None
6. Solve Problem 4 if the vane is moving with a velocity of 25 ft. per second in the same direction as the jet. 6. A 1^-in. nozzle has a coefficient of velocity of 0.97 and a coefficient of contraction of unity. The base of the nozzle has a diameter of 4 in., at which point the gage pressure is 80 Ibs. per square inch. The jet is 230 HYDRODYNAMICS deflected through an angle of 150° by a double-cusped vane that has a velocity, in the direction of the jet of 30 ft. per second. What is the pressure exerted on the vane and what is the amount of work done expressed in foot-pounds? 7. In Problem 6 determine the velocity the vane must have if the jet is to perform the maximum amount of work. What is the maximum work in foot-pounds? 8. If the jet, in Problem 6, strikes a series of vanes so arranged on the periphery of a wheel that the entire jet is deflected through an angle of 170°, what is the maximum amount of work that can be done? 9. A horizontal straight pipe gradually reduces in diameter from 12 in. to 6 in. If, at the larger end, the gage pressure is 40 Ibs. per square inch and the velocity is 10 ft. per second, what is the total longitudinal thrust exerted on the pipe?- Neglect friction. 10. A bend in a pipe line gradually reduces from 24 in. to 12 in. The deflection angle is 60°. If at the larger end the gage pressure is 25 Ibs. per square inch and the velocity is 8 ft. per second, determine the X- and F-com- ponents of the dynamic thrust exerted on the bend. Also determine the X- and F-components of the total thrust exerted on the bend, neglecting friction. 11. A 24-in. cast-iron pipe I in. thick and 6000 ft. long discharges water from a reservoir under a head of 80 ft. What is the pressure due to water hammer resulting from the instantaneous closure of a valve at the discharge end? 12. If the time of closing the valve, in Problem 11, is 6 sec., determine the resulting pressure due to water hammer, comparing results obtained by use of formulas by Joukovsky, Johnson, W^arren and Mead. ANSWERS TO PROBLEMS IN HYDRAULICS BY HORACE \V. KING AND CHESTER O. WISLER IECQND EDITION «^^^- PRICE lOc NET NEW YORK JOHN WILEY & SONS, INC. LONDON: CHAPMAN & HALL, LIMITED CHAPTER II. Pages 21-23 i. (a) 2496, (b) 17.3, (c) 4612, (d) 32.0. 2. (a) 566, (b) 3.93, (c) 2682, (d) 18.63. 3- («) 25,085 Ibs. gage pressure, (b) 158.8 Ibs. 4. (a) 27.12 ft., (b) 24.7 in. 5. 989. 6. (a) 16.97 ft., (b) 230.8 ft. 7. 11,520 Ibs. 8. 9.7. 9. 31.1 in. 10. 10.85 m IJ- 2I-3- I2« 1.15 ft. 13. 3.15 ft. 14. 10.6 Ibs. 15. (a) 9.38, (b) 24.08. 16. 25.4 in. 17. (a) 9.50, (6) —5.20. 18. 17.6 in. 19. 27.7 in. 20. 6.65. 21. (a) 9.50, (b) 7.46 Ibs. 22. (a) 30.0 ft., (b) 13.0, (c) 3200 ft. 23. (a) 5.23 Ibs. per sq. in., (6) 16.70 in. 24. 720 Ibs. 25. 5850 ft. 26. 14.76. 27. —1375. 28. 637 Ibs. per sq. in. 29. .00325. 30. 1.95 ft. CHAPTER III. Pages 37-39 i. 4493 Ibs., 2 ft. above base. 2. (a) 1.67 ft. from center, (b) 1.52. 3. (a) 6.93 ft., (b) 3.0, .