Aeronautical Engineer Data Book Episode 2 doc

11 Fundamental dimensions and units 10 bar 1 bar atmosphere 1MPa or 1 MN m 2 1 bar 1.013 bar 760 mm Hg 1.1097 kg/cm 2 10 5 N/m 2 or 10 5 Pa 10.3 m H 2 O 14.7 psi Rules of thumb: An apple ‘weighs’ about 1.5 newtons A meganewton is equivalent to about 100 tonnes An average car weighs about 15 kN Fig. 2.1 Pressure relationships KSI ϫ1000 ϫ 6.895.10 –3 ϫ 145.03 ϫ 1.0197 ϫ 0.9807 ϫ 10.0 ϫ 0.1 ϫ 0.09807 ϫ 10.197 ϫ 14.223 ϫ 0.06895 ϫ 0.0703 psi Bar Kg/cm 2 N/mm 2 (MPa) ϫ 14.503 Fig. 2.2 Pressure conversions 12 Aeronautical Engineer’s Data Book 0 K Volume –273.15˚C 0˚C 100˚C 32˚F 212˚F Fig. 2.3 Temperature 2.3.6 Heat and work The basic unit for heat ‘energy’ is the British thermal unit (BTU). Specific heat ‘energy’ is measured in BTU/lb (in SI it is joules per kilogram (J/kg)). 1 J/kg = 0.429923 ϫ 10 –3 BTU/lb Table 2.6 shows common conversions. Specific heat is measured in BTU/lb °F (or in SI, joules per kilogram kelvin (J/kg K)). 1 BTU/lb °F = 4186.798 J/kg K 1 J/kg K = 0.238846 ( 10 –3 BTU/lb °F 1 kcal/kg K = 4186.8 J/kg K Heat flowrate is also defined as power, with the unit of BTU/h (or in SI, in watts (W)). 1 BTU/h = 0.07 cal/s = 0.293 W 1 W = 3.41214 BTU/h = 0.238846 cal/s 2.3.7 Power BTU/h or horsepower (hp) are normally used or, in SI, kilowatts (kW). See Table 2.7. 2.3.8 Flow The basic unit of volume flowrate is US gallon/min (in SI it is litres/s). 1 US gallon = 4 quarts = 128 US fluid ounces = 231 in 3 13 Fundamental dimensions and units 1 US gallon = 0.8 British imperial gallons = 3.78833 litres 1 US gallon/minute = 6.31401 ϫ 10 –5 m 3 /s = 0.2273 m 3 /h 1 m 3 /s = 1000 litres/s 1 litre/s = 2.12 ft 3 /min ˚F 2500 2000 1500 1000 900 800 700 600 500 400 ˚C˚F 140 120 100 300 250 210 90 200 80 70 60 50 40 30 20 10 0 –10 190 180 170 160 150 140 130 120 110 100 90 80 70 60 50 40 +30 +20 –20 +10 0 0 –10 –30 –40 –50 –60 –70 –80 –90 –100 –20 –30 –40 –50 –60 –70 –80 –90 –100 –120 –140 ˚C –120 –140 –160 –180 –200 –250 Temperature conversions ˚C Fig. 2.4 1000 900 800 700 600 500 400 300 200 180 150 ˚F –160 –180 –200 –250 –300 –350 –400 Table 2.5 Pressure (p) Unit lb/in 2 (psi) lb/ft 2 atm in H 2 0 cmHg N/m 2 (Pa) 1 lb per in 2 (psi) 1 144 6.805 ϫ 10 –2 27.68 5.171 6.895 ϫ 10 3 1 lb per ft 2 6.944 ϫ 10 –3 1 4.725 ϫ 10 –4 0.1922 3.591 ϫ 10 –2 47.88 1 atmosphere (atm) 14.70 2116 1 406.8 76 1.013 ϫ 10 5 1 in of water at 39.2°F (4°C) 3.613 ϫ 10 –2 5.02 2.458 ϫ 10 –3 1 0.1868 249.1 1 cm of mercury at 32°F (0°C) 0.1934 27.85 1.316 ϫ 10 –2 5.353 1 1333 1 N per m 2 (Pa) 1.450 ϫ 10 –4 2.089 ϫ 10 –2 9.869 ϫ 10 –6 