26 27 28 29 30 K Ga Ge As Se Br Kr Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn 39.1 40.0 44.9 47.9 50.9 52.0 54.9 55.8 58.9 58.7 63.5 65.3 69.7 72.5 74.9 78.9 79.9 83.8 0 2 9 2 6 0 0 8 6 0 4 0 4 5 3 0 5 8 37 49 50 51 52 53 54 38 39 40 41 42 43 44 45 46 47 48 Rb In Sn Sb Te I Xe Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd 85.4 87.6 88.9 91.2 92.9 95.9 97.9 101. 102. 106. 107. 112. 114. 118. 121. 127. 126. 131. 7 8 7 8 6 9 3 2 1 2 1 4 1 9 4 9 4 56 55 81 82 83 84 85 86 57 72 73 74 75 76 77 78 79 80 Cs Tl Pb Bi Po At Rn Ba La Hf Ta W Re Os Ir Pt Au Hg 132. 137. 138. 178. 180. 183. 186. 190. 192. 195. 197. 200. 204. 207. 209. (209) (210) (222) 9 4 2 0 3 9 5 9 8 2 2 2 1 0 6 87 88 89 104 105 106 107 108 109 Fr Ra Db Sg Bh Hs Mt Ac Rf (223) 226. 227. (261) (262) (266) (264) (265) (268) 0 0 58 62 67 68 69 70 71 59 63 60 64 61 65 66 Ho Er Tm Yb Lu Ce Sm Pr Eu Nd Gd Pm Tb Dy 140. 140. 144. (145) 150. 152. 157. 158. 162. 164. 167. 168. 173. 175. 9 3 9 0 0 1 4 9 0 2 3 9 5 90 95 96 97 98 99 100 101 102 103 91 92 93 94 Es Fm Md No Lr Th Am Cm Bk Cf Pa U Np Pu 232. 231. 238. 237. (244) (243) (247) (247) (251) (252) (257) (258) (259) (262) 0 0 0 0 - 79 - - 80 - Formulas and Conversions 6.2 Resistor Color Coding Color Value Black 0 Brown 1 Red 2 Orange 3 Yellow 4 Green 5 Blue 6 Violet / Purple 7 Grey 8 White 9 Courtesy: Dick Smith Electronics, Australia - 81 - ABOUT IDC Technologies As one of the world’s leading engineering and technology training, consulting and publishing companies, IDC Technologies’ strength lies in providing practical and useful technical training for engineers, technicians and other technical personnel. Your business grows by developing the skills and expertise of your most important asset – your people. For the past 12 years, we have helped our clients in achieving their business objectives by developing their people. We specialize in the fields of electrical systems, industrial data communications, telecommunications, automation and control, mechanical engineering, project and financial management and are continually adding to our portfolio of over 140 different workshops. Our instructors are highly respected in their fields of expertise and in the last ten years have trained over 140,000 engineers, technicians and other technical personnel. With offices conveniently located worldwide, IDC Technologies has an enthusiastic team of professional engineers, technicians and support staff who are committed to providing the highest quality of training, publishing and consultancy services. Our worldwide offices are located in: Australia Canada Ireland New Zealand Singapore South Africa United Kingdom USA For more information visit our website: www.idc-online.com or email us on idc@idc-online.com Training Workshops and Books Data Communications & Networking Practical Data Communications & Networking for Engineers and Technicians Practical DNP3, 60870.5 & Modern SCADA Communication Systems Practical Troubleshooting & Problem Solving of Ethernet Networks Practical FieldBus and Device Networks for Engineers and Technicians Practical Fiber Optics for Engineers and Technicians Practical Troubleshooting & Problem Solving of Industrial Data Communications Practical Industrial Networking for Engineers & Technicians Practical TCP/IP and Ethernet Networking for Industry Practical Fundamentals of Telecommunications and Wireless Communications Practical Radio & Telemetry Systems for Industry