(c) from 4500 Ibs. per sq. ft. at toe to o at heel. 4. 10.9 ft. 5. 5.2 Ibs. and 26.0 Ibs. in each of upper and lower screws respectively. 6. 46.8 Ibs. and 100.2 Ibs. in each of upper and lower screws respectively. 7. 7000 Ibs., 1.81 ft. from bottom. 8. 12,580 Ibs., 2.54 ft. from base. 9. (a) 41.6 Ibs., (b) 62.4 Ibs. 10. 1498 Ibs., 2| ft. below vertex, n. 499 Ibs. 12. 1467 Ibs. 13. 3120 Ibs. and 6240 Ibs. in top and bottom hoops respectively. 14. 1580 Ibs. and 4740 Ibs. in top and bottom hoops respectively. 15. 34.6 Ibs. and 138.3 Ibs. in top and bottom hoops respectively. 16. 1404 Ibs. 17. 4.78 ft. from toe. 18. 2.3 ft. from toe. 19. 21.1 ft. 20. 10.4 ft. 21. 17.0 ft. 22. 18.9 ft. 23. 10,650 Ibs. 24. 21.3 ft. 25. 6.0 ft. 26. 31,980 Ibs. 27. 2910 Ibs. 28. 6990 Ibs., 6.43 ft. 29. 1764 Ibs., 0.14 ft. below center. 30. 12,720 Ibs. 31. 2250 Ibs. and 5625 Ibs. in top and bottom hoops respectively. CHAPTER IV. Pages 45-47 i. 2.6? ft. 2. 224,600 Ibs. 3. Remains stationary. 4. 2.0 ft. 5. (a) 4! ft. above bottom and 2.5 ft. from center line, (b) 4.71 ft., (r) righting couple = 909, 800 ft. -Ibs. 6. 16.83 ft. 7.. 12.76 ft. above m. 8. 28,650 Ibs. 9. 0.77 in. 10. 27.6 ft. n. 156 Ibs. 4 Ibs. 13. 264 Ibs. 14. 3.72 ft. 15. This would be, in effect, a pump lifting water from the lowest compartment, when it opens, to the highest compartment, when it closes. 16. Horizontal force opposing motion is equal to vertical, buoyant force. 17. Axis nearly horizontal. 18. Axis horizontal. 19. £ = 4.7 ft., ^ = 51.5 Ibs., ver- tical. 20. (a) 11.3 ft., (b) 4.7 ft. CHAPTER V. Pages 52-53 i. Horizontal. 2. Inclined from horizontal, 17° 15'. 3. (a) 1.31 ft., (b) 1420 Ibs., (c) 392 Ibs. 4.108.3. 5. 132.6. 6.153.2. 7. (a) o, —4.26, (b) 4.26,0. 8. 0.51 ft. 9. 2.i8cu.ft. '10. 1.43. u. 38.3. 4.2. CHAPTER VI. Pages 75-78 i. (a) From A to B, (b) 1.9 ft. 2. 4.50. 3. Pressure at B is .156 11). PIT sq. in. greater than at .1. 4. 6.14. 5. 29.3 ft. 6. 3.4. 8. (a) From K to .1, (b) 1.69 ft. 9. 6.93, 4.71 c. f. s., 3.05 M. (,. I). 10. 4.71 c. f. s., 2.70. ii. 3.70 c. f. s., o.o. 12. 7% in. 13. 8.67, 4.1 Ibs. per sq. in., absolute. 14. 7.95 in. 15. 13.2 <. f. s., o.o abs. 16. 7.23 in. 17. 4.11 in. 18. 14.75 ft- Per sec- 19. 14. 4 ^ mi. per hr. 20. 0.46. 21. 4.0 in. 22. 8.64. 23. 6.29. 24.6.1. 25. 10.4 in. 26. 4.0 in. 27. 4.45. 28. 4.33 in. CHAPTER VII. Pages 105-108 i. 1.58 in., 31.4 ft. per sec., 0.42 c. f. s. 2. 21.5 ft. 3. 5.71 ft. 4. 2.4 Ibs. per sq. in. 5. 0.64. 6. 3.75 in. 7. 23.0 ft. 8. 22.1 ft. 9. 3.35 in. 10. 18.2 ft. ii. 0.530. f.'s. 12. 10.9 ft. 13. Cc = .6i9, Cv = .979, C = .6o6. 14. 38.0 c. f. s. 15. 43.8 c. f. s. 16. 0.7 ft. 17. 2.0 ft. 18. 0.35 ft. 19. 2.57 c. f. s, 1,660,000 gals, per day. 20. 0.97 ft., 9.0. 21. 2.54, 4.32 in. 22. Theoretically infinite. 23. Theoretically infinite. 24. 25.6. 25. 5.15 ft. 26. 1.96 ft. 27. 226. 28. 108.6 ft. 29. 45.5 Ibs. per sq. in. 30. 217 ft. 31. 74.9 Ibs. per sq. in. 32. 69.2 ft. 33. (a) 1.25 c. f. s., (b) 1.33 c. f. s. 34. 