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Full text of "Elements of hydraulics"

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About Google Book Search Google's mission is to organize the world's information and to make it universally accessible and useful. Google Book Search helps readers discover the world's books while helping authors and publishers reach new audiences. You can search through the full text of this book on the web at |http : //books . google . com/ "^"W^^ Digitized by VnOOQlC Digitized by VnOOQlC Digitized by VnOOQlC Digitized by LjOOQIC ELEMENTS OF HYDRAULICS Digitized by VnOOQlC McGraw-Hill CookCompaiiy PujStis/iers c^3ooI^/c^ ElGCtrical World TheEiigingerii^aiidMning Journal LngiaeGriiig Record Engineering News Railway A^ Gazette American Machinist Signal Engineer AraericanEngjneer Electric Railway Journal Coal Age Mptallui-gical anl Chemical Engineering Power Digitized by VjOOQiC Digitized by VnOOQlC Digitized by Google ELEMENTS OF HYDRAULICS BY S. E. SLOCUM, B. E., Ph. D., PROFESSOR OF APPLI10D MATHEMATICS IN THE UNIVERSITY OF CINCINNATI Second Edition Revised and Enlarged McGRAW-HILL BOOK COMPANY, Inc. 239 WEST 39TH STREET. NEW YORK LONDON: HILL PUBLISHING CO., Ltd, 6 & 8 BOUVERIE ST., E. C. 1917 Digitized by VnOOQlC copymght, 1916, 1917, by the McGraw-Hill Book Company, Inc. . • • • .• t T. ; • • • ; : • T H K M A I* I. K I» K K K S Y <) K JC !» X Digitized by VnOOQlC PREFACE TO SECOND EDITION The present revision is based largely on the experience of those who have used the book in the class-room, and is intended to make it more efficient as a text as well as a more complete work for general reference. To effect this several changes in typog- raphy have been made, as well as important alterations in the text. For instance, each paragraph has been numbered for ease of reference, while the classical terms hydrostatics, hydrokinetics, and hydrodynamics have been replaced by the more familiar and descriptive names "pressure of water,'^ "flow of water,'' and "energy of flow.'' The principal additions to the text consist of a more complete and up-to-date discussion of the flow of water in pipes, with special reference to the exponential formula and its graphical solution; a summary of the principal formulas for the strength of pipe; a more extended discussion of weir formulas; a fuller presentation of the modern use of siphons on a large scale; recent developments in the theory of water hammer; and the modern solution of penstock and surge tank problems. Numerous minor changes have also been made wherever such changes seemed indicated for additional clearness or proper emphasis. For in- stance, the applicability of Chezy's formula to pipe flow as well as to open channels has been pointed out; the turbine constants have been revised to date; the principles of draft tube design indicated; the mathematical discussion of back-water has been replaced by a simple application of Chezy's formula; while special care has been taken to correct all misprints as well as inaccuracies of statement. A complete set of answers to problems has been prepared, and is printed separately. S. E. Slocum. December, 1916. 357398 n i Digitized by VnOOQlC Digitized by LnOOQlC PREFACE TO FIRST EDITION The remarkable impetus recently given to hydraulic develop- ment in this country has caused the whole subject to assume a new aspect. Not only is this apparent in new and improved con- struction details, but in the scientific study which is beginning to be given a subject which seemed to have crystallized into a set of empirical formulas. Such comprehensive plans as those recently undertaken by the State of New York and the Dominion of Canada for the system- atic development of all their available water power, indicates the extent of the field now opening to the hydraulic engineer. The extent and cheapness of the natural power obtained not only from the development of existing streams but also from the artificial pondage of storm water is sufficient to convince even the most casual observer that no phase of conservation will have a more immediate effect on our industrial development or be more far reaching in its consequences. The present text is intended to be a modern presentation of the fundamental principles of hydraulics, with applications to recent important works such as the Catskill aqueduct, the New York State barge canal, and the power plants at Niagara Falls and Keokuk. Although the text stops short of turbine design, the recent work of Zowski and of Baashuus is so presented as to en- able the young engineer to make an intelligent choice of the type of development and selection of runner. In order to make the book of practical working value, a col- lection of typical modern problems is added at the end of each section, and a set of the most useful hydraulic data has been compiled and is tabulated at the end of the volume. Cincinnati, Ohio, S. E. Slocum. January, 1916. . Digitized by VnOOQlC Digitized by VnOOQlC CONTENTS Preface Page V, vi SECTION I PRESSURE OF WATER Art. 1. Properties op a Perfect Fluid 1-3 1. Definition of fluid — 2. Distinction between liquid and gas — ^3. Elasticity of water — 4. Fluid pressure nor- mal to surface — ^5. Viscosity — 6. Density of water — 7. Specific weight. 2. Pressttbe of Water 3-8 8. Equal transmission of pressure — 9. Pressure propor- tional to area — 10. Hydraulic press — 11. Frictional re- sistance of packing — 12. Efficiency of hydraulic press. 3. Simple Pressure Machines 8-13 13. Hydraulic intensifier — 14. Hydraulic accumulator — 15. Hydraulic jack — 16. Hydraulic crane — 17. Hy- draulic elevator. 4. Pressure on Submerged Surfaces 13-17 18. Change of pressure with depth — 19. Pressure on submerged area — 20. Center of pressure — ^21. General formula for center of pressure — 22. Application. 5. Strength of Pipes under Internal Pressure . . . 17-19 23. Thin cylinder — ^24. Lame's formula — 25. Barlow's formula — 26. Clavarino's formula — 27. Birnie's formula. 6. Equilibrium of Two Fluids in Contact 19-22 28. Head inversely proportional to specific weight — ^29. Water barometer — ^30. Mercury barometer — ^31. Piezo- meter — 32. Mercury pressure gage — ^33. Differential gage. 7. Equilibrium of Floating Bodies 22-26 34. Buoyancy — ^35. Floating equilibrium — 36. Theo- rem of Archimedes — 37. Physical definition of specific weight — 38. Determination of specific weight by experi- ment — 39. Application to alloy — 40. Zero buoyancy. 8. Mbtagbnter 26-30 41. Stability of floating body — 42. Metacenter — 43. Codrdinates of metacentei^— 44. Metacentric height — 45. Period of oscillation— 46. Rolling and pitching. Applications 30-46 xi Digitized by VnOOQlC xu CONTENTS SECTION II FLOW OF WATER Page Abt. 9. Flow op Watbb fbom Reservoirs and Tanks . . . 47-51 47. Stream line — 48. Liquid vein — 49. Ideal velocity head — 50. Torricelli's theorem — 51. Actual velocity of flow — 52. Contraction coefficient — 53. Efflux coeffi- cient — 54. Effective head — 55. Discharge from large rectangular orifice — 56. Discharge of a rectangular notch weir. 10. Discharge through Sharp-edged Orifice 51-53 67. Contraction of jet — 58. Complete contraction — 69. Partial contraction — 60. Velocity of approach. 11. Rectangular Notch Weirs 53-60 61. Contracted weir — 62. Suppressed weir — 63. Sub- merged weir — 64. Triangular weir — 65. Trapezoidal or Cippoletti weir — 66. Formulas for rectangular notch weirs. 12. Standard Weir Measurements 60-63 67. Construction of weir — 68. Hook gage — 69. Pro- portioning weirs. 13. Time Required for Filling and Emptying Tanks. . 63-67 70. Change in level under constant head — 71. Varying head — 72. Canal lock — 73. Rise and fall in connected tanks — 74. Mariotte's flask. 14. Flow through Short Tubes and Nozzles 67-70 75. Standard mouthpiece — 76. Stream Kne mouthpiece — 77. Borda mouthpiece — ^78. Diverging conical mouth- piece — ^79. Venturi adjutage — 80. Converging conical mouthpiece — 81. Fire nozzles. 15. Kinetic Pressure in a Flowing LiQino 71-74 82. Kinetic pressure — 83. Bernoulli's theorem — 84. ^ Kinetic pressure head — 85. Application to standard mouthpiece. 16. Venturi Meter 74-78 86.' Principle of operation — 87. Formula for flow — 88. Commercial meter — 89. Catskill Aquaduct meter — 90. Rate of flow controller. 17. Flow op Water in Pipes 78-83 91. Critical velocity — 92. Viscosity coefficient — ^93. Parallel (non-sinuous) flow — ^94. Average velocity of flow in small pipes — 95. Loss of head in small pipes — ^96. Ordinary pipe flow. 18. Practical Formxtlas for Loss op Head in Pipe Flow. 83-97 97. Effective and lost head — 98. Loss at entrance — 99. Friction loss — 100. Wood stave pipe — 101. Graphical Digitized by VnOOQlC CONTENTS xiu Paob solution — 102. Cast iron pipe — 103. Deterioration with age — 104. Riveted steel pipe — 105. Concrete pipe — 106. Bends and elbows — 107. Enlargement of section — 108. Contraction of section — 109. Gate valve in circular pipe — 110. Cock in circular pipe — 111. Throt- tle valve in circular pipe — 112. Summary of losses — 113. Application. 19. Hydraulic Gradient 97-99 114. Kinetic pressure head — 116. Slope of hydraulic gradient — 116. Peaks above hydraulic gradient. 20. Hydraulic Radius 99-102 117. Definition of hydraulic radius — 118. Chezy's formula for pipe flow — 119. Kutter's and Bazin's for- mulas for pipe flow — 120. Williams and Hazen's expo- nential formula. 21. Divided Flow 102-105 121. Compound pipes — 122. Branching pipes. 22. Fire Streams 105-106 123." Freeman's experiments — 124. Formulas for dis- charge — 125. Height of effective fire stream — 126. Fleming's experiments. 23. Experiments on the Flow op Water 107-111 127. Verification of theory by experiment — 128. Method of conducting experiments — 129. Effect of sudden contraction or enlargement — 130. Disturbance pro- duced by obstacle in current — 131. Stream line motion in thin film— 132. Cylinder and flat plate— 133. Ve- locity and pressure. 24. Modern Siphons 111-120 134. Principle of operation — 135. Siphon spillways — 136. Siphon lock — 137. Siphon wheel settings. 25. Flow in Open Channels 120-123 138. Open and closed conduits — 139. Steady imiform flow — 140. Kutter's formula — 141. Limitations to Kut- ter's formula — 142. Bazin's formula — 143. Kutter's simplified formula. 26. Channel Cross-section 123-126 144. Condition for maximum discharge — 145. Maxi- mum hydraulic efi&cienxjy — 146. Regular circumscribed polygon — 147. Properties of circular an^ oval sections. 27. Flow in Natural Channels 12^129 148. Stream gaging — 149. Current meter measure- ments — 150. Float measurements — 151. Variation of velocity with depth — 152. Calculation of discharge. 28. The Pitot Tube 129-138 153. Description of instrument — 154. Darcy's modifi- Digitized by Google xiv CONTENTS Faob cation of Pitot's tube — 155. Pitometer — 156. Pitot re- corders — 157. Theory of the impact tube — 158. Con- struction and calibration of Pitot tubes — 159. DuBuat's paradox. 29. Non-uniform Flow; Backwater . ^ 138-139 160. Surface elevation. Applications 139-156 4 SECTION III ENERGY OF FLOW Art. 30. Pressure op Jet against Stationary Deflecting Surface 157-162 161. Normal impact on plane surface — 162. Relation of static to dynamic pressure — 163. Oblique impact on plane surface — 164. Axial impact on surface of revolu- tion — 165. Complete reversal of jet — 166. Deflection of jet — 167. Dynamic pressure in pipe bends and elbows. 31. Pressure Exerted by Jet on Moving Vane. . . . 162-167 168. Relative velocity of jet and vane — 169. Work done on moving vane — 170. Speed at which work be- comes a maximum — 171 . Maximum efficiency for single vane — 172. Maximum efficiency for continuous succes- sion of vanes — 173. Impulse wheel; direction of vanes at entrance and exit — 174. Work absorbed by impulse wheel. 32. Reaction op a Jet 167-169 175. Effect of issuing jet on equilibrium of tank — 176. Energy of flow absorbed by work on tank — 177. Prin- ciple of reaction turbine — 178. Barker's mill. 33. Types op Hydraulic Motors 169-171 179. Current wheels — 180. Impulse wheels — 181. Re- action turbines — 182. Classification of reaction tur- bines — 183. Classification of hydraulic motors. 34. Current and Gravity Wheels 171-173 184. Current wheels — 185. Undershot wheels — 186. Poncelet wheels — 187. Breast wheels — 188. Overshot wheels. 35. Impulse Wheels and Turbines 173-185 189. Pelton wheel— 190. Efficiency of Pelton wheel— 191. Characteristics of impulse wheels — 192. Girard impulse turbine — 193. Power and efficiency of Girard turbine. 36. Reaction Turbines 185-202 194. Historical development — 195. Mixed flow or Digitized by LnOOQlC CONTENTS XV Page - American type — 196. Use of draft tube — 197. Draft tube design— 198. Profile of draft tube— 199. Time of flow through draft tube — 200. Recent practice in tur- bine setting. 37. Characteristics of Rbaction Turbines 202-214 201. Selection of type — 202. Action and reaction wheels — 203. Speed criterion — 204. Capacity criterion — ^205. Characteristic speed — 206. Specific discharge — 207. Specific power — 208. Specific speed — 209. Rela- tion between characteristic speed and specific speed — 210. Classification of reaction turbines — 211. Numer- ical application — 212. Normal operating range — 213. Selection of stock runner. 38. Power Transmitted through Pipe Lines 214-217 214. Economical size of penstock — 215. Numerical application. 39. Effect of Translation and Rotation 217-220 216. Equilibrium under horizontal linear acceleration — 217. Equilibrium under vertical linear acceleration — 218. Free surface of liquid in rotation — ^219. Depres- sion of cup below original level in open vessel — ^220. Depression of cup below original level in closed vessel — 221. Practical applications 40. Water-hammer in Pipes 221-227 222. Maximum-water hammer — 223. Velocity of com- pression wave — 224. Ordinary water-hammer — ^225. Joukovsky's formula — 226. Allievi's formula — 227. Occurrence of water-hammer in supply systems. 41. Surge Tanks 228-236 228. Surge in surge tanks — 229. Differential surge tank. 42. Hydraulic Ram 236-238 230. Principle of operation — ^231. Efficiency of ram. 43. Displacement Pumps 238-246 232. Pump types — ^233. Suction pump — 234. Maxi- mum suction lift — 235. Force pump — 236. Stress in pump rod — 237. Direct driven steam pump — 238. Cal- culation of pump sizes — 239. Power required for opera- tion — ^240. Diameter of pump cyUnder — 241. Steam pressure required for operation — 242. Numerical application. 44. Centrifugal Pumps 246-255 243. Historical development — 244. Principle of opera- tion — 245. Impeller forms — 246. Conversion of kinetic energy into pressure — 247. Volute casing — 248. Vortex chamber — 249. Diffusion vanes — 250. Stage pumps. Digitized by LnOOQlC xvi CONTENTS Paob 45. Pressube Developed in Cbntripugal Pump .... 255-258 251. Pressure developed in impeller — ^252. Pressure de- veloped in diffusor — ^253. General expression for pres- sure head developed. 46. Centrifugal Pump Characteristics. 258-265 254. Effect of impeller design on operation — ^255. Ris- ing and drooping characteristics — ^256. Head developed by pump— 257. Effect of throttling the discharge — 258. Numerical illustration. 47. Eppicibncy and Design op Centrifugal Pumps . . 265-266 259. Essential features of design — 260. Hydraulic and commercial efficiency. 48. Centrifugal Pump Applications 26^273 261. Floating dry docks— 262. Deep wells— 263. Mine drainage— 264. Fire pumps — 265. Hydraulic dredging — ^266. Hydraulic mining. Applications 273-289 SECTION IV HYDRAULIC DATA AND TABLES Table 1. Properties of water 290 2. Head and pressure equivalents 291 3. Discharge equivalents 292 4. Weigbts and measures 293 5. Specific weights of various substances 294 6. Standard dimensions of pipes 295 7. Capacity of reciprocating pumps 29^297 8. Circumferences and areas of circles 298-302 9. Efflux coefficients for circular orifice 303 10. Efflux coefficients for square orifice 304 11. Fire streams 305-307 12. Coefficients of pipe friction 308 13. Friction bead in pipes 309-312 14. Bazin's values of Chezy's coefficient 313 15. Kutter's values of Cbezy's coefficient 314-315 16. Discbarge from wood-stave pipe 316-317 17. Discbarge coefficients for rectangular notcb weirs ..... 318 18. Discbarge per incb of lengtb over rectangular notcb weirs . 319 19. Discbarge per foot of lengtb over rectangular notcb weirs . 320 20. Discbarge per foot of lengtb over suppressed weirs .... 321 21. Principles of mecbanics 322-323 22. Submerged weir coefficients 324 Index 325 Digitized by Google ELEMENTS OF HYDRAULICS SECTION 1 PRESSURE OF WATER I. PROPERTIES OF A PERFECT FLUID 1. Definition of Fluid. — ^A fluid is defined, in general, as a substance which offers no resistance to change in form provided this deformation is not accompanied by change in volume. The fundamental property of a fluid is the perfect mobility of all its parts. Most of the applications of the mechanics of fluids relate to water, and this domain of mechanics is therefore usually called hydraulics or hydromechanics. It is customary to subdivide the subject into hydrostaiics, relating to water at rest; hydro^ kinetics, relating to water in motion; and hydrodynamicSf relating to the inertia forces exerted by fluids in motion, and the energy available from them. 2. Distinction between Liquid and Gas. — ^A liquid such as water has a certain degree of cohesion, causing it to form in drops, whereas a gaseous fluid tends to expand indefinitely. A gas is therefore only in equilibrium when it is entirely enclosed. In considering elastic fluids such as gas and steam, it is always necessary to take account of the relation between volume and pressure. For a constant pressure the volume also changes greatly with the temperature. For this reason the mechanics of gases is concerned chiefly with heat phenomena, and forms a separate field called thermodynamics, l3dng outside, and sup- plementary to, the domain of ordinary mechanics. 3. Elasticity of Water. — Water, like other fiuids, is elastic, and under heavy pressure its volume is slightly diminished. It has been found by experiment that a pressure of one atmosphere, or 14.7 lb. per sq. in., exerted on each face of a cube of water at 1 Digitized by VnOOQlC 2 ELEMENTS OF HYDRAULICS 32®F. causes it to diminish about 0.00005 in volume. Conse- quently, the bulk modulus of elasticity of water, B, defined as „ unit stress "" unit volume deformation' has as its numerical value As the change in volume is so small it is sufficiently accurate for most purposes to assume that water is incompressible. An ex- ception to this rule will be found in Art. XL. 4. Fluid Presstire Normal to Surface. — Since a perfect fluid is one which offers no resistance to change in form, it follows that the pressure on any element of surface of the fluid is everywhere normal to the surface. To prove this proposition, consider any small portion of a fluid at rest, say a small cube. Since this cube is assumed to be at rest, the forces acting on it must be in equi- librium. In the case of a fluid, however, the general conditions of equilibrium are necessary but not sufficient, since they take no account of the fact that the fluid offers no resistance to change in form. Suppose, therefore, that the small cube under con- sideration undergoes a change in form and position without any change in volume. Since the fluid offers no resistance to this deformation, the total work done on the elementary cube in producing the given change must be zero. In particular, suppose that the cube is separated into two parts by a plane section, and that the deformation consists in sliding one of these parts on the other, or a shear as it is called. Then in addition to the forces acting on the outside of each part, it is necessary to consider those acting across the plane section. But the total work done on each part separately must be zero inde- pendently of the other part, and also the total work done on the entire cube must be zero. Therefore, by subtraction, the work done by the forces acting across the plane section must be zero. But when any force is displaced it does work equal in amount to the product of this displacement by the component of the force in the direction of the displacement. Therefore if the work done by the force acting across the plane section of the cube is zero, this force can have no component in the plane of the section, and must therefore be normal, i.e., perpendicular, to the section. Digitized by vnOOQlC PRESSURE OF WATER 3 6. Viscosity. — This absence of shear is only rigorously true for an ideal fluid. For water there is a certain amount of shear due to internal friction, or viscosity, but it is so small as to be practi- cally negligible. The greater the shear the more viscous the fluid is said to be, and its amount may be taken as a measure of viscosity. It is found by experiment that the internal friction depends on the difference in velocity between adjacent particles, and for a given difference in velocity, on the nature of the fluid. The viscosity of fluids is therefore of great importance in consider- ing their motion, but does not affect their static equilibrium. For any fluid at rest, the pressure is always normal to any element of surface. 6. Density of Water. — In hydraulic calculations the unit of weight may be taken as the weight of a cubic foot of water at its temperature of greatest density, namely, 39°F. or 4°C. It is found by accurate measurement that a cubic foot of water at 39°F. weighs 62.42 lb. This constant will be denoted in what follows by the Greek letter 7. In all numerical calculations it must be remembered therefore that y = 62.4 lb. per cu. ft. (2) The density and volume of water at various temperatures are given in Table 1. 7. Specific Weight. — The weights of all substances, whether liquids or solids, may be expressed in terms of •the weight of an equal volume of water. This ratio of the weight of a given volume of any substance to that of an equal volume of water is called the specific weight of the substance, and will be denoted in what follows by s. For instance, a cubic foot of mercury weighs 848.7 lb., and its specific weight is therefore 848.7 s = 62 ^= 13.6 approximately. Its exact value^t 0°C. is s = 13.596, as may be found in Table 5. The weight of 1 cu. ft. of any substance in terms of its specific weight is then given by the relation Weight = 7S = 62.4s lb. per cu. ft. n. PRESSURE OF WATER 8. Equal Transmission of Pressure. — The fundamental princi- ple of hydrostatics is that when a fluid at rest has pressure applied Digitized by VnOOQlC ELEMENTS OF HYDRAULICS Fig. 1. to any portion of its surface, this pressure is transmitted equally to all parts of the fluid. To prove this principle consider any portion of the fluid limited by a bounding surface of any form, and suppose that a small cylin- drical portion is forced in at one point and out at another, the rest of the boundary remaining un- changed (Fig. 1). Then if AA de- notes the cross-sectional area of one cylinder and An its height, its volume is AA-An. Similarly, the volume of the other cylinder is AA'An', and, since the fluid is as- sumed to be incompressible, AAAn = AA'-Aw'. Now let p denote the unit pressure on the end of the first cylinder, i.e., the intensity of pressure, or its amount in pounds per square inch. Then the total pressure normal to the end is pAA, and the work done by this force in moving the distance An is pAA-An. Similarly, the work done on the other cylinder is p'AA'-An'. Also, if 7 denotes the heaviness of the fluid per unit volume, the work done by gravity in moving this weight yAA-An through the distance A, where h denotes the difference in level between the two elements considered, is yAA-An-h. Therefore, equating the work done on the fluid to that done by it, we have pAA'An + yAA'Anh = p'-AA' An'. Since AA-An = AA'-An', this reduces to p' = p + 7h. (3) If A = 0, then p' = p. Therefore the pressure at any point in a perfect fluid is the same in every direction. Also the pressure at the same level is everywhere the same. Moreover, if the intensity of pressure p at any point is increased by an amount w, so that it becomes p + w, then by Eq. (3) the intensity of pressure at any other point at a difference of level h becomes p" = (p + y>) + yh. But since p' = p + yh, we have by subtraction. // p' + w, Digitized by LnOOQlC PRESSURE OF WATER 5 that is, the intensity of pressure at any other point is increased by the same amount w, A pressiu'e applied at any point is therefore transmitted equally to all parts of the fluid. For fluids such as gas and steam the term yh is negligible, and consequently for such fluids the intensity of pressure may be assumed to be everywhere the same. 9. Pressure Proportional to Area. — To illustrate the applica- tion of this principle consider a closed vessel or tank, filled with water, having two cylindrical openings at the same level, closed by movable pistons (Fig. 2). If a load P is applied to one piston. — D— Fig. 2. Fig. 3. then in accordance with the result just proved there is an in- crease of pressure throughout the vessel of amount P where A denotes the area of the piston. The force P' exerted on the second piston of area A' is therefore whence p a' The two forces considered are therefore in the same ratio as their respective areas. This relation remains true whatever shape the ends of the pistons may have, the areas A and A' in any case being the crossHsectional areas of the openings. For instance, an enlargement of the end of the piston, such as shown in Fig. 3, has no effect on the force transmitted, since the upward and Digitized by LnOOQlC ELEMENTS OF HYDRAULICS ir(D^ - d^) cancel, leaving downward pressures on the ring of area -7- as the effective area. 4 10. Hydraulic Press. — An important practical application of the law of hydrostatic pressure is found in the hydraulic press. In its essential features this consists of two cylinders, one large and one small, each fitted with a piston or plunger, and connected by a pipe through which water can pass from one cylinder to the Fig. 4. other (Fig. 4). Let p denote the intensity of pressure within the fluid, Z>, d the diameters of the two plungers, P the load applied to one and W the load supported by the other, as indicated in the figure. Then and consequently W = -j-'p, W ^ P d2* (4) If the small plunger moves inward a distance A, the large one will be forced out a distance H such that each will displace the same volume, or Td% tD^H whence h = H 4 D2 Digitized by LnOOQlC PRESSURE OF WATER Neglecting friction, the work done by the force P in moving the distance h is then Ph=p(H^)=H(p^)=HW, and is therefore equal to the work done in raising W the dis- tance H. 11. Frictional Resistance of Packing. — Usually, however, there is considerable frictional resistance to be overcome, since for high pressures, common in hy- draulic presses, heavy packing is necessary to prevent leakage. One form of packing extensively used is the U leather packing shown in Fig. 5. In this form of packing the water leaking past the plunger, or ram as it is often called, enters the leather cup, pressing one side against the cylinder and the other against the ram, the pressure preventing leakage being pro- portional to the pressure of the water. To take into account the fric- tional resistance in this case, let /i denote the coefficient of friction between leather and ram, C, .c the depths of the packing on the large and small rams (Fig. 4), and p the intensity of water pressure. Then the area of leather in contact with the large ram is tDC and its frictional resistance is therefore TtDCpfx. Similarly, the frictional resistance for the small ram is irdcpii. Consequently W = ^v - fiprDC, and whence P = rd2 p 4" iMpirdc, (6) Digitized by VnOOQlC 8 ELEMENTS OF HYDRAULICS 12. Efficiency of Hydraulic Press. — The efficiency of any ap- paratus or machine for transforming energy is defined as T?ffi • Useful work or effort ^ "" Total available work or effort' and is therefore always less than unity. In the present case if there were no frictional resistance, the relation between W and P would be given by Eq. (4). The efficiency for this type of press is therefore the ratio of the two equations (5) and (4), or Efficiency = 1+4^ (6) m. SIMPLE PRESSURE MACHHTES m HiffhPresrare Outlet £ JDl HT P. m ^ B % IHL nm A *—-\ LowPressuia Intake ±^ Fig. 6. — Intensifier. form another cylinder C, fitted fixed to the yoke at the top. 13. Hydraulic Intensifier. — Besides the hydraulic press de- scribed in Art. II there are a number of simple pressure ma- chines based on the principle of equal distribution of pressure throughout a liquid. Four types are here illustrated and de- scribed, as well as their combiaa- tion in a hydraulic installation. When a hydraulic machine such as a punch or riveter is finishiQg the operation, it is re- quired to exert a much greater force than at the beginniag of the stroke. To provide this in- crease in pressure, an intensifier is used (Fig. 6). This consists of several cylinders telescoped one inside another. Thus in Fig. 6, which shows a simple form of intensifier, the largest cylinder A is fitted with a ram B. This ram is hollowed out to with a smaller ram D, which is Digitized by Google PRESSURE OF WATER 9 In operation, water at the ordinary pump pressure enters the cylinder A through the intake, thereby forcing the ram B upward. This has the effect of forcing the ram D into the cylinder C, and the water in C is thereby forced out through D, which is hollow, at an increased pressure. Let pi denote the pressure of the feed water, pt the intensified pressure in C and di, dt the diameters of the cylinders B and C respectively, as indicated in the figure. Then irdi^ whence Pi A ^ EZ) CD to, ■^ Weiflrhts ■^ The intensity of pressure in the cylinders is therefore inversely pro- portional to the areas of the rams. When a greater intensification of pressure is required a compound intensifier is used, consisting of three or four cylinders and rams, nested in telescopic form, the general arrangement and principle of operation being the same as in the simple intensifier shown in Fig. 6. 14. Hydraulic Accumulator. — ^A hydraulic accumvldlor is a pressure regulator or governor, and bears somewhat the same relation to a hydraulic system that the flywheel does to an engine; that is, it stores up the excess pump delivery when Fia. 7.-— Accumulator, the pumps are delivering more than is being used, and delivers it again under pressure when the demand is greater than the supply. There are two principal types of hydraulic accumulator, in one of which the ram is fixed or stationary, and in the other the cylin- der. The latter type is shown in Fig. 7. When the delivery of the pumps is greater or less than required by the machine, water enters or leaves the cylinder of the accumulator through the ^3 Water Digitized by vnOOQlC 10 ELEMENTS OF HYDRAULICS pipe A, The ram is thereby raised or lowered, and with it the weights suspended by the yoke from its upper end. The pressure in the system is thereby maintained constant and free from the pulsations of the pump. The capacity of the accumulator is equal to the volume of the ram displacement, and should be equal to the delivery from the pmnp in five or six revolutions. The diameter of the ram should be large enough to prevent a high speed in descent, so as to avoid the inertia forces set up by sudden changes in speed. 15. Hydraulic Jack. — The hydraulic jack is a lifting apparatus operated by the pressure of a liquid under the action of a force '^^^ Fig. 8. — Hydraulic jack, pump. Thus in Fig. 8 the hand lever operates the pump piston B, which forces water from the reservoir A in the top of the ram through the valve at C into the pressure chamber D under the ram. The force exerted is thereby increased in the direct ratio of the areas of the two pistons. Thus if the diameter of the pump piston is 1 in. and the diameter of the lifting piston or ram is 4 in., the area of the ram will be sixteen times that of the pump piston. If then a load of, say, 3 tons is applied to the pump Digitized by VnOOQlC PRESSURE OF WATER 11 piston by means of the lever, the ram will exert an upward lift- ing force of 48 tons. 16. Hydraulic Crane. — The hydraulic crane, shown in Fig. 9, consists essentially of a ram and cylinder, each carrying a set of pulleys. A chain or rope is passed continuously over the two sets of pulleys as in the case of an ordinary block and tackle, the I Cylinder ^ i ^1 Fig. 9. — Hydraulic crane. free end passing over guide pulleys to the load to be lifted. When water is pumped into the cylinder under pressure, the two pulley blocks are forced apart, thereby lifting the load at the free end of the chain or rope. 17. Hydraulic Elevator. — In hydraulic installations two or more of these simple pressure machines are often combined, as in the hydraulic elevator shown in Fig. 10. In this case the ac- cumulator serves to equalize the pump pressure, making the operation of the system smooth and uniform. Digitized by VnOOQlC 12 ELEMENTS OF HYDRAULICS The main valve for starting and stopping is operated, in the type shown, by the discharge pressure, maintained by means of an elevated discharge reservoir. A pilot valve, operated from Pump to BotQrD Yalvti Dt^chiifee Fia. 10. — Hydraulic elevator. the elevator cab, admits this low-pressure discharge water to opposite sides of the main valve piston as desired, thereby either admitting high-pressure water from the pump and accumulator. Digitized by VnOOQlC PRESSURE OF WATER 13 or opening the outlet valve into the discharge. The other details of the installation are indicated on the diagram. IV. PRESSURE ON SUBMERGED SURFACES 18. Change of Pressure with Depth. — For a liquid at rest in an open vessel or tank, the free upper surface is perfectly level. Let the atmospheric pressure on this surface be denoted by p. Then, from Eq. (3), the pressure p' at a depth h below the surface is given by ?>' = P + yh. Since the atmospheric pressure is practically constant, the free surface of the liquid may be assumed as a surface of zero pressure when considering only the pressure due to the weight of the liquid. In this case p = 0, and the pressure p' at any depth h, due to the weight of the liquid, becomes P' = yh. (7) Hence the pressure at any point in a liquid due to its own weight is directly proportional to the depth of this point below the free upper surface. Moreover, let AA denote any element of a submerged surface. Then the pressure on it is p'AA = yhAA. Therefore, the pressure on any element of area of a submerged surface is equal to the weight of a column of water of cross-sec- tion equal to the element con- sidered, and of height equal to the depth of this element below the surface. 19. Pressure on Submerged Area. — Consider the pressure on any finite area A in the side of a tank a reservoir contain- ing a liquid at rest (Fig. 11). Let AA denote any element of this area and x its distance below the surface of the liquid. Then, by what precedes, the pressure on this elementary area is yxAA, Digitized by Google 14 ELEMENTS OF HYDRAULICS and consequently the total pressure P on th^ entire area A is given by the summation P = J^yxAA = yl^xAA. But from the ordinary formula for finding the center of gravity of an area, the distance Xq of the center of gravity of A below the surface is given by Axq = J^xAA and consequently P = tAxo. (8) Therefore^ the pressure of a liquid on any submerged plane surface is equal ta the weight of a column of the liquid of cross-section equal to the given area and of height equal to the depth of the center of gravity of this area below the free surface of the liquid. 20. Center of Pressure. — The point of application of the resultant pressure on any submerged area is called the center of Fig. 12. pressure, and for any plane area which is not horizontal, lies deeper than the center of gravity of this area. For instance, consider the water pressure against a masonry dam with plane vertical face (Fig. 12). By Eq, (7) the pressure at any point A is proportional to the depth of A below the water surface. If, then, a length AB is laid off perpendicular to the wall and equal to the depth of A below the surface, that is, AB = AO, then AB will represent to a certain scale the normal pressure at A. If the same is done at various other points of the wall, their ends, jB, D, etc., will lie in a straight line inclined at 45° to the horizontal. Digitized by VnOOQlC PRESSURE OF WATER 15 For a portion of the wall of length 6, the pressure acting on it will then be equal to the weight of the water prism OEF, namely, and the center of pressure will coincide with the center of gravity of this prism. It therefore lies at a distance of %h below the water surface, which is below the center of gravity of the rectan- gular area under pressure since the latter is at a distance of ^ from the surface. 21. General Formula for Center of Pressure. — To obtain a general expression for the location of the center of pressure, con- sider any plane area inclined at an angle a to the horizontal (or Fig. 13. water surface) and subjected to a hydrostatic pressure on one side. Let 00' denote the line of intersection of the plane in which the given area lies with the water surface (Fig. 13). Also, let AA denote any element of the given area, h the depth of this element below the surface, and x its distance from 00', as indi- cated in the figure. Then from Eq. (7), the pressure AP acting on this element is AP = yhAA, and the moment of this force with respect to the line 00' is xAP = yhxAA. Now let P denote the total resultant pressure on the area A and Xe the distance of the point of application of this resultant from Digitized by VnOOQlC 16 ELEMENTS OF HYDRAULICS 00', i.e., Xe represents the x coordinate of the center of pressure. Then since the sum of the moments of aJl the elements of pres- sure with respect to any axis 00' is equal to the moment of their resultant with respect to this axis, we have XxAP = Pxc But AP = yhAA and P = Xyh^. Consequently this becomes XyhxAA = Xc^yhAA. Also, since h = xmi a, this may be written y sin aXx^AA = y sin aXe'SxAA or, cancelling the common factor 7 sin a, Tlx^AA = Xc^xAA. The left member of this expression is by definition the moment of inertia, I, of the area A with respect to the line 00', that is I = Sx«AA, while by the formula for the center of gravity of any area A we also have SxAA = xoA where Xo denotes the x coordinate of the center of gravity of A. The X coordinate of the center of pressure is therefore determined by the general formula _ _I_ _ Moment of inertia " Axo ~" Statical moment ' 22. Application. — In applying this formula it is convenient to use the familiar relation I = la + Ad« where J. = moment of inertia of A with respect to the axis 00'; Ig = moment of inertia of A with respect to a gravity axis parallel to 00'; d = distance between these two parallel axes. For example, in the case of the vertical dam under a hydrostatic head h, considered above, we have for a rectangle of breadth b and height h, 12' /a = Digitized by VnOOQlC PRESSURE OF WATER 17 Consequently the moment of inertia I with .respect to its upper edge is and therefore the depth of the center of pressure below the sur- face is bh* Xc = Axo 3 _2fc HI) which, of course, agrees with the result obtained geometrically. V. STRENGTH OF PIPES UNDER INTERNAL PRESSURE 23. ThiA Cylinder. — In the case of a pipe flowing fuH, the pressure of the liquid produces stress in the pipe walls. Assum- ing the internal pressure of the liquid to be constant at any sec- tion, let / = unit fiber stress in pipe (hoop stress) in pounds per square inch; w = imit internal fluid pressure in pounds per square inch; t = thickness of pipe walls; d = inside diameter of pipe; D = outside diameter of pipe. Now suppose that the pipe is divided longitudinally by a plane through its axis, and consider a section cut out of either half by two planes perpendicular to the axis, at a distance apart denoted by c (Fig. 14). Then the total internal pressure on the strip under consideration is cwd, and the total resisting tension in the pipe walls is 2ctf. Consequently cwd = 2ctf, whence wd f = 2t (10) This formula applies primarily to a thin cylinder or pipe, that is, one for which -3 ^ 0.025. 2 Digitized by VnOOQlC 18 ELEMENTS OF HYDRAULICS 24. Lamp's Formula. — For thick cylinders, in which the thickness of the cylinder is not negligible in comparison with the t diameter, or in other words cylinders for which ^ > 0.025; the formula commonly used is that due to Lam6, namely, ^ w(D» + d») i, ^ D« - d« ^ ^ 26. Barlow's Formula. — ^Another formula which is widely used because of its simplicity is that due to Barlow. The derivation of this formula is based on the assmnptions that the area of cross-section of the tube remains constant under the strain, and that the length of the tube also remains unaltered. As neither of these assumptions is correct, the resulting formula is only approximate. Using the same notation as above. Barlow's formula is Evidently it is of the same form as the formula derived above for the hoop stress in a thin cylinder, except that it is expressed in terms of the oviside diameter of the pipe instead of its inside diameter. From the results of their experience in the manufacture and testing of tubes, however, the National Tube Co. asserts that for any ratio of ^ < 0.3, Barlow's formula "is best suited for all ordinary calculations pertaining to the bursting strength of commercial tubes, pipes and cylinders." For certain classes of seamless tubes and cylinders, however, and for critical examination of welded pipe for which the least thickness of wall, yield point of the material, etc., are known with accuracy and close results are desired, they recommend that the following formulas due to Clavarino and Birnie be used rather than Barlow's. 26. Clavarino's Formula. — In the derivation of Clavarino's formula each particle of the tube is assumed to be subjected to radial stress, hoop stress and longitudinal stress, due to a uni- form internal pressure acting jointly on the walls of the tube and its closed ends. The derivation also involves Poisson's ratio ' For the derivation of this formula see Slocum, "Resistance of Mate- rials," p. 123 (Ginn & Co.). Digitized by VnOOQlC PRESSURE OF WATER 19 of lateral contraction, and is theoretically correct provided the maximum stress does not exceed the elastic limit of the material. Assuming the value of Poisson's ratio to be 0.3 and using the same notation as above, Clavarino's formula is f = w(13D« + 4d«) whence also 10(D« - d«) ♦ /lQf-13w. lOf + 4w (13) (14) 27. Bimie's Formula. — The derivation of Birnie's formula is based upon the same assumption as Clavarino's except that the longitudinal stress is assumed to be zero. Assuming Poisson's ratio for steel to be 0.3 and using the same notation as previously, Birnie's formula is w(13D* + 7d*) f = whence also 10(D* - d») = D^ lOf - 18w lOf + 7w* (15) (16) VL EQUILIBRroM OF TWO FLUIDS IN CONTACT 28. Head Inversely Proportional to Specific Weight— If two open vessels containing the same fluid, say water, are connected ^^,s^ -=-— :^j^ =r=~^t ~«fi :=-_=-:Lr^ '- ^ ^^^^^^l- r i-. Fig. 16. by a tube, the fluid will stand at the same level in both vessels (Fig. 15). If the two vessels contain different fluids which are of different weights per unit of volume, that is to say, of different specific gravities, then since the fluid in the connecting tube Digitized by VnOOQlC 20 ELEMENTS OF HYDRAULICS must exert the same pressure in either direction, the surface of the lighter fluid will be higher than that of the heavier. For instance, let «i and 8% denote the specific gravities of the two fluids in the apparatus shown in Fig. 16, and let A denote the area of their surface of contact. Then for equilibrium ySiAh = ySzAH whence h Ss Si (17) 1 I m A 1 ~ 7? / V i - -f 1 — 1 The ratio of the heights of the two fluids above their surface of separation is therefore inversely proportional to the ratio of their specific gravities. 29. Water Barometer. — ^If one of the fluids is air and the other water, we have what is called a water barometer. For example, suppose that a long tube closed at one end is filled with water and the open end corked. Then if it is placed cork downward in a vessel of water and the cork removed, the water in the tube will fall until it stands at a cer- tain height h above the surface of the water in the open vessel, thus leaving a vacuum in the upper end of the tube. The absolute pressure in the top of the tube, A (Fig. 17), is there- fore zero, and at the surface, B, is equal to the pressure of the atmosphere, or approximately 14.7 lb. per square inch. But from Eq. (3) we have Pb =^ Pa + yh where in the present case Pb = 14.7 lb. per square inch; Pa = 0; y = 62.4 lb. per cubic foot, and by substitution of these values we find that A = 34 ft. approximately. This is the height, therefore, at which a water column may be maintained by ordinary atmospheric pressure. It is therefore also the theoretical height to which water may be raised by means of an ordinary suction pump. As it is impossible in practice to secure a perfect vacuum, however, the actual working lift for a suction pump does not exceed 20 or 25 ft. Fig. 17. Digitized by LnOOQlC PRESSURE OF WATER 21 30. Mercury Barometer. — ^If mercury is used instead of water, since the specific gravity of mercury is 8 = 13.5956, we have 14.7 X 144 13.6 X 62.4 = 30 in., approximately. which is accordingly the approximate height of an ordinary mercury barometer. 31. Piezometer. — ^When a vessel contains liquid under pressure, this pressure is conveniently measured by a simple device called a piezometer. In its simplest form this consists merely of a tube inserted in the side of the vessel, of suflScient height to prevent overflow and large enough in diameter to avoid capillary action, say over J^ in. inside diameter. The height of the free surface of the liquid in the tube above any point B in the vessel then measures the pressure at B (Fig. 18). Since the top of the tube is open to the atmosphere, the absolute pressure at jB is that due to a head of A + 34 ft. i Fia. 18.— Piezometer. Fia. 19. — Pressure gage. 32. Mercury Pressure Gage. — In general it is convenient to use a mercury column instead of a water column, and change the form of the apparatus slightly. Thus Fig. 19 shows a simple form of mercury pressure gage, the difference in level, h, of the two ends of the mercury column measuring the pressure at B. Let 8 denote the specific weight of mercury and s' the specific weight of the fluid in the vessel. The pressure at any point C in the vessel is then p. = 7sh - Ts'h'. (18) Digitized by VnOOQlC 22 ELEMENTS OF HYDRAULICS For example, if the fluid in the vessel is water, then y = 62.4, «' = 1, « = 13.596, and consequently Po = 62.4(13.596A - h'). ^ c = = ' ' " \ ^' : ^_L -■ ^ Fig. 20. — Vacuum gage. In case there is a partial vacuum in the vessel, the gage may be of the form shown in Fig. 20. The pressure on the free surface AB in the reservoir is then the same as at the top of the barometer col- umn, C, namely, Pc = 14.7 — 7s/i. 33. Differential Gage.— When differences in pressure are to be measured, the gage commonly used is the U-tube differential gage, one form of which is shown in Fig. 21. In this form the lower part of the U-tube is filled with mercury, or some other heavy fluid, and from the differ- ence in elevation of the two ends of the mercury column the difference in pressure in the connecting tubes may be calculated. Vn. EQUILIBRIUM OF FLOATING BODIES 34. Buoyancy. — ^When a solid body floats on the water partially submerged, as in the case of a piece Digitized by VnOOQlC PRESSURE OF WATER 23 of timber or the hull of a ship^ each element of the wetted surface experiences a unit normal pressure of amount p = yh where h denotes the depth of the element in question below the water surface. Since the body is at rest, the total pressure acting on the wetted surface together with the weight of the body, which in this case is the only other external force, must then form a system in equilibrium. Since the weight of the body acts vertically downward, the water must therefore exert an upward pressure of the same amount. This resultant upward pressure of the water is called the buoyant effort, or buoyancy ^ and the point of application of this upward force is called the center of T^ W-B FiGL 22. buoyancy. For equilibrium, therefore, the buoyancy must be equal to the weight of the body and act vertically upward along the same line, since otherwise these two forces would form a couple tending to tip or rotate the body (Fig. 22). 36. Floating Equilibrium. — To. calculate the buoyancy, sup- pose that the solid body is removed and the space it occupied below the water line refilled with water. Then since the lateral pressure of the water in every direction must be exactly the same as before, the buoyancy must be equal to the weight of this vol- ume of water. The buoyancy is therefore equal to the weight of the volume of water displaced by the floating body, and the center of buoyancy coincides with the center of gravity of the displacement. For equilibrium, therefore, a solid body must Digitized by VnOOQlC 24 ELEMENTS OF HYDRAULICS sink until the weight of the water it displaces is equal to the weight of the body, and the centers of gravity of the body and its dis- placement must lie in the same vertical. These conditions also apply to the case when a body is entirely submerged. As the density of water increases with the depth, if a solid is slightly heavier than the water it displaces, it will sink until it reaches a depth at which the density is such that the weight of the water it displaces is exactly equal to its own weight. 36. Theorem of Archimedes. — If a solid is heavier than the weight of water it displaces, equilibrium may be maintained by suspending the body in water by a cord (Fig. 22), in which case the tension, T, in the cord is equal to the difference between the weight of the body and its buoyancy, that is, the weight of the water it displaces. A solid immersed in a liquid therefore loses in weight an amount equal to the weight of the liquid displaced. This is known as the Theorem of ArchimedeSf and was discovered by him about the year 260 B. C. 37. Physical Definition of Specific Weight. — Consider a solid completely immersed in a liquid, and let V denote the volume of the solid, and y the weight of a cubic unit of the liquid, say 1 cu. ft. Then the buoyancy, J?, of the body is y B^yV. Also, if 7i demotes the weight of a cubic unit of the solid, regarded as uniform and homogeneous, its weight is W==yiV. The ratio !=?=• (") is called the specific weight of the solid with respect to the liquid in which it is immersed (compare Art. I). In general, the liquid to which the specific weight refers is assumed to be water at a temperature of 39^F. The specific weight of any substance is then that abstract number which expresses how many times heavier it is than an equal volume of water at 39®F. The specific weight of water is therefore unity; for lighter substances such as wood or oil it is less than unity; and for heavier substances like lead and mercury it is greater than unity. 38. Determination of Specific Weight by Experiment. — The specific weight of a body may be determined by first weighing it Digitized by VnOOQlC PRESSURE OF WATER 26 in air and again when immersed in water. The actual weight of the body in air is then W = yiV =- ysV where 8 denotes its specific weight, and its apparent weight T when immersed (Fig. 22) is T = W -B ^yaV -yV ^ yV(8 - 1), that is, T = 7V(s - 1). (20) Therefore, by division, W ^ 8 T s- 1 whence s = ^. (21) The specific weight of a body is therefore equal to its weight in air divided by its loss in weight when immersed in water. 39. Application to Alloy. — If a body is an alloy or mixture of two different substances whose specific weights are known, the volume of each substance may be determined by weighing the body in air and in water. Thus let Vi denote the volmne, and «i the specific weight, of one substance, and Vs, 8% of the other. Then the weight of the body in air is . and its apparent weight T when immersed is, from Eq. (20), T =:77i(si - 1) + 772(«2 - 1). Solving these two equations simultaneously for V\ and V%, the result is T-(l^-)w Vl = T 72 = - (r:-')^ -{^-y c^')• This method of determining relative volumes was invented by Archimedes in order to solve a practical problem. Hiero, King Digitized by VnOOQlC 26 ELEMENTS OF HYDRAULICS of Syracuse, had furnished a quantity of gold to a goldsmith to be made into a crown. When the work was completed the crown was found to be of full weight, but it was suspected that the gold- smith had kept out a considerable amoimt of gold and substi- tuted an equal weight of silver. To test the truth of this sus- picion Archimedes first balanced the crown in air against an equal weight of gold, and then immersed both in water, when the gold was found to outweigh the crown, proving the goldsmith to be dishonest. 40. Zero Buoyancy. — When a body lies flat against the bottom of a vessel filled with water, fitting the bottom so closely that no water can get under it, its buoy- ancy is zero. In this case if W denotes the weight of the body, A the area of its horizontal cross- section, and h the depth of water on it, the force T required to lift it is (Fig. 23) r = TT + yAh. That is to say, the force T is the same as would be necessary to lift the body itself and the entire col- FiQ. 23. umn of water vertically over it. This same principle underlies the action of a leather sucker or vacuum-tipped arrow, the fluid in that case being air. Vm. METACENTER 41. Stability of Floating Body. — ^When a floating body is shoved to one side it remains in this position and is therefore in neutral equilibrium as regards lateral translation. In deter- mining the stability of a floating body it is therefore only neces- sary to consider its equilibrium as regards rotation. After a floating body has been tipped or rotated a small amoimt from its position of equilibrium, the buoyancy, in general, no longer passes through the center of gravity of the body. Conse- quently the weight and buoyancy together form a couple tending to produce rotation or tip the body. If this couple tends to right the body the equilibrium is stable, whereas if it tends to tip it over it is unstable. This evidently depends on the form of the Digitized by VnOOQlC PRESSURE OF WATER 27 wetted surface, and abo on the form of the part immersed by the rotation. 42. Metacenter. — For example, consider a floating box of rectangular cross-section, injmersed to a depth d below the surface A A (Fig. 24), and suppose it is tipped by an external couple until the water line becomes A'A\ In this new position the displacement is trapezoidal, and the .center of buoyancy B is the center of gravity of this trapezoid. But since the buoy- ancy is of the same amount as before the box was tipped and the triangle of immersion mno is equal to the triangle of emersion opq, the lines AA and A'A' intersect on the vertical axis CC. The intersection M of the line of action of the buoyancy with the vertical axis CC is called the metacenter. Evidently the loca- tion of the metacenter depends on the angle of tip and is different Fig. 24. for each position. It is also apparent that the equilibrimn is stable if the metacenter M lies above the center of gravity of the body, and unstable if M lies below 0. It is also shown in what follows that the metacenter moves higher as the angle of tip, a, increases. Its lowest position is called the true metacenter. 43. Coordinates of Metacenter. — ^For the special case of the rectangular cross-section shown in Fig. 24, let x, y denote the coordinates of the center of gravity of the trapezoid, and a, 6, c the lengths of three sides (Fig. 26). Then from geometry, _ c(2a + b) __ a^ + ab + b^ ^ ~ 8(a + 6) ' ^ ~ 3(a + 6) From Fig. 24 the sides a and b of the trapezoid expressed in terms of d and a are c c a ^ d " - tan a; 6 = d + 5 tan a. Digitized by VnOOQlC 28 ELEMENTS OF HYDRAULICS Inserting these values of a and b in the expressions for x and y, the result is c(3d - I tan a) Sd^ + ^ tan^ a ^ " 6d ' ^ " 6d Also, from Fig. 24, the total height H of the metacenter above the bottom of the vessel or box is ff = y + (| - «j cot a. Hence by inserting the above values for x and y in this expression for H and reducing, we obtain the relation H = | + ^(tan*« + 2). (22) The height H therefore increases with a; that is, the greater the angle of tip, the higher the meta- center M. Moreover, by sub- stituting a = in Eq. (22) the position of the true metacenter, or limiting position of M, is found to be at a height H' above the bottom given by H' = | + & W To prevent a ship from capsizing, it is necessary to so design and load it that the height of its center of gravity above the bottom shall be less than H'. 44. Metacentric Height. — To consider the general ease of equiUbrium of a floating body, take a vertical cross-section through the center of gravity of the body (Fig. 26), and suppose that by the application of an external couple it is slightly tipped or rotated about an axis OY, drawn through perpendicular to the plane of the paper. Then the volume displaced remains unchanged, but the center of buoyancy B is moved to some other point B\ To find the metacentric height Ay, or distance from the center of gravity G of the body to the metacenter 3f , let V = volume of liquid displaced; A = cross-sectional area of body in plane of flotation; 6 = distance from center of gravity G to center of buoyancy B; ky = radius of gyration of area of flotation A about the axis OY. Digitized by VnOOQlC PRESSURE OF WATER 29 Then it can be shown that^ h, = ^*-b. (24) Similarly, for rotation about the axis OX, the metacentric height hs is given by h. * ^ - b, (26) where fc, denotes the radius of gyration of the area of flotation A about the axis OX, Evidently the metacentric height is greater for a displacement about the shorter principal axis of the section A. For instance, it is easier to make a ship roll than to cause it to tip endwise or pitch. Fig. 26. The locus of the centers of buoyancy for all possible displace- ments is called the surface of buoyancy, and the two metacenters given by Eqs. (24) and (25) are the centers of curvature of its principal sections. 46. Period of Oscillation. — When a floating body is tipped and then released, it will oscillate, or roll, with a simple harmonic motion. To find the period of the oscillation, the general ex- pression for the period of oscillation of a solid body rotating about a fixed axis may be applied, namely,^ ^^ym' (26) where P = period or time of a complete oscillation; 1 Webster, " Dynamics of Particles," p. 474 (Teubner). * Sloctjm, "Theory and Practice of Mechanics," p. 302 (Holt & Co.). Digitized by VnOOQlC 30 ELEMENTS OF HYDRAULICS W = weight of the body; I = moment of inertia of the body with respect to the axis of rotation; h = distance from the center of gravity of the body to the axis of rotation. Since I = MK^, where M denotes the mass of the body and K its radius of gyration, and also W = Mg, Eq. (26) for the period may be written P = 2.JMK^=2^. (27) \Mgh Vgh 46. Rolling and Pitching. — In the present case, consider rota- tion about the two principal axes OX and OF of the section A in the plane of floatation, and let X,, Ky denote the radii of gyration of the solid with respect to these axes, and Px, Py the correspond- ing periods, or times of performing a complete oscillation about these axes. Then from Eq. (27), p — ^Ka ^ p _ 2fjrKy VgK' * Vghy Substituting in these expressions the values of hx and hy given by Eqs. (24) and (26), they become 1, /"7A M ' I, /"^A bT (28) For a body shaped like a ship, K and k increase together, and consequently the larger value of k corresponds to the smaller period P. A ship therefore pitches more rapidly than it rolls. For further applications of the metacenter the student is re- ferred to works on naval architecture. APPLICATIONS 1. The ram of a hydraulic press is 10 in. in diameter and the plunger is 2 in. in diameter. If the plunger is operated by a handle having a leverage of 8 to 1, find the pressure exerted by the ram, neglecting friction, when a force of 150 lb. is applied to the handle. 2. In a hydraulic press the diameter of the ram is 15 in. and of the plunger is % in. The coefficient of friction may be assumed as 0.12 and the width of the packing on ram and plunger is 0.2 Digitized by VnOOQlC PRESSURE OF WATER 31 of their respective diameters. What pressure will be exerted by the ram when a force of 200 lb. is applied to the plunger? 3. Water in a pipe AB is to be kept at a constant pressure of 1,200 lb. per square inch by forcmg in a plimger of diameter d (Fig. 27). This is operated by a piston of diameter D, whose lower surface is subjected to the pressure of a colunm of water 75 ft. high. Find the ratio of the two diameters d and D. 4. In a hydraulic pivot bearing, a vertical shaft carrymg a total load W is supported by hydraulic pressure (Fig. 28). The pivot is of diameter Z>, and is surrounded by a 17 leather packing B 3 jU — o — Fig. 27. of width c. Show that the frictional moment, or resistance to rotation, is given by the relation M = 2/icTr where /x denotes the coefficient of friction. 