Practical TCP/IP Troubleshooting & Problem Solving for Industry Practical Wireless Networking Technologies for Industry Practical Routers & Switches (Including TCP/IP & Ethernet) for Engineers and Technicians Best Practice in Industrial Data Communications Systems Practical Fundamentals of VOICE over IP (VoIP) for Engineers & Technicians Practical Troubleshooting, Design & Selection of Fiber Optic Systems for Industry Troubleshooting Industrial Ethernet & TCP/IP Networks Back to Basics Wireless Networking & Telemetry Systems for Industry Wireless Networking & Radio Telemetry Systems for Industry Electrical Power Practical Electrical Network Automation & Communication Systems Practical Troubleshooting of Electrical Equipment and Control Circuits Practical Grounding/Earthing, Bonding, Lightning & Surge Protection Practical High Voltage Safety Operating Procedures for Engineers and Technicians Practical Power Distribution Practical Power Quality: Problems & Solutions Practical Power Systems Protection for Engineers and Technicians Practical Variable Speed Drives for Instrumentation and Control Systems Practical Electrical Wiring Standards - IEE BS7671 - 2001 Edition Practical Wind & Solar Power - Renewable Energy Technologies Practical Distribution & Substation Automation (incl. Communications) for Electrical Power Systems Safe Operation & Maintenance of Circuit Breakers and Switchgear Troubleshooting, Maintenance and Protection of AC Electrical Motors & Drives Practical Power Transformers - Operation and Maintenance Lightning, Surge Protection and Earthing of Electrical & Electronic Systems Electronics Practical Digital Signal Processing Systems for Engineers and Technicians Practical Embedded Controllers: Troubleshooting and Design Practical EMC and EMI Control for Engineers and Technicians Practical Industrial Electronics for Engineers and Technicians Practical Image Processing and Applications Practical Shielding, EMC/EMI, Noise Reduction, Earthing and Circuit Board Layout of Electronic Systems Practical Power Electronics & Switch Mode Power Supply Design for Industry Practical Safety Instrumentation & Emergency Shutdown Systems for Process Industries using IEC 61511 and IEC 61508 Practical Fundamentals of E-Manufacturing, Manufacturing Execution Systems (MES) and Supply Chain Management Practical Industrial Programming using 61131-3 for Programmable Logic Controllers (PLCs) Control Valve Sizing, Selection and Maintenance Best Practice in Process, Electrical and Instrumentation Drawings & Documentation Practical Distributed Control Systems (DCS) Mechanical Engineering Information Technology Industrial Network Security for SCADA, Automation, Process Control and PLC Systems Practical Web-Site Development & E-Commerce Systems for Industry Chemical Engineering Practical Fundamentals of Chemical Engineering Instrumentation, Automation & Process Control Practical Analytical Instrumentation in On-Line Applications Practical Alarm Systems Management for Engineers and Technicians Troubleshooting Programmable Logic Controller's for Automation and Process Control Practical Batch Management & Control (Including S88) for Industry Practical Boiler Control and Instrumentation for Engineers and Technicians Practical Programming for Industrial Control - using ( IEC 1131-3 and OPC ) Practical Troubleshooting of Data Acquisition & SCADA Systems for Engineers and Technicians Practical Industrial Flow Measurement