4 min. 8 sec. 35. 6 min. 34 sec. 36. 9 min. 24 sec. 37. 5 min. 55 sec. CHAPTER VIII. Pages 138-140 i. (27) 54.6, (28) 54.6, (31) 55.2, (33) 55.3 (All values in c. f. s.). 2. (27) 79.9, (28) 80. i, (31) 83.6, (33) 83.6. 3. (27) 5-oi, (28)5.01, (31) 5.38, (33) 5.26. 4. i .06 ft. 5. 20.1 ft. 6. 3.34 ft. 7. 11.73 ft. 8. i. 80 ft. 9. 4.6 ft. 10. 2 hrs. 17 min. n. 0.97. 12. 21. 4 ft. 13. 5.91 ft. to 6.42 ft. 14. ii. i c. f. s. 15. 1.54 ft. 16. 0.7%, 1.7%. 17. 5.24 ft. 18. 22.7 ft. 19. 28 min. 40 sec. 20. 0.26 ft. 21. 3.59. 22. I.25/ 23. 1188.3. 24. (43) 62.3 C. f. S., (44) 63.4 'c. f. s. 25. (44) 0.93 ft. CHAPTER IX. Pages 179-183 i. 3-7.7 ft. 2. 5.23 c. f. s. 3. 12 in. 4. 45 in. 5. 46.5 c. f. s. 6. (fl) .32 ft., (b) .87 ft., (c) 1.20 ft, (fl/) .35 Ib. per sq. in., (b') .28 Ib. per sq. in, (cr) .17 Ib. per sq. in. (all pressures being greater after change). 7. (a) .23 ft, (b) .58 ft, (c) .68 ft, (a') .59 Ib. per sq. in. (bf) .91 Ib. per sq. in, (c') .99 Ib. per sq. in. (all pressures being less after change). 8. 19.6 ft. 9. 18.7 ft. 10. 41 in. ii. 88 c. f. s. 12. .77 Ib. per sq. in. greater at lower point. 13. 35 in. 14. 80.7 c. f. s. 15. 3.2 ft. 16. 2.1 ft. 17. Loss will be less for 6 ft. radius 18. 31.5 Ibs. per sq. in. 19. 20.3 c. f. s. 20. i.oo c. f. s. 21. —6.5 Ibs. per sq. in. 22. — 5.2 Ibs. per sq. in. 23. 42 ft. 24. .79 c. f. s. 25. 10 in. 26. 30.2 c. f. s. 27. 23.9 ft. 28. 20.4 c. f. s. 29. 48 c. f. s. 30. in ft. 31. 78 ft. 32. 27.8 c. f. s. 33. 6 in. 34. 21.8 ft. below A. 35. .82 c. f. s. flows into A. 36. 15 in. 37. 82 ft. 38. 71.4 Ibs. per sq. in. 39. 2 hrs. 30 min. 40. 1.93 ft. CHAPTER X. Pages 216 219 i. .000312 for « = .O225 or .000385 for # = .025. 2. 172 c. f. s. for « = .0225 or 155 c. f. s. for n = .o2$. 3. 4.02 ft. for ^ = .0225. 4. 2.75 ft. for « = .025. 5. 3.25 ft. for ^ = .0225. 6. 31 c. f. s. for # = .015. 7. 21 ft. per mi. for « = .oi5. 8. 7.78 ft. for ^=.015 and .030. 9. 2.34 ft. per mi. for ^=.0225. 10. 0.12 sq. ft. n. 579.0 for ;/=.o225 and .015. 12. 6.2 ft. for ^ = .015 and .0225. 13. 3.75 ft. for w = .o225 and .012. 14. 380 c. f. s., .0252, 2.24. 15. 0.95 ft. per mi. for ^ = .0225 or 1.17 ft. per mi. for n = .o2$. 16. .26 ft. below bottom of canal. 17. 4.22 ft. for ^ = .035 and .015, 0.88 ft. 18. 4.44 ft. 19. 441 c. f. s. (Entrance conditions govern.) 20. 404 . (KntraiKe conditions govern.) 21. 40700. f. s. for n = . 0225 and .04. 22. (<;) 3.14 ft. above level of weir crest, (b) 3.43 ft. 23. H"..= 13.32 ft., Il'i =4.17 ft., If '-2 =3.49 ft., H'min. =3-13 ft., for« = .oi5 and C, .=0.82. 24. (</) .61 ft., (b) .00019, (c) .20 ft., 9.5 ft., (</) 1.90 ft., (e) 769.39; 761.40; 760.09; 745-09; 743-59- CHAPTER XI. Pages 239-240 i. 6.61 Ibs. 2. 2.38 Ibs. 3. 100.2. 4. 7^ = 33.9 Ibs., ^ = 58.7 5. J*x = 1 9.0 Ibs., 7^ = 33.0 Ibs. 6. 261 Ibs., 7830 ft.-lbs. 7. 115 450 ft.-lb. 8. 2550 ft. -Ibs. 9. 3220 Ibs. 10. Fx=— 390 ibs. 7\= —1350 11)-. (negative signs indicating thrust, opposite in direction to that of How i. 7^ = 9870 Ibs., Ry = 3ijo Ibs. n. 585 Ibs. per sq. in. or 1 350 ft. head. 12. 