See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/326985877 Engineers' Practical Databook: A Technical Reference Guide for Students and Professionals (Chapter 8: Fluid Mechanics) Preprint · August 2018 DOI: 10.13140/RG.2.2.32518.06722 CITATIONS READS 0 16,321 1 author: Jay Smith 2 PUBLICATIONS 0 CITATIONS SEE PROFILE Some of the authors of this publication are also working on these related projects: HALO Gateway - High Altitude Laser from Orbit Gateway View project Engineers' Practical Databook View project All content following this page was uploaded by Jay Smith on 12 August 2018. The user has requested enhancement of the downloaded file. ENGINEERS’ PRACTICAL DATABOOK First Edition ENGINEERS’ PRACTICAL DATABOOK First Edition in SI Units This Data Book is provided for the teaching of engineering and conforms to typical teaching structure for selected modules within HNC, HND, Foundation Degree and Bachelor’s Degree qualifications in Engineering. Cover: “Azure Blue McLaren P1” by Axion23, cropped, is licenced under CC BY 2.0. “F-35 Front profile in flight” by MSgt John Nimmo Sr. public domain cropped, from defenseimagery.mil. "The Severn Bridge" by Martin Edwards is licensed under CC BY-SA 2.0. “Gear with terminology” is licenced under CC BY-SA 3.0. “Sine and cosine” is public domain. “Mohr’s circle plane stress” by Sanpaz is licenced under CC BY-SA 3.0. Unit Circle Angles by Gustavb is licenced under CC BY-SA 3.0. Material properties derived or calculated from various sources, including Materials Handbook (ASM Vol.2, 1979), th Granta Design (2018), Thermodynamics: An Engineering Approach (Çengel & Boles, 6 ed., 2007), www.engineeringtoolbox.com (2018), www.matweb.com (2018). Avogadro constant (2018 definition) and atomic elements from IUPAC (2018). The author accepts no liability for any injuries or damages caused that may result from the reader's acting upon or using the content contained in the publication. Always consult a professional. Whilst every care has been taken to include accurate information, the author would appreciate any corrections to be sent to EngineersDatabook@gmail.com, quoting the page number. All rights reserved. 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, or otherwise without either the prior written permission of the author or a licence permitting restricted copying in the United Kingdom issued by the aforementioned. All formulae are written in SI units (m, kg, s) unless otherwise explicitly stated. Full Textbook: USA: https://www.amazon.com/dp/1980619344 UK: https://www.amazon.co.uk/dp/1980619344 Copyright © 2018 Jay Smith All rights reserved. ISBN: 978-198-061934-5 Brief Contents Symbols ................................................................................................................ 1 Units of Measurement .......................................................................................... 2 Mathematical Constants ....................................................................................... 4 Physical Constants ................................................................................................ 4 Conversion Factors ................................................................................................ 5 1. Mathematics .................................................................................................. 9 2. Materials ..................................................................................................... 55 3. Mechanics .................................................................................................... 83 4. Structures .................................................................................................... 91 5. Machines and Mechanisms ........................................................................ 101 6. Electrical and Electronics ............................................................................ 111 7. Thermodynamics ....................................................................................... 127 8. Fluid Mechanics ......................................................................................... 137 9. Systems ..................................................................................................... 149 10. Project Management ................................................................................. 153 Properties of Water .......................................................................................... 157 Atmosphere ...................................................................................................... 158 Contents Symbols ...................................... 1 Units of Measurement ................ 2 Prefixes ........................................ 