for Engineers and Technicians Practical Hazops, Trips and Alarms Practical Hazardous Areas for Engineers and Technicians A Practical Mini MBA in Instrumentation and Automation Practical Instrumentation for Automation and Process Control Practical Intrinsic Safety for Engineers and Technicians Practical Tuning of Industrial Control Loops Practical Motion Control for Engineers and Technicians Practical Fundamentals of OPC (OLE for Process Control) Practical Process Control for Engineers and Technicians Practical Process Control & Tuning of Industrial Control Loops Practical SCADA & Telemetry Systems for Industry Practical Shutdown & Turnaround Management for Engineers and Managers Practical Fundamentals of Heating, Ventilation & Air-conditioning (HVAC) for Engineers & Technicians Practical Boiler Plant Operation and Management for Engineers and Technicians Practical Cleanroom Technology and Facilities for Engineers and Technicians Practical Hydraulic Systems: Operation and Troubleshooting Practical Lubrication Engineering for Engineers and Technicians Practical Safe Lifting Practice and Maintenance Practical Centrifugal Pumps - Optimizing Performance Practical Machinery and Automation Safety for Industry Practical Machinery Vibration Analysis and Predictive Maintenance Practical Pneumatics: Operation and Troubleshooting for Engineers and Technicians Practical Pumps and Compressors: Control, Operation, Maintenance and Troubleshooting Project & Financial Management Practical Financial Fundamentals and Project Investment Decision Making How to Manage Consultants Marketing for Engineers and Technical Personnel Practical Project Management for Engineers and Technicians Practical Specification and Technical Writing for Technical Professionals PAST PARTICIPANTS SAY: “Excellent instructor with plenty of practical knowledge.” Ian Kemp, ANSTO “Excellent depth of subject knowledge displayed.” Hugh Donohue, AMEC “Saved hours of trial and error.” Mario Messwa, DAPS “I’ve gained more useful info from this seminar than any I’ve previously attended.” Jim Hannen, Wheeling-Misshen Inc. “This is the 2nd IDC Technologies class I have taken – both have been excellent!” John Harms, Avista Corporation “A most enjoyable and informative course. Thank you.” Pat V Hammond, Johnson Matthey PLC “Written material was about the best I’ve seen for this type of course. The instructor was able to set an excellent pace and was very responsive to the class.” John Myhill, Automated Control Systems “Excellent, I have taken a TCP/IP Class before and didn’t understand it. After this course, I feel more confident with my newfound knowledge.” John Armbrust, Phelps Dodge “This was one of the best courses I have ever been on. The instructor was excellent and kept me fully interested from start to finish. Really glad I attended.” Chris Mercer, Air Products “Very competent and great presenter.” David Wolfe, Acromag “Well presented, excellent material” Stephen Baron, Air Products “Excellent presentation! Well done.” Brett Muhlhauser, Connell Wagner “Well compiled technical material.” Robert Higgenbotham, Yallourn Energy “Well presented and the instructor obviously has the practical knowledge to back things up.” Mike Mazurak, ANSTO “Great refresher on current practice. Also helped to bring me up to date on new technology.” E. Burnie, Sellotape “I like the practicality of the workshop.” Karl Armfield, Joy Mining TECHNICAL WORKSHOPS TECHNOLOGY TRAINING