300 Ibs. per sq. in., or 690 ft. head. INDEX Absolute and gage pressure, 10 Absolute velocity, 214 Acceleration, 45, 48, 181, 202 Accuracy of computations, 5 Air in pipes, 164, 165 Algebraic transformation of orifice formula, 80 of weir formula, 113 Allievi formula for water hammer, 228 Angle of heel, 40, 42 Angular velocity, 48 Approach, channel of, 71, 76, 103, 108 Approach, velocity of, 71, 76, 80, 103, 108, 112, 120, 128 Archimedes, principle of, 38 Atmospheric pressure, 9, 15, 17 Axis of moments, 25 Backwater, 204, 207 Barometer, mercury, 15 water, 16 Barr's experiments, 123 Base formula, 52 Bazin experiments, 111 open-channel formula, 191 submerged-weir formula, 121 weir formula, 116 Bends in pipes, loss of head, 143, 162 coefficients for, 163 thrust at, 222 Bernoulli's equation, 59, 63, 74, 87, 141 Bernoulli's theorem, 57 application to hydrostatics, 60 in practice, 60 Boiling point of water, 2 Borda's mouthpiece, 91 Bridge piers, 195 Broad-crested weirs, 129 Buoyancy, center of, 39 Buoyant force, 39, 40 Canals, 199 Capillary action, 4, 16 Cast-iron pipes, coefficients, 148, 149, 152, 153 Center of buoyancy, 39, 42 Center of gravity, 31, 39, 42 Center of pressure, 24, 29, 31, 32 and center of gravity, 31, 32 Channel entrance, 131, 202, 203 Channel of approach, 71, 76, 103, 108 Characteristics of jet, 71 Chezy formula, open channels, 185 pipes, 146 Chutes, 129 Cippoletti weir, 124 Circular channels, 176, 201 orifice, 71 Coefficient of contraction, nozzles, 89 orifices, 77, 78 tubes, 86, 88, 92, 93 weirs, 110 Coefficient of discharge, nozzles, 89 orifices, 78, 80 submerged orifice, 94 tubes, 86, 93 Coefficient of roughness, 185 Coefficient of velocity, nozzles, 90 orifices, 77, 78 tubes, 86, 93 weirs, 111 Coefficients of discharge, nozzles, 90 orifices, 79 tubes, 86 231 2.32 INDEX Coefficients, hose, 150 open channels, 187, 190, 191 pipes, 148, 149, 153, 161, 162, 163 weirs, 108 Coefficient, weir, 111 Components of pressure, 33, 35 Compound pipes, 168 Compressibility of water, 3 ( \ imputations, accuracy of, 5 Concrete pipes, coefficients, 14S. 1 !'.». 153 Conduits, 135, 199 Conical tubes, 88 Conservation of energy, 59 Converging tubes, 88 Continuity of discharge, 53 of flow, :,:{, 140, ISO, 202 Contraction, coefficient of, 77, 78, 89, 92, 110 (rest, weirs, 101 end, weirs, 102 gradual, pipes, 143, 159 in channels, 183, 192 of jet, 72, 76, 77, 88, 92 of nappe, 103, 108, 110 Midden, pipes 143, 158 suppressed, 84, 102 surf 102 vertical, weirs, 103 ( 'on verging tubes, 88 Crest contraction, 101 Critical velocity, 13<i Cross-section, most efficient, 198 Curves, backwater. L'I i;, in open channels. Is:'. I '.Ml in pi|,«->. 1 i:i, 162 Curved MM-: ure on, 33 Dams, coefficients for. TJ7 p.essurr ; I, :U • nmeiiN on Pilot tube. ti.~>. M modification of Pitot \\I\K, 68 pipe formula. 