Handbook of Formulae and Physical Constants pdf

Handbook of Formulae and Physical Constants For The Use Of Students And Examination Candidates Duplication of this material for student in-class use or for examination purposes is permitted without written approval 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 www.powerengineering.ca Printed July 2003 Table of Contents TOPIC PAGE SI Multiples Basic Units (distance, area, volume, mass, density) Mathematical Formulae .5 Applied Mechanics .10 Thermodynamics .21 Fluid Mechanics 28 Electricity 30 Periodic Table .34 Names in the Metric System VALUE EXPONENT 000 000 000 000 000 000 000 000 000 000 100 10 0.1 0.01 0.001 0.000 001 0.000 000 001 0.000 000 000 001 SYMBOL 1012 109 106 103 102 101 10-1 10-2 10-3 10-6 10-9 10-12 PREFIX T G M k h da d c m µ n p tera giga mega kilo hecto deca deci centi milli micro nano pico Conversion Chart for Metric Units To Centi- To Deci- To Metre, Gram, Litre To Deca- To Hecto- Kilo- x 106 x 105 x 104 x 103 x 102 x 101 Hecto- x 105 x 104 x 103 x 102 x 101 DecaTo Convert To Milli- x 104 x 103 x 102 x 101 Metre, Gram, Litre x 103 x 102 x 101 Deci- x 102 x 101 Centi- x 101 Milli- To Kilo- x 10-1 x 10-2 x 10-1 x 10-2 x 10-3 x 10-1 x 10-2 x 10-3 x 10-4 x 10-1 x 10-1 x 10-1 x 10-2 x 10-3 x 10-4 x 10-5 x 10-2 x 10-3 x 10-4 x 10-5 x 10-6 Page BASIC UNITS SI IMPERIAL DISTANCE metre (1 m) = 10 decimetres (10 dm) = 100 centimetres (100 cm) = 1000 millimetres (1000 mm) 12 in ft 5280 ft 1760 yd = = = = ft yd mile mile decametre (1 dam) = 10 m hectometre (1 hm) = 100 m kilometre (1 km) = 1000 m Conversions: in ft mile yd 1m = = = = = 25.4 mm 30.48 cm 1.61 km 0.914 m 3.28 ft Area sq metre (1 m2) = 10 000 cm2 = 000 000 mm2 sq hectometre (1 hm2) = 10 000 m2 = hectare (1 ha) ft2 = 144 in.2 yd2 = ft2 sq mile = 640 acre = section sq km (1 km2) = 000 000 m2 Conversions: in.2 m2 acre sq mile = = = = 6.45 cm2 = 645 mm2 10.8 ft2 0.405 2.59 km2 Page SI IMPERIAL Volume m3 = 000 000 cm3 = x 109 mm3 dm3 litre mL m3 = = = = ft3 = 1728 in.3 yd3 = 27 ft3 litre 1000 cm3 cm3 1000 litres 1(liquid) U.S gallon = = U.S barrel (bbl) = imperial gallon = 231 in.3 (liquid) quarts 42 U.S gal 1.2 U.S gal Conversions: in.3 m3 litre U.S.gal U.S bbl litre/s = = = = = = 16.4 cm3 35.3 ft3 61 in.3 3.78 litres 159 litres 15.9 U.S gal/min Mass and Weight kilogram (1 kg) = 1000 grams 1000 kg = tonne 2000 lb = ton (short) long ton = 2240 lb Conversions: kg (on Earth) results in a weight of 2.2 lb Density mass density = ρ= mass volume weight density = m ⎛ kg ⎞ ⎜ ⎟ V ⎝ m3 ⎠ ρ= weight volume w ⎛ lb ⎞ ⎜ ⎟ V ⎝ ft ⎠ Conversions: (on Earth) a mass density of kg results in a weight density of 0.0623 lb m3 ft Page SI Imperial RELATIVE DENSITY In SI R.D is a comparison of mass density to a standard For solids and liquids the standard is fresh water water In Imperial the corresponding quantity is specific gravity; for solids and liquids a comparison of weight density to that of 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 Water (sea average) 1.03 Aluminum 2.56 Antimony 6.70 Bismuth .9.80 Brass 8.40 Brick 2.1 Calcium .1.58 Carbon (diamond) .3.4 Carbon (graphite) 2.3 Carbon (charcoal) .1.8 Chromium 6.5 Clay 1.9 Coal 1.36-1.4 Cobalt 8.6 Copper 8.77 Cork 0.24 Glass (crown) 2.5 Glass (flint) .3.5 Gold 19.3 Iron (cast) 7.21 Iron (wrought) 7.78 Lead 11.4 Magnesium 1.74 Manganese 8.0 Mercury 13.6 Mica 2.9 Nickel 8.6 Oil (linseed) 0.94 Oil (olive) 0.92 Oil (petroleum) 0.76-0.86 Oil (turpentine) 0.87 Paraffin .0.86 Platinum 21.5 Sand (dry) 1.42 Silicon .2.6 Silver .10.57 Slate 2.1-2.8 Sodium 0.97 Steel (mild) 7.87 Sulphur .2.07 Tin .7.3 Tungsten 19.1 Wood (ash) 0.75 Wood (beech) .0.7-0.8 Wood (ebony) 1.1-1.2 Wood (elm) .0.66 Wood (lignum-vitae) 1.3 Wood (oak) .0.7-1.0 Wood (pine) 0.56 Wood (teak) 0.8 Zinc 7.0 Page Greek Alphabet α β γ ∆ ε ζ η θ Alpha Beta Gamma Delta Epsilon Zeta Eta Theta Iota Kappa Lambda Mu Nu Xi Omicron Pi ι κ λ µ ν ξ Ο π Rho Sigma Tau Upsilon Phi Kai Psi Omega ρ Σ, σ τ υ Φ, φ χ ψ Ω, ω MATHEMATICAL FORMULAE Algebra 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) Quadratic Equation If ax2 + bx + c = 0, Then x = - b ± b − 4ac 2a Page Trigonometry Basic Ratios Sin A = y , h cos A = x , h tan A = y x Pythagoras' Law x2 + y2 = h2 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° Solution of Triangles a Sine Law a b c = = Sin A Sin B Sin C b Cosine Law c2 = a2 + b2 - ab Cos C a2 = b2 + c2 - bc Cos A b2 = a2 + c2 - ac Cos B Page Geometry Areas of Triangles a All Triangles Area = base x perpendicular height Area = bc Sin A ab Sin C ac Sin B = = 2 and, Area = s (s - a) (s - b) (s - c) where, s is half the sum of the sides, or s = a+b+c b Equilateral Triangles Area = 0.433 x side2 Circumference of a Circle C = πd Area of a Circle A = πr2 = π circumference x r = d = 0.7854d2 4 Area of a Sector of a Circle A= arc x r A= θ° x π r2 360 A= θ°r 2 (θ = angle in degrees) (θ = angle in radians) Page Thermal Expansion of Solids Increase in length where L α (T2 – T1 ) Increase in volume Where V β (T2 – T1 ) = = = = = = = = L α (T2 – T1 ) original length coefficient of linear expansion rise in temperature V β (T2 – T1 ) original volume coefficient of volumetric expansion rise in temperature coefficient of volumetric expansion = coefficient of linear expansion x β = 3α Page 26 Chemical Heating Value of a Fuel ( Chemical Heating Value MJ per kg of fuel = 33.7 C + 144 H C H2 O2 S O2 ) + 9.3 S is the mass of carbon per kg of fuel is the mass of hydrogen per kg of fuel is the mass of oxygen per kg of fuel is the mass of sulphur per kg of fuel Theoretical Air Required to Burn Fuel Air (kg per kg of fuel) = [8 C + (H - O2 ) + S] 100 23 Air Supplied from Analysis of Flue Gases Air in kg per kg of fuel = C N2 CO2 CO N2 ×C 33 (CO + CO) is the percentage of carbon in fuel by mass is the percentage of nitrogen in flue gas by volume is the percentage of carbon dioxide in flue gas by volume is the percentage of carbon monoxide in flue gas by volume Boiler Formulae Equivalent evaporation = Factor of evaporation = Boiler efficiency = where m s h1 h2 mf = = = = m s (h - h ) 2257 kJ/kg (h - h ) 2257 kJ/kg m s (h - h ) m f x calorific value of fuel mass flow rate of steam enthalpy of steam produced in boiler enthalpy of feedwater to boiler mass flow rate of fuel Page 27 FLUID MECHANICS Discharge from an Orifice Let A and Ac then Ac = = = or Cc = cross-sectional area of the orifice = (π/4)d2 cross-sectional area of the jet at the vena conrtacta = ((π/4) d c CcA Ac ⎛ dc ⎞ =⎜ ⎟ A ⎝ d ⎠ where Cc is the coefficient of contraction 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 = A cv or Q = C cAC v 2gh The coefficients of contraction and velocity are combined to give the coefficient of discharge, Cd i.e C d = C cC v and Q = C dA 2gh Typically, values for Cd vary between 0.6 and 0.65 Circular orifice: Q = 0.62 A 2gh Where Q = flow (m3/s) A = area (m2) h = head (m) Rectangular notch: Q = 0.62 (B x H) 2gh Where B = breadth (m) H = head (m above sill) Triangular Right Angled Notch: Q = 2.635 H5/2 Where H = head (m above sill) Page 28 Bernoulli’s Theory P v2 + w 2g H = total head (metres) h = height above datum level (metres) P = pressure (N/m2 or Pa) H = h+ w = force of gravity on m3 of fluid (N) v = velocity of water (metres per second) Loss of Head in Pipes Due to Friction Loss of head in metres = f L v d 2g L = length in metres d = diameter in metres pipes v = velocity of flow in metres per second f = constant value of 0.01 in large pipes to 0.02 in small Note: This equation is expressed in some textbooks as Loss = 4f L v where the f values range from 0.0025 to 0.005 d 2g Actual Pipe Dimensions Page 29 ELECTRICITY Ohm's Law I = E R or E = IR where I = current (amperes) E = electromotive force (volts) R = resistance (ohms) Conductor Resistivity L a ρ = specific resistance (or resistivity) (ohm metres, Ω·m) L = length (metres) a = area of cross-section (square metres) R = ρ where Temperature correction Rt = Ro (1 + αt) where Ro = resistance at 0ºC (Ω) Rt = resistance at tºC (Ω) α = temperature coefficient which has an average value for copper of 0.004 28 (Ω/ΩºC) R2 = R1 (1 + αt ) (1 + αt ) where R1 = resistance at t1 (Ω) R2 = resistance at t2 (Ω) α Values Ω/ΩºC copper platinum nickel tungsten aluminum 0.00428 0.00385 0.00672 0.0045 0.0040 Page 30 Dynamo Formulae Average e.m.f generated in each conductor = 2Φ NpZ 60c where Z = total number of armature conductors c = number of parallel paths through winding between positive and negative brushes where c = (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 where EG EB Ia Ra = = = = generated e.m.f generated back e.m.f armature current armature resistance Alternating Current R.M.S value of sine curve = 0.707 maximum value Mean value of sine curve = 0.637 maximum value R.M.S value 0.707 Form factor of sinusoidal = = = 1.11 Mean value 0.637 Frequency of alternator = pN cycles per second 60 Where p = number of pairs of poles N = rotational speed in r/min Page 31 Slip of Induction Motor Slip speed of field - speed of rotor x 100 Speed of field Inductive Reactance Reactance of AC circuit (X) = 2πfL ohms where L = inductance of circuit (henries) 1.256T µA henries Inductance of an iron cored solenoid = L x 10 where T µ A L = = = = turns on coil magnetic permeablility of core area of core (square centimetres) length (centimetres) Capacitance Reactance Capacitance reactance of AC circuit = where ohms 2πfC C = capacitance (farads) ⎞ ⎛ Total reactance = ⎜ 2πfL ⎟ohms 2π fC ⎠ ⎝ Impedence (Z) = = (resistance) + (reactance) R + (2π fL - ) ohms π fC Current in AC Circuit Current = impressed volts impedance Page 32 Power Factor p.f = true watts volts x amperes also p.f = cos Φ, where Φ is the angle of lag or lead Three Phase Alternators Star connected Line voltage = x phase voltage Line current = phase current Delta connected Line voltage = phase voltage Line current = x phase current Three phase power P = EL IL cos Φ EL = line voltage IL = line current cos Φ = power factor Page 33 Page 34 ION NAMES AND FORMULAE MONATOMIC Ag+ Al3+ Au+ and Au2+ Be2+ Ca2+ Co2+ and Co3+ Cr2+ and Cr3+ Cu+ and Cu2+ Fe2+ and Fe3+ K+ Li+ Mg2+ Na+ Zn2+ silver ion aluminum ion gold ion beryllium ion calcium ion cobalt ion chromium ion copper ion iron ion potassium ion lithium ion magnesium ion sodium ion zinc ion POLYATOMIC BO33C2H3O2ClOClO2ClO3ClO4CNCO32C2O42CrO42Cr2O72HCO3H3O+ HPO42H2PO4HSO3HSO4MnO4N3NH4+ NO2NO3O22OCNOHPO33PO43SCNSO32SO42S2O32- borate ion acetate ion hypochlorite ion chlorite ion chlorate ion perchlorate ion cyanide ion carbonate ion oxalate ion chromate ion dichromate ion hydrogen carbonate or bicarbonate ion hydronium ion hydrogen phosphate ion dihydrogen phosphate ion hydrogen sulphite or bisulphite ion hydrogen sulphate or bisulphate ion permanganate ion azide ion ammonium ion nitrite ion nitrate ion peroxide ion cyanate ion hydroxide ion phosphite ion phosphate ion thiocyanate ion sulphite ion sulphate ion thiosulphate ion Page 35 This material is owned by Power Engineering Training Systems and may not be modified from its original form Duplication of this material for student use in-class or for examination purposes is permitted without written approval Address all inquiries to: Power Engineering Training Systems 1301 – 16 Ave NW, Calgary, AB Canada T2M 0L4 1-866-256-8193 Printed in Canada on Recycled Paper ... Value of a Fuel ( Chemical Heating Value MJ per kg of fuel = 33.7 C + 144 H C H2 O2 S O2 ) + 9.3 S is the mass of carbon per kg of fuel is the mass of hydrogen per kg of fuel is the mass of oxygen... p = number of pairs of poles N = rotational speed in r/min Page 31 Slip of Induction Motor Slip speed of field - speed of rotor x 100 Speed of field Inductive Reactance Reactance of AC circuit... cos Φ = power factor Page 33 Page 34 ION NAMES AND FORMULAE MONATOMIC Ag+ Al3+ Au+ and Au2+ Be2+ Ca2+ Co2+ and Co3+ Cr2+ and Cr3+ Cu+ and Cu2+ Fe2+ and Fe3+ K+ Li+ Mg2+ Na+ Zn2+ silver ion aluminum