2 S.I. Units ........................................ 2 Derived Units ................................. 3 Mathematical Constants ............. 4 Physical Constants ...................... 4 Conversion Factors ..................... 5 1. Mathematics........................... 9 1.1. Algebra ............................ 9 Laws of Exponents ........................... 9 e (the base of the natural logarithm)9 Laws of Logarithms ........................ 10 Polynomials ................................... 11 Partial Fractions ............................. 12 1.2. Sequences and Series .... 13 Sum of First n Natural Numbers .... 13 Sum of First n Squared Natural Numbers ........................................ 13 Sum of Arithmetic Progression ...... 13 Sum of Geometric Progression ...... 14 1.3. Coordinate Systems ....... 15 1.4. Complex Numbers ......... 16 Cartesian Form .............................. 16 Polar Form ..................................... 16 De Moivre’s Theorem .................... 17 1.5. Power series .................. 18 Taylor Series .................................. 19 Maclaurin Series ............................ 19 1.6. Fourier Series ................. 19 Fourier Series ................................. 20 1.7. Trigonometry................. 21 Definitions ..................................... 21 Unit Circle Identities ...................... 21 Unit Circle Angles........................... 22 Trigonometric Identities ................ 23 Sinusoidal Waveforms ................... 24 Expansion of R sin ωt+θ ............... 25 Area of a Triangle........................... 25 Law of Sines and Cosines ............... 25 1.8. Hyperbolic Functions ..... 26 1.9. Vectors .......................... 27 Notation ........................................ 27 Vector Multiplication ..................... 28 Triple Products .............................. 29 Vector Calculus .............................. 29 1.10. Matrices ........................ 30 Notation ........................................ 30 Rotation Matrices .......................... 31 Screw Matrix ................................. 31 Euler Angles ................................... 31 Determinant .................................. 32 Eigenvectors and Eigenvalues ........ 32 Matrix Inverse ............................... 33 1.11. Calculus ......................... 34 Derivatives ..................................... 34 Integrals......................................... 34 Table of Derivatives ....................... 35 Surface of Revolution .................... 36 Volume of Revolution .................... 36 Time Average of a Function ........... 36 1.12. Laplace Transforms ....... 37 Table of Laplace Transforms .......... 38 1.13. Statistics ........................ 40 Discrete Random Variables ............ 40 Grouped Frequency Distribution ... 40 Linear Regression (Least Squares Regression Line) ............................ 41 Pearson Product Moment Correlation Coefficient .................. 42 Spearman’s Rank Correlation Coefficient ..................................... 42 Combinatorics ............................... 43 Binomial Formula .......................... 44 Binomial Distribution ..................... 44 Poisson Distribution....................... 47 Normal Distribution ....................... 50 2. Materials ...............................55 2.1. The Elements ................. 55 Periodic Table ................................ 59 Pure Metallic Solids – Mechanical . 60 Pure Metallic Solids – Thermal and Electrical ........................................ 62 2.2. Steels and Alloys ............ 64 2.3. Alloys ............................. 67 2.4. Polymers ........................ 71 2.5. Specific Heat Capacity ... 74 2.6. Ashby Charts .................. 75 2.7. Composites .................... 78 Classifications................................. 78 Two-Phase Composites .................. 79 Fibre-Reinforced Composites ......... 79 Short Fibres, Random Orientation . 81 Discontinuous and Aligned Fibre Composites .................................... 81 3. Mechanics ............................. 83 Stress-Strain Relationship .............. 83 Uniaxial Stress ................................ 84 Shear .............................................. 84 Triaxial Stress ................................. 85 3.1. Mohr’s Circle .................. 87 For Plane Stress.............................. 87 Principle Stress ............................... 87 General Three-Dimensional Stress . 90 Yield Criteria................................... 90 4. Structures .............................. 91 4.1. Bending of Beams .......... 91 Second Moment of Area ................ 91 Radius of Gyration ......................... 91 4.2. The Bending Formula ..... 92 Plane Sections ................................ 