117 Density of \MI' Depth of flotation, 39 Differential gage. Is tion of resultant pressui. Discharge, 52 continuity of, 53, 140, 180 head lost at, 143, 155, 157, 183 under falling head, 96 Discharge coefficient, nozzles, 89 orifices, 78 tubes, 86, 93 Discussion of open-channel formulas, 191 of pipe formulas, 154 of weir formulas, 118 Distance, unit of, 1 Distilled water, properties of, 2 Distribution of velocities, 108, 138, 176 Diverging tube, 90 Divided flow, 169, 206 Dynamic force, 214-229 Efficient channel section, 198 Elasticity of water, 3 Elevation and air pressure, 9 Emptying vessel, 97 End contractions, weirs, 102 suppressed, 102 Energy and head, 54 kinetic, 54, 57 of position, 55, 57 of water in channel, 109, 179 of water in pipe, 139 per pound of water, 57, 140 potential, 55, 57 pressure, 56 English system of units, 1 Enlargement of section, channels, 193 pipes, 143 Knt ranee losses, channels, 183 pipes, 143, 155, 156 Entrance to channels, 131 K< juat ion of continuity, 53, 181, 202 Falling head, discharge under, 96 FalK 117 Floating bodies, 38 stability of, 39 Flow, continuity of, 53, 140, 180, 202 in open channels, 176-209 in rivers, 196 INDEX 233 Flow, non-uniform, 53, 201 over weirs, 101-134 steady, 53 stream line, 54, 136 through nozzles, 89 through orifices, 71-99 through pipes, 135-175 through tubes, 85-93 turbulent, 54, 136 uniform, 53 Fluids, definition, 1 Flumes, 199 Force, acting on mass, 212, 213 exerted on pipe, 220-223 exerted on vane, 213-218 Francis coefficient for weirs, 111 experiments, 111 formula for end contractions, 112 weir formula, 115 Free surface of liquid, 9 Freezing temperature of water, 2 Friction, 52 and distribution of velocities, 108, 138, 176 Bernoulli's equation with, 59, 141 coefficients for hose, 150 coefficients for open channels, 187, 190, 191 coefficients for pipes, 148, 149, 153 formulas for open channels, 185- 191 formulas for pipes, 146-154 independent of pressure, 145 in open channels, 176, 184 in orifices, 76, 77 in pipes, 138-146 in Venturi meter, 64 of flowing water, 59, 138, 176 over weirs, 108 Fteley and Stearns, experiments, 111 weir formula, 116 Fundamental orifice formula, 73 Gage, differential, 18 hook, 131 mercury, 17 oil, 19 water stage, 132 Gage pressure, 10, 16 Ganguillet and Kutter, 185 Gases, 1, 9 Gates, 95, 195 and valves, 143, 162 coefficients for, 96 pressure on, 26, 28, 32 Grade line, hydraulic, 144, 154, 164, 184, 191 Gradient, hydraulic, 144, 154, 164, 184, 191 Hazen- Williams formula, 150 Head, definition of, 54 energy and, 54 lost at bends or curves, 143, 162, 183, 196 lost at discharge, 143, 155, 157, 194 lost at entrance, 143, 155, 156, 193 lost at obstructions, 143. 