Handbook of Formulae and Constant

Handbook of Formulae and Physical Constants For The Use Of Students And Examination Candidates Approved by the Interprovincial Power Engineering Curriculum Committee and the Provincial Chief Inspectors' Association's Committee for the standardization of Power Engineer's Examinations n Canada. Duplication of this material for student in-class use or for examination purposes is permitted without written approval. ring.ca Printed July 2003 www.powerenginee Table of Contents TOPIC PAGE SI Multiples..........................................................................................1 Basic Units (distance, area, volume, mass, density) ............................2 Mathematical Formulae .......................................................................5 Applied Mechanics .............................................................................10 Thermodynamics.................................................................................21 Fluid Mechanics..................................................................................28 Electricity............................................................................................30 Periodic Table .....................................................................................34 Names in the Metric System VALUE EXPONENT SYMBOL PREFIX 1 000 000 000 000 1012 T tera 1 000 000 000 109 G giga 1 000 000 106 M mega 1 000 103 k kilo 100 102 h hecto 10 101 da deca 0.1 10-1 d deci 0.01 10-2 c centi 0.001 10-3 m milli 0.000 001 10-6 µ micro 0.000 000 001 10-9 n nano 0.000 000 000 001 10-12 p pico Conversion Chart for Metric Units To Milli- To Centi- To Deci- To Metre, Gram, Litre To Deca- To Hecto- To Kilo- Kilo- x 106 x 105 x 104 x 103 x 102 x 101 To C on ve rt Hecto- x 105 x 104 x 103 x 102 x 101 x 10-1 Deca- x 104 x 103 x 102 x 101 x 10-1 x 10-2 Metre, Gram, Litre x 103 x 102 x 101 x 10-1 x 10-2 x 10-3 Deci- x 102 x 101 x 10-1 x 10-2 x 10-3 x 10-4 Centi- x 101 x 10-1 x 10-2 x 10-3 x 10-4 x 10-5 Milli- x 10-1 x 10-2 x 10-3 x 10-4 x 10-5 x 10-6 Page 1 BASIC UNITS SI IMPERIAL DISTANCE 1 metre (1 m) = 10 decimetres (10 dm) 12 in. = 1 ft = 100 centimetres (100 cm) 3 ft = 1 yd = 1000 millimetres (1000 mm) 5280 ft = 1 mile 1760 yd = 1 mile 1 decametre (1 dam) = 10 m 1 hectometre (1 hm) = 100 m 1 kilometre (1 km) = 1000 m Conversions: 1 in. = 25.4 mm 1 ft = 30.48 cm 1 mile = 1.61 km 1 yd = 0.914 m 1 m = 3.28 ft Area 1 sq metre (1 m2) = 10 000 cm2 1 ft2 = 144 in.2 = 1 000 000 mm2 1 yd2 = 9 ft2 1 sq mile = 640 acre = 1 section 1 sq hectometre (1 hm2) = 10 000 m2 = 1 hectare (1 ha) 1 sq km (1 km2) = 1 000 000 m2 Conversions: 1 in.2 = 6.45 cm2 = 645 mm2 1 m2 = 10.8 ft2 1 acre = 0.405 ha 1 sq mile = 2.59 km2 Page 2 SI IMPERIAL Volume 1 m3 = 1 000 000 cm3 1 ft3 = 1728 in.3 = 1 x 109 mm3 1 yd3 = 27 ft3 1 dm3 = 1 litre 1(liquid) U.S. gallon = 231 in.3 1 litre = 1000 cm3 = 4 (liquid) quarts 1 mL = 1 cm3 1 U.S. barrel (bbl) = 42 U.S. gal. 1 m3 = 1000 litres 1 imperial gallon = 1.2 U.S. gal. Conversions: 1 in.3 = 16.4 cm3 1 m3 = 35.3 ft3 1 litre = 61 in.3 1 U.S.gal = 3.78 litres 1 U.S. bbl = 159 litres 1 litre/s = 15.9 U.S. gal/min Mass and Weight 1 kilogram (1 kg) = 1000 grams 2000 lb = 1 ton (short) 1000 kg = 1 tonne 1 long ton = 2240 lb Conversions: 1 kg (on Earth) results in a weight of 2.2 lb Density volume mass density mass = volume weight densityweight = ⎟⎠ ⎞⎜⎝ ⎛= 3m kg V m ρ ⎟⎠ ⎞⎜⎝ ⎛= 3ft lb V w ρ Conversions: (on Earth) a mass density of 1 kgm3 results in a weight density of 0.0623 lb ft3 Page 3 SI Imperial RELATIVE DENSITY In SI R.D. is a comparison of mass density In Imperial the corresponding quantity is to a standard. For solids and liquids the specific gravity; for solids and liquids a standard