6. For an ordinary flat pivot bearing of the same diameter D and for the same coefficient of friction /a as in the preceding prob- lem, the frictional moment is given by the relation^ M = HWDn. Show that the hydraulic pivot bearing is the more efficient of the two provided that c < D 6" Calculate their relative efficiency when c = 0.2D. ^ Slocttm, ''Theory and Practice of Mechanics," p. 194 (Holt). Digitized by VnOOQlC 32 ELEMENTS OF HYDRAULICS 6. An instrument for measuring the depth of the sea consists of a strong steel flask, divided into two compartments which are connected by a valve. The upper compartment is filled with 920 grams of distilled water and the lower compartment with mercury (Fig. 29). When lowered to the bottom, the outside pressure forces the sea water through a small opening in the side of the flask and thereby forces the mercury through the valve into the upper compartment. Assuming that the depth of the sea in certain parts of the Pacific ocean is 9,429 meters, and that the ratio of the densities of distilled and salt water is 35: 36, find how many grams of mercury enter the upper compartment.^ The modulus of compressibility of water is 0.000047, that is, an increase in pressure of one atmos- phere produces this decrease in volume. Note. — ^Assuming that a pressure of one at- mosphere is equal to a fresh-water head of 10J{ meters, the corresponding salt water head >- 10^ X 8^0 . 10.045 meters. At a depth of 9,429 9 429 meters the pressure is therefore = ^'q., « 938.7 Fig. 29. atmospheres. 7. A hydraulic jack has a 3-in. ram and a %-in. plimger. If the leverage of the handle is 10 : 1, find what force must be applied to the handle to lift a weight of 5 tons, assuming the efliciency of the jack to be 76 per cent. 8. A hydraulic intensifier is required to raise the pressure from 600 lb. per square inch to 2,600 lb. per square inch with a stroke of 3 ft. and a capacity of 4 gal. Find the required diameters of the rams. 9. In a hydraulic intensifier like that shown in Fig. 6, the diameters are 2 in., 6 in. and 8 in., respectively. If water is sup- plied to the large cylinder at a pressure of 500 lb. per square inch, find the pressure at the high-pressure outlet. 10. How would the results of the preceding problem be modi- fied if the frictional resistance of the glands, or packing, is taken into account, assuming that the frictional resistance of one stuff- ing box is 0.05 pd, where p denotes the water pressure in pounds per square inch, and d is the diameter of the ram in inches? 11. A hydraulic crane has a ram 10 in. in diameter and a iWiTTENBAUEB, "Aufgaben ausder Technischen Mechanik,'' Bd. III. Digitized by VnOOQlC PRESSURE OF WATER 33 velocity ratio of 1 : 12, that is, the speed of the lift is twelve times the speed of the ram. Assuming the efficiency of the crane to be SO per cent., find what load it will lift with a water pressure of 1,500 lb. per square inch. 12. A hydraulic crane has a velocity ratio of 1:9 and is re- quired to lift a load of 4 tons. Find the required size of the ram for a pressure in the mains of 750 lb. per square inch, a loss of head due to friction of 75 lb. per square inch, and a mechanical efficiency of 70 per cent. 13. How many foot-pounds of work can be stored up in a hydraulic accumulator having ^P T T rr" -I i T\B CL. i- w ._l_i— a ram 10 in. in diameter and a lift of 12 ft., with a water pressure of 800 lb. per square inch? 14. Find the energy stored in an accumulator which has a ram 10 in. in diameter, loaded to a pressure of 1,000 lb. per square inch, and hav- ing a stroke of 25 ft. If the full stroke is made in 1 min. find the horsepower available during this time. 16. The stroke of a hy- draulic accumulator is fifteen times the diameter of the ram and the water pressure is 1,200 lb. per square inch. Find the diameter of the ram for a capacity of 125 hp.-min. 16. The ram of a hydraulic accumulator is 20 in. in diameter, the stroke 25 ft., and the water pressure, 1,050 lb. per square inch. If the work during one full downward stroke is utilized to operate a hydraulic crane which has an efficiency of 50 per cent, and a lift of 35 ft., find the load raised. 17. An accumulator is balanced by means of a chain of length I passing over two pulleys A and B (Fig. 30) and carrying a coun- terweight W equal to the total weight of the chain. Find the distance apart of the pulleys and the required weight of chain per unit of length in order that this arrangement may balance the difference in pressure during motion. 3 Fig. 30. Digitized by LnOOQlC 34 ELEMENTS OF HYDRAULICS Hint, — Let A denote the area of the ram and w the weight of the chain per unit of length. Then for the dimensions shown in the figure, we have the relations wx — wz ^ yACf x + y+z^h x + k ''b + Cf whence yAc 18. A hydraulic accumulator has a ram 15 in. in diameter and carries a load of 60 tons. Assuming the total frictional resistance to be 3 tons, find the required water pressure when the load is being raised and when it is being lowered. 19. Show that the depth of the center of pressure below the surface for a vertical rec- tangle of breadth b and depth d, with upper edge immersed to a depth hi and lower edge to a depth ht (Fig. 31) is hi* - hi\ jj_. Fig. 31. given by the expression -<i^^ Fig. 32. 20. Show that the center of pressure for a vertical plane tri- angle with base horizontal and vertex at a distance hi below the^ surface (Fig. 32) is given by the expression = H( 3/i2^ -f- 2hih2 + hi\ 2h2 + hi / 21. From the results of the preceding problem show that if the Digitized by VnOOQlC PRESSURE OF WAJER 35 vertex of the triangle lies in the surface, the depth of the center of pressure is Xc = %d, and if the base of the triangle lies in the surface Xc = Hd. 22. Show that the depth of the center of pressure below the sur- face for a vertical circular area of radius r, immersed so that its center lies at a depth h below the surface is given by 23. A circular opening, 2 ft. in diameter in the vertical side of a tank is closed by a circular cover held on by two bolts, one 14 in. above the center of the cover and the other 14 in. below its center. When water stands in the tank at a level of 20 ft. above the center of the opening, find the stress in each bolt. 24. A pipe of 4 ft. inside diameter flows just full, and is closed by a valve in the form of a flat circular plate balanced on a horizontal axis. At what distance from the cen- . ter should the axis be placed in order that the valve may bal- ance about it? 26. An automatic movable flood dam, or flashboard, is made of timber and pivoted to a back stay at a certain point C, as shown in Fig. 33. The point C is so located that the dam is stable pro- vided the water does not rise above a certain point A, but when it rises above this point the dam automatically tips over. Deter- mine where the point C should be located. 26. An opening in a reservoir wall is closed by a plate 2}4 ft. square, hinged at the upper edge, and inclined at 60^ to the hori- zontal. The plate wei^ 250 lb., and is raised by a vertical chain attached to the middle point of its lower edge. If the center of the plate is 15 ft. below the surface, find the pull on the chain required to open it. 27. A rectangular cast-iron sluice gate in the bottom of a Fio. 33. Digitized by VnOOQlC 36 ELEMENTS OF HYDRAULICS dam is 3 ft. high, 4 ft. wide and 3 in. thick. The head of water on the center of the gate is 35 ft. Assuming the coefficient of friction of the gate on the slides to be H> ai^d that there is no water on the lower side of the gate, find the force required to lift it. Weight of cast iron is 450 lb. per cubic foot. 28. Flow from a reservoir into a pipe is shut off by a flap valve, as shown in Fig. 34. The pivot A is so placed that the weight of the valve and arm balance about this point. Calculate the pull P in the chain required to open the valve for the dimensions given in the figure and a head of 16 ft. on the center of the valve. 29. The waste gate of a power canal is 8 ft. high and 5 ft. wide, and when closed there is a head of 10 ft. of water on its center. If the gate weighs 1,000 lb. and the coefficient of friction between gate and seat is 0.4, find the force required to raise it. 30. A lock gate is 30 ft. wide and the depth of water on the two sides is 28 ft. and 14 ft. respectively. Find the total pressure on the gate and its point of application. 31. A lock is 20 ft. wide and is closed by two gates, each 10 ft. wide. If the depth of water on the two sides is 16 ft. and 4 ft. respectively, find the resultant pressure on each gate and its point of application. 32. A dry dock is 60 ft. wide at water level and 52 ft. wide at floor, which is 40 ft. below water level. The side walls have a straight batter. Find the total pressure on the gates and its point of application when the gates are closed and the dock empty. 33. A concrete dam is 6 ft. thick at the bottom, 2 ft. thick at top and 20 ft. high. The inside face is vertical and the outside face has a straight batter. How high may the water rise without causing the resultant pressure on the base to pass more than 6 in. outside the center of the base? Note.— A dam may fail either by sliding or by overturning. In general, however, if a well-laid masonry dam is stable against overturning, it will not fail by sliding on a horizontal joint. This kind of failure could occur only when the shearing stress at any joint exceeded the joint friction. Ordinarily the resultant pressure on any joint makes only a small angle with Fig. 34. Digitized by Google PkESSUkE OP WAtER 37 a I^^rjieiidiculai' to its plane," and since the angle of repose for masonry is largei failure by shear of this kind is not likely to occur. As a criterion against failure by shear^ it may be assumed that when the resultant pressure on any Joint makes an angle less than 30** with the normal to the joints it is safe against sliding at that joint. To guard against sUding on the base an anchorage should be provided by cutting steps or trenches in the foundation if it is of rock, or in the case of clay and similar material, by making the dam so massive that the angle which the resultant pressure on the base makes with the vertical is well within the angle of friction. Usually if the dam is heavy enough to satisfy the condition for stability against overturning, as explained belowi, it will also be safe against sliding on the base. In order for a dam to fail by overturning, one or more joints must open at the face, in which case this edge of the joint must be in tension. Al- though a well-laid masonry joint has considerable tensile strength, it is customary to disregard this entirely in designing, in which case the condi- tion necessary to assure stability against overturning is that every joint shall be subjected to compressive stress only. Fia. 35. Assuming a H^mmx distribution of pressure over the joint, as indicated by the trapezoid ABCD in Fig. 35, the resultant pressure R must pass through the eeater of gravity of this trapezoid. Consequently when the compres- sion at one face, Z>, becomes zero, as indicated in Fig. 35 the trapezoid be- comes a trian^^e, and the resultant is applied at a distance » from the opposite face C, where h denotes the width of the joint. Moreover, the resultant cannot approach any nearer to C without producing tensile stress at Z> as indicated in Fig. 35. For stability against overturning, therefore, the resultant pressure must ciU the base (or joint) within the middle third. If water is allowed to seep under a dam, it will exert a lifting effort equal to the weight of a column of water of height equal to the static head at fbhts point. To assure stability it is, therefore, essential to prevent seepage by means of a cutoff wall, as indicated in Figs. 36 and 37. In investigating the stabiUty of a dam, however, the best practice allows for accidental seep- age by making allowance for an upward pressure on the base due to a hydrostatic head of two-thirds the actual depth of water back of the dam. 34. Figure 36 shows a typical section of the Kensico Dam, forming part of the Catskill Water System of the City of New Digitized by LnOOQlC 38 ELEMENTS OF HYDRAULICS amS. UM //Atf/W f/J^jJ WASTE WEIR r^-r^"-^ — c— ^^ MAXIMUM SECTJON u_ J 3^-3"* KENSICO DAM Fig. 36. Digitized by VjOOQiC PRESSURE OF WATER 39 York. The Kensico Reservoir covers 2,218 acres, with a shore line 40 miles in length, and has a storage capacity of 38,000,000,- 000 gal. The dimensions of the main dam are length 1,843 ft.; height 300 ft.; thickness at base 230 ft.; thickness at top 28 ft. Investigate the stability of this dam in accordance with the conditions stated in the note to Problem 33. 36, Figure 37 shows a section of the Olive Bridge Dam and typical dyke section of the Ashokan Reservoir, which forms part a.^ft f /tag /tihf-^^ I TYPICAL SECTION OF DIKE Cats^U aqueduct system. i-*:ra OLIVE BRIDGE DAM MAXIMUM MASONRY SECTION Fig. 37. of the Catskill Water System of the City of New York. This reservoir covers 8,180 acres, with a shore line 40 miles in length and a storage capacity of 132,000,000,000 gal. The principal dimensions of the main dam are, length 4,650 ft.; height 220 ft.; thickness at base 190 ft.; thickness at top 23 ft. Investigate the stability of this dam as in the preceding prob- lem. 36. In the $25,000,000 hydraulic-power development on the Mississippi River at Keokuk, Iowa, the dam proper is 4,650 ft. long, with a spillway length of 4,278 ft. The power plant is designed for an ultimate development of 300,000 hp., and consists Digitized by LnOOQlC 40 ELEMENTS OF HYDRAULICS of vertical shaft turbines and generators in units of 10,000 hp. each. Transmission lines convey the current at 110,000 volts to St. Louis, 137 miles distant, and to other points.^ A notable feature of the plant is the ship lock which is of un- usual size for river navigation, the lock chamber being 400 ft. long by 110 ft. wide with a single Uft of from 30 to 40 ft., the total water content of the lock when full being about 2,200,000 cu. ft. The locks at Panama are the same width but the maximum lift on the Isthmus is 32 ft., the average lift being about 28 ft. Find the maximmn pressure on the lock gates at Keokuk and its point of application. (See frontispiece.) Oage L^mp ^......-.,....„^,_..5^;^?_ liMM Fig. 38. 37. The side walls of the Keokuk lock are monoUthic masses of concrete, with a base width of 33 ft., a top width of 8 ft., and an outside batter of 1: 1.5, as shown in Fig. 38. If the water stands 48 ft. above the floor of the lock on the inside and 8 ft. on the outside, find the point where the resultant pressure on the side walls intersects the base, neglecting the weight of the road- way on top and the arches which support it. 38. The lower lock gates at Keokuk are of the mitering type, as shown in Figs. 39 and 40, and are very similar to those in the Panama canal locks. The gates are 49 ft. high and each leaf consists of 13 horizontal ribs curved to a radius of 66 ft. 4^^ 1 Eng, News, Sept. 28, 1911. Digitized by VnOOQlC PRESSURE OF WATER 41 in. on the center line, framed together at the ends of the quoin and miter posts, and also having seven lines of intermediate framing. The chord length over the posts is 66 ft. 4^ in. and the rise of the curve is 10 ft. 8J^ in.^ Each leaf contains a buoyancy chamber to relieve the weight on the top hinge. This consists of a tank of about 3,840 cu. ft. capacity, placed between the curve of the face and the chord line of the bracing. The total weight of each gate in air is about 240 tons. Find how much the buoyancy chamber reUeves the weight on the top hinge. Fig. 39; 39. The upper gates of the Keokuk lock are of a floating type never before used, and consist essentially of floating tanks moving in vertical guides and sinking below the level of the sill (Fig. 41). To close the lock, compressed air is admitted to an open-bottom chamber in the gate, which forces out the water and causes the gate to rise. To open the lock, the air in this chamber is al- lowed to escape, when the weight of the gate sinks it to its lower position. The flotation of the gate is controlled by two closed displace- ment chambers, one at each end, and one open buoyancy cham- ber. Each of the former is 42 ft. long, 4 ft. deep and 16 ft. wide. The buoyancy chamber is 2}^ ft. high beneath the displacement chambers and 63^ ft. high in the 28-ft. space between them, its capacity being 6,000 cu. ft. » Eng. News, Nov. 13, 1913. Digitized by VnOOQlC 42 ELEMENTS OF HYDRAULICS With the gate floating and its bottom just clear of the sill, the weight of the part above water is 190 tons, which is increased by the ballast in the displacement chambers to 210 tons. The dis- ^-rl?. ^^ Section of 6fl+fi ^ Ci-rs+cr(E-F) Sectional Plan C-D Fig, 40. — Mitering gate, Keokuk lock. placement of the submerged part of the gate is 12 tons so that the buoyant efifort required is 198 tons. Find the equivalent displacement in cubic feet, from this result Digitized by LnOOQlC PRESSURE OF WATER 43 Digitized byCnOOQlC 44 ELEMENTS OF HYDRAULICS subtract the volume of the displacement chambers, and then find the required air pressure in the buoyancy chamber. In raising the gate it is actually found that this pressure varies from 2 lb. per square inch to as high as 12 lb. per square inch when the gate is leaving its lower seat. 40. A gas tank is fitted with a mercury gage as shown in Fig. 19. The height h of the mercury column is 20 in. Find the excess of pressure in the tank above atmospheric. 41. A piece of lead weighs 20 lb. in air. What will be its ap- parent weight when suspended in water, assuming the specific weight of lead to be 11.4? 42. A pail of water is placed on a platform scales and found to weigh 12 lb. A 6-lb. iron weight is then suspended by a Ught cord from a spring balance and lowered into the water in a pail Fig. 42. until completely immersed. Find the reading on the spring balance and on the platform scales. 43* A brass casting' (alloy of copper and zinc) weighs 200 lb. in air and 175 lb. in water. If the specific weight of copper is 8.8 and of zinc is 7, how many pounds of each metal does the casting contain? 44. One end of a wooden pole 12 ft. long, floats on the water and the other end rests on a wall so that 2 ft. project inward beyond the point of support (Fig. 42). If the point of support is 18 in. above the water surface, find how much of the pole is immersed. 46. A floating platform is constructed of two square wooden beams each 16 ft. long, one 18 in. square and the other 1 ft. square. On these is laid a platform of 2-in. plank, 10 ft. wide. Find where a man weighing 160 lb. must stand on the platform to make it float level, and how high its surface will then be above Digitized by VnOOQlC PRESSURE OF WATER 45 the water (Fig. 43). The weight of timber may be assumed as 50 lb. per cubic foot. 46, A piece of timber 4 ft. long and 4 in. square has a weight W attached to its lower end so that it floats in water at an angle of45MFig. 44). FindTT. 47. A rectangular wooden barge is 30 ft. long, 12 ft. wide and 4 ft. deepi outside measurement, and is sheathed with plank Fig. 43. 3 in. thick, the frame weighing half as much as the planking. Find the position of the water line when the barge floats empty, and also the load in tons it carries when the water line is 1 ft. from the top. Assume the weight of wood as 50 lb. per cubic foot. Fig. 44 48. A prismatic wooden beam 10 ft. long, 1 ft. wide and 6 in. thick floats flat on the water with 4 in. submerged and 2 in. above water. Find its specific weight. 49. A dipper dredge weighs 1,200 tons and floats on an even keel with bucket extended and empty. When the bucket carries a load of 3 tons at a distance of 50 ft. from the center line of the scow, a plumb line 15 ft. long, suspended from a vertical mast, swings out 5 in. Find the metacentric height. Digitized by LnOOQlC 46 ELEMENTS OF HYDRAULICS NoTB. — Let W denote the weight of the veBsd and O its center of gravity (Fig. 45). When an eccentric load P is added at a distance d from OM^ the center of gravity moves to G\ where for a small displacement: p GG' a i^y approximately. If B denotes the angle of heel, then also QQ' = GAT tan ^, whence GM GG' Pd tan $ W cot e. 50. A steamer is of 14,000 tons displacement. When its life boats on one side are filled with water, a plmnb line 20 ft. long suspended from a mast is found to swing out 9}4 "!• If the total weight of water in the boats is 75 tons and their distance from the center line of the vessel is 26 ft., find the metacentric height and period with which the ship will roll. Digitized by VnOOQlC SECTION 2 FLOW OF WATER IX. FLOW OF WATER FROM RESERVOIRS AND TANKS 47. Stream Line, — In the case of a flowing liquid, the path fol- lowed by any particle of the liquid in its course is called a stream line. In particular, if a reservoir or tank is filled with water and a small opening is made in one side at a depth h below the surface, the water flows out with a certain velocity depending on the depth, or head, h. Since the par- ticles of water flowing out converge at the opening, the stream lines inside the vessel are, in general, comparatively far apart, but be- come crowded more closely to- gether at the orifice. 48. Liquid Vein. — Under the conditions just considered, suppose that a closed curve is drawn in any horizontal cross-section of the vesr sel and through each point of the closed curve draw a stream line. The totality of all these stream lines will then form a tube, called a liquid vein (Fig. 46). From the definition of a stream line it is evident that the flow through such a tube or vein is the same as though it were an actual material tube. In particular, the same amount of liquid will flow through each cross-section of the vein and therefore the velocity of flow will be greatest where the cross-section of the vein is least, and vice versa, 49. Ideal Velocity Head. — ^In any particular vein let v denote the velocity of flow at a distance h below the surface, and Q the quantity of water per second flowing through a cross-section of the vein at this depth. Then the weight of water flowing through the cross-section per second is yQ and its potential energy at the 47 ¥io. 46. Digitized by Google 48 ELEMENTS OF HYDRAULICS height h is yQh. The kinetic energy of this quantity of water flowing at the velocity v is -^— . Therefore, by equating the potential energy lost to the kinetic energy gained and neglecting all frictional and other losses we have whence V = V^. (29) This relation may also be written in the form The quantity h is therefore called the ideal velocity head, since it is the theoretical head required to produce a velocity of flow v. 60. TorriceUi's Theorem. — The relation V = y/2gh is known as TorricelWs Theorem. Expressed in words, it says that the ideal velocity of flow under a static head h is the same as would be acquired by a solid body faUing in a vacuum from a height equal to the depth of the opening below the free surface of the liquid. 61. Actual Velocity of Flow. — The viscosity of the liquid, as well as the form and dimensions of the opening, have an important effect in modifying the discharge. Considering viscosity first, its effect is to reduce the velocity of the issuing liquid below the ideal velocity given by the relation V = y/2gh. It is therefore necessary to modify this relation so as to conform to experiment by introducing an empirical constant called a velocity coefficient. Denoting this coefficient by C„, the expression for the velocity becomes V = C.\/2^. (30) For water the value of the velocity (or viscosity) coefficient for an orifice or a nozzle is approximately Cv = 0.97. 62. Contraction Coefficient. — In the case of flow through an orifice or over a weir, the obUque pressure of the water approach- ing from various directions causes a contraction of the jet or stream so that the cross-section of the jet just outside the orifice Digitized by VnOOQlC FLOW OF WATER 49 is somewhat less than the area of the opening. Consequently the discharge is also less than it would be if the jet were the full size of the opening. If the area of the orifice is denoted by A, the area of the jet at the contracted section will be some fraction of this amoimt, say CcA, where Ce is an empirical constant called a contraction coeffi- cient, which must be determined experimentally for openings of various forms and dimensions. 63. EflSiux CoeflSicient. — ^Taking into account both the viscosity of the liquid and the contraction of the jet, the formula for dis- charge becomes Q = actual velocity X area of jet = (C,V2^) X (CcA) = CvCcA\/2gh, where A denotes the area of the orifice. Since there is no object in determining C» and Ce separately, they are usually replaced by a single empirical constant K = CvCc, called the coefficient of efflux, or discharge. In general, therefore, the formula for the actual discharge becomes Q = KAV2^. (31) 64. Effective Head. — The head h may be the actual head of water on the orifice; or if the vessel is closed and the pressure is produced by steam oi: compressed air, the effective head is the height to which the given pressure would sustain a colunm of water. The height of the equivalent water column corresponding to any given pressure may be determined by calculating the weight of a colunm of water 1 ft. high and 1 sq. in. in section, from which it is found that 1 ft. head =: 0.434 Ib./in.^ pressure, and conversely, 1 Ib./in.^ pressure = 2.304 ft. head. For an orifice in the bottom of a vessel, the head A is of course the same at every point of the opening, but if the orifice is in the side of the vessel, the head h varies with the depth. However, if the depth of the opening is small in comparison with h, as is frer quently the case, the head may be assiuned to be constant over 4 Digitized by LnOOQlC 50 ELEMENT S.OF HYDRAULICS the entire orifice and equal to the distance of its center of gravity from the free surface of the liquid (Fig. 47(a)). If an orifice is entirely submerged, as shown in Fig. 47(6), the effective head on it is the difference in level between the water surfaces on the two sides of the opening. V f k \f L _. y r^j=^z ^ — •■S-. -.^ Fig. 47. (t>l 66. Discharge from Large Rectangular Orifice. — In the case of a comparatively large orifice, the effective head is not the depth of itfi center of gravity below the surface, and the discharge must be determined in a different manner. To illustrate the method of procedure consider the particular case of an orifice in the form of a rectangle of breadth &, the upper Fig. 48. and lower edges being horizontal and at depths of h and H re- spectively below the surface, as shown in Fig. 48. Let this rec- tangle be divided up into narrow horizontal strips, each of breadth h and depth dy. Then the ideal velocity of flow in any one of these strips at a distance y below the surface is r = \/2gfj/, and Digitized by VnOOQlC FLOW OF WATER 61 since its area is bdy, the ideal discharge dQ through this elemen* tary area per second is dQ = bdyV2gy. The total discharge per second, Q, through the entire orifice is therefore Q = Kb jT^V^dy = %KbV2g(H'^ - h^). (32) This expression inay also be written in the form Q = %KbHV2gH - %KbhV2^ which makes it easier to remember from analogy with the weir formula which follows. 66. Discharge of a Rectangular Notch Weir. — ^If the upper edge of the rectangular orifice just considered coincides with the water surface, the opening is called a rectangular notch weir. In this case A = and the preceding formula for discharge becomes Q = %KbH\/2gH = %KAV2gH (33) where H denotes the head on the crest of the weir, and A is the area of that part of the opening which lies below the surface. X. DISCHARGE THROUGH SHARP- tZT-I EDGED ORIFICE 67. Contraction of Jet. — ^In con- sidering the flow of water through an orifice it is assumed in what follows that a sharp-edged orifice is meant, that is, one in which the jet is in contact with the wall of - the vessel along a Une only (Fig. 49). When this is not the case, the opening is called an adjutage or mouthpiece, and the flow is modified, owing to various causes, as explained in Par. 75. The value of the constant K in Eq. 33 depends on the form of the opening and also on the nature of the contraction of the jet. The contraction is said to be complete when it takes place on all sides of the jet; that is to say, when the size of the opening is small in comparison with its distance from the sides and botr Digitized by VnOOQlC 52 ELEMENTS OF HYDRAULICS torn of the vessel and from the water surface. The contraction is called incomplete when one or more of the edges of the orifice is continuous with the sides of the vessel. 68. Complete Contraction. — ^For a sharp-edged orifice with complete contraction the mean value of the efflux coefficient K is K = 0.62. The actual value of this coefficient varies slightly with the size of the orifice and effective head on it. The value given, however, is sufficiently accurate for all ordinary practical calculations. More exact values are given in Tables 9 and 10. 69. Partial Contraction. — ^In the case of incomplete contraction, let P denote the entire perimeter of the orifice, and nP that frac- tion of the perimeter which experiences no contraction. Then denoting the coefficient of efflux by Ki, its value as determined by experiment for sharp-edged orifices is as follows: • Rectangular orifice, Ki = K(l + 0.16n) \ (34) Circular orifice, Ki = K(l + O.lSn) j Assmning K = 0.62, the following table gives the corresponding values of Ki as determined from these relations.^ n = i n = i n = i Re^rtaninilar orifice Ki =0.643 Ki =0.667 Ki =0.690 Circular orifice Ki =0.640 Ki =0.660 Ki =0.680 60. Velocity of Approach. — So far it has been assumed that the effective head h in the formula for discharge through an orifice, namely, Q = KAV2gh, is simply the static head, measured from the center of the orifice, if it is small, to the surface level. If the velocity of approach is considerable, however, the velocity head must also be included in the effective head. Thus let A = area of orifice; A' = cross-section of channel of approach; V = ideal velocity corresponding to the total head H; V = velocity of approach; h' = velocity head = ^; 1 Lauenstoin, " Mechanik," p. 173. Digitized by LnOOQlC FLOW OF WATER 63 h = static head; H = effective head = -s— 2g Sin^e the total flow through the channel of approach must equal the discharge through the orifice, we have A'v = Q = KAV whence KAV Also the effective head H = h + h', or, expressed in terms of the velocities, Substituting v = ., in this relation, it becomes] whence 2g "+ 2gA'^ ^-4: 2gh The expression for the discharge Q is then ^ = ^^ = ^^/FwAv• (36) K« m From this relation it is evident that if the area A of the orifice is small in comparison with the cross-section A' of the channel, say A' not less than fifteen times A, the error due to neglecting the velocity of approach will be negligible; that is, the term K^ \-j-,) in Eq. (35) may be neglected, in which case the formula for the discharge simplifies into the original expression given by Eq. (33), namely, Q = KAV2gh. XI. RECTANGULAR NOTCH WEIRS 61, Contracted Weir. — ^The most common type of weir consists of a rectangular notch cut in the upper edge of a vertical wall, Digitized by Google 54 ELEMENTS OF HYDRAULICS and is called a contracted weir (Fig. 50(a)). In order that the con- traction shall be complete, there should be a clearance of not less than ih from the sides of the notch to the sides of the channel, and from the bottom of the notch (called the crest of the weir) to the bottom of the channel. 62. Suppressed Weir. — ^If the sides of the notch are continuous with the sides of the channel, it is called a suppressed weir (Fig. 50(6)). For both types of weir it has been found by experiment that the velocity of approach may be neglected when the product bh is less than one-sixth Ihe cross-section of the channel. For a suppressed weir this is equivalent to saying that the height of the weir crest above the bottom should be at least five times the head on the weir. Fia. 50. 63. Submerged Weir. — ^When the water level on the down- stream side of a weir is higher than the crest of the weir, the latter is called a submerged weir (Fig. 51(a)). At present the sub- merged weir is seldom used for measuring flow because of the lack of reliable data from which to determine the experimental constants involved. The principal experiments on submerged weirs are six made by Francis in 1848; 22 made by Fteley and Steams in 1877; and a more extensive set made by Bazin about 1897. From the experiments made by Francis and by themselves, Fteley and Steams derived the following formula for sharp- crested submerged weirs having no end contractions: Q = kl(h + |')Vh^=^ (36) Digitized by VnOOQlC FLOW OF WATER 55 where b = breadth of weir (Fig. 51(a)); h = depth of water on upstream side above level of crest; ft' = depth of water on downstream side above level of crest; k = empirical coefficient depending on the ratio -r- Values of the coefficient k are given in Table 22. Trianatilar Weir ib) Fia. 51. From a later study of the experiments made by Francis and by Fteley and Steams, Clemens Herschel proposed the following simple formula for sharp-crested submerged weirs: I, Q = 3.33b(nh)^, (37) where n is an empirical constant depending on the ratio -j-' Values of the coefficient n are given in Table 22. The formula proposed by Bazin for sharp-crested submerged weirs was as follows: Q = m (l.05 + 0.21^') (^^^) \hV2gh, (38) Digitized by VnOOQlC 56 ELEMENTS OF HYDRAULICS where d denotes the height of the crest of the weir above the bot- tom of the channel, and m is the coefficient for a similar sharp- crested weir with free overfall and full crest contraction, having the same values of h and d. 64. Triangular Notch Weir. — ^This form of weir is best adapted to the calculation of small discharge. To obtain the formula for discharge the notation indicated in Fig. 51 (6) will be used. Then the element of area may be taken as xdy, the theoretical velo- city of flow through this area is >/2gy, and consequently the actual total discharge is Q = c ( xdyV2gy, where c denotes the contraction coefficient to be determined by experiment. By similar triangles we have x = jih — y). Sub- stituting this value in the expression under the integral sign and performing the integration, the result is Q = c ^5& V2g ft^ Denoting the constant part of this expression by a single letter fc, the formula becomes simply Q = kh^. (39) The following values of the coefficient k for use in this formula were obtained experimentally by Professor James Thompson of Glasgow in 1860: Thompson's Values op Coeppicibnt k in Formula Q = kh!^ for 90° Triangular Weir Head h in inches measured from vertex Discharge Q in Coefficient k of notch to still-water cu. ft./min. surface of pond 2 1.748 0.3088 3 4.780 0.3067 4 9.819 0.3068 5 17.07 0.3053 6 26.87 0.3048 7 39.69 0.3061 More recently an accurate series of experiments on triangular notch weirs of 54° and 90° angle has been made by James Barr^ ^London Engineering^ 1910, pp. 435, 473. Digitized by VnOOQlC FLOW OF WATER 57 at the Watt Engineering Laboratories of the University of Glas- gow, with the following results, among others: Barb's Values of Cobfficibnt k in Fobmxjla Q = hl^ for 90® Triangular Weir Head h in inches measured from vertex of notch Discharge Q Discharge Q Coefficient Jk to still-water in cu. ft./min. in gal./min. surface of pond 2 1.755 10.94 0.3104 2i 3.045 18.97 0.3084 3 4.782 29.79 0.3068 3J 7.002 43.63 0.3057 4 9.75 60.74 0.3047 4i 13.05 81.29 0.3038 5 16.95 105.6 0.3032 5i 21.46 133.7 0.3026 6 26.63 166.0 0.3021 6i 32.49 202.4 0.3017 7 39.05 243.0 0.3013 7i 46.34 288.7 0.3009 8 54.06 339.9 0.3006 8i 62.92 392.0 0.3003 9 72.90 454.2 0.3000 9J 83.33 519.2 0.2998 10 94.70 590.0 0.2995 The results obtained by using a 54° triangular weir, as well as other results, are summarized in graphical form in the paper mentioned. 65. Trapezoidal or Cippoletti Weir. — A form of weir frequently used in irrigation practice is the trapezoidal weir, shown in Fig. Fia. 52. — Cippoletti weir. 52. It is evident that the discharge from a trapezoidal weir may be computed by considering it as equivalent to a rectangular suppressed weir and a triangular weir. Denoting the slopes of Digitizedby VnOOQlC 58 ELEMENTS OF HYDRAULICS I s I •s ^ i i L p s2 O w 1° Q I H Us + Si CO CO CO CO II I ' + ^^ — ^^ CO ^ CO ^ ^ CO S • ^ JH ^ O S o "** --; I-* «o CO CO s d I CO CO s ! II d II si lo CO ^* ? 61 II u^ o > S co' n « ,2 + 1 ?■ II ss I d Is O a « ^ Digitized by VnOOQlC FLOW OF WATER 59 the sides or ends by « so that the top width is b + 2$/i} and neglecting end contractions, the discharge from a trapezoidal weir will therefore be The total discharge is therefore greater, in general, than from a rectangular contracted weir of width h. An Italian engineer named Cippoletti proposed giving the sides such a slope that this increase would just equal the decrease in discharge through a con- tracted weir due to end contractions, and found that a slope of 1:4 would accomplish this result. A trapezoidal weir of side slope 1 horizontal to 4 vertical is therefore called a Cippoletti weir, and the formula proposed by Cippoletti for calculating its discharge is = 3.367bh^. (40) In using this type of weir the effect of end contractions may thus be neglected altogether, which makes it especially convenient for use under varying heads. 66. Formulas for Rectangular Notch Weirs. — ^Numerous ex- periments have been made on the flow of water over weirs for the purpose of deriving an empirical formula for the discharge. The most important of these results, including the formulas in common use, are tabulated on page 58. Although these for- mulas apparently differ somewhat in form, they are f oimd irf prac- tice to give results which agree very closely, A rational formula for the discharge over a rectangular weir was derived in Par. 56 as expressed by Eq. (33), namely, Q = %KA\/2gh. For a sharp-edged opening the mean value of the efflux coefficient is X = 0.62, as stated in Par. 58. In the present case, therefore, KA = 0.626A, and if h and h are expressed in feet, the above formula becomes Q = %(0.62bh)v^ih; " s 3.3bh^ cu. ft per sec. (41) It is often convenient to express h and h in inches, and the dis- charge Q in cubic feet per minute. Expressed in these units, the formula becomes Q = %(0.62^)^^X60. Digitized by VnOOQlC 60 ELEMENTS OF HYDRAULICS or, reducing and simplifjdng, Q = OAhh^ cu. ft per min. (42) where b and h are both expressed in inches. These formulas are the basis of many of the weir tables used in practical work, such as Tables 18 and 19 in this book. Xn. STANDARD WEIR MEASUREMENTS 6T. Constructioa of Weir. — ^From the experiments summarized in the preceding article it was found that any empirical weir formula could only be relied upon to give accurate results when Fia. 53. the conditions under which the measurement was made were approximately the same as those imder which the formula was deduced. To obtain accurate results from weir measurements it is therefore customary to construct the weir according to cer- tain standard specifications, as follows: 1. A rectangular notch weir is constructed with its edges sharply beveled toward the intake, as shown in Fig. 53. The bottom of the notch, called the crest of the weir, must be per- fectly level and the sides vertical. Digitized by LnOOQlC FLOW OF WATER 61 2. The length, or width, of the weir should be between four and eight times the depth of water flowing over the crest of the weir. 3. The channel or pond back of the weir should be at least 40 per cent, wider than the notch, and of sufficient depth so that the velocity of approach shall not be over 1 ft. per second. In general it is sufficient if the area feft is not over one- sixth the area of the channel section where fe denotes the width of the notch and h the head of water on the crest. 4. To make the end contractions complete there must be a clearance of from 2ft to 3ft between each side of the notch and the corresponding side of the channel. 5. The head ft must be accurately measured. This is usually accom- plished by means of an instrument called B, hook gage (Fig. 54), located as explained below. For rough work, however, the head may be measured by a graduated rod or scale, set back of the weir at a distance not less than the length of the notch, with its zero on a level with the crest of the weir (Fig. 53). 68. Hook Gage. — ^As usually con- structed, the hook gage consists of a wooden or metal frame carrying in a groove a metallic sliding scale gradu- ated to feet and hundredths, which is raised and lowered by means of a milled head nut at the top (Fig. 54). By means of a vernier attached to the frame, the scale may be read to thousandths of a foot. The lower end of the frame carries a sharp-pointed brass hook, from which the instrument gets its name. In use, the hook gage is set up in the channel above the weir and leveled by means of a leveling instrument so that the scale Fia. 54. — Hook gage. Digitized by LnOOQlC 62 ELEMENTS OF HYDRAULIC^ reads zero when the point of the hook is at the exact level of the crest of the weir. The hook is then raised until its point just reaches the surface, causing a distortion in the reflection of Hght from the surface of the water. If slightly lowered the distortion disappears, thus indicating the surface level with precision. The reading of the vernier on the scale then gives the head on the crest to thousandths of a foot. To avoid surface oscillations, and thereby obtain more precise readings, the hook gage should be set up in a still box communi- cating with the channel. The channel end of the opening or pipe leading into the still box must be flush with a flat surface set parallel to the direction of flow, and the pipe itself must be normal to this direction. The channel end of the pipe must be set far enough above the weir to avoid the slope of the surface curve, but not so far as to increase the head by the natural slope of the stream. If the formula of any particular experimenter is to be used, his location for the still box should be duplicated. 69. Proportioning Weirs. — ^To illxistrate the method of propor- tioning a weir, suppose that the stream to be measured is 5^ ft. wide and 13^ ft. deep, and that its average velocity, determined by timing a float over a measured distance or by using a current meter or a Ktot tube (Pars. 149 and 153), is approximately 4 ft. per second. The flow is then approximately 1,980 cu. ft. per minute. To determine the size of weir which will flow approxi- mately this amount, try first a depth of say 10 in. From Table 17 it is found that each inch of length for this depth will deliver 12.64 cu. ft. per minute. The required length of weir would then 1 980 be TKgT = 156.6, which is fifteen and tworthirds times the depth and therefore too long by Rule 2 of the specifications. Since the weir must evidently be deeper, try 18 in. From 'the table the discharge per linear inch for this depth is 30.54 cu. ft. . per minute, and consequently the required length would be 1 980 or^KA = 64.8 in., which is now only 3.6 times the depth and therefore too short. By further trial it is found that a depth of 15 in. gives a length 1 980 of 2323 ~ ^^'^ ^'* which is 5.7 times the depth and therefore comes within the limits required by Rule 2. ^ Digitized by LnOOQlC FLOW OF WATER 63 Suppose then that the notch is made 7 ft. long and say 20 in. deep, so that the depth may be increased over the calculated amount if necessary. If then the width of the pond back of the weir is not 50 per cent, greater than the width of the notch, or if the velocity of flow should be in excess of 1 ft. per second, the pond should, if possible, be enlarged or deepened to give the desired result. With the weir so constructed suppose that the depth of water over the stake back of the weir is found to be 15^ in. From the table the discharge per linear inch corre- sponding to this head is found to be 23.52 cu. ft. per minute, and this multiplied by 84, the length of the* weir in inches, gives 1,975.7 cu. ft. per minute for the actual measured discharge. XDI. TIME REQUIRED FOR FILLING AND EMPTYING TANKS 70. Change m Level under Constant Head. — ^To find the time required to raise or lower the water level in a tank, reservoir, or ^^^^^^^^^ Fia. 55. lock, let A denote the area of the orifice through which the flow takes place and K its coefficient of discharge or efflux. Several simple cases will be considered. The simplest case is that in which the water level in a tank is raised, say from AB to CD (Fig. 56), by water flowing in under a constant head h. Let V denote the total volume of water flow- ing in, represented in cross-section by the area ABCD in the figure. Then since the discharge Q through the orifice per second is Q:^[KAV2^, Digitized by Google 64 ELEMENTS OF HYDRAULICS the time t in seconds required to raise the surface to the levd CD is V t ='^ = Q KAv^ (43) 71. Varjring Head. — It is often necessary to find the time re- quired to empty a tank or reservoir, or raise or lower its level a certain amount. A common case is that in which the level is' to be raised or lowered from AB to CD (Fig. 56) by flow through Level Lowered Fig. 56. a submerged orifice, the head on one side, EFy of the orifice being constant. If the cross-section of the tank is variable, let Y denote its area at any section mn. In the time dt the level changes from the height y to y — dy, and consequently the vol- ume changes by the amoimt dV = Ydy. But by considering the flow through the orifice, of area A, the volume of flow in the time dt is dV = KAV2gidt. Digitized by VnOOQlC FLOW OF WATER 65 Hence, by equating these values of dV, we obtain the relation KAV2gydt = Ydy whence t = zaV: 1 n Ydy (44) 72. Canal Lock. — A practical application of Eq. (44) is in find- ing the time required to fill or empty a canal lock. For an ordinary rectangular lock of breadth b and length Z, the cross- section is constant, namely Y = 6f, and consequently the ex- pression for the time integrates into rHjdy ^ bl\/2 ^2gji Vy t = bl KA\/2g KAVg (VH - Vh). (46) 73. Rise and Fall in Connected Tanks. — When one tank dis- charges into another without any additional supply from outside, Fig. 57. the level in one tank falls as that in the other rises. If both tanks are of constant cross-section, then when the level in one tank has been lowered a distance y, that in the other tank will have been raised a distance y' (Fig. 57), such that if M and N denote their sectional areas, respectively. My = Ny\ In the interval of time dt suppose y changes to y + dy. Then considering the flow through the orifice of area A, as in the pre- ceding case, we have Mdy = KA\^2g[H - {y + y')]dt, Digitized by VnOOQlC t = (46) 66 ELEMENTS OF HYDRAULICS or, since j/' = -j^, this may be written ^^ ^ MdyVN KAV2g[NH - y(M + N)] Simplifying this expression and integrating, the resulting expres- sion for the time t is found to be MVN r ^ dy ^ KAV2gl VNH - y{M + N) Substituting the given limits, the time t required to lower the level a distance D is ' = ^^[mTn{^^ - VNH - D(M + N) j]. When the level becomes the same in both tanks, since the volume discharged by one is received by the other, we have MD = N{H - D), or D = ^^^. M + N Substituting this value of D in Eq. (46), it becomes t = 2 mnVh_ KAV2g(M + N)' which is therefore the length of time required for the water in the tanks to reach a common level. 74. Mariotte's Flask. — It is sometimes desirable in measuring flow to keep the head constant. It is difficult to accomplish this by keeping the supply constant, a more convenient method being by the arrangement shown in Fig. 58, which is known as Mariotte's Flask. This consists of putting an air-tight cover on the tank, having a corked orifice holding a vertical pipe open to the atmos- phere. Since the pressure at the lower end A of the tube is always atmospheric, the flow is the same as though the water level was constantly maintained at this height. As water flows out, air enters through the tube and takes its place so that the effective pressure remains constant, the pressure of air in the Digitized by VnOOQlC FLOW OF WATER 67 tank and of the column of water above the lower end of the tube together being constantly equal to the pressure of one atmosphere. Therefore as long as the water level does not sink below the bot- tom of the pipe, the effective head on the orifice is its distance h below the bottom of the pipe, and the discharge is given by the formula Q = KA V2gh. (48) Fig. 58.-rMariotte's flask. XIV. FLOW THROUGH SHORT TUBES AND NOZZLES 75. Standard Mouthpiece. — When a short tube (adjutage, mouthpiece or nozzle) is added to an orifice, the flow through the opening is changed both in velocity and in amount. In general the velocity is diminished by the mouthpiece, due to increased frictional resistance, whereas the quantity discharged may be either increased or diminished, depending on the form of the mouthpiece. ' What is called the standard mouthpiece consists of a circular tube projecting outward from a circular orifice, and of length equal to two or three diameters of the orifice (Fig. 60). At the inner end of the tube the jet is contracted as in the case of a standard orifice, but farther out it expands and fills the tube. The velocity of the jet is reduced by this form of mouthpiece to V = 0.82\/2^ which is considerably less than for a standard orifice, but since there is no contraction, the quantity discharged is Q;= 0.82AV2gh, Digitized by VnOOQ iC 68 ELEMENTS OF HYDRAULICS where A denotes the area of the orifice. The discharge is there- fore nearly one-third larger than for a standard orifice of the same area with complete contraction (Fig. 59). 76. Stream-line Mouthpiece. — By rounding the inner edge of the mouthpiece so that its contour approximates the form of a stream line, the velocity of the jet is greatly increased, its value for the relative dimensions shown in Fig. 61 being about V = 0M\/2gh; and since the jet suffers no contraction, the quantity discharged is Q = 0MAV2ghy the area A, as before, referring to the area of the orifice. 77. Borda Mouthpiece. — ^A mouthpiece projecting inward and having a length of only half a diameter is called a Borda movih- piece (Fig. 62). The velocity is greatly increased by this form of mouthpiece, its value being about V = 0.99\/2^, but the contraction of the jet is more than for a standard orifice, so that the discharge is only Q = 0.53il\/2^, where A denotes the area of the orifice. If, however, the length of the mouthpiece is increased to two or three diameters (Fig. 63) the discharge is increased nearly 50 per cent., becoming Q = 0,72AV2gh. 78. Diverging Conical Mouthpiece. — For a conical diverging tube with sharp edge at entrance (Fig. 64) the jet contracts at the inner end as for an orifice, but farther on expands so as to fill the tube at outlet provided the angle of divergence is not over 8°. The discharge is therefore greater than for a standard mouth- piece, its amount referred to the area A at the smallest section being Q = 0,95AV2gh. 79. Venturi Adjutage. — If the entrance to a diverging conical mouthpiece has a stream-line contour, it is called a Venturi Digitized by VnOOQlC FLOW OF WATER 69 w a ■»■ gj » j" * *^ g ^ 4r «=•' Sharp Edffed Orifice Q=.64 Av «^62 J^sfloh ^^r^:^ i b?j^^ ! j ! j^;^;^^i^!ji^^ Standard Mouthpiece Area A Measured on Section AB Fia. 59. Fig. 60. =!- Streamtine Contour ■^>] ^[Area A Measured I Section AB Borda'i Moatitp iece Q-.64 Ay- ,^AVZgh Area A Measamd ~on Sectlen 1^-.^^ AB FiQ. 61. [FiQ. 62. itrant Tube \Q=Ay^.72^2gh Area A Measured on Section AB . ■g ggJ T j E -j - ie'^ Conical Diversing Tube Area A Measaied on Section AB Fig. 63. Venturi Adjutage Anjrle a = G^'to 8** Q=1.6 A\/2gh Area A Measured on A Bestion AB Conical Convererinflr Tube An^le a= 6° to 10* Q— .98 A\/2gh Area A Measured on Section AB i Fig. 65. Fig. 66. Digitized by VnOOQlC 70 ELEMENTS OF HYDRAULICS aSjiUage (Fig. 65). In experiments by Venturi and Eytelwein with diverging mouthpieces of the relative dimensions shown in Fig. 65; a discharge was obtained nearly two and one-half times as great as for a standard orifice of the same diameter as that at the smallest section, or about twice that for a standard short tube of this diameter, the formula for discharge referred to the area A at the smallest section being Q = 1.55AV2gh, 80. Converging Conical Mouthpiece. — ^In the case of a conical converging tube with sharp corners at entrance (Fig. 66) the jet contracts on entering and then expands again until it fills the Fire Hoee; Smooth G>ne Nozzle Q=.91AVTgh~ Fig. 67. Fire Hoee; Smooth Convex Nozzle Fig. 68. Fire Hose; Square "Ring Nozzle ^.74AV2^ Fig. 69. Fire Hose; Undercut Ring Nozzle Fig. 70. tube, the most contracted section being just beyond the tip, and the greatest discharge occurring for an angle of convergence of approximately 13®. 81. Fire Nozzles. — The fire nozzles shown in Figs. 67, 68, 69 and 70 are practical examples of converging mouthpieces. The smooth cone nozzle with gradually tapering bore has been found to be the most efficient, the coefficient of discharge for the best specimen being 0.977 with an average coefficient for this type of 0.97. For a square ring nozzle like that shown in Fig. 69 the coefficient of discharge is 0.74; and for the undercut type shown in Fig. 70 the coefficient of discharge is 0.71. Digitized by VnOOQlC FLOW OF WATER 71 XV. KINETIC PRESSURE IN A FLOWING LIQUID 82. Kinetic Pressure. — For a liquid at rest, the normal pres- sure exerted by it on any bounding surface is called the hydro- static pressure and is given by the expression deduced in Par. 8, namely, P = p' + yh. If a liquid is in motion, however, the normal pressure it exerts on the walls of the vessel containing it, or on the bounding sur- face of a liquid vein or filament, follows an entirely different law, as shown below. To distinguish the hydrostatic pressure from the normal pres- sure exerted on any bounding surface by a liquid in motion, the latter will be called the kinetic pressure. 83. Bernoulli's Theorem. — To determine the kinetic pressure at any point in a flowing liquid, consider a small tube or vein of the liquid bounded by stream lines, as explained in Par. 48, and follow the motion of the liquid through this tube for a brief interval of time. Let A and A' denote the areas of two normal cross-sections of the vein (Fig. 71). Then since the liquid is assumed to be in- compressible, the volume Ad displaced at one end of the tube must equal the volume A'd' displaced at the other end. If p denotes the average unit pressure on A, and p' on A', the work done by the pressure on the upper cap, A, is + pAd, and that on the lower cap. A', is - p'A'd\ the negative sign indicating that the element of work is of opposite sign to that at the other end of the vein. Also, if h denotes their difference in static head, as indicated in Fig. 71, the work done by gravity in the displacement of the volume Ad a distance h is yAdh. Digitized by VnOOQlC 72 ELEMENTS OF HYDRAULICS Since the forces acting on the lateral surface of the vein are normal to this surface they do no work. Assuming, then, the case of steady flow, that is to say, assuming that each particle arriving at a given cross-section experiences the same velocity and pressure as that experienced by the preceding particle at this point, so that the velocities v and t;' through the caps A and A' are constant, the change in kinetic energy between these two positions is Therefore, equating the total work done to the change in energy, the result is pAd - p'A'd' + yAdh = ^{v'^ - v^), or, since Ad = A'd\ this reduces to p' + ^* = P + ^' + 7h. (49) This result is known as Bernoulli's Theorem, and shows that in the case of steady parallel flow of an ideal liquid, an increase in velocity at any point is accompanied by a corresponding decrease in kinetic pressure, or vice versa, in accordance with the relation just obtained. 84. Kinetic Pressure Head. — If the theoretic heads corre- sponding to the velocities v and v' are denoted hy H and H\ respectively, then in accordance with Torricelli's theorem (Par. 50) we have J7 — . IJf ^ ^ ' "-2-g' " -2^' and consequently Eq. (49) may be written in the form p' = p -f- 7(h + H - HO, (50) which is a convenient form from which to compute the kinetic pressure at any given point. If this relation is written in the form L' + H' = 5 + H + h, (51) then since p/.y is the head corresponding to the hydrostatic pressure p, each term is a length, and Bernoulli's theorem may be expressed by saying that: In the case of steady, parallel flow of an ideal liquid, the sum of Digitized by VnOOQlC FLOW OF WATER 73 5=4^^:^-=^ yyyyy^yyyy^yy^yy^^y^^y^^^^^^ mP^^^^^^^' the pressure head, velocity head and potential head is a constant quantity for any particle throughout its course. 85. Application to Standard Mouthpiece. — An illustration of Bernoulli's theorem is afforded by the flow through a standard mouthpiece. At the con- tracted section A (Fig. 72) the velocity is evidently greater than at the outlet B. Therefore, by Ber- noulli's theorem, the kinetic pressure must be less at A than at B, Thus if a piezometer is inserted in the mouthpiece at A, the liquid in it will rise, shew- ing that the pressure in the jet at this point is less than atmospheric. It was found by Venturi, and can also be proved theoretically, that for a standard mouthpiece the negative pressure head at A is approximately three-fourths of the static head on the opening, or, referring to Fig. 72, hi = ^h. To prove this relation apply Bernoulli's equation between a point at the surface and one in the contracted section A. Then if p denotes the unit atmospheric pressure at the surface, we have ■f ^1 Fig. 72. Va' Pa + ^-f-;i = -7^--|-^ + + head lost at entrance. Assuming the coefficient of contraction at A as 0.64, we have Va X 0.64A = VbXA, or 1 Va = 0.64 Vb. 0.82 Since Vs = O.S2\/2gh, we have Va = ^rrwTy/2gh = 1.28\/2gh From Art. 98 we have head lost at entrance = (0072 "" ^) "y~"~ ^ 063 2g Digitized by LnOOQlC 74 ELEMENTS OF HYDRAULICS Substituting all these values in Bernoulli's equation, it becomes ^ + fc = (1.28)% + ^ + 0.063 (1.28)2A 7 7 whence ^ = £-0.741fc. 7 7 Consequently the negative pressure head at A is 0.74fc, or f^/i approximately. XVI. VENTURI METER 86. Principle of Operation. — The Venturi meter, invented by Clemens Herschel in 1887 for measuring flow in pipe lines, illustrates an important commercial application of Bernoulli's theorem. This device consists simply of two frustums of conical tubes with their small ends connected by a short cylindrical section, inserted in the pipe line through which the flow is to be measured (Fig. 73). If a pressure gage is inserted in the pipe Fig. 73. line at any point A and another at the throat of the meter JB, as indicated in the figure, it will be found that the pressure at B is less than at A. 87. Formula for Flow. — ^Let va and Vb denote the velocities at A and JB, ahd Pa and pa the. kinetic pressures at these points, respectively. Then since both points are under the same static head, Bernoulli's theorem, disregarding frictional losses, gives the relation ^ iPa ^Vb^ .Pb 2g ^ y 2g'^ y' If a and b denote the cross-sectional areas at A and B, the dis- charge Q is given by Q = avA = bvB whence Q Q Va--; VB-y Digitized by VnOOQlC FLOW OF WATER 75 n Chafftbef drarnbcp Pfpet H^giitttf-' Fig. 74. — Venturi meter and recording gage manufactured by the Builders Iron Foundry. Digitized by VnOOQlC 76 ELEMENTS OF HYDRAULICS Digitized by VjOOQIC FLOW OF WATER 77 Substituting these values of Va and Vb in the preceding equation and solving for Q, the result is ab « - ^^biv/? <- - -)■ If Ha and hs denote the static piezometer heads corresponding to the kinetic pressures Pa and psj respectively, this formula may be written ^ = :;;;^pV2g(hA - hB)- K^) Ordinarily the throat diameter in this type of meter is made one-third the diameter of the main pipe, in which case a = 96. If, then, h denotes the difference in piezometer head between the upstream end and the throat, the formula for discharge, ignoring frictional losses, becomes Q = 1.062b\/2^. (53) By experiment it has been found that ordinarily for all sizes of Venturi meters and actual velocities through them, the actual discharge through the meter is given by the empirical formula Q = (0.97 ± 0.03)b\/2^. (54) 88. Commercial Meter. — A typical arrangement of meter tube and recording apparatus is shown in Fig. 74, the lower dial indi- cating the tate of flow, and the upper dial making a continuous autographic record of this rate on a circular chart. 89. Catskill Aqueduct Meter. — The Venturi meter affords the most accurate method yet devised for measuring the flow in pipe lines. Fig. 75 shows one of the three large Venturi meters built on the line of the Catskill Aqueduct, which is part of tlie water supply system of the City of New York. Each of these meters is 410 ft. long and is built entirely of reinforced concrete except for the throat castings and piezometer ring, which are of cast bronze. Provision is also made in connection with the City aqueduct for the installation of a Venturi meter upon each connection between the aqueduct and the street distribution pipes. ' 90. Rate of Flow Controller. — Figure 76 illustrates a rate of flow controller operated by the difference in pressure in a Venturi tube. This apparatus is designed for use in a water pipe or conduit through which a constant discharge must be maintained regard- Digitized by vnOOQlC 78 ELEMENTS OF HYDRAULICS less of the head on the valve. It consists of a perfectly balanced valve operated by a diaphragm which is actuated by the differ- ence in pressure between the full and contracted sections of a Venturi tube. The valve and diaphragm are balanced by an adjustable counterweight, which when set for any required rate of flow will hold the valve discs in the proper position for that flow. Fia. 76.