162, 195 lost by contractions, 143, 158, 192 lost by enlargements, 143, 160, 193 lost by friction, 59 lost by friction in open channels, 181, 183, 184 lost by friction in pipes, 140, 143, 144 lost in nozzles, 90 lost in orifices, 81 lost in tubes, 87 measurement of, 131 pressure, 13, 16, 56 velocity, 55 Height of weir, 103 Herschel, Venturi meter, 61 Hook gage, 4, 131 Horizontal and vertical components of pressure, 32, 34 Horizontal orifice, 71, 75 Horton's values of "n," 187 weir coefficients, 127 Hose, 150 coefficients, 150 Hydraulic grade line, 144, 154, 164, 184, 191 gradient, 144, 154, 164, 184, 191 jack, 13 radius, 135, 176 234 INDEX Hydraulics, definition, 1 or rivers, 196 Hydrodynamics, 211-230 definition, 1 fundamental principles, 211 Hydrokinetics, 1, 51-210 Hydrostatic pressure, 7-37 Hydrostatics, 1, 7-44 Immersed and floating bodies, 38-44 Impurities in water, 2 Incomplete contractions, 84, 103 Intensity of pressure, 7, 8, 10, 30, 35 Inversion of jet, 73 Inward-projecting tubes, 93 Irregular sections of channels, 197 Jack, hydraulic, 13 Jet, characteristics of, 71 contraction of, 72, 76, 77, 88, 92 definition, 71 deflected by curved vane, 216 force of, 213-218 forms assumed by, 72 impinging on fixed plate, 214 impinging on moving plate, 215 inversion of, 73 path of, 82 pressure in, 72 velocity with respect to vane, 217 work done by, 218 Johnson water-hammer formula, 229 Joukovsky, water-hammer experi- ments, 228 formula, 228 Kinetic energy, 54, 57 in channel of approach, 109 in open channels, 109, 179 in pipos, 139 King pipe formula, l.VJ submerged-weir formula, 121 weir formula, 117 Kutter formula, 185 table for solving, 186 Lea, investigation of pipes, 152 Linear acceleration, 45 Liquid, buoyant force of, 38 definition, 1 free surface of, 9 in motion, 51 intensity of pressure within, 7, 8 relative equilibrium, 45 rotating, 47 surface tension, 4 uniformly accelerated, 45 water most common, 51 Long pipes, 166 Loss of head, at discharge, 143, 155, 157, 183, 194 at entrance, 143, 155, 156, 183, 194 due to bends or curves, 143, 162, 183, 196 due to contractions, 143, 158, 192 due to enlargements, 143, 160, 193 due to friction in channels, 181, 184 due to friction in pipes, 140-146 due to obstructions, 143, 162, 195 in nozzles, 90 in open channels, 181 in orifices, 81 in pipes, 140 in tubes, 87 Manning formula, 188 Mead water-hammer formula, 229 Mean velocity over weirs, 107 Measurement of head, 131 Mercury barometer, 15 gage, 17 Metacenter, 40 Metacentric height, 40 Meter, Venturi, 61 Modifications of fundamental weir formula, 112 Modulus of elasticity, 3 Moment of inertia, 25 Momentum, definition, 211 Motion, stream line, 54, 136 turbulent, 54, 136 Mouthpiece, Borda's, 91 Nappe, 101 contracted section of, 103 contraction of, 108, 1 10 depth of mean velocity in, 108 INDEX 235 Nappe, pressure in, 104 velocities in, 104, 107, 111 Negative pressure, 10 Newton's laws of motion, 211 Non-uniform flow, 53, 201 Nozzles, 89 Numerical computations, 5 Obstructions in channels, 183, 195 in pipes, 143, 162 Oil-differential gage, 19 Open-channel friction formulas, 185- 191 Bazin formula, 191 Chezy formula, 185 Kutter formula, 185, 188 Manning formula, 188 Open channels, backwater in, 204 bends or curves in, 183, 196 contractions in section, 183, 192 description and definition, 