4.015 ϫ 10 –3 7.501 ϫ 10 –4 1 Table 2.6 Heat BTU ft-lb hp-h cal J kW-h 1 British thermal unit (BTU) 1 777.9 3.929 ϫ 10 –4 252 1055 2.93 ϫ 10 –4 1 foot-pound (ft-lb) 1.285 ϫ 10 –3 1 5.051 ϫ 10 –7 0.3239 1.356 3.766 ϫ 10 –7 1 horsepower-hour (hp-h) 2545 1.98 ϫ 10 6 1 6.414 ϫ 10 5 2.685 ϫ 10 6 0.7457 1 calorie (cal) 3.968 ϫ 10 –3 3.087 1.559 ϫ 10 –6 1 4.187 1.163 ϫ 10 –6 1 joule (J) 9.481 ϫ 10 –4 0.7376 3.725 ϫ 10 –7 0.2389 1 2.778 ϫ 10 –7 1 kilowatt hour (kW-h) 3413 2.655 ϫ 10 6 1.341 8.601 ϫ 10 5 3.6 ϫ 10 6 1 14 15 Table 2.7 Power (P) BTU/h BTU/s ft-lb/s hp cal/s kW W 1 BTU/h 1 2.778 ϫ 10 –4 0.2161 3.929 ϫ 10 –4 7.000 ϫ 10 –2 2.930 ϫ 10 –4 0.2930 1 BTU/s 3600 1 777.9 1.414 252.0 1.055 1.055 ϫ 10 –3 1ft-lb/s 4.62 1.286 ϫ 10 –3 1 1.818 ϫ 10 –3 0.3239 1.356 ϫ 10 –3 1.356 1 hp 2545 0.7069 550 1 178.2 0.7457 745.7 1 cal/s 14.29 0.3950 3.087 5.613 ϫ 10 –3 1 4.186 ϫ 10 –3 4.186 1 kW 3413 0.9481 737.6 1.341 238.9 1 1000 1 W 3.413 9.481 ϫ 10 –4 0.7376 1.341 ϫ 10 –3 0.2389 0.001 1 Table 2.8 Velocity (v) Item ft/s km/h m/s mile/h cm/s knot 1 ft per s 1 1.097 0.3048 0.6818 30.48 0.592 1 km per h 0.9113 1 0.2778 0.6214 27.78 0.5396 1 m per s 3.281 3.600 1 2.237 100 1.942 1 mile per h 1.467 1.609 0.4470 1 44.70 0.868 1 cm per s 3.281 ϫ 10 –2 3.600 ϫ 10 –2 0.0100 2.237 ϫ 10 –2 1 0.0194 1 knot 1.689 1.853 0.5148 1.152 51.48 1 16 Aeronautical Engineer’s Data Book 2.3.9 Torque The basic unit of torque is the foot pound (ft.lbf) (in SI it is the newton metre (N m)). You may also see this referred to as ‘moment of force’ (see Figure 2.5) 1 ft.lbf= 1.357 N m 1 kgf.m = 9.81 N m 2.3.10 Stress Stress is measured in lb/in 2 – the same unit used for pressure although it is a different physical quantity. In SI the basic unit is the pascal (Pa). 1 Pa is an impractically by small unit so MPa is normally used (see Figure 2.6). 1 lb/in 2 = 6895 Pa 1 MPa = 1 MN/m 2 = 1 N/mm 2 1 kgf/mm 2 = 9.80665 MPa 2.3.11 Linear velocity (speed) The basic unit of linear velocity (speed) is feet per second (in SI it is m/s). In aeronautics, the most common non-SI unit is the knot, which is equivalent to 1 nautical mile (1853.2 m) per hour. See Table 2.8. 2.3.12 Acceleration The basic unit of acceleration is feet per second squared (ft/s 2 ). In SI it is m/s 2 . 1 ft/s 2 = 0.3048 m/s 2 1 m/s 2 = 3.28084 ft/s 2 Standard gravity (g) is normally taken as 32.1740 ft/s 2 (9.80665 m/s 2 ). 2.3.13 Angular velocity The basic unit is radians per second (rad/s). 