# Full text of "Handbook of Formula and Physical constants"

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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 10 12 T tera 1 000 000 000 10 9 G giga 1 000 000 10 6 M mega 1 000 10 3 k kilo 100 10 2 h hecto 10 10' da deca 0.1 10" 1 d deci 0.01 10" 2 c centi 0.001 10" 3 m m i Mi 0.000 001 10" 6 H micro 0.000 000 001 10" 9 n nano 0.000 000 000 001 10 12 P pico Conversion Chart for Metric Units CD > O O To Milli- To Centi- To Deci- To Metre, Gram, Litre To Deca- To Hecto- To Kilo- Kilo- x10 6 x10 5 x10 4 x10 3 x10 2 x10 1 Hecto- x10 5 x10 4 x10 3 x10 2 x10 1 x10" 1 Deca- x10 4 x 10 3 x10 2 x10 1 x10" 1 x10" 2 Metre, Gram, Litre x 10 3 x 10 2 x10 1 x10" 1 x10" 2 x10" 3 Deci- x 10 2 x 10 1 x10" 1 x10" 2 x10" 3 x10 4 Centi- x 10 1 x10" 1 x10" 2 x10" 3 x10 4 x10" 5 Milli- x 10" 1 x10" 2 x10" 3 x10 4 x10" 5 x10" 6 Page 1 BASIC UNITS SI IMPERIAL DISTANCE 1 metre (1 m) = 10 decimetres (10 dm) 12 in. = 1 ft = 100 centimetres (100 cm) 3ft = 1yd = 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: lin. = 25.4 mm 1ft = 30.48 cm mile = 1.61km 1yd = 0.914 m lm = 3.28 ft Area 1 sq metre (lm 2 ) = 10 000 cm 2 = 1 000 000 mm 2 1 sq hectometre (1 hm 2 ) = 10 000 m 2 = 1 hectare (1 ha) 1 sq km (1 km 2 ) = 1 000 000 m 2 lft 2 lyd 2 1 sq mile 144 in. 2 9 ft 2 640 acre 1 section Conversions: lin. 2 lm 2 1 acre 1 sq mile 6.45 cm 2 10.8 ft 2 0.405 ha 2.59 km 2 645 mm Page 2 SI IMPERIAL Volume lm' = 1 000 000 cm' = lxl0 9 mm' 1ft' 1yd' = 1728 in.' = 27 ft 3 1dm' 1 litre 1 mL lm' = 1 litre = 1000 cm' = 1 cm' = 1000 litres 1 (liquid) U.S. gallon 1 U.S. barrel (bbl) 1 imperial gallon = 231 in.' = 4 (liquid) quarts = 42 U.S. gal. = 1.2 U.S. gal. Conversions: lin.' = 16.4 cm' lm' = 35.3 ft' 1 litre = 61 in.' lU.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 kg 1000 grams 1 tonne 2000 lb 1 long ton 1 ton (short) 2240 lb Density Conversions: 1 kg (on Earth) results in a weight of 2.2 lb mass density : m fkg V m mass volume weight density w V jb ft 3 weight volume Conversions: (on Earth) a mass density of 1 — 3 results in a weight density of 0.0623 — Page 3 SI 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. Imperial 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 4 Greek Alphabet Alpha a Beta P Gamma Y Delta A Epsilon s Zeta c Eta Tl Theta e Iota i Kappa K Lambda X Mu n Nu V Xi S Omicron Pi 71 Rho P Sigma I, a Tau X Upsilon u Phi ®A Kai i Psi V Omega Q, co MATHEMATICAL FORMULAE Algebra 1. Expansion Formulae (x + y) = x + 2xy + y (x - y) = x - 2xy + y x 2 -y 2 = (x-y) (x + y) (x + y) = x + 3x y + 3xy + y x + y = (x + y) (x - xy + y ) (x-y) = x" - 3x y + 3xy - y x 3 -y 3 = (x - y) (x 2 + xy + y 2 ) 2. Quadratic Equation Ifax 2 + bx + c = 0, Then x ■b±Vb 2 -4ac 2a Page 5 Trigonometry 1. Basic Ratios Sin A = — , cos A = — tan A: y 2. Pythagoras' Law 2,2 i 2 x +y = n 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 Sin A Sin B Sin C b. Cosine Law a 2 + b 2 - 2 ab Cos C b 2 + c 2 - 2 be Cos A a 2 + c 2 - 2 ac Cos B Page 6 Geometry 1. Areas of Triangles a. All Triangles base x perpendicular height Area: Area be Sin A ab Sin C ac Sin B and, Area = ^J 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 side 2 2. Circumference of a Circle C = 7id 3. Area of a Circle A = TOf circumference x r % d 2 =0.7854d 2 4. Area of a Sector of a Circle arcxr A = — — x 71 r 2 (9 = angle in degrees) A : 360 9°r 2 (9 = angle in radians) Page 7 5. Area of a Segment of a Circle A = area of sector - area of triangle Also approximate area = — h 2 J— - 0.608 6. Ellipse A=*Dd Approx. circumference = it (D + d) t d Triangle Segment 7. Area of Trapezoid H a — H A : a + b' -H 8. Area of Hexagon A = 2.6s 2 where s is the length of one side 9. Area of Octagon A = 4.83s 2 where s is the length of one side 10. Sphere Total surface area A =4tuc Surface area of segment A s = 7tdh 4 3 Volume V= — 7ir 3 Volume of segment V s = ^(3r-h) V. = -^-(h 2 + 3a 2 ) where a = radius of segment base Page 8 11. Volume of a Cylinder V = — d 2 L where L is cylinder length 12. Pyramid Volume V = - base area x perpendicular height Volume of frustum Frustum V F = — (A + a + v Aa) where h is the perpendicular height, A and a are areas as shown 13. Cone Area of curved surface of cone: A : 7i DL Area of curved surface of frustum A F 7c (D + d)L Volume of cone: _ base area x perpendicular height Volume of frustum: _ perpendicular height x tt (R 2 + r 2 + Rr) Page 9 APPLIED MECHANICS Scalar - a property described by a magnitude only Vector - a property described by a magnitude and a direction , T , mjL , A displacement Velocity - vector property equal to — —. The magnitude of velocity may be referred to as speed In SI the basic unit is ™, in Imperial ^ Other common units are ^H, ^ h h ^ . , m . _„ ft Conversions: 1 — = 3.28 — , km „ ^„, mi 1 = 0.621 — h h Speed of sound in dry air is 33 1 ^ at 0°C and increases by about 0.61 ^ for each °C s rise Speed of light in vacuum equals 3 x 10 8 m Acceleration - vector property equal to s change in velocity time In SI the basic unit is — -, in Imperial — - s s -. . , m _ _„ ft Conversion: 1— = 3.28 — s s Acceleration due to gravity, symbol "g", is 9.81 — or 32.2 — - s s Page 10 LINEAR VELOCITY AND ACCELERATION u initial velocity v final velocity t elapsed time s displacement a acceleration v = u + at 'v + u t v 1 9 s = ut + y at 2 - u 2 + 2 as Angular Velocity and Acceleration 9 angular displacement (radians) co angular velocity (radians/s); coi = initial, co 2 = final a angular acceleration (radians/s 2 ) co 2 = coi + at 9 = coi + co 2 x t 2 9 = coit + V^ar co 2 2 = coi 2 + 2a9 linear displacement, s = r 9 linear velocity, v = r co linear, or tangential acceleration, a T = r a Page 1 1 Tangential, Centripetal and Total Acceleration Tangential acceleration ax is due to angular acceleration a a T = roc Centripetal (Centrifugal) acceleration a c is due to change in direction only a c = v 2 /r = r co 2 Total acceleration, a, of a rotating point experiencing angular acceleration is the vector sum of a T and a c a = ax + a c 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 , s 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/s 2 In Imperial, the mass of an object (rarely used), in slugs, can be calculated from the known weight in pounds Weight ft m = — g — § = 32 V 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 = ^ a, where w is weight) Torque Equation T = I a where T is the acceleration torque in Nm, I is the moment of inertia in kg m 2 and a is the angular acceleration in radians/s 2 Momentum Vector quantity, symbol p, p = mv (Imperial p = ^ v, where w is weight) . „ x . . kg m in SI unit is — ^ — 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 Page 13 Kinetic Energy Energy due to motion E v = imv 2 In Imperial this is usually expressed