- -Venturi rate of flow controller manufactured by the Simplex Valve and Meter Co. XVII. FLOW OF WATER IN PIPES 91. Critical Velocity. — ^Innumerable experiments and investi- gations have been made to determine the laws governing the flow of water in pipes, but so far with only partial "success, as no gen- eral and universal law has yet been discovered. Experiments made by Professor Osborne Reynolds have shown that for a pipe of a given diameter there is a certain critical velocity, such that if the velocity of flow is less than this critical value, the flow proceeds in parallel filaments with true stream- line motion; whereas if this critical value is exceeded, the flow becomes turbulent, that is, broken by whirls and eddies and similar disturbances. The results of Professor Reynolds' experi- ments showed that at a temperature of 60°F. this critical velocity occurred when Dva = 0.02 where D denotes the diameter of the pipe in feet and Va is the average velocity of flow in feet per second. Digitized by VnOOQlC FLOW OF WATER 79 For parallel, or non-sinuous, flow it is possible to give a theoret- ical explanation of what occurs and deduce the mathematical law governing it, as shown below. No one, however, has yet ex- plained why the flow suddenly becomes turbulent at the critical velocity, or what law governs it subsequently. 92. Viscosity Coefficient. — ^The loss of energy accompanying pipe flow is due to the internal resistance arising from the viscosity of the liquid. This shear or drag between adjacent filaments is analogous to ordinary friction but follows entirely different laws.- Unlike friction between the surfaces of solids, fluid friction has been found by experiment to be dependent on the tempera- ture and the nature of the liquid; independent of the pressure; and, for ordinary velocities at least, approximately proportional to the difference in velocity between adjacent filaments. When this difference in velocity disappears, the f rictional resistance also disappears. The constant of proportionahty required to give a definite numerical value to fluid friction is called the viscosity coefficient and will be denoted by fi. This coefficient /i is an empirical con- stant determined by experiment, the values tabulated below being the result of experiments made by 0. E. Meyer. Temperature in degrees Fahr. 50* 60* 65* 70* Viscosity coefficient /i in lb. sec. ft.* 32 X 10-« 28 X 10-« 26 X 10-« 24 X 10-« The dimensions of /* are, of course, such as to make the equa- tion in which it appears homogeneous in the units involved, as will appear in what follows. 93. Parallel (non-sinuous) Flow.^ — Consider non-sinuous flow in a straight pipe of uniform circular cross-section, that is, at a velocity less than the critical velocity and therefore such that the filaments or stream lines are all parallel" to the axis of the pipe. By reason of symmetry the velocity of any par- ticle depends only on its distance from the axis of the pipe. Let V denote the velocity of any particle and x its distance from the center (Fig. 77). Then if x changes by an amount * The following derivation is substantially that given by FOppl in his "Dynamik." Digitized by VnOOQlC 80 ELEMENTS OF HYDRAULICS dx, the velocity changes by a corresponding amount dVj and since the velocity is least near the pipe walls, v decreases as x increases and consequently the rato -r- is negative. Since the pipe is assumed to be of constant cross-section and the flow uniform and parallel, the forces acting on any element of volume must be in equilibrium. Considering therefore a small water cyUnder of radius x and length dt/, in order to equiUbrate the frictional resistance acting on the convex surface of this cyUn- der there must be a difference in pressure on its ends. This ex- plains the fall in pressure along a pipe, well known by experiment. Fig. 77. Let dp denote the difference in pressure in a length dy. Then the difference in pressure on the ends of a cyhnder of radius x is {Trx^)d'p and the shear on its convex surface is {2jrxdy)fA^' dv Equating these two forces and remembering that -r- is negative, we obtain the relation (7rx2)dp = - {2Txdy)tJL^' Also, since the difference in pressure on the ends of any cyUnder is proportional to its length, we have dp _ pi - p2 _ , dy - I " ^' where pi and p2 denote the unit pressures at two sections at a distance I apart, and the constant ratio is denoted by k for convenience. Substituting dp = kdy from the second equation in the first and cancelling common factors, we have finally dvM_ k^ dx^~ 2/' Digitized by LnOOQlC FWW OF WATER 81 whence, by integration, fee* , where c denotes a constant of integration. To determine c assume that the frictional resistance between the pipe wall and the liquid follows the same law as that between adjacent filaments of the liquid. Then it follows that the hquid in contact with the pipe must have zero velocity, as otherwise it would experience an infinite resistance. This seems also to be confirmed by the experiments of Professor Hele-Shaw, who showed that in the case of turbulent flow there was always a thin fiilm of Uquid adjoining the pipe walls which showed true stream-line motion, proving that its velocity was certainly less than the critical velocity and therefore small. Furthermore, the walls of commercial pipes are comparatively rough and conse- quently a thin sldn or layer of liquid must be caught in these roughnesses and held practically stationary. Assuming then that t; = when x = r, and substituting this pair of simultaneous values in the above equation, the value of c is found to be 4m c = and consequently = |;(r»-x»). (56) This is the equation of a parabola, and therefore the velocity dia- gram is a parabolic arc with its vertex in the axis of the pipe; that is, the velocity is a maximum at the center where a; = 0, its value being ^-' = 4^7- 94. Average Velocity of Blow in Small Pipes. — Let the dis- charge through any cross-section of the pipe be denoted by Q. Then if the velocity at any radius x is denoted by v, we have -X or, since Q = I 2Trxdxv, , = |^(r« - »*), Digitized by LnOOQlC 82 ELEMENTS OF HYDRAULICS this becomes irfcr* '>-W."^-''''"t But if Va denotes the average velocity of flow we also have Q = Vaiirr^) whence by substituting the above value for Q, we have ^'^ -KT^ 7rr2 8^- Comparing this expression with that previously obtained for the maximum velocity, it is evident that the maximum velocity is twice the average velocity of flow. 96. Loss of Head in Small Pipes. — ^The loss in pressure in a length I is given by the relation obtained above, namely, or, if the difference in head corresponding to this difference in pressure is denoted by 7i, then, since p = yh, we have loss in head, h = — — = — 7 7 Substituting in this relation the value of k in terms of the average velocity of flow, the result is For small pipes, therefore, the loss of head is proportional to the first power of the average velocity, and inversely proportional to the square of the diameter of the pipe. This result has been verified experimentally for small pipes by the experiments carried out by Poiseuille. 96. Ordinary Pipe Flow. — Under the conditions usually found in practice the velocity of flow exceeds the critical velocity and consequently the flow is turbulent and a greater amoimt of energy is dissipated in overcoming internal resistance than in the case of parallel flow. The result of Professor Rejoiolds' experiments indicated that the loss of head in turbulent flow was given by the relation , Va'-'H hoz — z— . Digitized by VnOOQlC FLOW OF WATER 83 In commercial pipes the degree of roughness is a variable and un- certain quantity, so that the exact loss of head cannot be pre- dicted with accuracy. Practical experiments have shown, how- ever, that ordinarily the loss in head is proportional to the square of the average velocity, so that the relation becomes a Since the theoretical head corresponding to a velocity v is A = ^, the expression for the loss in head for a circular pipe nmning full may in general be written 2gd or, denoting the constant of proportionality by/, this becomes Here /is an empirical constant, depending on the condition of the inner surface of the pipe, and is determined by experiment. Eq. (57) is identical with Chezy's well-known formula as will be shown in Par. 118. XVm. PRACTICAL FORMULAS FOR LOSS OF HEAD IN PIPE FLOW 97. Eflfective and Lost Head. — ^In the case of steady flow through long pipes, much of the available pressure head disap- pears in frictional and other losses, so that the velocity is greatly diminished. Thus if h denotes the static head at the outlet and hi the head lost in overcoming frictional and other resistances to flow, the velocity v at the outlet is given by the relation or its equivalent, The lost head h is the sum of a number of terms, which will be considered separately. Digitized by VnOOQlC 84 ELEMENTS OF HYDRAULICS 98. Loss at Entrance. — ^A certain amount of head is lost at the entrance to the pipe, as in the case of a standard adjutage. If V denotes the velocity ilue to the head h with no losses, then h - ^' whereas if Va denotes the actual velocity of flow the head corre- sponding to this velocity is h' = ^*. • 2g The head, hi, lost at entrance, is therefore h, = h-h' = '^-'/. If Cv denotes the velocity coefficient for the entrance, then Va = CvV, and consequently the expression for the head lost at entrance may be written ^' 2g 2g 2g 2g ^2^(c?"^^)' 2g For the standard short tube C„ = 0.82 (Par. 75) and therefore ^ — 1 = .^ ^o\2 — 1 = 0.5 The head lost at entrance is therefore hi = 0.6g. (69) If the pipe projects into the reservoir, C = 0.72 (Par. 77), and the head lost at entrance is thereby increased to hi = 0.93 ^. ^g For ordinary service taps on water mains it may be assumed as A. = 0.62|. 99. Friction Loss. — ^In flow through long pipes the greatest loss in head is that due to the friction between the liquid and the Digitized by VnOOQlC FLOW OF WATER 86 walls of the pipe. Until recently this loss in head was assumed to be given exactly by the formula h, = f|iXg, (60) where d = internal diameter of pipe; I = length of pipe; / = empirical constant determined by experiment. Values of the constant / as determined by experiment for various kinds of pipe are given in Table 12. Average values ordinarily assumed for cast-iron pipe are: for new smooth pipes f - 0.024, \ /g^x for old rusty pipes f = 0.03. / ^ ^ More recent and accurate experiments have shown, however, that the loss in head due to pipe friction does not vary exactly as the square of the velocity nor inversely as the first power of the diameter, the results of such experiments leading to what is known as the exponential formula. This formula is of the form h2 = mj, (62) in which ^2 denotes the loss in head for a given length of pipe, say, 1,000 ft., thereby eUminating the length I, and m is an empirical constant which replaces the combination d- in the older formula. I'he essential diBFerence between the two formu- las consists in the fact that the exponent z is not 2 but varies between 1.7 and 2, and the exponent z is not unity but is ap- proximately 1.25 for all kinds of pipe. 100. Wood Stave Pipe. — From accurate and comprehensive experiments made by Moritz^ on wood stave pipe, ranging from 4 in. to 55^ in. in diameter, it was found that the formula in this case should read h2 = 0.38^ (63) where hi = friction head in feet per 1,000 ft. of pipe; V = mean velocity of flow in feet per second; d = diameter of pipe in feet. ^ E. A. MoRiTZ, Assoc. M. Am. Soc. C. E., Engineer U. S. Reclamation Service, "Experiments on the Flow of Water in Wood Stave Pipe," Trans. Am. Soc. C. E., vol. 74, pp. 411-482. Digitized by VnOOQlC 86 ELEMENTS OF HYDRAULICS Solving this relation for v, we have V = 1.72d«ni2^"» (64) and consequently the discharge Q in cubic feet per second is Q = 1.36d« W "* (66) From these formulas, the velocity and discharge have been calculated by Moritz^ for pipes ranging from 6 to 120 in. in diame- ter and for a large number of hydraulic slopes, and the results are summarized in Table 16 at the end of this book. If it is necessary to find the discharge or velocity for slopes not given in Table 16, it wiU in general be sufficiently accurate to interpolate between the next lower and next higher slopes. Greater accuracy may be obtained by calculating the value of h^o.hii Qj^^ multiplying it by the value of l.SbcP'^ given in column 4 of the table, which has been inserted especially for this purpose. 101. Graphical Solution. — ^Any exponential formula is par- ticularly adapted to graphical solution by plotting on logarithmic paper, as in this case the exponential curves are transformed into straight lines and therefore require the plotting of only two points to determine each. Such a logarithmic diagram has been prepared by Moritz for his formula for wood stave pipe, and is shown in Kg. 78. On this diagram all the essential factors, namely diameter, area, velocity, discharge and friction head, are shown at a glance. As a numerical example, the diagram shows that a 16-in. pipe has an area of 1.4 sq. ft., and with a velocity of 3 ft. per second will discharge 4.2 cu. ft. per second with a loss of head of 1.9 ft. per 1,000 ft. length of pipe. As another instance of the use of the diagram, suppose that it is required to obtain a discharge of 3 cu. ft. per sec. with an avail- able fall of 2 ft. per 1,000. Starting from the bottom scale, follow the vertical line for H = 2 to its intersection with the horizontal line representing Q = 3. Since the point of inter- section lies almost on the 14-in. pipe line, it shows that this size pipe is required. Interpolating between the lines representing velocities of 2 ft. per second and 3 ft. per second, the velocity of flow is found to be 2.8 ft. per second. 102. Cast-iron Pipe. — Many experiments have been made to determine the friction loss in cast-iron pipes, and a number of •.formulas have been proposed. The most reliable of these results 1 Eng, Record, vol. 68, No. 24, Dec. 13, 1913. Digitized by VnOOQlC FLOW OF WATER 87 100,0 I?-^!^W.J'-Vf"^*^ Loss of Head in Feet per 1000 Fia 78. — Graphical solution of exponential formula for flow of water in wood stave pipe. {MorUz,) Digitized by vnOOQlC 88 ELEMENTS OF HYDRAULICS are the experimental coefficients C detennined by Smith for use in the Chezy formula V = CVrSf and the exponential formula proposed by Gardner S. Williams, which has the form V = eT.Tdo^'V"* (66) From a careful investigation of available data and a comparison of the results of 16 formulas, including the two just given, Moritz^ obtained the exponential formula V = 77do-7sO"5' (67) whence Q = 1.31d2TiO"5 (68) This formula differs from that for wood stave pipe only in the value of the constant coefficient. The velocity and discharge for cast-iron pipe may therefore be obtained from Table 16 for wood stave pipe by simply multiplying the results there given 1.31 by the ratio of these coefficients, namely, j-^ = 0.97. 103. Deterioration with Age. — ^The above formula appUes'to new cast-iron pipe with smooth alignment and profile. To make allowance for deterioration with age, Moritz, adopts the assump- tion made by Williams and Hazen in their hydrauUc tables, namely, that the friction head increases 3 per cent, per year due to tuberculation, and that the diameter decreases 0.01 in. per year from the same cause. Applying these assumptions to the equation Q = 1.31(P-7Ao,565^ ^nd denoting by K the ratio of discharge after n years of service to the discharge when new, we have [n n2.7 r 1 -10.566 For example, this equation shows that a 12-in. pipe 10 years old will carry only 85 per cent, as much as the same pipe when new. 104. Riveted Steel Pipe. — Experiments on riveted steel pipes are too few to make an accurate formula possible. The formula proposed by Moritz for new asphalted pipes having smooth alignment and profile is Q = 1.18d2 Tio "5 (70) which gives results about 10 per cent, less than the formula for cast-iron pipe. * Eng. Record, Dec. 13, 1913. Digitized by VnOOQlC FLOW OF WATER 89 The discharge and velocity of flow for riveted steel pipes may be obtained from Table 16 for wood-stave pipes by multiplying the results there given by the ratio of the constants, namely, 106. Concrete Pipe. — Concrete pipe is manufactured in various ways and by different methods: namely, the dry-mix pipe which is built in short sections and laid Uke clay sewer pipe; the wet- mix pipe, also built and laid in short sections; and the wet-mix pipe built continuously in the trench. In the absence of reliable experiments with concrete pipe, Moritz assumes that the dis- charge may be closely approximated by classing the dry-mix pipe with riveted steel pipe; the wet-mix built in short sections with cast-iron pipe; and the continuous wet-mix pipe with wood stave pipe. 106. Bends and Elbows. — ^Bends and elbows in a pipe also greatly diminish the effective head. Until recently the formulas _>jLgg_^ — x_ Fig. 79. obtained by Weisbach from experiments on small pipe were generally accepted for lack of better authority. According to Weisbach the lost head due to a sharp elbow of angle a (Fig. 79) is given by the formula h3 = mg. (71) where w is a function of the angle a, given by the equation m = 0.9457 sin^ (|) + 2.047 sin* (|) • Values of m, calculated from this formula for various values of the angle a, are tabulated as follows: a = 1 20'' 30*' 1 40° 50° 60° 70° 1 80° 90° m = 1 .046 .073 .139 .234 .364 .533 .740 .984 Digitized by VnOOQlC 90 ELEMENTS OF HYDRAULICS For a curved elbow of radius 22 and central angle a (Fig. 80) Weisbach's formula for lost head is *»="©£ (72) d\»» 2g' where the coefficient n has the value n = 0.131 +0.163(1) Values of n calculated from this formula for various values of the ratio ^ are tabulated below for convenience in substitution. d .2 .8 .4 .6 .6 .8 1.0 1.2 1.26 1.3 1.4 1.6 1.8 2.0 It- |.13li.l33|.138|.145|.168|.206| .294| .440| .487| .639| .66l| .977| 1.40| 1.98| Weisbach's formula if it was of general application would imply that the greatest loss of head occurs in bends of smallest radius, and conversely, as the radius of the bend increases, the loss in head diminishes. Experiments made by Williams, Hubbell and Fenkell^ at Detroit on pipes of 12, 16 and 30 in. diameter, however, indicated that the loss of head is a minimum for bends with radii of about two and one-half times the diameter of the pipe. Further ex- periments made by Schoder* at Cornell on 6-in. pipe; by Bright- more* in England on 3- and 4-in. pipe; and by Davis* and Balch* at the University of Wisconsin on 2- and 3-in. pipes have shown that the Weisbach formula is not valid for larger ^Gardner S. Williams, Clarence W. Hubbell and George H. Fenkbll, "Experiments at Detroit, Mich., on the Effect of Curvature upon the Flow of Water in Pipes," Trans, Am. Soc. C. E., vol. 47. •Ernest W. Schodbr, "Curve Resistance in Water Pipes," Trans, Am. Soc. C. E., vol. 62. »A. W. Brightmore, "Loss of Pressure in Water Flowing Through Straight and Curved Pipes," Minutes of Proc. Inst. C. E., vol. 169, p. 323. ♦George Jacob Davis, Jr., "Investigation of Hydraulic Curve Resist- ance. Experiments with 2-in. Pipe," BuU, Univ. of Wis., No. 403, January, 1911. *L. R. Balgh under direction of George Jacob Davis, Jr., "Investiga- tion of Hydraulic Curve Resistance. Experiments with 3-in. Pipe, BvU, Univ. of Wis., No. 578, 1913. Digitized by VnOOQlC FLOW OF WATER 91 pipes under ordinary conditions of service. The Wisconsin ex- periments, however, did not confirm the Detroit experiments as to the minimum loss of head occurring with bends of two and one-half pipe diameters. The conclusions reached from the Wisconsin experiments were as follows: (a) The total loss of heads in bends decreases with an increase in radius until the radius of the bend equals about four pipe diameters. For bends with radii greater than five pipe diame- ters, the total loss in head increases with an increase in radius. (6) The constant / in the relation as applied to curve resistance is independent of the diameter of the pipe, but varies inversely as the radius of the bend. (c) The net curve resistance, that is, the loss of head due to the bend alone and not including pipe friction, decreases to a mini- mum for a radius of bend equal to about six pipe diameters, then increases until the radius is about 14 pipe diameters, after which it again decreases. (d) The net curve resistance per unit length of bend is inde- pendent of the diameter of the pipe; decreases with an increase in the radius of curvature; and varies approximately as the square of the velocity of flow. It has been pointed out by W. E. Fuller, ^ Consulting Engineer of New York City, that in all these experiments it was assumed that the loss of head in bends in different sizes of pipes should be the same when the radius of the bend in terms of the diameter of the pipe were alike, whereas with so many different factors contributing to the loss, there seems to be no adequate grounds for assuming such a relation to exist. From a careful. comparison of all available data. Fuller found that the loss is more nearly the same for different sizes of pipes with bends of the same actual radius than for bends of the same radius in pipe diameters. From this comparison the formula for loss of head was found to be hs = kv2-26 (73) ^ W. E. Fuller, "Loss of Head in Bends," Jour. New Eng. Water Works Assoc, Vol. 27, No. 4, December, 1913. Digitized by LnOOQlC 92 ELEMENTS OF HYDRAULICS where hi — loss of head in excess of the loss for an equal length I lot straight pipe; Jfc = coefficient depending on the radius of the bend; V = velocity of flow in feet per second. Values of the coefficient k are given in Fig. 81. .016 .014 •012 •:£ .010 |.006 > .006 .004 • .002 — ,/ ^ L_ / 10 15 40 46 50 55 20 26 80 35 Radius of Bend in Ft. Value of coeflBcient k in formula for curve resistance, hfln**^ FlQ. 81. The following table shows the loss of head for ordinary 90** bends of the New England Water Works Association standard. Loss OF Head Due to 90® Bends of the New England Water Works " Association Standard Size of pipe, inches Radius of bend, feet Excess loss over loss in straight pipe of length equal to tangents » = 3 ft. /sec. V = 5 ft./sec. »=10ft./sec- 4 1.33 0.021 0.073 0.37 6 1.33 0.025 0.082 0.40 8 1.33 0.026 0.086 0.41 10 1.33 0.027 0.089 0.42 12 1.33 0.028 0.090 0.43 16 2.0 0.026 0.085 0.41 20 2.0 0.027 0.086 0.41 24 2.5 0.026 0.085 0.41 30 3.0 0.026 0.083 0.41 36 4.0 0.026 0.083 0.40 For bends less than 90**, the use of the following values for loss of head is recommended: Digitized by VnOOQlC FLOW OF WATER 93 For 45** bends, use three-fourths of that due to 90° bends of same radius. For 223^** bends, use one-half of that due to 90** bends of same radius. For a F-branch, use three-fourths of that due to a tee. For velocities of 3 to 6 ft. per second the loss of head in bends is approximately proportional to the velocity head, and for rough approrimations the following rules may be used: For 90** bends of radius greater than 1 J^ ft. and less than 10 ft., A, = 0.25 ^• For tees, that is, bends of zero radius, ;i. = 1.25|- For sharp 90** bends of 6-in. radius A, = 0.5 2g 107. Enlargement of Section. — ^A sudden enlargement in the cross-section of a pipe decreases the velocity of flow and causes a loss of head due to eddying in the corners, etc. (Fig. 82). If the I mmmmmmmmmm/ 1^^^^^^=-- = - ^^2 Fig. 82. velocity is decreased by the enlargement from V\ to v^^ it has been found by experiment, and can also be proved theoretically, 4;hat the head lost in this way is given by the formula h4 = (Vl - V2)^ 2g (74) To prove this relation let A\ denote the smaller area of diameter d (Fig. 82) and A^ the larger area of diameter JD, so that from the law of continuity of flow we have Axvx = A%^)%. Digitized by LnOOQlC 94 ELEMENTS OF HYDRAULICS Applying Bernoulli's equation to points on opposite sides of the change in section^ we have I" +?- 1' +?+'""'-<'■ whence lost head = ^' - ^V ^i-^:^^ Since the dijBFerence in pressure is a force of amount P2A2 — P1A2, we have by the principle of impulse and momentum PzAi - P1A2 = "T ('^i " ^i) V where W = v^A^j and consequently this reduces to Substituting this in the expression for lost head it becomes Io8thead = |*-|*-^*(.x-r,) _ (»i - fs)' -—2r~' ^»^^<^<^<^^^sg<9g^^t^»t^<<^g^^?^^^^<^^s^^^g^<^^<^t^^^^^^^ i ^^"^.^ 1 __^ -^ C^ ^mvmv^^^^^^^^^^ r^^mW^M'M'MMMW^^: > ; tl^v^^\vvvvv^vvv\^v^^^^^ Fia. 83. To obtain a more convenient expression for hi, let a denote the area of cross-section of the smaller pipe and A of the larger. Then Via = V2A, whence V2A and consequently the expression for A4 may be written ■.-|■(^»)■ m 108. Contraction of Section. — ^A sudden contraction in section also causes a loss in head, similar to that due to a standard orifice Digitized by Google FLOW OF WATER 95 or adjutage (Fig. 83). The lost head in this case has been found by experiment to be given by the equation h5 = q^. (76) where q denotes an empirical constant, determined experimen- tally. The following tabulated values of the coefficient q are based on experiments by Weisbach, A being the cross-sectional area of the larger pipe and a of the smaller. ^^ a A .1 .2 .3 A .6 .6 .7 .8 .9 1.0 « .362 .338 .308 .267 .221 .164 .105 .053 .0151.000 109. Gate Valve in Circular Pipe.— The loss in head due to a partly closed gate valve (Fig. 84) has been determined by experiment for different ratios of height of opening to diameter of pipe with the following results.^ In this Table, x denotes the height of the opening, d the diameter of the pipe, ^6 the loss in head and f the empirical coefficient in the formula Ae = f o"' Fio. 84. X d i i \ J f i \ r 97.8 17.0 5.52 2.06 0.81 0.26 Q.07 Fig. 85. 110. Cock in Circular Pipe.— For a cock in a cylindrical pipe (Fig* 85) the coefficient f has been de- termined in terms of the angle of closure with the following results. e f 5° |io*» 15^ 20'' 25** 30*" 35** 40** 45** 50« 55« eo'' 65** 82* .05 .29 .75 1.56 3.1 5.47 9.68 17.3 31.2 52.6 106 206 486 Valve closed ^HosKiNS, "rXext-book^on^Hydraulics/'^p. 74. * The coefficient for losses at valves are based on experiments by Weis- bach and are given in most standard texts on ''Hydraulics." See for example Wittbnbaxjbr, ''Aufgabensammlung/' Bd. Ill, S. 318; Gibson, "Hydraulics and its Applications," pp. 249, 250. Digitized by Google 96 ELEMENTS OF HYDRAULICS / 111. Throttle Valve in Circular Pipe. A — ^The coefficient f in the formula h^ = for a throttle valve of the butterfly ' type (Fig. 86) for various angles of Fig. 86. closure with results as follows: / 1 5» 1 10° 20° 1 30° 40° 45° 50° 60° 70° f 1 0.21 1 0.52 1.54 3.91 10.80 18.70 32.6 118 751 112. Summary of Losses. — ^The total head, hi, lost in flow through a pipe line is then the sum of the six partial losses in head mentioned above, namely, hi = hi + h2 + hi + hi + hs + Ae- The values of these six terms may be tabulated as follows: Loss of head in pipe flow | Head lost at entrance ».-o.»| Coefficient modified by nature of entrance and varies from 0.5 to 0.9 Friction head ht-mj. Table 12 Head lost at bends and el- bows h, - *»»•«• Head lost at sudden enlarge- ment Head lost at sudden con- traction For values of coeffi- cient, see table, p. 95 Head lost at partially closed valve '•-^S See tabular values of f, pages 95 and 96 From Eq. (58) we have h=\^^+ht = ^^+hi + hi + h + Jn + hi + ft,, Digitized by VnOOQlC FLOW OF WATER 97 For a short straight pipe, where by a short pipe is meant one for I _ which 2 < 4,000, the loss in head is simply h^'^ + h + h and inserting in this the values of hj and h^ given, above, we have ^l + 0.6+f(J) Combining this with the relation Q = At; = -j- and solving for d, we have as the diameter required to furnish a given dis- charge Q, d = 0.4789[ (l.6d + fl) ^*]^ (77) which is best solved by trial. For a long straight pipe, where -j > 4,000, all other losses may be neglected in comparison with friction loss, in which case the above formula simplifies into d = 0.4789 (^')K (78) 113. Application. — To give a simple illustration of the applica- tion of the formula, suppose it is required to find the velocity of flow for a straight new cast-iron pipe, 1 ft. in diameter and 5,000 ft. long, with no valve obstructions, which conducts water from a reservoir the surface of which is 150 ft. above the outlet of the pipe. In this case ^l+0.5+/y) -^1+0.5 + 0.024 (^^) per sec. and the discharge is Q = At; = ^ X 8.9 X 60 = 419.4 cu. ft. per min. XDL HYDRAULIC GRADIBlf T 114. Kinetic Pressure Head. — ^In the case of steady flow through a long pipe, if open piezometer tubes are inserted at different points of its length and at right angles to the pipe, the height Digitized by VnOOQlC 98 ELEMENTS OF HYDRAULICS at which the water stands in any tube represents the kinetic pressure head at this point. Assuming that the pipe is straight and of uniform cross-section, the velocity head is constant through- out, and therefore as the frictional head increases the pressure head decreases. The head lo^t in friction between any two points m and n (Fig. 87) as given by Eq. 60 Par. 99 is h "■•^U2? 2g' and is therefore proportional to the distance I between these points. Consequently, the drop in the piezometer column be- tween any two points is proportional to their distance apart, and therefore the tops of these columns must lie in a straight line. Fig. 87. This line is called the hydraulic gradient^ or virtual slope of the pipe. Evidently the vertical ordinate between any point in the pipe and the hydraulic gradient measures the kinetic pressure head at the point in question. 116. Slope of Hydraulic Gradient. — When a pipe is not straight, successive points on the hydraulic gradient may be determined by computing the loss of head between these points from the relation taking as successive values of I the length of pipe between the points considered. In water mains the vertical curvature of the pipe line is gen- erally small, and its effect on the hydraulic gradient is usually neglected. When, however, a valve or other obstruction occurs in a pipe there is a sudden drop in the hydraulic gradient at the obstruction, due to the loss of head caused by it. It should be noted that the upper end of the hydraulic gradient Digitized by VnOOQlC FLOW OF WATER 99 lies below the water level in the reservoir a distance equal to the head lost at entrance plus the velocity head. The slope of the hydraulic gradient is usually defined, however, as oi ^ 1. J T J' X static head Slope of hydraulic gradient = length of pipe ' which is equivalent to neglecting the velocity head and head lost at entrance, thereby making the assumed hydraulic gradient slightly steeper than it actually is. 116. Peaks above Hydraulic Gradient. — When part of the pipe line rises above the hydraulic gradient (Fig. 88), the pressure in this portion must be less than atmospheric since the pressure Fig. 88. head V becomes negative. If the pipe is air-tight and filled be- fore the flow is started this will not affect the discharge. If the pipe is not air-tight, air will collect at the summit above the hydraulic gradient, changing the slope of the latter from AB to AC as indicated in Fig. 88, thereby reducing the head to V with a corresponding diminution of the flow. Before laying a long pipe line the hydraulic gradient should therefore be plotted on the profile to make sure there are no summits projecting above the gradient. In case such summits are imavoidable, provision should be made for exhausting the air which may collect at these points, so as to maintain full flow. XX. HYDRAULIC RADIUS 117. Definition of Hydraulic Radius. — That part of the boundary of the cross-section of a channel or pipe which is in contact with the water in it is called the wetted perimeter, and the area of the cross-section of the stream divided by the wetted perimeter is called the hydraulic radius, or hydraulic mean depth. In what follows the hydraulic radius will be denoted by r, defined as TT J 1. J. Area of flow ,_^v Hydraulic radius, r = Netted perimeter <^»> Digitized by VnOOQlC 100 ELEMENTS OF HYDRAULICS Some writers apply the term hydraulic radius only to circular pipes, and use the term hydraulic mean depth for flow in channels. For a channel of rectangular cross-section having a breadth b and depth of water h, the hydraulic radius is ^ ' b + 2h In a circular pipe of diameter d, running full, the hydraulic radius is For the same pipe running half full, ird^ -A ^ ^ " Td "4' ~2 and is therefore the same as when the pipe is full. Other examples of the hydraulic radius are shown in Figs. 108 to 114. 118. Chezy's Formula for Pipe Flow. — The formula proposed by Chezy for the velocity of flow in a long pipe is V = CVrs, (80) where s denotes the slope of the hydraulic gradient, defined in the preceding article; r is the hydraulic radius, defined above; and C is an empirical constant which depends on the velocity of flow, diameter of pipe, and roughness of its lining. For a circular pipe flowing full Chezy's formula is identical with the formula for friction loss in a pipe, given by Eq. (60), Par. 99, namely. To show this identity, substitute in Cheasy's formula the values d , h r = -7 and s = t* 4 { n 1^ Then it becomes (dh ITI' whence, by squaring and solving for A, it takes the form Digitized by VnOOQlC FLOW OF WATER 101 C!onsequently, if the constant term ^ is denoted by /, that is, Chezy's formula assumes the standard form, Chezy's formula also applies to flow in open channels, as ex- plained in Art. XXV. 119. Kutter and Bazin Formulas for Pipe Flow. — ^The use of I v^ Weisbach's formula '^ = /^ ' o^* ^ i^^ shown, is equivalent to using Chezy's formula v = Cy/ri under the assumption that the coefficient C is constant. It has been found by experiment, however, that the coefficient C in Chezy's formula is not strictly constant for any particular pipe or channel, nor dependent only on the roughness of the pipe or channel lining, but that it also varies with the slope and the hydraulic radius. Expressions for C in terms of these variables have been proposed by various engineers, the two formulas most widely used being those due to Kutter and to Bazin, given in Pars. 140 and 142. Although Kutter's and Bazin's formulas were intended primarily to apply to flow in open channels, they are now also used extensively for calculating flow in pipes and conduits. Kutter's and Bazin's values of Chezy's coefficient are also given and tabulated in Tables 14 and 15. 120. Williams and Hazen's Exponential Formula. — ^The varia- tion in Chezy's coefficient may also be taken into account by writing Chezy's formula in the exponential form V = Cr*s», (81) in which the exponents m and n as well as the coefficient C depend on the roughness of the channel lining, and to a certain extent on the form of the channel. An exponential formula of this type is more flexible as well as simpler than the Kutter and Bazin formulas, and is coming to be generally accepted as the standard type for calculating flow in pipes as well as in open channels. At present the most generally used formula of this type is that due to Williams and Hazen, namely, V = Cro "s« «H).001-« ?* (82) Digitized by VnOOQlC 102 ELEMENTS OF HYDRAULICS where v = velocity of flow in feet per second; 8 = slope; r = hydraulic radius in feet; C = 100. To obviate the inconvenience of using this formula, Williams and Hazen have prepared extensive hydraulic tables, as well as a special slide rule. A brief table giving friction head in pipes computed by means of this formula is given at the end of this volume as Table 13. XXI. DIVIDED FLOW 121, Compound Pipes. — In water works calculations the prob- lem often arises of determining the flow through a compound system of branching mains. WJ?WJWM??WW/^J. Profile M ^^^^^^m U ^1 B V2 S\ h •iiii^i^^^i^^^ii^^^^^^ti^ HiD Vi V-i N Vi Plan Fig. 89. To illustrate the method of finding the discharge through the various branches, consider first the simple case of a main tapped by a branch pipe which later returns to the main, as indictated in Fig. 89. The solution in this case is based on the fimdamental relation deduced in Par. 97, namely, Digitized by VnOOQlC FLOW OF WATER r03 where h denotes the static head, and h the head lost in friction. Using the notation indicated on the figure and considering the two branches separately, we obtain the following equations: For line ABMCD, trance + resistance at F-branch. (83) For line ABNCD, trance + resistance at F-branch. (84) By subtraction of these two equations we have ^(Dl'=4:)l'. <") which shows that the frictional head lost in the branch BMC is equal to that lost in BNC. Since the total discharge through the branches is the same as that through the main before dividing and after uniting, we also have the two relations aiOi = a2V2 + a^vz = 04^4. (86) By assuming an average value for the frictional coefficient /, the four equations 83, 84, 85 and 86 may then be solved for the four unknowns Vi, V2, Vz, Va. Having found approximate values of the velocities, corresponding values of / may be sub- stituted in these equations and the solution repeated, thus giving more accurate values of the velocities. Having found the velocities, the discharge through the various pipes may be obtained from the relations Qi = Q4 = aivi = 04^4; Q2 = a2V2; Qz = asv». The solution for more complicated cases is identical with the above, except that more equations are involved. 122. Branching Pipes. — ^Another simple case of divided flow which is often met is that in which a pipe AB of diameter d divides at some point B into two other pipes, BC and BD, of diameters di and da respectively, which discharge into reservoirs or into the air (Fig. 90). If any outlet, as C, is higher than the Digitized by VnOOQlC 104 ELEMENTS OF HYDRAULICS junction B, then in order for flow to take place in the direction BC, the hydraulic gradient must slope in this direction; that is to say there must be a drop in pressure between the junction B and the level of the outlet reservoir C, or, in the notation of the figure, the condition for flow in the direction BC is hi > h. Assuming this to be the case, the solution is obtained from the same fundamental relation as above, namely. h-'^ + h,. ga^j^g S i^gigg ssss^^ss^^^ D ^ Fig. 90. Using the notation indicated on the figure for length, diameter and velocity in the various pipes and considering one line at a time, we thus obtain the following equations: For line ABC For line ABD, (87) (88) Also, from the condition that the discharge through the main pipe must equal the sum of the discharges through the branches, denoting the crossnsectional areas by a, ai, aa respectively, we have av = aiVi + a2V2. (89) By assimiing an average value for the frictional coefficient /, these three equations may then be solved for the thr^e unknowns V, vx and v^. Having thus found approximate values of the veloc- ities, the exact value of / corresponding to each velocity may be substituted in the above equations and the solution repeated. Digitized by VnOOQlC FLOW OF WATER 105 giving more accurate values of the velocities. Having found the velocities, the discharge from each pipe is obtained at once from the relations The method of solution is the same for any number of branches, there being as many equations in any given case as there are un- known velocities to be determined. Other simple cases of divided flow are illustrated in the numer- ical examples at the end of the chapter. XXn. FIRE STREAMS 123. Freeman's Experiments. — ^Extensive and accurate experi- ments on discharge through fire hose and nozzles were made by John R. Freeman at Lawrence, Mass., in 1888 and 1890.^ From these experiments it was found that the smooth cone nozzle with simple play pipe is the most efficient for fire streams, the coefficient of discharge being nearly constant for the various types tried and having an average value of 0.974 for smooth cone nozzles and 0.74 for square ring nozzles. The friction losses for fire hose were found to be given approxi- mately by the empirical formula _ P , . P ^ Fld^K» "^ ^ 6472 with notation as given below. For fire hose laid in ordinary smooth curves but not cramped or kinked, the friction loss was found to be about 6 per cent, greater than in perfectly straight hose. 124. Eormulas for Discharge. — The following formulas for discharge were deduced by Freeman from these experiments. Notation: j Q = discharge in cubic feet per second; ' Q = discharge in gallons per minute = 448.83Q; h == piezometer reading at base of nozzle in feet of water; p == pressure at base of nozzle in lb. per sq. in. = 0.434A; P = hydrant pressure in lb. per sq. in.; K = coefficient of discharge =- 0.974 for smooth cone nozzles and 0.74 for square ring nozzles; C« = coefficient of contraction; » Tran». Am. Soc. C. E., vol. 21, pp. 303-482; voL 24, pp. 492-527. Digitized by VnOOQlC 106 ELEMENTS OP HYDRAULICS d ^ diameter of nozzle orifice in inches; D = diam. of channel, where pressure is measured, in in.; I = length of hose in feet; H = total hydrostatic head in feet = ,,, .• F = 30 for unlined linen hose 2)4 ui- in diam.; = 26 for inferior rubber-lined hose 2}4 in* in diam.; = 13 for best rubber-lined hose 2}4 in. in diam. Then and Q » 0.01374Ed' 0.0664eKd< G = 19.636Kd' = 29.83 Kd* d\* P d\« P H H (90) d\* -■^•S) H 91 126. Height of Effective Fire Stream. — It was also found that the height, y, of extreme drops in still air from nozzles ranging in size from % in. to 1% in. in diameter was given by the formula y = H -0.00136-^' • (92) The height of a first-class fire stream will then be a certain frac- tion of y as indicated in the following table: When y Height of first class fire stream = 50 ft. 0.82 y 75 ft. 0.79 y 100 ft. 0.73 1/ 125 ft. 0.67 y 150 ft 0.63 y Table 11 is abridged from a similar table computed by Freeman from these and other formulas, not here given, and will be found convenient to use in solving fire-stream problems. 126. Fleming's Experiments. — A series of experiments on fire streams from small hose and nozzles was made by Virgil R. Flem- ing at the University of Illinois in 1911. The results of these experiments are also summarized in Table 11. Digitized by VnOOQlC FLOW OF WATER 107 XXm. EXPERIMENTS ON THE FLOW OF WATER 127, Verification of Theory by Experiment. — The subject of hydraulics as presented in an elementary text book is necessarily limited to simple demonstrations of the fundamental principles. It should not be inferred from this that the subject, is largely ex- perimental and not susceptible of mathematical analysis. As a matter of fact, hydrodynamics is one of the most diflBlcult branches of applied mathematics, and its development has absorbed the best efforts of such eminent mathematical physicists as Poinsot, Kirchhoflf, Helmholz, Maxwell, Kelvin, Stokes and Lamb. Nat- urally the results are too technical to be generally appreciated, but aflford a rich field for study to those with sufficient mathe- matical preparation. Some of the results concerning the fiow of liquids derived by mathematical analysis have been verified experimentally by the English engineers. Professor H. S. Hele-Shaw and Professor Osborne Reynolds. The chief importance of these experiments is that they serve to visualize difficult theoretical results. 128. Method of Conducting Experiments. — In Par. 47 a stream line was defined as the path followed by a particle of liquid in its motion. A set of stream lines distributed through a fiowing liquid therefore completely determines the nature of the fiow. To make such stream lines visible, so as to make it possible to actually trace the motion of the particles of a clear fiuid, both experimenters named above allowed small bubbles of air to enter Sudden contraction. Sudden'enlargement. Fig. 91. a fiowing stream. These bubbles do not make the motion directly visible to the eye, but by making the pipe or channel of glass and projecting a portion of it on a screen by means of a lantern, its image on the screen as viewed in this transmitted Ught clearly shows certain characteristic features. 129. Effect of Sudden Contraction or Enlargement. — Figures 91 and 92, reproduced by permission of Professor Hele-Shaw, Digitized by Google 108 ELEMENTS OF HYDRAULICS show the effect of a sudden- contraction or enlargement of the channel section. It is noteworthy that the disturbance or eddy- ing is much greater for a sudden enlargement than for a sudden Ck)ntraction. Fig. 92. Enlargement. contraction. This is due to the inertia of the fluid which pre- vents it from immediately fiUing the channel after passing through the orifice. This also confirms what has already been observed in practice, namely, that the loss of energy due to a sudden en- largement in a pipe is much greater than that due to a corre- sponding contraction. Digitized by Google FLOW OF WATER 109 130. Disturbance Produced by Obstacle in Current. — If the channel is of considerable extent and a small obstacle is placed in it, the stream lines curve around the obstacle, leaving a small space behind it, as shown in Fig. 93. If the object is a square block or flat plate this effect is greatly magnified, as shown in Fig. 94. The water is prevented from closing at once behind the obstacle by reason of its inertia. This indicates why the design of the stem of a ship is so much more important than that of the bow, since if there are eddies in the wake of a ship, the pressure of the water at the stern is decreased, thereby increasing by juat this much the effective resistance to motion at the bow. Fia. 94. 131. Stream-line Motion in Thin Film. — In these experiments it was also observed that there was always a clear film of liquid, or border Une, on the sides of the channel and around the obstacle. This observed fact was accounted for on the ground that by reason of the friction between a viscous liquid and the sides of the channel or obstacle, the thin film of liquid affected was not mov- ing with turbulent motion but with true stream-line motion, as in an ideal fluid. To isolate this film so as to observe its motion, water was allowed to flow between two plates of glass in a sheet so thin that its depth corresponded to the thickness of the border line previously observed. When this was done it immediately became apparent that the flow was no longer turbulent but a steady stream-line motion. The flow of a viscous fluid Uke glyc- Digitized by vnOOQlC 110 ELEMENTS OF HYDRAULICS Fig. 95. Fig. 96. Digitized by Google FLOW OF WATER 111 erine in a thin film thus not only eliminates turbulent flow, but also to a certain extent the inertia effects, thereby resulting in true stream-line flow. 132. Cylinder and Flat Plate. — To make the stream lines vis- ible, colored liquid was injected through a series of small openings, the result being to produce an equal number of colored bands or stream lines in the liquid. Fig. 95 shows these stream lines for a cylinder, and Fig. 96 for a flat plate placed directly across the current, while Fig. 97 shows a comparison of theory and experi- ment for a flat plate inclined to the current. 133. Velocity and Pressure. — The variation in thickness of the bands is due to the difference in velocity in various parts of the channel, the bands of course being thinnest where the velocity is — ► Fig. 98. greatest. Since a decrease in velocity is accompanied by a cer- tain increase in pressure, the wide bands before and behind the obstacle indicate a region of higher pressure. This accounts for the standing bow and stern waves of a ship, whereas the narrow- ing of the bands at the sides indicates an increase of velocity and reduction of pressure, and accounts for the depression of the water level at this part of a ship. In the case of a sudden contraction or enlargement of the chan- nel section, the true stream-line nature of the flow was clearly apparent, as shown in Fig. 98, the stream lines following closely the form derived by mathematical analysis for a perfect fluid. XXIV. MODERN SIPHONS 134. Principle of Operation. — In its simplest form, a siphon is merely an inverted U-shaped tube, with one leg longer than the other, which is used for emptying tanks from the top when no Digitized by VnOOQlC 112 ELEMENTS OF HYDRAULICS outlet is available below the surface. In use, the tube is filled with liquid and the ends corked, or otherwise closed. The short end of the tube is then placed in the reservoir to be lowered, so that the level of the end outside is lower than the surface of the reservoir (Fig. 99). When the ends of the tube are opened, the liquid in the reservoir begins to flow through the tube with a head, hy equal to the difference in level between the surface of the reservoir and the lower, or outer, end of the tube. If the inner end of the siphon is placed close to the bottom of the reservoir Fig. 99 it can be practically emptied in this manner. For emptying small tanks a siphon can conveniently be made of a piece of ordi- nary tubing or hose. 136. Siphon Spillways. — The siphon principle is now being applied on a large scale in the construction of spillways, locks and wheel settings. In many cases the common overflow spillway requires such a great length for proper regulation of the pond or forebay as to make its use undesirable. This form of spillway is also ineffi- cient because of the low head under which it operates. This often makes it necessary to use flashboards and automatic gates to increase the head and consequently the velocity of discharge. The available head, however, is the total head between the water surfaces above and below the dam, and this may be utilized by building water passages through the dam and submerging the downstream end, thus forming a siphon. Such siphon spillways have been in use in Italy for a niunber of years. Until recently it was supposed that they could not be used in colder countries on account of the impossibility of keep- Digitized by VnOOQlC FLOW OF WATER 113 ing them free from ice. The first siphon spillways to be built outside of Italy were in Zurich, Switzerland, where for several years a number have been in successful operation on streams where ice forms for 2 or 3 months every year. In designing siphons there are at least two important prin- ciples which must be observed.^ , (a) The upper part must be so designed that as soon as the water rises above the level to be maintained, the siphon intake is sealed and remains sealed until the water level is brought down again to normal. The air openings must then be large enough to admit sufficient air to break the siphonic action immediately. Both of these features may be secured by having long and sharp edges on the intake to the siphon at the normal water level. CourteBy Eng. Record, Fig. 100. — Siphon spillway in use at Seon, Switzerland. (6) The lower edge of the siphon must be submerged deep enough to secure a constant seal. The upper edge of this open- ing must also be as sharp as possible to permit of an easy escape of the enclosed air. A siphon spillway in use at Seon, Switzerland, is shown in Fig. 100. The action of such a siphon is as follows: The pond rises until the water seals the upper sharp-edged slots of the in- take. As soon as this happens, the water flowing through the siphon carries the air with it, which escapes around the sharp lower edge, and the siphon is primed. The siphon then con- tinues in full action until the pond level is lowered sufficiently to admit air under the upstream edge. 1 HUiLBBRG, Eng. Record, May 3, 1913, p. 488. 8 Digitized by VnOOQlC 114 ELEMENTS OF HYDRAULICS It might be supposed that siphons could be used only where the difference in elevation of the water surfaces was less than the suction head for the siphon, but this condition has been found to be not essential. The siphon spillway at Gibswil, Switzerland, shown in section in Fig. ^101, operates under a head of 52.48 ft. It consists of a riveted steel pipe }^ in. thick, tapering from 31.5 to 23.6 in., this taper being intended to^keep the water column from parting under the high head. At^the upper water level the pipe is cut on a- horizontal plane and covered by a reinforced-concrete hood, projecting 3.28 ft. below normal water level, the purpose of this hood being to prevent ice from clogging Courtesy Eng, Record. Fig. 101. — High-head siphon spillway in use at Gibfwil, Switzerland. the siphon. To prevent the water from being lowered to the edge of the hood by siphonic action, long, narrow slots are cut through it on three sides at the normal level of the pool, these slots closing as the water rises above normal. A test of this siphon gave a discharge of 98.9 cu. ft. per second, but as the air slots were not all fully closed, it was estimated that the marimum discharge would be about 123.6 cu. ft. per second. As the end area is 3.03 sq. ft., this would mean a vdocity of 40.8 ft. per second. The friction head in the siphon itself was 10.2 ft., leaving a net effective head of 62.48 - 10.2 = 42.28 ft. Since the theoretical velocity of flow due to this head is V = 'V2gh = 62.18 ft. per second, the coefficient of discharge, 40.8 or efficiency, of the siphon is koTq = 78 per cent. Digitized by VnOOQlC FLOW OF WATER 115 The first siphon spillways to be constructed in this country were the three located on the champlain division of the New York State Barge Canal. At the place shown in Fig. 102 it was necessary to provide for a maximum outflow of about 700 cu. ft. per second and to limit the fluctuation of water surface to about 1 ft. The ordinary waste weir of a capacity sufficient to take care of this flow, with a depth of only 1 ft. of water on the crest, would require a spill- way 200 ft. long. The siphon spillway measures only 57 ft. Fig. 102. — Siphon spillway, Champlain division, New York State Barge Canal. between abutments and accomplishes the same results. This particular structure consists of four siphons, each having a^ cross-sectional area of 7% sq. ft. and working under a lOj^-ft.' head. There is also a 20-ft. drift gap to carry off floating debris. The main features of construction are shown in Fig. 103. The| siphon spillway was designed and patented by Mr. George F. Stickney, one of the Barge Canal engineers. Another instance is furnished by the second hydro-electric development of the Tennessee Power Co. on the Ocoee River, Tenn., where a spillway consisting of a battery of eight siphons has been constructed. The general features of the design are shown in Fig. 104. The entrance area is located 5}4 ft. below Digitized by VnOOQlC 116 ELEMENTS OF HYDRAULICS 09 O .a I I I 'a Digitized by VnOOQlC FLOW OF WATER 117 water surface to insure freedom from floating debris, and is 33^ by 6 ft. in area, protected by ^-in. screen bars spaced 4 in. apart on centers. The entrance area gradually diminishes in the upper leg to 8 by 1 ft. at the top or throat, the larger dimen- sion being horizontal. The lower leg is rectangular in cross- section and 8 sq. ft. in area throughout, but gradually changes in shape to a point 12.8 ft. below the crest, whence its section is 4 by 2 ft. to the outlet. Four of the siphons operate under a head of 27.2 ft., and the other four under a head of 19.2 ft. Hori- zontal air inlets 6 by 18 in. in section are provided for each siphon unit, extending through the throat casting to the upstream face of the dam. In a test of these siphons it was found that two of them dis- charged 422.8 cu. ft. per second, giving a velocity of flow of ewjjfl I rr ffl Fig. 104. — Siphon spillway constructed at Ocoee River, Tennessee. 422.8 2 y g = 26.425 ft. per second. Since the average head acting on the siphon during the test was 26.65 ft., the theoretical veloc- ity of flow is t; = '\r2gh = 40.54 ft. per second. The efficiency in this case is therefore 26.425 40.54 = 65 per cent. 136. Siphon Lock. — The siphon lock on the New York State Barge Canal is located in the City of Oswego, and is the only lock of this type in this country and the largest ever built on this principle. It consists of two siphons, as shown in Fig. 105 the crown of each being connected by a 4-in. pipe to an air tank in which a partial vacuum is maintained. To start the flow, the air valve is opened, the vacuimi in the tank drawing the air from the siphon and thereby starting the flow. When the siphon is discharging fully, its draft is such that the air is sucked out of the tank, thus restoring the partial vacuum. To stop the flow, Digitized by vnOOQlC 118 ELEMENTS OF HYDRAULICS o OS I I 44 d T o Digitized by VjOOQIC FLOW OF WATER 119 outer air is admitted to the crest of the siphon by another valve, thereby breaking the flow, as indicated in Fig. 106. The operat- ing power is thus self-renewing, and, except for air leakage, lock- ages can be conducted by merely manipulating the 4-in. air Digitized by LnOOQlC 120 ELEMENTS OF HYDRAULICS valves. However, to avoid the necessity of refilling the -tank when traffic is infrequent, it is customary to close the 20-in. outlet valve, thus holding the water in the tank. Using both siphons, the lock chamber can be filled in 4J^ to 5 min., and emptied in 5}4 to 6 min. 137. Siphon Wheel Settings. — The siphon principle has been utilized in several instances for waterwheel intakes. Fig. 107 shows the type of siphonic wheel setting used in the pump house r:\ Courtesy Eng. Record. Fig. 107.— Siphonic wheel setting in the pump house at Geneva, Switzerland. at Geneva, Switzerland. The chief advantage of this type is that it eliminates the use of headgates, which in design and opera- tion are one of the most difficult details of a hydro-electric development. The design and operation of such an intake is very similar to that for the siphonic locks at Oswego, described above. XXV. FLOW IN OPEN CHANNELS 138. Open and Closed Conduits. — Conduits for conveying water are usually classified as open and closed. By a closed Digitized by VnOOQlC FLOW OF WATER 121 conduit is meant one flowing under pressure, as in the case of ordi- nary pipe flow discussed in Art. XVII. Water mains, penstocks, draft tubes and fire hose are all examples of closed conduits. Open channels, or conduits, are those in which the upper sur- face of the liquid is exposed to atmospheric pressure only, the pressure at any point in the stream depending merely on the depth of this point below the free surface. Rivers, canals, flumes, aqueducts and sewers are ordinarily open channels. A river or canal, however, may temporarily become a closed channel when covered with ice, and an aqueduct or sewer may also be- come a closed channel if flowing full under pressure. 139. Steady Unifonn Flow. — The fundamental laws applying to flow in open and closed channels are probably identical, and in the case of steady, uniform flow the same formulas ajpply to both. For steady flow in an open channel the quantity of water passing any transverse section W the stream is constant, and for uniform flow the mean velocity is also constant. Under these conditions the cross-sectional area of the stream is constant throughout its length, and the hydrauUc gradient is the slope of the surface of the stream. The formula for velocity of flow is then the one given in Par. 118 under the name of Chezy's formula, namely V = CVSre. (93) 140. Eutter's Formula. — ^Numerous experiments have been made to determine the value of the coefficient C for open channels. In 1869, E. Ganguillet and W. R. Kutter, two Swiss engineers, made a careful determination of this constant, the result being expressed in the following form: V = 41.6. + »=»^ + J:Mi + [u.« + <^]^. Vri (94) in which « = hydraulic gradient, or slope of channel; V J ,. J. area of flow r = hydraulic radius = — tt— j — —-. — r— ; •^ wetted penmeter' n = coefficient of roughness. The coefficient of roughness, w, depends on the nature of the channel lining. Approximate values of n for various surfaces are given in the following table: Digitized by VnOOQlC 122 ELEMENTS OF HYDRAULICS Nature of Channel Lining Planed timber carefully joined, glazed or enameled siirfaces Smooth clean cement Cement mortar, one-third sand Unplaned timber or good new brickwork Smooth stonework, vitrified sewer pipe and ordinary bridcwork . Rough ashlar and good rubble masonry Urm gravel Ordinary earth Earth with stones, weeds, etc Earth or gravel in bad condition 0.009 0.010 0.011 0.012 0.013 0.017 0.020 0.025 0.030 0.035 141. Limitations to Eutter's Fonnula. — Kutter's formula, Eq. (94), is widely used and is reliable when applied to steady, uni- form flow under normal conditions. From a study of the data on which this formula is based, its use has been found to be sub- ject to the following Umitations: It is not rehable for hydrauUc radii greater than 10 ft., or veloci- ties greater than 10 ft. per second, or slopes flatter than 1 in 10,000. Within these Umits a variation of about 5 per cent, may be expected between actual results and those computed from the formula. Table 15 gives numerical values of the coefficient C calculated from Eq. (94). 142. Bazin's Fonnula. — In 1897, H. Bazin also made a careful determination of the coefficient C from all the experimental data then available, as the result of which he proposed the following formula: 87 V = 0.552 + -^ V r V: rs (95) where r = hydraulic radius; m = coefficient of roughness. Bazin's formula has the advantage of being simpler than Kutter's, and is independent of the slope s. Values of the coeffi- cient of roughness, m, for use with this formula are given in the following table: Nature of Channel Lining m Planed timber or smooth cement Unplaned timber, well-laid brick or concrete Ashlar, good rubble masonry or poor brickwork . Earth in good condition Earth in ordinary condition Earth in bad condition 0.06 0.16 0.46 0.86 1.30 1.75 Digitized by vnOOQlC FLOW OF WATER 123 Table 14 gives numerical values of the coefficient C calculated from Eq. (95). 