THAT WORKS We deliver engineering and technology training that will maximize your business goals. In today's competitive environment, you require training that will help you and your organization to achieve its goals and produce a large return on investment. With our "Training that Works" objective you and your organization will: • Get job-related skills that you need to achieve your business goals • Improve the operation and design of your equipment and plant • Improve your troubleshooting abilities • Sharpen your competitive edge • Boost morale and retain valuable staff • Save time and money EXPERT INSTRUCTORS We search the world for good quality instructors who have three key attributes: 1. Expert knowledge and experience – of the course topic 2. Superb training abilities – to ensure the know-how is transferred effectively and quickly to you in a practical hands-on way 3. Listening skills – they listen carefully to the needs of the participants and want to ensure that you benefit from the experience Each and every instructor is evaluated by the delegates and we assess the presentation after each class to ensure that the instructor stays on track in presenting outstanding courses. HANDS-ON APPROACH TO TRAINING All IDC Technologies workshops include practical, hands-on sessions where the delegates are given the opportunity to apply in practice the theory they have learnt. QUALITY MANUALS A fully illustrated workshop manual with hundreds of pages of tables, charts, figures and handy hints, plus considerable reference material is provided FREE of charge to each delegate. ACCREDITATION AND CONTINUING EDUCATION IDC workshops satisfy criteria for Continuing Professional Development for most engineering professional associations throughout the world (incl. The Institution of Electrical Engineers and Institution of Measurement and Control in the UK, Institution of Engineers in Australia, Institution of Engineers New Zealand) CERTIFICATE OF ATTENDANCE Each delegate receives a Certificate of Attendance documenting their experience. 100% MONEY BACK GUARANTEE IDC Technologies’ engineers have put considerable time and experience into ensuring that you gain maximum value from each workshop. If by lunch time of the first day you decide that the workshop is not appropriate for your requirements, please let us know so that we can arrange a 100% refund of your fee. ON-SITE TRAINING On-site training is a cost-effective method of training for companies with several employees to train in a particular area. Organizations can save valuable training dollars by holding courses onsite, where costs are significantly less. Other benefits are IDC's ability to focus on particular systems and equipment so that attendees obtain the greatest benefit from the training. All on-site workshops are tailored to meet with our client's training requirements and courses can be presented at beginners, intermediate or advanced levels based on the knowledge and experience of the delegates in attendance. Specific areas of interest to the client can also be covered in more detail. CUSTOMIZED TRAINING In addition to standard on-site training, IDC Technologies specializes in developing customized courses to meet our client's training needs. IDC has the engineering and training expertise and resources to work closely with clients in preparing and presenting specialized courses. You may select components of current IDC workshops to be combined