as E k = ^v 2 where w is weight Kinetic Energy of Rotation E R = — mk 2 (D 2 where k is radius of gyration, co is angular velocity in rad/s or Ep = — Icd 2 where I = mk 2 is the moment of inertia R 2 CENTRIPETAL (CENTRIFUGAL) FORCE F c = where r is the radius or F c = m co 2 r where co is angular velocity in rad/s Potential Energy Energy due to position in a force field, such as gravity E p = m g h In Imperial this is usually expressed E p = 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 10 6 ft-lb Power A scalar quantity, equal to the rate of doing work In SI the unit is the Watt W (or kW) lW=l| In Imperial, the units are: Mechanical Power - ~ — , horsepower h.p. Thermal Power - =^± Electrical Power - W, kW, or h.p. Conversions: 746 W = 1 h.p. 1 h.p. = 550 &^ 1 kW = 0.948 =^= Page 15 Pressure A vector quantity, force per unit area In SI the basic units of pressure are pascals Pa and kPa lPa=l4 m 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 mH 2 = 9.81 kPa Other pressure unit conversions: 1 bar= 14.5 psi = 100 kPa .2 1 kg/cm =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 = oo a/R 2 -x : Acceleration of P = (D xm/s m s 2 2% The period or time of a complete oscillation = — seconds CO General formula for the period of S.H.M. T = 2jc. (displacement acceleration Simple Pendulum T = 2ti / — T = period or time in seconds for a double swing V § L = length in metres The Conical Pendulum (elevation) force diagram R/H = tan 9= F c /W = co 2 R/g Page 17 Lifting Machines W = load lifted, F = force applied MA. load effort W F V.R. (velocity ratio) n efficiency effort distance load distance M.A. V.R. 1. Lifting Blocks V.R. = number of rope strands supporting the load block 2. Wheel & Differential Axle Velocity ratio 2tiR 27i(r - r, ) 2 2R r-r, 2R Or, using diameters instead of radii, „ . . 2D Velocity ratio (d-d,) 3. Inclined Plane V.R length height 4. Screw Jack V.R circumference of leverage pitch of thread Gearing S= n Snatch block Pitch A * Effort Load Page 18 Indicated Power LP. = P m A L N where LP. is power in W, P m is mean or "average" effective pressure in Pa, A is piston area in m , L is length of stroke in m and N is number of power strokes per second Brake Power B.P. = Lcd where B.P. is brake power in W, L is torque in Nm and co is angular velocity in radian/second STRESS, STRAIN and MODULUS OF ELASTICITY Direct stress Direct strain load _ P area A extension A£ original length L Modulus of elasticity „ direct stress P/A PL direct strain AiVL AAl Shear stress x Shear strain force area under shear L Modulus of rigidity _ shear stress (j shear strain Force Force x Force Page 19 General Torsion Equation (Shafts of circular cross-section) T J G0 1. For Solid Shaft T % 4 7id 4 J = — r = 2 32 2. For Hollow Shaft J = f(r: 4 -r 2 4 ) 32 (d 4 -d 4 ) T = torque or twisting moment in newton metres J = polar second moment of area of cross-section about shaft axis, x = shear stress at outer fibres in pascals r = radius of shaft in metres G = modulus of rigidity in pascals = 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 (7 y E R 1. For Rectangle D - BD 3 M I c y E R external bending moment in newton metres second moment of area in m bending stress at outer fibres in pascals distance from centroid to outer fibres in metres modulus of elasticity in pascals radius of currative in metres 12 2. For Solid Shaft 7iD 4 64 Page 20 THERMODYNAMICS Temperature Scales °C = -(°F-32) °F=-°C + 32 9 5 °R = °F + 460 (R Rankine) K = °C + 273 (K Kelvin) Sensible Heat Equation Q = mcAT m is mass c is specific heat AT 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 PiV, = P 2 V 2 where P is absolute pressure and V is volume 2. Charles' Law V When gas pressure is constant, — = constant or -=i = -=!