143. Eutter's Simplified Fonnula. — ^A simplified form of Eut- ter's formula which is also widely used is the following: where 6 is a coefficient of roughness which varies from 0.12 to 2.44. For ordinary sewer work the value of this coefficient may be assumed as 6 = 0.35. XXVI. CHANNEL CROSS-SECTION 144. Condition for Maximum Discharge. — From the Chezy formula for fiow in open channels, namely, Q = Av = ACVrSy it is evident that for a given stream section A and given slope s, the maximum discharge wiU be obtained for that form of cross- section for which the hydraulic radius r is ja maximum. Since area of fiow r = wetted perimeter this condition means that for constant area the radius r, and therefore the discharge, is a maximum when the wetted perimeter is a minimum. The reason for this is simply that by making the area of contact between channel lining and water as small as possible, the f rictional resistance is reduced to a minimum, thus giving the maximum discharge. 145. Maximum Hydraulic Efficiency. — ^In consequence of this, it follows that the maximum hydraulic efficiency is obtained from fiC semicircular cross-section, since for a given area its wetted perimeter is less than for any other form (Fig. 108). For rec- tangular sections the half square has the least perimeter for a given area, and consequently is most efficient (Fig. 109). Simi- larly, for a trapezoidal section the half hexagon is the most effi- cient (Fig. 110). In each case the hydraulic radius is half the water depth, as proved below. In the case of unUned open channels it is necessary to use the trapezoidal section, the slope of the sides being determined by the nature of the soil forming the sides. This angle having been Digitized by VnOOQlC 124 ELEMENTS OF HYDRAULICS determined, the best proportions for the section are obtained by making the sides and bottom of the channel tangent to a semi- circle drawn with center in the water surface (Fig. 111). 146. Regular Circumscribed Polygon. — ^Any section which forms half of a regular polygon of an even number of sides, and has each of its faces tangent to a semicircle having its center in Semicircle Fig. 108. 4R tan aO** Half Square Fig. 109. Half Octagon Fig. 112. Triangle Fig. 113. the water surface, will have its hydraulic radius equal to half the radius of this inscribed circle (Figs. 108-113). To prove this, draw radii from the center of the inscribed circle to each angle of the polygon. Then since the area of each of the triangles so formed is equal to one-half its base times its altitude, and since the altitude in each case is a radius of the inscribed circle, the total area is Area = ^ X perimeter. Digitized by VnOOQlC FLOW OF WATER 125 :l| 1"^ 1 il a 5 it =3 •5 o 1^ 8 1 •3 « 1^ SIS oo eo eo ao •9 00 eo 00 la u> >0 «^ « >0 u> ^ r5 05 • n7 1 ^ ^ ^ ^ r ^ ^ > *-4 CO CO 1-H t<. wi s .^ o^ 1-H CO CO CI r^ ^ 1-H 1-4 CO w ^ CO XJ< 1-i d d ^i ci CO « CO n II » O^ > f > > ^ 1 ^ ^ V 1 In. ^ S^ s 1-4 o oo *o o a C4 1-4 CO l^ r>. t^ t- t^ 00 00 r>. > d d d d d d d d «k a^ ft; 05 ft; 05 ~05 ft; ll^l"" ft; ^ 1 1-4 i ^ >^ *^ *. d II d d d d d d d W k. w 05 ft; 05 ~i~ 05 ft; 05 g <N CO a ^. ^ 1-4 Wetted perimete] 1-4 * CO ^ « ^ S CO '^* »o d '^* »o d t^ " II ^ fi. ft; 05 05 ft; 05 M 05 "^ 1-4 (N CI s CO XJ< ater ition A b- 00 "^ a »o t^ o o o ^. »o 1-4 c^* CO CO CO* Tjj •*' •** ^8 H II Q Q ti: Jji ^ Water depth h Q s 5 fe s ^. 00 oa « 00 o> o d d Q d d d tq II II < .< o o o r2 »o o o »o to o S' -e- g l^ s S 1-4 CO CO wi S g CO ^ d .2 .1 § 1 O 1 ■ i 5^ Digitized by Google 126 ELEMENTS OF HYDRAULICS Consequently the hydraulic radius r is 12 r = area of flow X perimeter wetted perimeter perimeter B 2* 147. Properties of Circular and Oval Sections. — For circular and oval cross-sections, the maximum velocity and maximum dis- charge are obtained when the conduit "f^ is flowing partly f uU, as apparent from the table on page 125, which is a collection of the most important data for circular and oval sections,^ as shown in Figs. 114 and 115. Theoretically, the maximum discharge for a circular pipe occurs when the pipe is filled to a depth of 0.949D, but if it is attempted to maintain flow at this depth, the waves formed in the pipe strike against the top, filhng it at periodic intervals and thus producing impact losses. To obtain the maxiipum discharge without danger of impact, the actual depth of flow should not exceed %D. Circular Section Fig. 114. l_ I Standard Oval Section Fig. 115. XXVn. FLOW IN NATURAL CHANNELS 148. Stream Gaging. — ^In the case of a stream flowing in a natural channel the conditions determining the flow are so vari- » Wbyrauch, '^Hydrauliches Rechnen," S. 61. Digitized by VnOOQlC FLOW OF WATER 127 able that no formula for computing the discharge has been de- vised that can claim to give results even approximately correct. To obtain accurate results, direct measurements of cross-sections and velocities must be made in the field. The two methods of direct measurement in general use are as follows: Either Fig. 116. — ^Electric current meter. 1. The construction of a weir across the stream, and the cal- culation of the discharge from a weir formula; or, if this is not feasible, 2. The measurement of cross-sections of the stream by means of soundings taken at intervals, and the determination of ^ average velocities by a current meter or floats. Digitized by VnOOQlC 128 ELEMENTS OF HYDRAULICS The first of these methods is explained in Art. XII. 149. Current Meter Measurements. — ^The current meter, one type of which is shown in Fig. 116, consists essentially of a bucket wheel with a heavy weight suspended from it to keep its axis horizontal, and a vane to keep it directed against the cur- rent, together with some form of counter to indicate the speed at which the wheel revolves. The meter is first rated by towing it through still water at various known velocities and tabulating the corresponding wheel speeds. From these results a table, or chart, is constructed ^ving the velocity of the current corre- sponding to any given speed of the wheel as indicated by the counter. This method of calibration, however, is more or less inaccurate, as apparent from Du Buat's paradox, explained in Par. 159. 160. Float Measurements. — ^When fioats are used to determine the velocity, a uniform stretch of the stream is selected, and two Water Surface 5 Velocity ^ 1 s :^ ;^ y wmmmmmmmmA Fig. 117. Bed of > Stream cross-sections chosen at a known distance apart. Floats are then put into the stream above the upper section and their times of transit from one section to the other observed by means of a stop watch. A subsurface float is commonly used, so ar- ranged that it can be run at any desired depth, its position being located by means of a small surface float attached to it. If the cross-section of the stream is fairly uniform, rod floats may be used. These consist of hollow tubes, so weighted as to float upright and extend nearly to the bottom. The velocity of the float may then be assumed to be equal to the mean velocity of the vertical strip through which it runs. 161. Variation of Velocity with Depth. — ^The results of such measurements show, in general, that the velocity of a stream is Digitized by VnOOQlC FLOW OF WATER 129 greatest midway between the banks and just beneath the surface. In particular, the velocities at different depths along any vertical are found to vary as the ordinates to a parabola, the axis of the parabola being vertical and its vertex just beneath the surface, as indicated in Fig. 117. From this relation it follows that if a float is adjusted to run at about 0.6 of the depth in any vertical strip, it will move with approximately the average velocity of all the particles in the vertical strip through which it runs. 162. Calculation of Discharge. — ^In order to calculate the dis- charge it is necessary to measure the area of a cross-section as well as the average velocities at various points of this section. The total crosshsection is therefore subdivided into parts, say Ai, A^i Azf etc. (Fig. 118), the area of each being determined by measuring the ordinates by means of soundings. The average velocity for each division is then measured by one of the methods explained above, and finally the discharge is computed from the relation Q = AiVi + A2V2 + AzVz + XXVm. THE PITOT TUBE 163. Description of Instrument. — ^An important device for measuring the velocity of flow is the instrument known as the Pilot tvbe. In 1732 Pitot observed that if a small vertical tube, open at both ends, with one end bent at a right angile, was dipped in a current so that the horizontal arm was directed against the current as indicated in Fig. 119^1, the liquid rises in the vertical arm to a height proportional to the velocity head. The height of the column sustained in this way, or hydrostatic head, is not exactly equal to the velocity head on account of the disturbance created by the presence of the tube. No matter how small the tube may be, its dimensions are never negligible, and its presence 9 Digitized by LnOOQlC 130 ELEMENTS OF HYDRAULICS has the effect of causing the filaments of liquid, or stream lines, to curve around it, thereby considerably modifying the pressure. Since the column of liquid in the tube is sustained by the impact of the current, this arrangement is called an impact tvbe. If a straight vertical tube is submerged, or a bent tube having its horizontal arm directed transversely, that is, perpendicularly, to the current, the presence of the tube causes the stream lines to turn their concavity toward the orifice, thereby producing a suc- tion which is made apparent by a lowering of the water level in this tube, as shown in Fig. 119B. In the case of the bent tube, if Impact Tube Suction or Pressure ' Trailing Tube Tube At ■I-.I.I 1 ^ . c = r 1" 1= z -= 1 ■£ - -E = - r _: . - Z - ~ = — = 1^ z :z r — — - ~ z - z -'z 1 1 'z i \ .z - -.-. M -E ^^^ -^ 7 : " ^^^n- - Direction of Flow Fig. 119. the horizontal arm is directed with the current, as shown in Fig. 119C, the effect is not so pronounced as when the tube is turned at right angles to the current, as for ordinary velocities the suc- tion effect due to viscosity predominates over that due to change in energy. When the horizontal arm of a bent tube is directed with the current, the arrangement is called a suction or trailing tvbe. It is practically impossible, however, to obtain satisfactory numerical results with this simple type of Pitot tube, as in the case of flow in open channels the free surface of the liquid is Digitized by VnOOQlC FLOW OF WATER 131 i: usually disturbed by waves and ripples and other variations in level, which are often of the same order of magnitude as the quan- tities to be measured; while in the case of pipe flow under pressure there are other conditions which strongly aflfect the result, as will appear in what follows. 164. Darcy's Modification of Ktot's Tube —In 1850 Darcy modified the Pitot tube so as to adapt it to general current measurements. This modification consisted in combining two Pitot tubes, as shown in Fig. 120, the orifice of the impact tube being directed upstream, and the orifice of the suction tube transverse to the current. In some forms of this apparatus, the suction tube is of the trailing type, that is, the hori- zontal arm is turned directly downstream. To make the readings more accurate, the difference in elevation of the water in the two tubes is magnified by means of a differential gage, as shown in Fig. 120. Here A denotes the impact tube and B the suction tube (often called the pressure tube), connected with the tubes C and Z>, between which is a graduated scale. After placing the apparatus in the stream to be gaged, the air in both tubes is equally rarified by suction at F, thereby causing the water level in both to rise proportional amounts. The valve at F is then closed, also the valve at E, and the apparatus is lifted from the water and the reading on the scale taken. It was assumed by Pitot and Darcy that the difference in level in the tubes was proportional to the velocity head «-, where v denotes the velocity of the current. CaUing hi and hi the dif- ferences in level, that is, the elevation or depression of the water in the impact and suction tubes respectively, and mi, mi the con- stants of proportionality, we have therefore [ T" J„, D >0^ B Fig. 120. WlAl = Off ~ ^2^i* Digitized by VnOOQlC 132 ELEMENTS OF HYDRAULICS If, then, h denotes the difference in elevation in the two tubes (Fig. 120), we hSve The velocity v is therefore given in terms of h by the equation v = mV^ (97) where '4, ifhi + wis The coefficient m depends, like mi and m2, on the form and dimen- sions of the apparatus, and when^ properly determined is a con- stant for each instrument, provided that the conditions imder which the instrument is used are the same as those for which m was determined. The vMue of m in this formula has been found to vary from 1 to as low as 0.7; the value m = 1 corresponding to A = «-; and the value m = 0.7 to A ?= — . The explanation of this apparent discrepancy is given below under the theory of the impact tube. In the case of variable velocity of flow it has been shown by Rateau^ that the Pitot, or Darcy, tube measures not the mean velocity but the mean of the squares of the velocities at the point where it is placed during the experiment. To obtain the mean velocity it is necessary to multiply ^r by a coefficient which ^ . . varies ietccording to the rate of change of the velocity with respect to the time. From Rateau's experiments this coefficient was found to vary from 1.012 to 1.37, having a mean value of 1.16. This corresponds to a mean value for m of 0.93. 166. Pitometer. — ^A recent modification of the Pitot tube is an instrument called the Pitometer (Fig. 121). The mouthpiece of this apparatus consists of two small orifices pointing in oppo- site directions and each provided with a cutwater, as shown in the figure. When in use, these are set in line parallel to the current, so that one points directly against the current and the other with it. The differential gage used with this instrument consists of a U-tube, one arm of which is connected* with one mouthpiece and the other arm with the other mouthpiece, and which is about ^ Anndlea dea Mines, Mars, 1898. Digitized by VnOOQlC { FLOW OF WATF!R 133 half filled with a mixture of gasoline and carbon tetrachloride, colored dark red. The formula for velocity as measured by this instrument is given in the form V = k[2gi8 - l)cq^ _J. Fig. 121. where k = empirical constant = 0.84 for the instrument as manufactured and calibrated; 8 = specific weight of the tetrachloride mixture = 1.26; d = difference in elevation in feet between the tops of the two colunms of tetrachloride. Inserting these numerical values, the formula reduces to 't; = 3M8Vdf It is claimed that velocities as low as 3^ ft. per second can be measm^ed with this instrument. Digitized by VnOOQlC 134 ELEMENTS OF HYDRAULICS 166. Pitot Recorders. — ^The Pitot meter is used in power houses, pumping stations and other places where a Venturi tube cannot be installed, and is invaluable as a water- works instrument to determine the pipe flow in any pipe of the system. Fia. 122. — Pilot recording meter. Simplex valve and meter Co. A recent portable type, especially adapted to this purpose is shown in Fig. 122. This instrument is 34 in. high, weighs 75 lb., and fiu'nishes charts of the Bristol type which are averaged with a special planimeter furnished with the instrument. A 1-in. tap in the water main is required for inserting the Pitot mouthpiece. Digitized by VnOOQlC I FLOW OF WATER 135 It is claimed that these instruments have a range from }4 ft- per second to any desired maximum. 167. Theory of the Impact Tube. — ^The wide variation in the range of coeflBcients recommended by hydraulic engineers for use with the Pitot tube can be accounted for only on the ground bf a faulty understanding of the hydraulic principles on which its action is based. The most important of these are indicated below, without presuming to be a complete exposition of its action. It will be shown in Par. 162 that the force produced by the impact of a jet on a flat plate is twice as great as that due to the hydrostatic head causing the flow. That is to say, if the theo- retical velocity of a jet is that due to a head h, where the force exerted on a fixed plate by the impact of this jet is equal to that due to a hydrostatic head of A' = 2h, in which case The orifice in a Pitot tube is essentially a flat plate subjected to the impact of the current. Considering only the impact efifect, therefore, the head which it is theoretically possible to attain in a Pitot tube is 9 which corresponds to a value of m of 0.7 in the formula V = niy/2gh' There are other considerations, however, which often modify this result considerably. The efifect of immersing a circular plate in a uniform parallel current has been fully analyzed theo- retically and the results confirmed experimentally. The results of such an analysis made by Professor Prasil, as presented in a paper by Mr. N. W. Akimofif,^ are shown in Mg. 123. The diagram here shown represents a vertical section of a current flowing vertically downward against a horizontal circular plate. The stream lines S, shown by the full lines in the figure, are curves of the third degree, possessing the property that the vol- umes of the cyUnders inscribed in the surface of revolution gen- ^ Jour. Amer. Water Works Assoc., May, 1914. Digitized by VnOOQlC 136 ELEMENTS OF HYDRAULICS erated by each stream line are equal. For instance, the volume of the circular cylinder shown in section by AA'BB' is equal to that of the cylinder CC'DD\ etc. It may also be noted that the size of the plate does not affect the general shape or properties of the curves shown in the diagram. The surfaces of equal velocity are ellipsoids of revolution hav- ing the center of the plate as center, and are shown in section in the figure by the eUipses marked EV. In general, each of these ellipses intersects any stream line in two points, such as F and G. Therefore somewhere between F and G there must be a point of minimum velocity, this being obviously the point of contact of the corresponding ellipse with the stream hne. The locus of these points of minimum velocity is a straight Une OH in section, inclined to the plate at an angle of approximately 20°. The surface of minimum velocity is therefore a cone of revolution with center at 0, of which OH is an element. Digitized by vnOOQlC FLOW OF WATER 137 The surfaces of equal pressure are also ellipsoids of revolution with common center below 0, and are shown in section by the ellipses marked PP in the figure. The surface of maximum pres- sure is a hyperboloid of revolution of one sheet, shown in section by the hyperbola YOY. It should be especially noted that the cone of minimum veloc- ity is distinct from the hyperboloid of maximum pressure so that in this case minimum velocity does not necessarily imply maxi- mum pressure, as might be assumed from a careless appUcation of Bernoulli's theorem. This analysis shows the reason for the wide variation in the results obtained by different experimenters with the Pitot tube, and makes it plain that they will continue to differ imtil the hydraulic principles imderlying the action of the impact tube are generally recognized and taken into account. 168. Construction and Calibration of Pitot Tubes. — ^The im- pact end of a Pitot tube is usuaUy drawn to a fine point with a very small orifice, whereas the vertical arm is given a much larger diameter in order to avoid the effect of capillarity. The tubes used by Darcy had an orifice about 0.06 in. in diameter which was enlarged in the vertical arm to an inside diameter of about 0.4 in. In his well-known experiments for determining the velocity of fire streams (Par. 123), Freeman used for the mouthpiece of his impact tube the tip of a stylographic pen, having an aperture 0.006 in. in diameter. With this apparatus and for the high velocities used in the tests, the head was found to be almost ex- actly equal to s;-, corresponding to a value of m = 1.0 in the formula v = m\/2gh. It is also important that the impact arm should be long enough so that its orifice is clear of the standing wave produced by the current flowing against the vertical arm. The cutwater used with some forms of apparatus (see Fig. 121) is intended to elimi- nate this effect but it is doubtful just how far it accomplishes its piu'pose. The most proUfic source of error in Pitot-tube measurements is in the calibration of the apparatus. The fundamental prin- ciple of calibration is that the tube must be caUbrated under the same conditions as those for which it is to be used. Thus it has been shown in Art. XVII that flow below the critical velocity fol- lows an entirely different law from that above this velocity. Digitized by VnOOQlC 138 ELEMENTS OF HYDRAULICS Flow in a pipe under pressure is also essentially different from flow in an open channel. 169. Du Buat's Paradox. — Furthermore, the method of calibra- tion is of especial importance. This is apparent from the weD- known hydrauUc principle known as Du BuaVs 'paradox. By ex- periment Du Buat has proved that the resistance, or pressure, offered by a body moving with a velocity v through a stationary liquid is quite different from that due to the Uquid flowing with the same velocity v past a stationary object. The pressure of the moving liquid on the stationary object was found by him to be greater than the resistance experienced by the moving object in a stationary liquid in the ratio of 13 to 10. All methods of calibration which depend on towing the instrument through a liquid at rest therefore necessarily lead to erroneous and mis- leading results. Since the Pitot tube is so widely used for measuring velocity of flow, its construction and caUbration should be standardized, so that results obtained by different experimenters may be subject to comparison, and utiUzed for a more accurate and scientific construction of the instrument. XXIX. NON-UNIFORM FLOW; BACKWATER 160. Surface Elevation. — The case of most practical impor- tance is that in which the level of a stream is to be raised by means of a dam or weir, and it is required to determine the new surface elevation at any given distance back of the dam or weir. As the mathematical solution of the problem is somewhat compUcated, the method commonly followed in practice is to obtain the hy- draulic gradient by a series of approximations. Thus having given the discharge and the dimensions of the channel cross- section, the velocity of flow, v, and the hydraulic radius, r, become known. Then assuming a value for the hydraulic gradient, s, the value of C is computed from Chezy^s formula V = Cy/rSj and also from Kutter's formula C = ,, ^^ , 0.00281 , 1.811 41.66 -\ H + (41.65 +«:°^)i Digitized by VnOOQlC FLOW OF WATER 139 and the two values compared. If these values are not equal, a new value of s is assumed and the process repeated until both formulas give the same value of C. The corresponding value of s is then taken to be the correct hydrauUc gradient, from which the actual elevation of the water surface at any point may be computed if the slope of the bed of the stream is known. The hydraulic gradient, s, may also be computed directly trom a formula of the exponential type such as that of Williams and Hazen, namely, 'V = Cr»-6V ^^0.001 - ®-®^ provided the engineer's experience warrants him in assuming a value for C. As the channel of an ordinary stream varies con- siderably, giving rise to non-uniform flow, an exact solution of the problem is impossible and the assumption of C is usually accurate enough to satisfy all practical considerations. APPLICATIONS 61. A device used by Prony for measuring discharge consists of a fixed tank A (Fig. 124) containing water, in which floats a cylinder C which carries a second tank B, Water flows through the opening D from A into B. Show CAB that the head on the opening D, and consequently the velocity of flow through this opening, remains con- stant (Wittenbauer). 62. A cylindrical tank of 6-ft. in- side diameter and 10 ft. high con- tains 8 ft. of water. An orifice 2 in. in diameter is opened in the bottom, Fig. 124. and it is found that the water level is lowered 21 in. in 3 min. Calculate the coefiicient of discharge. 63. Water flows through a circular sharp-edged orifice }4 in. in diameter in the side of a tank, the head on the center of the opening being 6 ft. A ring slightly larger than the jet is held so that the jet passes through it, and it is then found that the center of the ring is 8.23 ft. distant from the orifice horizontally, and 3 ft. below it. In 5 min. the weight of water discharged is 301 lb. Calculate the coefficients of velocity, contraction and discharge for this orifice. c A r-.- -"- D ^ N., xa - - ■ Digitized by VnOOQlC 140 ELEMENTS OF HYDRAULICS Note. — ^This is an interesting method of determining the coefficients by experiment but is not very accurate. If the velocity of the jet at exit is denoted by v, its abscissa x after t sec. will be approximately X = vt and the ordinate of the same point; considering the water as a freely falling body, will be Eliminating t between these two relations, the equation of the path followed by the jet is foimd to be x^ = > g which represents a parabola with axis vertical and vertex at the orifice. Having found the actual velocity v from this equation, the velocity coefficient is obtained from the relation V2ih The effiux coefficient K is then calculated from the measured dis- charge Q from the relation Q = KAv, and the contraction coefficient from 64. Find^the velocity with which water will flow through a hole in a steam boiler shell at a point 2 ft. below the surface of the water when the steam pressure gage indicates 70 lb. per square inch. 56. A reservoir having a superficial area of 0.6 sq. mile has an outlet through a rectangular notch weir 8 ft. long. If the head on the crest when the weir is opened is 2.6 ft., how long will it take to lower the level of the reservoir I ft.? 66. A rectangular notch weir 12 ft. long has a head of 16 in. of water on the crest. The cross-sectional area of the approach channel is 60 sq. ft. Calculate the flow. 67. A suppressed weir 6 ft. long has its crest 3 ft. above the bottom of the channel, and the head on the crest is 18 in. Com- pute the discharge. Digitized by Google t = FLOW OF WATER 141 68. A lock chamber 500 ft. long and 110 ft. wide is emptied through a submerged opening 6 ft. long by 3 ft. high, having a coeflBcient of discharge of 0.68. If the depth of water on the center of the opening is initially 30 ft. on the inside and 8 £t. on the outside, find how long it will take to lower the water in the lock to the outside level. 69. A hemisphere filled with water has a small orifice of area A at its lowest point. Calculate the time required for it to empty. Note. — If x denotes the depth of water at any instant, the area X of the water surface is X = v{2rx — x*), where r denotes the radius of the hemisphere. The time required to empty the hemisphere is therefore 1 r^Xdx ^ 14 m^ K^^jo V5 15 KAV2g 60. A tank 10 ft. square and 12 ft. deep is filled with water. A sharp-edged circular orifice 3 in. in diameter is then opened in the bottom. How long will it take to empty the tank through this opening? 61. Compute the discharge through a Borda mouthpiece 1.6 in. in diameter under a head of 12 ft., and determine the loss of head in feet. 62. Compute the discharge through a reentrant short tube 2 in. in diameter under a head of 20 ft., and determine the loss of head in feet. 63. Compute the discharge through a standard short tube of 1.76 in. inside diameter under a head of 6 ft., and also find the negative pressure head at the most contracted section of the vein. 64. Find the discharge in gallons per minute through a 1.6-in. smooth fire nozzle attached to a 2.6-in. play pipe under a pres- sure at base of nozzle of 90 lb. per square inch. 66. Water flows through a 6-in. horizontal pipe at 200 ft. per minute under a pressure of 30 lb. per square inch. If the pipe gradually tapers to 4 in. diameter, find the pressure at this point. 66. A 12-in. horizontal pipe gradually tapers to a diameter of 6 in. If the flow is 60,000 gal. per hour, calculate the difference in pressure at two sections having these diameters. 67. A Venturi meter in an 18-in. main tapers to 6 in. at the throat, and the difference in pressure in main and throat is equivalent to 11 in. of mercury. Find the discharge in gallons per minute. Digitized by VnOOQlC 142 ELEMENTS OF HYDRAULICS 68^ The Ashokan Venturi meter on the line of the Catskill Aqueduct is 7 ft. 9 in. inside diameter at the throat, the diameter of the main being 17 ft. 6 in. (see Fig. 76). Find the difference in pressure between main and throat for the estimated daily- flow of 600,000,000 gal. 69. The velocity of flow in a water main 4 ft. in diameter is 3.6 ft. per second. Assuming the coefficient of friction to be 0.0216, find the frictional head lost in feet per mile. 70. Two cylindricar tanks each 8 ft. in diameter are connected near the bottom by a 2-in. horizontal pipe 26 ft. long. If the water level in one tank is initially 12 ft. and in the other 3 ft. above the center line of the pipe, find how long it will take for the water to reach the same level in both tanks. 71. Find the frictional head lost in a pipe 2 ft. in diameter and 6 miles long which discharges 200,000 gal. per hour, assuming the coefiicient of friction to be 0.024. 72. Find the required diameter for a cast-iron pipe 10 miles long to discharge 60,000 gal. per hour under a head of 200 ft. 73. A house service pipe is required to supply 4,000 gal. per hour through a 1.6-in. pipe and a 1-in. tap. The total length of the service pipe is 74 ft., including the tap which is 1.6 ft. long. Find the total pressure required in the main. Solution, — In the solution of water-supply problems of this type, it is recommended by W. P. Gerhard^ that the following formulas be used. Head lost in tap, hi = 0.024 l^j w^ ; V ,2 Head lost at entrance, hi = 0.62 i, , Head lost at stopcock = }4 liead lost in tap; Head lost in pipe by Prony's formula. G = [M!^]^1.20032) where, in this last formula, d = diameter of pipe in inches, H = head in feet, L = length in yards, G = discharge in U. S. gallons per minute. 1 Discharge of Water through Street Taps and House Service Pipes," Cassier's Mag., November, 1905. Digitized by VnOOQlC FLOW OF WATER 143 Using these formulas we obtain in the present case the following numerical results: Pressure lost in tap = 2.24 lb. per square inch. Pressure lost at stopcock = 1.12 lb. per square inch. Pressure lost at entrance = 3.08 lb. per square inch. Pressure lost in 72.5 ft. of 1-in. pipe = 17.63 lb. per square inch. Total pressure required in main = 23.97 lb. per square inch. 74. A building is to be supplied with 2,600 gal. of water per hour through 180 ft. of service pipe at a pressure at the building line of 15 lb. per square inch. The pressure in the main is 35 lb. per square inch. Find the required size of service pipes and taps. Solution, — The total drop in pressure in this case is 20 lb. per square inch. Therefore, using the formulas given in the pre- ceding problem and assuming different sizes of service pipes, the results are as follows: One 1.26-in. full-size pipe 180 ft. long discharges, 1,715 gal. per hour. Two 1-in. full-size pipes discharge together, 1,920 gal. per hour. Two 1.25-in. pipes with %-in. taps discharge together 2,880 gal. per hour. One 1.5-in. pipe with 1-in. tap discharges 2,619 gal. per hour. The last has sujfficient capacity and is cheapest to install, and is therefore the one to be chosen. 76. A pipe 1 ft. in diameter connects two reservoirs 3 miles apart and has a slope of 1 per cent. Assuming the coeflBcient of friction as 0.024, find the discharge and the slope of the hydrau- lic gradient when the water stands 30 ft. above the inlet end and 10 ft. above the outlet end. 76. Two reservoirs 5 miles apart are connected by a pipe line 1 ft. in diameter, the difference in water level of the two reservoirs being 40 ft. Assuming the value of Chezy's constant in feet and second units to be 125, find the discharge in gallons per hour. 77. A 12-in. main 5,000 ft. long divides into three other mains, one 6 in. in diameter and 6,000 ft. long, one 10 in. in diameter and 7,000 ft. long, and one 8 in. in diameter and 4,000 ft. long. The- total static head lost in each line between reservoir and outlet is the same and equal to 100 ft. Find the discharge in gallons per 24 hr. at each of the three outlets. Digitized by VnOOQlC 144 ELEMENTS OF HYDRAULICS SoltUion. — ^The head lost in friction in the length I is given by the relation »-/(!) and the discharge by Eliminating v between these relations, we have A, = whence 2gdir*d*' o-V^vs^ Assuming/ = 0.02 and I = 1,000 ft., the discharge Q for pipes of various sizes in terms of the head lost per 1,000 ft. is given by the following relations: Diameter of pipe in Discharge in gal. per 24 hours in terms of head inches lost per 1000 ft. 4 = 58,430 V;iz 6 = 161,000V;ii 8 = 330,500V/n 10 = - 577,500 V/u 12 = 911,000V^ 16 = l,870,000V^z 20 = 3,266,000V;ii 24 = 5,147,000V/ij 30 = 9,002,000V/u 36 = 14,200,000V^i 48 = 29,150,000V/ii 56 = 42,850,000V^ 60 = 50,920,000V^ 66 - 64,600,000 V^ii 72 = 80,320,000V;iz In the present case let the flow in gallons per 24 hr. be denoted by Q with a subscript indicating the size of pipe. Then Qn = Qe + Qs + Qio. Also if ft with the proper subscript denotes the head lost in each pipe per 1,000 ft., we have from the above relations 0» = 911,000Vft^ Digitized by Google FLOW OF WATER 146 Q, = 161,000 Vftii^ Qb =330,500V^, . Qio = 577,500VAi^ and since from the conditions of the problem the head lost in each line amounts to 100 ft., we also have the relations 5fti2 + efte = 100, 5fti2 + 7Aio = 100, 5/ii2 + 4ft8 = 100. From these relations we find fts = H^h] hio = ^A«; hn = ^ -> and substituting these values in the first equation, the result is 161,000\/Ai + 330,500\/^ + 577,500\/^ = 911,000 /lOO - 6*6 5 whence Ae = 7.52 and consequently hs = 12.28; hio = 6.446; hn = 10.976. Substituting these values of h in the formulas for discharge, the results are Qe = 444,360 gal. per 24 hr. Qs = 1,110,480 gal. per 24 hr. Qio = 1,465,700 gal. per 24 hr. Qe + Qs + Qio = 3,020,540 gal. per 24 hr. The actual calculated value of Q12 is Q12 = 3,015,400 gal. per 24 hr., the discrepancy between these results being due to slight inac- curacy in extracting the square roots. 78. A pipe of constant diameter d discharges through a number of laterals, each of area A and spaced at equal distances I apart (e.g., street main and house service connections). Find the rela- tion between the volume of flow in three successive segments of the maini (Fig. 125). ij. P. Fbizell, Jour FranUin Inst., 1878. 10 Digitized by VnOOQlC 146 ELEMENTS OF HYDRAULICS Solution. — The discharge at A is where vi denotes the velocity of flow at this point. Also the head lost in friction in the segment AB is ^-'(^'^ and consequently wKere 2ff hi = oQi* 8fl E ffjr«d» D C B \ \ \ M N R FiQ. 125. At B the pressure head \&h + hi and the discharge is 0« - Qi = KA y/2Q V/i + Ai = i£:A\/2^VA+"aQr*. Similarly, for the discharge at C and D we obtain the relations whence by elimination (Q4 - QzY - (Qs - Q2)» = 6U* where ,..i^»x4?. The general relation is therefore Digitized by VjOOQIC FLOW OF WATER 147 The following geometrical construction may be used for deter- mining Qn» Determine an angle ^ such that b = tan and lay off Q».i and Qn-s on a straight line so that OM » Q«_2 and ON = Qn-i as shown in Fig. 125. At N erect a per- pendicular NP to ON J and then lay off NR = MP. Then OR 79. A reservoir discharges through a pipe line made up of pipes of different sizes, the first section being 4,000 ft. of 24-in. pipe, followed by 5,000 ft. of 20-in. pipe, 6,000 ft. of 16-in. pipe and 7,000 ft. of 12-in. pipe. The outlet is 100 ft. below the level of the reservoir. Find the discharge in gallons per 24 hr. Sohdion. — Using the same notation as in Problem 77, we have in the present case 4^24 + 5hio + 6Ai6 + 7hi% = 100. Also, since 8/7 »'-(i^)«'. the loss in head per 1,000 ft. varies inversely as the fifth power of the diameter, and consequently fti. = (^)'*"= 7.594A,«, /24\» ^20 = Q hu = 2.488^,4. Solving these three equations simultaneously with the first one, the results are hu = 0.35; hio = 0.871; hu = 2.658; hit = 11.20. As a check on the correctness of these results we have 4X 0.35 = 1.400 5 X 0.871 = 4.355 6 X 2.658 = 15.948 7 X 11.20 = 78.400 100.103 Digitized by VjOOQIC 148 ELEMENTS OF HYDRAULICS The discharge may be found from the formulas given in Problem 77, the results being as follows: Qi2 = Qi6 = Q20 = Q24 = 3,046,000 gal. per 24 hr. 80. A new cast-iron pipe AB, 1,000 ft. in length, divides at B into two branches, BC which is 600 ft. long and BD which is 900 ft. long. The fall for AB is 26 ft., for BC is 10 ft. and for BD is 20 ft., and the velocity of flow in AB is 4 ft. per second. Find the required diameters of the three pipes to deliver 500 gal. per minute at C and 300 gal. per minute at D. 81. A reservoir empties through a pipe AB (Fig. 126) which branches at B into two pipes BC and BD, one of which discharges 20,000 gal. per hour at C and the other 30,000 gal. per hour at D. FiQ. 126. The lengths of the pipes are AB = 1,200 ft., BC = 900 it, BD = 600 ft., and the depths of the outlets below the surface of the reservoir are hi = 25 ft., hi = 60 ft. The pipes are of cast iron, and the velocity of flow in AB is to be 3 ft. per second. Calculate the diameters of all three, and the velocity of flow in BC and BD. 82. Two reservoirs empty through pipes which unite at C (Fig. 127) into a single pipe which discharges at D. The lengths of the pipes are h = 1,500 ft., h = 900 ft. and I = 2,400 ft. The diameters of the pipes are di = 6 in., ^2 = 4 in., and d = 9 in., and the depths of the outlet below the levels of the reservoirs are fti = 75 ft., hi = 100 ft. Find the velocity of flow in each pipe and the total discharge in gallons per hour. 83. A water main 3 ft. in diameter divides into two smaller mains of the same diameter and whose combined area equals Digitized by VnOOQlC FLOW OF WATER 149 that of the large main. If the velocity of flow is 3 ft. per second, compare the heads lost per mile in the large and small mains. 84. The head on a fire hydrant is 300 ft. Find its discharge in gallons per minute through 400 ft. of inferior rubber-lined cotton hose 2.5 in. in diameter and a 1.5-in. smooth nozzle. 86. What head is required at a fire hydrant to discharge 250 gal.' per minute through a 1.25-in. ring nozzle and 500 ft. of 2.5-in. best rubber-lined cotton hose? 86. A fire stream is deUvered through 100 ft. of 2-in. rubber- lined cotton hose and a nozzle l}i in. in diameter. The hy- drant pressure is 75 lb. per square inch. Find pressure at nozzle, discharge in gallons per minute and height of effective fire stream. Fig. 127. 87. Two reservoirs are connected by a siphon 16 in. in diameter and 50 ft. long. If the difference in level in the reservoirs is 25 ft., calculate the discharge, assuming the coefficient of pipe friction to be 0.02 and considering only friction losses. 88. A cast-iron pipe 2 ft. in diameter has a longitudinal slope of 1 in 2,500. If the depth of water in the pipe is 18 in., calculate the discharge. 89. A rectangular flume 6 ft. wide, 3 ft. deep and 1 mile long is constructed of unplaned lumber and is required to deliver 120 cu. ft. per second. Determine the necessary gradient and the total head lost. 90. The Aqua Claudia, shown in Fig. 128, was one of the nine principal aqueducts in use in first century, A.D., for supply- ing Rome with water. The lengths and capacities of these nine aqueducts were as follows: Digitized by VnOOQlC 150 ELEMENTS OF HYDRAULICS Alti- Date of Length Length Discharge tude of springs Level Name construc- m in per day above Rome tion feet miles in cu. ft. sea level in feet in feet Aqua Appia 312 B. C. 53,950 . 10.2 4,072,500 98 65 Anio Vetus 272-269 209,000 39.6 9,814,200 918 157 Aqua Marcia... 144-140 299,960 56.8 10,465,800 1043 192 Aqua Tepula. . . 125 58,200 11.0 993,000 495 199 Aqua Julia 33 74,980 14.2 2,691,200 1148 209 Aqua Virgo 19 67,900 12.9 5,587,700 79 65 Aqua Absietina. # 107,775 225,570 20.4 874,800 685 54 Aqua Claudia. . . 38-52 A. D. 42.7 7,390,800 1050 221 Anio Novus 3a-52A.D. 285,330 54.0 10,572,900 1312 231 CatskiU. Croton . Old Croton. Aqua Claudia. Fig. 128. — Comparison of ancient and modem aqueducts. The con£i;ruction of the earliest aqueducts was the simplest, most of them being underground. In the Aqua Appia only 300 ft. were above Digitized by LnOOQlC FLOW OF WATER 161 ground, and in the Anio Vetus only 1,100 ft. were above ground. In the Aqua Marcia 7.5 miles were supported on arches; in the Aqua Claudia 10 miles were on arches, and in the Anio Novus 9.5 miles were on arches. The construction of the last shows the greatest engineering skill, as it fol- lows a winding course, at certain points tunndling through hills and at* other crossing ravines 300 ft. deep. The cross-flection of the channels {apecua) varied at different points of the course, that of the largest, the Anio I^vus, being 3 to 4 ft. wide and 9 ft. high to the top, which was of pointed shape. The channels were lined with hard cement (opiia signinum) containing fragments of broken brick. The water was so hard that it was necessary to clean out the calcareous deposits frequently, and for this purpose shafts or openings were constructed at intervals of 240 ft. Filtering and settling tanks (piaciruB limarioBf or " purgatories '0 were constructed on the line of the aqueduct just outside the city, and within the city the aqueducts ended in huge distributing reservoirs {CMteUa) from which the water was conducted to smaller reservoirs for distribution to the various baths and fountains. Supposing the population of Rome and suburbs to have then numbered one million, there was a daily water supply of nearly 400 gal per capita. Modem Rome with a population of half a million has a supply of about 200 gal. per capita. The volume of water may also be compared with that of the Tiber which discharges 342,395,000 gal. per day, whereas in the first cen- tury, A.D., the aqueducts carried not less than 392,422,500 gal. per day, which by the fourth century had been increased by additional supplies to 461,628,200 gal. per day. Assuming that the Aqua Claudia had an average i/vidth of 3 ft. with 6 ft. depth of water, and that the grade was uniform and the difference in head lost in friction, calculate from the values tabu- lated above the velocity of flow and Chezy's constant C in the formula v = Cy/rs. 91. A channel of trapezoidal section with side slopes of two horizontal to one vertical is required to discharge 100 cu. ft. per second with a velocity of flow of 3 ft. per second. Assuming Chezy's constant as 115, compute the required bottom width of channel and its longitudinal slope. 92. A channel of trapezoidal cross-section has a bottom width of 25 ft. and side slopes of 1:1. If the depth of water is 6 ft. and the longitudinal inclination of the bed is 1 in 5,000, find the discharge, assuming the coefficient of roughness, n, in Kutter's formula to be 0.02. 93. A channel of rectangular section has a bottom width of 20 ft., depth of water 6 ft. and longitudinal slope of 1 in 1,000. Calculate the discharge, assuming the coefficient of roughness, n, in Kutter's formula to be 0.01. Digitized by VnOOQlC 152 ELEMENTS OF HYDRAULICS 94. A reservoir A supplies another reservoir B with 400 cu. ft. of water per second through a ditch of trapezoidal section, with earth banks, 5 milies long. To avoid erosion, the flow in this channel must not exceed 2 ft. per second. From reservoir B the water flows to three other reservoirs, C, D, E. Prom B to C the channel is to be rectangular in section and 4 miles long, constructed of unplaned lumber, with a 10-ft. fall and a discharge of 150 cu. ft. per second. From B to D the channel is to be 5 miles long, semicircular in section and constructed of concrete, with 12-ft. fall and a dis- charge of 120 cu. ft. per second. From B to E the channel is to be 3 miles long, rectangular in section and constructed of rubble masonry, with 15-ft. fall and a discharge of 130 cu. ft. per second. Find the proper dimensions for each channel section. 1 i .2 0. 4 0. 6 0. 8 1 1. lo ^ 1 \ ) \ 08 \ \ ^ y 5 \ % y J f A P $ ^ X / % / / / n 1 "g y /^ / V / U.4 o / / ^ X A A S / A ^ ^ \....._^ / ^ ^ >^ 0.2 0.4 0.6 0.8 1.0 1.2 Ratio of Q and V to their Values when Pipe is Full Fig. 129. 95. The flow through a circular pipe when completely filled is 25 cu. ft. per second at a velocity of 9 ft. per second. How much would it discharge if filled to 0.8 of its depth, and with what velocity? Solution. — Fig. 129 shows a convenient diagram for solving a problem of this kind graphically.^ The curve marked v (velocity) is plotted from Kutter's simplified formula V = /lOOVr \ rjy/Ts \b + Vr ^ Imhoff, "Taschenbuch flir Kanalisations Ingenieure." Digitized by Google FLOW OF WATER 153 for a value of b of 0.35, and the discharge Q from the formula Q =lAVf the ordinates to the curves shown in the figure being the ratio of the depth of the stream to the diameter of the pipe, and the abscissas the ratios of Q and v respectively to their values when the pipe flows full. To apply the diagram to the problem under consideration, observe that for a depth of 0.8d the abscissa of the discharge curve is unity, and consequently the discharge for this depth is the same as when the pipe is completely filled. The abscissa of the velocity curve corresponding to this depth 0.8d {i.e., with abscissa 0.8) is 1.13, and consequently the velocity at this depth is 1.13 X 9 = 10.17 ft. per second. Similar diagrams have been prepared by Imhoff for a large variety of standard cross-sections and are supplemented by other diagrams or charts which greatly simplify ordinary sewer calculations. 96. In the Catskill Aqueduct, which forms part of the water supply system of the City of New York, there are four distinct types of conduit; the cut-and-cover type, grade tunnel, pressure tunnel, and steel pipe siphon. The cut-and-cover type, shown in section in Fig. 130, is 55 miles in length, and is constructed of concrete and covered with an earth embankment. This is the least expensive type, and is used wherever the elevation and nature of the ground permits. The hydraulic data for the standard type in open cut' is as follows: Digitized by VnOOQlC 154 ELEMENTS OF HYDRAULICS s = 0.00021 Depth of Area of Wetted Hydraulic flow flow perimeter "^ radius mfeet in sq. ft. in feet in feet 17.0 241.0 67.4 4.20 FuU, 16.2 237.7 50.8 4.67 Max. cap. 15.3 230.9 47.7 4.84 14.0 217.6 44.1 4.92 12.0 192.6 39.4 4.88 10.0 163.2 35.0 4.65 8.0 130.7 30.9 4.24 6.0 97.1 26.8 3.61 4.0 62.2 22.8 2.72 2.0 27.0 18.8 1.47 In the preliminary calculations the relative value of Chezy's coefficient for this type was assumed to be C = 125. Using this value, calculate the maximum daily discharge. 97. Where hills or mountains cross the line of the Aqueduct, tunnels are driven through them at the natural elevation of the Aqueduct (Kg. 131). There are 24 of these grade tunnels, aggre- gating 14 miles. The hydraulic data for the standard type of grade tunnel is as follows: 8 = 0.00037 1 Depth of Area of Wetted Hydraulic flow flow perimeter radius in feet in sq. ft. in feet in feet 17.0 198.6 52.2 3.80 FuU, 16.26 195.6 46.0 4.25 Max. cap. 15.3 188.5 42.7 4.41 14.0 175.7 39.3 4.46 12.0 152.4 35.0 4.35 10.0 126.8 31.0 4.10 8.0 100.2 26.9 3.72 6.0 73.8 22.9 3.22 4.0 47.6 18.9 2.51 2.0 21.0 14.9 1.49 The relative value of Chezy'a coefficient for this type was assumed in the preliminary calculations to be C « 120. Using this value, calculate the maximum daily discharge and the corresponding velocity of flow. 98. Where the line of the Aqueduct crosses broad and deep valleys and there is suitable rock beneath them, circular tunnels Digitized by LnOOQlC FLOW OF WATER 155 are driven deep in the rock and lined with concrete (Fig. 132). There are seven of these pressure tunnels, with an aggregate FiQ. 131. Fig. 132. length of 17 miles. The hydraulic data fo/these pressure tunnels are as follows: Slope Diameter Area of waterway Wetted perimeter Hydraulic radius 0.00059 14 ft. 6 in. 165.1 sq. ft. 45.55 ft. 3.625 ft. Assuming the relative value Chezy's coefficient to be C = 120, calculate the velocity of flow and the daily discharge. 99. In valleys where the rock is not sound, or where for other reasons pressure tunnels are impracticable, steel pipe siphons are \- Width]- 3 --^ £»T ^MtfM^^f^^^ Fig. 133. used (Fig. 133). These are made of steel plates riveted together, from J4e to % in, in thickness, and are 9 ft. and 11 ft. in diameter respectively. These pipes are embedded in concrete and covered Digitized by VnOOQlC 156 ELEMENTS OF HYDRAULICS with an earth embankment, and are lined with 2 in. of cement mortar as a protection to the steel and also for the sake of smooth- ness. There are 14 of these siphons aggregating 6 miles in length, and three pipes are required for the full capacity of the Aqueduct. Assuming three mortar-lined 11-ft. pipes, having a relative coeffi- cient of C = 120 and a slope s = 0.00059, calculate the velocity of flow through them and the maximum daily discharge. 100. A broad shallow stream has naturally a depth of 3 ft. and a longitudinal slope of 5 ft. per mile. If a dam 8 ft. high is erected across the stream, determine the rise in level 1 mile up stream assuming the value of the constant C in Chezy's formula as 76. Digitized by VnOOQlC SECTION 3 ENERGY OF FLOW PRESSURE OF JET AGAINST STATIONARY DEFLECTING SURFACE 161. Normal Impact on Plane Surface. — When a jet of water strikes a stationary flat plate or plane surface at right angles, the water spreads out equally in all directions and flows along this plane surface, as indicated in Fig. 134. The momentum of the water after striking the surface is equal to the sum of the momenta of its separate particles, but since these flow off in opposite directions their algebraic sum is zero. Conse- quently the entire momentum of a jet is destroyed by normal impact against a stationary plane surface. To find the pressure, P, exerted by the jet on the surface, let A denote the cross section of the jet and v its velocity. Then the mass of water flowing per unit of time is M = yAv and, consequently, from the principle of impulse and momentum, yAv^ I Pdt=- Mv -- g For uniform or steady flow, P is constant, and if M denotes the quantity flowing per unit of time, then t is unity. In this case the above expression for the hydrodynamic pressure P of the jet on the surface becomes 7Av* P = g (98) 167 Digitized by VnOOQlC 158 ELEMENTS OF HYDRAULICS If h denotes tiie velocity head, then * = s"' ^^^ ^^' ^^^^ may be written P = 27Ah. (99) 162. Relation of Static to Dynamic Pressure. — ^If the orifice is closed by a cover or stopper, then the hydrostatic pressure P' on this cover is approximately equal to the weight of a column of water^of height h and cross-section A; and consequently. P' = yAh. (100) Comparing Eqs. (99) and (100), it is apparent that the normal hydrodynamic pressure of a jet on an external plane surface is twice as great as the hydrostatic pressure on this surface would be if it was shoved up against the opening so as to entirely close the orifice. In deriving this relation, the coefficient of efflux is assumed to be unity; that is, the area A of the jet is assumed to be the same as that of the orifice, and the velocity v to be the full value corresponding to the head h. Since the coefficient is actually less than unity, the hydrodynamic pressure never attains the value given by Eq. (99). For instance, in the case of flow from a standard orifice, if A denotes the area of the orifice and a the crossHsection of the jet, then from Arts. IX and X a = 0.62A, and v = 0.97 V2^. Therefore the expression for P becomes p = 1^ = 27(0.62A)(0.97*A) = l.lQy Ah instead of 2y Ah, as given by Eq. (99). Note, however, that this apparently large discrepancy is due chiefly to the fact that the area A in Eq. (99) denotes the cross-section of the jet, whereas in Eq. (100) it denotes the area of the oriflce. If the area A in both expressions denotes the cross-section of the jet, Eq. (99) is practically true, and the hydrodynamic pressure is approxi- mately twice the hydrostatic pressure on an equal area. 163. ObUque Impact on Plane Surface. — ^If a jet strikes a stationary plane surface obliquely, at an angle a (Kg. 135), the axial velocity!; of the jet may be resolved into two components, v sin a normal to the surface, and v cos a tangential to the surface. If the surface is perfectly smooth, the water flowing along the Digitized by VnOOQlC ENERGY OF FLOW 159 surface experiences no resistance to motion, and the pressure, P) exerted on the surface is that corresponding to the normal velocity component w' = i; sin a. The area to be considered, however, is no\r a right sec- tion, A, of the jet, but a sec- tion A' normal to the com- ponent t; sin a, as indicated in Fig. 135. The total pres- sure, P, exerted on the surface, is then or, since w' = t; sin a and A' = -, this may be written in the form Fia. 135. sma' P = - — sm a. g (101) If a = 90**, this reduces to Eq. (98). 164. Axial Impact on Surface of Revolution. — If the surface on which the jet impinges is a surface of revolution, coaxial with the jet (Fig. 136), then in this case also the particles spread out equally in ^ directions, and consequently the sum of the mo- menta of the particles in the direction perpendicular to the axis of the jet is zero. The velocity of any particle in a direction parallel to the axis of the jet, however, becomes v cos a, where a denotes the angle which the final direction taken by the particles Digitized by LnOOQlC 160 ELEMENTS OF HYDRAULICS makes with their initial direction, as indicated in Fig. 136. The total initial momentum is then 9 and the total final momentum is Mv cos a = cos a. 9 Therefore, equating the impulse to the change in momentum, we obtain the relation g (1 — cos q). (102) For a jet impinging normally on a plane surface, a = 90**, and this expression reduces to Eq. (98). 166. Complete Reversal of Jet — If a is greater than 90**, then cos a becomes negative and the pressure P is correspondingly Fig. 137. increased. For example, if the direction of flow is completely, reversed, as shown in Fig. 137, then a = 180**, cos a = — 1, and hence * P = 2l|vl (108) The hydrodynamic pressure in this case is therefore twice as great as the normal pressure on a flat surface, and four times as great as the hydrostatic pressure on a cover over an orifice of the same area as the cross-section of the jet. 166. Deflectioa of Jet. — When a jet is deflected in an oblique direction, the final velocity v may be resolved into components V cos a and v sin a, as indicated in Fig. 138. The component Digitized by LnOOQlC ENERGY OF FLOW 161 of the final momentum parallel to the initial direction of the jet is then Mv(l — cos a) = - — (1 — cos a), and the horizontal component, J?, exerted in this direction is (104) H = ^^'(1 - cos a). g Similarly, the component of the final momentimi perpendicular to the initial direction of the jet is Mv sin « = sm a, 9 Vcostf Fia. 138. and the vertical component, F, exerted in this direction is V = -^ sin a, (106) The total pressure of the jet on the deflecting surface, or reaction of the surface on the jet, is, then, P = VlP~+Y^ = ^^V(l - cos a)2 + sin2 a which simpUfies into P = ^^V2(l - cos a). (106) A more convenient expression for P may be obtained by using 11 Digitized by VnOOQlC 162 ELEMENTS OF HYDRAULICS the trigonometric relation ^- — cos a . 1 V r — = sin -a, by means of 2 ^ which Eq. (106) may be written in the form 27Av' sin ^. g 2 (107) 167. Dynamic Pressure in Pipe Bends and Elbows. — When a bend or elbow occurs in a pipe through which water is flowing, the change in direction of flow produces a thrust in the elbow, as in the case of the deflection of a jet by a curved vane, considered in the preceding paragraph. From Eqs. (106) and (107), the amount of this thrust P is P =^ V 2(1 — cos a) = — sm 2' Fig. 139. and the direction of the thrust evidently bisects the angle a, as indicated in Fig. 139. In the case of jointed pipe lines if the angle of deflection is large or the velocity of flow considerable, this thrust may be sufficient to disjoint the pipe unless provision is made for taking up the thrust by some form of anchorage, as, for example, by filling in with concrete on the outside of the elbow. XXXI. PRESSURE EXERTED BY JET ON MOVING VANE 168. Relative Velocity of Jet and Vane. — In the preceding article it was assumed that the surface on which the jet impinged was fixed or stationary. The results obtained, however, remain valid if the surface moves parallel to the jet in the same or op- Digitized by VnOOQlC ENERGY OF FLOW 163 posite direction, provided the velocity, v, refers to the relative velocity between jet and surface. Thus if the surface moves in an opposite direction to the jet with a velocity v\ the relative veloc- ity of jet and surface is t; + v' and the pressure is correspondingly increased, whereas if they move in the same direction, their rela- tive velocity is v— v', and the pressure is diminished. 169. Work Done on Moving Vane. — Consider, for example, the case of a jet striking a deflecting surface and assume first that this surface moves in the same direction as the jet with velocity v' (Fig. 140). Since the surface, or vane, is in motion, the mass of water, ilf ', reaching the vane per second is not the same as the mass of water, M, passing a given cross-section of the jet per second. That is, the mass, ilf , issuing from the jet per second is Fig. 140. whereas the mass, M', flowing over the vane per second is yA(v - v') M' = a (108) Therefore the components of the force acting on the vane, given by Eqs. (104) and (105), become in this case H = M'(v - vO(l - cos a) = ^ (v - vy{l - cos a), y yA V = M\v — t;') sin a = ^—{v — v'y sin a. Since the motion of the vane is assumed to be in the direction of the component H, the component V, perpendicular to this direc- tion, does no work. The total work, TF, done on the vane by the jet is therefore W = Hv' = 7AvXv - vV (J _ ^^g ^^^ ^lOg^ Digitized by VnOOQlC 164 ELEMENTS OF HYDRAULICS 170. Speed at which Work Becomes a Maximum. — ^The con- dition that the work done shall be a maximum is ^ = = ^ (1 - cos «)[(. - »')» - 2v'iv -t/)], dW _^_Ay 9 whence V = }. (110) Substituting this value of v' in Eq. (109), the maximum amount of work that can be realized under the given conditions is found to be '^3 7 v\^ 4rYAv^ TTma. = "7^ («' T gj (1 " COS a) = -^ (1 - COS a). 171. Maximum Efficiency for Single Vane. — ^The efficiency of a motor or machino is defined in general as -^ . Useful work .^-^v ^^"•'^''y = Total energy available ' ^^"^ Since, in the present case, the total kinetic energy of the jet is the efficiency, E, becomes 4yAv^ ,- V 2g The maximum efficiency occurs when a = 180**, in which case E«a. = 1^ = 69-2 per cent. (112) 172. Maximum Efficiency for Continuous Succession of Vanes. — If there is a series of vanes following each other in succession so that each receives only a portion of the water, allowing this portion to expend its energy completely on this vane before leav- ing it, then the mass M' in Eq. (108) is replaced by M, and the component H becomes H = ilf(v - i;')(l - cos a) = ^^^(^ - ^') (1 - cos a). Digitized by vnOOQlC MEROY of Flow 166 The work done on the series of vanes is therefore W = Hv' = 'yAw^C^ -^0 (1 _ cos a). (US) The condition for a maximum in this case is ^ = =^(1 - cos«)[(r - t/) - t;'], whence (114) Substituting this value of r' in' Eq. (113), the maximum work which can be realized from a series of vanes moving parallel to the jet is yAv^ ( _v\ ^ 2 V 2/,, , yAv\, . Wfnax = (1 - cos c) = ^Tr-(1 - COS o). Hence the efficiency in this case is ^ — (1 — cos a) ^ ''-^^-^^ i(l-cosa). The maximum efficiency therefore occurs when a = 180®, its value being E«a« = 2 (2) = 100 per cent (116) The actual efficiency of course can never reach this upper limit, as the conditions assumed are ideal, and no account is taken of frictional and other losses. 173. Impulse Wheel; Direction of Vanes at Entrance and Exit. — In general, it is not practicable to arrange a series of vanes so as to move continuously in a direction parallel to the jet. As usually constructed, the vanes are attached to the circumference of a wheel revolving about a fixed axis (Fig. 141). Let « denote the angular velocity of the wheel about its axis, and ri, rt the radii of the inner and outer edges of the vanes. Then the tan- gential or Unear velocities at these points, say ui and ti2, are Ui = rico; U2 = r2W. Now let Vi denote the absolute velocity of the jet at entrance to Digitized by vnOOQlC 166 ELEMENTS OF HYDRAULICS the vane, and V2 its absolute velocity at exit. Then by forming a parallelogram of velocities on ui and Vi as sides, the relative velocity, Wi, between jet and vane at entrance is determined, as indicated in Fig. 141. Similarly, the parallelogram on U2, W2 as sides determines the absolute velocity, V2, at exit. In order that the water may gUde on the vane without shock, the tip of the vane at entrance must coincide in direction with the vec- tor tVi. Shaft Fig. 141. 