with additional topics or we can design a course entirely to your specifications. The benefits to companies in adopting this option are reflected in the increased efficiency of their operations and equipment. ON-SITE & CUSTOMIZED TRAINING For more information or a FREE proposal please contact our Client Services Manager: Kevin Baker: business@idc-online.com SAVE OVER 50% SPECIALIST CONSULTING IDC Technologies has been providing high quality specialist advice and consulting for more than ten years to organizations around the world. The technological world today presents tremendous challenges to engineers, scientists and technicians in keeping up to date and taking advantage of the latest developments in the key technology areas. We pride our selves on being the premier provider of practical and cost-effective engineering solutions. PROFESSIONALLY STAFFED IDC Technologies consists of an enthusiastic and experienced team that is committed to providing the highest quality in consulting services. The company has thirty-five professional engineers; quality focused support staff, as well as a vast resource base of specialists in their relevant fields. CLIENT FOCUS IDC’s independence and impartiality guarantee that clients receive unbiased advice and recommendations, focused on providing the best technical and economical solutions to the client's specific and individual requirements. COMPANIES WHO HAVE BENEFITED FROM IDC TECHNOLOGIES’ TRAINING: AUSTRALIA AIR DUCTER • AIR SERVICES • ALCOA • ALINTA GAS • AMPOL REFINERIES •ANSTO • AUSTRALIAN COMMUNICATIONS AUTHORITY • AUSTRALIAN GEOLOGICAL SOCIETY • AUSTRALIAN RAIL ROAD GROUP • BHP BILLITON • BHP BILLITON PETROLEUM DIVISION • BHP IRON ORE • BOC GASES • BOEING CONSTRUCTORS INC • BRISBANE CITY COUNCIL • BRITISH AEROSPACE AUSTRALIA • CAMMS AUSTRALIA PTY LTD • CHK WIRELESS TECHNOLOGIES •CI TECHNOLOGIES • CITIWATER TOWNSVILLE • CITY WEST WATER • CIVIL AVIATION AUTHORITY • COMALCO ALUMINIUM • CSIRO • DELTA ELECTRICITY • DEPT OF DEFENCE • DEPT OF TRANSPORT AND WORKS • DSTO • DUKE ENERGY INTERNATIONAL • EMERSON PROCESS MANAGEMENT • ENERGEX •ERG GROUP • ERGON ENERGY • ETSA • FMC FOODTECH PTY LTD • FOOD SCIENCE AUSTRALIA • GHD CONSULTING ENGINEERS • GIPPSLAND WATER •GLADSTONE TAFE COLLEGE • GORDON BROTHERS INDUSTRIES LTD •GOSFORD CITY COUNCIL • GREAT SOUTHERN ENERGY • HAMERSLEY IRON •HEWLETT PACKARD • HOLDEN • HOLDEN LTD • HONEYWELL • I&E SYSTEMS PTY LTD • INTEGRAL ENERGY • KALGOORLIE NICKEL SMELTER • METRO BRICK• MILLENIUM CHEMICALS • MISSION ENERGY • MT ISA MINES • MURDOCH UNIVERSITY • MURDOCH UNIVERSITY • NABALCO • NEC • NHP ELECTRICAL •NILSON ELECTRIC • NORMANDY GOLD • NORTH PARKES MINES • NU-LEC INDUSTRIES AUSTRALIA LTD • PARKER HANNAFIN • PEAK GOLD MINES •PHARMACIA & UPJOHN • POWER & WATER AUTHORITY NT (PAWA) • POWERCOR • POWERLINK • PROSPECT ELECTRICITY • QETC • QUEENSLAND ALUMINA •RAAF AIRCRAFT RESEARCH AND DEVELOPMENT UNIT • RAAF BASE WILLIAMTOWN • RAYTHEON • RGC MINERAL SANDS • RLM SYSTEMS • ROBE RIVER IRON ASSOCIATES • ROYAL DARWIN HOSPITAL • SANTOS LTD •SCHNEIDER ELECTRIC • SHELL - CLYDE REFINERY • SNOWY MOUNTAIN HYDRO• SPC FRUIT STANWELL POWER STATION • TELSTRA • THOMPSON MARCONI SONAR • TIWEST • TRANSEND NETWORKS PTY LTD • UNCLE BENS • VISION FIRE & SECURITY • WESFARMERS CSBP • WESTERN POWER • WESTRAIL • WMC - KALGOORLIE NICKEL SMELTER • WMC FERTILIZERS • WOODSIDE • WORSLEY ALUMINA • WYONG SHIRE • YOKOGAWA AUSTRALIA BOTSWANA ACTIVEMEDIA INNOVATION