■ , where V is volume and T is absolute temperature Page 21 3. Gay-Lussac's Law When gas volume is constant, — = constant T P P Or — = — , where P is absolute pressure and T is absolute temperature lj l 2 4. General Gas Law PiV, _ P 2 V 2 _ T, T 2 = constant P V=mRT where P V T m R Also PV = nPvoT where P V T N Ro absolute pressure (kPa) volume (nr ) absolute temp (K) mass (kg) characteristic constant (kJ/kgK) absolute pressure (kPa) volume (m 3 ) absolute temperature K the number of kmoles of gas 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 T = C p /C v GAS or or kJ/kg °C kJ/kg °C 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 T - T Carnot Cycle r\ = —^-= — - where Ti and T 2 are absolute temperatures of heat source and lj sink Air Standard Efficiencies 1. Spark Ignition Gas and Oil Engines (Constant Volume Cycle or Otto Cycle) 1 . . cylinder volume T| = 1 - — — — where r v = compression ratio = r; Y " } v clearance volume specific heat (constant pressure) y ~ specific heat (constant volume) 2. Diesel Cycle (R Y — V) n = 1 — ^ — where r = ratio of compression r v T y(R-l) R = ratio of cut-off volume to clearance volume 3. High Speed Diesel (Dual-Combustion) Cycle kp y -l n = l- where r v k rr[(k-l) + yk(P-l)] cylinder volume P clearance volume absolute pressue at end of constant V heating (combustion) absolute pressue at beginning of constant V combustion volume at end of constant P heating (combustion) clearance volume 4. Gas Turbines (Constant Pressure or Brayton Cycle) 1 r, = l- Page 23 where r p = pressure ratio compressor discharge pressure compressor intake pressure W o a* o a © O u si O &o O <| c a u < Q O 4=WCo-^ K Ik 60 ,0 £ cH ~ K |K .£ a. £ .0 OS £ ^k~ a £ K" 5 E-T ft. £ ft. £ K- CL £ Change In Internal Energy K? 5 E-T £ E-T K* 5 £ Work Done 1W2 kJ a. K a; £ 2* E-T Cj £ k" k" -a cd ^ cu k" K 5 K~" ft. -2 K £ c £ a. 2 CO c +^ CD K. I 1 Is 1 f-1 tl II eC Is" V II ^|k" 0, 1 a, lo, 11 EC Ik" n<C|^ 11 kIk* II kIk" 1 0, * vc£5 II 11 O 1 8 - ?» s: C4— 1 8 E 2 Z 1 ^ C <u C 03 C © -t-J C r \ +■* S " ° CQ CO g 8 11 13 ■»-** * .— co g 11 CO E° a 11 o c > t3 O * 50 o o + rn I! 3 a, (2jj B (Dm ^ £h ? „ ^ s i J O [§ < ^ _ ^ ■ — . a -3 +_> S3 o o 60 M to •> s ^MHD> S to 3 ^ -.. D O kl J3 3 ra T3 rf> Q. c a <U m ■* J2 eft -ji o o Oh Cl, a. S3 C8 t-w ■Js 13 "Es *n i> a> i> M nJLuBt f^«q p*i«l ^^ o o o q^ O O O ^3 u w a> ±3 & 0, a, c C/5 &0 C<0 ttj S a 5 o X « ^ w w a, o ,y ^ a ^ <d o P ^ c J= 3 +-> 5- to R J » W c I- 1 ~h Ph cu Page 24 Heat Transfer by Conduction n= AAtAT V d where Q = heat transferred in joules A, = thermal conductivity or coeficient of heat transfer in J x m or W m 2 x s x °C m x °C A = area in m 2 t = time in seconds AT = temperature difference between surfaces in °C d = thickness of layer in m COEFFICIENTS OF THERMAL CONDUCTIVITY Material Coefficient of Thermal Conductivity W/m °C Air 0.025 Aluminum 206 Brass 104 Brick 0.6 Concrete 0.85 Copper 380 Cork 0.043 Felt 0.038 Glass 1.0 Glass, fibre 0.04 Iron, cast 70 Plastic, cellular 0.04 Steel 60 Wood 0.15 Wallboard, paper 