176 discussion of formulas, 191 divided flow in, 206 efficient section for, 198 energy of water in, 109, 179 enlargement in section, 183, 193 entrance to, 131, 183 friction in, 176 hydraulic gradient of, 184, 191 hydraulic radius of, 176 loss of head in, 181 non-uniform flow, 201 obstructions in, 183, 195 water surface of, 184, 191 wetted perimeter of, 176 Orifice, circular, 71 coefficients, 76 definition and description, 71 discharge under falling head, 96 flow through, 71-85, 93-98 fundamental formula, 73 head lost in, 81 horizontal, 71, 75 partially submerged, 94 rectangular, 71, 82 square, 71 submerged, 93 suppressed contraction, 84 Orifice, under low heads, 82 vertical, 71 with velocity of approach, 80 Parabolic weir, 101 Partially submerged orifices, 94 Pascal's law, 8 Path of jet, 82 Perfect liquid, 4 Piezometer tubes, 5, 16 Pipe-friction formulas, 146-154 Chezy formula, 146 Dracy formula, 147 Hazen- Williams formula, 150 King formula, 152 non-turbulent flow, 154 Pipes, bends or curves in, 143, 162 branching, 166 coefficients, 148, 149, 151, 153 compound, 168 continuity of flow, 140 contractions in, 143, 158 critical velocity in, 136 description and definition, 135 discharge from, 143, 155, 157 divided flow in, 169 energy of water in, 139 enlargements in, 143, 160 entrance to, 143, 155, 156 flow through, 135-175 force exerted upon, 220 friction coefficients, 148, 149, 153 friction in, 138, 144 hydraulic gradient, 144, 154, 164 hydraulic radius of, 135 loss of head in, 140 more than one diameter, 142, 172 obstructions in, 143, 162 special problems, 165 tensile stress in walls, 35 thrust at bends, 222 viscosity, effect on friction, 145 water hammer in, 223 wetted perimeter of, 135 Piston, 13, 56 Pitot tube, 65-69 Plate, jet impinging against, 214, 215 Potential energy, 55, 57 236 INDEX Pressure, absolute and gage, 10 atmospheric, 9, 15 center of, 24, 29, 31 energy, 56, 62 head, 13, 16, 56 intensity of, 7, 8, 10, 30, 32, 35 negative, 10 on curved surfaces, 33, 35 on plane areas, 23 on surfaces, 23-37 relative, 10 transmission of, 13 vapor, 14, 164 Principle of Archimedes, 38 Principles of equilibrium, 8 Principles of hydrokinetics, 51-70 Principles of hydrostatic pressure, 7- 22 Pumps, suction, 20 Radius, hydraulic, 135, 176 of bends in pipes, 163 Reaction of jet, 213 Rectangular channels, 176, 199 orifice, 71, 82 weir, 101 Relative and absolute velocities, 214 Relative equilibrium of liquids, 45-50 Helat ive pressure. 10 Resultant pressure, direction of, 7, 33 position of, 32 Rivers, hydraulics of, 196 Rotating vesM-1, 47 Salt water, :i Scow, stability of, 42 Se< -lion of greatest efficiency, 198 Semicircular channel, 198 Sewers, 135, 176 Ships, stability of, 40 Siphon, 20 Skin of water surface, 4 Slope, "*," 145, 184 Special pipe problems, 165 Specific gravity of mercury, 17 of water, 2 Square orifice, 71 Stability of scow, 42 Stability of ship, 40 Standard short tube, 85 Steady flow, 53 Stream line motion, 54, 