174. Work Absorbed by Impulse Wheel. — ^Let M denote the mass of water passing over the vane per second. At entrance the velocity of this mass in the direction of motion {i.e., its tan- gential velocity) is Vi cos a, and at exit is V2 cos j8, where a and P are the angles indicated in Fig. 141. The Unear momentum of the mass M at entrance is then Mvi cos a, and its angular momentum is MviVi cos a. Similarly, its Unear momentum at exit is Mv2 cos P and its angular momentum is Mv2r2 cos p. The total change in «,ngular momentum per second, that is, the amount given up by the water or imparted to the wheel, is then MviTi cos a — Mv2r2 cos p. For a continuous succession of vanes, as in the case of an ordinary impulse wheel, the mass M is the total amount of water supplied by the jet per second. Hence, if T denotes the total torque ex- erted on the wheel, by the principle of angular impulse and Digitized by LnOOQlC ENERGY OF FLOW 167 momentum, remembering that M is the mass of water flowing per unit of time, and consequently that the time is unity, T = M(viri cos a - V2r2 cos j8). (116) The total work imparted to the wheel is W = Tw, or, since M = -) Ui = r{(a, u% = r2w, the expression for the work becomes W = Tw = '^^ (uiVi cos o - U2V2 cos j3). (117) These relations will be applied in Art. XXXV to calculating the power and efficiency of certain types of impulse wheels. XXXn. REACTION OF A JET 176. Effect of Issuing Jet on Equilibrium of Tank. — Consider a closed tank containing water or other liquid, and having an orifice in one side closed by a cover. When the cover is removed Fig. 142. the equiUbrium of water and tank will be destroyed. At the instant of removal this is due to the disappearance of the pressure previously exerted on the cover considered as part of the tank. After the jet has formed and a steady flow has been set up, as- suming that the depth of water is maintained constant by supply- ing an amount equal to that flowing out, as indicated in Fig. 142, the pressure within the fluid and on the walls of the tank will not regain its original static value, since, in accordance with Ber- noulU's theorem, an increase in velocity must be accompanied by a corresponding decrease in pressiure. Digitized by VnOOQlC 168 ELEMENTS OF HYDRAULICS 176, Energy of Flow Absorbed by Work on Tank. — To calcu- late the effect of the flow on the equilibrium of the tank, suppose that the tank is moved in the direction opposite to that of the jet and with the same velocity, v, as that of the jet. Then the relative velocity of the jet with respect to the tank is still v, but the absolute velocity of the jet is zero and consequently its kinetic energy is also zero. If h denotes the head of water on the orifice (Fig. 142) and Q the quantity of water flowing per second, then its loss in potential energy per second is yQh. Moreover, while this volume of water Q moved with the tank, it had a velocity v, yQv^ and therefore possessed kinetic energy of amount 2~' '^^® total energy given up by the water in flowing from the tank is then .0A + ^««'*- t;2 or, since A = h" approximately, these terms are equal, and the 2g ' hese total energy lost by the water becomes Energy given up = ^*. Now let P denote the reaction of the jet, that is, the resultant of all the pressure exerted on the tank by the water except that due to its weight. Then, since the distance traversed by the force P in a unit of time is the .velocity v of the tank, by equating the work done by P to the energy givea up by the water, we have g whence g g The reaction P is therefore twice the hydrostatic pressure due to the head h. This is also apparent from the results of Par. 162, since the pressure of a jet on a fixed surface close to the orifice must be equal to its reaction on the vessel from which the jet issues. The actual reaction of the jet is of course somewhat less than its theoretical value, as given by the relation P = 2yAh, since there are various losses, due to internal friction, etc. 177, Principle of Reaction Turbine. — In order for the tank to retain its uniform velocity, v, a resistance of amount P must Digitized by VnOOQlC ENERGY OF FLOW 169 constantly be overcome, for if the resistance is less than this amount the motion will be accelerated. It is apparent, there- fore, that by a proper choice of the velocity, w, of the tank it is possible to utilize almost the entire energy of the jet in over- coming a resistance coupled up with the tank. This is the prin- ciple on which the reaction turbine is based, as explained in Art. XXXVI. It should be noted that if the water flowing out is continually replaced from above, half of the available energy must be used in giving the supply water the same ve- locity as the tank. The useful work is therefore reduced to one-half the previous amount, and the available energy is only that due to the velocity head h. 178, Barker's Mill,— The simplest practical appUcation of the reaction of a jet is the apparatus known as Barker's mill (Fig. 143). In this ap- paratus water flows from a tank into a hollow vertical arm, or spindle, pivoted at the lower end, and from this into a horizontal tubular arm, having two orifices near the ends on opposite sides. The jets issue from these orifices, and their reactions cause the horizontal arm to rotate, driving the central spindle from which the power is taken oflf by a belt and pulley. The steam turbine invented by Hero of Alexandria in the first century B.C., is an almost identical arrangement, the motive power in this case being due to the reaction of a jet of steam instead of a jet of water. \L Y 1] wmmmmmmm. SIDE ELEVATION a PLAN Fig. 143. XXXm. TYPES OF HYDRAULIC MOTORS 179* Current Wheels. — ^There are three general types of hy- draulic motors, namely: 1. Current and gravity wheels. 2. Impulse wheeds and turbines. Digitized by VnOOQlC 170 ELEMENTS OF HYDRAULICS 3. Reaction turbines. The current wheel is the oldest type of prime mover, and in its primitive form consisted of a large vertical wheel, with a set of paddles or buckets attached to its circumference, and so placed in a running stream that the current acting on the lower, or im- mersed, portion produced revolution of the wheel. A later im- provement consisted in placing the wheel at the foot of a waterfall and conducting the water by a flume to the top of the wheel, the action of the water in this case being due almost entirely to its weight (Art. XXXIV). 180. Impulse Wheels. — The impulse wheel, as its name indi- cates, is designed to utilize the impulsive force exerted by a jet moving with a high velocity and striking the wheel tangentially. The wheel, or runner, in this case carries a series of curved buckets or vanes which discharge into the atmosphere. A feature of this type is that the runner rotates at a high velocity and can there- fore be made of comparatively small diameter. The two prin- cipal types of impulse wheel are the Girard impulse turbine, which originated in Europe, and the Pelton wheel, which was developed in the United States (Art. XXXV). 181. Reaction Turbines. — The reaction turbine depends chiefly on the reaction exerted by a jet on the vessel from which it flows, which in this case is the passage between the vanes on the runner. In an impulse wheel the energy of the water as it enters the wheel is entirely kinetic, and as there is free circulation of air between the vanes and they discharge into the atmosphere, the velocity of the water is that due to the actual head. In a reac- tion turbine the energy of the water as it enters the wheel is partly kinetic and partly pressure energy, and as the water completely fills the passages between the vanes, its velocity at entrance may be either greater or less than that due to the static head at that point. A feature of the reaction turbine is that it will operate when completely submerged. 182. Classification of Reaction Turbines. — Reaction turbines are subdivided into four classes, according to the direction in which the water fiows through the wheel. These are: 1. Radial outward-fiow turbines. 2. Radial inward-flow turbines. 3. Parallel or axial-flow turbines. 4. Mixed-flow turbines, the direction of flow being partly radial Digitized by VnOOQlC ENERGY OF FLOW 171 and partly axial, changing from one to the other in passing over the vanes (Art. XXXVI). 183, Classification of Hydratilic Motors. — The following tabu- lated classification is useful as a basis for the description of the various types of hydraulic motors given in Arts. XXXIV, XXXV and XXXVI. Current and Gravity Wheels: Utilizes impact of current or weight of the water. Impuhe Wheels and Turbines: Utilizes kinetic energy of jet at high velocity. Suitable for limited amount of water imder high head. Ordinarily used for heads from 300 ft. to 3000 ft. Reaction Turbines: Utilizes both kinetic and pressure en- ergy of water. Suitable for large quantities of water under low or me- dium head. Ordinarily used for heads from 5 to 600 ft. Current wheel, Undershot wheel (Poncelet), Breast wheel, Overshot wheel. Girard turbine (European), Pelton wheel (American). Radial inward flow (Francis type), Radial outward flow (Foumeyron type), Parallel or axial flow (Jonval type), Mixed flow (American type). XXXIV. CURRENT AND GRAVITY WHEELS 184. Current Wheels. — The vertical current wheel, mentioned in Par. 179, was the earliest type of hydraulic motor, dating from prehistoric times, although they are still in use in China and Syria. 186. Undershot Wheels.— The first improvement consist- ed in confining the water in a sluice and delivering it directly on the vanes. This type was known as the Undershot wheel, and was in common use until about the year 1800 A.D. (Fig. 144). Flat radial vanes were used with this type, for which the maximum theoretical effi- ciency was 50 per cent., the velocity of the vanes to reaUze this efficiency being one-half the velocity of the stream, as explained in Art. XXX. The actual efficiency of such wheels was much lower, being only from 20 to 30 per cent. Fig. 144. — Undershot wheel. Digitized by VnOOQlC 172 ELEMENTS OF HYDRAULICS 186. Poncelet Wheels. — Undershot wheels were greatly im- proved by Poncelet who curved the vanes, so that the water entered without shock and was discharged in a nearly vertical direction (Fig. 145). The water thus exerted an impulse on the vanes during the entire time it remained in the wheel, thereby raising the actual efficiency to about 60 per cent. Undershot Fig. 145. — Poncelet wheel. Fia. 146.— Breast wheel. Head Race J wheels of the Poncelet type are adapted to low falls, not exceed- ing 7 ft. in height. 187. Breast Wheels. — ^A modification of the undershot wheel is the Brectst wheel, the water being deUvered higher up than for an ordinary undershot wheel, and retained in the buckets during the descent by means of a breast, or casing, which fits the wheel as closely as practicable (Fig. 146). Wheels of this type are known as high-breast, breast, and low-breast according as the water is deUvered to the wheel above, at, or below the level of the center of the wheel. The high- breast wheel operates almost en- i Race tirely by gravity, that is, by the unbalanced weight of the water Fig. 147.— Overshot wheel. in the buckets, its efficiency be- ing from 70 to 80 per cent. Breast and low-breast wheels operate partly by gravity and partly by impulse, the efficiency varying from about 50 per cent. for small wheels to 80 per cent, for large wheels. This type was in use until about 1850. 188. Overshot WheeU. — ^A more recent type is the Overshot wheel, the characteristic of this type being that the water is OT M Digitized by vnOOQlC. ENERGY OF FLOW 173 delivered at the top of the wheel by a sluice, as indicated in Fig. 147. For maximum efficiency the diameter of the wheel should be nearly equal to the height of the fall, the efficiency for well- designed overshot wheels ranging from 70 to 85 per cent., which is nearly as high as for a modem turbine. An overshot wheel at Troy, N, Y., is 62 ft. in diameter, 22 ft. wide, weighs 230 tons, and develops 550 h.p. Another on the Isle of Man is 72 ft. in diameter and develops 150 h.p. XXXy. IMPT7LSE WHEELS AND TURBINES 189, Pelton WheeL — ^The intermediate link between the old type of waterwheel and the modem impulse wheel was the Hurdy Pelton Bucket Doble Bucket Fio. 148. Giwdy, which was introduced into the mining districts of Cali- fornia about 1865. This somewhat resembled the old current wheel, being vertical with flat radial vanes, but differed from it in Digitized by Google 174 ELEMENTS OF HYDRAULICS L ^H ir IZX4J r -L^H H 1 liiJ.---J.^- _ ^ t^^^l^^^^H r ..-^.JlMH^^Hririfl^^l ■ Fig. 149. Fig. 150. Digitized by )^nOOQl6 ENERGY OFFLOW 176 that it was operated by a jet impinging on the vanes at high velocity. The maximum theoretical efficiency of the Hurdy Gurdy was 50 per cent. (Art. XXX), while its actual efficiency varied from 25 to 35 per cent. The substitution of curved buckets for the flat radial vanes was the great improvement which converted the Hurdy Gurdy into the Pelton wheel. The construction of the bucket is shown in Fig. 14S, the jet being divided by the central ridge and each half deflected through nearly 180°. Evidently the angle of de- flection must be slightly less than 180°, so that the discharge from one bucket may clear the one following. A later improve- ment* is the Doble bucket, also shown in Fig. 148, each .half of which is ellipsoidal in form, with part of the outer lip cut away so as to clear the jet when coming into action. Fig. 151. The relation of the jet to the wheel is shown in Fig. 149, the type there shown being arranged with a deflecting nozzle for eco- nomic regulation. A more recent type of Pelton wheel is shown in Fig. 150, the features of this type being the Doble buckets and the so-called chain type of attachment of the buckets. One of the most important features of construction in this type of impulse wheel is the needle valve for regulating the flow. The cross-section shown in Fig. 151 indicates the location of the needle valve with respect to the nozzle. The methods of oper- ating the valve and of elevating and depressing the nozzle are shown in Fig. 152. This form of nozzle under the high heads ordinarily used gives a very smooth and compact jet, as shown by the instantaneous photograph reproduced in Fig. 153. Digitized by VnOOQlC 176 ELEMENTS OF HYDRAULICS FiQ. 152. — ^Pelton.regiilating-needle Dozzle. Fia. 153. Digitized by Google ENEBOY OF FLOW 177 190. Efficiency of Pelton WheeL — ^If the jet was completely reversed in direction and the speed of the buckets was one-half that of the jet, the theoretical efficiency of the Pelton whed would be unity, or 100 per cent., as shown by Par. 172, Eq. (116). This is also apparent from other considerations; for if the velocity of the jet is v and that of the buckets is ^, then the velocity of the water relative to the lowest bucket is v — 2> or g (Fig. 154). Therefore, at exit the water is moving with velocity g relative Fig. 154. to the bucket while the bucket itself is moving in the opposite direction with velocity s- Hence the absolute velocity of the water at exit ia ^ — ^9 or zero, and therefore, since the total kinetic energy of the water has been utilized, the theoretical efficiency of the wheel is imity. As a matter of fact there are hydraulic friction losses to be taken into account and also the direction of flow is not completely reversed. The efficiency of the Pelton wheel has been found in a niunber of authentic tests to -exceed 86 per cent. The actual efficiency in operation de- pends of course on the particular hydrauKc conditions under which the wheel operates. A good idea of what may be ex- pected in practice, ^however, is given by the following data, ob- tained from a imit of 4,000 kw. normal capacity, operating under a head of 1,300 ft.: 12 Digitized by VnOOQlC 178 ELEMENTS OF HYDRAULICS ' Load in KW Percentage of nonnal capacity Wheel efficiency 5000 4000 3000 2000 125% 100% 76% 50% 81% 83% 82% 70% Since the Pelton wheel operates by utilizing the kinetic energy of the water, it is best adapted to a small discharge under a high head. 191. Characteristics of Impulse Wheels. — The performance of an impulse wheel may be judged from the value of a certain combination of wheel constants known as the "specific speed/' or better, as the "characteristic speed/' The nature of this quantity is explained in Art. XXXVII, under the discussion of re- action turbines, the form there given applying also to impulse wheels. In discussing its application the following notation will be used: Let h = effective head in feet; H.P. = horse power developed by wheel; D = diameter of runner at pitch circle in inches; n = speed of runner in r.p.m. ; d = diameter of jet in inches; Q = discharge in cubic feet per minute; e = hydraulic efficiency of wheel; Ns = characteristic (specific) speed. Then as explained in Art. Ne is defined as XXXVII the characteristic speed N. = h^ (118) This quantity, N„ may be used to classify the various types of impulse wheels as indicated in the following table: Impulse wheels N. 1 2 3 4 5 Efficiency at ?i Load 80% 79% 78% 77% 76% The numerical values of N, in this table refer to the maxi- Digitized by VnOOQlC ENERGY OF FLOW 170 mum power of one nozzle only. In case the characteristic speed lies beween 5 and 10 it is therefore necessary to use more than one nozzle. For example, suppose that an impulse wheel is required to develop 1,300 hp. under a 400-ft. head at an efficiency of not less than 78 per cent. From the preceding table it is apparent that it is necessary to use a wheel having a characteristic i^)eed of about 3. If a single wheel and nozzle is used, the speed in r.p.m. at which it must run is found from Eq. (118) to be n = N.hyj^^ = 3 X 400 ^^^ = 150 r.p.m. If two nozzles are used, each furnishes half of the power and the corresponding speed is 150\/2 = 212 r.p.m. With four nozzles acting on two runners the required speed would be ISOVi = 300 r.p.m., and for 6 nozzles acting on 3 runners, n = 150\/6 = 367 r.p.m. Since the value of iV, is the same in each case, the efficiency is practically 78 per cent, in each case although there is a wide difference in the speed and setting. A quantity equally valuable in determining the performance of an impulse wheel is the ratio obtained by dividing the pitch diameter of the wheel by the diameter of the jet.* An expression for this ratio in terms of the wheel constants may be obtained as follows: The peripheral velocity of the wheel on the pitch circle is some fraction, say <py of the jet velocity. Since this peripheral velocity is of amount To~o an f^- P®r second we have therefore vDn 12 X60 whence = <pV2gh j^ ^ 12 X 60 ipV^gh ^ 1 838^. Moreover, the horse power developed is „_ 62.37Qhe ^•^' — 66(r~ » Carpenter, Eng, Record, June 17, 1916, p, 795. Digitized by VnOOQlC < 180 ELEMENTS OF HYDRAULICS or, assuming the average efficiency as 80 per cent., this becomes Since Q = ^^ ..^ \/2gh cu. ft. per second, we have therefore whence d = p'-^"X^** = l6.9l^^E (120) / .P. X 11 > 4 The required ratio, say R, is then l,838»>h>* R == D ?L_^ ^ 116.6 -^. (121) ^ 16.91^ "^"^ The value of <pm this expression varies from about 0.42 for ordi- nary foundry finish to 0.47 for polished buckets. Its average value may be taken as ^ = 0.45. Since the characteristic speed is given by the expression for R may also be written in the form R ^ 116.6 ^ or its equivalent, RNt = 115.5^ = constant. The following table gives an idea of the comparative values of R and Ntj computed for <p = 0.45 and 6 = 80 per cent. Comparative values of Nu and R = D d fi 1 7 1 8 9 1 10 1 11 1 12 1 13 1 14 1 16 1 16 |17 18 19 20 i\r.|7.43 6.60 5.78 5.20 4.73|4.33|4.00i3.72|3.47|3.25 3.05|2.89 2.74|2.60| The distinction between these two quantities is that the characteristic speed N, indicates whether the runner is to be of a high- or low-speed type, whereas the ratio R conveys some idea Digitized by Google ENERGY OF FLOW 181 as to the size of the finished wheel and the number of buckets required. For a single jet acting on a single runner, the characteristic speed ranges from to about 5, or possibly 6 as a maximum. For an efficient unit, therefore, the value of R should not lie below 10. In the case of a runner with more than one jet, the comparative characteristic speed varies directly as the square root of the nimiber of jets, and the equivalent R varies inversely as the square root of the nimiber of jets. Thus a runner for ff = 10 but with two jets instead of one has an equivalent R of -y= = 7.07 and a corresponding characteristic speed of 5.20\/2 = 7.35. 192. Girard Impulse Turbine. — A type of impulse wheel has been developed in Europe, known as the Girard im- pulse turbine. In this type the shaft may be mounted either vertically or horizontally, and the flow may be either radial or axial. The type shown in Fig. 155 is arranged for radial flow,- with vertical shaft. The construction is practically the same in all cases, the water entering through a pipe J?, as shown in Fig. 155, and proceeding through one or more guide passages C, which direct the water onto the vanes. The quan- tity of water admitted to the vanes is regulated by some kind of gate, that indicated in Fig. 155 being a sliding gate operated by a rack and pinion not shown in the figure. As the vanes are more oblique at exit than at entrance, they are necessarily closer tpgether at exit. To prevent choking, it is therefore necessary to widen the vanes laterally at exit, as shown in elevation in Fig. 155. As the water discharges under atmospheric pressure, ventilating holes are made through the sides of the vanes at the back to allow free admission of air. 193. Power and Efficiency of Girard Turbine. — The power and efficiency of a Girard turbine may easily be calculated from the results of Art. XXXI. Using the same notation, as indicated on Fig. 141, from Eq. (117), Par. 174, the work done per second on the wheel is given by the relation Work per second = {uiVi cos a — u^vz cos fi). Since the water is under atmospheric pressure, the absolute velocity vi of the water at entrance is calculated from the effective Digitized by VnOOQlC 182 ELEMENTS OF HYDRAULICS Fig. 155. Fig. 156. Digitized by VnOOQlC ENERGY OF FLOW 183 head h by means of the relation vi = y/2gh. From Fig. 156 we have by geometry vi cos a = Ui + Wi cos $, V2 cos P = Ui — Wi cos ^, and from the law of cosines Vi^ = Wi* + toi* + 2iiiix;i cos $. v^ = ti2^ + W2^ — 2U2W2 cos ^. Now the total energy imparted to the wheel per second is the difference in kinetic energy at entrance and exit, namely, Therefore, equating this to the expression for the work done per second, as given above, we have }i{vi^ — V2^) = UiVi cos a — U2V2 cos j8. Substituting in this equation the values given above for Vi cos a, V2 cos j8, vi^ and V2^ and reducing, the result is finally Wi« - Wa^ = Ui^ - U2^ (122) It is evident that the efficiency will be greater the more nearly the jet is reversed in direction, that is, the smaller <p becomes, or what amounts to the same thing, the smaller the absolute velocity ^2 of the water at exit. However, <p cannot be decreased indefinitely as it is necessary to provide a sufficient area at exit to carry the discharge. For a given value of <p, V2 will attain nearly its minimum value when U2 = lOj. In this case, however, by Eq. (122) we have ui = Wi, in consequence of which and hence e = 2a, and | = 90*" - /5, ^2 = 2u2 sin ^, and Ui — ^ 2 ' ""^ ^^ 2 cos a Now the peripheral velocity of the inner and outer ends of the vanes in terms of the angular velocity w of the runner is given by the relations wi= riw, 1^2= r2w, whence ti2 r2 Uir2 — = ~, or ti2 = -^• Ui ri' ri Digitized by LnOOQlC 184 ELEMENTS OF HYDRAULICS Substituting this value of Ut in the expression given above for vt, we have 2uir2 sin ^ ViU sin ^ 2 ri ri cos a Consequently if Q denotes the quantity of water discharged per second, the energy utilized per second is O O J r2^sin2-|\ Energy per second = g (n* - .,*) = ^(^1 - ^:^^^J- Substituting y = 62.4"and dividing by 550, the expression for the horse power of the wheel is therefore H.P. = «2!*QlL* 2g(660) \ri/ cos* a (123) yQvi^ Since the total kinetic energy available is —5 — , the efficiency, E, of the wheel, as defined by Eq. (Ill), Par. 171, is ,sin»| E = 1 - (-') — ^. (124) \ri/ cos* a ^ ' Since E is less than unity, it is evident that the maximum theoretical efficiency is always less than 100 per cent., and also that the efficiency is greater the smaller the angles a and <p. In practice the angle a is usually between 20° and 25°; <p be- tween 15° and 20°; and the ratio — between 1,15 and 1.25. Assuming as average numerical values a = 20°, <p = 15°, ^ = 1.15, and substituting these values in Eq. (124), the theoretical effi- ciency of the wheel in this case is found to be E [sin 7 5 n ^ 1.15 ^ = 97.5 per cent, approximately. This efficiency is, of course, merely ideal as it takes no account of hydraulic friction losses. Digitized by VnOOQlC ENERGY OF FLOW 185 The Girard tjrpe of impulse turbine was formerly manufac- tured in this country by the Stilwell-Bierce and Smith-Vaile Co. (now the Piatt Iron Works Co., of Dayton, Ohio) under the trade name of the "Victor High Pressure Turbine." In a test of a 45-in. wheel of this type installed in the power plant of the Quebec Railway Light and Power Co., Montmorency Falls, Quebec, with a rated capacity of 1,000 h.p. under a head of 195 ft. at a speed of 286 r.p.m., a maximimi efficiency of 78.38 per cent, was attained. In another test of a 25-in. Victor wheel installed in the Napa- noch Power Station of the Honk Falls Power Co., Ellensville, N. Y., with a rated capacity of 500 h.p. under 145 ft. head at a speed of 480 r.p.m., a maximum efficiency of 84.2 per cent, was attained. The average efficiency of Victor wheels in plants installed is said by the manufacturers to vary from 70 to 80 per cent., de- pending on the design of unit. In this type of unit, no draft tube is used and consequently that portion of the head from the center Une of the wheel to the level of the tailrace is lost. Various attempts have been made, both with this and other types of impulse wheel, to regain at least part of this lost head by means of an automatically regu- lated draft tube, designed to keep the water at a certain fixed level beneath the runner, but this feature has never proved suc- cessful in operation. XXXVI. REACTION TURBINES 194. Historical Development — In Art. XXX it was shown that a jet flowing from a vessel or tank exerted a pressure or reaction on the vessel from which it flows. A simple appUcation of this principle was shown in Barker's mill, Fig. 143, in which the reac- tions of two jets caused a horizontal arm to revolve. Later this device was improved by curving the arms so that the jets issued directly from the ends of the arms instead of from orifices in the side, and in this form it was known as the Scotch mill. Subse- quently the number of arms was increased and the openings en- larged, until it finally developed into a complete wheel. In 1826 a French engineer, Foiu'neyron, placed stationary guide vanes in the center to direct the water onto the runner, or wheel, the result being the first reaction tiu'bine, now known as the Digitized by LnOOQlC 186 ELEMENTS OF HYDRAULICS Fourneyron or outward-flow type (Fig. 157). This type was introduced into the United States in 1843. Fig. 157. — Radial outward flow, Fourneyron type. ^m Fig. 158. — Axial flow, Jonval type. Fig. 159. — Radial inward flow, Francis type. A later modification of design resulted in the axial or parallel- flow turbine, known as the Jonval type, which was also of Euro- pean origin, and was introduced into the United States about 1860 (Fig. 168). Digitized by VnOOQlC ENERGY OF FLOW 187 Fig. 160. Fig. 161. Digitized by Google 188 ELEMENTS OF HYDRAULICS Digitized by VjOOQIC ENEBOY OF FLOW 189 A crude form of inward-flow turbine was built in the United States as early as 1838. Subsequently the design was greatly improved by the noted American hydraulic engineer, J. B. Francis, and it has since been known as the Francis type (Fig. 159). Figure 160 shows a runner of the Francis type used in the plant of the Ontario Power Co. at Niagara Falls. These are double central discharge, or balanced twin turbines, designed to deliver 13,400 h.p. per unit, imder 180 ft. head. The runners are of bronze, 82^ in. in diameter; the shafts 24 in. maximum diameter; and the housings of reinforced-steel plate 16 ft. in diameter, spiral in elevation, and rectangular in plan, as shown in sectional Fig. 163. detail in Fig. 161. A cross-section of the power house in which these turbines are installed is shown in Fig. 162. 196. Mixed-flow, or American, Type. — ^The mixed-flow tur- bine, or American type, is a modification of the Francis turbine resulting from a demand for higher speed and power under low heads. Higher speed could only be obtained by using nmners of smaller diameter, which meant less power if the design was un- altered in other respects. To increase the capacity of a runner of given diameter the width of the runner was increased, fewer vanes were used, and they were extended further toward the cen- ter. As this decreased the discharge area, the vanes were curved so as to discharge the water axially (Fig. 163). In a standard turbine of this type, the water from the conduit or penstock, Digitized by VnOOQlC 190 ELEMENTS OF HYDRAULICS after passing through the shut-off valves, enters a cast-iron or cast-steel casing of spiral form encircling the runner, by which it is delivered to the whole circumference of the runner at a uni- form velocity (Fig. 164). The detail of the gate work for regu- lating the admission of water to the runner is shown in Fig. 165, and the entire turbine unit is shown in perspective in Fig. 166. 196. Use of Draft Tube. — In a reaction, or pressure, turbine the passages between the vanes are completely filled with water, Digitized by VnOOQlC ENERGY OF FLOW 191 Fig. 165. — Gate mechanism Fig. 166 Digitized by VnOOQlC 192 ELEMENTS OF HYDRAULICS and since this is the case, it will run submerged. By the use of a draft tube or suction tube, invented by Jonval in 1843, it is possible, however, to set the turbine above the level of the tail- water without losing head (Fig. 167). This is due to the fact Fia. 167. that the pressure at the upper end of the draft tube is enough less than atmospheric to compensate for the loss of hydrostatic pressure at the point of entrance to the wheel. The chief advan- tage of a draft tube, however, is that its use permits of setting the turbine in a more accessible position without any sacrifice of head. Digitized by LnOOQlC ENEBOY OF FLOW 193 The effect of using a draft tube may be explained mathematic- ally as follows: In Fig. 168 let A refer to a point in the free water surface of the headrace; B the point at which water enters the turbine; C its point of exit into the draft tube; and D a point in the free water surface of the tailrace, the level of the latter being Head Race Fig. 168. assumed as datum. Then, neglecting friction and writing out Bernoulli's equation between the points A and B, we obtain the relation 2g y 2g 7 or, since Va = 0, Jia = K and Kb = ^2, this becomes V^B 7 7 ' 2g y 2g Similarly, writing Bernoulli's equation between the points C and Z>, we have 2^ + - + Ac-^ + - + A., or, since Vd = 0, Ai> = 0, and he = A2, this becomes 7 7 ^ 2sf' 13 Digitized by VnOOQlC 194 ELEMENTS OF HYDRAULICS Now the effective head on the runner is equal to the difference between the total heads at B and C. Consequently Effective head = (I + ^' + ^.) - (^ + ^^ + /ic) ^^ J- I. -J- J. ^^ = — + Ai + A2 - — • But since Ai + A2 = A, and Pa = Pdi since both are atmospheric, this expression for the effective head reduces to simply Effective head = h. This, however, includes all frictional losses in the intake and draft tubes. Including such losses the expression for the effective head becomes Effective head = A — friction head. Provided the head in the draft tube does not exceed the ordinary suction head, say about 25 ft., the use of a draft tube therefore causes no loss in the static head h except the small amount due to friction. 197. Draft-tube Design. — The design of the draft tube is an important element in any hydraulic installation as a considerable percentage of the efficiency of the plant depends on this feature. The main object of the draft tube is to gradually reduce the velocity of discharge so as to make the final velocity at exit as small as possible, thereby wasting as little as possible of the kinetic * energy of flow. As the upper end of the draft tube where it joins the turbine case is necessarily circular, the ideal form of tube would be horn-shaped, that is, having a gradual flare and keeping the cross-section circular. For practical reasons, however, the outlet must usually be oval or flattened, which leads to certain difficulties in calculating the profile and cross-section of the tube. The following is perhaps the simplest solution of these difficulties.^ Cross-section of Draft Tvbe. — At the upper end of the draft tube, its cross-section as well as the discharge, Q, and velocity of flow, Vj are known. Assimiing a tentative value for the velocity of 1 Given by R. Dubs, Zurich Switzerland, in "La Houille Blanche," abstract by A. G. Hillbebg, Eng, Record, Aug. 9, 1913. For an elaborate analysis of draft-tube design see articles by A. G. Hillberg, Eng. Record^ Nov. 13, 1915, p. 604; Nov. 20, 1915, p. 630. Digitized by LnOOQlC ENERGY OF FLOW 195 flow at the lower end, the cross-section area, A, at this end may- then be determined from the relation A ^9. V If the shape of the cross-section is assumed to be a rectangle of length B and depth Z>, with circular corner fillets, as indicated in Fig. 169, then if E denotes the area of one fillet, we have A = BZ> - iE. Since the fillet is a quadrant of a circle, its area is E ^r^-"^ = ^(0.8684) and consequently A = J5D - OMMr\ Elementa of Desisn of a Flattened Concrete Draft Tube Courtesy of Eng. Record, Fig. 169. 198. Profile of Draft Tube. — The profile or longitudinal sec- tion of the draft tube is assumed to be formed of the involutes of circles, as indicated in Fig. 169 in which the large and small Digitized by LnOOQlC 196 ELEMENTS OF HYDRAULICS cirdes are the generating circles of the two involute curves, respectively. In order to determine the elements of these curves it is necessary to have given the dimensions H and T. In applying this method, therefore, it is necessary to begin by assimiing H and T tentatively until the other elements of the design have been worked out. To determine an involute curve it is necessary only to fix the center and radius of its generating circle. Since any tangent to the involute is perpendicular to the corresponding tangent to the circle, it is evident first of all that both generating circles must touch the planes of the cross-sections at entrance and exit. Let B denote the radius of the large generating circle, and n the distance of its center from the tangent to its involute at entrance, as indicated in the figure. Then, ilT > H,ia order to make the tB draft tube as short as possible we may assume w = -s", in which case, frdln the properties of the involute, the radius at the lowest point will be r + JB = tB, From this relation we have B = -^ = 0.467r, ir — 1 and also n = ^ = 0.7334r, which together determine'the construction of the outer curve. For the special case when T = H these relations become 22 = 0.467H; n = 0.7334iJ. To determine the elements of the inner curve, let r denote the radius of the small generating circle, m the distance of its center from the tangent to its involute at entrance, and a the angle be- tween the planes of the cross-sections at entrance and exit. Then from the property of involutes and the dimensions given in Fig. 169 we have evidently Di + m + z = n + y, m + (gg^) a + D + {y -x) =z. Eliminating m between these two equations the result is n + 22/ - 2x + [^) a^z-D + Dx. Digitized by LnOOQlC ENERGY OF FLOW 197 Inserting in this relation the values of x, y, z and n, namely, X = r tan ^; y = /2 tan ^ and solving for r, the result is X = r tan g; y = B tan 2; « = n + \^^J a;n = -jf r^B- ^^-^ which gives the radius of the smaller generating circle. The loca- tion of its center is then determined from the relation m = n + y-x — Di = ^ + 22 tan ^-r tan ^ — Di = ^ + (/2-r)tan|-Di. For the special case when a = 90**, these relations become IT ^ "^ 0.4292,' 2 2 X = r. 199. Time of Flow through Draft Tube. — By the determina- tion of the cross-section and profile of the tube, the velocity of flow at entrance and exit, say Vi and V2, have been fixed, and also the mean length, L, of the draft tube. To determine the time of flow through the draft tube it is assumed that the kinetic energy of a particle of water decreases linearly as it flows through the tube. That is to say, if M denotes the mass of the particle and s its distance from some origin back of the entrance, then }^Mv^ varies as -, or, since Af is a constant, this may be written 8V^ = Jfc, where fc is a constant as yet xmdetermined. The law of variation in velocity is indicated in Fig. 169. If the distances of the par- ticle from the origin at entrance and exit are denoted by Si and 82 respectively, and the corresponding velocities by vi and 1^2, then si^i^ = K; 82^2^ = K. and consequently «.-ax=L=A;(l,-i) whence tl _ Lvih>2^ fl* — V2^ Digitized by Google 198 ELEMENTS OF HYDRAULICS which determines k and therefore also 81 and St- Now from the ordinary differential notation for velocity we have whence ds Ik dt = — 7=.Vsds. Integrating this between the known limits si and 82 the time of flow through the tube is found to be 200. Recent Practice in Turbine Setting. — A typical illustra- tion of recent development in the design and installation of reaction turbines in the United States is furnished by the plant of the Mississippi River Power Co. at Keokuk, Iowa (see frontis- piece). These turbines are of the Francis type, and develop 10,000 h.p. per unit at a speed of 57.7 r.p.m. under a head of 32 ft., with an overload capacity of 13,000 h.p. The Keokuk power station is designed to accommodate four auxiliary and 30 main power units, the sole function of the aux- iliaries being to generate power to drive exciter machines. One- half of the station has been fully completed and is in operation, that is, two auxiliaries and 15 main power units. For the re- maining half of the station, all under-water work has been fin- ished complete, including the draft tubes. Likewise the forebay protection wall, intakes and headgate masonry are entirely com- pleted, with sufficient masonry backing the latter to make it stable as a dam. Seven of the main turbines were built by Wellman-Seaver- Morgan Co. and eight by I. P. Morris Co. and all are identical in every respect excepting the runners only. The Wellman- Seaver-Morgan runners are 147 in. rated diameter and 16 ft. 6 in. diameter at outside of band. Six of the I. P. Morris runners are 1533^ in. rated diameter, two are 1393^ in. rated diameter and all eight are 17 ft. 4 in. diameter at outside of band. Both wheel builders were required to guarantee an efficiency of 83 per cent. When tested at Holyoke, the Wellman-Seaver-Morgan model developed 88 per cent, efficiency and the I. P. Morris models (type 1533^ in. diameter) 86 per cent, and (type 139}^ in. diame- ter) 90 per cent, efficiency. After several months of conunercial Digitized by VnOOQlC ENERGY OF FLOW 199 FiQ. 170. — Center section of main turbine on transverse axis of power house, Mississippi River Power Co. Digitized by VnOOQlC 200 ELEMENTS OF HYDRAULICS operation, one of the Wellman-Seaver-Morgan wheels and one of the I. P. Morris wheels (1533^ in. diameter) were tested in place at the power station and both developed about 90 per cent, effi- ciency. These tests were conducted by Mr. B. H. Parsons, of New York City. Each of the units is of the single-runner vertical-shaft type as shown in Fig. 170. The trend of present development of the reaction turbine seems to indicate a still wider application of this type to aU conditions of head and speed, and that the single-nmner vertical-shaft turbine will eventually supersede the multi-runner horizontal-shaft type (Fig. 161) and the multi- runner vertical-shaft type (Fig. 167). In the Keokuk plant the intakes and draft tubes are of con- crete moulded in the substructure of the power house. The water from the forebay reaches each turbine through four intake open- ings, the outer dimensions of which are 22 by 7^ ft., leading into a scroll chamber 39 ft. in diameter (Fig. 171). The draft tubes are about 60 ft. in length along the center line and contain a right angle bend to 24 ft. in. radius. They are 18 ft. in. diameter at the throat, from whence they flare in straight lines to the mouth, where the cross-section is oblong 22 ft. 8 in. by 40 ft. 2 in., bounded by two semicircles connected by straight lines. The velocity of flow is thereby diminished from 14 ft. per second at the top of the draft tube to 4 ft. per sec- ond at the outlet, the effect being to increase the hydraulic efficiency of the plant about 7 per cent. This type of^ construc- tion is representative of recent practice, which seems to favor the moulding of the volute casing directly in the substructure of the power house for all low-head work. Foj heads exceeding 100 ft., the amount of reinforcement in the concrete becomes so great as to warrant the use of cast-iron casings, and for heads exceeding 250 ft. the use of cast steel for turbine casings is stand- ard practice. In the vertical-shaft turbine the weight is carried on a thrust bearing, the design of which has been one of the most important considerations affecting the adoption of this type. In the Keokuk plant the turbine nmner is coupled to the generator above by a shaft 25 in. in diameter the total weight of the revolv- ing parts, amounting to 550,000 lb., being carried on a single thrust bearing 6 ft. in diameter. This bearing is of the oil- pressure type, a thin film of oil being maintained at a pressure Digitized by VnOOQlC ENEMY OF FLOW 201 Digitized by VjOOQIC 202 ELEMENTS OF HYDRA ULICS of 250 lb. per square inch between the faces of the bearings. As a momentary failure of the oil supply would result in the immediate destruction of the bearing, provision is made for such an emerg- ency by introducing an auxiliary roller bearing which is normally unloaded. A slight decrease in the oil supply, however, allows the weight to settle on this roller bearing, which although not intended for permanent use is sufficiently large to carry the weight temporarily until the turbine can be shut down. The oil pressure bearing, when taken in connection with the necessary pumps and auxiliary apparatus, is expensive to in- stall and maintain, and requires constant inspection. For this reason the roller bearing and the Kingsbury bearing are now being appUed to large hydro-electric imits. One of the first installations in which the roller bearing was appUed to large hydraulic units was at the McCall Ferry Plant of the Pennsyl- vania Water and Power Co., where both the roller and the Kings- bury type of bearing are now in satisfactory use. XXXVn.^CHARACT£RISTlCS OF REACTION TURBINES 201. Selection of Type. — The design of hydrauUc turbines is a highly speciaUzed branch of engineering, employing a relatively small number of men, and is therefore outside the domain of this book. On account of the rapid increase in hydrauUc devel- opment, however, every engineer should have a general knowl- edge of turbine construction and type characteristics so as to be able to make an inteUigent selection of type and size of turbine to fit any given set of conditions. For this reason the following explanation is. given of the use and significance of commercial turbine constants, such, for instance, as those given in the runner table in Par. 213. 202. Action and Reaction Wheels. — The two systems of hydrauUc-power development now in use in this country are the impulse wheel and the radial inward-flow pressure turbine. When an impulse wheel is used, the total effective head on the runner is converted into speed at entrance and this type is there- fore sometimes called an action wheel. In the case of a pressure turbine, however, the effective head on the runner is not all con- verted into speed at entrance, the entrance speed being smaller than the spouting velocity, so that the water flows through the runner under pressure, the effect of which is to accelerate the Digitized by VnOOQlC ENERGY OF FLOW 203 stream as it passes over the runner. A pressure turbine is there- fore called a reaction wheel. Reaction turbines are generally used for heads between 5 and 500 ft., and impulse wheels for heads between about 300 and 3,000 ft.^ While there is no doubt as to the system proper for very low or very high heads, there is a certain intermediate range, say from 300 to 500 ft., for which it is not directly ap- parent which system is most suitable. To determine the proper system within' this range, the criterion called the characteristic speed has been introduced, as explained in what follows. 203. Speed Criterion. — In determining the various criteria for speed, capacity, etc., the following notation will be used: Let h = net head in feet at turbine casing, = gross head minus all losses in headrace, conduit and tailrace; d = mean entrance diameter of nmner in feet; 6 = height of guide casing in feet; n = runner speed in r.p.m.; V = spouting velocity in feet per second; Ui — peripheral velocity of runner in feet per second; Ui = y = ratio of peripheral speed of runner to spouting velocity of jet. From Eq. (30), Par. 51, the spouting velocity in terms of the head is given by the relation where the constant C = 0.96 to 0.97. For maximum efficiency the peripheral velocity of the runner is some definite fraction of the ideal velocity of the jet, \/2gh, that is, Ui = (pV^ (126) where (p denotes a proper fraction. For tangential or impulse wheels the average value of ^ is from 0.45 to 0.51, whereas for 1 At the hydro-electric plant of the Georgia Railway and Power Co. at Tallulah Falls, Ga., the hydraulic head is 680 ft., which is probably the highest head that has been developed east of the Rocky Mountains, and the highest in this country for which the reaction type of turbine has been employed. See Fig. 181. (General Electric Review, June, 1914, pp. 608- 621.) Digitized by VnOOQlC 204 ELEMENTS OF HYDRAULICS reaction turbines its value ranges from 0.49 to 0.96, with an average range from about 0.57 to 0.87. The ratio Ui of peripheral speed of runner to spouting velocity- is therefore given by the expression V C and consequently Ui is about 3 per cent, more than (p. Since in Eq. (125) the factor (py/2g is a constant, this equa- tion may be written in the form til = KVh, i where the coefficient K may be called the speed constant. For a given runner for which d, h and n are known, this speed constant may be calculated from the relation k. = -*^ = -^- (126) Vh eovh By the use of the speed constant fc«, different types of runners may be compared as regards speed. In the case of reaction tur- bines if the speed constant is much in excess of 7, either the speed is too high for maximum efficiency or the nominal diameter of the runner is larger than its mean diameter. 204. Capacity Criterion. — The entrance area A of the runner is given by the relation A = Ciirdb where Ci denotes a proper fraction, since the open circumference is somewhat less than the total circumference by reason of the space occupied by the ends of the vanes or buckets. The ve- locity of the stream normal to this entrance area is the radial component of the actual velocity at entrance, say Ur, and like this velocity is a multiple of \/A, say Ur = C2\/A. Since the discharge Q is the passage area multiplied by the speed component normal to this area, we have Q = Aur = {ciTdb)ct'\/h. It is customary, however, to express the height of a runner in terms of its diameter as 6 = Czd, where the coefficient c$ is a constant for homologous runners of a given type. For American reaction turbines Cz varies from about Digitized by VnOOQlC ENERGY OF FLOW 205 0.10 to 0.30. Substituting this value for 6 in the expression for the discharge, it becomes Therefore, if the constant part of this expression is denoted by fcg, it may be written Q = kgd^Vh. (127) The coefficient k^ may be called the capacity constant of the run- ner, and for any given runner may be computed from the relation For American reaction turbines the capacity constant ranges in value from about 2 to 4. Since kg has approximately the same value for all runners of a given type, it serves as a criterion for comparing the capacity of runners of different types. 206. Characteristic Speed. — The speed constant and capacity constant taken separately are not sufficient to fix the require- ments of combined speed and capacity. That is to say, two runners may have diiGferent values of K and kg and yet be equivalent in operation. To fix the type, therefore, anothei criterion must be introduced which shall include both k^ and kg. The most convenient combination of these constants is that introduced by Professor Camerer of Munich and the well- known turbine designer, Mr. N. Baashuus of Toronto, Ontario. This criterion may be obtained as follows.^ The horse power of a turbine is given by the expression where e denotes the hydraulic efficiency of the turbine. If the horse power, discharge Q and head h are given, the efficiency may be calculated from this relation by writing it in the form ^ 550H.P. ^ 62.37 QA' If the efficiency is known, the constants in the above expression for the horse power may be combined into a single constant fc, and the equation written in the form H.P. = kQh. (129) ^S. G. ZowsKi, "A Comparison of American High-speed Runners for Water Turbines, Eng. News, Jan. 28, 1909, pp. 99-102. Digitized by VnOOQlC 206 ELEMENTS OF HYDRAULICS When the efficiency is not known it is usually assumed as 80 per cent., in which case fc = jr* From Eq. (126) the speed in r.p.m. is given by the relation n = ?^ (130) and from Eq. (128) the nominal diameter of the runner is given as -i. Q Eliminating d between these two relations, we have therefore eoKVKVhVh n = j=z • tVq Moreover, from Eq. (129) we have and substituting this value of Q in the preceding expression for n, we have finally ^ )vww: ^"^^ The expression in parenthesis is a constant for any given type and may be denoted by Ng, in which case we have hVh ^ / eokVi \ TT For any given type of turbine this constant iV, may be calculated from the relation Various names have been proposed for this constant N^ such as "type constant" and "type characteristic." In Germany, where its importance as determining the type and perfonnance of a turbine seems to have first been recognized, it is called the specific speed (spezifische Geschwindigkeit, or spezifische Um- laufzahl). This term, however, is not entirely satisfactory to American practice, as it seems desirable to use the term specific speed in another connection, as explained in what follows. The term for the constant Ns favored by the best authorities as more Digitized by VnOOQlC ENERGY OF FLOW 207 fully describing its meaning is characteristic speed, which is therefore the name adopted in this book.^ For impulse wheels the characteristic speed ranges in value from about 1 to 5, while for radial, inward-flow turbines its value Ues between 10 and 100. 206. Specific Discharge. — It is convenient to express the discharge, power, speed, etc., in terms of their values under a 1-ft. head. The discharge under a 1-ft. head is called the specific die- charge, and its value is found by substituting A = 1 in. Eq. (127). Consequently if the specific discharge is denoted by Qi, its value is and therefore Q = QiVh. (133) For reaction turbines the specific discharge ranges in value from 0.302(P for ahe slowest speeds, to 2.866d^ for the highest speeds, the diameter d being expressed in feet. 207. Specific Power. — Similarly, the power developed under 1-ft. head is called the specific power, and will be denoted in what follows by H.P.i. From Eq (129) we have H.P. = kQh and since from Eq. (127) Q = k^d^Vh, by eliminating Q between these two relations we have H.P. = kk^d^hVh. Substituting A = 1 in this equation, the specific power is there- fore given as H.P.1 = kk^d^, and consequently H.P. = H.P.ih\/h. (134) 208. Specific Speed. — ^By a nalogy with what precedes, the speed under 1-ft. head will be called the specific speed and denoted ^ The use of the term characteristic speed has been recommended to the author by the well-known hydraulic engineer Mr, W. M. White, who is using this term in preparing the American edition of the German handbook "de Htitte," and strongly advocates its general adoption in American practice. Digitized by VnOOQlC 208 ELEMENTS OF HYDRAULICS by ni. Substituting ft = 1 in Eq. (130), we have therefore 60fc, and consequently ^^ = Tr n = niVh. (136) For reaction turbines the specific speed ranges in value from 78 147 -^ for the lowest speeds, to — r- for the highest speeds,-the diame- ter d being expi:essed in feet. 209. Relation between Characteristic Speed and Specific Speed. — From the relation . nVHT. the characteristic speed N, may be defined in terms of the quan- tities defined above as specific. Thus, assmning ft = 1 and HJP. = 1, we have N, = n, expressed in r.p.m. Therefore, the characteristic speed is the speed in r.p.m. of a turbine diminished in all its dimensions to such an extent as to develop 1 h.p. when working under a head of 1 ft. Since it is apparent from Eq. (131) that N, stands for the combination ,, QOK\%Vk iV, = 9 T where fc is a function of the efficiency e, the characteristic speed Ns is an absolute criterion of turbine performance as regards speed, capacity and efficiency. From Eq. (132), however, it is evident ' that Ns may be calculated directly from the speed, power and head without knowing the actual dimensions of the runner, its discharge, or its efficiency. 210. Classification of Reaction Turbines. — The character- istic speed Ng may be used as a means of classifying not only the various types of impulse wheels (Par. 191) but also of reaction, or pressure, turbines. In the following table practically all the different kinds of pressure turbines of the radial inward-flow type are classified by their characteristic speeds, the corresponding efficiencies being also given in each case. Digitized by VnOOQlC ENERGY OF FLOW 209 Type of pres- sure turbine Charac- teristic speed, Na Efficiency Maximum At half power Low-speed Medium-speed.. . High-speed Very high-speed. 10- 20 30- 60 60- 80 90 -100 88-92 per cent, at ?i power 88-92 per cent, at ?i power 87-91 per cent, at 0.8 power 86-90 per cent, at 0.9 power 80-85 per cent. 78-82 per cent. 75-80 per cent. 73-76 per cent. The values of the constant iV, in this table refer to the maxi- mum power of one runner only. In case the characteristic speed is higher than 100, it is necessary to use a multiple unit. At maximum power, the efficiencies are slightly lower than the maximum efficiencies given above. From this table it is apparent that low-speed turbines show a favorable efficiency over a wide range of loads but are prac- tically limited to high heads, whereas high-speed turbines are efficient at about 0.8 load but show a notable decrease in effi- ciency at half load. The use of the latter is therefore indicated for low heads where the water supply is ample at all seasons. 211. Numerical Application. — ^To illustrate the use of the pre- ceding numerical data, suppose that it is required to determine the proper system of hydraulic development for a power site, with an available flow of 310 cu. ft. per second under an effective head of 324 ft. The power capacity in this case is H.P. = «?i^ = 9,100. Of this amoimt about 100 h.p. will be required for exciter and Ughting purposes. There would therefore be installed two exciter units running at 550 r.p.m., one of which would be a reserve unit. The characteristic speed for these units would then be ^ 650 flOO .^- Since this lies between 1 and 5, an impulse wheel would be used for driving the exciter generators. The main development of 9,000 h.p. would be divided into three units of 3,000 h.p. each, running at 500 r.p.m., with a fourth unit as a reserve. The characteristic speed for these main units would then be ^ _ 600 /3,000 ^ ^' "■ 3ioVi7:6 " ^^• 14 Digitized by VnOOQlC 210 ELEMENTS OF HYDRAULICS As N, lies between 10 and 100, a pressure turbine would be used for driving the main generators. 212. Normal Operating Range. — Having determined the proper type of development, it is necessary, in case a reaction ^periatink i JtaiflTA. r^ z. ! *^ K-K^^^ ,^ ^ ^.^ / I 4 fP y ;> \ :3 Cjv^ r OS d C;,i5 lev y / \ / ': / -^ '? ^ / ^ / \ ^ y / I 1 L 1 10 20 40 50 60 70 Per Cent Turbine Load Fia. 172. \ 90 10^ turbine is used, to determine the required size and type of runner to develop maximum efficiency under the given conditions of operation. For a turbine direct-connected to a generator, the capacity of the turbine, in general, should be such as to permit the full overload capacity of the generator to be developed and at the same time place the normal operating ^^^ "?^ range of the unit at the point of maxi- ^^5W mum efficiency of the turbine, as indi- ^^-"""'^ ^ cated in Fig. 172. The normal horse \^^ power, or full-load, here means the power at which the maximum efficiency is attained, any excess" power being regarded as an overload. When the supply of water is ample but the head is low, effi- ciency may to a certain extent be sacrificed to speed and capacity J.' V/X C» M IJ.X ILf AXX^ VI Digitized by Google ENERGY OF FLOW 211 in order that the greatest power may be developed from each runner, thereby reducing the investment per horse power of the installation. On the other hand, when the flow of water is insufficient to meet all power requirements, an increase in effi- FiG. 174. Fig. 175. ciency shows a direct financial return in the increased output of the plant. 213. Selection of Stock Runner. — Ordinarily it is required to select a stock runner which will operate most favorably under the Digitized by VnOOQlC 212 ELEMENTS OF HYDRAULICS given conditions. To explain how an intelligent selection of size and type of runner may be made from the commercial con- stants given by manufacturers, the following runner table of a standard make of turbine is introduced^ (page 213). The cut accompanjdng each of the six types given in the table shows the outline of runner vane for this t3rpe. To indicate its relation to the runner and to the turbine unit as a whole, Fig. 173 shows a typical cross-section of runner; Fig. 174 shows how this is related to the casing; and Fig. 175 shows a cross-section of the entire turbine unit. The runner is also shown in per- spective in Fig. 176. Fig. 176. From Eq. (132) it is evident that, other things being equal, the characteristic speed for high heads will be relatively small whereas for low heads it will be large. Thus in the runner table above, type "A," with a characteristic speed of 13.55 is adapted to high heads, running up to 600 ft., while at the other end of the series, type "F," with a characteristic speed of about 75, is adapted to effective heads as low as 10 ft. To give a numerical illustration of the use of the runner table, suppose it is required to determine the tjrpe of runner and the speed in r.p.m. to develop 750 h.p. under a head of 49 ft. In this case hy/h = 49\/49 = 343, and consequently ^■^" - AVi " 343 - ^•^' ^ The Allis-Chalmers Co., Milwaukee. Digitized by VnOOQlC ENERGY OF FLOW 213 TVPE"li"^RimilEII TYPE^'e'RIUliaR TYPE t Vrunjier ife^ \ 1^ \ K ol -^ i ^ 2 ^^ 11 N. - 18.56 N, - 20.3 Nm m 29.4 Ui - 0.585 Ui - 0.625 Ut m 0.668 Diam. R.P.M.1 H.P.1 Qi R.P.M.1 1 H.P.1 Ql R.P.M.1 1 H.P., 1 Qi 16 71.7 0.0358 0.394 76.6 0.0705 0.776 81.4 0.130 1.43 18 50.8 0.0514 0.565 63.8 0.105 1.155 67.8 0.187 2.06 21 51.2 0.0705 0.776 54.7 0.138 1.523 58.2 0.225 2.48 24 44.8 0.0915 1.007 47.8 0.182 2.00 51.0 0.333 3.66 27 39.8 0.116 1.276 42.5 0.229 2.52 45.2 0.423 4.65 90 35.8 0.142 1.562 38.3 0.284 3.12 40.7 0.520 5.72 34 31.6 0.184 2.024 33.8 0.363 3.99 35.9 0.668 7.35 38 28.3 0.230 2.53 30.2 0.453 4.98 32.2 0.835 9.19 42 25.6 0.280 3.08 27.4 0.551 6.06 29.1 1.016 11.18 46 23.4 0.336 3.69 25.0 0.665 7.32 26.6 1.225 13.48 50 21.5 0.398 4.38 23.0 0.79 8.69 24.4 1.450 15.95 55 19.5 0.480 5.28 20.9 0.95 10.45 22.2 1.745 19.20 60 17.9 0.573 6.30 19.1 1.13 12.43 20.4 2.08 22.88 66 16.5 0.672 7.39 17.7 1.33 14.63 18.8 2.44 26.84 70 15.4 0.785 8.64 16.4 1.53 16.83 17.5 2.82 31.00 TYPE^D^RUffltER TYPE'E'RUNJIER 1 type"f"rojuier S < S w 1 p ^ < ,) k N. - 40.7 N, - 61.7—60.5 i^. -71.4— 79 Ui - 0.70 Ui - 0.76 Ui - 0.85 Diam. 1 R.P.M.I iHJP.i Qi Diam. R.P.M.1 H.P.1 Qi R.P.M.1 H.P.1 1 Qi 14 98.4 0.277 3.05 111.5 0.410 4.51 16 86.1 0.367 4.04 97.7 0.541 5.95 18 76.5 0.471 5.18 86.8 0.704 7.74 15 85.7 0.226 2.49 20 69.0 0.597 6.57 78.1 0.912 10.03 18 71.4 0.324 3.56 22 62.6 0.731 8.04 71.0 1.133 12.46 21 61.3 0.442 4.86 24 57.4 0.883 9.70 65.1 1.375 15.13 24 53.6 0.577 6.35 26 53.0 1.055 11.60 60.1 1.62 17.85 27 47.6 0.731 8.04 28 49.2 1.243 13.67 55.8 1.93 21.25 30 42.8 0.902 9.92 30 46.0 1.436 15.80 52.1 2.20 24.20 34 37.8 1.158 12.74 32 43.0 1.65 18.15 48.8 2.55 28.10 38 33.9 1.444 15.88 34 40.5 1.89 20.80 46.0 2.82 31.10 42 30.6 1.765 19.4 36 38.3 2.15 23.65 43.5 3.14 34.55 46 28.0 2.12 23.3 38 36.3 2.42 26.60 41.1 3.52 38.70 50 25.7^ 2.50 27.5 40 34.4 2.75 30.25 39.1 3.93 43.20 55 23.4 3.04 33.4 42^ 32.4 3.09 34.0 36.8 4.33 47.60 60 21.4 3.61 39.7 45 30.6 3.53 38.8 34.7 4.92 54.10 65 19.8 4.22 46.4 47J4 29.0 4.01 44.1 32.9 5.66 62.25 70 18.4 4.90 53.9 50 27.6 4.45 49.0 31.2 6.13 67.40 5aH 26.3 4.95 54.5 29.8 6.75 74.25 55 25.1 5.52 60.7 28.4 7.60 82.50 57H 24.0 6.10 67.1 27.2 8.16 89.75 60 23.0 6.80 ' 74.8 26.0 8.94 98.30 Digitized by VnOOQlC 214 ELEMENTS OF HYDRAULICS which corresponds to a 30-in. type "F" runner. Referring to the table for this type and size, we have ni = 52.1, from which the required speed is found to be n = 52.lVh = 364.7, say 360 r.p.m. If twin turbines were used, we would have which corresponds to a 22-in. type "F" runner, having a speed of n = 71.0y/h = 497, say 500 r.p.m. As a second illustration, let it be required to find from the table the t3rpe of runner and speed to develop, 4,000 h.p.under an eflfective head of 300 ft. In this case h\/h = 300\/300 = 5,190, and consequently the specific power is H.P.. = ^ = i^ = 0.77. hVh 5,190 which corresponds to a 50-in. type "B" runner. Referring to the table for this type and size we have H.P.i = 0.79, and rii = 23, and consequently the power and speed for this type and size is H.P. = 0.79hVh = 4,100, and n = 23 Va = 395, say 400 r.p.m. XXXVm. POWER TRANSMITTED THROUGH PIPE LINES 214. Economical Size of Penstock. — In hydraulic develop- ments involving the construction of long pipe lines or pipes under a high head, the cost of the pipe line is often a considerable part of the total investment. In such cases the determination of the most economical size of penstock is of special importance. In discussing this problem the following notation will be used: a = cost of metal pipe per pound; b = cost of wood pipe per foot of diameter, per foot of length; c — constant in Chezy's formula v = cy/rs; d = inside diameter of pipe; Digitized by VnOOQlC ENERGY OF FLOW 215 / = allowable fiber stress in metal pipe in pounds per square inch; h = friction head per unit of length; i = income in dollars per year per foot of head; . , irWd k = numencal coefficient = -s— ; { = length of pipe; p = total percentage return on investment, including operat ing expenses, depreciation and profit; Q = volume of flow through pipe in cubic feet per second; r = hydraulic radius = j for pipe flowing full; 8 = hydraulic gradient = y-; t = thickness of metal pipe; V = velocity of pipe flow; w = internal water pressure in pipe in pounds per square inch; In the Chezy formula, v = c\/rs, if we put ^ = 7> « = r ^^d solve for the loss in head, h, the result is , 4fo2 or, since v = -tj, this may be written , _ 64ZQ« Consequently the loss in annual income due to this loss in head is QilQH Income lost = cVd^ Another factor which reduces the income is the annual fixed charge on the pipe. To calculate this for metal pipe, note that the thickness of pipe wall is given by Eq. (10) Par. 23, namely, and therefore the weight of pipe per foot of length is irdi. Conse- ir'uyd^fL quently the cost per foot of length for metal pipe is wdta = ~~o7~"» and hence finally Digitized by VnOOQlC 216 ELEMENTS Of HYDRAULICS Total cost of pipe = — sT" ~ ^h where for metal pipe and for wood stove pipe k may be determined from the relation Cost of pipe per foot of diameter per foot of length = kd. Adding the terms representing these two losses we have then Total annual loss = ^7: , + ^^^ . c^d^T^ 100 The most economical diameter of penstock is then the value of d which makes this expression a minimuni. The condition for such a minimum is obtained by equating to zero the first deriva- tive of the expression with respect to d. Performing this differ- entiation^ the condition for a minimum is then S2QIQH 2kdlp _ c«dV2 "^ 100 ~ "' * whence, solving for d, the formula for economical diameter of penstock is determined as c^kp 216. Numerical Application. — The following numerical exam- ple illustrates the use of the formula as applied to wood stave pipe.