0.076 Page 25 Thermal Expansion of Solids Increase in length where L a (T 2 - Ti ) L a (T 2 - Ti ) original length coefficient of linear expansion rise in temperature Increase in volume Where V P (T 2 - T, ) V P (T 2 - Ti ) original volume coefficient of volumetric expansion rise in temperature coefficient of volumetric expansion P coefficient of linear expansion x 3 3a SPECIFIC HEAT and LINEAR EXPANSION OF SOLIDS Mean Coefficient Mean Coefficient Specific Heat of Specific Heat of between 0°C Linear Expansion between 0°C Linear Expansion Solid and 100°C between 0°C and Solid and 100°C between 0°C and kJ/kgK or 100°C (Multiply by 10" 6 ) kJ/kgK or 100°C (Multiply by 10" 6 ) kJ/kg°C kJ/kg°C Aluminum 0.909 23.8 Iron (cast) 0.544 10.4 Antimony 0.209 17.5 Iron (wrought) 0.465 12.0 Bismuth 0.125 12.4 Lead 0.131 29.0 Brass 0.383 18.4 Nickel 0.452 13.0 Carbon 0.795 7.9 Platinum 0.134 8.6 Cobalt 0.402 12.3 Silicon 0.741 7.8 Copper 0.388 16.5 Silver 0.235 19.5 Glass 0.896 9.0 Steel (mild) 0.494 12.0 Gold. 0.130 14.2 Tin 0.230 26.7 Ice 2.135 50.4 Zinc 0.389 16.5 (between -20°C and 0°C) SPECIFIC HEAT and VOLUME EXPANSION FOR LIQUIDS Liquid Specific Heat (at20°C) kJ/kgKorkJ/kg°C Coefficient of Volume Expansion (Multiply by 10" 4 ) Liquid Specific Heat (at 20°) kJ/kgKorkJ/kg°C Coefficient of Volume Expansion (Multiply by 10-4) Alcohol (ethyl) 2.470 11.0 Olive Oil 1.633 Ammonia 0.473 Petroleum 2.135 Benzine 1.738 12.4 Gasoline 2.093 12.0 Carbon Dioxide 3.643 1.82 Turpentine 1.800 9.4 Mercury 0.139 1.80 Water 4.183 3.7 Page 26 Chemical Heating Value of a Fuel Chemical Heating Value MJ per kg of fuel = 33.7 C + 144 (h 2 - ^) + 9.3 S 8 C is the mass of carbon per kg of fuel H 2 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 Theoretical Air Required to Burn Fuel Air (kg per kg of fuel) = [-C + 8 (h 2 - ^_) + s] 100 23 Air Supplied from Analysis of Flue Gases Air in kg per kg of fuel = 33 (C( ^ 2 + CQ) x C C is the percentage of carbon in fuel by mass N 2 is the percentage of nitrogen in flue gas by volume C0 2 is the percentage of carbon dioxide in flue gas by volume CO is the percentage of carbon monoxide in flue gas by volume Boiler Formulae m,(h!-h 2 ) Equivalent evaporation Factor of evaporation = Boiler efficiency 2257 kJ/kg 2257 kJ/kg m s( h i- h 2 ) m f x calorific value of fuel where rh s = mass flow rate of steam hi = enthalpy of steam produced in boiler h 2 = enthalpy of feedwater to boiler rh, = mass flow rate of fuel Page 27 FLUID MECHANICS Discharge from an Orifice Let A and A c then A c or C c cross-sectional area of the orifice = (7t/4)d cross-sectional area of the jet at the vena conrtacta = ((tt/4) d ^ C C A A where C c is the coefficient of contraction Vena contracta At the vena contracta, the volumetric flow rate Q of the fluid is given by Q = area of the jet at the vena contracta x actual velocity = A c v or Q = C C AC V s/^gli The coefficients of contraction and velocity are combined to give the coefficient of discharge, C d i.e. (^j — ^A^„ and Q = C d A ^/^gh Typically, values for Cd vary between 0.6 and 0.65 Circular orifice: Q = 0.62 A y/2gh Where Q = flow (m 3 /s) A = area (m 2 ) h = head (m) Rectangular notch: Q = 0.62 (B x H) ^sfl$\ Where B = breadth (m) H = head (m above sill) Triangular Right Angled Notch: Q = 2.635 H 5/2 Where H = head (m above sill) Page 28 Bernoulli's Theory H H h P w 2g total head (metres) height