PTE LTD • FLOTECH CONTROLS • LAND TRANSPORT AUTHORITY • NGEE ANN POLYTECHNIC • OWER SERAYA LTD • WESTINGHOUSE • YOKOGAWA SINGAPORE SOUTH AFRICA AMATOLA DISTRICT COUNCIL • ANGLO AMERICAN • BATEMAN METALS • CALTEX REFINERIES • CHEVRON ANGOLA • COLUMBUS STAINLESS • DE BEERS KIMBERLEY • DE BEERS VENETIA MINE • DEBEERS DEBTECH • DURBAN METRO• EAST DRIEFONTEIN GOLD MINE • EASTERN CAPE TECH • EMERGENCY SERVICES, METRORAIL • ESKOM • GRINTEK EWATION • HIGHVELD STEEL •HILLSIDE • ILLOVO SUGAR • IMPALA PLATINUMS • ISCOR • IST • JOY MINING •KOEBURG POWER STATION • LEVER PONDS • METSO AUTOMATION •MIDDLEBURG FERROCHROME • MINTEK • MONDI KRAFT • MOSSGAS •NAMAQUA SANDS • NESTLE • NKOMATI MINE • OMNIA FERTILISERS • ORBICOM• OTB • PALABORA MINING • POTGIETERUS MUNICIPALITY • PROCONICS PTY LTD • RAND WATER BOARD • RDI • RICHARDS BAY MINERALS • SA NAVY • SABC• SALDANHA STEEL • SANS FIBRES • SAPPI DURBAN • SASOL COAL • SASOL MSM ROTATING EQUIPMENT • SASOL SYNTHETIC FUELS • SATRA • SILDANHA STEEL • SKILLTEC • SPOORNET • STEINMULLER AFRICA • TRANSTEL EASTERN REGION • UMGENI WATER • WATER UTILISATION CORPORATION • WESTERN PLATINUM • WITWATERSRAND TECHNIKON • YELLAND CONTROLS SWAZILAND SIMUNYE SUGAR TANZANIA GOLDEN PRIDE MINE UNITED ARAB EMIRATES EUROMATECH • PROMIS GROUP UNITED KINGDOM MASIBUS 24 SEVEN • ABB AUTOMATION LTD • AER RIANTA • AIR PRODUCTS • ALLEN STEAM TURBINES/ROLLS ROYCE • ALLIED COLLOIDS • ALLIED DISTILLERS • ALSTOM • AMEC DESIGN & MANAGEMENT • BAE SYSTEMS • BAILEY ICS • BBC ENGINEERING • BECHTEL • BNFL - MAGNOX GENERATION • BP CHEMICALS • BRITISH AMERICAN TOBACCO • BRITISH ENERGY • BRITISH GAS • BRITISH STEEL • CEGELEC • CERESTAR • COE LTD • CONOCO • CORBY POWER STATION • CORUS GROUP PLC • CRODA LEEK LTD • CRUICKSHANKS LTD • DARESBURY LABORATORIES • DATEL RAIL SYSTEMS • DRAX POWER STATION • ELF EXPLORATION UK PLC • ENERGY LOGISTICS • EURO TUNNEL • EUROTHERM • EUROTUNNEL • EVESHAM MICROS • EXPRO NORTH SEA LTD • EXULT LTD • FIRST ENGINEERING LTD • FISHER ROSEMOUNT • GEC METERS • GENESIS OIL & GAS CONSULTANTS • GLAXO CHEM • GLAXO SMITH KLINE • GLAXO WELLCOME • GRAMPION REGIONAL COUNCIL • GREAT YARMOUTH POWER • HALLIBURTON KBR • HAMWORHTY COMBUSTION • HONEYWELL - ABERDEEN • HONEYWELL BRACKNELL • ICI NOBEL ENTERPRISES • ICS TRIPLEX • IGGESUND PAPER BOARD • INMARSAT LTD • INSTEM LIMITED • JOHN BROWN ENGINEERING • JOHNSON MATTHEY • KODAK • KVAERNER ENERGY • LEVER FABRIGE • LINDSAY OIL REFINERY • LLOYDS • LOGICA • LUCAS AEROSPACE • MERSEY TUNNELLS • METHODE ELECTRONICS • METTLER TOLEDO • MILLTRONICS • MOBIL OIL • MONTELL • MWH GLOBAL • NDC INFRARED • NEC SEMICONDUCTORS • NISSAN UK • NORTHERN LIGHTHOUSE BOARD • OKI EUROPE LTD • ORGANON LABORATORIES LTD • PHARMA SITE ENGINEERING • PHILLIPS PETROLEUM • POWERGEN • QINETIQ • RAIL TRACK SYSTEMS • RIG TECH • ROBERTS & PARTNERS • ROLLS ROYCE • ROVER GROUP • RUGBY CEMENT • SCOTTISH COURAGE • SCOTTISH HYDRO ELECTRIC PLC • SCOTTISH POWER • SHELL CHEMICALS • SHELL UK EXPLORATION & PRODUCTION • SHOTTON PAPER PLC • SIEMENS - AUTOMATION & DRIVES • STRATHCLYDE WATER • SUN VALLEY POULTRY • SWALEK • TEXACO PEMBROKE • THAMES WATER • TMD TECHNOLOGIES LTD • TOTAL OIL MARINE • TOYOTA UK • TRANSCO • TRANSCO LOCKERLEY COMPRESSOR • TREND CONTROL SYSTEMS LTD • UKAEA • UNITED KINGDOM PAPER • VG GAS • VICTREX PLC • VSEC • WATER SERVICE • YARROW SHIPBUILDERS • YORKSHIRE ELECTRIC • YORKSHIRE ELECTRIC IRELAND USA DE BEERS - JWANENG MINE • DE BEERS - ORAPA MINE CANADA AECL • AIRCOM INDUSTRIES (76) LTD • ATCO ELECTRIC • BC GAS - CANADA •BC HYDRO • BOMBARDIER • CITY OF LONDON ONTARIO • CITY OF OTTAWA •CITY OF SASKATOON • CONOCO CANADA LIMITED • DEPT OF NATIONAL DEFENCE - CANADA • ENBRIDGE PIPELINES • ENMAX • FORD ELECTRONICS MANUFACTURING PLANT • GE ENERGY SERVICES • GENERAL MOTORS •GUILLEVIN AUTOMATION • HUSKY OIL • IMC LTD • IMPERIAL OIL • INCO LTD •KALPEN VACHHARAJANI • KEYANO COLLEGE • LABRADOR HYDRO • MANITOBA HYDRO • MANITOBA LOTTERIES CORP • MEMORIAL UNIVERSITY OF NEW FOUNDLAND • MILLTRONICS • NEW BRUNSWICK POWER • NOVA CHEMICALS •NXTPHASE CORPORATION - VANCOUVER • ONTARIO HYDRO • OTTAWA HYDRO• PETRO CANADA • POWER MEASUREMENT LTD • SASKATCHEWAN POWER •SPARTAN CONTROLS • STONE CONSOLIDATED • STORA • SUNCOR ENERGY •SYNCRUDE • TELUS • TRANS CANADA PIPELINES • TROJAN TECHNOLOGIES •WASCANA ENERGY • WEST COAST ENERGY • WEYERHAUSER FRANCE SCHLUMBERGER INDIA BAYER DIAGNOSTICS • ESB DISTRIBUTION • INTEL • IRISH CEMENT • JANNSEN PHARMACEUTICALS LTD • MICROSOL LIMITED • PFIZER • PILZ IRELAND •PROSCON ENGINEERING KOREA US DEPT OF THE ARMY MALAWI DWANGA SUGAR CORPORATION MALAYSIA GERMAN MALAYSIA INSTITUTE NAMIBIA NAMIBIAN BROADCASTING CORPORATION • NAMPOWER • NAMWATER ACW INCORPORATED • AERO SYSTEMS - NASA • AK STEEL CORPORATION • ALCATEL • ALLEN BRADLEY • AMERICAN ELECTRIC POWER/RADISSON AIRPORT HOTEL • AMGEN INCORPORATED • ANDERSEN CORPORATION • ARROW INTERNATIONAL • ASTRA ZENECA PHARMACEUTICALS • AVISTA CORPORATION • BOEING • BOWATER NEWSPRINT • CENTRAL MAINE POWER COMPANY • CHEVRON • CITY OF DETROIT • DAISHOWA PAPER MILL • DEGUSSA CORPORATION • DEPT OF ENERGY • DEQUESNE LIGHT • DETROIT WATER • EXXON MOBIL CHEMICAL COMPANY • FMC CORPORATION • GENERAL MONITORS • HARNISCHFEGER • HOME STAKE MINING CO • HONEYWELL • HUGHES AIRCRAFT • IDM CONTROLS • ISA • K-TRON INSTITUTE • LCRA • LIFESCAN • LONGVIEW FIBER • LOOP LLC • LUCAS BODY SYSTEMS • MCKEE FOODS • MILLTRONICS • NASA • PARKER COMPUTER • PEPPERL FUCHS • PHELPS DODGE • PHILIP MORRIS • PROCESS EQUIPMENT COMPANY • RALSTON PURINA • SAN DIEGO COUNTY WATER AUTHORITY • SAN FRANCISCO WATER DEPARTMENT • SANTA CLARA VALLEY WATER • SECURITIES INDUSTRY AUTOMATION CORP • SERANO LABORATORIES • SIEMENS POWER • SIEMENS WESTINGHOUSE • SPAWAR SYSTEMS CENTER • SPEEDFAM CORP • STILL WATER MINING CORPORATION • TOYOTA MOTOR MANUFACTURING • TUCSON ELECTRIC • UNITED TECHNOLOGIES CORP (UTC) • UNOCAL ALASKA RESOURCES • UTILITY ENGINEERING • VALTEK • WASHINGTON WATER POWER • WISCONSIN POWER • ZENECA ZIMBABWE TRIANGLE LIMITED NEW ZEALAND ACI PACKAGING • AJ GREAVES • ANCHOR PRODUCTS • AUCKLAND REGIONAL COUNCIL • BALLANCE AGRI NUTRIENTS • CONTACT ENERGY • ENZAFOODS NZ LTD • ERICCSON • FISHER & PAYKEL • GEC ALSTHOM • JAMES HARDIE • METHANEX NZ LTD • NATURAL GAS NZ • NZ MILK PRODUCTS • NZ WATER AND WASTE ASSOC • NORSKE SKOG • NZ ALUMINIUM SMELTERS • NZ REFINING CO • PAN PAC FOREST PRODUCTS • POWERCO • ROCKWELL NZ • ROTORUA DISTRICT COUNCIL • ROYAL NEW ZEALAND NAVY • THE UNIVERSITY OF AUCKLAND • SAUDI ARABIA SAUDI ELECTRIC COMPANY SINGAPORE COMPANY MISSION “To provide our clients with measurable and significant productivity gains through excellence in cutting edge, practical engineering and technology training” ...
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This note was uploaded on 11/24/2010 for the course STRUCTURAL 1 taught by Professor 2 during the Spring '10 term at Indian Institute of Technology, Roorkee.

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