93 4.3. Linear Elastic Beams ...... 94 4.4. Torsion of Shafts ............ 98 Max Shear Stress............................ 98 Torsional Stiffness .......................... 99 4.5. Euler’s Buckling Criterion100 Euler Buckling Force ..................... 100 Euler Buckling Stress .................... 100 5. Machines and Mechanisms . 101 5.1. Mechanisms ................. 101 Mobility (Gruebler’s Equation)..... 101 Grashof’s Criterion ....................... 101 5.2. Classical Mechanics ..... 102 Linear Terms ................................ 102 Rotary Terms................................ 102 Derivatives of Linear Position....... 102 Derivatives of Angular Position .... 102 Kinematics.................................... 103 Kinetics......................................... 106 5.3. Periodic motion ............ 109 Springs (Hooke’s Law) .................. 109 Dampers or Dashpots .................. 109 Simple Harmonic Motion ............. 109 Mass-Spring-Damper ................... 110 Simple Pendulum ......................... 110 5.4. Lagrange’s Equation .... 110 6. Electrical and Electronics ..... 111 6.1. Fundamentals of Electromagnetism ...................... 111 Ohm’s Law ................................... 111 Resistance .................................... 111 Inductance ................................... 111 Capacitance ................................. 112 Stored Energy .............................. 112 Reactance .................................... 112 Complex Impedance .................... 113 6.2. Transformers................ 114 Induced EMF ................................ 114 Ideal Transformers ....................... 114 6.3. Electrical Machines ...... 115 Lorentz Force ............................... 115 DC Machines ................................ 115 AC Machines ................................ 116 6.4. AC Power...................... 117 Single Phase ................................. 117 Balanced 3-Phase ......................... 117 Star-delta Conversions ................. 118 Complex Waveforms.................... 118 6.5. Electronics .................... 119 Circuit Diagram Symbols .............. 119 Diodes .......................................... 120 Field Effect Transistors ................. 120 Bipolar Junction Transistors ......... 121 Bipolar Amplifier Configurations .. 122 Operational Amplifiers ................. 123 Logic Gates ................................... 124 Boolean Algebra ........................... 125 Resistor Colour Code Table .......... 126 7. Thermodynamics ................. 127 7.1. First Law....................... 127 Internal Energy............................. 127 7.2. Second Law .................. 128 7.3. Ideal Gas Equations ..... 129 Ideal Gas Law ............................... 129 Combined Gas Law....................... 129 Specific Heat Capacities ............... 130 7.4. Isentropic Processes in Gas Turbines ..................................... 130 7.5. Cycle Efficiency ............ 130 Ideal Cycle Efficiency ................... 131 7.6. Heat Pumps and Refrigeration .............................. 131 Coefficient of Performance .......... 131 7.7. Carnot Cycle ................ 131 7.8. Heat Transfer .............. 132 Sensible Heat ............................... 132 Conduction .................................. 132 Convection ................................... 132 Composite Slab or Laminate with Fluid Boundaries .......................... 132 Solid Expansion ............................ 133 Work ............................................ 133 Power .......................................... 133 7.9. Flow Equations ............ 134 Change in Enthalpy ...................... 134 Steady Flow Energy Equation (open system) ........................................ 134 7.10. Diffusion ...................... 135 Fick’s First Law of Diffusion.......... 135 Fick’s Second Law of Diffusion ..... 135 The Arrhenius Equation ............... 135 8. Fluid Mechanics ...................137 8.1. Statics .......................... 137 Static Pressure ............................. 137 Buoyancy Force ........................... 137 Surface Tension ........................... 138 8.2. Dynamics ..................... 139 Stagnation Pressure ..................... 139 Bernoulli’s Equation..................... 139 Mass Continuity ........................... 139 Dynamic Viscosity ........................ 139 Reynolds Number ........................ 140 Pipe Friction - D’Arcy’s Formula ... 140 Laminar Friction Factor ................ 140 Stokes Drag .................................. 140 Pipe Roughness............................ 141 K-Factor of a Pipe Fitting ............. 143 Mass Continuity (Control Volume) ..................................................... 