136 Submerged bodies, 38 orifices, 93 weirs, 95, 118, 120, 195 Suction pumps, 20 Sudbury conduit, 177 Summit of pipe, 164, 165 of siphon, 20 Suppressed contraction, orifices, 84 Suppressed weirs, 102 Suppression of contraction, 84, 102 Surface contraction, 102 Surfaces, pressure on, 23-37 Surface tension, 4, 177 Tangential stress of water, 1, 4, 8 Temperature, 2 effect on critical velocity, 138 effect on viscosity, 138 effect on weight of water, 2 Tensile stress in pipe walls, 35 Theoretical discharge over weirs, 105 velocity over weirs, 107 velocity through orifices, 74 Thickness of pipes required, 35 Thompson's experiments, 123 Time of emptying vessel, 97 Time, unit of, 1 Torricelli's theorem, 76 Transmission of pressure, 13 Trapezoidal channels, 176, 199 weirs, 123 Triangular weirs, 121 Tube, definition of, 71 Tubes, capillary action in, 4 converging, 88 diverging, 90 immersed in flowing water, 65 standard short, 85 Turbulent motion, 54, 136 Uniform flow, 53 Unite in hydraulics, 1 I" us toady flow, 53 r-tubc, 17 INDEX 237 Vacuum, 10, 17 amount of, 10 perfect, 10 Vane, jet deflected by, 215, 216 fixed curved, 215 moving curved, 216 work done on, 218 Vapor pressure, 14, 164 variation with temperature, 15 Velocities, distribution of, 138, 176 in vertical, 177 Velocity, absolute and relative, 214 at any depth, over weirs, 104 critical, 136 curves of, 177 from orifices, 74 head, 55, 81 of approach, orifice, 71, 75, 76, 78, 80 weir, 103, 108, 112, 114, 120, 128 Velocity coefficient, nozzles, 90 orifices, 77, 78 tubes, 86, 93 weirs, 111 Vena contracta, 72 Venturi meter, 61 Vertical contraction, 103 Vertical jets, 75 Vertical orifice, 71 Vertical velocity curves, 177 Vessel, rotating, 47 time of emptying, 96 with constant acceleration, 45 Viscosity, 4, 137, 143 Warren water-hammer formula, 229 Water barometer, 16 boiling point of, 2 compressibility of, 3 dynamic force, 214-229 freezing point of, 2 hammer, 223 maximum density of, 2 physical properties of, 2-5 Water, weight of, 2 Weight of submerged bodies, 38 of mercury, 17 of water, 2 unit of, 1 Weir, broad crested, 129 channel of approach, 71, 76, 103, 108 Cippoletti, 124 coefficient, 111, 127 coefficients, 108, 110, 111 conditions for accurate use, 132 definition and description, 101 discharge formulas for sharp crest, 115 discussion of formulas, 118 experiments, 111, 115 formula for mean velocity, 105 fundamental formulas, 83, 107, 112 head measurement, 131 height of, 103 modification of fundamental for- mula, 112 not sharp-crested, 101, 103, 125 range of accuracy, 132 rectangular, 101 sections, 126 sharp-crested, 101 submerged, 95, 118 theoretical formula for discharge, 105 transformation of formula, 113 trapezoidal, 101, 123 triangular, 101, 121 velocity of approach to, 103 with end contractions, 102, 112 with end contractions suppressed, 102 Wetted perimeter, 135, 176 Wooden pipes, coefficients, 148, 149, 153 Work done on vane, 218 on mass of water, 58 0 f/ I JINDING DEPT.