above datum level (metres) pressure (N/m 2 or Pa) w V force of gravity on 1 m of fluid (N) velocity of water (metres per second) Loss of Head in Pipes Due to Friction Loss of head in metres = f- ?L v 2 L d pipes length in metres diameter in metres d 2g v = f = velocity of flow in metres per second constant value of 0.01 in large pipes to 0.02 in small Note: This equation is expressed in some textbooks as T v 2 Loss = 4f=^ ^— where the f values range from 0.0025 to 0.005 d 2g ° Actual Pipe Dimensions Scheduled (SI Units) Nominal Pipe Size (in) Outside Diameter (mm) Inside Diameter (mm) Wall Thickness (mm) Flow Area (m 1 ) l s 10.3 6.8 1.73 3.660 x 10 ~* i 4 13.7 9.2 2.24 6.717 x 10" 5 1 8 17.1 12.5 2.31 1.236 x 10"* 1 2 21.3 15.8 2.77 1.960 x KT 4 1 4 26.7 20.9 2.87 3.437 x 10"* 1 33.4 26.6 3.38 5.574 x 10 " 4 li 42.2 35.1 3.56 9.653 x 10 ~* ii 48.3 40.9 3.68 1.314 x 10" 3 2 60.3 52.5 3.91 2.168 x 10" 3 2i 73.0 62.7 5.16 3.090 x 10" 3 3 88.9 77.9 5.49 4.768 x 10" 3 8} 101.6 90.1 5.74 6.381 x 10 " 3 4 114.3 102.3 6.02 8.213 x 10 -3 5 141.3 128.2 6.55 1.291 x 10 -2 6 168.3 154.1 7.11 1.864 x KT 2 8 219.1 202.7 8.18 3.226 x 10 -2 10 273.1 254.5 9.27 5.090 x 10 " 2 12 323.9 303.2 10.31 7.219 x 10" 2 14 355.6 333.4 11.10 8.729 x 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 Page 29 ELECTRICITY Ohm's Law or h where I E R E R IR current (amperes) electromotive force (volts) resistance (ohms) Conductor Resistivity a where p = specific resistance (or resistivity) (ohm metres, Q-m) L = length (metres) a = area of cross-section (square metres) Temperature correction R t = Ro (1 + at) where R, = resistance at 0°C (Q) R t = resistance at t°C (Q) a = temperature coefficient which has an average value for copper of 0.004 28 (Q/Q°C) R 2 = Rl (l±^il (1 + atJ where Ri = resistance at ti (Q) R2 = resistance at t2 (Q) a Values Q/Q°C 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 = — rr r ° 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) O = useful flux per pole (webers), entering or leaving the armature p = number of pairs of poles N = speed (revolutions per minute) Generator Terminal volts = E G - I a R a Motor Terminal volts = E B + I a R a where E G = generated e.m.f. E B = generated back e.m.f. I a = armature current R a = 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 1 „ Form factor of sinusoidal = = = 1.11 Mean value 0.637 pN Frequency of alternator = cycles per second 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 , nn — - — xlOO Speed of field Inductive Reactance Reactance of AC circuit (X) = 27ifL ohms where L = inductance of circuit (henries) Inductance of an iron cored solenoid = — ; — henries LxlO 8 where T = turns on coil [i = magnetic permeablility of core A = area of core (square centimetres) L = length (centimetres) Capacitance Reactance Capacitance reactance of AC circuit = ohms 2jrfC where C = capacitance (farads) r Total reactance 27ifL ohms 2ti fC J Impedence (Z) = -^/(resistance) 2 + (reactance) 2 R 2 + (2tc fL - — *— ) 2 ohms I 27tfC Current in AC Circuit impressed volts Current ■ impedance Page 32 Power Factor p.f. true watts volts x amperes also p.f. = cos O, where O is the angle of lag or lead Three Phase Alternators Star connected Line voltage Line current Delta connected Line voltage Line current V3 x phase voltage phase current phase voltage V3 x phase current Three phase power P = V3 E l I l cosO E l = line voltage II = line current cos O = power factor \ E 2 P \ R E E \ / E I R \ / I 2 R 1 P I \ R E \ I \ R E / VPR / \y^ E 2 \^^ I R ^ / \ P / P p / I 2 i Page 33 Noble Gases < oo = 00 m o co o CO o Jg d cn co >; cri 00 8£8 ■<t a> co CM CO m §^S i- > <N X »- 2 cj 7- < co in X t- CD S J^ E c o CO &co o in o o> o < o 5^ o -a- cn in^oi CD CO CM 8<i& Of N 8^8. i- > cd ^t O) U. i- T-On CO CO |-~ m — f- A, r» >- t- 1 -T o t-- CD co 4, cn ?i o ^ CO oo O -r- o co .