1 rad/s = 0.159155 rev/s = 57.2958 degree/s The radian is also the SI unit used for plane angles. A complete circle is 2π radians (see Figure 2.7) A quarter-circle (90°) is π/2 or 1.57 radians 1 degree = π/180 radians 17 Fundamental dimensions and units Force ( N ) Radius ( r ) Torque = Nr Fig. 2.5 Torque Area 1 m 2 1 MN Fig. 2.6 Stress 2 π radians θ Fig. 2.7 Angular measure 18 Table 2.9 Area (A) Unit sq.in sq.ft sq.yd sq.mile cm 2 dm 2 m 2 a ha km 2 1 square inch 1 - – – 6.452 0.06452 – - – - 1 square foot 144 1 0.1111 - 929 9.29 0.0929 – - – 1 square yard 1296 9 1 – 8361 83.61 0.8361 – – – 1 square mile – – – 1 – – – – 259 2.59 1 cm 2 0.155 – – – 1 0.01 – – – – 1 dm 2 15.5 0.1076 0.01196 – 100 1 0.01 – – – 1 m 2 1550 10.76 1.196 – 10 000 100 1 0.01 – – 1 are (a) – 1076 119.6 – – 10 000 100 1 0.01 – 1 hectare (ha) – – – – – – 10 000 100 1 0.01 1 km 2 – – – 0.3861 – – – 10 000 100 1 19 Fundamental dimensions and units 2.3.14 Length and area Comparative lengths in USCS and SI units are: 1 ft = 0.3048 m 1 in = 25.4 mm 1 statute mile = 1609.3 m 1 nautical mile = 1853.2 m The basic unit of area is square feet (ft 2 ) or square inches (in 2 or sq.in). In SI it is m 2 . See Table 2.9. Small dimensions are measured in ‘micro- measurements’ (see Figure 2.8). The microinch (µin) is the commonly used unit for small measures of distance: 1 microinch = 10 –6 inches = 25.4 micrometers (micron ) Oil filter mesh 450µin Diameter of a hair: 2000µin Smoke particle 120µin A smooth-machined ‘mating’ –32µin 1 micron (µm) = 39.37µin A fine ‘lapped’ with peaks within 1µin surface with peaks 16 surface Fig. 2.8 Micromeasurements 2.3.15 Viscosity Dynamic viscosity (µ) is measured in lbf.s/ft 2 or, in the SI system, in N s/m 2 or pascal seconds (Pa s). 1 lbf.s/ft 2 = 4.882 kgf.s/m 2 = 4.882 Pa s 1Pas = 1Ns/m 2 = 1 kg/m s A common unit of viscosity is the centipoise (cP). See Table 2.10. 20 Aeronautical Engineer’s Data Book Table 2.10 Dynamic viscosity (␮) Unit lbf-s/ft 2 Centipoise Poise kgf/m s 1 lb (force)-s 1 4.788 4.788 4.882 per ft 2 ϫ 10 4 ϫ 10 2 1 centipoise 2.089 1 10 –2 1.020 ϫ 10 –5 ϫ 10 –4 1 poise 2.089 100 1 1.020 ϫ 10 –3 ϫ 10 –2 1 N-s per m 2 0.2048 9.807 98.07 1 ϫ 10 3 Kinematic viscosity (␯) is a function of dynamic viscosity. Kinematic viscosity = dynamic viscosity/ density, i.e. ␯ = µ/␳ The basic unit is ft 2 /s. Other units such as Saybolt Seconds Universal (SSU) are also used. 1 m 2 /s = 10.7639 ft 2 /s = 5.58001 ϫ 10 6 in 2 /h 1 stoke (St) = 100 centistokes (cSt) = 10 –4 m 2 /s 1 St > 0.00226 (SSU) – 1.95/(SSU) for 32 < SSU < 100 seconds 1 St  0.00220 (SSU) – 1.35/(SSU) for SSU > 100 seconds 2.4 Consistency of units Within any system of units, the consistency of units forms a ‘quick check’ of the validity of equations. The units must match on both sides. Example: To check kinematic viscosity (␯) = dynamic viscosity (µ) ᎏᎏᎏ = µ ϫ 1/␳ density (␳) ft 2 lbf.s ft 4 ᎏ = ᎏ ϫ ᎏ s ft 2 lbf.s 2 ft 2 s.ft 4 ft 2 Cancelling gives ᎏ = ᎏ = ᎏ s s 2 .ft 2 s OK, units match. [...]