^ A stream is to be developed by building a large reservoir to equalize the flow, and conducting the water to the surge tank and penstocks by a single wood stave pipe 13,000 ft. long. The avail- able head is 440 ft., and the average flow through the pipe while the plant is in operation is 112 cu. ft. per second, developing about 3,000 kw. As the plant is to be used as a base- load plant and the load is to be nearly constant, the value of Q is assumed as 112 cu. ft. per second. The plant develops annuaUy about 26,000,000 kw.-hr., which sold at 1 ct. per kilowatt-hour gives a gross income of $260,000 or $590 per foot of fall. To determine the most economical value of d, a velocity of 6 ft. per second is first assumed, which would require a 5-ft. pipe, the cost of which in place is estimated at $10 per foot, or $2 per foot » Wabrkn, Trans, Am. Soc. C. E., vol. kxix, p. 270. =# Digitized by VnOOQlC ENERGY OF FLOW 217 of diameter per foot of length, making h — % — 0.4. Sub- stituting these numerical values in the formula, namely, = 112; i = 590; C = 113 (assumed); p = 15; fc == 0.4, the resulting value of d is 625(112)2590 = 5.52 ft. : (113)45 A 5K-ft. penstock would therefore be assumed as most eco- nomical. XXXIX. EFFECT OF TRANSLATION AND ROTATION 216. Equilibrium under Horizontal Linear Acceleration. — Consider the equiUbrium of a body of water having a motion of translation as a whole but with its particles at rest relatively to one another, such, for example, as the water in the tank of a locomotive tender when in mo- tion on a straight level track. If the speed is constant, the forjses acting on any particle of the liquid are in equiUbrium, and conditions are the same as when the tank is at rest. If the motion is accelerated, however, every particle of the liquid must experience an inertia force pro- portional to the acceleration. Thus, if the acceleration is de- noted by a, the inertia force F acting on any particle of mass m^ according to Newton's law of motion, is given by the relation F = ma. For a particle on the free surface of the Uquid (Fig. 177), the inertia force F acting on this particle must combine with its weight W into a resultant R having a direction normal to the free surface of the liquid. From the vector triangle shown in the figure we have and by Newton's law F = TT tan a, whence by division F = tan< W r«; S (137) Digitized by Google 218 ELEMENTS OF HYDRAULICS 217. Equilibrium under Vertical Linear Acceleration. — ^If the tank is moving vertically upward or downward, the surface of the liquid will remain horizontal. If the motion is uniform, that is, with constant velocity, the conditions will be the same as though the tank was at rest. If it is moving upward with acceleration a, the surface will still remain horizontal but the pressure on the bottom of the tank will be increased by the W amoimt ma = — a, where W denotes the weight of a column of 9 water of imit cross-section and height equal to the depth of water in the tank. Thus if p denotes the pressure on the bottom W of the tank in pounds per square inch, then since ma = — a = — a, we have p = 7li + ma = 7li(^^)- (138) If the acceleration is vertically downward, the pressure on the bottom of the tank is diminished by the amount 7A(~)> its value being p = ^h(^). (139) 218. Free Surface of Liquid in Rotation. — If the tank is in the form of a circular cylinder of radius r,' and revolves with angular velocity o) about its vertical axis YY (Fig. 178), the free surface of the Uquid will become curved or dished. To find the form assumed by the surface, let P denote any parti- cle on the free surface at a dis- tance X from the axis of rota- tion. Then if m denotes the mass of this particle, the centrifugal force C acting on it is mx(a^ = W -zca'' From the vectof triangle shown in the figure we have C^ _x<a^ tan ^ = = = Digitized by VnOOQlC ENERGY OF FLOW 219 dy and since the slope of the surface curve is ^^ = tan^, we have as its differential equation 3^ = tan 6 = — I dx g whence, by integration, its algebraic equation is found to be y = 2g (140) The surface curve cut out by a diametral section is therefore a parabola with vertex in the axis of rotation, and the free surface is a paraboloid of revolution. 219. Depression of Cup below Original Level in Open Vessel. — Since the volume of a paraboloid is half the volume of the cir- cumscribing cylinder, the volume of liquid above the level OX of the vertex (Fig. 179) is 2 ' Vol. OCDEF = E Y . D J \ / B F \ -1 / \ i X Fig. 179. Fig. 180. 7rr2fc== But if AB is the level of the liquid when at rest, then 2 where k denotes the depth of the cup below the original level, and therefore 2,.,2 r^w k = - = ■ ^ 2 4g (141) Consequently the depth of the cup below the original level is proportional to the square of the angular velocity. 220. Depression of Cup below Original Level in Closed Vessel. —If the top of the tank is closed and the angular velocity Digitized by Google hr*-- = HR* R* = hr* H 220 ELEMENTS OF HYDRAULICS increased until the liquid presses against the top, as shown in Fig. 180, the surface will still remain a paraboloid. If the total depth of the cup is denoted by H and its greatest radius by R, then since its volume must be the same as that of the cup of depth h, we have the relation whence 03 X But from the equation of the surface curve, y = , by sub- stituting the simultaneous values y = H, x =^ R,we have „ _ o>'R' Jti = —^ — i and substituting in this relation the value of fi* from the previous equation, the result is whence H = «r^|. (142) Therefore after the liquid touches the top cover of the tank, the total depth of the cup is proportional to the first power of the angular velocity. 221. Practical Applications. — An important physical applica- tion of these results consists in the formation of a true parabolic mirror by placing mercury in a circular vessel which is then rotated with uniform angular velocity, the focus of the mirror depending on the speed of rotation. Another practical application has been found in the con- struction of a speed indicator. A glass cylinder containing a colored liquid is mounted on a vertical spindle which is geared to the shaft whose speed is required. The required speed is then obtained by noting the position of the vertex of the para- boloid on a vertical scale. From the level AB to the level CD (Fig. 180) the graduations on the scale are at unequal distances apart, as apparent from Eq. (141), but below this point they are equidistant, as shown by Eq. (142). Digitized by VnOOQlC ENERGY OF FLOW 221 XL. WATER-HAMMER IN PIPES 222. Mazimum Water-hammer. — If water is flowing in a pipe with uniform velocity and the flow is suddenly stopped, as by the closing of a valve, the pressure in the pipe is greatly increased, producing what is known as water-hanuner. The maximum pressure occurs of course when the flow is stopped instantane- ously. In this case all the kinetic energy of flow is expended in compressing the water and distending the pipe. The increase in pressure due to instantaneous closing may therefore be deter- mined by simply calculating the work done in compressing the water and in distending the pipe, and equating their sum to the kinetic energy of the water flowing in the pipe when shut off. This derivation assumes that all the energy of the water in any section of given length is expended in compressing the water and distending the pipe in this particular section. This assumption, however, can be shown to be true theoretically and is also verified by experiment.* As a notation for use in the derivation which follows, let: W = weight of water in the pipe in pounds; V = velocity of flow in feet per second; 7 = 62.4 = weight of water in pounds per cubic feet; A = area of cross-section of pipe in square feet; d ^ diameter of pipe in feet; I = length of pipe in feet; h = head in feet due to water-hammer alone in excess of static head; B = bulk modulus of elasticity of water = 294,000 lb. per square inch (Par. 3) ; E = Young'smodulusof elasticity of pipe material; h = thickness of pipe wall in feet. Then the kinetic energy of the water flowing in the pipe with velocity V is kinetic energy of flow -^ — = —k — ft.-lb. 1 All the formulas for maximum watei^hammer are the same however derived, although they appear in different forms. The derivation here given is probably the simplest and is a modification of that given by Minton M. Wabben in his paper on "Penstock and Surge Tank Problems," pub- lished in Trans, Am. Soc. C. E., vol. Ixxix, p. 238, 1915. The correct for- mula for maximum water-hammer seems to have been first obtained by JOTJKOWSKT. Digitized by VnOOQlC 222 ELEMENTS OF HYDRAULICS To calculate the work expended in compressing the water in the pipe, note that the total pressure on any crossnsection of the pipe is Ahy lb., and consequently the unit pressure is hy lb. per square foot. Since the bulk modulus of elasticity of water B is defined as (Par. 3) we have D _ unit pressure "" unit deformation hy imit deformation = -^« Therefore the total compression of a column of water { ft. in hyl length will be -^. Now the total pressure on any cross-section is * Ahy, but since this abnormal pressure starts at zero and increases imiformly to its full value, the average pressure on the cross-sec- tion is y^Ahy, Consequently the total work expended in com- pressing the water is work done on water = }4Ahy X -^ = ^J ft.-lb. To determine the amount of work expended in distending the pipe, note that the total tension on any longitudinal seam in the pipe is lb., and consequently the imit stress in the pipe is — rr or ~r^- Since by definition of Young's modulus jjf unit stress unit deformation we have unit deformation = ^r^> and consequently the increase in the length of the circumference is ''^^2&^" 2bE~' Since this tension starts at zero and increases uniformly to its full value, the work done in distending the pipe is , , . 1 ^dlhy^ 7rd%y Adlh^y\^ „ work done on pipe = ^ X -— X -^j^ = 2&g ^'^^' Digitized by VnOOQlC ENERGY OF FLOW 223 Equating the kinetic energy of flow to the work of deformation expended on water and pipe wall jointly, we have AlyV* Ah*yH Adlh*y* 2g 2B "•" 2bE whence, by reducing and solving for h, we have ,_ W^ » = V '+§f Inserting in this expression the numerical values of the constants, namely, B = 294,000 lb. per sq. in. = 42,336,000 lb. per sq. ft., E = 30,000,000 lb. per sq. in. = 4,320,000,000 lb. per sq. ft., 7 = 62.4 lb. per cu. ft. it becomes , 145.27 ^= / =y Jl + 0.0098,? or, with suflBicient exactness, ^. h = — =F=- / (143) ^1>P.(»| 223. Velocity of Compression Wave. — ^As mentioned above, what happens, when the flow in a pipe is suddenly cut off, is an increase in pressure, first exerted at the valve, which compresses the water and distends the pipe at this point. Beginning at the valve, this effect travels back toward the reservoir or supply, producing a wave of compression in the water and a wave of distortion in the pipe. When all the water in the pipe has been brought to rest, the total kinetic energy originally possessed by the flowing water is stored up in the elastic deformation of the water and pipe walls. Since this condition cannot be maintained under the actual head in the reservoir, the pipe then begins to contract and the water to expand, thereby forcing the water back into the reservoir until it acquires a velocity approximately equal to its original velocity but in the opposite direction, that is, back toward the reservoir. After this wave has traversed the pipe the water again comes to rest, but the kinetic energy acquired by the flow toward the reservoir will have reduced the Digitized by VnOOQlC 224 ELEMENTS OF HYDRAULICS pressure below normal. Consequently water again enters the pipe from the reservoir and flows toward the valve, beginning a new cycle of operations. Now, in addition to the notation previously adopted, let: Vc = velocity of compression wave; t =» time required for wave starting at the valve to reach upper end of pipe; F = total pressure on any crossnsectidH of pipe. Then, from the principle of impulse and momentum, we have W g ' or since F =» Ahy; W = Aly; < = -, by substituting these values we have which reduces to - \ n r^ •--? — '^ Since the first expression for h derived in the preceding paragraph may also be written in the form ^V'+i-f iyg by substituting this value of F in the formula just derived for Vc we have P hg \y Vc = " ^P"■ E b Inserting in this relation the numerical values of g, B, E and 7, as above, it reduces to the final form 4,674 V.= /j . nn^d (1«) /I + 0.01^ 224. Ordinary Water-hammer. — The rise in pressure due to the gradual shutting off of the flow, as by the closing of a gate Digitized by VnOOQlC ENERGY OF FLOW 225 or valve, is more difficult to determine than the maximum water- hammer due to instantaneous closing. It is evident that if the reflected waves from the reservoir return to the gate before it is entirely closed, the rise in pressure will be less than the maximum given by Eq. (143). Since the period of the compression wave, that is, the time required to make a round trip from the gate to 21 the reservoir and back again, is — > the statement just made is equivalent to saying that if the time of closure is greater than 21 —9 the rise in pressure will not reach the maximum given by Eq. (143). Several formulas have been proposed for the rise in pressure due to ordinary closure of a gate or valve. The one here derived is due to Minton M. Warren,^ and is the simplest and also seems to agree best with experiment. The assumption on which the formula is based is that the pressure head rises linearly from zero 21 to A in the time — and then remains constant until the gate is closed, after which it falls again. The derivation of the formula is based on the principle of impulse and momentum, namely, that the momentum destroyed must equal the product of the pressure produced by the length of time it acts. In applying this principle it is assumed to be sufficiently exact to use the momentum of the entire mass of water filling the pipe. This assumption would make no appreciable difference for ordi- nary closing, but might lead to errors in the case of very high velocities or extremely slow closing. Derivation. — Since the total pressure in the pipe is Ahy, the 21 impulse received in the time — > during which interval the pres- Ve sure head is assumed to be increasing linearly from zero to h, is HAhyxf' 21 During the remaining time of closure, namely, t — —» the impulse received is 2l\ Vc M'-^ Aly Since the mass of the water filling the pipe is -— , the total mo- ^ Trans. Am. Soc. C. E., voL Ixxix, p. 250. 15 Digitized by VnOOQlC 226 ELEMENTS OF HYDRAULICS Aly mentum destroyed is V. Equating the impulse received to the momentum destroyed we have therefore 21 , ^^ /, 2l\ _ Aly g HAhy.?^ + Ahy(t^^)=^V Vc \ Vcl g whence h = --!^-Tr (i«) g i'-v) This formula is subject to the limitations that it is liable to be in error for very slow closing or for very high velocities; also, that it does not give any value for the fall in pressure following the rise, and does not apply to opening the gates. However, it is probably the most accurate formula yet derived, and its simplic- ity commends it for ordinary use.* 226. Joukovsky's Formula. — The formula for maximum water- hammer derived above, namely, 9 ' was first derived by Joukovsky.^ From this result he obtained a formula for ordinary water-hammer by assuming that if the time of closure is greater than the period T of the compression wave, the excess pressure is inversely proportional to the time of closure, that is, actual pressu re _ T. maximum pressure t Substituting the proper values in this relation we have therefore 21 h _ Vc Vvc^ t whence 21V h = ^. (147) This formula is now generally regarded as inaccurate, and for slow closure may be as much as 100 per cent, in error. 1 Joukovsky's experiments were made in Moscow in 1897-8, and the re- sults published in the Memoirs of the Imperidl Academy of Sciences, St. Petersburg, vol. ix, 1897. This account was translated from the original Russian into English in 1904 by Miss Olqa Simin, and the results discussed and amplified by Mr. Boris Simin, Jour. Amer WcUer Works Assoc, 1904, p. 335. Digitized by VnOOQlC ENERGY OF FLOW 227 226. Allievi's Formula. — Another formula for ordinary water- hammer has been derived more recently by L. Allievi.^ The derivation of this formula is based on the assumptions that the area of the gate decreases linearly, and that the velocity of flow decreases at a constant rate throughout the interval of closure. 21 It also assumes that the time of closure, L is not less than — , and is good for such values of t only. The derivation is somewhat complicated, involving the solu- tion of a differential equation and other mathematical difficulties, and is not given here. The complete derivation may be found in Mr. Warren's article, cited above. The result of AUievi's discussion is embodied in the formula h = ^±'H-^' + N (148) 2 where N = (-7^) , H = normal pressure head. The minus sign in the formula is supposed to apply in opening the gates, and therefore, according to the formula, the drop in pressiffe can never exceed H but approaches it as a limit. In 21 this formula the time t >—, and it has been found by experiment Ve that the formula becomes inaccurate when t approaches the value Ve 227. Occurrence of Water-hammer in Supply Systems. — The question of water-hammer is not restricted to the closure of gates, and surge in penstocks, but is also of frequent occurrence in the water-supply systems of large manufacturing cities where a large number of factories shut down at about the same time, say 5 :30 or 6 p.ni. It has also been found that when a small branch pipe having a dead end leads ofif from a larger pipe in which water-hammer takes place, the small pipe may be subjected to double the pres- sure set up in the large pipe. This should be taken into account in designing distributing systems, and also in taking readings on water-hammer from a pressure gage set in the end of a small pipe leading from a penstock or wheel case. 1 Annali della Societa degli Ingegneri, Rome, vol. xvii, 1902. Digitized by VnOOQlC 228 ELEMENTS OF HYDRAULICS UL. SURGE TANKS 228. Surge in Surge Tanks. — To relieve the effects of water- hammer in penstocks, produced by suddenly starting up or shut- ting down the plant, it is customary to provide surge tanks so that Surge Tank — JffUTtinPbn Concrete surge tank, penstocks, turbines and draft tune for 680- foot developmeut, Tallulah Falls, Georgia. I4?ntun' Meter-^ 45 hy&otJicQi^ ^ , . operated Gate \kive '^ . -^*^.-^^^^ ' ,,,^_,__„._ . — .i'^of^,U»ivt .^-^-,-j, Surge Reservot -Area 2IX'*' El. 1529 Cowrteay Eng. Record, Fig. 181. — Changes in hydraulic gradient due toVater hammer in penstock. the shock is cushioned by the simple rise or fall of the water level in the tank. To make the calculations requisite for surge-tank design, it is necessary to know the height of the surge up or down in the tank. In discussing this problem the following notation will be used: A = cross-sectional area of surge tank in square feet; a = cross-sectional area of conduit in square feet; Digitized by VnOOQlC ENERGY OF FLOW 229 V = velocity of flow in main conduit at time / in feet per second; V = normal velocity of flow in main conduit in feet per second during period of steady flow; F == friction head in feet corresponding to velocity of flow V; f = friction head in feet corresponding to velocity of flow v, and consequently/ = Fy-^] q = volume of discharge through penstock at time t in cubic feet per second; Q == volume of discharge through conduit at time t in cubic feet per second, and consequently g = Q when t^ = F; H = total head in feet lost between forebay and surge tank corresponding to velocity of flow V; 8 = distance of surge-tank level at any time t from its initial level, i.e., below level of forebay when starting up, and above a point H ft. below level of forebay when shutting down; S = maximum surge up or down in feet, measured from the initial level the same as for «• l«vel at time t Forebay Head lost in conduit^JSrT SUtic Level X Initial level when / startinir up Draw down when startinar up^^S 8-= Surge when Ishuttinflr down =^ Initial level when shutting down Fig. 182. — Changes in level in surge tank. To begin the demonstration assume that the forebay has a spillway such that the elevation of the water surface in the fore- bay remains constant during the period under discussion. Before starting up the plant, the water is at rest in the entire system, the water in the surge tank being at the same level as in the forebay. When the gate is suddenly opened, full-load flow is immediately drawn from the surge tank and the level suddenly Digitized by VnOOQlC 230 ELEMENTS OF HYDRAULICS drops by an amount called the "draw-down," and then oscillates until constant normal flow is established in the conduit. Under normal operating conditions the level of the water in the surge tank is then at a distance H below the level of the forebay, where H denotes the total head lost in the conduit for normal flow. In shutting down the plant, a sudden closing of the gate causes an upward surge in the surge tank, and the water level again oscil- lates iip and down until the motion dies away and the water once more comes to rest. The problem is, then, to find the maximum draw-down when starting the plant, and the upward surge when shutting down. The expression for this change in level, or surge, will be derived by appljdng the principle of work and energy. Consider first the case when the plant is starting up, and calcu- late the changes in work and energy which take place when the water in the conduit flows a distance dx. The kinetic energy of flow in the conduit at velocity v is kinetic energy = }4 ''^^^ = — o — ' ^^• The rate of change of this kinetic energy with the time is, then, ^(K.E.) = ^?^.«*. at g dt Consequently the total change in kinetic energy in the time dt during which the particles move a distance dx = vdt, is change m K.E. = ^ (K.E.)d« = The work done on the water entering the conduit during a motion dx is equal to the kinetic energy this water acquires, namely, ,/ , 62.4adx „ y^rnv^ = — ^ • v^, 2g The work expended in overcoming friction is work expended in friction = 62Aafdx, but since the friction head is proportional to the square of the velocity, we have f = F — where F denotes the friction head at the velocity F, and conse- quently the above expression may be written work expended in friction = 62.4 adx - Fy^ Digitized by LnOOQlC ENERGY OF FLOW 231 Finally, remembering that the plant is assumed to be starting up, the potential energy given up by the water in the surge tank is potential energy used = 62.4 asdx. Therefore, by equating the total work done to the kinetic energy acquired, we have ^^ . , 62AaLvdv , 62Aadxv^ , ^cy a ^ i? ^^ 62Aasdx = ^ h 62AadxF ,7^ g 2g 72 Substituting dx = vdt and then dropping the common factors 62.4 av, this reduces to gdi~^ ^ V 2g To simplify this expression, note that in the case of steady flow V becomes ^> jT = 0? a^^d s takes the value ff , where H denotes the total head lost in the conduit. Substituting these simul- taneous values in the differential equation, it becomes whence 72 "-I' + W Using this relation, the differential equation simplifies into ^S -»-=;. (i4„ A second fundamental equation may be obtained from the condition for continuous flow, namely, that the volume flowing through the conduit in any small interval of time dt must equal the sum of the change in volume in the surge tank and the vol- ume flowing through the penstock in the same interval. Since the rate at which the level in the surge tank is rising or falling is -r* its change of volume in a unit of time is A -r. In starting up, assuming that the full-load flow is immediately drawn from the surge tank, the amount flowing out through the penstock per second is 5, while the amount coming in through the conduit per second is av, and consequently the amount drawn from the surge tank per second \^ q — av] whence A ds A^ = q-av. Digitized by VnOOQlC 232 ELEMENTS OF HYDRAULICS For full-load flow we have ff = Q, and therefore this relation may also be written Differentiating this equation with respect to the time t, we have . (Pa do and eliminating ^ between this relation and the original differ- ential eqtiation, the result is dt* -M'-^vil- Substituting in this relation the value of v obtained from the condition for continuous flow, namely, V = > the result is finally d^s _ Hg /^ A ^s\' , ags dt^ ALV^a (^-^sT+S-'- ("») This differential equation embodies all the conditions of the problem, and its integral is the required solution. So far, how- ever, no one has succeeded in obtaining its integral by any direct method.* The simplest solution, and in fact the only one ob- tained so far which does not involve an erroneous assumption, is that due to Professor I. P. Church of Cornell University, ^ This differential equation has been obtained independently by Db. D. L. Webster of the Physics Department of Harvard University, and Pbof. Fbanz Prasil, but neither of them succeeded in integrating it in its general form. To effect its integration both made the assumption that the friction head is proportional to the first power of the velocity instead of to its square. Under this assumption, the differential equation reduces to that for a damped harmonic oscillation, the solution of which is well known. For Webbtbb'b derivation see "Penstock and Surge Tank Problems," by M. M. Wabrbn, Trans. Am. Soc. C. E., vol. badx, 1916. For Pbasil's derivation see "Surge Tank Problems," authorized translation by E. R. Weinmann and D. R. Cooper, Canadian Engineer^ vol. xxvii, 1914. Also reprint in pamphlet form of this series of articles. Digitized by LnOOQlC ENERGY OF FLOW 233 which consists in plotting the integral curve from its differential equation by means of successive tangents and points of tangency.^ To carry out this method^ eliminate dt between the two rela^ tions and A^ = Q - at; = a{V - v) Ldv _ _ „ j^ Instead of the above differential equation we then have the equivalent relation Fia. 183. which contains only two variables, s and v. Integrating this between the limits zero and S for s and the corresponding limits zero and V for v, we have H r-s La — ^ I v^ds = — {V - v)dv, which becomes S' 2^ r 2^ LaV^ Ag In this relation the quadrature J]fv^ds can be evaluated only when the relation between v and s is known. To obtain such a relation between v and s let AC, Fig. 183, *See discussion of Warren's article on "Penstock and Surge Tank Problems," by I. P. Church, frans. Am. Soc. C. E., vol. Ixxix, 1915, p. 272. Digitized by VnOOQlC 234 ELEMENTS OF HYDRAULICS denote the initial level in the surge tank at the beginning of the surge, and BD the level at the end of the first downward surge; in which case the distance CD = S. At any instant during the surge let v denote the velocity of flow in the conduit, and 8 the distance of the surface level in the surge tank below its initial position AC. Then by putting the differential equation in the form ds La V — V dv Ag H ^ y2 2 for any pair of values of v and s, the value of the slope -r at this point is determined. In particular, at A, which represents the beginning of the surge, we have v = and s = and therefore tan 1^ = oc ; whereas at D, which represents the end of the first downward surge, we have s = S, v = V and tan t> = 0. Using numerical data, it was found by Professor Church that the en- velope of the tangents so determined was very approximately the quadrant of an ellipse. The equation of an ellipse referred to the given axes with origin A is Sv = V(S - \/S' - s2). Substituting this relation in the quadrature and integrating, its value is found to be i: Subsituting this value in the differential equation, its complete integral is S' - 0.1917HS = ^^y Ag whence S = 0m58H + >fe^ + 0.00918/^2. Ag As an approximation sufficiently accurate for all practical pur- poses we may therefore write s = roH + >^' (151) which is the required formula. Digitized by VnOOQlC ENERGY OF FLOW 23S itial Ports This formula gives the maximum draw-doWn at starting up due to a sudden full opening of the penstock gates. For a sudden shutdown the sign of s is reversed, and the upward surge attains the same value S as the maximum draw-down, but is measured from the initial level which in this case is at a distance H below the static level in the forebay. Referring both the upward and downward surge to the static level in the forebay, we may there- fore say that the maximmn rise in the surge tank above this level is less than the maximum draw- down below it by the amount H of the friction head lost in the con- duit at the normal velocity of flow V. 229. Differential Surge Tank.— The differential surge tank is a modification of the ordinary surge tank, its function being to throttle the surges.^ Fig. 184 shows in out- line a surge tank of this type erected by the Salmon River Power Co. near Altmar, N. Y. Its func- tion is to absorb the energy of a column of water 9,625 ft. long and from 11 to 12 ft. in diameter. It consists of an elevated steel tank, 50 ft. in diameter and 80 ft. high, with a hemispherical bottom which adds 25 ft. to its height, connected with the distributor for the power-house penstocks by a 12-ft. riser. Inside of the tank is an interior riser 10 ft. in diameter flaring to a diameter of 10 ft. 8 in. at bottom and 15 ft. at top. As the riser to the tank is 12 ft. in diameter, this leaves an annulai opening 8 in. wide between the risers at the bottom of the tank. This opening is divided into 12 spaces forming the differential ports. The action of the surge tank is therefore as follows: When part of the load is thrown oflf the power station and the upward wave begins in the riser to the tank, a part of the volume is deflected through these ports into the main tank, thereby ^ Invented by R. D. Johnson, Hydraulic Engineer of Ontario Power Co. See A. S. M. E. Pamphlet No. 1,204, 1908; also Eng, Recardy July, 18, 1914, p 82. Fig. 184. — Differential surge tank. Digitized by VnOOQlC 236 ELEMENTS OF HYDRAULICS reducing the ultimate height of the level in the riser. When the wave recedes, water flows through the ports into the riser thereby- decreasing the depth of the surge. The greater the difference in elevation between the levels in the tank and the riser, the larger will be the volume of flow through the ports. The action there- fore tends to throttle the surges. The best precaution against hydraulic shock of this nature has been foimd to be the use of slow-closing valves. Air chambers placed near the valves have also been found effective if kept filled with air, and safety valves of course reduce the shock to a pres- sure corresponding to the strength of spring used. XLH. HYDRAULIC RAM 230. Principle of Operation. — ^A useful application of water- hammer is made in the hydraulic ram. In principle, a hydraulic Drive Pipe Connection A Discharge Pipe F Air Chamber E Escape Valve C Delivery Valve D Air Feeder H Fig. 185. — Rife hydraulic ram. ram is an automatic pimip by which the water-hanuner produced by suddenly checking a stream of running water is used to force a portion of that water to a higher elevation. To illustrate the method of operation, a cross-section of a ram is shown in Fig. 185. The ram is located below the level of the Digitized by VnOOQlC ENERGY OF FLOW 237 supply water in order to obtain a flow in the drive pipe. If located some distance from the supply, the water is first con- ducted to a short standpipe, as shown in Fig. 186, and from here a drive pipe of smaller diameter than the supply pipe conducts the water to the ram. The object of this arrangement is to utilize the full head of water available without making the drive pipe too long for the capacity of the ram. Fig. 186. Referring to Kg. 185, the water flowing in the drive pipe A at first escapes around the valve C, which is open, or down. This permits the velocity of flow to increase imtil the pressure against C becomes sufficient to raise it against its seat B. Since the water can then no longer escape through the valve C, it enters the air chamber E through the valve D, thereby increasing the pres- sure precisely as in the case of water-hammer discussed in the preceding article. When the pressure in E attains a certain maxi- mimi value, the flow is checked and the valve D falls back into place, closing the opening and trapping the water which has already entered the chamber E. The pressure in E then forces this water into the supply pipe F, which delivers it at an elevation proportional to the pressure in E. Hydraulic rams are also so built that they can be operated from one source of supply and pump water from a different source (Fig. 187). Muddy or impure water from a creek or stream may thuig be used to drive a ram, and the water pumped from a pure spring to the delivery tank. Digitized by VnOOQlC 238 ELEMENTS OF HYDRAULICS 231. Efficiency of Ram. — The mechanical efficiency of a ram depends on the ratio of fall to pumping head, ranging from 20 per cent, for a ratio of 1 to 30, up to 75 per cent, for a ratio of 1 to 4. Its efficiency as a pump is of course very small, as only a small fraction of the water flowing in the drive pipe reaches the delivery pipe. The advantages of the hydraulic ram are its small first cost, simplicity of operation, and continuous service day and night without any attention. To obtain an expression for the mechanical efficiency of a ram, let: .H: = supply head; h = effective delivery head including friction; q = quantity delivered; Q = quantity wasted at valve. Then the total input of energy to the ram is (Q + q)H, and the total output is qh. Consequently the mechanical efficiency is given by the ratio W - g^ (Q + q)H This is known as d'Aubuisson's efficiency ratio. The hydraulic efficiency, however, is the ratio of the energy required for delivery to the energy of the supply. Consequently its value is ^^ QH ' The latter expression is known as Rankine's formula. XLin. DISPLACEMENT PUMPS 232. Pump Tjrpes. — There are two types of pumps in general use; the displacement, or reciprocating type, and the centrifugal type. In the displacement pump the liquid is raised by means of a bucket, piston, or plunger, which reciprocates backward and forward inside a cylindrical tube called the pump barrel or cylinr der. In the centrifugal pump, as its name indicates, the opera- tion depends on the centrifugal force produced by rotation of the liquid. 233. Suction Pump. — One of the simplest forms of displace- ment pump is the ordinary suction pump shown in Fig. 188. Here the essential parts are a cylinder or barrel C containing a Digitized by LnOOQlC ENERGY OF FLOW 239 bucket B, which is simply a piston provided with a movable valve, permitting the water to pass through in one direction only. This bucket is made to reciprocate up and down inside the barrel 'by means of a rod E. A suction pipe S leads from the lower end of the barrel to the liquid to be raised, and a delivery pipe D discharges the liquid at the desired elevation. In operation, the bucket starts from its lowest position, and as it rises, the valve m closes of its own weight. The closing of this valve prevents the air from en- tering the space below the bucket, and consequently as the bucket rises the increase in volume below it causes the air confined in this space to ex- pand and thereby lose in pressure. As the pressure inside the suction pipe S thus becomes less than atmospheric, the pressure outside forces some of the liquid up into the lower end of the pipe. When the bucket reaches the top of its stroke and starts to descend, the valve n closes, trapping the liquid already in the suction pipe S and also that in the barrel, thereby lifting the valve m as the bucket descends. When the bucket reaches its lowest position, it again rises, repeating the whole cycle of operations. At each repetition the water rises higher as it replaces the air, until finally it fills the pump and a continuous flow is set up through the delivery pipe. 234. Maximum Suction Lift. — Since atmospheric pressure at sea level is 14.7 lb. per square inch, a pump operating by suc- tion alone cannot raise water to a height greater than the head corresponding to this pressure. Since a cubic foot of water weighs 62.4 lb., the head corresponding to a pressure of one atmos- phere is 14 7 14 7 Fig. 188. — Suction pump. h = 62.4 144 0.434 which is therefore the maximum theoretical height to which water can be lifted by suction alone. As there are frictional and other Digitized by VnOOQlC 240 ELEMENTS OF HYDRAULICS losses to be considered, the actual suction lift of pumps is only about two-thirds of this amount, the practical lift for different attitudes and pressures being as given in the following table. Altitude Barometric Equivalent head Practical suction pressure of water lift of pumps Sea level 14.701b. per sq. in. 33.95 ft. 22 ft. 1/4 mUe... 14.021b. per sq. in. 32.38 ft. 21ft. 1/2 mUe... 13.331b. per sq. in. 30.79 ft. 20 ft. 3/4 mUe... 12.661b. per sq. in. 29.24 ft. 18 ft. 1 mile 12.02 1b. persq, in. 27.76 ft. 17 ft. 1-1/4 miles. 11.421b. persq. in. 26.38 ft. 16 ft. 1-1/2 miles. 10.881b. persq. in. 25.13 ft. 15 ft. 2 miles 9.881b. persq. in. 22.82 ft. 14 ft. 235. Force Pump. — When it is necessary to pump a liquid to a height greater than the suction lift, or when it is desired to equal- ize the work between the up and down strokes, a combination suction and force pump may be used, as shown in Fig. 189. In the simple type here illus- trated, the bucket is replaced by a solid piston, tlie movable valves being at m and n as shown. On the up stroke of the piston the valve m closes and the pump operates like a simple suction pump, filling the barrel with liquid. When the piston starts to descend, the valve n closes, and the liquid in the bairel is there- fore forced out through the valve m into the delivery pipe D. By making the suction and pressure heads equal, the piston ca'n therefore be made to do the same amoimt of work on the down as on the up stroke; or the entire suction head may be utilized and the pressure head made whatever may be necessary. 236. Stress in Pump Rod. — To find the pull P on the pump rod E for the type shown in Fig. 188, let A denote the area of the bucket and hi, A2, the heads above and below the bucket, as FiQ. 189. — Combined suction and force pump. Digitized by VnOOQlC ENERGY OF FLOW 241 indicated in the figure. Then the downward pressure Pi on top of the bucket is Pi = 14.7A + 62AhiA, and the upward pressure P2 on the bottom of the bucket is P2 = U.7A - 62AhiA. Therefore the total pull P in the rod is P = Pi - P2 = 62.4ii(Ai + hi) = 62.4Ah. If I denotes the length of the stroke, the work done per stroke is then work per stroke = PI = 62.4Ahl. For the combined suction and pressure type shown in Fig. 189, the pressure in the rod on the down stroke is P = 62.4iift2, and the tension in the rod on the up stroke is P = 62.4iifti. Fig. 190. — Direct acting steam pump. 237. Direct-driven Steam Pump. — The modern form of reciprocating power pump of the suction and pressure type is the direct-driven steam pump, illustrated in Figs. 190 and 191. In this type the steam and water pistons are on opposite ends of the same piston rod and therefore both have the same stroke, Id Digitized by vnOOQlC 242 ELEMENTS OF HYDRAULICS Fio. 191. \^=J Steam lulet Fig. 192, Digitized by VnOOQlC ENERGY OF FLOW 243 although their diameters are usually different. Until recently this was the standard type of general service pump, being used for all pressures and capacities, from boiler-feed pumps to muni- cipal pumping plants. Although the centrifugal type is rapidly taking its place for all classes of service, the displacement pump is the most efficient where conditions demand small capacity at a high pressure, as in the operation of hydraulic machinery. Fig. 192 illustrates the use of a displacement pimip in connection with a hydraulic press. The best layout in this case would be to use a high-pressure pump and place an accumulator (Par. 14) in the discharge line between pump and press. The press cylin- der can then be filled immediately at the maximum pressure and the ram raised at its greatest speed, the pmnp running meanwhile at a normal speed and storing excess p>ower in the accumulator. 238. Calculation of Pump Sizes. — To illustrate the calculation of pump sizes, suppose it is required to find the proper size for a duplex (i.e., two cylinder, Fig. 191) boiler-feed pump to supply a 100-h.p. boiler. For large boilers the required capacity may be figured as 34 J^ lb. of water evaporated per hour per horse power. For small boilers it is customary to take a larger figiu-e, a safe practical rule being to assume Ko gal. per minute per boiler horse power. In the present case, therefore, a 100-h.p. boiler would require a supply^ of 10 gal. per minute. Assuming 40 strokes per minute as the limit for boiler feed pumps, the required capacity is j^ = 0.25 gal. per stroke. Therefore, assuming the efficiency of the pump as 50 per cent., the total capacity of the pump per stroke should be ^-^ = 0.5 gal. per stroke. Since we are figuring on a duplex, or two-cylinder, pmnp, the required capacity per cylinder is -^ = 0.25 gal. per stroke per cylinder, and consequently the displacement per stroke on each side of the piston must be 25 -^ = 0.125 gal. piston displacement. Referring to Table 7 it is found that a pump having a water cylin- Digitized by VnOOQlC 244 ELEMENTS OF HYDRAULICS der 2^ in. in diameter, with a 6-in. stroke, will have the required capacity. 239. Power Reqtiired for Operation. — ^To find expressions for the horse power and steam pressure required to operate a dis- placement pump, let: Q = discharge of pump in gallons per minute; h = total pumping head in feet (including friction and suction head if any) ; D ^ diameter of steam piston in inches; d = diameter of water piston in inches; p = steam pressure in pounds per square inch; w = water pressure in pounds per square inch; n = number of full strokes (i.e., roimd trips) per minute; c = number of pump cylinders {e.g., for duplex pump, Fig. 191, c = 2); I = length of stroke in inches; E == efficiency of pump. Since a gallon of water weighs 8.328 lb., the total work per minute required to raise the given amount Q to the height h is work = 8.328QA ft.-lb. per minute. Taking into account the efficiency of the pump, the actual horse power required is therefore 240. Diameter of Pump Cylinder. — If the pump makes n full strokes per minute, the piston displacement per minute for each cylinder is and the actual effective displacement of the pump per minute is (/?2 \ -T-lj cu. in. per minute. Equating this to the required discharge Q, expressed in cubic inches per minute, we have 2ncE(^)^231Q, whence the required diameter of the pump cylinder in terms of the speed is found to be Digitized by VnOOQlC whence ENERGY OF FLOW 245 241. Steam Pressure Required for Operation. — Since the total pressure on the steam piston cannot be less than that on the water piston, the minimum required steam pressure, p, is given by the relation p(f).0.43«(f), p = 0.434h(^)*. (164) 242. Numerical Application. — ^To illustrate the application of these results, suppose it is required to determine the indicated horsepowerlio operate a fire engine which delivers two streams of 250 gal. per minute each, to an effective height of 60 ft. Since the height of an effective fire stream is approximately four-fifths that of the highest drops in stiU air, the required head at the nozzle is jh = Jx 60 = 75 ft. 4 4 To this must be added the friction head h/ lost in the hose between pump and nozzle, which is given by the relation (Par. 99) where I is the length of the hose and d its diameter, both expressed in inches, v is the velocity of fiow through the hose, and / is an empirical constant. For the best rubber-lined hose, / = 0.02 for the first 100 ft. of hose and 0.0025 for each additional 100 ft. whereas for unlined hose / = 0.04 for the first 100 ft. and 0.005 for each additional 100 ft. In the present case, assmning 100 ft. of the best 2J^-in. rubber-lined hose, we have/ = 0.02, and since the quantity of water delivered is Q = 250 X Y72Q ^^* ^^- P®^ minute, and the area of the hose is . xd* ir(2.5)2 ' rt^Q . A = -^ = ^. ^ = 4.908 sq.m., the velocity of flow in the hose is V = 144X60 = 16.3 ft. per second. Digitized by LnOOQlC S46 ELEMENTS OF HYbRAULtCS Consequently the friction head h/ is^ f A, = 0.02ig?-^ = 39.6,8ay40ft., and therefore the total pumping head H is H = 76 + 40 = 115 ft. From Eq. (103) the total horse power required, assuming a pimip efficiency of 60 per cent., is then found to be t; H.P. = 0.00025 ^^^— = 28.76. Assiuning the efficiency of the engine to be 60 per cent., the total indicated horse power required would be XLIV. CENTRIFUGAL PUMPS 243. Historical Development. — The centrifugal pump in its modern form is a development of the last 16 years although as a type it is by no means new. The inventor of the centrifugal pimip was the celebrated French engineer Denis Papin, who brought out the first pump of this type in Hesse, Germany, in 1703. Another was designed by Euler in 1754. These were regarded as curiosities rather than practical machines until the type known as the Massachusetts pump was produced in the United States in 1818. From this time on, gradual improve- ments were made in the centrifugal pump, the most important being due to Andrews in 1839, Bessemer in 1845, Appold in 1848, and John and Henry Gynne in England in 1851. Experiments ' seemed to show that the best efficiency obtainable from pumps of this type ranged from 46 to 64 per cent, under heads varjdng from 4^ to 15 ft., and 40 ft. was considered the maximum head for practical operation. About the year 1901 it was shown that the centrifugal pump was simply a water turbine reversed, and when designed on simi- lar lines was capable of handling heads as large, with an efficiency as high, as can be obtained from the turbines themselves. Since this date, great progress has been made in both design and con- struction, the efficiency of centrifugal pumps now ranging from 55 to over 90 per cent., and it being possible to handle heads as Digitized by LnOOQlC ENERGY OF FLOW 247 high as 300 ft. with a single-stage turbine pump and practically any head with a multi-stage type.^ The advantages of the centrifugal over the displacement type are its greater smoothness of operation, freedom from water- hammer or shock, absence of valves, simplicity and compactness, and its adaptability for driving by belt or by direct connection to modern high-speed prime movers, such as steam turbines, gas engines and electric motors. Under favorable conditions the first cost of a high-lift centrifugal pimip may be as low as one* third that of a displacement pump, and the floor space occupied one-fourth that required by the latter. However, for small quantities of water discharged under a high liead the displace- ment pmnp is preferable to the centrifugal type, as the latter requires too much compounding imder such conditions. 244. Principle of Operation. — The principle on which the original centrifugal pumps of Papin and Euler operated was simply that when water is set in rotation by a paddle wheel, the centrifugal force created, forces the water outward from the cen- ter of rotation. Appold discovered that the eflSciency depended chiefly on the form of the blade of the rotary paddle wheel, or impeller, and the shape of the enveloping case, and that the best form for the blade was a curved surface opening in the opposite direction to that in which the impeller revolved, and for the case was a spiral form or volute. The first engineer to discover the value of compounding, that is, leading the discharge of one cen- trifugal pimip into the suction of another similar pump, was the Swiss engineer Sulzer of Winterthur, who was closely followed by A. C. E. Rateau of Paris, France, and John Richards and Byron Jackson of San Francisco, Cal. In its modern form, the power appUed to the shaft of a cen- trifugal pump by the prime mover is transmitted to the water by means of a series of curved vanes radiating outward from the center and mounted together so as to form a single member called the impeller (Fig. 193). The water is picked up at the inner edges of the impeller vanes and rapidly accelerated as it flows between them, until when it reaches the outer circumference of the impeller it has absorbed practically all the energy applied to the shaft. 'Rateau found by experiment that with a single impeller 3.15 in. in diameter, rotating at a speed of 18,000 r.p.m., it was possible to attain a head of 863 ft. with an efficiency of approximately 60 per cent. Engineer^ Mar. 7, 1902. Digitized by VnOOQlC 248 ELEMENTS OF HYDRAULICS Hollow arm impeller. Concave arm impeller. Sand pump impeller. Open impeller used in sewage pumps. Enclosed side suction impeller. Enclosed double suction impeller. Fig. 193. — Impeller types. {Courtesy Morris Machivs Works,) Digitized by LnOOQlC ENERGY OF FLOW 249 246. Impeller Forms. — There are two general forms of impel- ler, the open and the closed types. In the former the vanes are attached to a central hub but are open at the sides, revolving between two stationary side plates. In the closed type, the vanes are formed between two circular disks forming part of the impeller, thus forming closed passages between the vanes, extending from the inlet opening to the outer periphery of the impeller. The friction loss with an open impeller is considerably more than with one of the closed type, and consequently the design of pumps of high efficiency is Umited to the latter. 246. Conversion of Kinetic Energy into Pressure. — ^As the water leaves the impeller with a high velocity, its kinetic energy Fia. 194. forms a considerable part of the total energy and the efficiency of the pimip therefore depends largely on the extent to which this kinetic energy is converted into pressure in the pump casing. In some forms of pump no attempt is made to utilize this kinetic energy, the water simply discharging into a concentric chamber surrounding the impeller, from which it flows into a discharge pipe. The result of such an arrangement is that only the pressure generated in the impeller is utilized and all the kinetic energy of the discharge is dissipated in shock and eddy formation. 247, Volute Casing. — ^This loss of kinetic energy may be par* Digitized by VnOOQlC 250 ELEMENTS OF HYDRAULICS tially avoided by making the casing spiral in section, so that the sectional area of the discharge passage increases uniformly, making the velocity of flow constant (Fig. 194). This type of casing is called a volute chamber (Fig. 195). When the volute is properly designed, a high efficiency may be obtained with this type of casing.^ Fig. 195. — Double-suction volute pump, Piatt Iron Works Co. 248. Vortex Chamber* — ^An improvement on the simple volute chamber is that known as the whirlpool chamber, or vortex chamber, suggested by Professor James Thomson. In this type the impeller discharges into a concentric chamber considerably larger than the impeller, outside of and encircling which is a volute chamber. In its original form this necessitated exces- sively large dimensions, but in a modified form it is now very generally used (Figs. 196 and 197). The effectiveness of this arrangement depends on the principle of the conservation of angular momentum. Thus, after the water leaves the impeller no turning moment is exerted on it (neglecting frictional resistance) and consequently as a given * With the De Laval volute type of centrifugal pump shown in Fig. 196> efficiencies as high as 85 per cent, have been obtained under favorable conditions. Digitized by VnOOQlC ENERGY OF FLOW 261 Fig. 196. — Longitudinal section of De Laval single-Btage double -suction volute pump. Fio. 197.— Longitudinal section of Alberger volute pump. Digitized by vnOOQlC 252 ELEMENTS OF HYDRAULICS mass of water moves outward, its speed decreases to such an extent as to keep its angular momentum constant. For a well- designed vortex chamber, the velocity of the water at the outside of the diffusion space is less than the velocity of the water as it leaves the impeller in the inverse ratio of the radii of these points, and if this ratio is large, a large part of the kinetic energy of the discharge may therefore be converted into pressure head in this manner. This method of diffusion is therefore well adapted to the small impellers of high-speed pumps, since the ratio of the outer radius of the diffusion chamber to the outer radius of the impeller may be made large without unduly increasing the size of the casing. Fig. 198.— Diffusion ring. 249. Diffusion Vanes. — ^Another method for converting the kinetic energy of discharge into pressure head consists in an application of Bernoulli's law as illustrated in the Venturi tube; namely, that if a stream flows through a diverging pipe the initial velocity head is gradually converted into pressure head without appreciable loss. To apply this principle to a centrifugal pump, the impeller is surrounded by stationary guide vanes, or diffusion vanes (Fig. 198), so designed as to receive the water without Digitized by VnOOQlC ENERGY OF FLOW 253 shock on leaving the impeller and conduct it by gradually diverg- ing passages into a vortex chamber or volute casing. This type of construction is therefore essentially a reversed turbine, and is commonly known as a turbine pump (Fig. 199). END SECTIONAL VIEW S^DE SECTIONAL VIEW FiQ. 199. — Alberger two-stage turbine pump. The angle which the inner tips of the diffusion vanes make with the tangents to the discharge circle is calculated exactly as in the case of the inlet vanes of a turbine, that is, so that 'they shall be parallel to the path of the water as it leaves the impeller. Digitized by VnOOQlC 254 ELEMENTS OF HYDRAULICS As this angle changes with the speed, the angle which is correct for one speed is incorrect for any other and may actually obstruct the discharge. A turbine pump must therefore be designed for a particular speed and discharge, and when required to work under variable conditions loses considerably in efficiency. If the conditions are very variable, the vortex chamber type is preferable, both by reason of its greater average efficiency under such conditions and also on account of its greater simplicity and cheapness of construction. 260. Stage Pumps. — Single impellers can operate efficiently against heads of several hundred feet, but for practical reasons it Fia. 200. — Worthington two-stage turbine pump. is desirable that the head generated by a single impeller should not exceed about 200 ft. When high heads are to be handled, therefore, it is customary to mount two or more impellers on the same shaft within a casing so constructed that the water flows successively from the discharge of one impeller into the suction of the next. Such an arrangement is called a stage pump j and each impeller, or stage, raises the pressure an equal amount. Fig. 200 shows a multi-stage pump of the turbine type and Fig. 201 one of the volute type. A single impeller pump may be either of the side-suction or double-suction type. In the latter, half of the flow is received on Digitized by VnOOQlC ENERGY OF FLOW 265 each side of the impeller which is therefore perfectly balanced against end thrust (Fig. 196). A side-suction pump, however, is simpler in construction, and it is also possible to balance them hydraulically against end thrust (Fig. 197). In stage pumps the device sometimes used for balancing is to arrange the impellers in pairs so that the end thrust of one impeller is balanced by the equal and opposite end thrust of its mate. Fig. 201. — De Laval three^stage volute pump. XLV. PRESSURE DEVELOPED IN CENTRIFUGAL PUMP 251. Pressure Developed in Impeller. — The pressure produced in a centrifugal pump must be sufficient to balance the static and frictional heads. When there is no volute, vortex chamber or diffusor, the kinetic energy of the discharge is all dissipated and the entire change in pressmre is produced in the impeller. If, however, the velocity of discharge is gradually reduced by means of one of these devices, a further increase in pressmre is produced in the casing or diffusion space, and if a diverging discharge pipe is used the pressure is still further increased. The change in pressure which is produced in passing through the impeller may be deduced by applying Bernoulli's theorem. For this purpose it is convenient to separate the total difference in pressure between the inlet and discharge circles into two com- ponents; one due to the rotation of the water in a forced vortex with angular velocity w, and the other due to the outward flow, i.e., the relative motion of the water with respect to the vanes of the impeller. Let the subscripts 1 and 2 refer to points on the inlet and discharge circles respectively. Then the radii of Digitized by VnOOQlC 256 ELEMENTS OF HYDRAULICS these circles will be denoted by ri, r»; the pressure at any point on these circles by pi, p2, etc. Also let « denote the angular Fig. 202. — Detail of labyrinth rings in piunp showD in Fig. 201. Fig. 203. velocity of rotation of the impeller, and ui, u^ the tangential velocities of the vanes at their inner and outer ends (Fig. 203), in which case u\ = riw and u% = rtw. Digitized by LnOOQlC ENERGY OF FLOW 267 Applying Bemoull's theorem to the change in pressure pro- duced by rotation alone, we have therefore 7 ^ " V ^' Consequently the total change in pressure due to rotation, say Pr where Pr = Pi — Pu ^ given by the relation 7 " 7 " 2/^^* "^^ ^ 2(7 This expression is often called the centrifugal head. By similar reasoning the change in pressure produced by the outward flow is given by the relation y + tot* 2g y + 2g' P't — P'l Wi' — w»* whence _ T 2(7 If the water enters radially, ^1 = 50° and consequently it?!