is fresh water. comparison of weight density to that of water. Conversions: In both systems the same numbers hold for R.D. as for S.G. since these are equivalent ratios. RELATIVE DENSITY (SPECIFIC GRAVITY) OF VARIOUS SUBSTANCES Water (fresh)...............1.00 Mica............................2.9 Water (sea average) ....1.03 Nickel .........................8.6 Aluminum...................2.56 Oil (linseed) ................0.94 Antimony....................6.70 Oil (olive) ...................0.92 Bismuth.......................9.80 Oil (petroleum) ...........0.76-0.86 Brass ...........................8.40 Oil (turpentine) ...........0.87 Brick ...........................2.1 Paraffin .......................0.86 Calcium.......................1.58 Platinum....................21.5 Carbon (diamond).......3.4 Sand (dry) ...................1.42 Carbon (graphite)........2.3 Silicon.........................2.6 Carbon (charcoal) .......1.8 Silver.........................10.57 Chromium...................6.5 Slate ............................2.1-2.8 Clay.............................1.9 Sodium........................0.97 Coal.............................1.36-1.4 Steel (mild) .................7.87 Cobalt .........................8.6 Sulphur .......................2.07 Copper ........................8.77 Tin...............................7.3 Cork ............................0.24 Tungsten ...................19.1 Glass (crown)..............2.5 Wood (ash) .................0.75 Glass (flint) .................3.5 Wood (beech) .............0.7-0.8 Gold ..........................19.3 Wood (ebony).............1.1-1.2 Iron (cast)....................7.21 Wood (elm).................0.66 Iron (wrought) ............7.78 Wood (lignum-vitae) ..1.3 Lead ..........................11.4 Wood (oak).................0.7-1.0 Magnesium .................1.74 Wood (pine)................0.56 Manganese..................8.0 Wood (teak) ................0.8 Mercury ....................13.6 Zinc.............................7.0 Page 4 Greek Alphabet Alpha α Iota ι Rho ρ Beta β Kappa κ Sigma Σ, σ Gamma γ Lambda λ Tau τ Delta ∆ Mu µ Upsilon υ Epsilon ε Nu ν Phi Φ, φ Zeta ζ Xi ξ Kai χ Eta η Omicron Ο Psi ψ Theta θ Pi π Omega Ω, ω MATHEMATICAL FORMULAE Algebra 1. Expansion Formulae (x + y)2 = x2 + 2xy + y2 (x - y)2 = x2 - 2xy + y2 x2 - y2 = (x - y) (x + y) (x + y)3 = x3 + 3x2y + 3xy2 + y3 x3 + y3 = (x + y) (x2 - xy + y2) (x - y)3 = x3 - 3x2y + 3xy2 - y3 x3 - y3 = (x - y) (x2 + xy + y2) 2. Quadratic Equation If ax2 + bx + c = 0, Then x = 2a ac4b b- 2 −± Page 5 Trigonometry 1. Basic Ratios h y A Sin = , h x A cos = , x y A tan = 2. Pythagoras' Law x2 + y2 = h2 3. Trigonometric Function Values Sin is positive from 0° to 90° and positive from 90° to 180° Cos is positive from 0° to 90° and negative from 90° to 180° Tan is positive from 0° to 90° and negative from 90° to 180° 4. Solution of Triangles a. Sine Law CSin c BSin b A Sin a == b. Cosine Law c2 = a2 + b2 - 2 ab Cos C a2 = b2 + c2 - 2 bc Cos A b2 = a2 + c2 - 2 ac Cos B Page 6 Geometry 1. Areas of Triangles a. All Triangles 2 heightlar perpendicu x base Area = Area 2 BSin ac 2 CSin ab 2 ASin bc === and, c) - (s b) - (s a)-(s s Area = where, s is half the sum of the sides, or s = 2 c b a ++ b. Equilateral Triangles Area = 0.433 x side2 2. Circumference of a Circle C = πd 3. Area of a Circle A = πr2 = 2 r x ncecircumfere = 2d 4 π = 0.7854d2 4. Area of a Sector of a Circle A = 2 r x arc A = 2r x π 360 θ° (θ = angle in degrees) A = 2 rθ 2° (θ = angle in radians) Page 7 5. Area of a Segment of a Circle A = area of sector – area of triangle Also approximate area = 0.608- h d h 3 4 2 6. Ellipse A = Dd 4 π Approx. circumference = ( ) 2 d D π + 7. Area of Trapezoid A = h 2 b a ⎟⎠ ⎞⎜⎝ ⎛ + 8. Area of