144 Momentum Continuity (Control Volume) ....................................... 144 Navier-Stokes Equation ............... 144 8.3. Aerodynamics .............. 145 Lift and Drag Coefficient .............. 145 Induced Drag Coefficient ............. 145 Speed of Sound ............................ 145 Mach Number.............................. 145 Ram Air Recovery ........................ 146 8.4. Jet Engines ................... 147 Propulsive Power ......................... 147 Thrust Specific Fuel Consumption 147 Propulsive Efficiency .................... 147 Thermal Efficiency ....................... 147 Overall Efficiency ......................... 147 9. Systems ...............................149 Transfer Function ........................ 149 First Order Systems ..................... 149 Response to a Step Input ............. 149 Second Order Systems ................. 150 10. Project Management .........153 Earned Value Analysis .................. 153 Performance Indices .................... 154 Properties of Water .................157 Atmosphere .............................158 International Standard Atmosphere 158 Troposphere Model ......................... 158 Standard Atmosphere Table ............ 159 SYMBOLS Name Uses alpha Angles, angular acceleration, thermal expansion coefficient beta Angles, coefficients gamma Heat capacity ratio, kinematic viscosity, shear strain š¤ Gamma Circulation (fluid dynamics) delta ā Delta Difference, damping Difference, determinant (matrix) epsilon Strain, permittivity (electromagnetism), electromotive force (EMF), random error (regression), emissivity (thermodynamics) eta Efficiency theta Angle, temperature (thermodynamics) κ kappa Curvature (=1/R) Thermal conductivity, wavelength, eigenvalue, parameter in the lambda Poisson Distribution (mean, variance) mu Friction coefficient, dynamic viscosity nu Kinematic viscosity, Poisson’s ratio rho Mass density, resistivity (electrical), curvature (alternative to r) š sigma š tau š phi Normal stress, standard deviation Shear stress, torque, time constant (electronics) Angles, heat flow, potential energy, magnetic flux psi Helix angle (gears), stream function (fluid dynamics) omega Angular velocity Ω Omega Electrical resistance (ohm) ζ š“ zeta Damping ratio Area (m2), current (A) Diameter (m) Young’s modulus (Pa) Shear modulus (Pa) Mass moment of inertia (kg m2), area moment of inertia (m4) Polar moment of inertia (m4) Radius (m) Volume flow rate (m3/s), heat flowrate (kW) Volume (m3), velocity (m/s) UNITS OF MEASUREMENT PREFIXES Symbol Prefix Multiplication factor P peta 1 000 000 000 000 000 T tera 1 000 000 000 000 G giga 1 000 000 000 M mega k kilo 1 000 d deci 0.1 c centi 0.01 m milli 0.001 μ micro 0.000 001 n nano 0.000 000 001 p pico 0.000 000 000 001 9 1 000 000 − − − − −9 − S.I. UNITS Symbol Unit Quantity m metre Length kg kilogram Mass [M] s second Time [T] A ampere Electric current [A] K kelvin Temperature [θ] cd candela Luminous intensity [I] mole Amount of substance [N] mol Dimension [L] DERIVED UNITS Quantity Unit Symbol Force newton N Pressure and Stress pascal Pa Torque newton-metre Energy, Work, Heat joule J Power watt W Frequency hertz Hz Plane angle radian rad Solid angle steradian sr Luminous flux lumen lm Illuminance lux lx Kinematic Viscosity stokes St Dynamic Viscosity poiseuille Pl Magnetic Flux weber Magnetic Flux Density tesla T Electrical Capacitance farad F Electrical Charge coulomb C Electrical Conductance siemens S Electrical Inductance henry H Electrical Resistance ohm ļ Potential difference/ Electromotive force volt V Base Units − − Nām − − − − − − = − = − − Wb š“ − − − − − − š“− š“− š“ā − − š“ − š“− š“− − − š“− MATHEMATICAL CONSTANTS Symbol š Description Value . … Base of the natural logarithm . … Ratio of circumference to diameter of a circle √− °/ = . Imaginary unit …° Radian PHYSICAL CONSTANTS Description Symbol Value − . . . . − 9 × × × . . Speed of light in vacuum − − − − / × . − − − − × − š“− Elementary charge ( + Electric constant (permittivity of free space, or vacuum permittivity) š“ Magnetic constant (permeability of free space, or vacuum permeability) Standard gravity − Gravitational constant Universal gas constant − Specific gas constant for dry air − Avogadro constant 1. FLUID MECHANICS 1.1. STATICS Static Pressure The change in static pressure in a barotropic, compressible fluid is given by: ā Δ =− ∫ ā For an incompressible fluid, this simplifies to: = = ā= š š š [ ā š ā š ā š − ] š . Δ =− − ā Δā [ ] A barotropic fluid is one whose pressure and density are related by an equation of state that does not contain temperature as a dependent variable. The static pressure p2 in the manometer below is given by: = + Δā + š = = ′ š − [ ] š š Δā − . Buoyancy Force = š š [ ] = Δā š š„ā š 138 Fluid