* = Vi* + til*. In this case, denoting the difference in pressure at inlet and exit due to the flow by p/, where p/ = p'2 — p'l, we have Pf ^ p'i - p\ ^ vi^ + ^i' - t^2*^ 7 T 2flf * The total increase in pressure in the impeller between the inlet and discharge ends of the vanes is therefore given by the relation Pr + Pf _ Vl^ + Ul* -• W%^. + U2* -^ til* _ Pi* + ti2* — t£?2* 7 " 2flf " 2g ' 262. Pressure Developed in Difihisor. — ^Besides the increase in pressure produced in the impeller, the use of a suitable diffusion chamber permits part of the kinetic energy at exit, due to the absolute velocity V2 of the discharge from the impeller, to be converted into pressure. Thus if k denotes the fraction of this kinetic energy which is converted into pressure in the diffusor, va* the expression derived above is increased by the term k^r-. When ^g diffusion vanes are used, as in a turbine pump, the value of k may be as high as 0.75, and for a vortex chamber it may reach 0.60. 263. General Expression for Pressure Head Developed. — Combining the terms derived above, the total pressure head H developed by the pump is given by the simple expression ^^kv,* + Vx* + «,»-w.»^ (166) 17 Digitized by VnOOQlC 258 ELEMENTS OF HYDRAULICS In applying this formula it is convenient to note that the total head H developed in the pump consists of three terms, as follows: U2 2 _ W2^ 2g kv£_ 2g = head at eye (entrance) of impeller; = head developed in impeller; = head developed in casing or diffusor. XLVI. CENTRIFUGAL PUMP CHARACTERISTICS 264. Effect of Impeller Design on Operation. — The greatest source of loss in a centrifugal pump is that due to the loss of the Velocity jelabVe^ *o Blade .Wa Fig. 204. kinetic energy of the discharge. As only part of this kinetic energy can be recovered at most, it is desirable to reduce the velocity of discharge to as low a value as is compatible with effi- ciency in other directions. This may be accomplished by curving the outer tips of the impeller vanes backward so as to make the Digitized by vnOOQlC ENERGY OF FLOW 259 discharge angle less than 90**. The relative velocity of water and vane at exit has then a tangential component acting in the oppo- site direction to the peripheral velocity of the impeller, which therefore reduces the absolute velocity of discharge. This is apparent from Fig. 204 in which the parallelogram of velocities in each of the three cases is drawn for the same peripheral velocity U2 and radial velocity at exit W2 sin 62. A comparison of these diagrams indicates how the absolute velocity at exit V2 increases as the angle ^2 increases. The backward curvature of the vanes also gives the passages a more imiform cross-section, which is favorable to efficiency. The average value of ^2 at exit is about 30^ The effect which the design of the impeller has on the operation of the pump is most easily illustrated and imderstood by plotting curves showing the relations between the variables under con- sideration. Assuming the speed to be constant, which is the usual condition of operation, three curves are necessary to com- pletely illustrate the operation of the pump; one showing the relation between capacity and head, one between capacity and power, and one between capacity and efficiency. The first of these curves is usually termed the charctcteristic. 266. Rising and Drooping Characteristics. — The principal fac- tor influencing the shape of the characteristic is the direction of the tips of the impeller blades at exit, although there are other factors which affect this somewhat. If the tips are curved for- ward in the direction of rotation the characteristic tends to be of the rising type, whereas if they curve backward the characteristic tends to be of the drooping type (Figs. 205 and 206). For a ris- ing characteristic the head increases as the delivery increases 3,nd consequently the power curve also rises, since a greater discharge against a higher head necessarily requires more power (Fig. 207). Digitized by VnOOQlC 260 ELEMENTS OF HYDRAULICS -u A u ■^ ^ "^ ^ ^ £fA«Lniv - ^^ ::s ^ ^_ " T ■ ( A ^ ^ /" V** ■^ N, ^ ^ "1*^ **^ s ^ i n // ^ **7 ^V N \ / 1 V s, y i / S \ ^ \ \ 2 _4 2- ( 80 lf)0 120 N LO CAPAcnr CAPACITY 1AA— li iO £J ^ - ^ -^ ■^ T i ■"" — — (J ^ Uj \ ^ ^ ^ .J::r ■^ ■^lOO ou $&< i4jC c5- =^ ■N J L ^ *^ "^ A iO y /^' / / 2 __4 0_ __j _^ e i JL ji k. 1 to Jj iO CAPACITY FiQ. 206. — Characteristics and efficiency curves obtained from De Laval centrifugal pumps. Digitized by VnOOQlC ENEROY OF. FLOW 261 A drooping power curve may be obtained by throttling at the eye of the impeller, -but a greater efficiency results from designing the impeller so as to give this form of curve normally. — — — — ' — — — — J — — — ' — — — — ' — ' — — — ISO J — 140 - j !»(» h- _ — 1 — — — — ^ ^ — — = ^ — . — — — — — — — 1 — — m — ■ — ■ - — — — — — 1 — — — — — — 1 — ■ — -^ *- ■* ■^ 5— 1 r^ 110 jEn^ ^ ^ = — — L— . — 1 ^ « ^^ ^ ^J^ -^ — — — — — lOti :=^ ^ — „ — — Z-^ F^- ^^^^ ■^ r^ ^ ^ ^T^ =fc: — -4 — — — — 90 fc^ ■!►- -^T^ r^ 1^ '^ 1 .-^ T' \ W ^ .-^^ i:f'> \ ^. TO ^ ^5C5 . 1 s - f^ i^ ^^ flfl j — ^ t^ ^ ^^"^ , , ; ^H _^ p— 1 _ DO — — — 1 — ■—3 * i^ -Vii — ■ — — — — — — h -^ — ' — — r^:^ — — 1^ ^ \\ 40 ^ ^ ^ ^ , . ' f\K- f^— 7^ .• — — 1 — — " — ~^ — — 2^ r ' — ^T\ / ■ ■ \\ SO ^. i. ^^ I- .^ 1 F 3t^ 1^ ^ ■ 1 \ _ . 1 i 1 i' 1 . I , ^ -iV k^— 10 20 30 40 GO 60 TO 80 90 100 110 120 m UQ 1^ 160 170 Percentage of Nonnal Oapacltj FiQ. 207. For a high-lift pump under an approximately constant head, as in the case of elevator work, a pump with radial vanes is most suitable as the discharge may be varied with a small alteration in deUvery head. This is also true for a pump working under a — — ~~ ^" ~" — r— n^ ' [ ICO 1 1 m 130 ^r: ~ ^:r — ■^1 ■^ ,^_; 4^« Irf- _^ - __ - 120 ' V • — — — — ■ — ^ h ^ — ~ — — ■ — — • iin n ^ 110 ; 100 ^ ^ ^^^ P^ •^^5^ « .vf^ m -- 1 1^^ ---^ - ,..u ^^t^ « \!^l [ev % N, m s s. '^ 132!! i ■ J*r ^ 1Q ^-*-^ .\^'^ Nj— ^^.4_ .r-^ \ ^. 60 U- , \ — ^ \ N GO J y* X ] ■' \ \ 40 ^ ' s \ y ^ ^ 30 ^^ .1 \ ■! ^ _j 1"" w / L 10 ^ y. f y xn L- — _ , \ I K) 20 30 40 50 60 70 80 90 100 110 120 130 140 16Q 160 Percentage Of Nonnal OapaoLty Fig. 208. falling head, as in the case of emptying a lock or dry dock, as it makes it possible to obtain a large increase in the discharge as the head diminishes, thereby saving time although at a loss of efficiency. Digitized by VnOOQlC 262 ELEMENTS. OF HYDRAULICS One of the most important advantages of a drooping char- acteristic (Fig. 208) is that it is favorable to^ a drooping power- delivery curve, making it impossible for the pump to overload the driving motor. For an electrically driven pump, in which the overload is limited to 20 per cent., or at most 25 per cent., of the normal power, backward-curved vanes are therefore essential. Moreover, with a pump designed initially to work against a certain head, if the vanes at exit are radial, or curved forward, the possible diminution in speed is very small, the discharge ceasing altogether when the speed falls slightly below normal. As the backward curvature of the vanes increases the range of speed also increases, and consequently when the actual working head is not constant, as in irrigation at different levels, or in delivering cooUng water to jet condensers in low-head work, where the level of the intake varies considerably, a pump with drooping characteristic is much better adapted to meet varying conditions without serious loss of efficiency. 266. Head Developed by Pump. — These facts may be made more apparent by the use of the expression for the head developed by the pump, derived in the preceding article. Considering only . the head developed in the impeller and casing, and omitting that due to the velocity of flow at entrance, Vi, which does not depend on the design of the pump, the expression for the head developed is a. = zz • Since v^ is the geometric resultant of Ui and w^, we have by the law of cosines, V2^ = t*2^ + t02* — 2t*2t^2 cos ^2. V2^ For an ideal pump, that is, one in which all the velocity head ^ - is converted into pressure head in the diffusor, k is unity. As- suming A; = 1 and substituting the expression for v^^ in the equa- tion for Hj the result is Tj ^2* — t*2t^2 cos 62 Q For constant speed of rotation, U2 is constant. For forward-curved vanes 62 is greater than 90® and therefore cos 02 is negative. In this case as W2 increases H also increases; i.e., the greater the delivery the greater the head developed. Digitized by VnOOQlC mSR^Y OF FLOW 26S For radial-tipped vanes, 62 = 90** and cos 62 = 0. In this case H = — , which is constant for all deliveries. For backward-curved vanes dt is less than 90® and cos 62 is positive. Consequently in this case as the delivery increases the head diminishes. Although these relations are based on the assumption of a perfect pump, they serve to approximately indicate actual conditions, as is evident by inspection of the three types of characteristic. 267. Effect of Throttling the Discharge. — It is always neces- sary to inake sure that the maximum static head is less than the head developed by the jpiunp' ^,t no discharge. This is self-evi- dent for the drooping characteristic, but the rising characteristic is misleading in this respect as the head rises above that at shut- off. Since for a certain range of head two different outputs are possible, it might seem that the operation of the pimip under such conditions would be imstable. This instability, however, is coimteracted by the frictional resistance in the suction and delivery pipes, which usually amoimts to a considerable part of the total head. Any centrifugal pump with rising characteristic will therefore work satisfactorily if the maximum static head is less than the head produced at shut-off. If the frictional resist- ance is small it may be increased by throttling the discharge, so that by adjusting the tlirottle it is possible to operate the pump at any point of the curve with absolute stability. 258. Numerical Illustration. — ^The particular curves shown in Fig. 209 were plotted for an 8-in., three-stage turbine fire piunp built by the Alberger Co., New York, and designed to deliver 760 gal. per minute against an effective head of 290 ft., the pump being direct connected to a 75-h.p. 60-cycle induction motor op- erating at a synchronous speed of 1,200 r.p.m. The head curve shows that this pump would deliver two fire streams of 260 gal. per minute each, at a pressure of 143 lb. per square inch; three streams of 260 gal. per minute each, at a pres- sure of 125 lb. per square inch; four streams of 25Crgal. per minute each, at a pressure of nearly 100 lb. per square inch; or even five fairly good streams at a pressure of 80 lb. per square inch. With the discharge valve closed the pump delivers no water but pro- duces a pressmre equivalent to a head of 308 ft. If the head against which the pump operates exceeds this amount, it is of Digitized by VnOOQlC 264 ELEMENTS OF HYDRAULICS course impossible to start the discharge. The head for which this particular piunp was designed was 290 ft., which corresponds to the point of maximum efficiency. It is therefore appar- ent that the operating head must be carefully ascertained in advance, for if it is higher than that for which the pump was designed, both the efficiency and the capacity are diminished, whereas if it is lower, the capacity is increased but the efficiency is diminished. 500 600 700 800 900 1000 1100 1200 Gallons-per Minute Fig. 209. 1800 The power curve shows that under low heads the power rises. Also that the overload in the present case is confined to about 12 per cent, of the normal power. Consequently the motor could only be overloaded 12 per cent, if all the hose lines should burst, whereas the head curve shows that if all the nozzles were shut oflF no injurious pressure would result. The efficiency curve always starts at zero with zero capacity, as the pump does no useful work until it begins to discharge. The desirable features of an efficiency curve are steepness at the Digitized by VnOOQlC ENERGY OF FLOW 265 two ends, a flat top and a large area. Steepness at the beginning shows that the efficiency rises rapidly as the capacity increases, whereas a flat top and a steep ending show that it is maintained at a high value over a wide range. Since the average efficiency is obtained by dividing the area enclosed by the length of the base, it is apparent that the greater the area for a given length, the greater will be the average efficiency. XLVn. EFFICIENCY AND DESIGN OF CENTRIFUGAL PUMPS 269. Essential Features of Design. — The design of centrifugal pumps like that of hydraulic turbines requires practical ex- perience as well as detailed mathematical analysis. The general principles of design, however, are simple and readily understood, as will be apparent from what follows: Three quantities are predetermined at the outset. The inner radius of the impeller, fi, is ordinarily the same as the radius of the suction pipe or slightly less; the outer radius, r2, is usually made twice n; and the angular speed o) at which the impeller is designed to rim is fixed by the particular type of prime mover by which the pump is to be operated. The chief requirement of the design is to avoid impact losses. In order therefore that the water shall glide on the blades of the impeller without shock, the relative velocity of water at entrance must be tangential to the tips of the vanes. Assiuning the direction of flow at entrance to be radial, which is the assiunption usually made although only approximately realized in practice, the necessary condition for entrance without shock is (Fig. 203) Vi = Wi tan ^1, which determines the angle ^i. The relative velocity of water and vane at entrance is then Wi = Vwi* + Vi^. The direction of the outer tips of the vanes, or angle 6%, Fig. 203, is determined in practice by the purpose for which the pump is designed, as indicated in Art XL VI. For an assigned value of $2, the absolute velocity of the water at exit is Vi^ = tia* + t»2* — 2uk%wt cos $2 and consequently as 0% increases, the absolute velocity at exit, vt, also increases. Digitized by VnOOQlC 266 ELEMENTS OF HYDRAULICS Let sif 82, Fig. 203| denote the radial velocity of flow at en- trance and exit, respectively, and Ai, A2 the circumferential areas of the impeller at these points. Then for continuous flow 81A1 = «ailj. Usually Si = 82, in which case Ai = A2. If 6] and bs denote the breadth of the impeller at inlet and outlet respectively, then Ai = 2Tri&i and A2 — 2irr2b2, and conse- quently for ill = A 2 we have biri = ftgrj. Assuming the radial velocity of flow throughout the impeller to be constant, the breadth b at any radius r is given by the relation br = biVi. 260. Hydraulic and Commercial Eflkiency. — ^Let H' denote the total effective head against which the piunp operates, including suction, friction, delivery and velocity heads. Then if w denotes the velocity of the water as it leaves the delivery pipe, h the total lift including suction and delivery heads, and h/ the friction head, we have H' = A + A, + |. The total theoretical head H developed by the pump, as derived in ArtXLV, is Vi^ + kV2^ + W2* - W2^ H = 2g Consequently the hydravlic efficiency of the pump is the ratio of these two quantities, that is, H' Hydraulic efficiency = =-• (156) The commercial efficiency of the pump is the ratio of the work actually done in lifting the water through the height h to the total work expended in driving the impeller shaft, and is of course less than the hydraulic efficiency. XLVm. CENTRIFUGAL PUMP APPLICATIONS 261. Floating Dry Docks. — To illustrate the wide range of ap- plications to which centrifugal pumps are adapted, a few typical examples of their use will be given. The rapid extension of the world's commerce in recent years has created a demand for docking facilities in comparatively isolated ports, which has given rise to the modern floating dry dock (Fig. 210). In docks of this type the various compartments into which they are divided are provided with separate pumps so that Digitized by VnOOQlC ENERGY OF FLOW 267 they may be emptied in accordance with the distribution of weight on the dock. Provision is usually made for handling one short vessel, two short vessels, or one extremely long ship, the balancing of the dock on an even keel being accomplished by emptying the various compartments in proportion to the weight sustained. The number of pumps in docks of this type varies from 6 to 20, depending on the number of compartments. The centrifugal pump is widely used and particularly suitable for this class of work, where a large quantity of water has to be discharged in a short time against a changing head which varies from zero. FIG. 210. when the piunping begins, to 30 or 40 ft. when the dock is nearly dry (Fig. 211).i 262« Deep Wells. — In obtaining a water supply from deep wells, the problem is to secure a pump which will handle a large quantity of water efficiently in a drilled well of moderate diame- ter, the standard diameters of such wells being 12 to 15 in. To meet this demand, centrifugal pumps are now built which will de- liver from 300 to 800 gal. per minute from a 12-in. well, and from 800 to 1,500 gal. per minute from a 15-in. well, with efficiencies ranging from 55 to 75 per cent. The depth from which the water is pumped may be 300 ft. or more, the pumps being built in several stages according to the depth (Fig. 212). ^ Figs. 201-214 are reproduced by permission of the Piatt Iron Works Co., Dayton, Ohio. Digitized by VnOOQlC ELEMENTS OF HYDRAULICS z\ jm fc-* t^ mrm^ Fig. 211. Digitized by VnOOQlC ENERGY OF FLOW 269 ,--^^h Fio. 212. Fio. 213. Digitized byCjOOQlC 270 ELEMENTS OF HYDRAULICS 263. Mine Drainage. — The extensive use of electric power for operating mining machinery has led to the employment of cen- trifugal pumps for mine drainage. The advantages of this type of pump when direct-connected to a high-speed motor are its compactness, simplicity and low first cost. Fig. 213 illustrates a mine-sinking turbine pump which operates against a 1,250-ft. head in a single lift. Pumps of similar design are in operation in nearly all the important mining regions of the United States and Mexico. The turbine pump is used to best advantage where it is required to unwater a flooded mine shaft. For actual sinking work a displacement pump is preferable unless an ample smnp is provided in order to keep the turbine pump well supplied with water so that it will not take air. 264. Fire Pumps. — The use of centrifugal pumps for fire pro- tection has been formally approved by the Fire Insurance Under- writers, who have issued specifications covering the essential Fia. 214. features of a pump of this type to comply with their requirements. In the case of fire boats the centrifugal pump has been foimd to fully meet all demands. The New York fire boats "James Duane" and "Thomas Willett" are equipped with turbine pumps, each of which has a capacity of 4,500 gal. per minute against 150 lb. per square inch pressure. For automobile fire en- gines, the great range of speed for gas engines gives the cen- trifugal pump a great advantage, making it possible to throw streams to a great height by merely increasing the speed of the motor. This type can also be readily moimted on a light chassis and driven from the driving shaft of the machine, making a light, compact, flexible and efficient unit (Fig. 214). Digitized by VnOOQlC ENERGY OF FLOW 271 266. Hydraulic Dredging. — The rapid development and im- provement of internal waterways in the United States has demon- strated the efficiency of the hydraulic or suction dredge. The advantage of the hydraulic dredge over the dipper and ladder types is that it not only dredges the material but also delivers it at the desired point with one operation. Its cost for a given capacity is also less than for any other type of dredge, while its capacity is enormous, some of the Government dredges on the Mississippi handling over 3,000 cu. yd. of material per hour. In operation the dredging pump creates a partial vacuum in the suction pipe, sufficient to draw in the material and keep it moving, and also produces the pressure necessary to force the discharge to the required height and distance. Hundreds of such pumps, ranging from 6 to 20 in. in diameter, are used on Western rivers for dredging sand and gravel for building and other purposes. The dredge for this class of service is very sim- ple, consisting principally of the dredging pump with its driving equipment moimted on a scow, the suction pipe being of suffi- cient length to reach to the bottom, and the material being delivered into a flat deck scow with raised sides, so that the sand is retained and the water flows overboard. For general dredging service where hard material is handled, it is necessary to use an agitator or cutter to loosen the material so that it can be drawn into the suction pipe. In this case the suction pipe is mounted within a structural steel ladder of heavy proportions to stand the strain of dredging in hard material, and of sufficient length to reach to the depth required. The cutter is provided with a series of cutting blades, and is moimted on a heavy shaft supported on the ladder, and driven through gearing by a separate engine (Fig. 215). Usually two spuds are arranged in the stern of the dredge to act as anchors and hold the dredge in position. The dredge is then swimg from side to side on the spuds as pivots by means of lines on each side controlled by a hoisting engine, thus controlling the operation of the dredge. Suction dredges are usually equipped with either 12-, 15-, 18- or 20-in. dredging piunps, the last-named being the standard size. For most economical operation as regards power, the ve- locity through the pipe line should not be greater than just sufficient to carry the material satisfactorily. With easily handled material the delivery pipe may be a mile Digitized by LnOOQlC 272 ELEMENTS OF HYDRAULICS Digitized by VjOOQIC ENERGY OF FLOW 273 or more in length, but with heavy material requiring high velocity the length should not exceed 4,000 ft. The practical maximum discharge pressure is about 50 lb. per square inch. For long pipe lines it therefore becomes necessary to use relay pumps, the dredging pump delivering through a certain length of pipe into the suction of the relay pxunp, and the latter delivering it through the remainder of the line. For high elevations or very long lines, several relay pumps may have to be used. The efficiency of a dredging pump is usually only 40 or 50 per cent., a high efficiency in this case not being so important as the ability to keep going. 266. Hydraulic Mining. — The centrifugal pump is also suc- cessfully used in hydraulic mining, where a high-pressure jet is used to wash down a hill. A number of centrifugal pumps are used for this purpose in the phosphate mines of Florida. Other uses for centrifugal pumps besides those described above are found in municipal water-works, sewage and drainage plants, sugar refineries, paper mills and irrigation works. APPLICATIONS 101. A jet 2 in. in diameter discharges 5 cu. ft. of water per second which impinges on a flat vane moving in the same direc- tion as the jet with a velocity of 12 ft. per second. Find the horsepower expended on the vane. 102. A fireman holds a hose from which a jet of water 1 in. in diameter issues at a velocity of 80 ft. per second. What force will the fireman have to exert to support the jet? 103. A small vessel is propelled by two jets each 9 in. in diame- ter. The water is taken from the sea through a vertical inlet pipe with scoop facing forward, and driven astern by a centrifugal pump 2 ft. 6 in. in diameter running at 428 r.p.m. and delivering approximately 2,250 cu. ft. of water per second. If the speed of the boat is 12.6 knots (1 knot = &,080 ft. per hour), calculate the hydraulic efficiency of the jet. 104. In the preceding problem, the efficiency of the pump was 48 per cent, and efficiency of engine and shafting may be assumed as 80 per cent. Using these values, calculate the total hydraulic efficiency of this system of propulsion. Note. — The jet propeller is more efficient than the screw pro- peller, the obstacle preventing the adoption of this system in 18 Digitized by VnOOQlC 274 ELEMENTS OF HYDRAULICS the past being the low efficiency obtainable from centrifugal pumps. 106. A locomotive moving at 60 miles per hour scoops up water from a trough between the rails by means of an L-shaped pipe with the horizontal arm projecting forward. If the trough is 2,000 ft. long, the pipe 10 in. in diameter, the opening into the tank 8 ft. above the mouth of the scoop, and half the available head is lost at entrance, find how many gallons of water are lifted into the tank in going a distance of 1,600 ft. Also find the slowest speed at which water will be delivered into the tank. 106. A tangential wheel is driven by two jets each 2 in. in diameter and having a velocity of 75 ft. per second. Assuming the wheel efficiency to be 85 per cent, and generator efficiency 90 per cent., find the power of the motor in kilowatts (1 hp. = 746 watts = 0.746 kilowatt). 107. In a commercial test of a Pelton wheel the diameter of the jet was found to be 1.89 in., static head on runner 386.5 ft., head lost in pipe friction 1.8 ft., and discharge 2.819 cu. ft. per second. The power developed was found by measurement to be 107.4 hp. Calculate the efficiency of the wheel. 108. A nozzle having an efflux coefficient of 0.8 delivers a jet 13^-in. in diameter. Find the amount and velocity of the dis- charge if the jet exerts a pressure of 200 lb. on a flat surface nor- mal to the flow. 109. A jet 2 in. in diameter is deflected through 120^ by striking a stationary vane. Find the pressure exerted on the vane when the nozzle is discharging 10 cu. ft. per second. 110. A power canal is 50 ft. wide and 9 ft. deep, with a velocity of flow of 13^ ft. per second. It supplies water to the turbines under a head of 30 ft. If the efficiency of the turbines is 80 per cent., find the horsepower available 111. It is proposed to supply 1,200 electrical hp. to a city 25 miles from a hydraulic plant. The various losses are estimated as follows: Generating machinery, 10 per cent. ; line, 8 per cent. ; trans- formers at load end, 9 per cent. ; turbine efficiency, 80 per cent. The average velocity of the stream is 3 ft. per second, available width 90 ft., and depth 6 ft. Find the net fall required at the dam. 112. The head race of a vertical water wheel is 6 ft. wide and the water 9 in. deep, flowing with a velocity of 5 ft. per second. Digitized by LnOOQlC ENERGY OF FLOW 275 If the total fall is 20 ft. and the efficiency of the wheel is 70 per cent., calculate the horse power available from it. 113. A stream is 150 ft. wide with an average depth of 4 ft. and a velocity of flow of 1 ft. per second. If the net fall at the dam is 20 ft. and the efficiency of the wheel is 75 per cent., find the horsepower available. 114. Eighty gallons of water per minute are to be pumped from a well 12 ft. deep by a pump situated 50 ft. from the well, and delivered to a tank 400 ft. from the pump and at 80 ft. elevation. The suction pipe is 3 in. in diameter and has two 3-in. elbows. The discharge pipe is 2}4 in. in diameter and has three 23^-in. elbows. Find the size of engine required. Note. — The lift is 92 ft. and the friction head in pipe and elbows amounts to about 25 ft., giving a total pumping head of 117 ft. The pump friction varies greatly, but for a maximum may be assumed as 50 per cent, of the total head, or, in the pres- ent case, 583^ ft. 115. A single-acting displacement pump raises water 60 ft. through a pipe line 1 mile long. The inside diameter of the pump barrel is 18 in., the stroke is 4 ft., and the piston is driven by a connecting rod coupled to a crank which makes 30 r.p.m. The velocity of flow in the pipe line is 3 ft. per second. Assum- ing the mechanical efficiency of the pump to be 75 per cent., and the slip 5 per cent., find the horse power required to drive the pump and the quantity of water delivered. 116. A 6-in. centrifugal pump deUvers 1,050 gal. per minute, elevating 20 ft. The suction and discharge pipes are each 6 in. in diameter and have a combined length of 100 ft. Find the friction head, total horse power required, and speed of pump for 50 per cent, efficiency. Note. — The velocity of flow in this case is 12 ft. per second and the corresponding friction head for 100 ft. of 6-in. pipe is 8.8 ft. The total effective head is therefore 28.8 ft., requiring 15.26 h.p. at a speed of 410 r.p.m. 117. In the preceding problem show that if an 8-in. pipe is used instead of 6-in. there will be a saving in power of over 22 per cent. 118. A hydraulic ram uses 1,000 gal. of water per minute under a 4-ft. head to pump 40 gal. per minute through 300 ft. of 2-in. pipe into a reservoir at an elevation of 50 ft. above the ram. Digitized by VnOOQlC 276 ELEMENTS OF HYDRAULICS Calculate the mechanical and hydraulic efficiencies of the ram, assuming the coefficient of pipe friction as 0.024. 119. An automobile booster fire pump, used for making a quick initial attack on a fire, is required to deliver two streams through ^-in. nozzles and 250 ft. of 1-in. hose. The pump is of the centrifugal type and is geared up to a speed of 3,500 r.p.m. from the gas engine which drives the machine. Calculate the discharge in gallons per minute and the horsepower required to drive the pump, assuming 50 per cent, efficiency. Note. — For this size nozzle, the maximum discharge is reached with a nozzle pressure of about 68 lb. per square inch correspond- ing to a velocity of about 100 ft. per second. 120. Feed water is pumped into a boiler from a round vertical tank 23^ ft. in diameter. Before starting the pimap the water level in the boiler is 38 in., and in the tank 22 in., above the floor level, and when the pump is stopped these levels are 40 in. and 15 in. respectively. If the steam pressure in the boiler while the pump is at work is 100 lb. per square inch, find the number of foot-pounds of work done by the pump. 121. A fire pump delivers three fire streams, each discharging 250 gal. per minute under 80 lb. per square inch pressure. Find the horse power of the engine driving the pump if the efficiency of the engine is 70 per cent, and of the piunp is 60 per cent. 122. A mine shaft 580 ft. deep and 8 ft. in diameter is full of water. How long will it take a 6-h.p. engine tounwater the shaft if the efficiencies of pump and engine are each 75 per cent.? 123. A fire engine pumps at the rate of 500 gal. per minute against a pressure of 100 lb. per square inch. Assuming the overall efficiency to be 50 per cent., calculate the indicated horse power of the engine. 124. A water-power plant is equipped with tangential wheels having an efficiency of 80 per cent. The water is delivered to the wheels through a cylindrical riveted-steel penstock 5 miles long with a total fall of 900 ft., practically the entire penstock being under this head. The cost of power house and equipment is estimated at $50,000, penstock 6 cts. per pound, operating expenses $5,000 per annum, and interest on total investment 4 per cent, per annum. The income is to be derived from the sale of power at $12 per horse power per annum. A constant supply of water of 100 cu. ft. Digitized by VnOOQlC ENERGY OF FLOW 277 per minute is available. Find the diameter of penstock for which the net income is a maximum. 126. A hydraulic pipe line is required to transmit 150 h.p. with a velocity of flow not greater than 3 ft. per second and a delivery pressure of 900 lb. per square inch. Assuming that the most economical size of pipe is one which allows a pressure drop of about 10 lb. per square inch per mile, determine the required size of pipe. 126. Find the maximum horse power which can be transmitted through a 6-in. pipe 4 miles long assuming the inlet pressure to be 800 lb. per square inch and the coefficient of pipe friction to be 0.024. Also determine the velocity of flow and outlet pressure. 127. A 6-in. pipe half a mile long leads from a reservoir to a nozzle located 350 ft. below the level of the reservoir and dis- charging into the air. Assuming the coefficient of friction to be 0.03, determine the diameter of nozzle for maximum power. Solution. — The discharge is IT 7)2 Q = ^ 62.4r where D = inside diameter of pipe and V = velocity of flow through pipe. Also the horse power delivered at the nozzle is H.P. at nozzle = -^^^^ ((h-f-^ ^) where h ^ static head at nozzle and I = length of pipe. The value of V for which the horse power is a maximum is found from the calculus condition. dV "" whence, solving for F, we find V Now let A = area of cross-section of pipe; a = area of cross-section of nozzle; V = velocity of flow through nozzle; p = pressure before entering nozzle. Then AV = av, and therefore from Bernoulli's theorem 62.4 ■^2fif 2g \a) 2g' Digitized by LnOOQlC 278 ELEMENTS OF HYDRAULICS and also 62.4 ^2(7 ^^ D2g "' whence, by subtraction and reduction, a^ AV^ l2ghD-flV^ Substituting in this relation the value of V obtained above for maximum power, the result is ** = ^ W For a circular pipe and nozzle this becomes which gives the required size of nozzle for maximum output of power. 128. A 10-in. water main 900 ft. long is discharging 1,000 gal. of water per minute. If water is shut oflF in 2 sec. by closing a valve, how much is the pressure in the pipe increased? 129. In a series of experiments made by Joukowsky on cast- iron pipes, the time of valve closure in each case being 0.03 sec, the following rises in pressures were observed.^ Show that these results give the straight line formula, p = 57v. Cast-iron Pipe, Diameter 4 In., Length 1,050 Ft. Vel. in ft./sec 0.6 2.0 3.0 4.0 9.0 Observed pressure in Ib./in.*... . 31 119 172 228 511 Cast-iron pipe, diameter 6 in., length 1,066 ft. Vel. in ft./sec 0.6 2.0 3.0 7.5 Observed rise in pressure in lb. /in.* 43 113 173 426 130. It is customary in practice to make allowance for possible water-hammer by designing pipes to withstand a pressure of 100 lb. per square inch in excess of that due to the static head. Show that this virtually allows for an instantaneous stoppage at a velocity of 1.6 ft. per second. 131. A bowl in the form of a hemisphere, with horizontal rim, is filled with liquid and then given an angular velocity w about its vertical axis. How much liquid flows over the rim (Fig. 216) ? ^ Gibson, "Hydraulics and Its Applications," p. 239. Digitized by VnOOQlC ENERGY OF FLOW 279 132. A closed cylindrical vessel of height H is three-fourths full of water. With what angular velocity co must it revolve around its vertical axis in order that the surface paraboloid shall just touch the bottom of the vessel (Fig. 217). Fig. 216. Fig. 217. 133. A closed cylindrical vessel of diameter 3 ft. and height 6 in. contains water to a depth of 2 in. Find the speed in r.p.m. at which it must revolve about its vertical axis in order that the water shall assume the form of a hollow truncated paraboloid for which the radius of the.upper base is 1 per cent, greater than the radius of the lower base; or, referring to Fig. 218, such that ri = l.Olrz. 134. The test data for a 19-in. New American turbine runner are as follows: Head 25 ft.; speed 339 r.p.m.; discharge 2,128 cu. ft. per minute; power developed 80 h.p. Calculate the turbine constants including the characteristic speed. Solution, — In this case, from Art. XXXVII, ^19. Tdn -r^r- FiG. 218. Ivy """ <P = Ky VTg 6OV25 5.62 2,128 = 5.62; = 0.7; O - ^ - 60 V25 = 7.0933; Digitized by VjOOQIC 280 ELEMENTS OF HYDRAULICS _Qi_7_J0m_ \i2/ i,. = !^^ = ??^0 = 54.24. 136. Two types of turbine runner, A and S, are to be compared. From tests it is known that runner A will develop a maximum of 2,080 h.p. at 500 r.p.m. under 100-ft. head, and runner B will develop 4,590 h.p. at 580 r.p.m. under 150-ft. head. Determine which of these types is the higher speed. Solution.-Type A, N. = ^OOV^jSO ^ ^^.n, loov^lob Type 5, iV, = ^?5^P? = 74.86. 150\/l50 136. Show that to transform the characteristic speed iV, from the EngHsh to the metric system it is necessary to multiply by the coefficient 4.46; that is to say, if the horse power and head are expressed in foot-pound units, andiV, in the metric system, we have the relation Vhjp. Ns = 4.46n- /i^ 137. Five two-runner Frajicis turbines installed in the power house of the Pennsylvania Water and Power Co. at McCalPs Ferry on the Susquehanna River are rated at 13,500 h.p. each under a head of 53 ft. at a speed of 94 r.p.m. The quantity of water required per turbine is 2,800 cu. ft. per second. Calculate from this rating the characteristic speed, efficiency, and other turbine constants. 138. Four two-runner Francis turbines operating in the Little Falls plant of the Washington Water Power Co. have a nominal power capacity of 9,000 h.p. each under a head of 66 ft. at a speed of 150 r.p.m. The quantity of water required per tiu-bine is 1,500 cu. ft. per second. From this rating calculate the char- acteristic speed, efficiency, and specific constants for these units. 139. The upper curve shown in Fig. 219 is the official efficiency test curve of the 9,000-h.p. turbines, built by the I. P. Morris Co. for the Washington Water Power Co. These wheels are of the Digitized by VnOOQlC ENERGY OF FLOW 281 horizontal shaft, two-runner, central discharge type, with volute casings. Head 66 ft., speed 150 r.p.m., and rated runner diameter 6 ft. 2 in. The lower curve shown in the figure is derived from a test at Holyoke of a homologous experimental runner having a rated diameter of 2 ft. 8 i%4 in. These curves are almost identical in shape, the eflSciency of the large units exceeding by a small margin that of the experimental runner. Calculate the discharge and characteristic speed at maximum efficiency, and from these results compute the specific constants. m - - - - - - - - - ^ ~ - '^ - - - - - _ = s B = « ^ ^ - ^ ■^ ^ ■^ s a ^ ^^ 's ? 70 J ^ ^ H ^ ^ 1 n fiO A / .^ ? ^ A 7 " f 40 ^ A _J so f 1 J ' 70 f " \ J ^ 10 f „ > h ~ I] _ _ _ _ _ _ . SO 00 50& 30^ 6u^M)ooJ60oaM^ffia^aal035oo*0(*4500£«)06«0(KM^ 1500 tjooo aGQO 9uaj seoo Fig. 219. 140. In testing a hydraulic turbine it was found by measure- ment that the amount of water entering the turbine was 8,000 cu. ft. per minute with a net fall of 10.6 ft. The power devel- oped was measured by a friction brake clamped to a pulley. The length of brake arm was 12 ft., reading on scales 4001b., and speed of pulley 100 r.p.m. Calculate the efficiency of the turbine. 141. One of a series of 65 tests of a 31-in. Wellman-Seaver- Mopgan turbine runner gave the following data:^ Gate opening 75 per cent.; head on runner 17.25 ft.; speed 186.25 r.p.m.; discharge 63.12 cu. ft. per second; power developed 111.66 h.p. Calculate the efficiency and the various turbine constants. 142. One of a series of 82 tests of a 30-in. Wellman-Seaver- Morgan turbine runner gave the following data:^ ^ "Characteristics of Modem Hyd. Turbines," C. W. Larnbb, Trans, Am. Soc. C. E., vol. Ixvi (1910), pp. 306-386. « Ibid, Digitized by VnOOQlC 282 ELEMENTS OF HYDRAULICS Gate opening 80.8 per cent.; head on runner 17.19 ft.; speed 206 r.p.m.; discharge 85.73 cu. ft. per second; power developed 146.05 h.p. Calculate the efficiency and the other turbine constants. 143. Four of the turbines of the Toronto Power Co. at Niagara Falls are of the two-runner Francis type, with a nominal develop- ment of 13,000 h.p. each under a head of 133 ft. at a speed of 250 r.p.m. The quantity of water required per turbine is 1,060 cu. ft. per second. Calculate the efficiency, characteristic speed and specific turbine coefficients for these units. 144. The upper curve shown in Fig. 220 is the official test curve of the 6,000-h.p. turbines designed by the I. P. Morris Co. 90 J m^ =^ ^ a 90 ,0' •^ ' ^ ««■ •*•« ^ 80 ^ ^ -- *• ■■ n hi 80 ^ .«* 70 ■ ^ f^ -* " ~ ~ /" ,^ /^ 70 60 / / ' /' / 50 / / 60 y i 40 r r iO / J 30 f, f J / 80 20 / f 20 7 10 ^ 10 _ __ _ L. 400 800 12(X) 1600 2000 2400 280082009G00400044004800&200Se00600064W Hone Power Fig. 220. for the Appalachian Power Co. The rated runner diameter is 7 ft. 6>i in., head 49 ft., and speed 116 r.p.m. These turbines are of the single-runner, vertical-shaft type. The lower curve is derived from'a test at Holyoke of the small, homologous, experimental runner, having a rated diameter of 27^ in. The curves are identical in shape, but owing to the better arrangement of water passages in the large plant, its efficiency considerably exceeds that of the experimental runner. It may also be noted that the efficiency shown on this diagram is the highest ever recorded in a well-authenticated test. Calculate the discharge and characteristic speed at maximum efficiency, and from these results compute the specific turbine constants. Digitized by VnOOQlC ENERGY OF FLOW 283 146. The following data, taken from the official Holyoke test reports, give the results of tests made on a 35-in. vertical Samson turbine built by the James Leflfel Co. of Springfield, Ohio. Calculate the turbine constants and characteristic speeds. Tbsts of 35-in. Vertical Samson Tubbinb Head on Speed Discharge Horse Efficiency Gate opening wheel in in rev. in. cu. ft. power in per feet per min. per sec. . developed cent. Full gate 16.57 187 120.61 188.27 83.06 0.9 gate 16.69 191 114.35 188.88 87.26 0.8 gate 16.78 189 105.10 179.87 89.93 0.75 gate 16.86 187 100.29 172.57 89.99 0.7 gate 17.08 188 92.83 160.03 88.99 0.6 gate 17.23 185 77.15 128.22 85.05 0.5 gate 17.47 188 66.89 108.72 82.03 146. The speed and water consumption of a turbine vary as the square root of the head (\/^), and the power varies as the square root of the cube of the head ( V^)- Thus if the head on a wheel is multiplied by 4, the speed and discharge will be multi- plied by 2 and the power by 8. Given that a 12-in. turbine under 12-ft. head develops 14 h.p. at 480 r.p.m. using 762 cu. ft. of water per minute, find the power, speed and discharge for the same tiu'bine under 48-ft. head. 147. On page 284 is given a rating table of turbines manu- factured by the S. Morgan Smith Co. of York, Pa., computed from actual tests of each size turbine under the dynamometer at the Holyoke testing flume. Calculate the nominal efficiency and characteristic speed for each size runner, and determine whether it is of the low-, medium- or high-speed type. Note. — ^^Data of this kind may be used by the instructor as problem material for an entire class without duplicating results, the final results being collected and tabulated, thus serving as a check on the calculations and also showing the range gf the constants involved. 148. On pages 285, 286, and 287 is given a rating table of Victbr Turbines manufactured by the Piatt Iron Works Co., Dayton, Ohio. Calculate the nominal efficiency, characteristic speed, and speed and capacity constants for each diameter and head. 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O CO tH CO Q CO OS lO »0 CO OS 04 1-1 1-1 04 CO 5 .fH CO 00 CO 04 00 »0 00 iH 04 o <5 00 \o in t^ t^ OS 1-t 1-1 00 04 iH fH O CD '^t* CO fH t* fH CO FH ^ S CO OS OS CO lO fH tH 04 04 t^ 00 CO t^ lO CO 00 00 fH 00 CO Q 00 l> 00 00 OS 04 N 1-i fH 04 04 »0 04 CO 1-1 lO 04 04 O l> CO 04 »-• lO 04 lO O CO CO 00 "««*< 1-i CO b» fH lO '^ t^ »0 00 iH 1-1 t^ 04 s 04 t^ lO 2|- CO CO O OS O oO OS OS 04 t^ 00 CO OS t« CO Tf iH Tt< -^ OS OS CO CO ^ -^ iH ^P4 Ti« Q O CO O CO "^ ^O 04 Tf p »o 00 ^ 1-1 =3 £ & 8.2 ill 0? „ IIJ 6.* -g 3 -^ > WO(3 Digitized by VnOOQlC 288 ELEMENTS OF HYDRAULICS 149. Fig. 221 shows a vertical section of the 10,800-h.p. tur- bines designed by the I. P. Morris Co. for the Cedar Rapids Mfg. and Power Co. The rated diameter of these turbines is 11 ft. lOJ^ in., head 30 ft., and speed 65.6 r.p.m. These turbines are at present the largest in the world, and it may be noted that all the latest features have been incorporated in the design, namely, volute casings and draft tubes molded in the concrete; cast-iron speed rings supporting the concrete, gen- FiG. 221. erator and thrust-bearing loads from above; lignum vitae turbine guide bearing; thrust-bearing support located above the genera- tor; Kingsbury thrust bearing with roller auxiliary; and pneu- matic brakes acting on the rotor of the generator. Calculate the characteristic speed from the rating given above, and from the table on page 209 determine to which speed type it belongs. 150. The following table gives the results of 20 tests out of a total of 66 made Sept. 3 and 4, 1912, at the testing flmne of the Digitized by VnOOQlC ENERGY OF FLOW 289 Holyoke Water Power Co. on a 24-in. Morris turbine type "0" runner. Calculate the characteristic speed for each test, and note its high value in test number 10. Note. — Two of the wheels for the Keokuk installation and nine wheels for the Cedar Rapids plant are built with this type of runner, the large wheels being geometrically similar to the experimental wheel tested at Holyoke. Report of Tests op a 24-in. Morris Turbine, Type "O" Runner made IN THE Testing Flume op the Holyoke Water Power Co. Number of experi- ment Open- ing of speed gate in inches Per cent, of full dis- charge of wheel Head on wheel in feet Speed in rev. per min. Dis- charge in cu. ft per sec. Horse power devel- oped Effi. ciency in per cent. 1 3.0 0.792 17.39 344.25 84.66 103.98 62.31 2 3.0 0.779 17.40 298.00 83.35 126.01 76.66 3 3.0 0.775 17.39 275.75 82.91 133.26 81.55 4 3.0 0.778 17.36 257.50 83.13 139.99 85.59 5 3.0 0.779 17.83 249.20 83.13 143.01 87.58 6 3.0 0.782 17.30 241.25 83.35 145.73 89.17 7 3.0 0.779 17.30 235.50 83.13 146.53 89.89 8 3.0 0.781 17.28 238.25 83.24 146.80 90.05 9 3.0 0.783 17.27 «40.20 83.46 146.55 89.71 10 3.0 0.781 17.23 236.80 83.13 146.62 90.32 11 3.0 0.762 17.31 214.00 81.28 138.97 87.15 12 3.0 0.782 17.24 237.20 83.24 146.51 90.08 13 3.0 0.744 17.34 185.50 79.44 128.86 82.54 14 3.5 0.902 17.15 357.20 95.73 107.89 57.98 15 3.5 0.900 17.10 320.50 95.38 135.52 73.31 16 3.5 0.903 17.06 287.75 95.61 156.44 84.62 17 3.5 0.892 17.14 265.75 94.70 160.53 87.26 .18 3.5 0.871 17.17 243.25 92.55 154.29 85.67 19 3.5 0.903 17.15 280.20 95.84 160.80 86.32 20 3.5 0.904 17.19 278.60 96.07 164.09 87.67 19 Digitized by Google SECTION 4 HYDRAULIC DATA AND TABLES Table 1. — ^Pbopbbtibs of Wateb Density and Volume of Water Temp, in Volume of Temp, in Volume of degrees Density 1 gram in degrees Density 1 gram in Centigrade cu. cm. Centigrade cu. cm. 0.999874 1.00013 24 0.997349 1.00266 1 0.999930 1.00007 26 0.996837 1.00317 2 0.999970 1.00003 28 0.996288 1.00373 2 0.999993 l.OOOOl 30 0.995705 1.00381 4 1.000000 1.00000 32 0.995087 1.00394 5 0.999992 1.00001 35 0.995098 1.00394 6 0.999970 1.00003 40 0.99233 1.00773 •7 0.999932 1.00007 45 0.99035 1.00974 8 0.999881 1.00012 1 50 0.98813 1.01201 9 0.999815 1.00018-! 55 0.98579 1.01442 10 0.999736 1.00026 j 60 0.98331 1.01697 11 0.999643 1.00036 1 65 0.98067 1.01971 12 0.999537 1.00046 ' 70 0.97790 1.02260 13 0.999418 1.00058 ! 75 0.97495 1.02569 14 0.999287 1.00071 80 0.97191 1.02890 16 0.998988 1.00101 85 0.96876 1.03224 18 0.998642 1.00136 90 0.96550 1.03574 20 0.998252 1.00175 95 0.96212 1.03938 22 0.997821 1.00218 ; 100 0.95863 1.04315 Weight of Water Temp, in Weight in Temp, in Weight in Temp, in Weight in degrees pounds per degrees pounds per degrees pounds per Fahrenheit cu. ft. Fahrenheit cu. ft. Fahrenheit cu. ft. 32 62.42 1 100 62.02 170 60.77 40 62.42 110 61.89 180 60.65 50 62.41 120 61.74 190 60.32 60 62.37 130 61.56 200 60.07 70 62.31 140 61.37 210 59.82 80 62.23 150 61.18 212 59.56 90 62.13 160 60.98 290 Digitized by VnOOQlC HYDRAULIC DATA AND TABLES 291 Table 2. — ^Head and I'bessukb Eqthvalbnts Head of Water in Feet and Equivalent Pressure in Pounds per Sq. In. Feet Pounds per Feet Pounds per Feet Pounds per head sq. in. head sq. in. head sq. in. 1 0.43 55 23.82 190 82.29 2 0.87 60 25.99 200 86.62 3 1.30 65 28.15 225 97.45 4 1.73 ■ 70 30.32 250 108.27 5 2.17 75 32.48 275 119.10 6 2.60 80 34.65 300 129.93 7 3.03 85 36.81 325 140.75 8 3.40 90 38.98 350 151.58 9 3.90 95 41.14 375 162.41 10 4.33 100 43.31 400 173.24 15 6.50 110 47.64 500 216.55 20 8.66 120 61.97 600 259.85 25 10.83 130 66.30 700 303.16 30 12.99 140 60.63 800 346.47 35 15.16 150 64.96 900 389.78 40 17.32 160 69.29 1000 433.09 45 50 19.49 21.65 170 180 73.63 77.96 Pressure in Pounds per Sq. In. and Equivalent Head of Water in Feet Pounds per sq. in. Feet head Pounds per sq. in. Feet head Pounds per sq. in. Feet head 1 2 3 4 5 6 7 8 9 10 15 20 25 30 35 40 45 50 2.31 4.62 6.93 9.24 11.54 13.85 16.16 18.47 20.78 23.09 34.63 46.18 57.72 69.27 80.81 92.36 103.90 115.45 55 60 65 70 75 80 85 90 95 100 110 120 125 130 140 150 160 170 126.99 138.54 150.08 161.63 173.17 184.72 196.26 207.81 219.35 230.90 253.98 277.07 288.62 300.16 323.25 346.34 369.43 392.52 180 190 200 225 250 275 300 325 350 375 400 500 415.61 438.90 461.78 519.51 577.24 643.03 692.69 750.41 808.13 865.89 922.58 1154.48 Digitized by Google 292 ELEMENTS OF HYDRAULICS Table 3. — Dischabqb Equivalbnts Gallons per min. Cubic feet per sec. Cubic feet per min. Gallons per hour Gallons per 24 hours Bbls. per minute, 42 gal. bbl. Bbls. per hour, 42 gal. bbl. Bbls. per 24 hours, 42 gal. bbl. 10 12 15 18 20 25 27 30 35 36 40 45 50 60 70 75 80 90 100 125 135 150 175 180 200 225 250 270 300 315 360 400 450 500 540 600 630 675 720 800 900 1,000 1,125 1,200 1,350 1,500 1,575 1,800 2,000 2,025 2,250 2,500 2,700 3,000 1.3368 1.6042 2.0052 2.4063 2.6733 3.342 3.609 4.001 4.678 4.812 5.348 6.015 6.684 8.021 9.357 10.026 10.694 12.031 13.368 16.710 18.046 20.052 23.394 24.062 26.736 30.079 33.421 36.093 40.104 42.109 48.125 53.472 60. 158 66.842 72.186 80.208 84.218 90.234 96.25 106.94 120.31 133.68 150.39 160.42 180.46 200.52 210.54 240.62 267.36 270.70 300.78 334.21 360.93 401.04 600 720 900 1,080 1,200 1,500 1,620 1,800 2.100 2,160 2,400 2,700 3,000 3,600 4,200 4,500 4,800 5,400 6,000 7,500 8,100 9,000 10,500 10,800 12,000 13,500 15,000 16,200 18,000 18,900 21,600 24,000 27,000 30,000 32.400 36,000 37,800 40.500. 43.200 48,000 64.000 60.000 67,500 72,000 81.000 90.000 94,500 108,000 120,000 121.500 135,000 150,000 162.000 180.000 14,400 17,280 21,600 25,920 28,800 36.000 38,880 43,200 50,400 51.840 57,600 64,800 72,000 86,400 100,800 108.000 115,200 129.600 144.000 180,000 194.400 216,000 252,000 259,200 288,000 324,000 360,000 388.800 432,000 453.600, 518.400 576.000 648.000 720,000 777,600 864,000 907.200 972.000 1.036,800 1,152,000 1,296,000 1,440,000 1,620,000 1.728,000 1,944,000 2,160,000 2,268,000 2,692,000 2,880,000 2,916,000 3.240.000 3,600,000 3.880.000 4,320,000 0.24 0.29 0.36 0.43 0.48 0.59 0.64 0.71 0.83 0.86 0.96 1.07 1.19 1.43 1.66 1.78 1.90 2.14 2.39 2.98 3.21 3.57 4.16 4.28 4.76 6.36 5.96 6.43 7.14 7.6 8.57 9.52 10.7 11.9 12.8 14.3 15.0 16.0 17.0 19.05 21.43 23.8 26.78 28.57 32.14 35.71 37.5 42.85 47.64 48.21 53.67 59.52 64.3 71.43 14.28 17.14 21.43 25.71 28.57 36.71 38.57 42.85 50.0 51.43 57.14 64.28 71.43 85.71 100.0 107.14 114.28 128.5 142.8 178.6 192.8 214.3 250.0 257.0 285.7 321.4 357.1 386.7 428.6 450.0 514.3 671.8 642.8 714.3 771.3 857.1 900.0 964.0 1,028.0 1,142.0 1.285.0 1,428.0 1,607.0 1,714.0 1,928.0 2,142.0 2,250.0 2,571.0 2,867.0 2,892.0 3,214.0 3,671.0 3,857.0 4,285.0 342.8 411.4 514.3 617.1 685.7 857.0 925.0 1,028.0 1,200.0 1.234.0 1.371.0 1,543.0 1,714.0 2,057.0 2,400.0 2,570.0 2,742.0 3.085.0 3.428.0 4,286.0 4,628.0 6,143.0 6,000.0 6,171.0 6,857.0 7.714.0 8,670.0 9.267.0 10.284.0 10,800.0 12,342.0 13.723.0 16,428.0 17,143.0 18.512.0 20.570.0 21,600.0 23,143.0 24,686.0 27,387.0 30,867.0 34,284.0 38,671.0 41.143.0 46,086.0 51,427.0 54,000.0 61,710.0 68,568.0 69,426.0 77.143.0 85,704.0 92,572.0 102,840.0 0.1 0.2 0.3 0.4 0.5 0.6 0.8 1.0 1.2 . 1.4 1.5 1.6 2.0 2.5 3.0 3.5 4.0 4.5 6.0 Digitized by Google HYDRAULIC DATA AND TABLES 293 H .O M S 8 S 00 .1 adard ht 4536 kil 8024 kil 0483 kil 1 1 fc fc s .^ CO «o i t^ s s |-?<=^S2 3 §2SI s « s * > 2 P ^-^^ •1 5;-^ « t:; B » n ti n H •5 1. .1 « ^11 III iii 1H 1-H T-4 ft O"^ .-1 f-l 1-4 ft O" M T-l ^ 1-4 m5 III CO .'SI CO « :S N 2j -; ft 3 S . a&S i 58S i 00 W t« Sz:>;is ■g ^ »o csi Tjl ^^^ 00 CO O tJ 'c ^ R n H >» d d (N II II y 1. , » Q S .1 .=5 -3 ll 1 am ogra tinea §11 •*J fc« *» b3 2 ^ ^ ^ ^ ^ ;S ri iH 1-H 1 2 2 « fe g mdard gth 9 centimete 4 centimete 3 meters 3 kilometer l22 § 3 3 & & a s s o 3 M II II n II ' 2^5 lO g? CO -* 6 00 o o d d .2 o II ja ^ -g o II II 11 il HE a .B^t « 1 »H ^ T-4 ^ S'S'S' 1 ^ »H ^ ,-4 .Sift 2 d ches ches ches J g'g'S' 1 .a .3 .3 3 3SS 1-( «o S © CO O M* lO O 00 1^ CO -* 00* *& d d d d « o o 2 csi 0) CO 1 " 3 •2 S „ „ 7 ^ n n II n •shJ n " « II h > Me millimeter centimeter meter kilometer o 3 1 --- -H s'g'S^ o 1 T-t 1-1 .-1 •^ Xi in 3 H p:; N p -< ^ Ed n s s p '^ 15 00 <i 3 X> <£ H UJ W o o I fa o DQ p a C) h gs .-co O 3 d d d 1-4 * o 8 o , ^ d ^ CO t* «o iC fe CO 04 to « u5 o CO o o o CO ^' d 00 ;:q' <N 1 "^l ' '"' •"• C4 CO ■ b. <<|4 00 ^ § o Sc =!oSo H d d d c i-d- o lO <N •^ 05 r^ t^ 00 »o lO o o »o o o o o o O N ^ o '"' t» -d CO o >o 3 CO O g CO C 00 d • (N iH (2 rH O O o o CO CO o «c »o o CO 1-1 Xi -4^500 00 O) O CO 3 d d d •^ o .-4 lO O »0 O CO CO A -* C3 a O <N O O b- o o o •H b. »H 00 b- -4 oo •s 00 t* (>♦• c< ^ <N (N b. o 3 o o ^ 'C ^ w CO O CO C 9* "flS « O O (N S MONO i-t o ^ o « o g g Ul o e^ o * c '(J* OO M O QQ »-4 ^ O t- O CO 00 O ^ D S N c CI 11- t 1 a|f,-2. g ol ilsSI OH JO Digitized by VnOOQlC 294 ELEMENTS OF HYDRAULICS Table 6. — Spbcifig Wbighto of Various Substancbs Air, press. 76 cm. 'Hg., 0»C Alcohol Aluminium, pure commercial Basalt Bismuth Brass Brick Cadmium Carbon, charcoal diamond graphite Coal, hard Copper, cast electrolytic wire Cork Earth Gold Glass Granite Hydrogen, press. 76 cm. Hg.,0»C Ice 0.001293 0.79 2.583 2.7 - 2.8 2.4 - 3.3 9.76- 9.93 7.8 - 8.7 1.4 - 2.3 8.54- 8.69 1.45- 1.70 3.4^ 3.53 2.17- 2.32 1.2 - 1.8 8.3 - 8.92 8.88- 8.95 8.93- 8.95 0.24 1.4 - 2.8 19.30-19.34 2.5 - 3.8 2.5 - 3.0 0.0000894 0.926 Iron, cast pure steel wrought Lead Lime mortar Limestone Magnesium ..... Marble Mercury at 0*C. Nickel Oil Platinum, cast . . wire and foil . . Quartz Rubber Sand Sandstone Seawater Silver Timber, oak fir poplar Tin Zinc .03- 7.73 .85- 7.88 .60- 7.80 .79- 7.85 .21-11.45 .6 - 1.8 .4 - 2.8 .6^ 1.75 .5 - 2.9 13.596 .57- 8.93 .91- 0.94 48^21.50 2 -21.7 3 - 2.7 0.93 2 - 1.9 9 - 2.7 02- 1.03 42-10.57 62- 1.17 5 - 0.9 35- 1.02 97- 7.37 85- 7.24 Digitized by VnOOQlC HYDRAULIC DATA AND TABLES 295 nil 1^ o 5 « i-« a» t» U3 ■^ N lo CO i-« C4 •« lO 00 r-l «C ^ lO M « ^ b- ^ us CO t^ t^ O CO r« M iH t^ o o o o o -^ •-I N N CO "3 O Oft S I 5 QQ OQ § S I o g o i 9 O W M Tf O CO CO ^ N O ^ q6 U5 b- I* lO CO t^ <^ CI Oi '^ ^ Oi t* "* CO N M ^ 11 Is «* o t» <o -^ S;0 t^ ^ -^ I* CO 00 «o CO csl ci fh »H U) h- Od 00 t» ^ t>. ^ ^ U3 C4 O O 00 h- eo 5 S N S CO ub Tf Tf CO iH iH O O O o o o o o 08 o fi OS II lO b- t* t* ■* o « s s OS I* •« -* CO 5 CO o < N N Csl 1 •H U3 O) <«i b. §>0 ^ « 00 OS OO b> CO ^ o d d d t* r-l CO b- »5 t« Q ^ OS u5 lO 25 Tf CO CO d d d d d 6 6 b- oj CO N I* O ii -• M CO iQ 00 t^ ^ 00 Oft O Oft t« o •^ to t^ O h- CO OS •* ■«* «0 ^ t» t» CO •-) e5| CO ^ I* CO I to CO -^ > C4 00 CO CI I OS CO O OS o o o o o N CI CO CO •* iQ CO 00 O ^ CO 1-t b- 00 ( I* ^ 1-1 ^ ( §22gl 00 t^ r-l 00 00 CO CO ( CO QO t^ OS ( o o o o o O 1-1 M CO Tf t^ OS CI kO OS ^ ^ ^ CI CO I r^ d CO I 00 -^ lO CI IQ (O CO CO OS CO »H 00 ^ ^ b* d M CO CO ■* ® "<1< I* I* o o o o o T-4 CI CI -* to > CI to OS <^ Tti I *H r^ i-l d CO ■ Si: -* CO ^5«5»ooo aacoSo8»« 00i-<>OOSiO cicoo-^t* > CO 00 CI OS i;ss2s > to CO t* > b» b- b» > o o -* CO "^ US CO b» o -^ c« -^ >o I CO CO < e« » 1-1 o) < I* OS w CO < CI CO *H (O < I 1-1 CI 1 to CO 1 -* o it^oob- eoiocoooci Ti<»o coOb» i-4Kdaeot« ^»0 •"Ib--^ OOOSOCIb- OS© li-iCICICO -^lOiOl^OS O CI •<«< lO b- l?5 O000i-<«CO ^ lO-Tjt^ l^COl^COOS ^r^CI"^* coooaoi-" co^^kOQ THCico-^ua oopci4'< ooo^^ i-i^^^55 cicicicici cic5co«< o o o o o o o o o o o o o o o o o o o o < ± ll ^ 8 S-^j^co-^ 00 ^t*oo i*ootooo»o kOcocit«os oscici "^oOfhcoco ©-^ciq^ tociooeo^ ClcO<^tOO0 OCOCOO^ O^OubO ooososo o o o o o 66661^ 00 •* 00 N •<«< »H fH CO -^ CO ^i-i^CIC« COCO"^-**© I kQ >0 I CO OS CO 00 i ^ f-I CI ci 10 o >o o >o CO -^ -^ »o "O I CO Tf -* I CO b- t^ 00 o Mi Hi Hi %ei CI CI CI CI »o CO CO CO CO b* CO b» 00 OS o CO to OD 01 e o o »H CI Digitized by VnOOQlC 296 ELEMENTS OF HYDRAULICS Table 7. — Capacttt of Reciprocating Pumps Capacity, or piston displacement, of reciprocating pumps in gallons per single stroke Diameter of cylindpr, inches Length of stroke in inches 2 3 4 5 6 1 7 8 9 10 1-1/4 0.0106 0.0159 0.0212 0.0266 0.0319 0.0372 0.0425 0.0478 0.0531 1-3/8 0.0128 0.0192 0.0256 0.0321 0.0385 0.0419 0.0513 0.0578 0.0642 1-1/2 0.0153 0.0229 0.0306 0.0382 0.0459 0.0535 0.0612 0.0688 0.0765 1-3/4 0.0208 0.0312 0.0416 0.0521 0.0625 0.0729 0.0833 0.0937 0.1041 2 0.0272 0.0408 0.0544 0.068 0.0816 0.0952 0.1088 0.1224 0.136 2-1/4 0.0344 0.0516 0.0688 0.086 0.1033 0.1205 0.1377 0.1548 0.1721 2-1/2 0.0425 0.0637 0.0850 0.1062 0.1275 0.1487 0.17 0.1912 0.2125 2-3/4 0.0514 0.0771 0.1028 D.1285 0.1543 0.1799 0.2057 0.2313 0.2571 3 0.0612 0.0918 0.1224 0.1530 0.1836 0.2142 0.2448 0.2754 0.306 3-1/4 0.0718 0.1077 0.1436 0.1795 .0.2154 0.2513 0.2872 0.3231 0.3594 3-1/2 0.0833 0.1249 0.1666 0.2082 0.2499 0.2915 0.3332 0.3748 0.4165 3-3/4 0.0956 0.1434 0.1912 0.239 0.2868 0.3346 0.3824 0.4302 0.478 4 0.1088 0.1632 0.2176 0.272 0.3264 0.3808 0.4352 0.4896 0.544 4-1/4 0.1228 0.1842 0.2456 0.307 0.3684 0.4298 0.4912 0.5526 0.6141 4-1/2 0.1377 0.2065 0.2754 0.3442 0.4131 0.4819 0.5508 0.6196 0.6885 4-3/4 0.1534 0.2301 0.3068 0.3835 0.4602 0.5369 0.6136 0.6903 0.7671 5 0.17 0.2550 0.34 0.425 0.51 0.595 0.68 0.765 0.85 5-1/4 0.1874 0.2811 0.3748 0.4685 0.5622 0.6559 0.7496 0.8433 0.9371 5-1/2 0.2057 0.3085 0.4114 0.5142 0.6171 0.7199 0.8228 0.9256 1.0285 5-3/4 0.2248 0.3372 0.4496 0.562 0.6744 0.7868 0.8992 1.011 1.124 6 0.2448 0.3672 0.4896 0.612 0.7344 0.8568 0.9792 1.1016 1.2240 6-1/4 0.2656 0.3984 0.5312 0.6640 0.7968 0.9296 1.062 1.195 1.328 6-1/2 0.2872 0.4308 0.5744 0.7182 0.8610 1.0052 1.1488 1.2926 1.4364 6-3/4 0.3098 0.4647 0.6196 0.7745 0.9294 1.084 1.239 1.394 1.549 7 0.3332 0.4998 0.6664 0.833 0.9996 1 . 1662 1.3328 1.4994 1.666 7-3/4 0.4084 0.6126 0.8168 1.021 1.225 1.429 1.633 1.837 2.042 - 8 0.4352 0.6528 0.8704 1.088 1.3056 1.5232 1.7408 1.9584 2.176 9 0.5508 0.8262 1.1010 1.377 1.6524 1.9278 2.2032 2.4786 2.754 10 0.68 1.02 1.36 1.7 2.04 2.38 2.72 3.06 3.4 11 0.8227 1.2341 1.6451 2.057 2.464 2.879 3.2911 3.7258 4.1139 12 0.9792 1.468 1.9584 2.448 2.9376 3.4222 3.9168 4.4064 4.896 13 1.149 1.723 2.297 2.872 3.445 4.022 4.596 5.170 5.745 14 1.332 1.998 2.665 3.331 3.997 4.664 5.33 5.996 6.663 15 1.529 2.294 3.059 3.824 4.589 5.354 6.119 6.884 7.649 16 1.74 2.61 3.48 4.35 5.22 6.09 6.96 7.83 8.703 18 2.202 3.303 4.404 5.505 6.606 7.707 8.808 9.909 11.01 20 2.720 4.08 5.440 6.8 8.16 9.52 10.88 12.24 13.6 Digitized by vnOOQlC HYDRAULIC DATA AND TABLES m Tisva 7. — Capacitt of REapROCATiNO Pumps. — {Continued^ Diameter of cylinder, inches Length of stroke in inchea 12 14 15 16 18 20 22 24 1-1/4 0.0637 0.0743 0.0797 0.0848 0.0955 0.1062 0.1168 0.1274 1-3/8 0.077 0.089 0.0963 0.1027 0.1156 0.1280 0.1408 0.1541 1-1/2 0.0918 0.1071 0.1147 0.1224 0.1377 0.1530 0.1683 0.1836 1-3/4 0.1249 0.1457 0.1562 0.1666 0.1874 0.2082 0.2290 0.2499 2 0.1632 0.1904 0.204 0.2176 0.2448 0.2720 0.2992 0.3264 2-1/4 0.2063 0.241 0.258 0.2754 0.3096 0.344 0.3784 0.4128 2-1/2 0.255 0.2975 0.3187 0.34 0.3825 0.4252 0.4677 0.61 2-3/4 0.3085 0.3598 0.3855 0.4114 0.4626 0.5142 0.5666 0.617 3 0.3672 0.4284 0.459 0.4896 0.6608 0.612 0.6732 0.7344 3-1/4 0.4312 0.503 0.6385 0.5748 0.6466 0.7182 0.79 0.8624 3-1/2 0.4998 0.5831 0.6247 0.6664 0.7497 0.833 0.9163 0.9996 3-3/4 0.5736 0.6692 0.687 0.7648 0.8605 0.9661 1.0517 1.147 4 0.6528 0.7616 0.816 0.8904 0.9792 1.088 1.1968 1.3056 4-1/4 0.7368 0.8596 0.921 0.9824 1.105 1.228 1.3508 1.473 4r-l/2 0.8262 0.9639 1.0327 1.1016 1.2393 1.377 1.6147 1.6524 4r-3/4 0.9204 1.073 1.15 1.227 1.380^ 1.534 1.6874 1.84 5 1.02 1.19 1.275 ' 1.36 1.53 1.7 1.87 2.04 5-1/4 1.124 1.311 1.405 1.499 1.686 1.874 2.0614 2.248 5-1/2 1.2342 1.4399 1.5427 1.6456 1.8513 2.067 2.2627 2.4684 5-3/4 1.348 1.573 1.686 1.789 2.022 2.248 2.4728 2.696 6 1.4688 1.7136 1.8362 1.9584 2.2032 2.448 2.6928 2.9376 6-1/4 1.593 1.859 1.992 2.124 2.39 2.666 2.9216 3.186 6-1/2 1.7955 2.0109 2.1546 2.2982 2.6885 2.8728 3.16 3.4473 6-3/4 1.858 2.168 2.323 2.479 2.788 3.098 3.4078 3.716 7 1.9992 2.3324 2.499 2.6656 2.9988 3.332 3.6662 3.9984 7-3/4 2.45 2.858 3.063 3.266 3.674 4.084 4.4924 4.9 8 2.6112 3.0464 3.264 3.4816 3.9168 4.352 4.7872 5.2224 9 3.3048 3.8556 4.131 4.4064 6.0572 5.608 6.0588 6.6096 10 4.08 4.76 5.1 5.44 6.12 6.8 7.48 8.16 11 4.9367 5.7595 6.1709 6.5823 7.4051 8.2279 9.0506 9.8735 12 5.8752 6.8544 7.344 7.833 8.8128 9.792 10.7712 11.7504 13 6.894 8.042 8.616 9.192 10.34 11.49 12.639 13.78 14 7.994 9.328 9.993 10.66 11.99 13.32 14.652 15.98 15 9.178 10.70 11.47 12.23 13.76 16.29 16.819 18.36 16 10.44 12.18 13.05 13.92 15.66 17.40 19.14 20.88 18 13.21 15.41 16.51 17.61 19.81 22.02 24.22 26.42 20 16.32 19.04 20.4 21.76 24.48 27.2 29.92 32.6 Digitized by VnOOQlC 298 ELEMENTS OF HYDRAULICS Table 8. — Gibcumfbbbnceb and Abeas of Cibclbs DUmeters, 1/16 in. up to and including 120 in. Advancing, 1/16 to 1 ; 1/8 to 60; 1/4 to 80, and 1/2 to 120 Diam- Circum- Area, Diame- Circum- Area, Diame- Circum- Area. eter, ference, square ter, ference, square ter. ference. square inches inches inches inches inches inches inches inches inches 1/16 0.19635 0.00307 4r-l/2 14.137 15.904 9-5/8 30.237 72.759 1/8 0.3927 0.01227 4-5/8 14.529 16.800 9-3/4 30.630 74.662 3/16 0.6890 0.02761 4-3/4 14.922 17.720 9-7/8 31.023 76.588 1/4 0.7854 0.04909 4-7/8 15.315 18.665 5/16 0.9817 0.07670 10 31.416 78.540 3/8 1.1781 0.1104 5 15.708 19.635 10-1/8 31.808 80.615 7/16 1.3744 0.1503 5-1/ 8 16.100 20.629 10-1/4 32.201 82.516 1/2 1.5708 0.1963 5-1/4 16.493 21.647 10-3/8 32.594 84.540 9/16 1.7771 0.2485 5-3/8 16.886 22.690 10-1/2 32.986 86.590 5/8 1.9635 0.3068 5-1/2 17.278 23.758 10-5/8 33.379 88.664 11/16 2.1598 0.3712 5-5/8 17.671 24.850 10-3/4 33.772 90.762 3/4 2.3562 0.4417 5-3/4 18.064 25.967 10-7/8 34.164 92.885 13/16 7/8 2.5525 2.7489 0.5185 0.6013 5-7/8 18.457 27.108 11 34.558 95.033 15/16 2.9452 0.6903 6 18.849 28.274 11-1/8 11-1/4 34.960 35.343 97.205 99.402 1 3.1416 0.7854 6-1/8 19.242 29.464 11-3/8 35.736 101.623 1-1/8 1-1/4 1-3/8 1-1/2 3.5343 3.9270 4.3197 0.9940 1.2271 1.4848 6-1/4 6-3/8 6-1/2 19.635 20.027 20.420 30.679 31.919 33.183 11-1/2 11-5/8 11-3/4 36.128 36.621 36.913 103.869 106.139 108.434 4.7124 1 . 