Hexagon A = 2.6s2 where s is the length of one side 9. Area of Octagon A = 4.83s2 where s is the length of one side 10. Sphere Total surface area A =4πr2 Surface area of segment As = πdh Volume V = 3r π 3 4 Volume of segment Vs = πh2 3 (3r – h) Vs = πh6 (h 2 + 3a2) where a = radius of segment base Page 8 11. Volume of a Cylinder V = Ld 4 π 2 where L is cylinder length 12. Pyramid Volume V = 3 1 base area x perpendicular height Volume of frustum VF = )Aa a (A 3 h ++ where h is the perpendicular height, A and a are areas as shown 13. Cone Area of curved surface of cone: A = 2 DL π Area of curved surface of frustum AF = 2 d)L (D π + Volume of cone: V = base area × perpendicular height3 Volume of frustum: VF = perpendicular height × π (R2 + r2 + Rr) 3 Page 9 APPLIED MECHANICS Scalar - a property described by a magnitude only Vector - a property described by a magnitude and a direction Velocity - vector property equal to displacementtime The magnitude of velocity may be referred to as speed In SI the basic unit is ms , in Imperial ft s Other common units are kmh , mi h Conversions: s ft 3.28 s m 1 = h mi 0.621 h km 1 = Speed of sound in dry air is 331 ms at 0°C and increases by about 0.61 m s for each °C rise Speed of light in vacuum equals 3 x 108 ms Acceleration - vector property equal to change in velocitytime In SI the basic unit is 2s m , in Imperial 2s ft Conversion: 1 2s m = 3.28 2s ft Acceleration due to gravity, symbol "g", is 9.81 2s m or 32.2 2s ft Page 10 LINEAR VELOCITY AND ACCELERATION u initial velocity v final velocity t elapsed time s displacement a acceleration v = u + at s = v + u2 t s = ut + 12 at 2 v2 = u2 + 2 as Angular Velocity and Acceleration θ angular displacement (radians) ω angular velocity (radians/s); ω1 = initial, ω2 = final α angular acceleration (radians/s2) ω2 = ω1 + α t θ = ω1 + ω2 x t 2 θ = ω1 t + ½ α t2 ω22 = ω12 + 2 α θ linear displacement, s = r θ linear velocity, v = r ω linear, or tangential acceleration, aT = r α Page 11 Tangential, Centripetal and Total Acceleration Tangential acceleration aT is due to angular acceleration α aT = rα 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 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 ms2 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 m = Weightg g = 32.2 ft s2 Page 12 Newton's Second Law of Motion An unbalanced force F will cause an object of mass m to accelerate a, according to: F = ma (Imperial F = wg a, where w is weight) 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 = wg v, where w is weight) in SI unit is kg ms 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 = F s 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 Page 13 Kinetic Energy Energy due to motion Ek = 12mv 2 In Imperial this is usually expressed as Ek = w2gv 2 where w is weight Kinetic Energy of Rotation 22 R ωmk2 1 E = where k is radius of gyration, ω is angular velocity in rad/s or 2 R Iω2 1 E = where I = mk2 is the moment of inertia CENTRIPETAL (CENTRIFUGAL) FORCE r mv F 2 C = where r is the radius or FC = m ω2 r where ω is angular velocity in rad/s Potential Energy 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 Page 14 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 A scalar quantity, equal to the rate of doing work In SI the unit is the Watt W (or kW) 1 W = 1Js In Imperial, the units are: Mechanical Power - ft – lbs , horsepower h.p. Thermal Power - Btus Electrical Power - W, kW, or h.p. Conversions: 746 W = 1 h.p. 1 h.p. = 550 ft – lbs 1 kW = 0.948 Btus Page 15 Pressure A vector quantity, force per unit area In SI the basic units of pressure are pascals Pa and kPa 1 Pa = 1 Nm2 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 1 atmosphere (atm) = 101.3 kPa = 14.7 psi Page 16 Simple