Mechanics Surface Tension = š = [ cos − ] ≈ š¹ ā š (capillary action) Needle supported by surface tension = SIDE cos š ā FRONT š š šæ Pressure of a drop or bubble Liquid Drop Hollow Bubble š¾ šš − šš = š šš − šš = (Two surfaces) šš šš š Surface tension [10-3 N m-1] Fluid Acetone (propanone) š¾ š C3H6O 23 Crude oil, light Hydrocarbons 32 Crude oil, heavy Hydrocarbons 37 Ethanol (Ethyl alcohol) C2H6O 22.3 Ethylene glycol C2H6O2 48.8 Mercury Hg 465 Water H2O 72.8 Properties are at 20ā. For comparison, water at 100ā has a surface tension of 59 × 10-3 N m-1. Fluid Mechanics 139 1.2. DYNAMICS Stagnation Pressure = š − [ š = š š ] + š š š¦ š š = + Bernoulli’s Equation + + ā= = + For inviscid, incompressible, steady, irrotational flow, the sum of stagnation pressure and elevation pressure is constant along a streamline. š +ā The total head (i.e. internal energy) of a fluid is comprised of the static pressure head, velocity head, and elevation head. Mass Continuity Ģ = Ģ = š“= ā= āā = š [ š š [ − ] ] š“ [ − = ā āĢ = ] ā š Dynamic Viscosity For an isotropic Newtonian fluid š= ā = = = š š š š [ š ] š [ ]=[ − [ ] [ ] − š= − ] š“ ā āĢ 140 Fluid Mechanics Reynolds Number = š = ā = ā š š š š š ā = Pipe Friction - D’Arcy’s Formula Head loss through a pipe ā = Pressure loss through a pipe = − š ā ā = š š š Head loss through a fitting = š š , āšæ = š š Laminar Friction Factor ā = Stokes Drag Flow past a sphere at Re<2 š ∞ = ∞ š· . Fluid Mechanics 141 Pipe Roughness Pipes (Material) Drawn tubing (glass, brass, plastic) Copper Absolute Roughness microns [ −š š] 1.5 1.5 Aluminium 1.5 PVC 1.5 Red brass 1.5 Fiberglass 5 Carbon steel or wrought iron 45 Stainless steel 45 Cast iron-asphalt dipped 120 Galvanized iron 150 Cast iron uncoated 250 Wood stave 100-200 Ductile iron 2,500 Concrete 300 – 3,000 Riveted steel 1,000 – 10,000 Fittings L/D Globe valve 340 Gate valve 8 Lift check valve 600 Swing check valve 50 – 100 Ball valve 6 Butterfly valve 35 Flush pipe entrance (sharp corner) K=0.5 Flush pipe entrance (radius >0.15) K=0.04 Pipe exit K=1 Tee through 20 Tee branch flow 60 Elbow (90 degrees) 30 Elbow (45 degrees) 16 142 Fluid Mechanics Fluid Mechanics 143 K-Factor of a Pipe Fitting Fitting 45° Elbow 90° Elbow Curved 90° Elbow Square or Mitred 180° Bend Tee, Run Through Tee, as Elbow Tee, as Elbow Tee, Branching Flow Gate valve Diaphragm valve Globe valve, Bevel Seat Plug valve Butterfly valve Check valve Types Standard (R/D = 1) Long Radius (R/D = 1.5) Standard (R/D = 1) Long Radius (R/D = 1.5) Close Return Branch Blanked Entering in run Entering in branch Fully Open 3/4 Open 1/2 Open 1/4 Open Fully Open 3/4 Open 1/2 Open 1/4 Open Fully Open 1/2 Open θ = 5° θ = 10° θ = 20° θ = 40° θ = 60° θ = 5° θ = 10° θ = 20° θ = 40° θ = 60° Swing Disk Ball š² 0.4 0.2 0.8 0.5 1.3 1.5 0.4 1 1 1 0.2 0.9 4.5 24 2.3 2.6 4.3 21 6 9.5 0.1 0.3 1.6 17 206 0.2 0.5 1.5 11 118 2 10 70 144 Fluid Mechanics Mass Continuity (Control Volume) Integral Form = ā= āā = š“= = š ∫ +∫ š ā ā āā š“= The mass change within a control volume plus the mass flowing out of the surface of the control volume equals zero. For incompressible flows, there is no mass change term: =∫ Differential Form +∇ā ā ā āā š“= š = For incompressible flows, the divergence equals zero: ∇ā = ā +∫ Momentum Continuity (Control Volume) ā = ∫ š Navier-Stokes Equation š ā ā ā āā š“= + š ā ∇š = −∇ + ∇ š + š The Navier-Stokes Equations are analogous to Newton’s Second Law applied to an infinitesimal unit volume. The LHS includes the material derivative and is analogous to mass times acceleration. The RHS is analogous to net force. The material derivative of an infinitesimal fluid packet is equal to the sum of the negative pressure gradient, shear forces, and body forces. Fluid Mechanics 145 1.3. AERODYNAMICS Lift and Drag Coefficient šæ = š š“= [ šæ = ∞ = ] š“ = šæ = š“ ∞ š“ = š“ Induced Drag Coefficient š“ = = š š š š š šæ = š“ e is the wing span efficiency value by which the induced drag exceeds that of an elliptical lift distribution, typically 0.95-0.99. Speed of Sound š ā š š ā š šš ā š š ā š . [ ] Mach Number = = š [ − ] [ − . ] − =√ š − š š = Air is generally considered incompressible below M = 0.3. 146 Fluid Mechanics Ram Air Recovery To calculate ideal ram air temperature and pressure recovery (e.g. NACA scoops, engine intakes): šš = š ( + ( šš = š š š ā [ ] š š š ā š = ( +( . š − − š ) ) š ) š¾ š¾− ) š The actual recovery pressure is slightly less than the total pressure due to losses: Fluid Mechanics 147 1.4. JET ENGINES Propulsive Power š= ā Ģ = ∞ = = [ ] [ ā š =šā − ] − ] [ š [ ∞ − = Ģ − ∞ ā ∞ ] Thrust Specific Fuel Consumption [ Ģ = − = ] Ģ š Propulsive Efficiency = Thermal Efficiency = = = š š š¾− š¾ = š [ [ −( ) ] ] Overall Efficiency = View publication stats š£ + š£š ∞ = −( ) −š¾ š¾