... 11/ 32 23/64 3/8 25 /64 13/ 32 27/64 7/16 29 /64 15/ 32 31/64 1 /2 33/64 17/ 32 35/64 9/16 37/64 19/ 32 39/64 5/8 41/64 21 / 32 43/64 11/16 45/64 23 / 32 47/64 3/4 49/64 25 / 32 51/64 13/16 53/64 27 / 32 55/64 7/8 57/64 29 / 32 59/64 15/16 61/64 31/ 12 63/64 1 0.015 62 0.03 125 0.04687 0.0 625 0 0.078 12 0.09375 0.10937 0. 125 00 0.140 62 0.15 625 0.17187 0.18750 0 .20 3 12 0 .21 875 0 .23 437 0 .25 000 0 .26 5 62 0 .28 125 0 .29 687 0.3 125 0... 0. 625 00 0.640 62 0.65 625 0.67187 0.68750 0.703 12 0.71875 0.73437 0.75000 0.765 62 0.78 125 0.79687 0.8 125 0 0. 828 12 0.84375 0.85937 0.87500 0.890 62 0.90 625 0. 921 87 0.93750 0.953 12 0.96875 0.98437 1.00000 13.09687 13.49375 13.890 62 14 .28 750 14.68437 15.08 125 15.478 12 15.87500 16 .27 187 16.66875 17.065 62 17.4 625 0 17.85937 18 .25 625 18.653 12 19.05000 19.44687 19.84375 20 .24 0 62 20.63750 21 .03437 21 .43 125 21 . 828 12. .. 0. 328 12 0.34375 0.35937 0.37500 0.390 62 0.40 625 0. 421 87 0.43750 0.453 12 0.46875 0.48437 0.50000 0.39687 0.79375 1.190 62 1.58750 1.98437 2. 38 125 2. 778 12 3.17500 3.57187 3.96875 4.365 62 4.7 625 0 5.15937 5.55 625 5.953 12 6.35000 6.74687 7.14375 5.540 62 7.93750 8.33437 8.73 125 9. 128 12 9. 525 00 9. 921 87 10.31875 10.715 62 11.1 125 0 11.50937 11.90 625 12. 303 12 12. 70000 0.515 62 0.53 125 0.54687 0.5 625 0 0.578 12 0.59375... 21 .03437 21 .43 125 21 . 828 12 22. 225 00 22 . 621 87 23 .01875 23 .415 62 23.8 125 0 24 .20 937 24 .60 625 25 .003 12 25.40000 M Mass L Length T Time For example, velocity (v) is represented by length divided by time, and this is shown by: L [v] = ᎏ : note the square brackets denoting T ‘the dimension of’ Table 2. 12 shows the most commonly used quantities 24 Aeronautical Engineer s Data Book Table 2. 12 Dimensional analysis... r = |z| x2 + y� r = � 2 ␪ is called the argument and this may be written ␪ = arg z y tan ␪ = ᎏᎏ x If z1 = r (cos␪1 + i sin ␪1) and z2 = r2 (cos 2 + i sin 2) z1z2 = r1r2 [cos(␪1 + 2) + i sin(␪1 + 2) ] = r1r2Є(␪1 + 2) r1[cos(␪1 – 2) + i sin(␪1 + 2) ] z1\z2 = ᎏᎏᎏᎏ r2 r1 = ᎏᎏ Є(␪1 – 2) r2 2. 8.6 Standard series Binomial series n(n – 1) (a + x)n = an + nan–1 x + ᎏᎏ an 2 x2 2! n(n – 1)(n – 2) + ᎏᎏ an–3... bracket: � kg 0.16 lb ␳=ᎏ ᎏ in3 2. 205 lb � Step 3: Then apply the ‘dimension’ unity brackets (cubed): � 0.16 lb kg ␳=ᎏ ᎏ 3 in 2. 