„ c\i S^fS CM .CD CM m H i- S£S CO Q E C CO cn t co co H t- CO IIS I-- OO 00 o co m < o cn in „ d cn co <J ■* ■^ £ CM cn C ~ ° co ,*-, CD § E te T- > r- 2 i- i- CL CO co < r- in c/) t- CO CO CM (D 111 i- en cn CM cn -l o ,-, c\i o ■* ; - oo in cm .9 c\i °,5™ CM XI O S O CD 8?£S idOt- t- CO CM co O r-. in CO -^ CO CL CM CO X T- 00 CM 00 ■* in cog 00 m CO i- cn CO -= CD ■■- < CN coSS ?^ r- ,_ O 00 I- CM ^ CM §cT2 SoS cm m 8 o c in cob?! CD ™d o °>o cn in p m feall CO N CD ■* O 7- CO X CM CD 1- -^ m cn o co ScSl j: co O) = CO CM O CO $.?§ cn 3 cn f- < t- SS^ co en CD CO = 00 co p o iri co j- cn o -, CM co sis i- > (A wZ in -a- o_ -^ I-. CL t- h- Z m cn > LLI CO cn CM ■<t ^-. ^ s UJ _i en cm O in CM cn £ °5 r o cm , s in CD CO i- cncL-S UJ CO LU CO m CO > X 1- u. O CD c o CO in CO co .cd in cm u. in 3cr° CM CD "> § h- O T- g8 CD CL C Stl ui "fr CM CM o m CO < 1- co cn m £ -* cm 5 in 3*1 co in & co O?? CO CM _ CO i~- > 1- f~ DC t- T- 3 CD Z i- cn 3 cm o Q o ■<t cn 0) ^^ CO CD > o DC UI o «* ,\ CM cm O m cn CM 5 in ^ 1 o> co "3- > 00 h- ■S- t- 11 O cn £- ■* m cl t- - « CM cn o. £i 0. ■* ,_ cn o m cn co ^ d cn T-5 W o 8& co <8 ? _ CM o p co m > cm > in *Z(JI h~ F *- •^ 3 mOT- cn 1— cm CO CM m co CD CO CD CO CM co -* 3" ■D *> CM ,_ |-~ CM 1- ■* 3 N cn ?!x^ °3 CO t3 < CO , — cn ^-^. _i * n = o> -> cn CM CO 3- cn O) . 00 co >■ co « CO h- CO CO in -J t- §3^^ * c co <N T — oo CM co < CM = 1€ S :* CO J < UJ 2 (D o •* CO O) CO C\j o 5 ^ i- 2 CM o o « d CM O ■* CD co CO oo co <$ co m co i- « to 00 ra CM 00 CC CM 00 „_ O) o f- o> ,-. ,_ < 8 7- X T^ CO Ij CO cn i- JS cm cn v , cri t- ^ CO r~ £> in co H co in « co in O t- co CL 3 o c s|Bia|A| ||B>||\ 1 O Page 34 ION NAMES AND FORMULAE MONATOMIC POLYATOMIC Ag + silver ion B0 3 3 " borate ion Al 3+ aluminum ion C2H3O2 acetate ion Au + and Au 2+ gold ion CIO" hypochlorite ion Be 2+ beryllium ion C10 2 " chlorite ion Ca 2+ calcium ion CIO3- chlorate ion Co 2+ and Co 3+ cobalt ion CIO4" perchlorate ion Cr 2+ and Cr 3+ chromium ion CN" cyanide ion Cu + and Cu 2+ copper ion CO3 2 " carbonate ion Fe 2+ and Fe 3+ iron ion C2O4 2 " oxalate ion K + potassium ion Cr0 4 2 " chromate ion Li + lithium ion Cr 2 7 2 " dichromate ion Mg 2+ magnesium ion HCO3" hydrogen carbonate or bicarbonate ion Na + sodium ion H 3 + hydronium ion Zn 2+ zinc ion HPO4 2 " hydrogen phosphate ion H2PO4" dihydrogen phosphate ion HSO3" hydrogen sulphite or bisulphite ion HSO4" hydrogen sulphate or bisulphate ion Mn0 4 " permanganate ion N 3 " azide ion NH 4 + ammonium ion N0 2 " nitrite ion N0 3 " nitrate ion 2 2 - peroxide ion OCN" cyanate ion OH" hydroxide ion P0 3 3 " phosphite ion P0 4 3 " phosphate ion SCN" thiocyanate ion SO3 2 " sulphite ion S0 4 2 " sulphate ion S2O3 2 " thiosulphate ion Page 35 power engineering TRAINING SYSTEMS This material is owned by Power Engineering Training Systems and may not be modified from its original form. 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