7671 6-5/8 6-3/4 20.813 21.205 34.471 35.784 11-7/8 37.306 110.753 1-5/8 5.1051 2.0739 6-7/8 21.598 37.122 12 37.699 113.097 1-3/4 1-7/8 5.4978 5.8905 2.4052 2.7621 7 21.991 38.484 12-1/8 12-1/4 38.091 38.484 115.466 117.869 2 6.2832 3.1416 7-1/8 22.383 39.871 12-3/8 38.877 120.276 2-1/8 6.6759 3.5465 7-1/4 22.776 41.282 12-1/2 39.270 122.718 2-1/4 7.0686 3.9760 7-3/8 23.169 42.718 12-5/8 39.662 126.184 2-3/8 7.4613 4.4302 7-1/2 23.562 44.178 12-3/4 40.055 127.676 2-1/2 7.8540 4.9087 7-5/8 23.954 45.663 12-7/8 40.448 130.192 2-5/8 2-3/4 8.2467 8.6394 5.4119 5.9395 7-3/4 7-7/8 24.347 24.740 47.173 48.707 13 13-1/8 40.840 41.233 132.732 136.297 2-7/8 9.0321 6.4918 13-1/4 41.626 137.886 8 25.132 50.265 13-3/8 42.018 140.500 3 9.4248 7.0686 8-1/8 25.515 51.848 13-1/2 42.411 143.139 3-1/8 9.8175 7.6699 8-1/4 25.91$ 53.456 13-6/8 42.804 145.802 3-1/4 10.210 8.2957 8-3/8 26.310 56.088 13-3/4 43.197 148.489 3-3/8 10.602 8.9462 8-1/2 26.703 66.745 13-7/8 43.689 161.201 a- 1/2 10.995 9.6211 8-5/8 27.096 58.426 3-5/8 11.388 10.320 8-3/4 27.489 60.132 14 43.982 163.938 3-3/4 11.781 11.044 8-7/8 27.881 61.862 14-1/8 44.376 166.699 3-7/8 12.173 11.793 14-1/4 44.767 169.486 9 28.274 63.617 14-3/8 46.160 162.295 4 12.566 12.566 9-1/8 28.667 65.396 14r-l/2 45.663 165.130 4-1/8 12.959 13.364 9-1/4 29.059 67.200 14-6/8 46.945 167.989 4r-l/4 13.351 14.186 9-3/8 29.452 69.029| 14-3/4 46.338 170.873 4-3/8 13.744 15.033 9-1/2 29.845 70.882 14-7/8 46.731 173.782 Digitized by VnOOQlC HYDRAULIC DATA AND TABLES 299 Table 8.— -Cibcumferbncbs and Abbas of Circles— {Continued) Diame> Ciroum- Area Diame- Circum- Area Diame- Circum- Area ter ferenoe square ter ference square ter ference square inches inches inches inches inches inches inches inches inches 15 47.124 176.715 21 65.973 346.361 27 84.823 672.556 15-1/8 47.616 179.672 21-1/8 66.366 360.497 27-1/8 86.215 577.870 15-1/4 47.909 182.654 21-1/4 66.769 354.657 27-1/4 86.608 583.208 15-8/8 48.302 185.661 21-3/8 67.161 358.841 27-3/8 86.001 588.571 15-1/2 48.694 188.692 21-1/2 67.544 363.061 27-1/2 86.394 593.968 15-6/8 49.087 191.748 21-6/8 67.937 367.284 27-5/8 86.786 699.370 1&-3/4 49.480 194.828 21-3/4 68.329 371.643 27-3/4 87.179 604.807 15-7/8 49.872 197.933 21-7/8 68.722 376.826 27-7/8 87.572 610.268 16 60.265 201.062 22 69.115 380.133 28 87.964 616.763 16-1/8 60.668 204.216 22-1/8 69.507 384.466 28-1/8 88.357 621.263 16-1/4 61.061 207.394 22-1/4 69.900 388.822 28-1/4 88.750 626.798 16-3/8 51.443 210.697 22-3/8 70.293 393.203 28-3/8 89.142 632.357 16-1/2 61.836 213.826 22-1/2 70.686 397.608 28-1/2 89.535 637.941 16-6/8 62.229 217.077 22-6/8 71.078 402.038 28-5/8 89.928 643.594 16-3/4 62.621 220.363 22-3/4 71.471 406.493 28-3/4 90.321 649.182 16-7/8 63.014 223.664 22-7/8 71.864 410.972 28-7/8 90.713 654.837 17 63.407 226.980 23 72.266 415.476 29 91.106 660.621 17-1/8 53.799 230.330 2^1/8 72.649 420.004 29-1/8 91.499 666.277 17-1/4 64.192 233.705 2^1/4 73.042 424.557 29-1/4 91.891 671.958 17-3/8 64.686 237.104 2^3/8 73.434 429.136 2^3/8 92.284 677.714 17-1/2 64.978 240.628 23-1/2 73.827 433.731 29-1/2 92.677 683.494 17-6/8 66.370 243.977 23-6/8 74.220 438.363 29-5/8 93.069 689.298 17-3/4 66.763 247.450 23-3/4 74.613 443.014 29-3/4 93.462 695.128 17-7/8 66.166 260.947 23-7/8 75.005 447.699 29-7/8 93.856 700.981 18 66.648 254.469 24 75.398 452.390 30 94.248 706.860 18-1/8 66.941 258.016 24-1/8 76.791 457.116 30-1/8 94.640 712.762 18-1/4 57.334 261.686 24-1/4 76.183 461.864 30-1/4 95.033 718.690 18-3/8 67.726 266.182 24-3/8 76.576 466.638 30-3/8 95.426 724.641 18-1/2 58.119 268.803 24r-l/2 76.969 471.436 30-1/2 95.818 730.618 18-5/8 68.612 272.447 24-5/8 77.361 476.259 30-5/8 96.211 736.619 18-3/4 68.905 276.117 24-3/4 77.754 481.106 30-3/4 96.604 742.644 18-7/8 69.297 279.811 24-7/8 78.147 485.978 30-7/8 96.996 748.694 19 69.690 283.629 26 78.640 490.875 31 97.389 764.769 19-1/8 60.083 287.272 26-1/8 78.932 495.796 31-1/8 97.782 760.868 19-1/4 60.476 291.039 25-1/4 79.325 500.741 31-1/4 98.175 766.992 19-3/8 60.868 294.831 26-3/8 79.718 505.711 31-3/8 98.567 773.140 19-1/2 61.261 298.648 26-1/2 80.110 510.706 31-1/2 98.968 779.313 19-6/8 61.663 302.489 26-6/8 80.503 515.725 31-5/8 99.353 785.510 19-3/4 62.046 306.366 26-3/4 80.896 520.769 31-3/4 99.745 791.732 19r7/8 62.439 310.246 25-7/8 81.288 525.837 31-7/8 100.138 797.978 20 62.832 314.160 26 81.681 530.930 32 100.531 804.249 20-1/8 63.224 318.099 26-1/8 82.074 636.047 32-1/8 100.924 810.546 20-1/4 63.617 322.063 26-1/4 82.467 541.189 32-1/4 101.316 816.866 20-3/8 64.010 326.061 26-3/8 82.869 83.262 546.366 32-3/8 101.709 823.209 20-1/2 64.402 330.064 26-1/2 551.647 32-1/2 102.102 829.578 20-5/8 64.795 334.101 26-6/8 83.645 656.762 32-6/8 102.494 835.972 20-3/4 65.188 338.163 26-3/4 84.037 562.002 32-3/4 102.887 842.390 20-7/8 66.680 342.250 26-7/8 84.430 567.267 32-7/8 103.280 848.833 Digitized by VnOOQlC 300 ELEMENTS OF' HYDRAULICS CiRCUMPBRBNCBS AND ArEAS OP Circles — (Cont inued) Diame- Circum- Area Diame- Circum- Area Diame- Circum- Area ter ^ ference square ter ference square ter ference square inches inches inches inches inches inches inches inches inches 33 103.672 855.30 39 122.622 1194.69 45 141.372 1590.43 33^1/8 104.055 861.79 39-1/8 122.915 1202.26 46-1/8 141.764 1599.28 33-1/4 104.458 868.30, 39-1/4 123.307 1209.95 46-1/4 142.157 1608.15 33-3/8 104.850 874.84 39-3/8 123.700 1217.67 46-3/8 142.660 1617.04 33-1/2 106.243 881.41; 39-1/2 124.093 1225.42 46-1/2 142.942 1625.97 33-6/8 105.636 888.00 39-5/8 124.485 1233.18 46-6/8 143.336 1634.92 33-3/4 106.029 894.61 39-3/4 124.878 1240.98 46-3/4 143.728 1643.89 33-7/8 106.421 901.25 39-7/8 125.271 1248.79 46-7/8 144.120 1662.88 34 106.814 907.92 40 126.664 1266.64 46 144.613 1661.90 34-1/8 107.207 914.61 40-1/8 126.066 1264.50 46-1/8 144.906 1670.95 34-1/4 107.699 921.32 40-1/4 126.449 1272.39 46-1/4 145.299 1680.01 34-3/8 107.992 928.06 40-3/8 126.842 1280.31 46-3/8 145.691 1689.10 34-1/2 108.386 934.82 40-1/2 127.234 1288.25 46-1/2 146.084 1698.23 34-5/8 108.777 941.60 40-6/8 127.627 1296.21 46-6/8 146.477 1707.37 34-3/4 109.170 948. 4i; 40-3/4 128.020 1304.20 46-3/4 146.869 1716.54 34-7/8 109.663 966.25 40-7/8 128.412 1312.21 46-7/8 147.262 1725.73 35 109.966 962.11 41 128.805 1320.25 47 147.666 1734.94 35-1/8 110.348 968.99 41-1/8 129.198 1328.32 47-1/8 148.047 1744.18 35-1/4 110.741 975.90 41-1/4 129.691 1336.40 47-1/4 148.440 1753.45 35-3/8 111.134 982.84 41-3/8 129.983 1344.61 47-3/8 148.833 1762.73 35-1/2 111.526 989.80 41-1/2 130.376 1362.65 47-1/2 149.226 1772.05 35-5/8 111.919 996.78 41-6/8 130.769 1360.81 47-5/8 149.618 1781.39 35-3/4 112.312 1003.78 41-3/4 131.161 1369.00 47-3/4 150.011 1790.76 35-7/8 112.704 1010.82 41-7/8 131.554 1377.21 47-7/8 150.401 1800.14 36 113.097 1017.88 42 131.947 1386.44 48 150.796 1809.66 36-1/8 113.490 1024.95 42-1/8 132.339 1393.70! 48-1/8 161.189 1818.99 36-1/4 113.883 1032.06! 42-1/4 132.732 1401.98' 48-1/4 151.682 1828.46 . 36-3/8 114.275 1039.19' 42-3/8 133.125 1410.29, 48-3/8 151.974 1837.93 36-1/2 114.668 1046.35' 42-1/2 133.518 1418.62' 48-1/2 162.367 1847.45 36-5/8 116.061 1053.52 42-6/8 133.910 1426.98 48-6/8 152.760 1866.99 36-3/4 116.463 1060.73 42-3/4 134.303 1436.36 48-3/4 153.153 1866.65 36-7/8 115.846 1067.95 42-7/8 134.696 1443.77 48-7/8 153.545 1876.13 37 116.239 1076.21 43 135.088 1462.20 49 153.938 1886.74 37-1/8 116.631 1082.48 43-1/8 135.481 1460.65 49-1/8 154.331 1895.37 37- 1/4 117.024 1089.79 43-1/4 135.874 1469.13 49-1/4 154.723 1905.03 37-3/8 117.417 1097.11 43-3/8 136.266 1477.63 49-3/8 155.116 1914.70 37-1/2 117.810 1104.46 43-1/2 136.669 1486.17 49-1/2 155.509 1924.42 37-5/8 118.202 1111.84 43-5/8 137.062 1494.72 1 49-5/8 166.901 1934.16 37-3/4 118.696 1119.24 43-3/4 137.445 1603.30 49-3/4 166.294 1943.91 37-7/8 118.988 1126.66 43-7/8 137.837 isii.oo 49-7/8 156.687 1953.69 38 119.380 1134.11 44 138.230 1520.53 50 167.080 1963.50 38-1/8 119.773 1141. 69| 44-1/8 138.623 1629.18 50-1/4 157.865 1983.18 38-1/4 120.166 1149.08 44-1/4 139.016 1537.86 50-1/2 158.660 2002.96 38-3/8 120.658 1156.61 44-3/8 139.408 1546.56 60-3/4 159.436 2022.84 38-1/2 120.961 1164.15 1 44-1/2 139.801 1665.28 51 160.221 2042.82 38-6/8 121.344 1171.73 44-5/8 140.193 1664.03 51-1/4 161.007 2062.90 38-3/4 121.737 1179.32 44-3/4 140.586 1672.81 i 51-1/2 161.792 2083.07 38-7/8 122.129 1186.94 44-7/8 140.979 '1581.61 1 51-3/4 162.577 2103.35 Digitized by VnOOQlC HYDRAULIC DATA AND TABLES 301 Circumferences AND Areas op Circles — {Continued) Diame- Ciroum- Area Diame- Circum- Area Diame- Circum- Area ter ference square ter ference square ter ference square inches inches inches inches inches inches inches inches inches 52 163.363 2123.72 63 197.920 3117.25 74 232.478 4300.84 52-1/4 164.148 2144.19 63-1/4 198.706 3142.04 74-1/4 233.263 4329.96 52-1/2 164.934 2164.75 6»-l/2 199.491 3166.92 74-1/2 234.049 4359.16 52-3/4 165.719 2185.42 63-3/4 200.277 3191.91 74r-3/4 234.834 4388.47 53 166.504 2206.18 64 201.062 3216.99 76 235.620 4417.86 53-1/4 167.490 2227.05 64-1/4 201.847 3242.17 76-1/4 236.405 4447.37 6a-l/2 168.075 2248.01 64-1/2 202.633 3267.46 76-1/2 237.190 4476.97 53-3/4 168.861 2269.06 64-3/4 203.418 3292.83 76-3/4 237.976 4606.67 54 169.646 2290.22 65 204.204 3318.31 76 238.761 4636.46 64-1/4 170.431 2311.48 : 65-1/4 204.989 3343.88 76-1/4 239.547 4666.36 54-1/2 171.217 2332.83 65-1/2 205.774 3369.56 76-1/2 240.332 4606.35 54-3/4 172.002 2354.28 64-3/4 206.560 13395.33 76-3/4 241.117 4626.44 55 172.788 2375.83 66 207.345 3421 . 19 77 241.903 4666.63 55-1/4 173.573 2397.48 66^1/4 208.131 3447.16 77-1/4 242.688 4686.92 55^1/2 174.358 2419.22 66-1/2 208.916 3473.33 77-1/2 243.474 4717.30 55^3/4 175.144 2441.07 66-3/4 209.701 3499.39 77-3/4 244.269 4747.79 56 175.929 2463.01' • 67 210.487 3525.66 78 246.044 4778.36 56-1/4 176.715 2485.05, 67-1/4 211.272 3552.01 78-1/4 245.830 4809.06 56-1/2 177.500 2507.19 67-1/2 212.058 3578.47 78-1/2 246.616 4839.83 56-3/4 178.285 2529.42 67-3/4 212.843 3605.03 78-3/4 247.401 4870.70 57 179.071 2551.76 68 213.628 3631.68 79 248.186 4901.68 57-1/4 179.856 2574.19 68-1/4 214.414 13658. 44| 7^1/4 248.971 4932.76 57-1/2 180.642 2596.72 1 68-1/2 215.199 3686.29 7^1/2 249.767 4963.92 57-3/4 181.427 2619.35 68-3/4 215.985 3712.24 7^3/4 260.642 4996.19 58 182.212 2642.08 69 216.770 3739.28 80 261.328 5026.56 58-1/4 182.998 2664.91 69-1/4 217.565 3766.43 80-1/2 262.898 6089.58 58-1/2 58-3/4 183.783 184.569 2687.83 2710.85 69-1/2 69-3/4 1 218.341 219.126 3793.67 3821.02 '81 81-1/2 264.469 266.040 5163.00 5216.82 59 185.354 2733.97 70 219.912 3848.45 82 82-1/2 267.611 269.182 5281.02 6346.62 59-1/4 59-1/2 186.139 186.925 2757.19 2780.51 70-1/4 70-1/2 220.697 221.482 3875.99 3903.63 5^3/4 187.710 2803.92 70-3/4 222.268 3931.36 83 260.752 6410.61 60 188.496 2827.43 71 223.053 3959.19 83-1/2 262.323 5476.00 60-1/4 189.281 2851.05 71-1/4 223.839 3987.13 84 263.894 6641.77 60-1/2 190.066 2874.76 71-1/2 224.624 4015.16 84-1/2 266.466 6607.96 60-3/4 190.852 2898.56 71-3/4 225.409 4043.28 86 267.036 6674.61 61 191.637 2922.47 72 226.195 4071.50 86-1/2 268.606 5741.47 61-1/4 192.423 2946.47 72-1/4 226.980 4099.83 86 86-1/2 270.177 271.748 6808.80 6876.66 61-1/2 61-3/4 193.208 193.993 2970.57 2994.77 72-1/2 72-3/4 227.766 228.551 4128.25 4156.77 62 62-1/4 194.779 195.564 3019.07 3043.47 73 73-1/4 229.336 230.122 4185.39 4214.11 87 87-1/2 273.319 274.890 6944.68 6013.21 62-1/2 196.350 3067.96 73-1/2 230.907 4242.92 88 276.460 6082.12 62-3/4 197.135 3092.56 73-3/4 231.693 4271.83 88-1/2 278.031 6161.44 Digitized by VnOOQlC 302 ELEMENTS OF HYDRAULICS OiRCUHFEBBNCES AND Arbab OF CiBCiJis — (Continued) Diame- ter inches Circum- ference inches Area square inches Diame- ter inches Circum- ference inches Area square inches Diame- ter inches Circum- ference inches Area square inches 89 89-1/2 90 90-1/2 91 91-1/2 92 92-1/2 93 93-1/2 94 94-1/2 95 95-1/2 96 96-1/2 97 97-1/2 98 98-1/2 99 99-1/2 279.602 281.173 282.744 284.314 285.885 287.466 289.027 290.598 292.168 293.739 295.310 296.881 298.452 300.022 301.593 302.164 304.734 306.306 307.876 309.446 311.018 312.588 6221 . 14 6291.25 6361.73 6432.62 6503.88 6573.56 6647.61 6720.07 6792.91 6866.16 6939.78 7013.81 7088.22 7163.04 7238.23 7313.84 7389.81 7474.20 7542.96 7620.12 7697.69 7775.64 100 100-1/2 101 101-1/2 102 102-1/2 103 103-1/2 104 104-1/2 105 105-1/2 106 106-1/2 107 107-1/2 108 108-1/2 109 109-1/2 110 110-1/2 314.159 315.730 317.301 318.872 320.442 322.014 323.584 325.154 326.726 328.296 329.867 331.438 333.009 334.580 336.150 337.722 339.292 340.862 342.434 344.004 345.575 347.146 7863.98 7938.72 8011.86 8091.36 8171.28 8251.60 8332.29 8413.40 8494.87 8676.76 8659.01 8741.68 8824.73 8908.20 8992.02 9076.24 9160.88 9245.92 9331.32 9417.12 9603.32 9589.92 111 111-1/2 112 112-1/2 113 113-1/2 114 114r-l/2 115 115-1/2 116 116-1/2 117 117-1/2 118 118-1/2 119 119-1/2 120^ 348.717 350.288 351.868 363.430 366.000 366.670 368.142 359.712 361.283 362.854 364.425 365.996 367.666 369.138 370.708 372,278 373.849 375.420 376.991 9766.89 9674.28 9852.03 9940.20 10028.76 10117.68 10207.03 10296.76 10386.89 10477.40 10668.32 10669.64 10751.32 10843.40 10936.88 11028.76 11122.02 11215.68 11309.73 Digitized by VnOOQlC HYDRAULIC DATA AND TABLES 303 Tablb 9.' — ^Epfltjx Cobfficibnts fob Circular Obificb Values of efflux coeffioient JC in Eq. (32). Par. 55. Q - 2/ZKby/2giHf^* - h*^*), for circular. ▼ertioal orifices, with sharp edges, full contraction and free discharge in air. For heads over 100 ft., use JT >■ 0.592. Head Diameter of orifice in feet on cen- ter of orifice 0.02 0.03 0.04 0.05 0.07 0.10 0.12 0.15 0.20 0.40 0.60 0.80 1.0 in feet 0.3 0.4 0.6 0.6 0.637 0.631 0.627 0.624 0.628 0.624 0.621 0.618 0.621 0.618 0.616 0.613 0.613 0.612 0.610 0.609 0.608 0.606 0.606 0.606 0.637 0.633 0.630 0.655 0.643 0.640 0.600 0.601 0.696 0.696 0.592 0.593 0.690 0.7 0.651 0.637 0.628 0.622 0.616 0.611 0.607 0.604 0.601 0.697 0.594 0.691 0.690 0.8 0.648 0.634 0.626 0.620 0.615 0.610 0.606 0.603 0.601 0.697 0.594 0.592 0.591 0.9 0.646 0.632 0.624 0.618 0.613 0.609 0.605 0.603 0.601 0.598 0.595 0.593 0.591 1.0 0.644 0.631 0.623 0.617 0.612 0.608 0.605 0.603 0.600 0.598 0.695 0.693 0.591 1.2 0.641 0.628 0.620 0.615 0.610 0.606 0.604 0.602 0.600 0.598 0.596 0.694 0.592 1.4 0.638 0.625 0.618 0.613 0.609 0.605 0.603 0.601 0.600 0.599 0.696 0.694 0.693 1.6 0.636 0.624 0.617 0.612 0.608 0.605 0.602 0.601 0.600 0.699 0.697 0.696 0.594 1.8 0.634 0.622 0.615 0.611 0.607 0.604 0.602 0.601 0.509 0.699 0.697 0.696 0.595 2.0 0.632 0.621 0.614 0.610 0.607 0.604 0.601 0.600 0.509 0.699 0.597 0.596 0.595 2.5 0.620 0.619 0.612 0.608 0.605 0.603 0.601 0.600 0.599 0.599 0.598 0.697 0.596 3.0 0.627 0.617 0.611 0.606 0.604 0.603 0.601 0.600 0.599 0.599 0.698 0.597 0.697 3.5 0.625 0.616 0.610 0.606 0.604 0.602 0.601 0.600 0.599 0.599 0.698 0.697 0.696 4.0 0.623 0.614 0.609 0.605 0.603 0.602 0.600 0.599 0.599 0.698 0.597 0.697 0.596 5.0 0.621 0.613 0.608 0.605 0.603 0.601 0.599 0.599 0.698 0.698 0.697 0.696 0.696 6.0 0.618 0.611 0.607 0.604 0.602 0.600 0.599 0.599 0.598 0.598 0.597 0.696 0.596 7.0 0.616 0.609 0.606 0.603 0.601 0.600 0.699 0.599 0.598 0.958 0.597 0.696 0.696 8.0 0.614 0.608 0.606 0.603 0.601 0.600 0.599 0.598 0.698 0.697 0.696 0.596 0.696 9.0 0.613 0.607 0.604 0.602 0.600 0.699 0.699 0.698 0.697 0.697 0.596 0.696 0.596 10.0 0.611 0.606 0.603 0.601 0.599 0.598 0.598 0.697 0.597 0.697 0.596 0.596 0.696 20.0 0.601 0.600 0.599 0.698 0.597 0.596 0.596 0.696 0.696 0.596 0.596 0.595|0.694 50.0 0.596 0.696 0.595 0.595 0.594 0.694 0.594 0.594 0.594 0.594 0.594 0.693 0.593 100.0 0.593 0.593 0.592 0.592 0.592 0.592 [0.592 0.592 0.692 0.592 0.502 0.592 0.692 1 From Hamilton Smith's "Hydraulics." Digitized by LnOOQlC 304 ELEMENTS OF HYDRAULICS Table 10. ^ — ^Effujx Cobfficibnts pob Square Orifice Values of eflBlux coefficient it in Eq. (32), Par. 56. Q - 2/^Khy/2o{H^'^ - h^'\ for square, vertical cnrifices, with sharp edges, full contraction, and free discharge in air. For heads over 100 ft. use IC - 0.598 Head on cen- ter of Side of square in feet orifice in feet 0.02 0.03 0.04 0.60. 0.07 0.10 0.12 0.15 0.20 0.40 0.60 0.80 1.0 0.3 0.4 0.5 0.6 0.642 0.637 0.633 0.630 0.632)0 624 0.617 0.616 0.614 0.613 0.612 0.611 0.610 0.610 0.643 0.639 0.636 0.628 0.626 0.623 0.621 0.619 0.617 0.660 0.648 0.645 0.605 0.605 0.601 0.597 0.6010.598 0.596 0.7 0.656 0.642 0.633 0.628 0.621 0.616 0.612 0.609 0.605 0.602 0.599 0.598 0.596 0.8 0.652 0.639 0.631 0.626 0.620 0.615 0.611 0.608 0.605 0.602 0.6000.698 0.697 0.9 0.650 0.637 0.629 0.623 0.619 0.614 0.610 0.608 0.605 0.603 0.60110.599 0.598 1.0 0.648 0.636 0.628 0.622 0.618 0.613 0.610 0.608 0.606 0.603 0.60l|0.600 0.599 1.2 0.644 0.623 0.625 0.620|0.616 0.611 0.609 0.607 0.605 0.604 0.602 0.601:0.600 1.4 0.642 0.630 0.623 0.618 0.614 0.610 0.608 0.606 0.605 0.604 0.602 0.6010.601 1.6 0.640 0.628 0.621 0.Q17 0.613 0.609 0.607 0.606 0.605 0.605 0.603 0.602 0.601 1.8 0.638 0.627 0.620 0.616 0.612 0.609 0.607 0.606 0.605 0.605 0.603 0.602 0.602 2.0 0.637 0.626 0.619 0.615 0.612 0.608 0.606 0.606 0.605 0.606 0.604 0.602 0.602 2.5 0.634 0.624 0.617 0.613 0.610 0.607 0.606 0.606 0.605 0.605 0.604 0.603 0.602 3.0* 0.632 0.622 0.616 0.612 0.609 0.607 0.606 0.606 0.605 0.605 0.604 0.603 0.603 3.5 0.630 0.621 0.615 0.611 0.609 0.607 0.606 0.606 0.605 0.605 0.604 0.603 0.602 4.0 0.628 0.619 0.614 0.610 0.608 0.606 0.606 0.605 0.605 0.605 0.603 0.603 0.602 5.0 0.626 0.617 0.613 0.610 0.607 0.606 0.605 0.605 0.604 0.604 0.603 0.602 0.602 6.0 0.623 0.616|0.612 0.609 0.607 0.605 0.605 0.605 0.604 0.604 0.603 0.602 0.602 7.0 0.621 0.615 0.611 0.608 0.607 0.605 0.606 0.604 0.604 0.604 0.603 0.602 0.602 8.0 0.619 0.613 0.610 0.608 0.606 0.605 0.604 0.604 0.604 0.603 0.603 0.602 0.602 9.0 0.618 0.612 0.609 0.607 0.606 0.604 0.604 0.604 0.603 0.603 0.602 0.602 0.601 10.0 0.616 0.611 0.608 0.606 0.605 0.604 0.604 0.603 0.603 0.603 0.602 0.602 0.601 20.0 0.606 0.605 0.604 0.603 0.602 0.602 0.602 0.602 0.602 0.601 0.601 0.601 0.600 50.0 0.602 0.601 0.601 0.601 0.601 0.600 0.600 0.600 0.600 0.600 0.699 0.599 0.699 100.0 0.599 0.598 0.598 0.598 0.598 0.598 0.698 0.59810.598 0.598 0.598 0.598 0.698 « From Hamilton Smith's "Hydraulics." Digitized by Google HYDRAULIC DATA AND TABLES 305 Table 11. — Fibb Streams From Tablet Published by John R. Freeman 3/4-in . Smooth Nossle Pressure in pounds per sq. in. re- Pressure at nossle in pounds per sq.in. Discharge in gallons per min. Height of effective fire stream • Horiaontal distance of stream quired at hydrant or pump to main- tain pressure at noaxle through vari- ous lengths of 2-1/2-in. smooth, rubber-lined hose. 50 100 200 300 400 500 600 800 1000 ft. ft. ft. ft. ft. ft. ft. ft. ft. 35 97 65 41 37 38 40 42 44 46 48 53 57 40 104 60 44 42 43 46 48, 60 53 55 60 65 46 110 64 47 47 48 51 54 67 69 62 68 73 50 116 67 50 52 54 57 60 63 66 69 75 81 55 122 70 52 58 69 63 66 69 73 76 83 89 60 127 72 54 63 65 68 72 76 79 83 90 97 65 132 74 56 68 70 74 78 82 86 90 98 106 70 137 76 58 73 75 80 84 88 92 97 105 114 75 142 78 60 79 81 85 90 94 99 104 113 122 80 147 79 62 84 86 91 96 101 106 111 120 130 • 85 151 80 64 89 92 97 102 107 112 117 128 138 90 156 81 65 94 97 102 108 113 119 124 135 146 95 160 82 66 99 102 108 114 120 125 131 143 154 100 164 83 68 105 108 114 120 126 132 138 150 163 7/8-i] a. Smooth N( )axle 35 133 66 46 38 40 44 48 52 56 60 68 76 40 142 62 49 43 46 50 55 59 64 68 78 87 45 150 67 52 49 51 57 62 67 72 77 87 97 60 159 71 55 54 57 63 69 74 80 86 97 108 55 166 74 58 60 63 69 76 82 88 94 107 119 60 174 77 61 65 69 75 82 89 96 103 116 130 65 181 79 64 71 74 82 89 96 104 111 126 141 70 188 81 66 76 80 88 96 104 112 120 136 152 75 194 83 68 82 86 94 103 111 120 128 145 162 80 201 85 70 87 91 101 110 119 128 137 155 173 85 207 87 72 92 97 107 116 126 136 145 165 184 90 2ia 88 74 98 103 113 123 134 144 154 174 195 95 219 89 75 103 109 119 130 141 152 163 184 206 100 224 90 76 109 114 126 137 148 160 171 194 216 1-in Smooth Noj isle 35 174 68 51 40 44 51 57 64 71 78 92 105 40 186 64 65 46 50 58 66 73 81 89 105 120 45 198 69 58 62 56 65 74 83 91 100 118 135 50 208 73 61 57 62 72 82 92 102 111 131 151 55 218 76 64 63 69 79 90 101 112 122 144 166 60 228 79 67 67 75 87 98 110 122 134 157 181 65 237 82 70 75 81 94 107 119 132 145 170 196 70 246 85 72 80 87 101 115 128 142 156 183 211 75 255 87 74 86 94 110 123 138 152 167 196 226 80 263 89 76 92 100 115 131 147 162 178 209 241 85 274 91 78 98 106 123 139 156 173 189 222 90 279 92 80 103 112 130 147 165 183 200 236 95 287 94 ' 82 109 118 137 156 174 193 211 249 100 295 96 83 115 125 144 164 183 203 223 20 Digitized by VnOOQlC 306 ELEMENTS OF HYDRAULICS Fire Streams — (Continued) l'l/8-inoh Smooth Nosale Pressure at nosale in pounds per sq. in. ■6& 40 45 50 £5 m 65 TO 75 SO &5 90 95 100 m 95 IQO Discharge in gallons per min. Height of effective fire stream 222 238 252 206 27D 29t ao3 314 325 336 340 350 S66 376 65 70 75 SO 83 80 90 02 94 03 90 Horisontal distanog of stream Pressure in pounds per sq. in. required at hydrant or pump to maintain pres- sure at nosale through various lengths of 2-1/2-in. smooth, rubber-lined hoee 200 ft. 300 ft 54 63 72 75 77 79 81 &3 S5 S7 SO 49 eo 400 ft. 102:118 500 ft. 600 ft 800 ft. aooo ft. 105J127 120 145 1^5 163 94 107 120 134 150'l81 95,ll2jl30| 147 105 200 lOa, 122; 141 160 1S012I8 195 '236 200,254 224 ... 239 1061 1 191 146 173' 200|237 254 112,126 155:133 2121241 112|]32|t53,174 120,143*165 187 93jl05il29 153 177.201 09'll2[l38 163ll88 2l4 , — ,1 lis 133^163,194 124!l40 172:204 224 254 230 . . . Ii49 171 192 213 235 256 l-l/4-ineh Smooth Noiile 35 277 40 2^6 45 314 SO 331 55 347 80 36^ 65 377 70 392 75 405 80 419 85 432 444 468 67 72 77 81 85 88 91 93 y.T 97 99 100 101 59 63 67 70 73 76 79 81 83 85 SS 90 92 93 48 55 62| 6S 75 82 89 96 103 110 57 65 73 81 89 07 105 113 121. 129 116 137 74 84 05 106 116 127 137 148 158 im\ 179| 190 210 211 1 01 109 104 117 130 143 156 124 140 155 170 198 186 169 201 182 '195 208 221 234 247 261 217 232 248 ]20|l^ll78 1441164 203 229 204 225 245 162' 184 180^ 216 234 252 254 212 243 35 40 45 50 55 60 65 70 75 80 85 90 95 100 340 363 3g5 406 426 445 463 480 497 514 529 545 560 574 l-S/S-inch Bmooth lioute 62 69 74 79 m 87 90 92 95 97 09 100 101 103 62 66 70 73 76 79 82 84 8i\ 88 00 92 94 96 54 62 70 78 67 77 87 06 106 116 04 107 120 134 147 120 146 137 166 154 187 171 208 I88l22ft 172 196 221 245 270 198 226 454 250 86 93 160 205 250 101 \n 174 222 109 135 145 154 164 187 339 117 201 2-^ lt>4 214 5>fl7 13S* 140 17:1940 148'l83 254 156 193 ... Digitized by VnOOQlC HYDRAULIC DATA AND TABLES 307 FiBB SmBAMa— (Continue*!) From ezperimenU made by Tirgil R, Heming 6/16-lN. NOMM Preasure base of nossle lb. per sq. in. Discharge gallons per Loss of head in 100 ft. of hose Vertical height of jet for good fire Horisontal distance of | Jet for good fire stream feet Extreme drops at level Rubber lined UnUned linen minute lb. per sq. in. lb. per sq. in. stream feet of nossle feet 20 12 0.7 1.3 28 15 53 30 15 1.1 1 9 32 18 63 40 17 1.5 2.6 34 21 71 50 19 1.8 3.2 35 23 78 60 21 2.2 3 9 36 26 84 70 23 2.6 4.5 37 28 90 80 24 2.9 5.2 38 29 96 90 26 3.3 5.9 39 30 102 100 28 3.7 6.5 40 31 107 7/16-In. Nossls 20 25 2.8 6.1 23 10 46 3a 30 4.2 7.7 27 13 54 40 35 5.6 10.2 30 16 63 60 39 7.0 12.8 32 18 70 60 43 8.5 15.3 33 20 77 70 47 9.8 17.8 34 21 84 80 50 11.1 20.3 35 23 94 90 53 12.7 22.9 36 24 99 100 56 14.1 25.5 37 25 lOtf 1/2-lN. NOMIJl 20 33 6.2 9.5 34 15 63 30 40 7.7 14.4 87 20 79 40 46 10.2 18.8 38 25 91 50 52 12.8 23.8 39 30 102 60 57 15.4 28.5 40 33 111 70 61 18.0 32.7 41 37 . 120 80 65 20.5 38.4 42 40 127 90 69 23.0 42.0 43 43 134 100 73 25.6 47.0 44 46 140 Digitized by VnOOQlC 308 ELEMENTS OF HYDRAULICS Table 12." — CosFnciiiNTB of Pipe Fwction Value of the frietion coeffioient /, in the formula Computed from the exponential formulas of Thrupp, Tutton and Unwin Material Diameter in inches Velocity of flow in feet per second ] 2 4 6 8 10 1 0.032 0.026 0.024 0.022 0.021 Lead pipe 2 0.030 0.026 0.023 0.021 0.020 3 0.029 0.024 0.022 0.020 0.019 4 0.028 0.023 0.021 0.020 0.019 Wood pipe 6 12 18 24 0.034 0.027 0.024 0.022 0.033 0.027 0.024 0.022 0.032 0.026 0.023 0.021 0.032 0.026 0.023 0.021 36 0.020 0.019 0.019 0.019 48 0.018 0.018 0.017 0.017 6 0.026 0.023 0.022 0.021 0.020 9 0.025 0.022 0.021 0.020 0.019 12 0.024 0.021 0.020 0.019 0.019 Asphalted pipe 18 0.023 0.020 0.019 0.018 0.018 24 0.022 0.020 0.018 0.017 0.017 36 0.021 0.019 0.017 0.017 0.016 48 0.020 0.018 0.017 0.016 0.015 3 0.024 0.021 0.019 0.018 0.017 6 0.022 0.019 0.017 0.016 0.016 12 0.019 0.017 0.015 0.014 0.014 Bare wrought iron 24 0.017 0.015 0.014 0.013 0.012 pipe 36 0.016 0.015 0.014 . 0.013 0.012 0.011 48 0.013 0.012 0.011 0.011 60 0.015 0.013 0.012 0.011 0.010 12 0.025 0.022 0.021 0.020 0.019 24 0.020 0.018 0.017 0.016 0.016 Riveted wrought iron 36 0.017 0.016 0.015 0.014 0.014 or steel pipe 48 0.016 0.014 0.014 0.013 0.013 60 0.015 0.013 0.013 0.012 0.012 72 0.014 0.013 0.012 0.011 0.011 3 6 0.028 0.024 0.026 0.022 0.025 0.022 0.025 0.021 New cast-iron pipe 9 12 0.021 0.020 0.020 o.ai9 0.020 0.018 0.019 0.018 18 0.018 0.017 0.017 0.016 24 0.017 0.016 0.016 0.015 36 0.015 0.015 0.014 0.014 3 6 0.059 0.050 0.058 0.050 0.058 0.050 0.058 0.049 Old cast iron pipe 9 12 0.046 0.043 0.045 0.042 0.045 0.042 0.044 0.042 18 0.039 0.039 0.038 0.038 24 0.037 0.036 0.036 0.036 36 0.033 0.033 0.033 0.032 I Compiled from data in Gibson's **Hydraulics." Digitized by VnOOQlC HYDRAULIC DATA AND TABLES 309 Table 13. — Friction Head in Pipes According to Exponential For- mula OF Williams and Hazen Friction head in feet for each 100 ft. of straight, clean, cast-iron pipe. For old pipes the tabular values of the friction head should be doubled. Computed from Williams and . Hasen's formula, v »• C7r»«»w»« (o.OOl) -oo«: v = velocity in feet per sec., « ■» slope; 9 » hydraulic radius in feet, C ■> 100. Inside diameter of pipe | 1/2 in. 3/4 in. lin. 1-1/2 in. 2 in. 1 1 li .a« li Friction head in feet per 100 ft. 1 .9 . ll Friction head in feet per 100 ft. Friction head in feet per 100 ft. 1 2 3 4 5 1.05 2.10 3.16 4.21 5.26 2.1 7.4 15.8 27.0 41.0 1.20 1.80 2.41 3.01 1.9 4.1 7.0 10.5 1.12 1.49 1.86 1.26 2.14 3.26 q.63 0.79 0.26 0.40 6 8 10 12 15 6.31 8.42 10.52 67.0 98.0 147.0 3.61 4.81 6.02 7.22 9.02 14.7 25.0 38.0 53.0 80.0 2.23 2.98 3.72 4.46 5.57 4.55 7.8 11.7 16.4 25.0 0.94 1.26 1.57 1.89 2.36 0.56 0.95 1.43 2.01 3.05 0.61 0.82 1.02- 1.23 1.53 0.20 0.33 0.60 0.70 1.07 20 25 30 35 40 60 60 70 80 90 12:03 136.0 7.44 9.30 11.15 13.02 14.88 42.0 64.0 89.0 119.0 152.0 3.15 3.93 4.72 5.51 6.30 5.2 7.8 11.0 14.7 18.8 2.04 2.55 3.06 3.57 4.08 1.82 2.73 3.84 -5.1 6.6 1 i! 1 ,' 7.87 9.44 11.02 12.59 14.17 28.4 39.6 53.0 68.0 84.0 5.11 6.13 7.15 8.17 9.19 9.9 13.9 18.4 23.7 29.4 1 100 120 140 160 180 15.74 18.89 22.04 102.0 143.0 190.0 10.21 12.25 14.30 16.34 18.38 35.8 50.0 67.0 86.0 107.0 1 200 250 1 1 j 20.42 25.53 129.0 196.0 II 1 Digitized by VnOOQlC 310 ELEMENTS OF HYDRAULICS Fbiction Head in Pipe»— (C(m«nue(i) i b a a Inside diameter of pipe | 2-1/2 in. 1 8 in. 1 4 in. 5 in. 6 in. 1 u it |5 1 > |.9§ 'J 11 10 15 20 25 30 0.65 0.98 1.31 1.63 1.96 0.17 0.37 0.61 0.92 1.29 0.45 0.68 0.91 1.13 1.36 0.07 0.15 0.25 0.38 0.54 0.51 0.64 0.77 0.06 0.09 0.13 0.49 0.04 35 40 50 60 70 2.29 2.61 3.27 3.92 4.58 1.72 2.20 3.32 4.65 6.2 1.59 1.82 2.27' 2.72 3.18 0.71 0.91 1.38 1.92 2.57 0.89 1.02 1.28 1.53 1.79 0.17 0.22 0.34 0.47 0.63 0.57 0.65 0.82 0.98 1.14 0.06 0.08 0.11 0.16 0.21 0.45 0.57 0.68 0.79 0.03 0.05 0.07 0.09 80 90 100 120 140 5.23 5.88 6.54 7.84 9.15 7.9 9.8 12.0 16.8 22.3 3.63 4.09 4.54 5.45 6.35 3.28 4.08 4.96 7.0 9.2 2.04 2.30 2.55 3.06 3.57 0.81 1.0 1.22 1.71 2.28 1.31 1.47 1.63 1.96 2.29 0.27 0.34 0.41 0.58 0.76 0.91 1.02 1.13 1.36 1.58 0.11 0.14 0.17 0.24 0.31 160 180 200 250 300 10.46 11.76 13.07 16.34 19.61 29.0 35.7 43.1 65.5 92.0 7.26 8.17 9.08 11.35 13.62 11.8 14.8 1,7.8 27.1 38^0 4.08 4.60 5.11 6.38 7.66 2.91 3.61 4.4 6.7 9.3 2.61 2.94 3.27 4.08 4.90 0.98 1.22 1.48 2.24 3.14 1.82 2.05 2.27 2.84 3.40 0.41 0.53 0.61 0.93 1.29 350 400 450 500 550 22.87 26.14 29.41 122.0 156.0 196.0 15.89 18.16 20.43 22.70 24.96 50.5 65.0 81.0 98.0 117.0 8.93 10.21 11.49 12:77 14.04 12.4 16.0 19.8 24.0 28.7 5.72 6.54 7.35 8.17 8.99 4.19 5.4 6.7 8.1 9.6 3.98 4.54 5.11 5.68 6.24 1.73 2.21 2.75 3.35 3.98 600 700 800 900 1000 27.23 137.0 15.32 17.87 20.42 22.98 33.7 44.9 57.0 71.0 9.80 11.44 13.07 14.71 16.34 11.3 15.1 19.4 24.0 29.2 6.81 7.95 9.08 10.22 11.35 4.68 6.24 7.98 9.93 12.04 1100 1200 1300 1400 1500 17.97 19.61 34.9 40.9 12.49 13.62 14.76 15.89 17.03 14.4 16.9 19.6 22.5 25.6 1 1 ' * * ' Digitized by VnOOQlC HYDRAULIC DATA AND TABLES 311 Friction Head in Fifeu— (Continued) i H p Inside diameter of pipe { Sin. 1 10 in. 12 in. 16 in. 20 in'. 1 1 .a h .3 . 1 .3 1 .a I' 1 .9 1^ .a n Ii 1! II 200 250 300 350 400 0.309 0.386 0.464 0.641 0.619 0.89 1.11 1.33 1.56 1.77 0.08 0.12 0.14 0.22 0.28 0.57 0.71 0.85 0.9SL 1.13 0.04 0.05 0.06 0.07 0.09 0.39 0.49 0.59 0.69 0.79 0.01 0.02 0.02 0.03 0.04 0.22 0.28 0.33 0.39 0.44 0.003 0.004 0.006 0.008 0.010 0.28 0.003 450 500 550 600 700 0.696 0.774 0.851 0.928 1.083 1.99 2.22 2.44 2.66 3.10 0.34 0.42 0.50 0.59 0.78 1.28 1.42 1.56 1.70 1.99 0.12 0.14 0.17 0.20 0.26 0.89 0.99 1.09 1.18 1.38 0.05 0.06 0.07 0.08 0.11 0.50 0.55 0.61 0.66 0.77 0.012 0.016 0.017 0.02 0.03 0.31 0.35 0.39 0.43 0.50 0.004 0.005 0.006 0.007 0.009 800 900 1,000 1.100 1,200 1.238 1.392 1.547 1.702 1.867 3.55 3.99 4.43 4.88 5.37 0.99 1.24 1.51 1.80 2.11 2.27 2.55 2.84 3.12 3.40 0.34 0.42 0.51 0.61 0.71 1.58 1.77 1.97 2.17 2.36 0.14 0.17 0.21 0.25 0.29 0.89 1.00 1.11 1.22 1.33 1 0.03 0.04 0.05 0.06 0.07 0.57 0.64 0.71 0.78 0.85 0.012 0.014 0.017 0.020 0.024 1.500 2.000 2.500 3.000 3,600 2.321 3.094 3.868 4.642 6.41 6.65 8.86 11.08 13.30 3.18 5.4 8.4 11.6 4.26 5.67 7.10 8.51 9.93 1.08 1.84 2.78 3.86 5.19 2.96 3.94 4.92 5.91 6.89 0.44 0.76 1.15 1.60 2.13 1.66 2.22 2.77 3.32 3.88 0.11 0.19 0.28 0.40 0.53 1.06 1.42 1.77 2.13 2.48 0.04 0.06 0.09 0.13 0.18 4.000 . 6.000 6.000 7,000 8,000 6.19 7.74 9.28 10.83 12.38 11.35 14.19 17.03 6.65 10.05 14.09 7.88 9.85 11.82 13.79 15.76 2.70 4.10. 5.8 7.7 9.9 4.43 5.54 6.65 7.76 8.86 0.68 1.02 1.43 1.90 2.42 2.84 3.55 4.26 4.96 6.67 0.23 0.34 0.48 0.64 0.82 1 9,000 10,000 11.000 12.000 15,000 13.92 15.47 17.02 18.67 23.21 17.73 19.70 12.2 15.0 9.97 11. Q8 12.19 13.30 16.62 3.02 3.68 4.40 6.2 7.8 6.38 7.09 7.80 8.51 10.64 1.02 1.24 1.48 1.74 2.62 ,. I 1 1 16.000 17.000 18.000 19.000 20.000 24.76 26.30 27.85 29.40 30.94 1 11.35 12.06 12.77 13.47 14.18 2.96 3.31 3.68 4.07 4.48 ' * : 1 t ........... 1 ' :--• •-;. 1 1 Digitized by VnOOQlC 312 ELEMENTS OF HYDRAULICS Friction Head in Pipes — (Continued) ij is i g H 1^ Inside diameter of pipe | 24 in. 1 30 in. 1 36 in. 1 42 in. 48 in. 1 .9 h .9^ 1! .9;g 31 1 .9 . |i ss •a ^ .n 1 .9 Is |i gl 1 II 1 ^ 1.0 1.6 2.0 2.5 3.0 1.547 2.321 3.094 3.868 4.642 0.49 0.74 0.98 1.23 1.48 0.007 0.015 0.026 0.039 0.065 0.32 0.47 0.63 0.79 0.95 0.002 0.005 0.009 0.013 0.018 0.22 0.33 0.44 0.55 0.66 0.002 0.003 0.004 0.005 0.008 1 1 0.48 0.004 3.6 4.0 4.5 5.0 6.0 5.41 6.19 6.96 7.74 9.28 1.72 1.97 2.22 2.46 2.96 0.07 0.09 0.12 0.14 0.20 1.10 1.26 1.42 1.58 1.89 0.025 0.032 0.039 0.048 0.067 0.77 0.88 0.99 1.09 1.31 0.010 0.013 0.020 0.027 0.56 0.64 0.72 0.80 0.96 0.005 0.006 0.007 0.009 0.013 0.49 0.55 0.62 0.74 0.003 0.004 0.005 0.007 7.0 8.0 9.0 10.0 12.0 10.83 12.38 13.92 15.47 18.57 3.45 3.94 4.43 4.92 5.91 0.26 0.34 0.42 0.51 0.71 2.21 2.52 2.84 3.15 3.78 0.09 0.11 0.14 0.17 0.24 1.53 1.75 1.97 2.19 2.63 0.036 0.047 0.058 0.071 0.099 1.13 1.29 1.45 1.61 1.93 2.25 2.57 2.89 3.22 3.53 0.017 0.022 0.027 0.033 0.047 0.86 0.98 1.10 1.23 1.48 0.009 0.012 0.014 0.017 0.024 14.0 16.0 18.0 20.0 22.0 21.66 24.76 27.85 30.94 34.04 6.89 7.88 8.86 9.85 10.83 0.95 1.22 1.52 1.83 2.19 4.41 5.04 5.67 6.30 6.93 0.32 0.41 0.51 0.62 0.74 3.06 3.50 3.94 4.38 4.82 0.13 0.17 0.21 0.25 0.30 0.06 0.08 0.10 0.12 0.14 1.72 1.97 2.22 2.46 2.71 0.032 0.042 0.052 0.063 0.075 24.0 26.0 28.0 30.0 32.0 37.13 40.23 43.32 46.42 49.51 11.82 12.80 13.79 14.77 2.59 2.99 3.42 3.90 7.56 8.20 8.83 9.46 10.09 0.87 1.01 1.16 1.32 1.48 5.25 5.69 6.13 6.57 7.00 0.36 0.41 0.48 0.54 0.61 3.86 4.18 4.50 4.82 5.16 0.17 0.20 0.22 0.26 0.29 2.96 3.20 3.45 3.69 3.94 0.09 0.10 0.12 0.13 0.15 34.0 36.0 38.0 40.0 50.0 52.6 55.7 58.8 61.9 77.4 1 10.72 11.35 11.98 1.66 1.84 2.04 7.44 7.88 8.32 8.76 10.95 0.68 0.76 0.84 0.92 1.39 5.47 5.79 6.11 6.45 8.04 0.32 0.36 0.40 0.44 0.66 4.19 4.43 4.68 4.92 6.16 0.17 0.19 0.21 0.23 0.34 ' 1 1 1 I 1 1 60.0 70.0 80.0 90.0 100.0 92.8 108.3 123.8 139.2 154.7 13.13! 1.96 9.65 11.26 12.86 0.92 1.22 1.57 7.39 8.62 9.86 11.08 0.48 0.64 0.82 1.02 1.24 1 1 ' 1 1 1 1 ' 112.31 Digitized by vnOOQlC HYDRAULIC DATA AND TABLES 313 Table 14. — Bazin's Values of Chezy's Coefficient Values of the coefficient C in Chesy's formmla t - CVri according to Basin's formula (Par. 142): ^87 0.552 + V" r Hydraulic radius r, in feet Coefficient of roughness, m | Planed tim- ber or smooth cement Unplaned timber, well laid brick, or concrete rubble mas- onry, or poor brick- work Earth in good condition Earth in ordinary condition Earth in bad condition m -0.06 m - 0.16 m - 0.46 m - 0.85 m - 1.30 m - 1.75 0.1 117 82 43 27 19 14 0.2 127 96 55 35 25 19 0.3 131 103 63 41 30 23 0.4 135 108 68 46 33 26 0.5 136 112 71 50 36 29 0.6 138 115 76 53 39 31 0.7 139 117 79 55 41 33 0.8 141 119 82 58 43 35 0.9 141 121 84 «r-* 45 36 1.0 142 122 86 62 47 38 1.25 143 125 90 66 51 41 1.50 145 127 94 70 54 44 1.75 145 129 97 73 57 47 2.00 146 131 99 75 59 49 2.5 147 133 104 80 63 53 3.0 148 135 106 83 67 57 4.0 150 a|s\ 111 89 72 61 5.0 160 140 115 93 77 65 6.0 151 141 118 97 80 69 7.0 152 142 120 100 83 72 8.0 152 143 122 102 86 74 9.0 152 144 123 104 88 77 10.0 152 145 ^ 125 106 90 79 12.0 153 145 127 109 94 82 15.0 153 147 130 113 98 86 20.0 154 148 133 117 103 92 30.0 155 150 137 123 110 100 40.0 ' 155 151 139 127 115 105 50.0 155 151 141 129 118 109 Digitized by VnOOQlC 314 ELEMENTS OF HYDRAULICS TABIiB 15. — ^KijTTBB'S VikLTJES OF ChEZT'S CoEFFICIBNT Value* of the eoefflcient C in Cheiy's formula « — Cs/ri ao«<»rdin( -to Kutten formula (Eq. (94) Par. 140): 41.65 + -?^5o??L + JL«i 0.00281 \ A -r V .oo T « J V r Slope. 9 Coefficient of rouBhnesfl, n Hydraulic radius r, in feet • | 0.1 0.2 0.4 0.6 0.8| 1 1.5 2 3 1 4 1 6 1 8 1 10 15 20 0.009 65 87 111 127 138|148 166 179 197209 226 238 246 262 271 ^ 0.010 57 75 97 112 122 131 148 160 177 188 206 216 225 240 249 S 0.011 50 67 87 100 109!ll8 133 144 160 172 188 199 207 222 231 0.012 44 59 78 90 99|l06 121 131 147 158 174 184 192 206 216 0.013 40 53 70 81 90 97 111 121 135 146 161 171 179 193 202 ih /-,» 0.017 28 38 51 60 66 72 83 91 103 113 126 135 142 155 164 5 fl S 0.020 23 31 42 49 55 60 69 77 88 96 108 117 124 136 144 6^6 0.026 ' 17 24 32 38 43 47 55 61 70 78 88 96 102 11* 121 III 0.030 14 19 26 31 35 38 45 50 59 65 74 82 87 98 106 0.035 12 16 22 26 30 32 38 43 50 56 64 71 76 86 94 0.009 78 100 124 139 150 158 173 184 198 207 220 228 234 244 250 ± 0.010 ' 67 87 109 122 133 140 154 164 178 187 199 206 212 220;228| 1 0.011 59 77 97 109 119 126 139 148 161 170 182 189 195 205 211 Pi 0.012 52 68 88 98 107 114 126 135 148 156 168 176 181 189 196 0.013 47 62 79 90 98 104 116 124 136 146 156 163 169 179 184 0.017 33 44 57 65 71 77 87 94 104 111 122 129 134 142 149 8.sS 0.020 26 35 46 53 59 64 72 79 88 95 105 111 116 126 131 6 r^ O 0.026 20 26 35 41 46 49 57 62 71 77 85 91 96 104 110 1 H R 0.030 16 21 28 33 37 40 47 51 59 64 72 78 82 90 96 0.035 13 18 24 28 31 34 40 44 50 56 63 68 72 79 85 V 0.009 90 112 136 149 158 166 178 187 198 206 215 221 226 233 237 1 . 0.010 78 98 119 131 140 147 159 168 178 186 195 201 205 212216 0.011 68 86 106 118 126 132 144 151 162 169 178 184 188 196 200 ^i 0.012 60 76 95 105 114 120 130 138 149 156 164 170 174 181 185 0001 in lO.OG 628 ft. 0.013 54 69 86 96 103 109 120 127 137 143 152 158 162 169 173 0.017 37 48 62 70 76 81 89 96 104 111 119 124 128 136 139 o ^ 6 0.020 30 39 50 57 63 67 75 81 89 94 102 107 111 118 122 H R R 0.025 22 29 38 44 48 52 59 64 71 76 84 88 92 98 102 «0 0.030 17 23 31 35 39 42 48 53 59 64 71 76 78 86 89 0.035 14 19 25 30 33 35 41 45 51 65 61 66 69 76 79 Digitized by VnOOQlC HYDRAULIC DATA AND TABLES 315 Kutteb'b Valdb op Chbzy's Coefficient — {Continued) Slope. 9 Coefficient of roughnesa, n Hydraulic radius r in feet \ 0.1 0.2 0.3|0.4 0.6 0.8 1.0 1.6 2 3 4 6 10 15 20 0.000 99 121 133 143 166 164 170 181 188 200 205 213 222 228 231 1 0.010 86 105 116 126 138 145 151 162 170 179 186 193 201 207 210 0.011 74 93 103 112 122 131 136 146 164 163 168 176 185 190 194 ).0002 Lin 5000 1.056 ft. per 0.012 66 83 92 100 111 118 123 133 140 149 155 162 170 176 180 0.013 69 74 83 91 100 107 113 122 129 137 143 150 158 164 168 0.017 41 52 69 66 73 79 83 91 97 106 111 117 126 131 134 0.020 32 42 48 63 60 66 69 77 82 89 94 100 108 113 117 1 1 R «0 0.026 24 31 36 40 46 60 54 60 64 72 76 82 89 96 98 0.030 18 25 29 32 37 41 44 49 64 69 63 69 76 82 86 0.036 16 21 24 27 31 34 37 42 45 51 66 60 67 72 76 0.009 104 126 138 148 167 166 172 183 190 199 ^ 211 219 224 227 1 0.010 89 110 120 129 140 148 164 164 170 179 184 191 199 203 207 0.011 78 97 107 116 126 133 138 148 164 162 168 175 183 187 190 1 O »H N 1 1 H 0.012 69 87 96 104 113 121 126 135 141 149 164 161 168 172 176 0.013 ^ 78 87 94 103 110 116 124 130 138 z 149 157 162 164 0.017 43 64 62 68 76 81 86 93 98 106 110 116 123 128 131 0.020 34 44 60 55 62 67 70 78 83 89 94 99 107 110 115 0.025 26 32 37 42 47 61 66 61 66 71 76 81 88 92 96 «0 0.030 19 25 30 33 38 42 45 60 64 69 63 69 76 80 83 0.035 16 21 24 27 31 36 37 42 45 191 51 199 65 204 60 211 66 218 70 222 73 225 0.000 IIO 129 141 150 161 169 175 184 A 0.010 94 113 124 1-31 142 160 156 166 171 179 184 190 197 202 206 - 0.001 - 1 in 1000 - 5.28 ft. per mil( 0.011 83 99 109 117 127 134 139 149 156 163 168 174 181 186 188 0.012 73 89 98 105 115 122 127 136 142 149 164 160 167 171 176 0.013 66 81 89 96 104 111 116 124 130 138 142 149 155 160 163 0.017 45 57 63 69 76 82 86 93 98 105 110 116 122 127 129 0.020 36 45 61 66 63 68 71 78 83 89 93 99 105 110 113 0.025 27 34 39 43 48 62 56 62 66 71 75 81 87 91 94 w 0.03d 21 27 30 34 39 42 45 50 64 69 63 68 74 78 81 0.035 17 22 25 28 32 36 38 43 46 61 64 69 65 68 72 0.009 110 130 143 161 162 170 175 185 191 199 204 210 217 222 225 0.010 95 114 126 133 143 161 166 165 171 179 184 190 196 200 204 1 0.011 83 100 111 119 129 135 141 149 155 162 167 173 180 184 187 0.012 74 90 100 107 116 123 128 136 142 149 154 160 166 170 173 y 100 8 ft. pe 0.013 66 81 90 98 106 112 117 125 130 138 142 148 154 169 161 0.017 46 57 64 70 77 82 87 94 99 105 109 115 121 126 128 III 0.020 36 46 62 57 64 68 72 79 83 89 93 99 105 108 112 0.025 27 34 39 44 49 53 66 62 66 71 76 81 86 90 93 ■> « 0.030 21 27 31 35 39 43 46 61 56 69 63 68 74 77 80 0.035 17 22 26 29 33 35 38 43 46 61 65 Iki 66 68 71 Digitized by VnOOQlC 316 ELEMENTS OF HYDRAULICS & n H H H I) e4e4eQeo'4t'4t^ioio««« CQ <o o CO <o o» ei >o t« ^ '^t <o o» ei '^t e« :fNesr'-. «(Oe<«iO(OiOC4Q0C0 <H ei c<4 CO 00 CO <«' '^t <^ lO lO lO it^Hoocoudteiot-oeoi^eoat^iOiO^ 3'4t<«io<ot»aoa»^e<«cou)SQOQC4a> ddddt^b-'t^t^aoaoaOQoddddd^'He^ aococ<4obt»^e«K coaoio^t«<«<«aooo^Oio dd^c^coaot^dc^id iOi-icoeiQO^<-ia>>o C<4 «0 O '^ 06 >0 fH 00 iQ ^ CO e<« CO U3 1» Q -^ a> iC C<4 W CO O) lO vH^oe^e^co^^toot^QOoOfHco^aot-ab^^oor* iHtHtHtHtHtHtHe^COCO^ aote'-4ieo>o>o»t»^^oocoQOe<«t»^'<ft«o^'<t<oo6d<-i'-i^'^ t«*Hio«^^QOOcoda)^'«Sa)^^oaO'Hcoiot»a»e«^oaora <H 09 04* ei CO eo CO <«' <«* <^ <^ lO iou)iQ(0<o<ocDt«t«t.t«t«aoQOoo«o>dd^^'e<« it»t«^CDiOO>'HOU3QO 25aoc4te«ococooO'Hcoiid ^ »H N ei CI CO* CO CO CO -^ ^' -^ .H>HiHt-l>H>-l<NC^C0^»O<O Q5J-*io«ooh.h.t*«b»*iA'*c4»Hd>coui(t^ 00 o C4 ^ <o 00 o C4^ ^ CO X o e<i ^ CD t. 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Ill > O <0 CO CO o o o o o O O O Q.O <6 d CO CO a» lo 00 ^ 1 CO S CO CO ( o o o o o o o o o o ' CO (b CO < o o o o o t^ ^ CO *-• I o o o o o t* CO I SSI o o o o o o o o o o o o o o d liiis o o o o o n i;nn o o o o o CO CO to CO CO odd I e3 S « J o o o o o tssg • CO CO CO o o o o o CO CO X 00 00 r-l *^ VH CO CO CO d d d isiii o o o o o i Sa S^ ^ ( o o o o o o o o o o o o o o o 11^ CO CO o o o o o d d I III (O CD d d ^ CO t* >o to CO CO SOI C< 04 CO CO CO CO CO CO CO d d d d iH ^ N M eo d d d d d "^ U3 CO t« 00 d o 6 G o a> o ^ CI CO O <^ vH >^ >^ § eo 3 ISI St: CO CO »0 "xl* CO w ^ o o < IH tH IH IH < CO CO CD CO < o o o o o o o o o o O O O O Q §111 d d d d «0 N ■ CD CO < O) CO CO to <* CO M VH «H IH «0 CO CO ISSSI I CO V V < o o o o o ooooo ooooo s^gss; ,3 d d d d d CO CO CO N ^ ooooo CO S d d « t* CO « ei M r« CO CO CO d d CO w V C> O G if to CO to d d lH CO FH CO CD < ooooo ;22l O CO < ooooo ooooo 1^2 I CD CO I >o CO < ooooo ooooo > A 00 00 > Q 53 !i ) CO CO CO S o < ooooo ooooo CO ^ CO 1 S CO CO < f-4 00 lO o o o >o d d o o o ooooo « oi 00 IH VH Q CO CO V odd ^ CD ( O lO I q5 CD O V o ^ ^ s s S S o s ooooo 00 CO lO ]f ooooo CO W OB t> CO "* M F^ SS3 ooooo ooooo X lo CO *-i a» s s s s s ooooo CO eo «H o) < ooooo ^ 01 Q d d d C« O t« I ooooo CI o eg "H d d lH lO ^ « s s lO O I SSI Digitized by vnOOQlC HYDRAULIC DATA AND TABLES 319 Table 18. — Dischabge per Inch of Length oyer Rectangular Notch Weirs Discharge over sharp-crested, vertical, rectangular notch weirs in cubic feet per minute per inch of length Computed from Eq. (42), Par. 66; Q - 0.46*'^' f or 6 >i 1 in. Depth on crest in inches 1/8 1/4 3/8 1/2 6/8 3/4 7/8 0.00 0.01 0.06 0.09 0.14 0.19 0.26 0.32 1 • 0.40 0.47 0.65 0.64 0.73 0.82 0.92 1.02 2 1.13 1.23 1.35 1.46 1.68 1.70 1.82 1.96 3 2.07 2.21 2.34 2.48 2.61 2.76 2.90 3.05 4 3.20 3.35 3.50 3.66 3.81 3.97 4.14 4.30 5 4.47 4.64 4.81 4.98 5.16 5.33 6.61 5.69 6 5.87 6.06 6.26 6.44 6.62 6.82 7.01 7.21 7 7.40 7.60 7.80 8.01 8.21 8.42 8.63 8.83 8 9.05 9.26 9.47 9.69 9.91 10.13 10.36 10.57 9 10.80 11.02 11.26 11.48 11.71 11.94 12.17 12.41 10 12.64 12.88 13.12 13.36 13.60 13.86 14.09 14.34 11 14.59 14.84 16.09 16.34 16.59 16.86 16.11 16.36 12 16.62 16.88 17.15 17.41 17.67 17.94 18.21 18.47 13 18.74 19.01 19.29 19.56 19.84 20.11 20.39 20.67 14 20.95 21.23 21.51 21.80 22.08 22.37 22.65 22.94 15 23.23 23.52 23.82 24.11 24.40 24.70 26.00 26.30 16 25.60 25.90 26.20 26.60 26.80 27. U 27.42 27.72 17 28.03 28.34 28.65 28.97 29.28 29.69 29.91 30.22 18 30.54 30.86 31.18 31.50 31.82 32.15 32.47 32.80 19 33.12 33.45 33.78 34.11 34.44 34.77 35.10 36.44 20 35.77 36.11 36.46 36.78 37.12 37.46 37.80 38.16 21 38.49 38.84 39.18 39.63 39.87 40.24 40.60 40.96 22 41.28 41.64 41.98 42.36 42.68 43.04 43.44 43.76 ' 23 44.12 44.48 44.84 46.20 46.66 45.96 46.32 46.68 24 47.04 47.40 47.76 48.12 48.52 48.88 49.28 49.64 25 50.00 50.40 50.76 51.08 51.52 61.88 62.28 52.64 26 53.04 53.40 53.80 64.16 54.66 54.96 55.36 65.72 27 56.12 66.52 56.92 67.32 67.68 58.08 68.48 68.88 28 59.28 69.68 60.08 60.48 60.84 61.28 61.68 62.08 29 62.48 62.88 63.28 63.68 64.08 64.62 64.92 65.32 30 66.72 66.16 66.56 66.96 67.36 67.80 68.20 68.64 Digitized by VnOOQ IC 320 ELEMENTS OF HYDRAULICS TaBLB 19. — DiBCHABOE PER FoOT OP LENGTH OVEB RECTANQULi^R NOTCH Weirs Disoharge over sharp crested, vertical, rectangular notch weirs in cubic feet per second per foot of length. Computed from Eq. (41), Far. 66: Q- S.Zbh*/* for b - 1ft. Depth on crest in feet 0.00 0.01 0.02 0.03 0.04 0.06 0.06 0.07 0.08 0.09 0.0 0.000 0.003 0.009 0.017 0.026 0.037 0.049 0.061 0.075 0.089 0.1 0.104 0.120 0.137 0.155 0.173 0.192 0.211 0.231 0.252 0.273 0.2 0.295 0.317 0.341 0.364 0.388 0.413 0.438 0.463 0.489 0.515 0.3 0.642 0.570 0.697 0.626 0.654 0.683 0.713 0.743 0.773 0.804 0.4 0.835 0.866 0.898 0.931 0.963 0.996 1.030 1.063 1.098 1.132 0.5 1.167 1.202 1.238 1.273 1.309 1.346 1.383 1.420 1.468 1.496 0.6 1.538 1.572 1.611 1.650 1.690 1.729 1.769 1.810 1.850 1.892 0.7 1.933 1.974 2.016 2.068 2.101 2.143 2.187 2.230 2.273 2.317 0.8 2.361 2.406 2.450 2.495 2.541 2.586 2.632 2.678 2.724 2.768 0.9 2.818 2.865 2.912 2.960 3.008 3.065 3.104 3.152 3.202 3.251 1.0 3.300 3,350 3.399 3.449 3.501 3.561 3.600 3.663 3.703 3.755 1.1 3.808 3.858 3.911 3.963 4.016 4.069 4.122 4.178 4.231 4.283 1.2 4.340 4.392 4.448 4.501 4.657 4.613 4.666 4.722 4.778 4.834 1.3 4.891 4.947 5.006 6.062 6.118 6.178 6.^ 5.293 6.349 5.409 1.4 5.468 5.524 6.684 6.643 6.702 6.762 6.821 5.881 6.940 6.003 1.5 6.062 6.126 6.184 6.247 6.306 6.369 6.428 6.491 6.664 6.617 1.6 6.679 6.742 6.805 6.867 6.930 6.993 7.059 7.121 7.187 7.250 1.7 7.316 7.379 7.446 7.608 7.573 7.639 7.706 7.772 7.838 7.904 1.8 7.970 8.036 8.102 8.171 8.237 8.303 8.372 8.438 8.607 8.673 1.9 8.643 8.712 8.778 8.847 8.917 8.986 0.066 9.126 9.194 9.263 2.0 9.332 9.405 9.474 9.544 9.616 9.686 9.768 9.827 9.900 9.969 2.1 10.042 10.115 10.187 10.260 10.332 10.406 10.478 10.660 10.623 10.695 2.2 10.768 10.841 10.916 10.989 11.065 11.138 11.213 11.286 11.362 11.435 2.3 11.510 11.586 11.662 11.738 11.814 11.887 11.966 12.042 12.118 12.194 2.4 12.269 12.345 12.425 12.500 12.576 12.656 12.731 12.811 12.890 12.966 2.5 12.935 13.124 13.200 13.279 13.368 13.438 13.617 13.696 13.675 13.764 2.6 13.834 13.916 13.995 14.075 14.154 14.236 14.316 14.-398 14.477 14.660 2.7 14.642 14.721 14.804 14.886 14.969 15.048 15.131 16.213 15.296 16.378 2.8 15.461 15.543 15.629 16.711 16.794 15.876 16.962 16.045 16. 130 16.213 2.9 16.299 16.381 16.467 16.650 16.636 16.721 16.807 16.889 16.975 17.061 3.0 17.147 17.233 17.318 17.404 17.490 17.679 17.665 17.761 17.837 17.926 3.1 18.011 18.101 18.186 18.275 18.361 18.460 18.636 18.625 18.714 18.803 3.2 18.889 18.978 19.067 19.157 19.246 19.335 19.424 19.613 19.602 19.694 3.3 19.784 19.873 19.962 20.064 20.143 20.236 20.326 20.414 20.606 20.699 3.4 20.688 20.780 20.873 20.962 21.054 21.146 21.239 21.331 21.424 21.616 22.447 3.5 21.608 21.701 21.793 21.886 21.978 22.074 22.166 22.269 22.354 3.6 22.542 22.635 22.730 22.823 22.919 23.011 23.107 23.202 23.296 23.390 3.7 23.486 23.682 23.678 23.773 23.869 23.965 24.060 24.156 24.252 24.347 3.8 24.446 24.642 24.638 24.734 24.833 24.928 26.027 25.123 25.222 25.318 3.9 25.417 25.616 25.611 25.710 25.809 26.905 26.004 26.103 26.202 26.301 4.0 26.400 26.499 26.698 26.697 26.796 26.895 26.997 27.096 27.196 27.298 Digitized by vnOOQlC HYDRAULIC DATA AND TABLES 321 Table 20.' — Discharqb per Foot of Length over Suppressed Weirs Difloharge ovev 8harp<ere«ted, vertical, suporesaed weirs in cubio feet per second per foot of length. Computed by Basin's formula (Art. XI) Q - (0.405 + 5:5^ ) (l + 0.55(^)*) bhVm. for 6 - 1 ft. — Head on crest, h, in feet Height of weir, d, in feet | 2 4 6 8 10 20 30 0.1 0.13 0.13 0.13 0.13 0.13 0.13 0.13 0.2 0.33 0.33 0.33 0.33 0.33 0.33 0.33 0.3 0.58 0.58 0.58 0.58 -0.58 0.58 0.58 0.4 0.88 0.88 0.87 0.87 0.87 0.87 0.87 0.5 1.23 1.21 1.21 1.21 1.21 1.20 1.20 0.6 1.62 1.59 1.58 1.58 1.57 1.57 1.57 0.7 2.04 1.99 1.98 1.98 1.97 1.97 1.97 0.8 2.50 2.43 2.41 2.41 2.40 2.40 2.40 0.9 3.00 2.90 2.88 2.86 2.86 2.85 2.85 1.0 3.53 3.40 3.36 3.35 3.34 3.33 3.33 1.1 4.09 3.92 3.87 3.86 3.85 3.84 3.84 1.2 4.68 4.48 4.42 4.40 4.38 4.36 4.36 1.3 5.31 5.07 4.99 4.96 4.94 4.91 4.91 1.4 5.99 5.68 5.58 5.54 5.52 5.49 5.48 1.5 6.68 6.30 6.20 6.16 6.13 6.10 6.09 1.6 7.40 6.97 6.84 6.78 6.74 6.69 6.69 1.7 8.14 7.66 7.49 7.42 7.39 7.33 7.32 1.8 8.93 8.37 8.18 8.09 8.05 7.98 7.96 1.9 9.75 9.11 8.89 8.79 8.74 8.65 8.63 2.0 10.58 9.87 9.62 9.51 9.44 9.34 9.32 2.1 11.45 10.65 10.37 10.25 10.17 10.05 10.02 2.2 12.34 11.46 11.14 10.99 10.91 10.78 10.75 2.3 13.24 12.29 11.93 11.77 11.66 11.52 11.48 2.4 14.20 13.15 12.75 12.56 12.45 12.28 12.24 2.5 15.17 14.03 13.59 13.38 13.26 13.06 13.01 2.6 16.16 14.92 14.44 14.20 14.07 13.85 13.80 2.7 17.18 15.83 15.31 15.04 14.92 14.65 14.60 2.8 18.23 16.79 16.21 15.92 15.76 15.48 15.42 2.9 19.29 17.77 17.11 16.79 16.63 16.33 16.25 3.0 20.39 18.74 18.06 17.71 17.52 17.18 17.10 3.1 21.50 19.74 19.02 18.64 18.42 18.04 17.96 3.2 22.64 20.77 19.98 19.58 19.34 18.93 18.83 3.3 23.81 21.80 20.98 20.55 20.27 19.82 19.73 3.4 24.98 22.89 21.99 21.52 21.24 20.76 20.63 3.5 26.20 24.00 23.01 22.48 22.22 21.69 21.60 3.6 27.41 25.09 24.06 23.52 23.20 22.62 22.48 3.7 28.64 26.22 25.14 24.56 24.20 23.59 23.43 3.8 29.94 27.38 26.22 25.60 25.23 24.56 24.39 3.9 31.21 28.53 27.33 26.65 26.26 25.53 25.34 4.0 32.54 29.74 28.45 27.74 27.32 26.55 26.35 4.1 33.85 30.95 29.59 28.83 28.36 27.55 27.33 4.2 35.22 32.18 30.75 29.96 29.48 28.59 28.36 4.3 36.59 33.43 31.93 31.10 30.58 29.62 29.37 4.4 37.99 34.70 33.12 32.24 31.70 30.66 30.42 4.6 39.40 35.98 34.33 33.39 32.83 31.74 31.47 4.6 40.83 37.29 35.56 34.58 33.98 32.84 32.53 4.7 42.29 38.62 36.82 35.75 35.13 33.93 33.61 4.8 43.75 39.96 38.07 37.00 36.33 35.05 34.70 4.9 45.22 41.30 39.35 38.20 37.49 36.15 35.77 5.0 46.71 42.67 40.62 .39.44_ 38.70 37.28 36.88 1 Compiled from extensive hydraulic tables by Williams and Hazek. 21 Digitized by GoQgle 322 ELEMENTS OF HYDRAULICS Table 21. — Principles Kinematics (motion) Linear motion Angular motion « " displacement 9 B displacement V - velocity <a - velocity a « acceleration a B acceleration vo - initial velocity wo - initial velocity F - force M « torque about fixed axis Nototion IF - F« - work W " Me " work m = mass J - 2mr« - moment of inertia ' v B my a. weight t - time 1 t - time 3ff - impulse ! Ft - impulse Ibi « momentum 1 mv " momentum Definitions ds dv d*8 " "df" ' dt - dt* de do» d*e " " dt' *" dt " d<8 Uniformly V " vo •\- at w * wo + «< accelerated motion « - ro< + ia/« <? - «o< + W^ acoel. « const. V* = i;o« + 2o« \ wo «« - «o« + 2ae Derivation of above formulas If « = 0, w - uo .'.Ci - d/« • «• dt " «' + ^' « - ia<« + Cit -f Ca If < =» 0, « - «o .'.Ci - uo If < - 0. « - .-.Ci = If < - 0, « - .-.c* - Relation between V " ru linear and at " ra at - tang. comp. of accel. angular motion On » v^/r - r«2 a* » normal comp. of accel. vB Arc AB » ds '^ rde u - rw Derivation of first two formulas .rY d« d(? dv do dt ~' "" dt dt '^ dt /^^ u u — r« at "■ ra ^4;'' If body at A were free, it would proceed in direction of tangent AB and in f /^ time t would reach B where AB « vf. Derivation of normal accel. for uniform circular motion lyt> J Since it is found at C instead of B it jyy^^^^ must have experienced a central acceleration. Let an denote this central acceleration. Then BC - \ant'^' By geometry BC X BD ■ .^B> and in the limit BD approaches 2r. Hence ian«« X 2r - v2««, from which Un « v^/r « wV. Digitized by VnOOQlC HYDRAULIC DATA AND TABLES 323 OF Mbchanics Dynsmicg (force) Fundamental law ; Linear motion I F '^ ma Angular motion la DiBcussion and j derivation I Principle of work and energy Derivation By experiment it is found that F oz a (Newton's 2nd Law) .'. F/a >- const., say m, whence F — ma. m ■■ intrinsic prop- erty of body called its mass. Mass " measured inertia. i If F » then a '^ and hence I V s or constant, which ex- | presses Newton's 1st Law. F » impressed force, ma » kin- etic reaction or inertia force, i Equality F -■ ma is dynamical expression of Newton's 3rd Law. Fa ir -F« - mv* mvo* 2 Consider rotation of rigid body about a fixed axis. - Then for a parti- cle of mass m at distance r from .azis of rotation, law F '^ ma becomes Fr -> mar^ or since a — ra, Fr ■■ mr*a. By summation 2 Fr ■> — mr*a But X Fr " Mt and X mr*' :.M - la Tf - Af a - J«i W 2 Principle of im- pulse Sc momentum mat «* ■■ so* + 2a« at ■■ — n — , and mwo* 2 Tf - F« - ma9 - 2 Ft . Derivation Power F - mo. V ■■ VO + a< •. (rt - V - wo and Ft - merf ■■ mw - - mwi - Ja, tr» — wi* + 2ao . . oo — s • ftnd W - Af - Jao ' I»« - !«.« Aft - J«, Jo>» Plower — Fw, h.p. Fy_ 650 Af — ~ Ja, tp -» tri + al .*. a< « w — wi, and Aft — lal " luf — etri Power — Mw, h.p. — gg^ Centrifugal force I Derivation ' D*Alembert's principle F. = w V* — - w*r dU F - «5^ - I F oi ma where F >- external impressed force and a » accel. produced. I Introduced another force P, given by P ■> — ma. Then by addition, I F + F "■ 0; i.e., the body is in equilibrium under the action of F and ; P. P is called the kinetic reaction, or reversed effective force, tinee [P » — F. By introducing this idea of the kinetic reactions equiU- Explanation and i brating the impressed forces, all problems in dynamics are reduced to statical problems. This is called d*Alembert's Principle, and is usually expressed in the form d*a F - m^^ - 0. Digitized by Google 324 ELEMENTS OF HYDRAULICS Tablb 22. — Submerged Weir Coefficients.^ Values of the coefficient n in Herschel's submerged weir formula h = depth of water measured to crest h' = depth of water measured to crest level on upstream side; level on downstream side. Hi 1 2 3 4 5 6 7 8 9 0.0 1.000 1.004 1.006 1.006 1.007 1.007 1.007 1.006 1.006 1.005 0.1 1.005 •1.003 1.002 i.oop 0.998 0.996 0.994 0.992 0.989 0.987 0.2 0.985 0.982 0.980 0,977 0.975 0.972 0.970 0.967 0.964 0.961 0.3 0.959 0.956 0.953 0.950 0.947 0.944 0.941 0.938 0.935 0.932 0.4 0.929 0.926 0.922 0.919 0.915 0.912 .908 0.904 0.900 0.896 0.5 0.892 0.888 0.884 0.880 0.875 0.871 0.866 0.861 0.856 0.851 0.6 0.846 0.841 0.836 0.830 0.824 0.818 0.813 0.806 0.800 0.794 0.7 0.787 0.780 0.773 0.766 0.758 0.750 0.742 0.732 0.723 0.714 0.8 0.703 0.692 0.681 0.669 0.656 0.644 0.631 0.618 0.604 0.590 0.9 0.574 0.557 0.539 0.520 0.498 0.471 0.441 0.402 0.352 0.275 Values of the coefficient k in Pteley and Steams submerged weir formula Q = Kl(k + ^) y/h-W h 1 2 3 4 5 6 7 8 9 0.0 3.33 3.33 3.34 3.34 3.36 3.37 3.37 3.37 3.37 0.1 3.37 3.36 3.35 3.34 3.34 3.33 3.32 3.31 3.30 3.29 0.2 3.29 3.28 3.27 3.26 3.26 3.25 3.24 3.23 3.23 3.22 0.3 3.21 3.21 3.20 3.19 3.19 3.18 3.18 3.17 3.17 3.16 0.4 3.16 3.15 3.15 3.14 3.14 3.13 3.13 3.12 3.12 3.12 0.5 3.11 3.11 3.11 3.10 3.10 3.10 3.10 3.10 3.10 3.09 0.6 3.09 3.09 3.09 3.09 3.09 3.09 3.09 3.09 3.09 3.09 0.7 1 3.09 3.09 3.10 3.10 3.10 3.10 3.11 •3.11 3.11 3.12 0.8 3.12 3.13 3.13 3.14 3.14 3.15 3.16 3.16 3.17 3.18 0.9 3.19 3.20 3.21 3.22 3.23 3.25 3.26 3.28 3.30 3.33 1 "Hydraulics," Hughes and Safford, pp. 228, 229. Digitized by VnOOQlC INDEX Accumulator, hydraulic, 9 Adjutage, 67 Venturi, 69 Age, deterioration with, 88 AUievi's formula for water hammer, 227 American type of reaction turbine, 189 Appalachian Power Co. turbines, 282 Aqueduct, Catskill, 153-156 Roman, 150 Archimedes, theorem of, 24 B Backwater, 138 Barge canal, N. Y. State, 115 Barker's mill, 169 Barlow's formula, 18 Barometer, mercury, 21 water, 20 Bazin's formula, 122 for pipe flow, 101 values of Chezy's coefficient, 313 Bends and elb6ws, resistance of, 89 Bernoulli's theorem, 71 Bimie's formula, 19 Borda mouthpiece, 68 Branching pipes, 103 Breast wheel, 172 Bulk modulus of water, 1 Buoyancy, 22 zero, 26 Capacity criterion, 204 Cast-iron pipe, flOw in, 86 Catskill aqueduct, 153-156 meter, 77 Cedar Rapids turbines, 288 Center of pressure, 14 Centrifugal pumps, 246 characteristics, 258 design of, 265 Characteristics of centrifugal pumps, 258 of impulse wheels, 178 Characteristic speed, 205 Chezy's coefficient, Bazin's values of, 313 Kutter's values of, 314 formula for open channels, 121 for pipe flow, 100 Cippoletti weir, 57 Circles, properties of, 298 Clavarino's formula, 18 Cock, head lost at, 95 Coefficient of pipe friction, 85 Complete contraction, 52 Compound pipes, 102 Concrete pipe, 89 Conduits, flow in, 120 Conical mouthpiece, 69-70 Contracted weir, 53 Contraction coefficient, 48 of jet, 51 of section, 94 partial and complete, 52 Crane, hydrauHc, 11 Critical velocity of flow, 78 Current meter measurements, 128 wheels, 169, 171 D Dams, Catskill aqueduct system, 39 Keokuk, 40 stability of, 37 Darcy's mod. of Pitot tube, 131 Deep well cent, pump, 267 Deflection of jet, 160 Density of water, 3, 290 325 Digitized by VnOOQlC 326 INDEX Design of centrifugal pumps, 265 Differential gage, 22 surge tank, 235 Diffusion vanes, 252 Diffusor, pressure developed in, 257 Discharge equivalents, 292 Displacement pumps, 238 Divided flow, 102 Doble bucket, 173 Draft tube design, 194 profile, 195 theory of, 193 time of flow through, 197 use of, 190 Draw down in surge tanks, 230 Dry dock, floating, 266 DuBuat's paradox, 138 Dynamic pressure, 158 Fourneyron type of reaction tur- bine, 186 Francis type of reaction turbine, 180 Free surface of liquid in rotation, 218 Freeman's experiments, 105 Friction head in pipes, 309 loss, 84 Fteley and Steams formula, 324 Fuller, W. E., 91 G Gage, differential, 22 pressure, 21 Gate valve, head lost at, 95 Girard turbine, efficiency of, 181 Graph of exponential formula, 87 E Effective head, 49 Efficiency, hydraulic, 123 of centrifugal pump, 266 of hydraulic press, 8 ram, 238 for moving vanes, 164 Efflux coefficient, 49 for circular orifice, 303 for square orifice, 304 Elasticity of water, 1 Elevator, hydraulic, 11 Energy of flow, 157 Enlargement of section, 93 Entrance, loss of head at, 84 Equilibrium of fluids in contact, 19 Exponential (ormula, 85, 88 friction head from, 309 Williams and Hazen, 101 Head and pressure equivalents, 291 developed by cent, pumps, 262 loss of, in pipes, 83 lost at bends, 90 varying, 64 Hele-Shaw's experiments, 107 Herschel, Clemens, 74 Hook gage 61 Hydraulic dredging, 271 efficiency, 123 gradient, 97 slope of, 98 mining, 273 motors, classification of, 171 types of, 169 press, 6 efficiency of, 8 radius, 99 ram, 236 Fire nozzles, 70 pimip, centrifugal, 270 streams, 105 table, 305 Fleming's experiments, 106 Fluid, properties of, 1 Force pump, 240 Impact on plane surface, 158 surface of revolution, 159 tube, theory of, 135 Impellers, centrifugal pump, 249 Impulse wheels, 170 characteristics of, 178 vane angles, 165 Digitized by VnOOQlC INDEX 327 Impulse wheels, work absorbed by, 166 Intensificr, 8 Jack, hydraulic, 10 Jet, pressure of, on surface, 157 reaction of, 167 Jonval type of reaction turbine, 186 Joukowsky's formula, 226 K Keokuk lock and dam, 39-43 turbines, 200 Kinetic pressure, 71 Kutter's formula, 121 for pipe flow, 101 simplified formula, 123 values of Chezy's coefficient, 314 Lamp's formula, 18 Liquid, definition of, 1 vein, 47 Loss of head in pipe flow, 83 M Mariotte's flask, 66 Mechanics, principles of, 322 Metacenter, 26 coordinates of, 27 Metacentric height, 28 Mine drainage, cent, pump for, 270 Mixed flow type of turbine, 189 Modulus of elasticity of water, 1 Moritz, E. A., 85 formula, table from, 316 Mouthpiece, standard, 67 N Natural channels, flow in, 126 Needle nozzle, 175 Non-sinuous flow, 79 Non-uniform flow, backwater, 138 Normal pressure of water, 2 Open channels, flow in, 120 Operating range, normal. 210 Orifice, circular, efllux coefficients for, 303 square, efflux coefficients for, 304 Oscillation, period of, 29 Overshot wheel, 172 Packing, frictional resistance* of , 7 Parallel flow, 79 Partial contraction, 52 Pelton wheel, 173 efficiency of, 177 Penstock, economical size of, 214 Piezometer, 21 Pipe flow, 78-83 friction, coefficient of, 85 lines, power transmitted through, 214 strength of, 17 Pipe friction, coefficient of, 308 Pipes, dimensions of, 295 Pitometer, 132 Pitot recorders, 134 tube, 129 calibration of, 137 Poncelet wheel, 172 Pressure, center of, 14 change with depth, 13 developed in cent, pump, 255 equal transmission of, 3 gage, mercury, 21 head developed in cent, pumps, 257 kinetic, 72 kinetic, 71, 97 normal to surface, 2 of jet on surface, 157 proportional to area, 5 Principles of mechanics, 322 Pump cylinders, diameter of, 244 Pumps, capacity of reciprocating, 296 centrifugal, 246 Digitized by VnOOQlC 328 INDEX Pumps, displacement, 238 Pump sizes, calculation of, 243 Ram, efficiency of, 238 hydraulic, 236 Rate of flow controller, 77 Reaction of jet, 167 turbines, 170 classification of, 170, 208 principle of, 168 Rectangular notch weirs, discharge * coefficients for, 318-320 discharge from, 51 orifice, discharge from, 50 Riveted steel pipe, 88 Rolling and pitching, 30 Roman aqueducts, 150 Rotation, liquid in, 218 S Selection of stock runner, 211 Service pipes, house, 143 Sharp edged orifice, discharge from, 51 Siphon lock, 117 modem, 111 spillways, 112 wheel settings, 120 Slope of hydraulic gradient, 98 Specific discharge. 207 power, 207 speed, 206-208 weight, 3 determination by experiment, 24 of various substances, 294 physical definition of, 24 Speed criterion, 203 Stage pumps, 254 Static pressure, 158 Steam pump, 241 Stock runner, selection of, 211 Stream gaging, 126 line, 47 mouthpiece, 68 Strength of pipe, 17 Submerged weir, 54 coefficients, 324 Suction lift, maximum, 239 pump, 238 Suppressed weir, 54 discharge from, 321 Surge in surge tanks, 228 tanks, 228 differential, 235 Tanks, filling and emptying, 63-65 Throttling discharge of cent, pump, 263 Throttle valve, head lost at, 96 Torricelli's theorem, 48 Translation, horizontal linear, 217 vertical linear, 218 Trapezoidal weir, 57 Triangular notch weir, 56 Turbine pumps, 253 setting, recent practice in, 198 U Undershot wheel, 171 V Vane, pressure of jet on, 162 work done on moving, 163 Varying head, 64 Velocity, critical, 78 head, ideal, 47 of approach, 52 Venturi adjutage, 69 meter, 74 Viscosity coefficient, 79 of water, 3 Volute casing, 249 Vortex chamber, 250 W y- Water hammer in pipes, 221 ordinary, 224 Warren, Minton M., 221 Water, properties of, 290 Digitized by VnOOQlC INDEX 329 Wave, compression, in pipe, 223 Weight of water, 290 Weights and measures, 293 Weir, Cippoletti, 57 construction of, 60 contracted, 53 formulas, empirical, 58-69 measurements, 60 Weirs, proportioning, 62 rectangular notch, discharge coefficients for, 318-320 Weirs, submerged, 64 coefficients, 324 suppressed, 64 discharge from, 321 trapezoidal, 57 triangular notch, 56 Williams and Hazen's exponential formula, 101 tables from, 309 Wood stave pipe, 85 discharge from, 316 Digitized by VnOOQlC Digitized by VnOOQlC Digitized by LnOOQlC Digitized by VnOOQlC p^ Digitized by VnOOQlC ( THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW AN INITIAL FINE OF 25 CENTS WILL BE ASSESSED FOR FAILURE TO RETURN THIS BOOK ON THE DATE DUE. 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