Harmonic Motion Velocity of P = s m x- R ω 22 Acceleration of P = ω2 x m/s2 The period or time of a complete oscillation = ω π2 seconds General formula for the period of S.H.M. T = 2π onaccelerati ntdisplaceme Simple Pendulum T = 2π g L T = period or time in seconds for a double swing L = length in metres The Conical Pendulum R/H = tan θ= Fc/W = ω 2 R/g Page 17 Lifting Machines W = load lifted, F = force applied M.A. = effort load = F W V.R. (velocity ratio) = distance load distanceeffort η = efficiency = V.R. M.A. 1. Lifting Blocks V.R. = number of rope strands supporting the load block 2. Wheel & Differential Axle Velocity ratio = 2 )r -(r π2 Rπ2 1 = 1r -r 2R 2 R Or, using diameters instead of radii, Velocity ratio = )d - (d 2D 1 3. Inclined Plane V.R. = height length 4. Screw Jack V.R. = threadofpitch leverage of ncecircumfere Page 18 Indicated Power I.P. = Pm A L N where I.P. is power in W, Pm is mean or "average" effective pressure in Pa, A is piston area in m2, L is length of stroke in m and N is number of power strokes per second Brake Power B.P. = Tω where B.P. is brake power in W, T is torque in Nm and ω is angular velocity in radian/second STRESS, STRAIN and MODULUS OF ELASTICITY Direct stress = A P area load = Direct strain = L length original extension A∆= Modulus of elasticity E = AA ∆=∆= A PL L/ P/A straindirect stressdirect Shear stress τ = shearunder area force Shear strain = L x Modulus of rigidity G = strainshear stressshear Page 19 General Torsion Equation (Shafts of circular cross-section) T J = τ r = G θ L )d (d 32 π )r - (r 2 π J 32 πdr 2 π J 4 2 4 1 4 2 4 1 4 4 −= = == Shaft HollowFor 2. Shaft SolidFor 1. T = torque or twisting moment in newton metres J = polar second moment of area of cross-section about shaft axis. τ = shear stress at outer fibres in pascals r = radius of shaft in metres G = modulus of rigidity in pascals θ = angle of twist in radians L = length of shaft in metres d = diameter of shaft in metres Relationship Between Bending Stress and External Bending Moment M I = σ y = E R 1. For Rectangle M = external bending moment in newton metres I = second moment of area in m4 σ = bending stress at outer fibres in pascals y = distance from centroid to outer fibres in metres E = modulus of elasticity in pascals R = radius of currative in metres I = 12 BD3 2. For Solid Shaft I = πD464 Page 20 THERMODYNAMICS Temperature Scales ° )32F ( 9 5 C −°= °F = 32 C 5 9 +° °R = °F + 460 (R Rankine) K = °C + 273 (K Kelvin) Sensible Heat Equation Q = mc∆T m is mass c is specific heat ∆T is temperature change 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 Gas Laws 1. Boyle’s Law When gas temperature is constant PV = constant or P1V1 = P2V2 where P is absolute pressure and V is volume 2. Charles’ Law When gas pressure is constant, constant T V = or V1T1 = V2T2 , where V is volume and T is absolute temperature Page 21 3. Gay-Lussac's Law When gas volume is constant, constant T P = Or 2 2 1 1 T P T P = , where P is absolute pressure and T is absolute temperature 4. General Gas Law P1V1 T1 = P2V2T2 = constant 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) 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 SPECIFIC HEATS OF GASES Specific Heat at Specific Heat at Ratio of Specific Constant Pressure Constant Volume Heats kJ/kgK kJ/kgK γ = cp / cv GAS or or kJ/kg oC kJ/kg oC 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 Helium 5.234 3.153 1.66 Hydrogen 14.235 10.096 1.41 Hydrogen Sulphide 1.105 0.85 1.30 Methane 2.177 1.675 1.30 Nitrogen 1.043 0.745 1.40 Oxygen 0.913 0.652 1.40 Sulphur Dioxide 0.632 0.451 1.40 Page 22 Efficiency of Heat Engines Carnot Cycle η = T1 – T2T1 where T1 and T2 are absolute temperatures of heat source and sink Air Standard Efficiencies