205 lb 1000 mm �ᎏᎏ� m 3 in � �ᎏᎏ� 25 .4 mm 3 3 22 Aeronautical Engineer s Data Book Step 4: Expand and cancel*: kg in3 0.16 lb ␳ = ᎏ ᎏ ᎏᎏ 3 in 2. 205 lb (25 .4)3 mm3 � � �� (1000)3 mm3 ᎏᎏ m3 � � 0.16 kg (1000)3 ␳ = ᎏᎏ 2. 205 (25 .4)3 m3 ␳ = 4 428 . 02 kg/m3 Answer *Take care to use... ax2 + bx + c = r – p(x + q )2 the maximum value of the function occurs when (x + q) = 0 and its value is r 2. 8.4 Cubic equations x3 + px2 + qx + r = 0 x = y – ᎏ1ᎏ p gives y3 + 3ay + 2b = 0 3 where 3a = –q – ᎏ1ᎏ p2, 3 2b = 2 p3 – ᎏ1ᎏ pq + r 27 3 On setting 3 S = [–b + (b2 + a )1 /2] 1/3 and T = [–b – (b2 + a3)1 /2] 1/3 the three roots are x1 = S + T – ᎏ1ᎏ p 3 x2 = – ᎏ1ᎏ(S + T) + �3 \2 i(S – T) – ᎏ1ᎏ p � 2. .. + (x2 < a2) The number of terms becomes inifinite when n is negative or fractional Fundamental dimensions and units 29 � � 1 bx b2x2 b3x3 (a – bx)–1 = ᎏᎏ 1 + ᎏᎏ + ᎏᎏ + ᎏᎏ + a a a2 a3 2 2 2 (b x < a ) Exponential series (x ln a )2 (x ln a)3 ax = 1 + x ln a + ᎏᎏ + ᎏᎏ + 2! 3! x2 x3 ex = 1 + x + ᎏᎏ + ᎏᎏ + 2! 3! Logarithmic series 1 1 ln x = (x – 1) – ᎏᎏ (x – 1 )2 + ᎏᎏ (x – 1)3 – (0 2 3 < x < 2) x–1... log a a n loga 1 = 0 loge N = 2. 3 026 log10 N 2. 8.3 Quadratic equations If ax2 + bx + c = 0 b2 – 4ac –b ± ��� x = ᎏᎏ 2a If b2 –4ac > 0 the equation ax2 + bx + c = 0 yields two real and different roots If b2 –4ac = 0 the equation ax2 + bx + c = 0 yields coincident roots If b2 –4ac < 0 the equation ax2 + bx + c = 0 has complex roots If ␣ and ␤ are the roots of the equation ax2 + bx + c = 0 then b sum of... T) – �3 \2 i(S – T) – ᎏ1ᎏ p � 2 3 For real coefficients all roots are real if b2 + a3 ≤ 0, one root is real if b2 + a3 > 0 At least two roots are equal if b2 + a3 = 0 Three roots are equal if a = 0 and b = 0 For b2 + a3 < 0 there are alternative expressions: 1 1 1 1 x1 = 2c cosᎏ3ᎏ␪ – ᎏ3ᎏ p x2 = 2c cosᎏ3ᎏ (␪ + 2 ) – ᎏ3ᎏ p 1 1 ᎏᎏ (␪ + 4π) – ᎏᎏ p x3 = 2c cos 3 3 b 2 where c = –a and cos␪ = – ᎏᎏ c3 2. 8.5 . 21 .43 125 23 /64 0.35937 9. 128 12 55/64 0.85937 21 . 828 12 3/8 0.37500 9. 525 00 7/8 0.87500 22 .22 500 25 /64 0.390 62 9. 921 87 57/64 0.890 62 22. 621 87 13/ 32 0.40 625 10.31875 29 / 32 0.90 625 23 .01875 27 /64. 0.78 125 19.84375 19/64 0 .29 687 5.540 62 51/64 0.79687 20 .24 0 62 15/16 0.3 125 0 7.93750 13/16 0.8 125 0 20 .63750 21 /64 0. 328 12 8.33437 53/64 0. 828 12 21.03437 11/ 32 0.34375 8.73 125 27 / 32 0.84375 21 .43 125 . 27 /64 0. 421 87 10.715 62 59/64 0. 921 87 23 .415 62 7/16 0.43750 11.1 125 0 15/16 0.93750 23 .8 125 0 29 /64 0.453 12 11.50937 61/64 0.953 12 24 .20 937 15/ 32 0.46875 11.90 625 31/ 12 0.96875 24 .60 625 31/64